production notes - university of...

Barry T. Coyle

Production Economics

This is a draft: please do not distributec Copyright; Barry T. Coyle, 2010

October 9, 2010

University of Manitoba

Foreword

This booklet is typed based on professor Barry Coly’s lecture notes for ABIZ 7940Production Economics in Winter 2008. I am responsible for all the errors and typos.

Winnipeg, October 2010 Ning Ma

v

Contents

1 Static Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Properties of c.w; y/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Corresponding properties of x.w; y/ solving problem (1.1) . . . . . . . 31.3 Second order relations between c.w; y/ and f .x/ . . . . . . . . . . . . . . . 41.4 Additional properties of c.w; y/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Applications of dual cost function in econometrics . . . . . . . . . . . . . . . 71.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Static Competitive Profit Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.1 Properties of �.w; p/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Corresponding properties of y.w; p/ and x.w; p/ . . . . . . . . . . . . . . . 132.3 Second order relations between �.w; p/ and f .x/, c.w; y/ . . . . . . . 142.4 Additional properties of �.w; p/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Le Chatelier principles and restricted profit functions . . . . . . . . . . . . . 172.6 Application of dual profit functions in econometrics: I . . . . . . . . . . . . 192.7 Industry profit functions and entry and exit of firms . . . . . . . . . . . . . . 212.8 Application of dual profit functions in econometrics: II . . . . . . . . . . . 23References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Static Utility Maximization and Expenditure Constraints . . . . . . . . . . . 273.1 Properties of V.p; y/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Corresponding properties of x.p; y/ solving problem (3.1) . . . . . . . . 313.3 Application of dual indirect utility functions in econometrics . . . . . . 323.4 Profit maximization subject to budget constraints . . . . . . . . . . . . . . . . 34References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Nonlinear Static Duality Theory (for a single agent) . . . . . . . . . . . . . . . . 394.1 The primal-dual characterization of optimizing behavior . . . . . . . . . . 394.2 Producer behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Consumer behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

vii

viii Contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Functional Forms for Static Optimizing Models . . . . . . . . . . . . . . . . . . . 475.1 Difficulties with simple linear and log-linear Models . . . . . . . . . . . . . 475.2 Second order flexible functional forms . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Examples of second order flexible functional forms . . . . . . . . . . . . . . 515.4 Almost ideal demand system (AIDS) . . . . . . . . . . . . . . . . . . . . . . . . . . 575.5 Functional forms for short-run cost functions . . . . . . . . . . . . . . . . . . . 58

5.5.1 Normalized quadratic: c� D c=w0, w� D w=w0. . . . . . . . . . 595.5.2 Generalized Leontief: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.5.3 Translog: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6 Aggregation Across Agents in Static Models . . . . . . . . . . . . . . . . . . . . . . . 616.1 General properties of market demand functions . . . . . . . . . . . . . . . . . 616.2 Condition for exact linear aggregation over agents . . . . . . . . . . . . . . . 646.3 Linear aggregation over agents using restrictions on the

distribution of output or expenditure . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.4 Condition for exact nonlinear aggregation over agents . . . . . . . . . . . . 75References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7 Aggregation Across Commodities: Non-index Number Approaches . . 777.1 Composite commodity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.2 Homothetic weak separability and two-stage budgeting . . . . . . . . . . . 797.3 Implicit separability and two-stage budgeting . . . . . . . . . . . . . . . . . . . 83References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8 Index Numbers and Flexible Functional Forms . . . . . . . . . . . . . . . . . . . . 878.1 Laspeyres index numbers and linear functional forms . . . . . . . . . . . . 878.2 Exact indexes for Translog functional forms . . . . . . . . . . . . . . . . . . . . 898.3 Exact indexes for Generalized Leontief functional forms . . . . . . . . . . 968.4 Two-stage aggregation with superlative index numbers . . . . . . . . . . . 978.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1008.6 Laspeyres and Paasche cost of living indexes . . . . . . . . . . . . . . . . . . . 1028.7 Fisher indexes (for inputs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

9 Measuring Technical Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059.1 Dual cost and profit functions with technical change . . . . . . . . . . . . . 1069.2 Non-Parametric index number calculations of changes in

productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109.3 Parametric index number calculations of changes in productivity . . . 1139.4 Incorporating variable utilization rates for capital . . . . . . . . . . . . . . . . 1179.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Chapter 1Static Cost Minimization

Consider a firm producing a single output y using N inputs x D .x1; � � � ; xN /

according to a production function y D f .x/. The firm is a price taker in its Nfactor markets, i.e. the firm treats factor prices w D .w1; � � � ; wN / as given. Weassume that all inputs are freely adjustable and perfect rental markets exist for allcapital goods, i.e. for now we ignore all costs of adjustment that can lead to dynamicbehavior.

A necessary condition for static profit maximization is static cost minimization,i.e. at the profit maximizing level of output y and input pricesw the firm necessarilysolves the following cost minimization problem:

minx�0

NXiD1

wixi D wx�

s.t. f .x/ � y

(1.1)

The minimum cost c � wx� to problem (1.1) depends on the levels of inputprices w and output y, and of course on the production function y D f .x/. Bysolving (1.1) using different value of .w; y/we can in principle trace out the relationbetween minimum cost c and parameters .w; y/, conditional on the firm’s particularproduction function y D f .x/. This relation c D c.w; y/ between minimum costand parameters .w; y/ is called the firm’s dual cost function.

1.1 Properties of c.w; y/

Property 1.1.

a) c.w; y/ is increasing (or, more precisely, non-decreasing) in all parameters.w; y/.

1

2 1 Static Cost Minimization

Fig. 1.1 c.w; y/ is concavein w

c(wA, wB, y)

wA

b) c.w; y/ is linear homogeneous in w. i.e. c.�w; y/ D �c.w; y/ for all scaler� > 0 (if all factor prices w increase by the same proportion. e.g. 10%, thenthe minimum cost of attaining the same level of output y also increases bythis proportion)

c) c.w; y/ is concave in w. i.e. c.�wA C .1 � �/wB ; y/ � �c.wA; y/C .1 ��/c.wB ; y/ for all 0 � � � 1. See Figure 1.1

d) If c.w; y/ is differentiable in w, then

xi .w; y/ [email protected]; y/

@wi

(Shephard’s Lemma)1

Proof. Property 1.1.a is obvious from Equation (1.1), and 1.1.b follows from thefact that an equiproportional change in all factor prices w does not change relativefactor prices and hence does not change the cost-minimizing level of inputs x� forproblem (1.1). 1.1.c is not so obvious. In order to prove it simply note that, forwC � �wA C .1 � �/wB and x�C solving (1.1) for .wC ; y/,

c.wC ; y/ � wCx�C

D �wAx�C C .1 � �/wBx

�C

� �c.wA; y/C .1 � �/c.wB ; y/

(1.2)

since wAx�C � c.wA; y/ and wBx�C � c.wB ; y/ (e.g. wAx�C cannot be less thanthe minimum cost for problem (1.1) given prices wA ¤ wC : wAx�C > c.wA; y/

unless x�C solves (1.1) for prices wA as well as for prices wC ). Numerous proofs ofShephard’s Lemma 1.1.d are available. Here we simply consider the most obviousmethod of proof (see Varian 1992 for alternative methods).

Expressing (1.1) in Lagrange form

1 Note that c.w; y/ can be differentiable in w even if, e.g. the production function y D f .x/ isLeontief (fixed proportions). In general differentiability of c.w; y/ is a weaker assumption thandifferntiability of y.

1.2 Corresponding properties of x.w;y/ solving problem (1.1) 3

minx;Q�

wx � Q� .f .x/ � y/ D wx� � Q��f .x�/ � y

�D c.w; y/ (1.1’)

with first order condition

wi � Q�� @f .x

�/

@xiD 0

f .x�/ � y D 0

Then @c.w;y/@wi

can be calculated by total differentiation as follows:

@c.w; y/

@wiD x�i C

NXjD1

@x�j

@wi

�wj � Q�

� @f .x�/

@xj

��@ Q��

@wi

�f .x�/ � y

�D x�i

(1.3)

by the first order conditions to problem (1.1’). ut

It is important to note that Shephard’s Lemma 1.1.d is simply an application ofthe envelope theorem (Samuelson 1947). The lemma states that, for an infinitesimalchange in factor price wi (all other factor prices and output remaining constant), thechange in minimum cost divided by the change in wi is equal to the equilibriumlevel of input i in the absence of any change in .w; y/. In other words, in the limit,zero changes in equilibrium x� in response to a change inwi are optimal. Obviouslysuch a lemma has no economic content, i.e. does not describe optimal response tofinite changes in wi . Nevertheless Shephard’s and analogous envelope theorems arecritical to the empirical and theoretical application of duality theory. This distinctionis easily missed in more complex models.

1.2 Corresponding properties of x.w; y/ solving problem (1.1)

Property 1.2.

a) x.w; y/ is homogeneous of degree 0 in w. i.e. x.�w; y/ D x.w; y/ for allscalar � � 0.

b)[email protected];y/@w

iN�N

is symmetric negative semidefinite.

Proof. 1.2.a simply states that the cost minimizing solution x� to problem (1.1) de-pends only on relative prices. In order to prove 1.2.b, note that the Hessian [email protected];y/@w@w

iN�N

is symmetric negative semidefinite by concavity and twice differen-

tiability of c.w; y/ inw, and then note that @c.w;y/@w

D x.w; y/ for allw (Shephard’s

Lemma))[email protected];y/@w@w

iN�N

D

[email protected];y/@w

iN�N

. ut


In order to test economics theories it is important to know all of the restrictionsthat are placed on observable behavior by particular theories. This is known as theintegrability problem in economics. It can easily be shown that 1.2.a-b exhaustedthe (local) properties that are placed on factor demands x.w; y/ by the hypothesisof cost minimization (1.1).

Proof. It has already been shown that the properties 1.1 of the cost function imply1.2. In order to show that 1.2 exhausts the the implications of cost minimization(1.1) for local properties of x.w; y/, first total differentiate c.w; y/ � wx.w; y/

with respect to wi ,

@c.w; y/

@wi� xi .w; y/C

NXjD1

wj@xj .w; y/

@wii D 1; � � � ; N: (1.4)

1.2.a implies (by Euler’s theorem2)PNjD1wj

@xj .w;y/

@wiD 0 (i D 1; � � � ; N ) and to-

gether with symmetry 1.2.b this reduces the identity (1.4) to @c.w;y/@wi

D xi .w; y/ �

0, (i D 1; � � � ; N ) (Shephard’s Lemma). In [email protected];y/@w

iN�N

1.2.b implies

that the system of differential equations xi .w; y/ [email protected];y/@wi

.i D 1; � � � ; N / inte-grates up to an underlying cost function c.w; y/ (Frobenius theorem3). Shephard’sLemma also implies (by simple differentiation of x.w; y/ D @c.w; y/=@w withrespect to w)

[email protected];y/@w

iN�N

D

[email protected];y/@w@w

iN�N

which is negative semidefinite

by 1.2.b. It can then be shown [email protected];y/@w@w

iN�N

negative semidefinite impliesy D f .x/ is quasiconcave at x D x.w; y/ (see (1.6) below). This establishesthe second order conditions on the production function y D f .x/ for competitivecost minimization. The first order condition follow from the fact that 1.2 establishesShephard’s Lemma for all @x.w;y/

@w, @�.w;y/

@w, satisfying 1.2 (see (1.3)). ut

1.3 Second order relations between c.w; y/ and f.x/

It is sometimes interesting to ask whether or not the firm’s production functiony D f .x/ can be recovered from knowledge of the firm’s cost function c.w; y/,i.e. can we construct y D f .x/ directly from knowledge of c.w; y/? The answer isessentially yes (the only qualification is that we cannot recover f .x/ at levels of xthat cannot be solutions to a cost minimization problem (1.1) for some .w; y/, i.e. atlevels of associated with locally non-convex isoquants). For example, given knowl-

2 Euler’s theorem states that ,if g.�x/ D �rg.x/ for all scaler � > 0 (i.e. the function g.x/ ishomogenous of degree r), then rg.x/ D

Pi xi

@g.x/

@xiand .r � 1/ @g.x/

@xjDPi xi

@2g.x/

@xi@xj3 The Frobenius theorem states that a system of differential equations gi .v/ D

@ .v/

@vi.i D

1; � � � ; T / has a solution .v/ if and only if @gi .v/@vj

D@gj .v/

@vi.i; j D 1; � � � ; T /.

1.3 Second order relations between c.w; y/ and f .x/ 5

edge of c.w; y/ and differentiability of c.w; y/, application of Shephard’s Lemma@c@w1D x1; � � � ;

@c@wN

D xN immediately gives the cost-minimizing levels of inputsx corresponding to .w; y/ (this assumes a unique solution x to problem (1.1)). Byvarying .w; y/ we can map out y D f .x/ from c.w; y/ in this manner.

A related point that will be important later (in a lecture on functional forms) isthat the first and second derivatives of f .x/ at x.w; y/ can be calculated directlyfrom knowledge of the first and second derivatives of c.w; y/. The first derivativescan be calculated simply as

@f .x.w; y//

@xiD

wi

@c.w; y/=@yi D 1; � � � ; N (1.5)

using the first order conditions @f .x/@xi� Q�wi D 0 .i D 1; � � � ; N / for cost minimiza-

tion (1.1’), where Q� � @c.w;y/@y

. The procedure for calculating the second derivativesof f .x/ from c.w; y/ is not quite as obvious. The corresponding formula in matrixnotation is"

@c.w;y/@y

˝@2f@x@x

@f@x

@f@x

0

#.NC1/�.NC1/

D

"@2c.w;y/@w@w

@2c.w;y/@w@y

@2c.w;y/@y@w

@2c.w;y/@y@y

#�1.NC1/�.NC1/

(1.6)

where it is assumed without loss of generality that the above inverse exists4.

Proof. Consider the case N D 2 so y D f .x1; x2/ and c D c.w1; w2; y/. The firstorder condition for cost minimization (1.1’) can be written as

w1 � cy.w; y/fx1 D 0

w2 � cy.w; y/fx2 D 0

y D f .x/

(1.7)

where for now cy.w; y/ �@c.w;y/@y

, fx1 �@f .x/@x1

, etc.Total differentiating these conditions (1.7) with respect to .w1; w2; y/ yields (us-

ing Shephard’s Lemma):

4 c.w; y/ linear homogenous in w and Euler’s theorem imply 0 DPNjD1wj

@2c.w;y/

@[email protected] D

1; � � � ;N/, which [email protected];y/

@w@w

iN�N

does not have full rank. Nevertheless the above bor-

dered matrix

"@2c.w;y/

@w@w

@2c.w;y/

@w@[email protected];y/

@y@w

@2c.w;y/

@y@y

#.NC1/�.NC1/

generally have full rank.


@

@w1)

8<:1 � cyw1fx1 � cyfx1x1cw1w1 � cyfx1x2cw1w2 D 0

�cyw1fx2 � cyfx1x2cw1w1 � cyfx2x2cw1w2 D 0

fx1cw1w1 � fx2cw1w2 D 0

@

@w2)

8<:�cyw2fx1 � cyfx1x1cw1w2 � cyfx1x2cw2w2 D 0

1 � cyw2fx2 � cyfx1x2cw1w1 � cyfx2x2cw2w2 D 0

fx1cw1w2 C fx2cw2w2 D 0

@

@y)

8<:�cyyfx1 � cyfx1x1cw1y � cyfx1x2cw2y D 0

�cyyfx2 � cyfx1x2cw1y � cyfx2x2cw2y D 0

fx1cw1y C fx2cw2y D 0

(1.8)

Writing (1.8) in matrix notation,24 cw1w1 cw1w2 cyw1cw1w2 cw2w2 cyw2cyw1 cyw2 cyy

3524 cyfx1x1 cyfx1x2 fx1cyfx1x2 cyfx2x2 fx2fx1 fx2 0

35 D241 0 0

0 1 0

0 0 1

35 (1.9)

ut

1.4 Additional properties of c.w; y/

Property 1.3.

a) y D f .x/ homothetic, c.w; y/ D �.y/ c.w; 1/ for some function �.b) y D f .x/ constant returns to scale, c.w; y/ D y c.w; 1/.c) All the partial elasticity of substitution between inputs i and j (output y

constant)

�ij .w; y/ �

@.xi .w;y/=xj .w;y//xi=xj

@.wi=wj /wi=wj

can be calculated simply as

�ij .w; y/ Dc.w; y/ @

2c.w;y/@wi@wj

@c.w;y/@wi

@c.w;y/@wj

(see Uzawa 1962, p. 291-9).d) Assuming a vector of outputs y D y1; � � � ; yM . The transformation functionf .x; y/ D 0 is disjoint (i.e. y1 D f1.x1/; � � � ; yM D fM .xM / where inputvector x1; � � � ; xM do not overlap) only if @2c.w;y/

@yi@yjD 0 for all i ¤ j , all

.w; y/.

1.5 Applications of dual cost function in econometrics 7

1.5 Applications of dual cost function in econometrics

The above theory is usually applied by first specifying a functional form �.w; y/

for the cost function c.w; y/ and differentiating �.w; y/ with respect to w in orderto obtain the estimating equations

xi D@�.w; y/

@wii D 1; � � � ; N (1.10)

(employing Shephard’s Lemma). Then the symmetry restrictions @2�@wi@wj

D@2�

@wj @wi

.i D 1; � � � ; N / are tested and the second order conditionh@2�@w@w

iN�N

negativesemidefinite is checked at all data points .w; y/.

For example a cost function could be postulated as having the functional form

c D y

NXiD1

NXiD1

aijw12

i w12

j

(a Generalized Leontief functional form with y D f .x/ showing constant returnsto scale), which leads to the following equations for estimation:

xi

yD

NXjD1

aij

�wj

wi

� 12

i D 1; � � � ; N (1.11)

Here the symmetry restriction @xi@wj

D@xj@wi

are expressed as aij D aj i .i D

1; � � � ; N / which are easily tested. Equation (1.10) can be interpreted as being de-rived from a cost function c.w; y/ for a producer showing static, competitive cost-minimizing behavior if and only if the symmetry and second order conditions aresatisfied.

The major advantage of this approach is that it permits the specification of asystem of factor demand equations x D x.w; y/ that are consistent with cost min-imization and with a very general specification of technology. In contrast, supposethat we wished to specify explicitly a solution to a cost minimization problem. Thenwe would estimate a production function directly with first order conditions for costminimization:


y D f .x/

@f=@xi

@f=@xjDwi

wj

@f=@xi

@f=@xkDwi

wk

� � �

(1.12)

However, unless a very restrictive functional form is specified for the productionfunction (e.g. Cobb-Douglas), then we can seldom drive the factor demand equa-tions x D x.w; y/ explicitly from (1.12). Since policy makers are usually moreinterested in demand and supply behavior than in production functions per se, thisgreater ease in specification of x.w; y/ is an important advantage of duality theory.

Two other advantages of a duality approach rather than a primal approach (1.12)to the estimation of producer behavior are apparent. First, the hypothesis of compet-itive cost minimization is more readily tested in the framework of equations (1.10)than (1.12). Second, variables that are omitted from the econometric model (butare observed by producers) influence both the error terms and production decisionsbut do not necessarily influence factor prices to the same degree (e.g. factor pricesmay be exogenous to the industry). This tends to introduce greater simultaneousequations biases into the estimation of (1.12) than of (1.10).

One disadvantage of (1.10) is that output y, as well as factor prices w, is treatedas exogenous. Since the firm generally in effect chooses y jointly with x , this mis-specification can lead to simultaneous equations biases in the estimators. So theextent that production is constant return to scale with a single output, this difficultycan be avoided by using 1.3.b to specify a unit cost function c.w/ D c.w; y/=y andapplying Shephard’s Lemma to obtain estimating equations

xi

[email protected]/

@wii D 1; � � � ; N (1.13)

for a given functional form c.w/ (e.g. (1.11)). Under constant returns to scalexi .w; y/=y depends on w but not on y, so that (1.13) is well defined and estimationis independent of whether y is endogenous or exogenous to the firm.

A second disadvantage of this duality approach to the specification of functionalforms for econometric models, and a disadvantage of primal approaches such as(1.12) as well, is that it is derived from the theory of the individual firm but isusually applied to market data that is aggregated over firms. Difficulties raised bysuch aggregation will be discussed in a later lecture.

1.6 Conclusion

The dual cost function approach offers many advantages in the estimation of pro-duction technologies. The estimated factor demands x D x.w; y/ measure factor

References 9

substitution along an isoquant and the effects of scale of output on factor demands,and first and second derivatives of the production function can be calculated. More-over the assumption of cost minimization is consistent with various broader theoriesof producer behavior.

However effective policy making depends more on a knowledge of producer be-havior than of production functions per se. By ignoring the effect of output priceson the firm’s input levels, the dual cost function approach generally is inappropriatefor the modeling of economic behavior.

Of course cost functions can be embedded within a broader behavioral model.For example, static competitive profit maximization implies a cost minimizationmodel such as 1.1 together with first order conditions

@c.w; y/

@yD p (1.14)

for optimal output levels (marginal cost equals output price). This equation implic-itly defines the optimal level of y as y D y.w; p/ provided of course that y en-ters (1.14), i.e. @c.w;y/

@yis not independent of y or equivalently f .x/ does not show

constant returns to scale. In this case equations 1.1.d, (1.10) and (1.14) can be esti-mated jointly, and the second order condition for profit maximization are expressedas c.w; y/ concave and @2c.w;y/

@y@y� 0.

Nevertheless there can be substantial disadvantages to this approach to model-ing competitive profit maximization. The assumption of constant returns to scale iscommonly employed in empirical studies, and the assumption of profit maximiza-tion is not so easily tested here (the homogeneity and reciprocity condition for costminimization do not imply integration up to a profit function). Therefore, for policypurposes, it is often better to model and test directly (using dual profit functions) thehypothesis of competitive profit maximization behavior.5

References

1. Samuelson, Paul A., 1947, Enlarged ed., 1983. Foundations of Economic Analysis, HarvardUniversity Press. ISBN 0-674-31301-1

2. Uzawa, H. (October 1962). Production Function with Constant Elasticities of Substitution,Review of Economics Studies, Vol. 29, pp. 291-299.

3. Varian, H. R. (1992). Microeconomic Analysis, Third Edition. W. W. Norton & Company, 3rdedition.

5 In passing, note that this indirect approach to the modeling of profit maximization (1.1.d, (1.10),(1.14)) may be superior to a direct approach (see next lecture) when there is substantially highermulticollinearity between factor prices w and output prices p than between w and output levelsy.

Chapter 2Static Competitive Profit Maximization

As before, consider a firm producing a single output y using N inputs x D.x1; � � � ; xN / according to a production function y D f .x/, and assume that thefirm is a price taker in all N factor markets. In addition, we now assume that thefirm takes output price p as given and chooses its levels of inputs to solve the fol-lowing static competitive profit maximization problem:

maxx�0

(pf .x/ �

NXiD1

wixi

)D pf .x�/ � wx�: (2.1)

The maximum profit � � pf .x�/�wx� to problem (2.1) depends on prices .w; p/and the firm’s production function y D f .x/. The corresponding relation � D�.w; p/ between maximum profit and prices is denoted as the firm’s dual profitfunction.

2.1 Properties of �.w;p/

Property 2.1.

a) �.w; p/ is decreasing in w and increasing in p.b) �.w; p/ is linear homogeneous in .w; p/. i.e., �.�w; �p/ D ��.w; p/ for

all scalar � > 0.c) �.w; p/ is convex in .w; p/. i.e.,

� Œ�wA C .1 � �/wB ; �pA C .1 � �/pB � � ��.wA; pA/C.1��/�.wB ; pB/

for all 0 � � � 1: See Figure 6.1.

11

12 2 Static Competitive Profit Maximization

Fig. 2.1 �.w;p/ is convexin .w;p/

p

π(w , p)

d) if �.w; p/ is differentiable in .w; p/, then

y.w; p/ D@�.w; p/

@p;

xi .w; p/ D �@�.w; p/

@wii D 1; � � � ; N:

(Hotelling’s Lemma)

Proof. Properties 2.1.a–b follow obviously from the definition of the firm’s maxi-mization problem (2.1). In order to prove 2.1.c simply note that, for .wC ; pC / ��.wA; pA/C .1 � �/.wB ; pB/ and x�C solving 2.1 for .wC ; pC /,

�.wC ; pC / � pCf .x�C / � wCx

�C

D ��pAf .x

�C / � wAx

�C

�C .1 � �/

�pBf .x

�C / � wBx

�C

�� .wA; pA/C .1 � �/�.wB ; pB/

(2.2)

since �.wA; pA/ � pAf .x�C /�wAx�C , �.wB ; pB/ � pBf .x�C /�wBx

�C . In order to

prove Hotelling’s Lemma 2.1.d simply total differentiate �.w; p/ � pf .x�/�wx�

with respect to .w; p/ respectively and then apply the standard first order conditionsfor an interior competitive profit maximum:

@�.w; p/

@pD f .x�/C

NXkD1

�p@f .x�/

@xk� wk

�@x�k

@p

D f .x�/

@�.w; p/

@wiD �x�i C

NXkD1

�p@f .x�/

@xk� wk

�@x�k

@wi

D �x�i i D 1; � � � ; N:

(2.3)

ut

2.2 Corresponding properties of y.w;p/ and x.w;p/ 13

Hotelling’s Lemma plays the same role in the theory of competitive profit max-imization as Shephard’s Lemma plays in the theory of competitive cost minimiza-tion. Hotelling’s Lemma is an envelope theorem. The Lemma applies only for in-finitesimal changes in a price and yet is critical to the empirical theoretical applica-tion of dual profit functions.

2.2 Corresponding properties of y.w;p/ and x.w;p/

Property 2.2.

a) y.w; p/ and x.w; p/ are homogeneous of degree 0 in .w; p/. i.e.,y.�w; �p/ D y.w; p/ and x.�w; �p/ D x.w; p/ for all scalar � � 0.

b)

"�@x.w;p/@w N�N

@y.w;p/@w N�1

�@x.w;p/@p 1�N

@y.w;p/@p 1�1

#.NC1/�.NC1/

is symmetric positive

semidefinite.

Proof. Property 2.2.a follows directly from the maximization problem (2.1).1 In or-der to prove 2.2.b, note that

h@2�.w;p/@w@p

i.NC1/�.NC1/

is symmetric positive semidef-

inite by 2.1.c and then apply 2.1.d to evaluate this matrix. ut

Moreover, 2.2.a–b exhaust the (local) properties that are placed on output supplyy.w; p/ and factor demand x.w; p/ relations by the hypothesis of competitive profitmaximization (2.1).

Proof. First total differentiate �.w; p/ � pf .x.w; p// � wx.w; p/ to obtain

@�.w; p/

@p� y.w; p/C p

@y.w; p/

@p�

NXkD1

[email protected]; p/

@p

@�.w; p/

@wi� �xi .w; p/C p

@y.w; p/

@wi�

NXkD1


@wii D 1; � � � ; N

(2.4)

Property 2.2.a implies (by Euler’s theorem)

1 Alternatively, �.w;p/ homogeneous of degree one in .w;p/ implies (by Euler’s theorem)@�.w;p/

@wiD �xi .w;p/ homogeneous of degree 0.



@pC

NXkD1


@wkD c

p@xi .w; p/

@pC

NXkD1

wk@xi .w; p/

@wkD 0 i D 1; � � � ; N

and property 2.2.b states the reciprocity relations

@y.w; p/

@wkD �

@xk.w; p/

@p

@xi .w; p/

@[email protected]; p/

@wii; k D 1; � � � ; N;

so properties 2.2.a–b jointly imply


@p�

NXkD1


@pD 0


@wi�

NXkD1


@wiD 0 i D 1; � � � ; N:

Substituting this into the identity (2.4) yields

@�.w; p/

@pD y.w; p/ � 0

@�.w; p/

@wiD �xi .w; p/ � 0 i D 1; � � � ; N:

(Hotelling’s Lemma)

The reciprocity relations 2.2.b imply that the system of differential equations@�.w;p/@p

D y.w; p/, @�.w;p/@wi

D �xi .w; p/, .i D 1; � � � ; N / integrates up to anunderlying function �.w; p/ (Frobenius theorem). The positive semidefiniteness re-striction 2.2.b implies positive semidefiniteness of the Hessian matrix of �.w; p/,and this in turn implies y D f .x/ is concave at all x� (see (2.6) below). This estab-lishes the second order conditions on the production f .x/ for competitive profitmaximization. The first order conditions follow from the fact that 2.2 establishHotelling’s Lemma for all @y.w;p/

@p, @x.w;p/

@p, @y.w;p/

@w, @x.w;p/

@wsatisfying properties

2.2 (see (2.3)). ut

2.3 Second order relations between �.w;p/ and f.x/, c.w; y/

As in the case of a cost function, the firm’s production function y D f .x/ canbe recovered from knowledge of the profit function �.w; p/. Given knowledge of

2.3 Second order relations between �.w;p/ and f .x/, c.w; y/ 15

�.w; p/ and differentiability of �.w; p/, application of Hotelling’s Lemma imme-diately gives a profit maximizing combination .x; y/ for prices .w; p/.

Likewise the first and second derivatives of f .x/ at x D x.w; p/ can be calcu-lated directly from the first and second derivatives of �.w; p/. The first derivativescan be calculated simply as

@f Œx.w; p/�

@xiDwi

pi D 1; � � � ; N (2.5)

using the first order conditions for an interior solution to problem (2.1). The secondderivatives can be calculated from the matrix equation

p ˝

�@2f .x.w; p//

@x@x

�N�N

D �

�@2�.w; p/

@w@w

��1N�N

(2.6)

assuming an inverse forh@2�.w;p/@w@w

i.

Proof. Simply total differentiate the first order conditions p @f .x�/

@xi� wi D 0 (i D

1; � � � ; N ) with respect to w to obtain

p

NXkD1

@2f .x�/

@xi@xk

@xk.w; p/

@wj� 1 D 0 i; j D 1; � � � ; N; (2.7)

Substitute @xk.w;p/@wj

D �@2�.w;p/@wk@wj

(by Hotelling’s Lemma) into (2.7) and expressthe result in matrix form. ut

This result (2.6) can easily be extended to the case of multiple outputs (Lau1976).

Since elasticities of substitution (holding output y constant) and scale effectsare easily expressed in terms of a dual cost function c.w; y/, it is useful to note that�.w; p/ also provides a second order approximation to c.w; y/. The first derivativesof c.w; y/ can be calculated simply as

@c.w; y�/

@wiD xi .w; y

�/

D �@�.w; p/

@wii D 1; � � � ; N

@c.w; y�/

@yD p

(2.8)

where y� � y.w; p/ D @�.w;p/@p

, the profit maximizing level of output given.w; p/. The second derivatives of c.w; y/ at y� D y.w; p/ can be calculated from�.w; p/ using the following matrix relations, here cww.w; y/ �

[email protected];y/@w@w

iN�N

,

cwy.w; y/ �[email protected];y/@w@y

iN�N

, �ww.w; p/ �h@2�.w;p/@w@w

iN�N

, �wp.w; p/ �

16 2 Static Competitive Profit Maximizationh@2�.w;p/@w@p

iN�N

,

cww.w; y�/ D ��ww.w; p/C �wp.w; p/ �pp.w; p/

�1 �wp.w; p/T

cwy.w; y�/ D ��wp.w; p/ �pp.w; p/

�1

cyy.w; y�/ D �pp.w; p/

�1

(2.9)

Proof. xi .w; p/ D xi Œw; y.w; p/� .i D 1; � � � ; N / implies (by Hotelling’s Lemmaand Shephard’s Lemma)

��wi .w; p/ D cwi .w; y�/ (2.10)

where y� D y.w; p/. Differentiating with respect to .w; p/,

��ww.w; p/ D cww.w; y�/C cwy.w; y

�/ �wp.w; p/T

��wp.w; p/ D cwy.w; y�/ �pp.w; p/

(2.11)

Combining (2.11),

cww.w; y�/ D ��ww.w; p/C �wp.w; p/ �pp.w; p/

�1 �wp.w; p/T

cwy.w; y�/ D ��wp.w; p/ �pp.w; p/

�1(2.12)

Finally differentiating the first order condition cy.w; y�/ D p (for profit maximiza-tion) with respect to p yields cyy.w; y�/ D �pp.w; p/�1.

2.4 Additional properties of �.w;p/

Property 2.3.

a) the partial elasticity of substitution �ij between inputs i and j , allowing forvariation in output, can be defined as

�ij .w; p/ D�.w; p/ @

2�.w;p/@wi@wj

@�.w;p/@wi

@�.w;p/@wj

i; j D 1; � � � ; N

and, in the case of multiple outputs y D .y1; � � � ; yM /, the partial elasticityof transformation between outputs i and j can be defined as

tij .w; p/ D�.w; p/ @

2�.w;p/@pi@pj

@�.w;p/@pi

@�.w;p/@pj

i; j D 1; � � � ; N

2.5 Le Chatelier principles and restricted profit functions 17

b) the transformation function f .x; y/ D 0 is disjoint in outputs only if

@2�.w; p/

@pi@pjD 0 for all i ¤ j , all .w; p/:

2.5 Le Chatelier principles and restricted profit functions

Samuelson proved the following Le Chatelier principle: fixing an input at its initialstatic equilibrium level dampens own-price comparative static responses, or moreprecisely

@y.w; p/

@p�@y.w; p; Nx1/

@p� 0

@xi .w; p/

@wi�@xi .w; p; Nx1/

@wi� 0

(2.13)

where Nx1 � x1.w; p/, i.e., @y.w;p/@p

, @x.w;p/@w

denote the comparative static changesin .y; x/ when the level of input 1 cannot vary from its initial equilibrium levelx1.w; p/ (Samuelson 1947). The following generalization of this result is easilyestablished using duality theory:"�@x.w;p/@w

@y.w;p/@w

�@x.w;p/@p

@y.w;p/@p

#.NC1/�.NC1/

�

"�@x.w;p; Nx1/

@[email protected];p; Nx1/

@w

�@x.w;p; Nx1/

@[email protected];p; Nx1/

@p

#.NC1/�.NC1/

(2.14)is a positive semidefinite matrix.

Proof. By the definition of competitive profit maximization (2.1),

�.w; p/ � maxx�0

(pf .x/ �

NXiD1

wixi

)� max

x�0

(pf .x/ �

NXiD1

wixi

)� �.w; p; xA1 /

s.t. x1 D xA1

(2.15)i.e., adding a constraint x1 D xA1 to a maximization problem (2.1) generally de-creases (and never increases) the maximum attainable profits. Then

�.w; p; xA1 / � �.w; p/ � �.w; p; xA1 /

(� 0 for all .w; p; xA1 /

D 0 for xA1 D x1.w; p/(2.16)

i.e., �.w; p; xA1 / attains a minimum (D 0) over .w; p/ at all xA1 D x1.w; p/. By thesecond order condition for an interior minimum,

18 2 Static Competitive Profit Maximization�@2�.w; p; xA1 /

@w@p

�.NC1/�.NC1/

�

�@2�.w; p/

@w@p

��

�@2�.w; p; xA1 /

@w@p

�(2.17)

is positive semidefinite at xA1 D x1.w; p/. (2.17) and Hotelling’s Lemma establish(2.14). ut

Samuelson’s Le Chatelier Principle (2.13)/(2.14) has often been given the fol-lowing dynamic interpretation: assuming that the difference between short-run, in-termediate run and long-run equilibrium can be characterized in terms of the numberof inputs that can be adjusted within these time frames, the magnitude of the firm’sresponse �xi (or �y) to a given change in price wi (or p) increases over time, andthe sign of these responses does not vary with the time frame.

However this characterization of dynamics in terms of a series of static modelswith a varying number of fixed inputs is unsatisfactory. In general dynamic behaviormust be analyzed in terms of truly dynamic models.

For example, it often appears that an increase in price for beef output leads to ashort-run decrease in beef output and a long-run increase in output, which contra-dicts the dynamic interpretation of Samuelson’s Le Chatelier Principle (2.13). Thiscan be explained in terms of the dual role of cattle as output and as capital inputto future production of output: a long-run increase in output generally requires anincrease in capital stock, and this can be achieved by a short-run decrease in output(Jarvis 1974). This illustrates the following point: a series of static models with avarying number of fixed inputs completely ignores the intertemporal decisions as-sociated with the accumulation of capital (durable goods).

Nevertheless, versions of profit functions conditional on the levels of certain in-puts can be useful in applied work. The following restricted dual profit function isconditional on the level of capital stocks K:

�.w; p;K/ � maxx�0

(pf .x;K/ �

NXiD1

wixi

): (2.18)

�.w; p;K/ has the same properties in its price space .w; p/ as does the unrestrictedprofit function �.w; p/, which implcitly treats capital (or services from capital) asa freely adjustable input. In addition @�.w;p;K/

@Kmeasures the shadow price of cap-

ital, and twice differentiability of �.w; p;K/ and Hotelling’s Lemma establish thefollowing reciprocity conditions:

@y.w; p;K/

@KD

@

@p

�@�.w; p;K/

@K

�@xi .w; p;K/

@KD �

@

@wi

�@�.w; p;K/

@K

�i D 1; � � � ; N:

(2.19)

There are three major advantages to specifying a restricted dual profit function.First, a restricted profit function �.w; p;K/ is consistent with short-run equilibriumfor a variety of dynamic models that dichotomize inputs as being either perfectly

2.6 Application of dual profit functions in econometrics: I 19

variable or quasi-fixed in the short-run. Second, it is more realistic to assume thatfirms are in a short-run equilibrium rather than in a long-run equilibrium, and mis-specifying an econometric model as long-run equilibrium (e.g. using an unrestrictedprofit function �.w; p/) implies that the resulting estimators of long-run equilibriumeffects of policy are unreliable. Third, long-run equilibrium effects can sometimesbe inferred correctly from the estimates of �.w; p;K/ using the following first ordercondition for a long-run equilibrium level of capital stock K��:

@�.w; p;K��/

@KD wK or

@�.w; p;K��/

@KD .r C ı/pK (2.20)

where wK � rental price of capital, pK � asset price of capital, r � an appropriatediscount rate and ı � rate of depreciation for capital. After estimating �.w; p;K/we would solve (2.20) for K��.2

2.6 Application of dual profit functions in econometrics: I

As in the use of a cost function c.w; y/, the above theory is usually applied by firstspecifying a functional form .w; p/ for a profit function �.w; p/ and differentiat-ing .w; p/ with respect to .w; p/ to obtain the estimating equations

y D@ .w; p/

@p

xi D �@ .w; p/

@wii D 1; � � � ; N

(2.21)

(employing Hotelling’s Lemma). Then the symmetry restrictions @y@wiD �

@xi@p

,@xi@wjD

@xj@wi

, (i; j D 1; � � � ; N ) are tested andh@2 @w@p

i.NC1/�.NC1/

is checked for

positive semidefiniteness. For example, a profit function could be postulated as

� D

NXiD1

NXjD1

aijpwipwj C

NXjD1

a0jpwjpp C ap

2 In general @�.w;p;K/@K

cannot be measured directly from the estimating equations (a) y D@�.w;p;K/

@p, (b) x D � @�.w;p;K/

@w, e.g. when � D K

Pi

Pj aijpvipvj C aKKK,

(v � .w;p/). Estimates of @�.w;p;K/@K

can be obtained either by estimating � D �.w;p;K/jointly with (a)–(b) and then calculating directly @�.w;p;K/

@K, or by estimating (a)–(b) and then

calculating @�.w;p/

@KD p @y.w;p/

@K�PNiD1wi

@xi .w;p/

@K. This last equation is derived as fol-

lows, �.�w;�p;K/ D ��.w;p;K/) @�.�w;�p;K/

@KD � @�.w;p;K/

@K)

@�.w;p;K/

@KD

p @2�.w;p;K/

@p@KCPNiD1wi

@2�.w;p;K/

@wi@K(Euler’s theorem), and then applying (2.19).


(this is a Generalized Leontief functional form, which imposes homogeneity of de-gree one in prices on the profit function). This leads to the following equations forestimation

y D aC

NXjD1

a0j

�wj

p

� 12

xi D

NXjD1

aij

�wj

wi

� 12

C ai0

�p

wi

� 12

i D 1; � � � ; N

(2.22)

(applying Hotelling’s Lemma). Here the symmetry restrictions are expressed asaij D aj i (i; j D 1; � � � ; N ) and a0i D �ai0 (i D 1; � � � ; N ). If and only ifthe symmetry and second order conditions are satisfied, then equations (2.21) canbe interpreted as being derived from a profit function �.w; p/ for a produces show-ing static, competitive profit maximizing behavior. Similar comments apply to theestimation of a restricted profit function �.w; p;K/.

Here we can note three major advantages of this approach to the modeling ofproducer behavior. First, it enables us to specify systems of output supply and factordemand equations that are consistent with profit maximization and with a generalspecification of technology. This cannot be achieved by estimating a productionfunction directly together with first order conditions for profit maximization. Sec-ond, modeling a dual profit function explicitly allows for the endogeneity of outputlevels to the producer. This is in contrast to cost functions c.w; y/ where outputgenerally is treated as exogenous. Third, aggregation problems are likely to be lesssevere for unrestricted dual profit functions than for dual cost functions. If all firmsf D 1; � � � ; F (a constant number of firms over time) face identical prices .w; p/then it is clear that a profit function can be well defined for aggregate market data:

FXfD1

�f .w; p/ D ˆ.w; p/ � ….w; p/: (2.23)

In the special case of profit maximization at identical prices, a cost function also iswell defined for data aggregated over firms even though the output level yf variesover firms (we shall use this in a later lecture); but in this case we may as wellestimate ….w; p/ directly.

The disadvantage of estimating a profit function �.w; p/ rather than a cost func-tion c.w; y/ is that the former imposes stronger behavioral assumptions which areoften very unrealistic. For example, at the time of production decisions, farmers gen-erally have better knowledge of input prices than of output prices forthcoming at thetime of marketing in the future. Thus risk aversion and errors in forecasting pricesare more likely to influence the choice of output levels rather than to contradict thehypothesis of cost minimization. In addition in the case of food retail industries,hypothesis of oligopoly behavior plus competitive cost minimization may be morerealistic than the hypothesis of competitive profit maximization.

2.7 Industry profit functions and entry and exit of firms 21

2.7 Industry profit functions and entry and exit of firms

As mentioned above, an industry profit function is well defined provided that allfirms face identical prices .w; p/ and the composition of the industry in terms offirms does not change. In this case the industry profit function is simply the sum ofthe profit functions of the firms f D 1; � � � ; F : ….w; p/ D

PFfD1 �f .w; p/ for all

.w; p/.A more realistic assumption is that changes in prices .w; p/ induce some estab-

lished firms to exit the industry and some new firms to enter the industry. Supposethat there is free entry and exit to the industry (all firms in the industry earn non-negative profits because all firms that would earn negative profits at .w; p/ are ableto exit the industry, and all firms excluded from the industry are also at long-runequilibrium). Then the industry profit function inherits essentially the some proper-ties as the individual firm’s profit function �f .w; p/.

Proposition 2.1 provides a characterization of the industry profit function �.w; p/,assuming (a) competitive behavior, (b) both input prices w D .w1; � � � ; wN / andoutput prices p are exogenous, and (c) a continuum of firms and free entry/exit tothe industry.

The role of the assumption of a continuum of firms deserves comment. As notedby Novshek and Sonnenschein (1979) in the case of marginal consumers, an in-finitesimal change in price will lead to entry/exit behavior only in the case of acontinuum of agents. Therefore it is necessary to assume that there exists a contin-uum of firms. Industry profits are calculated by integrating over the continuum offirms in the industry:

….w; p/ D

Z fm.w;p/

1

�.w; p; f /�.f / df

where �.f / is the density of firms f . �.w; p; f / denotes the individual firm’s profitfunction conditional on the firm being in the industry, and industry profits are ob-tained by integrating over those firms in the industry given prices .w; p/ and freeentry/exit.

Adapting the arguments of Novshek and Sonnenschein, the assumption of a con-tinuum of firms also establishes the differentiability of the industry profit function….w; p/, industry factor demands X.w; p/ and industry output supplies Y.w; p/with free entry/exit. The argument can be outlined as follows. Since the individualfirm’s profit function �.w; p; f / is conditional on firm f remaining in the indus-try, it is reasonable to assume that �.w; p; f / is twice differentiable in .w; p/, i.e.,�.w; p; f / is not kinked at � D 0 due to exit from the industry. �.w; p; f / is dif-ferentiable in f and the derivative �f .w; p; f / < 0 assuming a continuum of firmsindexed in descending order of profits. Then (by the implicit function theorem) amarginal firm f m D f m.w; p/ is defined implicitly by the zero profit condition�.w; p; f m/ D 0, and f m.w; p/ is differentiable. Under these assumptions it caneasily be shown that the industry profit function with free entry/exit is differentiable,and its derivatives …w.w; p/, …p.w; p/ can be calculated by applying Leibnitz’s


rule to the above equation for industry profits (note that the zero profit condition�.w; p; f m/ D 0 established Hotelling’s Lemma at the industry level). A similarprocedure establishes the differentiability of industry factor demands X.w; p/ andoutput supplies Y.w; p/ with free entry/exit.

Proposition 2.1. Assume that the industry consists of a continuum of firms such thateach individual firm’s profit function �.w; p; f / is twice differentiable in .w; p/,differentiable in f , �f .w; p; f / < 0, and is linear homogeneous and convexin .w; p/ and satisfies Hotelling’s Lemma. Also assume �.w; p; f m/ D 0 for amarginal firm f m. Then �.w; p/ is linear homogeneous and convex in .w; p/.Moreover, it also satisfies Hotelling’s Lemma, i.e.,

�Xi .w; p/ D@….w; p/

@wii D 1; � � � ; N

Yk.w; p/ D@….w; p/

@pkk D 1; � � � ; N

(2.24)

and industry derived demands X.w; p/ and output supplies Y.w; p/ are differen-tiable.

Proof. Given prices .w; p/ and free entry/exit, index firms in the industry in de-scending order of profits. Industry profits can be calculated as

….w; p/ D

Z fm.w;p/

1

�.w; p; f / �.f / df (2.25)

where �.f / is the density of firms f and a marginal firm f m satisfies the zero profitcondition

�.w; p; f m/ D 0: (2.26)

Assuming �.w; p; f / differentiable in .w; p; f / and �f .w; p; f / < 0, the zeroprofit condition (2.26) establishes (using the implicit function theorem) f m Df m.w; p/ is defined and differentiable. Differentiability of �.w; p; f / and f m.w; p/establishes (using (2.25) ….w; p/ is differentiable). Applying Leibnitz’s rule to(2.25), Pro:2.1

@….w; p/

@wiD

Z fm.w;p/

1

@�.w; p; f /

@wi�.f / df C � .w; p; f m.w; p// � .f m.w; p//

@f m.w; p/

@wi

D �Xi .w; p/ i D 1; � � � ; N

(2.27)

@….w; p/

@pkD

Z fm.w;p/

1

@�.w; p; f /

@pk�.f / df C � .w; p; f m.w; p// � .f m.w; p//

@f m.w; p/

@pk

D Yk.w; p/ k D 1; � � � ;M

(2.28)

2.8 Application of dual profit functions in econometrics: II 23

using Hotelling’s Lemma for the individual firm in the industry and the marginalcondition � .w; p; f m.w; p// D 0. Similarly

xi .w; p/ D

Z fm.w;p/

1

xi .w; p; f / �.f / df i D 1; � � � ; N (2.29)

yk.w; p/ D

Z fm.w;p/

1

yk.w; p; f / �.f / df k D 1; � � � ;M (2.30)

Differentiability of x.w; p; f /, y.w; p; f /, f m.w; p/ implies (using (2.29)–(2.30))X.w; p/, Y.w; p/ differentiable. ….w; p/ homogeneous of degree one in .w; p/follows from (2.25): �.w; p; f / is linear homogeneous in .w; p/ for all f , andthe continuum of firms 1; � � � ; f m.w; p/ is invariant to equiproportional changesin .w; p/. Convexity of ….w; p/ can be established as follows. Define a profitfunction N�.w; p; g/ for each firm g allowing for free entry/exit to the industry:N�.w; p; g/ � 0 for all .w; p; g/, and N�.w; p; g/ D 0 when the firm has exited theindustry in order to avoid negative profits. Thus ….w; p/ with entry/exit is the sumor integral over the continuum of profit functions N�.w; p; g/ for the fixed set of po-tential firms. N�.w; p; g/ is convex in .w; p/ by standard arguments (e.g. McFadden1978), and in turn ….w; p/ is convex in .w; p/. ut

2.8 Application of dual profit functions in econometrics: II

Our development of Hotelling’s Lemma when the number of firms is variable tothe industry has important implications for empirical studies. An industry profitfunction ….w; p/ satisfies Hotelling’s Lemma in two extreme cases: the numberof firms in the industry is fixed (the standard case) or there is free entry/exit tothe industry (Proposition 2.1). In intermediate cases, where the number of firms isvariable but entry/exit is not instantaneous and costless, the Lemma does not apply.This can be seen from equations (2.29) and (2.30) in the proof of Proposition 2.1:applying Leibnitz’s rule to the integral for industry profits over a continuum of firmswith entry/exit

….w; p/ D

Z fm.w;p/

1

�.w; p; f /�.f / df

yields Hotelling’s Lemma only if there exist marginal firms earning zero profits, i.e.,�.w; p; f m.w; p// D 0 (assume for the sake of argument that f m.w; p/ is differ-entiable in the intermediate case). This zero profit condition is characteristic of freeentry/exit with a continuum of firms but it is not characteristic of the intermediatecase.

The usual assumptions that are acknowledged in standard applications of Hotelling’sLemma to industry-level data are that each firm in the industry shows static com-petitive profit maximizing behavior (conditional on the firm being in the industry).In addition we must generally add the restrictive assumption that the composition


of the industry does not change over the time period of the data or there is freeentry/exit to the industry.

Nevertheless there is at least in principle a simple procedure for avoiding this ad-ditional restrictive assumption: the industry profit function can be defined explicitlyas conditional upon the number of firms of each different type. For example, supposethat an industry consists of two homogeneous types of firms in variable quantitiesF1 and F2. The industry profit function can be written as ….w; p; F1; F2/, whereF1 nd F2 are specified as parameters along with .w; p/. Then X.w; p; F1; F2/ D�…w.w; p; F1; F2/, Y.w; p; F1; F2/ D …p.w; p; F1; F2/ by standard argument(e.g. Bliss). Of course in practice reasonable data on the number of firms by typeis not always available, but whenever possible such modifications seem likely toimprove the specification of the model.

Thus, if we have time series data on the number of firms F1 and F2 in the twoclasses as well as data on total output, total inputs and prices, then we can postu-late an industry profit function ….w; p; F1; F2/ conditional on F1, F2 and applyHotelling’s Lemma to obtain the estimating equations

Y D@….w; p; F1; F2/

@p

Xi D �@….w; p; F1; F2/

@wii D 1; � � � ; N:

(2.31)

If there is free entry/exit to the industry, then (by 2.1) parameters F1 and F2 dropout of the system of estimating equations (2.31)—in this manner the assumptionof long-run industry equilibrium is easily tested. If there is not free entry/exit andif the number of firms F1 and F2 varies over time, then the parameters F1 and F2are significant in equations (2.31). The stocks of firms F1 and F2 at any time tprobably can be approximated as predetermined at time t (i.e., the number of firmsis essentially inherited from the past, given substantial delays in entry and exit).Then F1;t and F2;t do not necessarily covary with the disturbance terms at time tfor equations (2.31), so that equations (2.31) may be estimated consistently even ifthere is costly entry/exit to the industry.

However for policy purposes it may be desirable to estimate (2.31) jointly withequations of motion for the number of firms:

F1;tC1 � F1;t D F1.wt ; pt ; F1;t ; � � � /

F2;tC1 � F2;t D F2.wt ; pt ; F2;t ; � � � /(2.32)

Equations (2.31) can indicate the short-run impact of price policies or industry out-put and inputs levels .Y;X/, i.e., the impact of the price policies before there is anadjustment in the number of firms.

Equations (2.31) and (2.32) jointly indicate intermediate and long-run impacts ofprice policies. For example, at a static long-run equilibrium there is no exit/entry tothe industry, so the long-run equilibrium numbers of firms F ��1 , F ��2 for any .w; p/can be calculated from (2.32) by solving the implicit equations F1.wt ; pt ; F ��1;t / D

References 25

0, F2.wt ; pt ; F ��2;t / D 0. Obvious difficulties here are (a) problems in specifyingthe dynamics of entry and exit (equations (2.32)) correctly, and (b) dangers in using(2.31) to extrapolate to long-run equilibrium numbers of firms F ��1 , F ��2 that areoutside of the data set.

References

1. Fuss, M.& McFadden, D. (1978). Production Economics: A Dual Approach to Theory andApplications: The Theory of Production, History of Economic Thought Books, McMasterUniversity Archive for the History of Economic Thought

2. Jarvis, L. (1974). Cattle as Capital Goods and Ranchers as Portfolio Managers: An Applica-tion to the Argentine Cattle Sector, Journal of Political Economy, pp. 489–520

3. Lau, L. (1976). A Characterization of the Normalized Restricted Profit Function, Journal ofEconomic Theory, pp. 131–163.

4. Novshek, W. & Sonnenschein, H .(1979). Supply and marginal firms in general equilibrium,Economics Letters, Elsevier, vol. 3(2), pp. 109–113.

5. Samuelson, P. A. (1947). Enlarged ed., 1983. Foundations of Economic Analysis, HarvardUniversity Press.

Chapter 3Static Utility Maximization and ExpenditureConstraints

Here we model both consumer and producer behavior subject to expenditure con-straints. We begin with the case of consumer.

Consider a consumer maximizing utility u by allocating his income y amongN commodities x D .x1; � � � ; xN /, and denote his utility function as u D u.x/.Assume that the consumer takes commodity prices p D .p1; � � � ; pN / as given andsolves the following static competitive utility maximization problem:

maxx�0

u.x/ D u.x�/

s.t.NXiD1

pixi � y(3.1)

The maximum utility V � u.x�/ to problem (3.1) depends on prices and income.p; y/ and the consumer’s utility function u D u.x/. The corresponding relationV D V.p; y/ between maximum utility and prices and incomes is denoted as theconsumer’s dual indirect utility function.

A necessary condition for utility maximization (3.1) is that the consumer attainsthe utility level u.x�/ at a minimum cost. In other words, if x� does not minimizethe cost

PNiD1 pixi subject to u.x/ D u.x�/, then a higher utility level u > u.x�/

can be attained at the same costPNiD1 pix

�i D y (for this result we only need to

assume local nonsatiation of u.x/ in neighborhood of x�). Thus a solution x� tothe utility maximization problem (3.1) also solves the following cost minimizationproblem when the exogenous utility level u is equal to u.x�/:

minx�0

NXiD1

pixi D

NXiD1

pix��i

s.t. u.x/ � u

(3.2)

The minimum costE DPNiD1 pix

��i to problem (3.2) depends on prices and utility

level .p; u/ and the consumers utility function u.x/. The corresponding relation

27

28 3 Static Utility Maximization and Expenditure Constraints

between minimum expenditure and prices and utility level,E D E.p; u/, is denotedas the consumers dual expenditure function. Note that (3.2) is formally equivalent tothe producer’s cost minimization problem minx�0

PNiD1wixi s.t. f .x/ � y (1.1).

Thus the expenditure function E.p; u/ inherits the essential properties (1.1) of theproducers cost function c.w; y/:

Property 3.1.

a) E.p; u/ is increasing .p; u/.b) E.p; u/ is linear homogeneous in p.c) E.p; u/ is concave in p.d) If E.p; u/ is differentiable in p, then

xi .p; u/ [email protected]; u/

@pii D 1; � � � ; N: (Shephard’s Lemma)

Also note that a sufficient condition for utility maximization (3.1) is that theconsumer attains the utility level u.x�/ at a minimum cost equals to

PNiD1 pix

�i .

In other words, if xA solves (3.2) subject to u.x/ D uA, then xA also solves (3.1)subject to

PNiD1 pixi D y �

PNiD1 pix

Ai .

Proof. Assume the that u.x/ is continuous and xA solves (3.2) at the exogenouslydetermined utility level u � uA. Now suppose that x� rather than xA solves (3.1) atthe exogenously determined expenditure level yA �

PNiD1 pix

Ai > 0. This would

imply u.x�/ > u.xA/ and (given local nonsatiation)PNiD1 pix

�i D

PNiD1 pix

Ai .

Then continuity of u.x/would imply that there exits an Qx in the neighborhood of x�

such that u.x�/ � u. Qx/ � u.xA/ andPNiD1 pi Qxi <

PNiD1 pix

Ai , which contradicts

the assumption xA solves (3.2). Therefore xA solves (3.2) implies xA solves (3.1).ut

Thus xA solves (3.1) if and only if xA solves (3.2) subject to u � u.xA/. Thisimplies that the restrictions placed on Marshallian consumer demands x D x.p; y/(corresponding to problem (3.1)) by the hypothesis of utility maximization (3.1) canbe analyzed equivalently in terms of the restrictions placed on Hicksian consumerdemands x D xh.p; u/ (corresponding to problem (3.2)) by the hypothesis of costminimization (3.2). In other words, the hypothesis of cost minimization (3.2) ex-hausts the restrictions placed on Marshallian demands x D x.p; y/ by the hypoth-esis of utility maximization. Properties 3.1 of E.p; u/ imply

Property 3.2.

a) x.p; u/ are homogenous of degree 0 in p. i.e. x.�p; u/ D x.p; u/ for allscalar � > 0.

3.1 Properties of V.p; y/ 29

b)�@x.p; u/

@p

�N�N


And properties 3.2 exhausts the implications of cost minimization for the (local)properties of Hicksian demands xh.p; u/ (the proof is the same as in the case ofcost minimization by a producer). Therefore, by the proof immediately above, 3.2also exhausts the implications of utility maximization (3.1) for the (local) propertiesof Marshallian demands x.p; y/, where 3.2 is evaluated at a utility maximizationu D u .x.p; y//. Of course the characterization of utility maximization in termsof (3.2) is not immediately useful empirically in the sense that utility level u is notobserved (nor is u exogenous to the consumer).

This relation between utility maximization and cost minimization is very differ-ent from the relation between profit maximization and cost minimization for theproducer: profit maximization implies but is not equivalent to cost minimizationin any sense. The explanation is that in the consumer case, in contrast to the pro-ducer case, maximization is subject to an expenditure constraint defined over allcommodities.

3.1 Properties of V.p; y/

Property 3.3.

a) V.p; y/ is decreasing in p and increasing in y.b) V.p; y/ is homogenous of degree 0 in .p; y/. i.e. V.�p; �y/ D V.p; y/ for

all scalar � > 0.c) V.p; y/ is quasi-convex in p. i.e. fp W V.p; y/ � kg is a convex set for all

scalar k � 0. See Figure 3.1d) If V.p; y/ is differentiable in .p; y/, then

xi .p; y/ D �@V.p; y/=@pi

@V.p; y/=@yi D 1; � � � ; N

(Roy’s Theorem)

Proof. Properties 3.3.a–b follows simply from the definition of the consumer’s max-imization problem (3.1). For a proof of 3.2.c see Varian (1992, pp. 121–122). In or-der to prove 3.3.d note that, if u� is the maximum utility for (3.1) given parameters.p; y/, then u� � V.p; y�/ where y� � E.p; u�/, i.e. y� is the minimum expen-diture necessary to attain a utility level u� given prices p. Total differentiating this


Fig. 3.1 V.p; y/ is quasi-convex in p

identity u� � V .p;E.p; u�// with respect to pi ,

0 [email protected]; y�/

@[email protected]; y�/

@y

@E.p; u�/

@pii D 1; � � � ; N (3.3)

which yields 3.3.b since @E.p; u�/=@pi D xhi .p; u�/ D xi .p; y

�/, i D 1; � � � ; N .ut

Roy’s Theorem and V.p; y/ quasi-convex in p can also be derived as follows.Since V.p; y/ � maxx u.x/ s.t. px D y,

V.p; y/ � u.x/ � 0 for all .p; y; x/ such that px D y (3.4)

or equivalently

V�p;PNiD1 pixi

�� u.x/ � 0 for all .p; x/ (3.5)

where y is now defined implicitly as y DPNiD1 pixi . By (3.5),

G.p; x/ � V�p;PNiD1 pixi

�� u.x/ � 0 for all p

G.p; x/ � V�p;PNiD1 pixi

�� u.x/ D 0 at p such that x D x.p; y/

for y �PNiD1 pixi

(3.6)

In other words, given that xA solves (3.1) conditional on .pA; yA/, then G.p; xA/attains a global minimum over p at pA. This implies the following first and secondorder conditions for maximization:

@G.pA; xA/

@piD 0 i D 1; � � � ; N�

@2G.pA; xA/

@p@p

�N�N

is symmetric positive semidefinit.(3.7)

Since G.p; x/ � V�p;PNiD1 pixi

�� u.x/, conditions (3.7) imply

3.2 Corresponding properties of x.p; y/ solving problem (3.1) 31

@G.p; x/

@pi�@V.p; y/

@[email protected]; y/

@yxi D 0 i; j D 1; � � � ; N (3.8a)

@2G.p; x/

@pi@pj�@2V.p; y/

@pi@[email protected]; y/

@pi@yxj C

@2V.p; y/

@y@pixi C

@2V.p; y/

@y@yxjxi

(3.8b)

(3.8a) is Roy’s [email protected];x/@p@p

iN�N

symmetric positive semidefinite (3.7) im-plies semidefinite—these are the second order restrictions implied by V.p; y/ quasi-convex in p.

3.2 Corresponding properties of x.p; y/ solving problem (3.1)

Property 3.4.

a) x.p; y/ is homogeneous of degree 0 in .p; y/. i.e. x.�p; �y/ D x.p; y/ forall scalar � � 0.

b)@xi .p; y/

@pjD@xhi .p; u

�/

@pj�@xi .p; y/

@yxj .p; y/ i; j D 1; � � � ; N

(Slutsky equation) where u� � V.p; y/ [email protected];u�/

@p

iN�N

is symmetricnegative semidefinite.

Proof. Property 3.4.a is obviously true. In order to prove 3.4.b, let u� be the max-imal utility for problem (3.1) conditional on .p; y/. Then we have proved the fol-lowing identity (see pages 27–28): xi .p; y/ � xhi .p; u

�/ where y � E.p; u�/,i.e.

xhi .p; u�/ � xi

�p;E.p; u�/

�i D 1; � � � ; N (3.9)

In words, the Hicksian and Marshallian demands xh.p; u/ and x.p; y/ are equalwhen xh.p; u/ are evaluated at a utility level u D u� solving (3.1) for prices p anda given income y and x.p; y/ are evaluated at p and a level y � E.p; u�/ solving(3.2) given p and the level u D u�. Differentiating (3.9) with respect to pj (holdingutility level u� constant) yields

@xhi .p; u�/

@pjD@xi .p; y/

@pjC@xi .p; y/

@y

@E.p; u�/

@pji; j D 1; � � � ; N: (3.10)

Since (using Shephard’s Lemma) @E.p; u�/=@pj D xj .p; u�/ D xj .p; y/ .j D

1; � � � ; N / [email protected];u�/

@p

iN�N

is symmetric negative semidefinite from 3.2.b, (3.10)establishes 3.4.b. ut


The integrability problem has traditionally been prosed as follows: given a set ofMarshallian consumer demand relations x D x.p; y/, what restrictions on x.p; y/exhaust the hypothesis of competitive utility maximizing behavior by the consumer?We can now construct an answer as follows. Since utility maximization (3.1) and ex-penditure minimization subject to u.x/ D u� are equivalent (see pages 27–28), wecan rephrase the question as: what restrictions on x.p; y/ exhaust the hypothesis ofcompetitive cost minimizing behavior by the consumer? Using the Slutsky equation3.4.b we can recover the matrix of substitution effects

[email protected];u�/

@p

iN�N

for the cost

minimization demands xh.p; u�/ from the Marshallian demands x.p; y/. xh.p; u�/homogeneous of degree 0 in p and symmetry of


@p

iN�N

imply the set

of differential equations xhi .p; u�/ D @E.p; u�/=@pi , i D 1; � � � ; N (Shephard’s

Lemma) (see page 4). By the Frobenius theorem, these differential equations can beintegrated up to a macrofunction E.p; u�/ if and only if


@p

iN�N

is sym-

metric. [email protected];u�/

@p

iN�N

negative semidefinite implies (by Shephard’s

Lemma)[email protected];u�/@p@p

iN�N

negative semidefinite, which in turn implies E.p; u�/concave and u.x/ quasi-concave at x.p; y/. Thus Marshallian demand relationsx D x.p; y/ can be interpreted as being derived from competitive utility maximiz-ing behavior if and only if�

@x.p; y/

@p

�N�N

C

�@x.p; y/

@y

�N�1

Œx.p; y/�1�N �

"@xh.p; u�/

@p

#N�N

is symmetric negative semidefinite,(3.11a)

x.p; y/ homogenous of degree 0 in .p; y/.1 (3.11b)

3.3 Application of dual indirect utility functions in econometrics

The above theory is usually applied by first specifying a functional form .p; y/

for the indirect utility function V.p; y/ and differentiating .p; y/ with respect to.p; y/ in order to obtain the estimating equations

xi D �@ .p; y/=@pi

@ .p; y/=@yi D 1; � � � ; N (3.12)

1 x.p; y/ homogenous of degree 0 in .p; y/ implies xh.p; u�/ homogenous of degree 0 in p,since x.p; y/ � xh .p;V.p; y//.

3.3 Application of dual indirect utility functions in econometrics 33

(using Roy’s Theorem). .p; y/ is usually specified as being homogeneous of de-gree 0 in .p; y/.

The symmetry conditions

@2 .p; y/

@pi@pjD@2 .p; y/

@pj @pi;@2 .p; y/

@pi@yD@2 .p; y/

@y@pii; j D 1; � � � ; N

(3.13)are satisfied if and only if corresponding Hicksian demand relations xhi .p; u

�/ D

@E.p; u�/=@pi .i D 1; � � � ; N / integrate up to a macrofunction E.p; u�/.

Proof. By the Frobenius theorem, the Hicksian demands xhi .p; u�/ D @E.p; u�/=@pi

.i D 1; � � � ; N / integrate up to a macrofunction if and only if xhi .p; u�/=@pj D

xhj .p; u�/=@pi .i; j D 1; � � � ; N /. Differentiating xi D �

@V.p;y/=@[email protected];y/=@y

(Roy’s Theo-rem) with respect to .p; y/ and substituting into the Slutsky equation 3.4.b yields

�@xhi .p; u

�/

@pjDVpipj Vy � Vypj Vpi

VyVyCVpiyVy � VyyVpi

VyVy�

��Vpj

Vy

�(3.14)

where Vpipj �@@pj

�@V.p;y/@pi

�, Vypj �

@@pj

�@V.p;y/@y

�, Vpjpi �

@@pj

�@V.p;y/@pi

�,

etc. Inspection of (3.14) shows that @xhi .p; u�/=@pj D @x

hj .p; u

�/=@pi if and onlyif Vpipj D Vpjpi , Vypi D Vpiy .i; j D 1; � � � ; N /. ut

Thus the hypothesis of competitive utility maximization is verified by (a) testingfor the symmetry restrictions (3.13) and (b) checking the second order conditions(3.8b) at all data points .p; y/ or (equivalently) using Slutsky relations to recoverthe Hicksian matrix @xh=@p and checking for negative semidefiniteness.

For example an indirect utility function could be postulated as having the func-tional form V D y=

Pi

Pj aijp

1=2i p

1=2j (a Generalized Leontief reciprocal in-

direct utility function with homotheticity). Applying Roy’s Theorem leads to thefollowing functional form for the consumer demand equations:

xi DyPNjD1 aij .pj =pi /

1=2PNjD1

PNkD1 ajkp

1=2j p

1=2

k

i D 1; � � � ; N: (3.15)

Here the symmetry restrictions (3.13) are expressed as aij D aj i .i; j D 1; � � � ; N /which are easily tested. Equations (3.15) can be interpreted as being derived from anindirect utility function V.p; y/ D y=

Pi

Pj aijp

1=2i p

1=2j for a consumer show-

ing static competitive utility maximizing behavior if and only if the symmetry andsecond order conditions are satisfied.

The major advantage of this approach to modeling consumer behavior is that itpermits the specification of a system of Marshallian commodity demand equationsx D x.p; y/ that are consistent with utility maximization and with a very gen-eral specification of the consumer’s utility function.2 In contrast, even if an accept-

2 The high degree of flexibility of V.p; y/ in representing utility functions (consumer preferences)u.x/ follow from the fact that the first and second derivatives of V.p; y/ determine the first and


able measure of utility is defined and the first order conditions @u.x�/=@[email protected]�/=@xj

D pi=pj

.i; j D 1; � � � ; N / are estimated, the corresponding Marshallian demand equationsx D x.p; y/ can be recovered explicitly only under very restrictive functional formsfor u.x/ (e.g. Cobb-Douglas).

A serious problem with this and all other consumer demand models derived frommicrotheory (behavior of the individual consumer) is that the data is usually aggre-gated over consumers. Difficulties raised by the use of such market data will bediscussed in a later lecture.

3.4 Profit maximization subject to budget constraints

The above model (3.1) of consumer behavior subject to a budget constraint is for-mally equivalent to the following model of producer behavior subject to a budgetconstraint

maxx�0

(pf .x/ �

NXiD1

wixi

)� �.w; p; b/

s.t.NXiD1

wixi D b

(3.16)

since (3.16) andmaxx�0

pf .x/ � R.w; p; b/

s.t.NXiD1

wixi D b(3.17)

have identical solutions x� assuming that the budget constraint is binding. Herepurchases of all inputs are assumed to draw upon the same total budget b, and thereis a single output with production function y D f .x/. In this case the solution x�

to (3.16) also solves the cost minimization problem

minx�0

wixi � c.w; y�/

s.t. f .x/ D y�(3.18)

where y� D f .x�/. Note that cost minimization (conditional on y�) is sufficient aswell as necessary for a solution to (3.16) (see pages 27–28), in contrast to the caseof profit maximization without a budget constraint.

second derivatives of the corresponding utility function u.x/ at x.p; y/. This can be seen by dif-ferentiating the first order conditions @u.x

�/

@xiD

@V.p;y/

@ypi ,

PNiD1 pix

�iD y .i D 1; � � � ;N/

for utility maximization and employing Roy’s Theorem, in a manner analogous to the derivationof (1.6).

3.4 Profit maximization subject to budget constraints 35

The envelop relations and second order conditions for (3.16) and (3.17) are easilyderived as follows. (3.16) implies

.w; p; b; x/ � �.w; p; b/ � fpf .x/ � wxg � 0

for all .w; p; b; x/ such that wx D b

.w; p; b; x.w; p; b// D 0

(3.19)

or, substituting wx for b in �.�/,

Q .w; p; x/ � �.w; p;wx/ � fpf .x/ � wxg � 0 for all .w; p; x/Q .w; p; x.w; p; b// D 0

(3.19’)

Thus Q �w;p; x.w0; p0; b0/

�attains a minimum over .w; p/ at .w0; p0/. This im-

plies the following first order conditions for a minimum:

@ Q .w0; p0; x0/

@p�@�.w0; p0; b0/

@p� f .x0/ D 0 i D 1; � � � ; N

@ Q .w0; p0; x0/

@wi�@�.w0; p0; b0/

@wiC@�.w0; p0; b0/

@bx0i C x

0i D 0

(3.20)

i.e. (3.16) implies

y.w; p; b/ D@�.w; p; b/

@p

xi .w; p; b/ D �@�.w; p; b/=@wi

1C @�.w; p; b/=@bi D 1; � � � ; N

(3.21)

Note that (3.21) reduces to Hotelling’s Lemma if @�.w; p; b/=@b D 0 , i.e. if thebudget constraint is not binding. The second order conditions for a minimum ofQ �w;p; x.w0; p0; b0/

�over prices .w; p/ are

�Q w;p.w; p; x

0/�

positive semidefi-nite. Twice differentiating the identity Q .w; p; x/ � �.w; p;wx/� fpf .x/ � wxgwith respect to prices .w; p/ yields

@2 Q .w; p; x/

@wi@wj�@2�.w; p; b/

@wi@wjC@2�.w; p; b/

@wi@bxj C

@2�.w; p; b/

@b@wjxi C

@2�.w; p; b/

@b@bxixj

@2 Q .w; p; x/

@wi@p�@2�.w; p; b/

@wi@pC@2�.w; p; b/

@b@pxi

@2 Q .w; p; x/

@p@wi�@2�.w; p; b/

@p@wiC@2�.w; p; b/

@p@bxi

@2 Q .w; p; x/

@p@p�@2�.w; p; b/

@p@pi; j D 1; � � � ; N

(3.22)


Thus profit maximization subject to a budget constraint wx D b implies that the.N C 1/-dimensional matrix defined in (3.22) is symmetric positive semidefinite.

Likewise (3.17) implies

�.w; p; x/ � R.w; p;wx/ � pf .x/ � 0 for all .w; p; x/� .w; p; x.w; p; b// D 0

(3.23)

Proceeding as above we obtain the envelop relations

y [email protected]; p; b/

@p

xi D �@R.w; p; b/=@wi

@R.w; p; b/=@b

(3.24)

and second order conditions analogous to (3.22) symmetric positive semidefinite.Nevertheless, assuming an expenditure constraint over all inputs and a single

output, the simplest approach is to estimate a cost function c.w; y/ using Shephard’sLemma to obtain estimating equations

xi [email protected]; y/

@wii D 1; � � � ; N: (3.25)

Under the above assumptions the hypothesis of cost minimization conditional ony� exhausts the implications of the hypothesis of profit maximization subject toa budget constraint, although of course this approach (3.25) still mis-specifies themaximization problem (3.16) by treating output y as exogenous.

As a second and more interesting example of modeling budget constraints inproduction, suppose that expenditures on only a subset of inputs are subject to abudget constraint, (e.g. different inputs may be purchased at different times andcash constraints may be binding only at certain times, or alternatively credit may beavailable for the purchase of some but not all inputs). In this case the firm solves theprofit maximization problem

maxxA;xB�0

(pf .xA; xB/ �

NAXiD1

wAi xAi �

NBXiD1

wBi xBi

)� �.w; p; b/

s.t.NACNBXiDNAC1

wBi xBi D b

(3.26)

(3.26) implies

.w; p; x/ � �.w; p;wBxB/ � fpf .xA; xB/ � wAxA � wBxBg

� 0 for all .w; p; x/D 0 for .w; p; x.w; p; b//

(3.27)

3.4 Profit maximization subject to budget constraints 37

Proceeding as before leads to the envelop relations

y D@�.w; p; b/

@p

xAi D �@�.w; p; b/

@wAii D 1; � � � ; N

xBi D �@�.w; p; b/=@wBi1C @�.w; p; b/=@b

i D NA C 1; � � � ; NA CNB

(3.28)

and to the second order relations

@2 .w; p; x/

@wBi @wBj

�@2�.w; p; b/

@wBi @wBj

C@2�.w; p; b/

@wBi @bxBj C

@2�.w; p; b/

@b@wBjxBi C

@2�.w; p; b/

@b@bxBi x

Bj

i; j D 1; � � � ; NB

@2 .w; p; x/

@wBi @wAj

�@2�.w; p; b/

@wBi @wAj

C@2�.w; p; b/

@b@wAjxBi i D 1 � � � ; NB j D 1; � � � ; NA

@2 .w; p; x/

@wBi @p�@2�.w; p; b/

@wBi @pC@2�.w; p; b/

@b@bxBi i D 1 � � � ; NB j D 1; � � � ; NA

@2 .w; p; x/

@wAi @wAj

�@2�.w; p; b/

@wAi @wAj

i D 1; � � � ; NA

@2 .w; p; x/

@wAi @p�@2�.w; p; b/

@wAi @pi D 1; � � � ; NA

@2 .w; p; x/

@p@p�@2�.w; p; b/

@p@p(3.29)

The .NA C NB C 1/-dimensional matrix defined by (3.29) should be symmetricpositive semidefinite, assuming profit maximization subject to a binding budget con-straint wBxB D b.

The above theory of profit maximization subject to a budget constraint can beapplied by first specifying a functional form .w; p; b/ for the profit function�.w; p; b/ and differentiating .w; p; b/with respect to .w; p; b/ in order to obtainthe estimating equation

y D@ .w; p; b/

@p

xAi D �@ .w; p; b/

@wAii D 1; � � � ; NA

xBi D �@ .w; p; b/=@wBi1C @ .w; p; b/=@b

i D NA C 1; � � � ; NA CNB

(3.30)


where only inputs xB are subject to a budget constraint wBxB D b. The symmetryconditions @2

@wi@wjD

@2 @wj @wi

, @2 @wi@p

D@2 @p@wi

, .i; j D 1; � � � ; NACNB/ imply thatthe output supply and factor demand equations (3.30) integrate up to a macrofunc-tion .w; p; b/, and satisfaction of the second order conditions (3.29) (replacing� with in the derivatives) implies that .w; p; b/ can be interpreted as a profitfunction �.w; p; b/ corresponding to the behavioral model (3.26). Note that modelssuch as (3.30) may be useful in testing whether expenditures on a particular inputdraw on a binding budget, i.e. in testing whether the shadow price for the budgetconstraint @ .w; p; b/=@b is significant in the demand equation for a particular in-put.

For example, the functional form for �.w; p; b/ could be hypothesized as

� D b

�Xi

Xjaijw

12

i w12

j C

Xia0ip

12w

12

i

�(3.31)

(a Generalized Leontief functional form with constant returns to scale). This leadsto the following functional form for the output supply and factor demand equations:

y D b

NACNBXiD1

a0i

�wi

p

� 12

xAi D �b

NACNBXjD1

aij

�wj

wi

� 12

i D 1; � � � ; NA

xBi D �bPNACNBjD1 aij

�wj =wi

� 12

1CPNACNBjD1

PNACNBkD1

ajkw12

j w12

kCPNACNBjD1 a0jp

12w

12

i

i D NA C 1; � � � ; NA CNB(3.32)

Note that the particular functional form (3.31) implies @�.w; p; b/=@b D �.w; p; b/=b,so that the demand equations for inputs xB can be simplified to

xBi D �

�b2

b C �

�NACNBXjD1

aij

�wj

wi

� 12

i D NA C 1; � � � ; NA CNB (3.33)

However, in the absence of constant returns to scale, the demand equations for inputssubject to a budget constraint generally will be nonlinear in the parameters to beestimated.

References

1. Varian, H. R. (1992). Microeconomic Analysis, Third Edition. W. W. Norton & Company, 3rdedition.

Chapter 4Nonlinear Static Duality Theory (for a singleagent)

4.1 The primal-dual characterization of optimizing behavior

In previous lecture we have constructed primal-dual relations such as

G.w; p; x/ � �.w; p/ � fpf .x/ � wxg (page 33) (4.1a)QG.p; y; x/ � V.p; y/ � u.x/ ((3.6) on page 30) (4.1b)QQG.p; u; x/ � E.p; u/ � p x (4.1c)

where the first term denotes the optimal value of profits/utility/expenditure for anagent solving a profit maximization/utility maximization/cost minimization problemand the second term denotes a feasible level of profits/utility/expenditure, respec-tively. It is obvious that the primal-dual relations (4.1a)–(4.1b) attain a minimumvalue (equal to zero) at an equilibrium combination of choice variables x and pa-rameters: .w; p; x.w; p///.p; y; x.p; y//. Likewise the primal-dual relations c forcost minimization attains a maximum value (equal to zero) at an equilibrium com-bination of choice variables x and parameters: .p; u; x.p; u//. This implies that,given equilibrium levels of x� of the agent’s choice variables x, the primal-dual re-lations (4.1a)–(4.1b) attain a minimum over possible values of the parameters at theparticular level of the parameters for which x� solves the profit/utility maximizationproblem:

given x0 � x.w0; p0/ solving a profit maximization problem (2.1):

.w0; p0/ solves minw;p

G.w; p; x0/ � �.w; p/ �˚pf .x0/ � wx0

D 0 (4.2)

given x0 � x.p0; y0/ solving a utility maximization problem (3.1):

39

40 4 Nonlinear Static Duality Theory (for a single agent)

.p0; y0/ solves minp;yQG.p; y; x0/ � V.p; y/ � u.x0/ D 0 s.t. px0 D y

(4.3)

and similarly for case (4.1c), given x0 D x.p0; u0/ solving a cost minimizationproblem (3.2):

p0 solves maxp

QQG.p; u0; x0/ � E.p; u0/ � p x0 D 0 (4.4)

where u0 � u.x0/. By analyzing the first order conditions for an interior solutionto these problems, we easily derived Hotelling’s Lemma (page 12), Roy’s Theorem(page 29) and we can easily derive Shephard’s Lemma, respectively.

To repeat, it is obvious that the hypotheses of profit maximization, utility max-imization and cost minimization imply that an equilibrium combination of choicevariables and parameters are obtained by solving (4.2)–(4.4), respectively. In thissense a necessary condition for x0 to solves a profit maximization problem (2.1)conditional on prices .w0; p0/ is that .w0; p0/ solve (4.2), and similarly for utilitymaximization and cost minimization.

A further question is: does x0 solve a profit maximization problem (2.1) (e.g.)conditional on .w0; p0/ if and only if (w0; p0) solves (4.2) conditional on x0? Theanswer is yes and this implies that problems (4.2)–(4.4) exhaust the implications ofbehavioral models (2.1), (3.1) and (3.2), respectively, for local comparative staticanalysis of changes in prices.

The intuitive explanation of this result is surprisingly simple (given the confusionthat has been raised in the recent past over this matter, especially Silberberg 1974,pp. 159–72). For example, note that the profit maximizing derived demands x.w; p/solving maxx fpf .x/ � wxg � �.w; p/ also solve

minxG.w; p; x/ � �.w; p/ � fpf .x/ � wxg D 0 (4.5)

and note that any x such that G.w; p; x/ D 0 also solves maxx fpf .x/ � wxgconditional on .w; p/. Since the minimum value ofG.w; p; x/ over .w; p/ is also 0,it follows that any combination .w0; p0; x0/ solving minw;pG.w; p; x0/ also solvesminxG.w0; p0; x/, and conversely any .w0; p0; x0/ solving minxG.w0; p0; x/ alsosolves minw;p G.w; p; x0/.

Thus solving minw;p G.w; p; x0/ is equivalent to solving the profit maximiza-tion problems maxx

˚p0f .x/ � w0x

. Also note that an (interior) global solution

.w0; p0/ to minw;p G.w; p; x0/ implies for local comparative static purposes onlythat

@G.w0; p0; x0/

@pD 0;

@G.w0; p0; x0/

@wiD 0 i D 1; � � � ; N (4.6a)�

@2G.w0; p0; x0/

@p@w

�.NC1/�.NC1/

symmetric positive semidefinite (4.6b)

4.2 Producer behavior 41

i.e. only the first and second order conditions for an interior minimum are relevant.1

The implication of the above argument is that (4.6) exhaust the restrictions placedon the comparative static effects of (local) changes in prices .w; p/ by the hypoth-esis of competitive profit maximization for the individual firm. Similar conclusionshold for the cases of utility maximization (4.3) and cost minimization (4.4).

More generally, consider an optimization problem

maxxf .x; ˛/ � �.˛/

s.t. g.x; ˛/ D 0(4.7)

where both the objective function f .x; ˛/ and constraint (or vector of constraint)g.x; ˛/ D 0 are conditional on a vector of parameters ˛ (e.g. prices or parametersshifting the price schedules facing the agent). Then x0 solves (4.7) conditional on˛0 if and only if ˛0 solves the following problem conditional on x0:

min˛G.x0; ˛/ � �.˛/ � f .x0; ˛/

s.t. g.x0; ˛/ D 0(4.8)

Therefore the first and second order conditions for a solution to problem (4.8) inparameter (˛) space exhaust the implications of the behavioral model (4.7) for com-parative static effects of (local) changes in parameters ˛.2

4.2 Producer behavior

Consider the following general profit maximization problem

maxxfR.x; ˛A/ � c.x; ˛B/g � Q�.x

�; ˛/ � �.˛; b/

s.t. c.x; ˛C / D b(4.9)

where all prices may be endogenous to the producer and there is a budget constraint(or vector of constraints) c.x; ˛C / D b limiting expenditures on at least some in-puts. Hence R.x; ˛A/ denotes total revenue as a function of the output level y (orequivalent the inputs levels x, given a single output production function y D f .x/)and parameters ˛A shifting the price schedules p.y; ˛A/ facing the firm. c.x; ˛B/denotes total costs as a function of the input levels x and the parameters ˛B shifting

1 The condition G.w0; p0; x0/ D 0 for a global solution to minw;pG.w;p; x0/ implies onlythat �.w0; p0/ D p0f .x0/�w0x0, which is not directly relevant to local comparative statics.2 Likewise, if we are only interested in the comparative static effects of changes in a sub-set ˛B of parameters ˛, then x0 solves (4.7) conditional on ˛0 if and only if ˛0B solvesmin˛B G.x

0; ˛0A; ˛B/ s.t. g.x0; ˛0A; ˛B/ D 0. In turn the first and second order conditionsin the parameter ˛B subspace for this minimization problem exhaust the implications of (4.7) forthe comparative static effects of changes in parameters ˛B .


the factor supply schedules wi .x; ˛B/ .i D 1; � � � ; N / facing the firm. �.˛; b/ de-notes the dual profit function for (4.9), i.e. the relation between maximization attain-able profits and parameters .˛; b/. Formally (4.9) allows for traditional monopoly/monopsony behavior, the specification of financial constraints in terms of both fixedcash constraints (at level b) and costs of borrowing the vary with the level of bor-rowing (and the firm’s debt-equity ratio), the endogeneity of the opportunity cost(value of forgone leisure) of farm family labor, etc.

Consider the corresponding minimization problem

min˛;b

G.˛; b; x/ � �.˛; b/ � fR.x; ˛A/ � c.x; ˛B/g

s.t. c.x; ˛C / D b(4.10)

or equivalently (substituting out the budget constraint c.x; ˛C / D b)

min˛QG.˛; x/ � � .˛; c.x; ˛C // � fR.x; ˛A/ � c.x; ˛B/g (4.11)

The analysis in the previous section implies that the following first and second or-der conditions for an (interior) solution to (4.11) exhaust the implications of profitmaximization for the effects of local changes in parameters ˛: (using obvious vectornotation):

QG˛.˛; x�/ D 0

QG˛˛.˛; x�/ symmetric positive semidefinite

(4.12)

where x� D x.˛; b/ solves (4.9), and QG˛ �h@ QG@˛1� � �

@ QG@˛´

i1�Z

, QG˛˛ �h@2 QG@˛@˛

iZ�Z

.

Since QG.˛; x�/ � � .˛; c.x�; ˛C //� Q�.x�; ˛/, restrictions (4.12) can be rewrittenas

QG˛.˛; x�/ � �˛.˛; b/C �b.˛; b/c˛.x

�; ˛/ � Q�˛.x�; ˛/ D 0 (4.13a)

QG˛˛.˛; x�/ � �˛˛.˛; b/C �˛b.˛; b/c˛.x

�; ˛/C �b˛.˛; b/c˛.x�; ˛/

C �bb.˛; b/c˛.x�; ˛/c˛.x

�; ˛/C �b.˛; b/c˛˛.x�; ˛/

� Q�˛˛.x�; ˛/ symmetric positive semidefinite.

(4.13b)

In order to derive a system of estimating equations for a behavioral model of thetype (4.9), we can begin by specifying a functional form .˛; b/ for the firm’s dualprofit function �.˛; b/. Then we calculate corresponding envelop relations (4.13a).Note, however, that calculation of these envelop relations also requires knowledge ofthe derivatives c˛.x; ˛/, R˛.x; ˛/, C˛.x; ˛/,3 i.e. we must specify functional formsfor R.x; ˛/, C.x; ˛/ and c.x; ˛/ in addition to the functional form for �.˛; b/.This does not appear to cause any serious problems: although these functions arenot independent of the functional form for �.˛; b/, a flexible specification (see next

3 In the competitive case R.x; ˛A/ � pf.x/, C.x; ˛B/ � wx, c.x; ˛C / � wCxC ; So thatR˛A D y, C˛B D x, c˛C D xC .

4.3 Consumer behavior 43

lecture) of these functions and of �.˛; b/ need not severely restrict the implicitspecification of the production technology y D f .x/.

After specifying the functional forms of �.˛; b/ and R.x; ˛/, C.x; ˛/ andc.x; ˛/, we then solve the envelope relations in the form

Q�˛.x�; ˛/ � Q�b.˛; b/ c˛.x

�; ˛/ D �˛.˛; b/ at b � c.x�; ˛/ (4.14)

we see (using the Frobenius theorem) that these estimating equations integrate upto a macrofunction � .˛; c.x�; ˛// if and only if the symmetry condition (4.13b) issatisfied. Thus we test the symmetry conditions QG˛˛.˛; x�/ � �˛˛.˛; b/ symmet-ric and check the second order conditions QG˛˛.˛; x�/ positive semidefinite, using(4.13b).

4.3 Consumer behavior

Consider the following general utility maximization problem

maxxu.x/ � V.˛; y/

s.t. c.x; ˛/ D y(4.15)

where these is a nonlinear budget constraint c.x; ˛/ D y, i.e. in general prices ofthe commodities x can depend upon the levels of the commodities purchased by theconsumer as well as upon the parameters ˛. Also consider the corresponding costminimization problem

minxc.x; ˛/ � E.˛; u/

s.t. u.x/ D u(4.16)

Comparative static effects for the maximization problem (4.15) can be analyzedin term of the primal-dual relation

min˛;y

G.˛; y; x/ � V.˛; y/ � u.x/

s.t. c.x; ˛/ D y(4.17)

or equivalently (substituting out the budget constraint)

min˛QG.˛; x/ � V.˛; c.x; ˛// � u.x/ (4.18)

The first and second order conditions for a solution to (4.18) yield

QG˛.˛; x�/ � V˛.˛; y/C Vy.˛; y/ c˛.x

�; ˛/ D 0 (4.19a)


QG˛˛.˛; x�/ �V˛˛.˛; y/C V˛y.˛; y/ c˛.x

�; ˛/C Vy˛.˛; y/ c˛.x�; ˛/

C Vyy.˛; y/ c˛.x�; ˛/ c˛.x

�; ˛/C Vy.˛; y/ c˛˛.x�; ˛/

symmetric positive semidefinite

(4.19b)

Equation (4.19a) c˛.x�; ˛/ D �V˛.˛; y/=Vy.˛; y/ are a generalization of Roy’sTheorem.

Likewise comparative static effects for the minimization problem (4.16) can beanalyzed in terms of the primal-dual relation

max˛H.˛; x/ � E.˛; u.x// � c.x; ˛/ (4.20)

The first and second order conditions for a solution to (4.20) yield

H˛.˛; x�/ � E˛.˛; u

�/ � c˛.x�; ˛/ D 0 (4.21a)

H˛˛.˛; x�/ � E˛˛.˛; u

�/ � c˛˛.x�; ˛/ symmetric negative semidefinite

(4.21b)

(4.21a) c˛.x�; ˛/ D E˛.˛; u�/ is a generalization of Shephard’s Lemma.In order to derive a generalization of the Slutsky equation, note that the Hick-

sian demands xh.˛; u/ solving the cost minimization problem (4.16) for param-eters .˛; u/ also solve the utility maximization problem (4.15) for parameters.˛; b/ D .˛;E.˛; u//, i.e.

xhi .˛; u/ D xMi .˛;E.˛; u// i D 1; � � � ; N for all ˛ (4.22)

Differentiating (4.22) with respect to ˛ (holding utility level u constant),

@xhi .˛; u�/

@ j

D@xMi .˛; y/

@ j

C@xMi .˛; y/

@y

@E.˛; u�/

@ j

i D 1; � � � ; N j D 1; 2

(4.23)Substituting the generalized Shephard’s Lemma (4.21a) into (4.23),

@xMi .˛; y/

@ j

D@xhi .˛; u

�/

@ j

�@xMi .˛; y/

@y

@c.x�; ˛/

@ j

i D 1; � � �N j D 1; 2

(4.24)where @c.x�; ˛/=@ j ¤ xj .˛; y/ except in the competitive case, where c.x; ˛/ �PNiD1 ˛ixi .˛i � wi /. Also note that the second order conditions for cost minimiza-

tion (4.21b) do not reduce toh@xh.˛;u�/

@˛

isymmetric negative semidefinite except in

the competitive case.An alternative generalization of the Slutsky equation that directly relates Mar-

shallian demands to the second order conditions for cost minimization (4.21b) canbe derived as follows. The first order conditions for cost minimization E˛.˛; u�/ �c˛.x

�; ˛/ D 0 (4.21a) can be rewritten as

E˛.˛; u�/ D c˛.x

�; ˛/ for all ˛ at x� � xM .˛;E.˛; u�// (4.25)

References 45

using (4.22). Differentiating (4.25) with respect to ˛,

E˛˛.˛; u�/ D c˛˛.x

�; ˛/Cc˛x.x�; ˛/

hxM˛ .˛; y/C x

My .˛; y/E˛.˛; u

�/i

(4.26)

substituting (4.25) into (4.26) and rearranging,

c˛;x.x�; ˛/

hxM˛ .˛; y/C x

My .˛; y/ c˛.x

�; ˛/iD E˛˛.˛; u

�/ � c˛˛.x�; ˛/

symmetric negative semidefinite(4.27)

by the second order conditions (4.21b) for cost minimization.In order to derive a system of estimating equations for a behavioral model of the

type (4.15), we can begin by specifying a functional form for the consumer’s indirectutility function V.˛; y/ and the budget constraint c.x; ˛/ D y. Then we calculatecorresponding envelope relations c˛.x�; ˛/ D �V˛.˛; y/=Vy.˛; y/ (4.19a) whichwe then solve for consumer demand relations x� D x.˛; y/. Since utility maxi-mization and cost minimization are equivalent, the Marshallian demand equationsintegrate up to a macrofunction V.˛; y/ representing utility maximization behav-ioral if and only if the corresponding envelope relations c˛.x�; ˛/ D E˛.˛; u

�/

for cost minimization satisfy the symmetry and second order conditions (4.21b) forintegration up to a macrofunction E.˛; u�/ representing cost minimizing behavior.These symmetry and second order conditions are related to the parameters of theestimated Marshallian demand equations using the generalized Slutsky equations(4.27).

References

1. Silberberg, E. (1974). A revision of comparative statics methodology in economics. Journalof Economic Theory, pages 159–72

Chapter 5Functional Forms for Static Optimizing Models

5.1 Difficulties with simple linear and log-linear Models

The main purpose of this lecture is to discuss the concept of flexible functional formsand to present specific functional forms that are commonly employed with staticduality theory. In order to appreciate the potential value of such functional forms,we begin with a discussion of the problem in using simpler linear and log-linearmodels. For simplicity we restrict this discussion to cost-minimizing behavior.

First suppose that a simple linear model of a cost function c.w; y/ is formulated:

c D a0 C

NXiD1

aiwi C ayy: (5.1)

If a producer minimizes his total cost of production, then Shephard’s Lemma appliesto the cost function. Applying Shephard’s Lemma to (5.1),


@wiD ai i D 1; � � � ; N: (5.2)

i.e. the cost-minimizing factor demands x D x.w; y/ are in fact independent offactor prices and the level of output. Alternatively suppose that factor demands areestimated as a linear function of prices and output:

xi D ai0 C

NXjD1

aijwj C aiyy i D 1; � � � ; N: (5.3)

However if factor demands are homogeneous of degree 0 in prices then (by Euler’stheorem)

PNjD1

@xi .w;y/@wj

wj D 0 .i D 1; � � � ; N:/ (see footnote 2 on page 3). So(5.3) is consistent with cost minimizing behavior only if (5.3) reduces to

xi D ai0 C aiyy i D 1; � � � ; N: (5.4)

47

48 5 Functional Forms for Static Optimizing Models

i.e. cost minimizing factor demands are independent of factor prices (implying aLeontief production function).

Of course these difficulties can be circumvented by first normalizing prices on anumeraire price:

xi D ai0 C

NXjD2

aij

�wj

w1

�C aiyy i D 1; � � � ; N: (5.5)

Here the hypothesis that factor demands are homogenous of degree 0 in prices w isimposed a priori by linearizing on normalized prices; so homogeneity cannot placeany further restrictions on the functional form (5.5) for factor demands. Thus, to theextent that factor demands are homogenous of degree 0 in prices, (5.5) clearly is abetter specification of factor demands than is (5.3). Nevertheless, note that (5.5) doesimpose an symmetry on the effects of the numeraire pricew1 relative to the effects ofother prices on factor demands. Responses @c.w;y/

@wjwhere (j ¤ 1) can be calculated

directly form (5.5) are simply @c.w;y/@wj

Daijw1

, but responses @c.w;y/@w1

must be calcu-

lated indirectly from the homogeneity condition @c.w;y/@w1

w1CPNjD2

@c.w;y/@w1

wj D 0

as @c.w;y/@w1

D �PNjD2 aij

�wjw1

�.i D 1; � � � ; N /.

Next consider simple log-linear models of cost minimizing behavior. A log-linearcost function (corresponding to a cobb-Douglas technology)

ln c D a0 CNXiD1

ai lnwi C ay lny (5.6)

implies that @ ln c@ lnwi

D ai , but @ ln c@ lnwi

�@c.w;y/@wi

= cwiD

wixic

using Shephard’sLemma. Then the share of each input in that costs is independent of prices andoutput:

si �wixi

cD ai i D 1; � � � ; N: (5.7)

Alternatively consider log-linear factor demands:

ln xi D ai0 CNXjD1

aij lnwj C aiyy i D 1; � � � ; N: (5.8)

Homogeneity impliesPNjD1

@xi .w;y/@wj

= 1wjD 0 .i D 1; � � � ; N / and @ lnxi

@ lnwjD

@xi .w;y/@wj

= xiwj

using Shephard’s Lemma; so homogeneity of factor demands does not

imply the restrictionPNjD1 aij D 0 .i D 1; � � � ; N /. On the other hand, in the case

of log-linear consumer demands x D x.p; y/ conditional on prices p and incomey, the adding up constraint

PNiD1

@xi .p;y/@y

pi D 1 (derived by differentiating thebudget constraint px D y with respect to y) is generally satisfied only if all incomeelasticities are equal to 1 (see Deaton and Muellbauer 1980, pp. 16–17).

5.2 Second order flexible functional forms 49

5.2 Second order flexible functional forms

The previous section illustrated the severe restrictions implied by linea or log-linearmodels of cost functions or (by extension) profit functions or indirect utility func-tions. These functional forms imply extremely restrictive production functions andbehavioral relations. Likewise the most commonly employed production functions,i.e. Cobb-Douglas or CES, place significant restrictions on behavioral relations (allelasticities of substitution equal 1 or all elasticities of substitutions are equal, re-spectively). Also note that a Cobb-Douglas production function is equivalent to aCobb-Douglas functional form for the associated cost function and profit function(e.g. Varian 1984, p. 67).

The concept of second order flexible functional forms has been employed in or-der to generate less restrictive functional forms for behavioral models (see Diewert1971, pp. 481–507).

Suppose that the true production function, cost function, profit function or indi-rect utility function for an agent is represented by the functional form f .x/. Then,taking a second order Taylor series approximation of f .x/ about a point x0,

f .x0C�x/ � f .x0/CX

i

@f .x0/

@xi�xi C

1

2

Xi

Xj

@2f .x0/

@xi@xj�xi�xj : (5.9)

Thus for a small �x, f .x0 C �x/ can be closely approximated in terms of thelevel of f at x0 (f .x0/) and its first and second derivatives at x0

�@f .x0/@x

; @2f .x0/@x@x

�.

In turn, any other function g.x/ whose level at x0 can equal the level of f at x0.g.x0/ D f .x0// and whose first and second derivatives at x0 can equal the firstand second derivatives of f at x0

�@g.x0/@xD

@f .x0/@x

; @2g.x0/@x@x

D@2f .x0/@x@x

�can closely

approximate f .x/ for small variations in x about x0.To be more formal, g.x/ provides a second order (differential) approximation to

f .x/ at x0 if and only if

g.x0/ D f .x0/ (5.10a)@g.x0/

@xiD@f .x0/

@xii D 1; � � � ; N: (5.10b)

@2g.x0/

@xi@xjD@2f .x0/

@xi@xji; j D 1; � � � ; N: (5.10c)

In general the true functional form f .x/ is unknown. Thus g.x/ provides a secondorder flexible approximation to an arbitrary function f .x/ at x0 if conditions (5.10)can be satisfied at x0 for any function f .x/. In other words, g.x/ is a second orderflexible functional form if, at any point x0, any combination of level g.x0/ andderivatives @g.x0/

@x, @

2g.x0/@x@x

can be attained. Thus a general second order flexiblefunctional form g.x/must have at least 1CN C N.NC1/

2free parameters (assuming

@2g@x@x

symmetric). If g.x/ is linear homogeneous in x, then (using Euler’s theorem)


there are 1CN restrictions.

g.x0/ D

NXiD1

@g.x0/

@xixi0

0 D

NXjD1

@2g.x0/

@xi@xjxj0 i D 1; � � � ; N

(5.11)

on g.x0/ and its first and second derivatives. Then a linear homogeneous second or-der flexible functional form g.x/ has N.NC1/

2free parameters. Note that the number

of free parameters N.NC1/2

increase exponentially with the dimension N of x.Consider a functional form c.w; y/ for a producer’s cost function. c.w; y/ is a

second order flexible functional form if c.w0; y0/,@c.w0;y0/

@w, @c.w0;y0/

@y,[email protected];y0/@w@y

i.NC1/�.NC1/

are not restricted a priori (except for homogeneity restrictions (5.11)) at .w0; y0/.Moreover a second order flexible approximation c.w; y/ to a true cost function im-plies a second order flexible approximation to a true production function (see equa-tion (1.5), (1.6) on page (1.5)).

Next consider a functional form �.w; p/ for a firm’s profit function. �.w; p/is a second order flexible functional form if the combination �.w0; p0/,

@�.w0;p0/@w

,@�.w0;p0/

@p,h@2�.w0;p0/@w@p

i.NC1/�.NC1/

is not restricted a priori (except for homogene-

ity restrictions). In addition a second order flexible approximation to a true profitfunction implies a second order flexible approximation to a true production function(see equation (2.5), (2.6) of page (2.5)). Likewise a second order approximationV.p; y/ to a true indirect utility function implies a second order approximation to atrue utility function (structure of preferences).

Thus, to the extent that �w, �y is small over the data set .w; y/, a second or-der flexible functional form for a producer’s cost function c.w; y/ provides a closeapproximation to the true cost function and to the underlying production function.Similar comments apply to profit functions and indirect utility functions. Note thatif prices w are highly co-linear over time then in effect the variation �w in pricesmay be small. Thus second order flexible functional forms may be useful in model-ing behavior over data sets with very high multicollinearity. On the other hand, fora cost function, changes in output .�y/ are likely to be substantial over time andnot perfectly correlated with changes in factor prices .�w/. In this case it may bedesirable to provide a third order or even higher order of approximation in outputy to the true cost function, e.g. by specifying a cost function c.w; y/ such that anycombination of the following cost and derivatives can be attained at a point .w0; y0/(subject to homogeneity restrictions):

5.3 Examples of second order flexible functional forms 51

c.w0; y0/ (5.12a)@c.w0; y0/

@wi

@c.w0; y0/

@yi D 1; � � � ; N (5.12b)

@2c.w0; y0/

@wi@wj

@2c.w0; y0/

@wi@y

@2c.w0; y0/

@y@yi; j D 1; � � � ; N (5.12c)

@3c.w0; y0/

@wi@y@y

@3c.w0; y0/

@y@y@yi D 1; � � � ; N (5.12d)

Notice, however, that if third derivatives (5.12d) as well as (5.12a)–(5.12c) to befree then a greater number of free parameters must be estimated in the model, whichimplies a lost of degrees of freedom in the estimation.

In sum, there are two serious problems in the application of flexible functionalforms. First, the number of parameters to be estimated increases exponentially withthe number of variations (e.g. prices, outputs) included in the functional form andwith the order of the Taylor series approximation. Thus flexible functional formsgenerally require a high level of aggregation of commodities; but consistent aggre-gation of commodities is possible only under strong restrictions on the underlyingtechnology or preference structure (see next lecture). Second, when there is sub-stantial variation in the data, the global properties of flexible functional forms (howwell these forms approximate the unknown true function f .x/ over large variationin prices and output) become very important. Unfortunately the global properties ofmany common flexible functional forms are not clear. It is often difficult to discernwhether these functional forms impose restrictions over large�x that are unreason-able on a priori grounds and hence seriously bias econometric estimates (see pages232–236 of M. Fuss, D. McFadden, Y. Mundlak, “A Survey of Functional Formsin the Economic Analysis of Production,” in M. Fuss, D. McFadden, ProductionEconomic: A Dual Approach to Theory and Applications, 1978 for a good earlydiscussion of this problem). Nevertheless the concept of flexible functional formsappears to be useful in practice.

5.3 Examples of second order flexible functional forms

The most common flexible functional forms are the Translog and Generalized Leon-tief, so we focus on these plus the Normalized Quadratic (which is the most obviousflexible functional form). First consider dual profit functions �.w; p/, which wenow write as �.v/ where v � .w; p/.

The most obvious candidate for a flexible functional form for a dual profit func-tion �.v/ is the quadratic:

�.v/ D a0 C

NXiD1

aivi C1

2

NXiD1

NXjD1

aij vivj (5.13)


Note that this quadratic can be viewed as a second order expansion of � in pow-ers of v. In the absence of homogeneity restrictions, (5.13) obviously is a sec-ond order flexible functional form: differentiating twice yields @2�=@vi@vj D aij.i; j D 1; � � � ; N / i.e. each second derivative is determined as a free parameter aij ;differentiating once yields @�=@vi D ai C

PNjD1 aij vj .i D 1; � � � ; N / i.e. there

is a remaining free parameter ai to determine @�=@vi at any level; and finally theparameter a0 is free to determine � at any level. However linear homogeneity of�.v/ in v implies (by Euler’s theorem)

PNjD1

@2�@vi@vj

vj D 0 .i D 1; � � � ; N / and in

turn (by Hotelling’s Lemma)PNjD1 aij vj D 0. Thus the quadratic profit function

(5.13) reduces to a linear profit function:

� D a0 C

NXiD1

aivi (5.14)

which implies (using Hotelling’s Lemma) that output supplies and factor demandsare independent of prices v.

This problem is circumvented by defining the quadratic profit function in termsof normalized prices:

z�. Qv/ D a0 C

NXiD1

ai Qvi C1

2

NXiD1

NXjD1

aij Qvi Qvj (5.15)

where Qvi � vi=v0 .i D 1; � � � ; N /, i.e. the inputs and outputs of the firm are indexedi D 0; � � � ; N and v0 is chosen as the numeraire. By construction (5.15) satisfiesthe homogeneity condition (i.e. only relative prices Qv matter) so homogeneity doesnot place any further restrictions on the functional form (5.15). Since Qvi � vi=v0implies d Qvi D 1

v0dvi (for v0 fixed), the derivatives of z�. Qv/ and the correspond-

ing �.v/ can be related simply as follows, @z�.Qv/@Qvi

D@�.v/@vi

, @2 z�.Qv/@Qvi@Qvj

D1v0

@2�.v/@vi@vj

.i; j D 1; � � � ; N /. The derivatives of �.v/ with respect to v0 can then be recoveredfrom (5.15) using the homogeneity conditions (5.11). Often calculating the data Qvfrom the data .v0; vi ; � � � ; vN / we can assume without loss of generality (since onlyrelative prices matter) that v0 � 1 and specify the estimating equations as

yi D@z�. Qv/

@ QviD ai C

NXjD1

aij Qvj

xi D �@z�. Qv/

@ QvjD �ai �

NXjD1

aij Qvj i D 1; � � � ; N

(5.16)

for outputs y and inputs x. The corresponding equation for the numeraire com-modity can be recovered from (5.16) using homogeneity. These equations (5.16)integrate up to a macrofuction z�. Qv/ if @2 Q�.Qv/

@Qv@Qvis symmetric, i.e. if aij D aj i


.i; j D 1; � � � ; N /, and profit maximization implies that the matrix @2 z�.Qv/@Qv@Qv

D�aij�

is symmetric positive semidefinite.Next consider a Generalized Leontief dual profit function:

�.v/ D a0 C

NXiD1

aipvi C

1

2

NXiD1

NXjD1

aijpvipvj (5.17)

This is a second order expansion of � in powers ofpv. Note that

Pj aijp�vi

p�vj D

�Pj aijpvipvj whereas

Pi aip�vi D

p�Pi aipvi ; so �.�v/ D ��.v/ re-

quires a0 D 0, ai D 0 .i D 1; � � � ; N /. Thus, imposing the restriction of linearhomogeneity, the Generalized Leontief �.v/ can be rewritten as

�.v/ D1

2

NXiD1

NXjD1

aijpvipvj (5.18)

Differentiating �.v/ (5.18) once yields @�@viD aij C

Pj¤i aij

�vjvi

�1=2.i D

1; � � � ; N /, and differentiating �.v/ twice yields

@2�.v/

@vi@vjD

aijpvipvj

j ¤ i

@2�.v/

@vi@viD �

Xj¤i

aij

pvj

v3=2i

i; j D 1; � � � ; N

(5.19)

This provider a second order flexible form for �.v/ subject to homogeneity restric-tions. The corresponding estimating equations are

yi D@�.v/

@viD ai i C

Xj¤i

aij

�vj

vi

�1=2

xi D �@�.v/

@viD �ai i �

Xj¤i

aij

�vj

vi

�1=2i D 1; � � � ; N

(5.20)

for outputs y and inputs x. These equations integrate up to a macrofunction �.v/ if@2�.v/@v@v

is symmetric, i.e. aij D aj i .i; j D 1; � � � ; N / (see (5.19)). Profit maximiza-

tion implies that the matrix @2�.v/@v@v

is symmetric positive semidefinite.Third consider a Translog dual profit function

ln�.v/ D a0 CNXiD1

ai ln vi C1

2

NXiD1

NXjD1

aij ln vi ln vj (5.21)


This is a second order expansion of ln� in powers of ln v. In order to determinehomogeneity restrictions, note that �.�v/ D ��.v/ implies ln�.�v/ D ln� Cln�.v/. Multiplying prices v by � in (5.21).

ln�.�v/ D a0 CX

iai ln.�vi /C

Xi

Xjaij ln.�vi / ln.�vj /

D a0 C ln�X

iai C

Xiai ln vi C .ln�/2

Xi

Xjaij

C 2 ln�X

i

Xjaij ln vi C

Xi

Xjaij ln vi ln vj

D ln�.v/C ln�X

iai C 2 ln�

Xi

�Xjaij

�ln vi C .ln�/2

Xi

Xjaij

(5.22)

Thus the linear homogeneity condition ln�.�v/ D ln�C ln�.v/ is satisfied if

NXiD1

ai D 1

NXjD1

aij D 0 i D 1; � � � ; N

(5.23)

Rearranging @ ln�@ lnvi

D@�@vi= �vi

as @�@viD

@ ln�@ lnvi

��vi

and differentiating with respect tovj ,

@2�.v/

@vi@vjD

@2 ln�@ ln vi@ ln vj

@ ln vj@vj

��

viC@ ln�@ ln vi

@�

@vj�1

vi

D�

vivj

�aij C

@ ln�@ ln vi

@ ln�@ ln vj

�i ¤ j

@2�.v/

@vi@viD

@2 ln�@ ln vi@ ln vi

@ ln vi@vi

��

viC@ ln�@ ln vi

@�

@vi�1

vi�@ ln�@ ln vi

�

.vi /2

D�

.vi /2

"ai i C

�@ ln�@ ln vi

�2�@ ln�@ ln vi

#i; j D 1; � � � ; N

(5.24)

since @2 ln�@ lnvi@ lnvj

D aij (5.21), @ lnvj@vj

D1vj

and @�@vjD

@ ln�@ lnvj

��vj

. Thus, as in thecost of the normalized Quadratic and Generalized Leontief, the differential equa-tions yi .v/ D

@�.v/@vi

(for outputs), xi .v/ D �@�.v/@vi

(for inputs) integrate up to amacrofunction �.v/ if aij D aj i .i; j D 1; � � � ; N /. Profit maximization furtherrequires that the matrix @2�.v/

@v@vdefined by (5.24) is symmetric positive semidefinite.

In contrast to the normalized Quadratic and Generalized Leontief, the Translogmodel (5.21) is more easily estimated in terms of share equations rather than outputsupply and factor demand equations per se. The elasticity formula @ ln�

@ lnviD

@�@vi= �vi


and Hotelling’s Lemma yield yi D @ ln�@ lnvi

�vi

, xi D � @ ln�@ lnvi

�vi

; so substituting for@ ln�@ lnvi

and � from (5.21) yields estimating equations for y and x that are nonlinear inthe parameters .a0; � � � ; aN ; a11; � � � ; aNN / which are to be estimated. On the otherhand, the elasticity formula and Hotelling’s Lemma directly imply piyi

�D

@ ln�@ lnvi

and wiyi�D �

@ ln�@ lnvi

. Thus the Translog model (5.21) can be most easily estimatedin terms of equations for “profit shares”:

si �piyi

�D ai C

NXjD1

aij ln vj

si �wixi

�D �ai �

NXjD1

aij ln vj i D 1; � � � ; N

(5.25)

The homogeneity condition (5.23) and reciprocity conditions aij D aj i .i; j D

1; � � � ; N / can easily be tested, and the restriction @2�.v/@v@v

positive semidefinite canbe checked at data points .v; �/ using (5.24).

The only complication in this procedure for estimating Translog models arisesfrom the fact that the sum of the output shares minus the sum of the input sharesequals 1 :

Pipiyi��Piwixi�D

��D 1. Writing these equations as si D

˙.ai CPNjD1 aij ln vj / C ei .i D 1; � � � ; N / where ei denotes the disturbance

for equation i , we see that this linear dependence between shares implies a lineardependence between disturbances, i.e. ei for any equation must be a linear combi-nation of the disturbances for all other equations. This in turn implies that one ofthe share equations must be dropped from the econometric model for the purposesof estimation, but this is not a serious problem since there are simple techniquesfor insuring that econometric results are invariant with respect to which equation isactually dropped.

The above functional forms are easily generalized to cost functions c.w; y/, andthe homogeneity, reciprocity and concavity conditions are calculated in an analo-gous manner. The normalized quadratic cost function can be written as

Qc. Qw; y/ D a0C

NXiD1

ai QwiCayyC1

2

NXiD1

NXjD1

aij Qwi QwjC

NXiD1

aiy QwiyCayyy2 (5.26)

where Qwi �wiw0

, and the corresponding factor demand equations are (using Shep-hard’s Lemma)

xi D@ Qc. Qw; y/

@ Qwi

D ai C

NXjD1

aij Qwj C aiyy i D 1; � � � ; N

(5.27)


The effects of changes in the numeraire pricew0 can be calculated from (5.27) usingthe homogeneity restrictions (5.11). Note that the parameters ay and ayy are notincluded in the factor demand equations, so that @ Qc. Qw; y/=@y cannot be recovereddirectly from (5.27). Nevertheless @c.w; y/=@y can be calculated indirectly from(5.27) using the homogeneity condition @c.�w; y/=@y D �@c.w; y/=@y and thereciprocity relations @2c.w; y/=@wi@y D @2c.w; y/=@y@wi .i D 1; � � � ; N / plusShephard’s Lemma (See Footnote 2 on page 19). Alternatively the cost function(5.26) can be estimated directly along with N � 1 of the factor demand equations(5.27).

The Generalized Leontief cost function is often written as

c.w; y/ D1

2y

NXiD1

NXjD1

aijpwipwj C y

2

NXiD1

aiywi (5.28)

where aiy D 0 i D 1; � � � ; N / implies that the production function is constantreturns to scale. The corresponding factor demand equations are


@wi

D ai iy CXj¤i

aijy

�wj

wi

� 12

C aiyy2 i D 1; � � � ; N

(5.29)

The Translog cost function is

ln c.w; y/ D a0 CNXiD1

ai lnwi C ay lny C1

2

NXiD1

NXjD1

aij lnwi lnwj

C

NXiD1

aiy lnwi lny C ayy.lny/2(5.30)

and the corresponding factor share equations are

si �wixi

cD ai C

NXjD1

aij lnwj C aiy lny i D 1; � � � ; N (5.31)

As in the case of the Translog profit function,PNiD1 si D 1, so that one share

equation must be deleted from the estimation.Finally, we briefly consider indirect utility functions V.p; y/ where p denotes

prices for consumer goods and y now denotes consumer income. The NormalizedQuadratic indirect utility function can be written as

zV . Qp/ D a0 C

NXiD1

ai Qpi C1

2

NXiD1

NXjD1

aij Qpi Qpj (5.32)

5.4 Almost ideal demand system (AIDS) 57

where Qpi � pi=y, so that (5.32) is by construction homogeneity of degree 0 in.p; y/. @V.p; y/=@y can be calculated from (5.32) using the condition

PNiD1

@V.p;y/@pi

[email protected];y/@y

y D 0 implied by zero homogeneity. Then the consumer demand equations

can be calculated using Roy’s identity xi D �@V.p;y/@pi

= @V.p;y/@y

.i D 1; � � � ; N /.Roy’s identity generally leads to equations that are nonlinear in the parameters (a)to be estimated. A simple example of a Translog indirect utility function is

lnV. Qp/ D a0 CNXiD1

ai ln Qpi C1

2

NXiD1

NXjD1

aij ln Qpi ln Qpj (5.33)

where again Qpi � pi=y .i D 1; � � � ; N /. (see Varian 1984, pp. 184–186 for furtherexamples of functional forms for indirect utility functions).

5.4 Almost ideal demand system (AIDS)

Suppose consumer expenditure function in the form

logE.p; u/ D a.p/C b.p/ u p D .p1; � � � ; pM / (5.34)

where

a.p/ D ˛0 CX

i˛i logpi C

1

2

Xi

Xjr�ij logpi logpj

b.p/ D ˇ0Y

ipiˇi ( ˇ0 p1ˇ1 � � � pMˇM

(5.35)

Since E.�p; u/ D �E.p; u/, the following relation apply,

MXiD1

˛i D 1

MXiD1

r�ij D

MXjD1

r�ij D

MXiD1

ˇi D 0

(5.36)

By Shephard’s Lemma.

@ logE@ logpi

�@E

@pi�pi

EDxipi

E� si (cost share i ) (5.37)

So that cost share equations are attained from Shephard’s Lemma by differentiating(5.34)–(5.35):


si D@ logE@ logpi

D ˛i CX

jr�ij logpj„ ƒ‚ …

@a@ logpi

C u„ƒ‚…logE�a.�/b.�/

by (??)

�@b

@ logpi(5.38)

This (5.38) simplifies to

si D ˛i C

NXiD1

rij logpi C ˇi logY=P (5.39)

where Y is consumer expenditure .E/ and P is a price index given by

logP D ˛0 CMXjD1

logpi C1

2

MXiD1

MXjD1

rij logpi logpj (5.40)

andrij �

1

2.r�ij C r

�j i / for all i; j (5.41)

Except for the price indexP , demands (5.39) are linear is coefficients. Homogeneityand symmetric imply.

MXjD1

rij D 0 i D 1; � � � ;M (homogeneity) (5.42a)

rij D rj i for all i; j D 1; � � � ;M: (5.42b)

In practice P is usually approximated by an appropriate arbitrary pure index, e.g.

logP �MXjD1

sj logpj (5.43)

and then (5.39) is estimated.

5.5 Functional forms for short-run cost functions

c.w; y;K/ CRTS f .�x; �K/ D �c.x;K/ ) c.w; �y; �K/ D �c.w; y;K/

The issue: form for c.w; y;K/ should be 2nd-order flexible functional form withand without CRTS.

References 59

5.5.1 Normalized quadratic: c� D c=w0, w� D w=w0.

e.g.

c� D

�a0 C

Xiaiw

�i C

1

2

Xi

Xjaijw

�i w�j

�y C

�b0 C

Xibiw

�i

�y2

C

�c0 C

Xiciw�i

�K C

�d0 C

Xidiw

�i

�K2 C

�e0 C

Xieiw�i

�pKy

) (under CRTS)

c� D

�a0 C

Xiaiw

�i C

1

2

Xi

Xjaijw

�i w�j

�yC

�c0 C

Xiciw�i

�KC

�e0 C

Xieiw�i

�pKy

5.5.2 Generalized Leontief:

c D

�1

2

Xi

Xjaijpwipwj

�y C

�Xibiwi

�y2 C

�Xiciwi

�K C

�Xidiwi

�K2

C

�Xieiwi

�pKy

) (under CRTS)

c D

�1

2

Xi

Xjaijpwipwj

�yC

�Xiciwi

�KC

�Xieiwi

�pKy

5.5.3 Translog:

log c D a0 CX

iai .logwi /C

1

2

Xi

Xjaij .logwi /.logwj /C b0 logy C b1.logy/2

C

Xibi .logwi /.logy/C c0 logK C c1.logK/2 C

Xici .logwi /.logK/C e.logy/.logK/

) (under CRTS)

log c.w; �y; �K/ D log�C log c.w; y;K/

References

1. Deaton and Muellbauer (1980). Economics and Consumer Behavior. pp. 16–17


2. Diewert W. (1971), An Application of the Shephard Duality Theorem: A Generalized LeontiefProduction Function, Journal of Political Economy, 1971, pp. 481–507

3. Fuss M., McFadden D. (1978). Production Economic: A Dual Approach to Theory and Ap-plications

4. Varian H. (1984), Microeconomic Analysis, second edition,WW Norton & Co. pp. 67

Chapter 6Aggregation Across Agents in Static Models

6.1 General properties of market demand functions

In previous lecture we characterized the behavior of an individual producer or con-sumer at a static equilibrium. However in practice we often have data aggregatedover agents rather than data for the individual producers or consumers. This leads tothe following question: do the behavioral restrictions that apply to data at the firmor consumer level also apply to data that has been aggregated over agents?

In several simple behavioral models the answer to this question is “yes”. Themost obvious example is the simple theory of static profit maximization, where allinput are free variable and all firms face the same prices w, p. Then the marketprices w, p correspond exactly to the parameter facing the individual firms so thatindustry output supply and factor demand relations Y D Y.w; p/, X D X.w; p/,where Y D

PFfD1 y

f and X DPf x

f , do not misrepresent the parameters facingindividual firms. More precisely,

�f .w; p/ �

(pyf �

NXiD1

wixfi

)� 0 for all .w; p/ f D 1; � � � ; F

)

Xf

�f .w; p/ �Xf

(pyf �

NXiD1

wixfi

)� 0 for all .w; p/

i.e. ….w; p/ �

(pY �

NXiD1

wiXi

)� 0 for all .w; p/

(6.1)

By (6.1), we can develop the properties of the industry profit function ….w; p/and industry output supplies Y D Y.w; p/ and derives demands X D X.w; p/

61

62 6 Aggregation Across Agents in Static Models

in essentially the same manners as in the case of data for the individual firm (seesection 4.1 of lecture 4 on nonlinear duality).1

As a second example consider the case where each consumer maximizes utilitysubject to the value

Pi piw

fi of his initial endowments wf D .w

f1 ; � � � ; w

fN / of

the N commodities rather than an income yf that is independent of commodityprices p, i.e. each firm f solves

maxxf

uf .xf /

s.t.NXiD1

pixfi D

NXiD1

piwfi :

(6.2)

Then the solution xf�

to (6.2) is conditional on .p;wf / and in turn aggre-gate market demands

Pf x

f � are conditional on .p;w1; � � � ; wF /. If the en-dowments wf of consumer f are constant over time for each consumer f D1; � � � ; F , then for purposes of estimation the market demands

Pf x

f � are ineffect conditional only on the prices p which are common to each consumer:x� D x�.p/ D

Pf x

f �.p/. Then the aggregate market demand relation x D x.p/are well defined and inherit all linear restrictions on xf .p/. This includes thehomogeneity restrictions xf .�p; yf / D xf .p; yf /, which can be expressed asxf .�p;wf / D xf .p;wf /, but apparently excludes the nonlinear second or-der conditions

hxfi pj

�p; yf

�� x

fi yf

�p; yf

�xj

i�

hxHi pj

�p; uf �

�isymmetric

negative semidefinite.In general the answer to the above question is “no”, i.e. the behavioral restric-

tions that apply to data at the level of the individual agent generally do not apply todata that has been aggregated over agents. This question has been addressed in thecontext of utility maximization by consumers:

maxxf

uf .xf /

s.t.NXiD1

pixfi D y

f! xf

�D xf .p; yf /

which implies the Slutsky relations

xfp .p; yf /C x

f

yf.p; yf /xf .p; yf / symmetric negative semidefinite (6.3)

(property (3.4).b on page 31). It has been shown that, in the absence of specialrestrictions on utility functions uf .xf / or on the distribution of income yf overconsumers, the above restrictions (6.3) on demands of the individual consumer donot carry over to aggregate demands X D X.p; Y / where X �

Pf x

f and Y �

1 The one exception concerns whether these relations �.w;p/, y.w;p/, x.w;p/ are well de-fined in cases of free entry and exit to the industry (see footnote 2 on page 19).

6.1 General properties of market demand functions 63Pf y

f . Indeed, if the number of consumers is equal to or greater than the numberof goods N , then any continuous function X.p; Y / that satisfies Walras’ Law

Walras Law is a principle in general equilibrium theory asserting that when con-sidering any particular market, if all other markets in an economy are in equilib-rium, then that specific market must also be in equilibrium. Walras Law hinges onthe mathematical notion that excess market demands (or, conversely, excess mar-ket supplies) must sum to zero. That is,

PXD D

PXS D 0. Walras’ Law is

named for the mathematically inclined economist Leon Walras, who taught at theUniversity of Lausanne, although the concept was expressed earlier but in a lessmathematically rigorous fashion by John Stuart Mill in his Essays on Some Unset-tled Questions of Political Economy (1844). (in particular the adding up propertiesPNiD1

@xi .p;y/@y

pi D 1,PNiD1

@xi .p;y/@pj

pi C@xi .p;y/@y

y D 0, j D 1; � � � ; N in thedifferentiable case) can be generated by some set of utility maximizing consumerswith some distribution of income (see Debrew 1974, pp. 15–22; Sonnenschein 1973,pp. 345–354).

There is a relatively simple way of addressing the above question. FollowingDiewect (1977, page 353–362) write the Slutsky relations (6.3) in the form

xfp .p; yf /

N�N

D KfN�N

� xf

yf.p; yf /

N�1

xf .p; yf /1�N

(6.4)

where Kf � xhfp .p; uf �/ is the Hickscian substitution matrix which is symmetric

negative semidefinite. In generalhxf

yf.p; yf /xf .p; yf /

iis neither symmetric nor

negative semidefinite. Choose N � 1 linearly independent vector v1; v2; � � � ; vN�1such that xf

1�NvnN�1D 0 for n D 1; � � � ; N � 1, and write v1; � � � ; vN�1 in matrix

form as VN�.N�1/

. Pre and post-multiplying (6.4) by V ,

V T.N�1/�N

xfpN�N

VN�.N�1/

D V TKf V � V T xfy xf V D V TKf V (6.5)

since xf V D 0. Since Kf � xhfp .p; uf �/ is symmetric negative semidefinite,

(6.5) implies that V fTxfp .p; y

f /V f is symmetric negative semidefinite.Now sum (6.4) over consumers f D 1; � � � ; F to obtain the matrix of partial

derivatives of aggregate demands X DPf x

f with respect to prices p as

Xp.p; yf / D

Xf

Kf �Xf

xf

yf.p; yf /xf .p; yf /: (6.6)

Let Qv1; � � � ; QvN�F be N �F linearly independent vectors, each of which is orthog-onal to x1; � � � ; xF , the set of initial demand vectors of the F consumers. Define theN � .N �M/ matrix QV D Œ Qv1; � � � ; QvN�M �. Pre and postmultiplying (6.6) by QV ,

QV TXp QV.N�M/�.N�M/

D QV TXf

Kf QV � QV TXf

Kf xf

yfxf QV D QV T

Xf

Kf QV (6.7)


since QV T xf D 0N�F for all f D 1; � � � ; F . Therefore Kf symmetric negativesemidefinite (f D 1; � � � ; F ) implies

QV TXp QV is a symmetric negative semidefinite matrix of dimension.N�F /�.N�F /:(6.8)

(6.8) can be viewed as restrictions places on aggregate consumer demandsX.p; y1; � � � ; yF / by symmetry negative semidefiniteness of Hicksian demand re-sponses xhfp .p; uf �/ .f D 1; � � � ; F /. Moreover these are essentially the only re-strictions on the first derivatives of market demand functions (aside from addingup constraints) (Mantel 1977). Note that if the number of consumers F is equal tothe number of goods N , then (6.8) places 0 restrictions on aggregate demands (thedimensions of the matrix of restrictions are .N � F / � .N � F / equals 0 � 0 inthis case). This is consistent with the result of Debreu and Sonnenschein that sym-metry negative semidefiniteness of Hicksian demand responses xhfp .p; uf �/ � Kf

places no restrictions on aggregate demands when the number of consumers is equalto or greater than number of goods.

In sum, in general (i.e. in the absence of restrictions on production/utility func-tions or on the distribution of exogenous parameters across agents) the microthe-ory of the individual agent cannot be applied to data that has been aggregated overagents. In the next two sections we investigate how restrictions on production/utilityfunctions and on the distribution of parameters across agents can influence this con-clusion.

6.2 Condition for exact linear aggregation over agents

Here we consider restrictions on production and utility functions that imply consis-tent aggregation over agents for any distribution of the exogenous parameters thatvary over agents. Fist, consider the conditional factor demands

xfi D x

fi

�w; yf

�i D 1; � � � ; N f D 1; � � � ; F (6.9)

where w � vector of factor prices (which are common to each firm) and yf �output level of firm f . Aggregate factor demands

Pf x

f D X.w;Pf y

f / exist ifand only if

Xi

�w;Pf y

f�DPf x

fi

�w; yf

�i D 1; � � � ; N (6.10)

for all data .w; y1; � � � ; yF /. Differentiating (6.10) w.r.t. yf ,

6.2 Condition for exact linear aggregation over agents 65

@Xi .w; y/

@y

@y

@yfD@xfi .w; y

f /

@yf

i.e.@Xi .w; y/

@yD@xfi .w; y

f /

@yfi D 1; � � � ; N f D 1; � � � ; F

(6.11)

since @y=@yf D 1 in the case of linear aggregation y DPf y

f .Thus, in the absence of restrictions on the distribution of total output y over firms,

aggregate factor demands X.w;Pf y

f / exist if and only if @xf .w; yf /=@yf areconstant across all firms (for a given w). This condition implies that conditionalfactor demand equations for firms are of the following form:

xfi D ˛i .w/y

fC ˇ

fi .w/ i D 1; � � � ; N f D 1; � � � ; F (6.12)

where the function ˛i .w/ is invariant over firms. (6.12) is called a “Gorman PolarForm”

Gorman polar form is a functional form for indirect utility functions in eco-nomics. Imposing this form on utility allows the researcher to treat a society ofutility-maximizers as if it consisted of a single individual. W. M. Gorman showedthat having the function take Gorman polar form is both a necessary and sufficientfor this condition to hold.. Here the scale effects @xf .w; yf /=@yf are independentof the level of output, which implies that the production function yf D yf .xf / is“quasi-homothetic”:

Fig. 6.1 quasi-homotheticproduction function

0xf2

y0fy1f

xf1

Here, as output yf expands and factor pricew remain constant, the cost minimizinglevel of inputs increase along a ray (straight line) in input space. Condition (6.12)implies that aggregate demands have a Gorman Polar Form2:

2 The more restrictive assumption of hornatheticity implies that the expansion path 4xf .w;yf /

4yfis

a ray through the origin. Given the Gorman Polar Form (6.12), homotheticity requires ˇf .w/ D 0(which in turn implies the stronger assumption of constant restrun to scale).


Xi DXf

xfi .w; y

f / D ˛i .w/Xf

yf CXf

ˇfi .w/ D ˛i .w/Y C ˇi .w/

i D 1; � � � ; N:

(6.13)

Condition (6.12) implies the existence of aggregate conditional factor demandssatisfying (6.10), irrespective of whether producers are minimizing cost. The nextquestion is: under what conditions do the restrictions on cost minimizing factordemands at the firm level generalize to aggregate factor demands X.w;

Pf y

f /?The simplest way to answer this question is as follow: a Gorman Polar Form foraggregate demands (6.13) implies that identical demand relations could be gener-ated by a single producer with an output level equal to

Pf y

f , and by assumptionthis producer is a cost minimizer. Therefore aggregate demands of a Gorman Po-lar Form necessarily inherit the properties of cost minimizing factor demand for asingle agent.

To answer the above question more rigorously, first note that the assumption ofcost minimization and condition (6.12) for existence of aggregate factor demandsX.w;

Pf y

f / are jointly equivalent to the assumption of cost minimization plus aGorman Polar Form for the cost function of firm f :

cf .w; yf / D a.w/yf C bf .w/; f D 1; � � � ; F: (6.14)

Sufficiency is obvious: (6.14) and Shephard’s Lemma imply xfi .w; yf / D @cf .w;yf /

@wiD

@a.w/@wi

yf C @bf .w/@wi

where @a.w/@wiD ˛.w/, @b

f .w/@wi

D ˇf .w/, and necessity followssimply by integrating (6.12) up to a cost function using Shephard’s Lemma. Thisimplies the existence of an industry cost function which is also a Gorman PolarForm: X

fcf .w; yf / D

Xf

�a.w/yf C bf .w/

�D a.w/

Xfyf C

Xfbf .w/

D a.w/X

fyf C b.w/

D C�w;X

fyf�

(6.15)

Moreover this industry cost function C.w;Pf y

f / D a.w/Pf y

f C b.w/

inherits all the properties of cost minimization by a firm. To see this, note that


cf�w; yf .xf /

��

NXiD1

wixfi � 0 for all w f D 1; � � � ; F

)

Xf

cf�w; yf .xf /

��

Xf

NXiD1

wixfi � 0 for all w

, a.w/Xf

yf .xf /CXf

bf .w/ �Xf

NXiD1

wfi x

fi � 0 for all n

, a.w/Xf

yf C b.w/ �

NXiD1

wixi � 0 for all w

, C�w;X

fyf��

NXiD1

wixi � 0 for all w

(6.16)

and likewise

cf�w; yf .xf /

��

NXiD1

wixfi D 0 f D 1; � � � ; F

) C�w;X

fyf��

NXiD1

wixi D 0

(6.17)

Thus the aggregate primal-dual relation C.w;Pf y

f / �PNiD1wixi has the same

properties as the cost minimizing firm’s primal-dual cf .w; yf /�PNiD1wix

fi .f D

1; � � � ; F /, and in turn C.w;Pf y

f / has the same properties as the cost minimizingcf .w; yf /:

Property 6.1.

a) C.w;Pf y

f / is increasing in w,Pf y

f ;

b) C.�w;Pf y

f / D �C�w;Pf y

f�

;

c) C.w;Pf y

f / is concave in w;d) C.w;

Pf y

f / satisfies Shephard’s Lemma3

Xi

�w;X

fyf�D

@C�w;Pf y

f�

@wii D 1; � � � ; N

3 Conditions Prop.6.1.a–c are satisfied forC.w;Pf y

f / � a.w/Pf y

f Cb.w/ if a.w/ > 0and both a.w/ and b.w/ are increasing, linear homogeneous and concave in w .


In turn the aggregate demands satisfy the restrictions as the cost minimizing de-mands for a firm:

Property 6.2.

a) X��w;

Xfyf�D X

�w;X

fyf�

b)

"@X.w;

Pf y

f /

@w

#N�N


Similar results hold for linear aggregation of consumer demand equations xfi Dxfi .p; y

f / where p price of consumer goods (which are assumed to be identicalfor all consumers) and yf � exogenous income of consumer f . The conditions forexistence of aggregate demands are

Xi

�p;X

fyf�D

Xfxfi

�p; yf

�i D 1; � � � ; N (6.18)

and the conditions are equivalent to Gorman Polar Forms

xfi

�p; yf

�D ˛i .p/y

fC ˇ

fi .p/ i D 1; � � � ; N f D 1; � � � ; F (6.19)

which imply

Xi

�p;X

fyf�D ˛i .p/

Xfyf C ˇi .p/ i D 1; � � � ; N: (6.20)

In order to determine rigorously the restrictions that permit aggregate demands(6.20) to inherit the properties of utility maximization, first note that utility max-imization by a consumer is essentially equivalent to cost minimization (see page28):

minxf

NXiD1

pixfi D E

f .p; uf /

s.t. uf .xf / D uf�

(6.21)

where uf�

is the maximum attainable utility level for the consumer given his budgetconstraint pxf D yf . An aggregate cost function E.p;

Pf u

f / exists if and onlyif

E�p;X

fuf�D

XfEf .p; uf / (6.22)

and this condition is equivalent to the existence of Gorman Polar Forms

Ef .p; uf / D a.p/uf C bf .p/ f D 1; � � � ; F (6.23)


which in turn implies

E�p;X

fuf�D a.p/

Xfuf C b.p/ (6.24)

(6.23) and (6.24) imply

Ef�p; uf .xf /

��

NXiD1

pixfi � 0 for all p f D 1; � � � ; F

) E�p;X

fuf��

NXiD1

pixi � 0 for all p

Ef�p; uf .xf /

��

NXiD1

pixfi D 0 f D 1; � � � ; F

) E�p;X

fuf��

NXiD1

pixi D 0:

(6.25)

ThereforeE.p;Pf u

f / inherit the properties of cost minimization, and Shephard’sLemma implies that

Xhi

�p;X

fuf�D

@E�p;Pf u

f�

@pi

[email protected]/

@pi

Xfuf C

@b.p/

@pii D 1; � � � ; N

(6.26)

In order to relate (6.26) to Marshallian demands, note that (6.23) and utility maxi-mization by consumers f imply

yf D a.p/uf�C bf .p/

) uf�Dyf � bf .p/

a.p/f D 1; � � � ; F

(6.27)

i.e. the consumer’s inherit utility function has the Gorman Polar Form

V.p; yf / D Qa.p/yf C Qbf .p/ (6.28)

where Qa.p/ D 1=a.p/, Qb.p/ D �bf .p/=a.p/. Substituting (6.27) into (6.26),


Xi [email protected]/

@pi

Xf

yf � bf .p/

a.p/

[email protected]/

@pi

1

a.p/

Xf

yf �@a.p/

@pi

Xf

bf .p/

a.p/[email protected]/

@pii D 1; � � � ; N

(6.29)

(6.29) implies the existence of aggregate Marshallian demands of a Gorman PolarForm:

Xi

�p;X

fyf�D QQa.p/

Xfyf C

QQb.p/ i D 1; � � � ; N (6.30)

where a.p/ D @a.p/@pi

1a.p/

, b.p/ D � @a.p/@pi

b.p/a.p/C

@b.p/@pi

.Thus cost minimization by individual consumers and a Gorman Polar Form

(6.18/6.30) that satisfy restrictions analogous to restriction on Marshallian demandsby individual utility maximizing consumers. The restriction (6.1) implied by costminimization are satisfies for E.p;

Pf u

f / (6.24) if

a.p/ > 0I (6.31a)

a.p/, b.p/ increasing and linear homogeneous in P ; (6.31b)�@2a.p/

@p@p

�,�@2b.p/

@p@p

�symmetric, negative semidefinite. (6.31c)

These restrictions can be tested or imposed on the Gorman Polar FormX.p;Pf y

f /

(6.30)Alternatively we could try to incorporate the utility maximization restrictions

into aggregate demands X.p;Pf y

f / directly by specifying consumers’ indirectutility functions V f .p; yf / as Gorman Polar Form:

V f .p; yf / D QQQa.p/yf CQQQbf .p/: (6.32)

However inverting this relation yields

yf Duf��QQQb.p/

QQQa.p/(6.33)

i.e.

Ef .p; uf / D1

QQQa.p/uf �

QQQbf .p/

QQQa.p/f D 1; � � � ; F: (6.34)

Thus an indirect utility function V f .p; yf / has a Gorman Polar Form if and onlyif the corresponding cost function Ef .p; uf / has a Gorman Polar Form. Thereforethe above analysis in term of cost minimizing behavior exhaust the restrictions per-


mitting the existence of aggregate Marshallian demands X.p;Pf y

f / that satisfythe same restrictions as Marshallian demands xf .p; yf / for an individual utilitymaximizing consumer.

Note that these restrictions (6.23–6.24, 6.30–6.31) on preference structure doimply mild restriction on the distribution of expenditures over consumers:

Ef D a.p/uf C bf .p/ a.p/ > 0

) yf > bf .p/ f D 1; � � � ; F(6.35)

Aside from this restriction, Gorman Polar Forms permit aggregate demands to in-herit the properties of utility maximizing demands irrespective of the distribution ofexpenditure over consumers. bf .p/ can be viewed as the agent’s “committed ex-penditure” at prices p (since it is independent of utility level uf ), and yf � bf .p/can be defined as the corresponding “uncommitted expenditure”.

Aggregation problems also arise when there are variations in prices over agents,and these problems can be severe. For example, suppose that profit maximizingcompetitive firms in an industry are distributed across different regions of the coun-try and as a result firms face different output prices pf . Aggregate demands canbe defined as

Pf x

f D X.w;Pf f p

f /,Pf y

f D Y.w;Pf f p

f / wherePf f p

f denotes a (weighted) average output price Np. Aggregate demands andsupplies exist if

Xi

�w;X

f f p

f�D

Xfxfi .w; p

f / i D 1; � � � ; N

Y�w;X

f f p

f�D

Xfyf .w; pf /

(6.36)

These conditions are satisfied if the aggregate demands and supplies have GormanPolar Forms:

Xi D ˛i .w/X

f f p

fC ˇi .w/ i D 1; � � � ; N

Y D ˛0.w/X

f f p

fC ˇ0.w/

(6.37)

Now suppose further that these aggregate relation are to inherit the propertiesof profit maximizing demand and supply relations for the individual firm. In theabsence of any restrictions on the distribution of prices pf over firms, this requiresthe firm and industry profit functions to have the following Gorman Polar Forms

�f .w; pf / D a.w/pf C bf .w/ f D 1; � � � ; F

…�w;X

f f p

f�D a.w/

Xf f p

fC b.w/

(6.38)

However Hotelling’s Lemma now implies that output supplies are independent ofoutput prices:


yf .w; pf / D@�f .w; pf /

@pfD a.w/ f D 1; � � � ; F (6.39)

using (6.38).In sum, Gorman Polar Forms (GPF) are both necessary and sufficient for ex-

act linear aggregation over agents. Unfortunately these functional forms are fairlyrestrictive. For example, GPF conditional factor demands x D x.w; y/ imply lin-ear expansion paths, and GPF consumer demands x D x.p; y/ imply linear Engelcurves (and income elasticilties tend to unity as total expenditure incuaser). Theseassumptions may or may not be realistic over large changes in output or expenditure(see Deaton and Muellbauer 1980, pp. 144–145, 151–153).

On the other hand it should be noted that Gorman Polar Form cost and indi-rect utility functions are flexible functional forms. These former provide secondorder approximations to arbitrary cost and indirect utility functions (Diewert 1980,pp. 595–601). Therefore Gorman Polar Forms provide a local first order approx-imation to any system of demand equations (except, of course, for cases such as(6.38)).

6.3 Linear aggregation over agents using restrictions on thedistribution of output or expenditure

The analysis in the previous section was aimed at achieving consistent aggregationover agents while imposing essentially zero restrictions on the distribution of outputor expenditure over agents. However these distribution are in fact highly restrictedin most cases, and these restrictions may help to achieve consistent aggregation.

Unfortunately there are few results on the relation between restrictions on thedistribution of “exogenous” variables such as output or income and consistent ag-gregation. This section simply summaries several example that illustrate the effectsof alternative restrictions on the distribution of such variables.

First, output or expenditure may be choice variables for the agent rather thanexogenous variables, and it is very important to incorporate this fact into the analysisof possibilities for consistent aggregation. For example, suppose that producers arecompetitive profit maximizers and face the same output price p. Then the first ordercondition in the output market for profit maximization is

@cf .w; yf�/

@yfD p f D 1; � � � ; F (6.40)

where cf .w; yf / D minxfPNiD1wix

fi s.t. yf .xf / D yf . This implies that the

marginal cost is identical across firms at all observed combinations of output levels.y1�; � � � ; yF

�/ for given prices w, p.

Now remember from our earlier discussion that identical marginal cost acrossfirms is the condition for consistent linear aggregation of cost functions across firms

6.3 Linear aggregation over agents using restrictions on the distribution of output or expenditure73

(irrespective of the distribution of output across firms) (see pages 64–66). Thus,the assumption of competitive profit maximization and identical price imply thatan aggregate cost function C.w;

Pf y

f �/ exists over all profit maximizing lev-els of output

Pf y

f � DPf y

f .w; p/. Moreover this aggregate cost functionC.w;

Pf y

f /, wherePf y

f is restricted to the equilibrium levelsPf y

f .w; p/,also inherits the properties of cost minimization.4

As a second example, supposed that cost functions of individual consumers areof the form

Ef .p; uf / D af .p/uf C bf .p/ f D 1; � � � ; F: (6.41)

this is slightly more general than the Gorman Polr Form (6.23) in the sense thathave the function af .p/ can vary over consumers. Nevertheless preference are stillquasi-homothetic.

Solving yf D af .p/uf�C bf .p/ (6.41) for the consumer’s indirect utility

function yields

V f .p; yf / Dyf � bf .p/

af .p/f D 1; � � � ; F; (6.42)

and applying Roy’s Theorem to this result (6.42) yields

xfi .p; y

f / D�

h�@bf .p/@pi

af .p/ � @af .p/@pi

�yf � bf .p/

�i.�af .p/

�21ıaf .p/

D@bf .p/

@piC@af .p/

@pi

�yf � bf .p/

�,af .p/

(6.43)

Thus, in contract to the Gorman Polar Form,

xfi .p; y

f /

@yfD@af .p/

@pi

,af .p/ i D 1; � � � ; N f D 1; � � � ; F (6.44)

i.e. Engel curves can vary over consumers (although these curves are still linear).Now suppose that the distribution of “uncommitted expenditure” remains pro-

portionally constant over consumers, i.e. consumer incomes y1; � � � ; yF satisfy thefollowing restrictions for all variations in commodity prices p:

4 On the other hand, endogenizing consumer expedition yf , as the wage ratew times the amountof labor supplied by the agent, is not sufficient for consistent linear aggregation in the case ofutility maximization. Endogenizing yf in this manner implies the additional first order condition

@uf .xf�; xLe

�/=@xLe D

h@V f .p; yf

�/=@yf

iw where xLe � le of leisure for con-

sumer f , .f D 1; � � � ; F /. Since the marginal utility of leisure @uf .xf�; xLe

�/=@xLe will

generally vary over consumers, the marginal utility of income @V f .p; yf�/=@yf also varies

over consumers.


yf � bf .p/ D �f�XF

fD1

�yf � bf .p/

��> 0 f D 1; � � � ; F

where �f > 0,XF

fD1�f D 1.

(6.45)

If the individual cost functions are of the form (6.41) and if the distribution of ex-penditure over agents satisfies the restriction (6.45), then aggregate Marshallian de-mand functions exist, have Gorman Polar Form and inherit the properties of utilitymaximization.

Proof. Summing (6.43) over consumers and substituting in (6.45),

Xfxfi D

Xf

@bf .p/

@piC

Xf

@af .p/

@pi�f

hXf

�yf � bf .p/

�i,af .p/

D

Xf

@bf .p/

@piC

Xf�f@af .p/=@pi

af .p/

Xfyf �

Xf�f@af .p/=@pi

af .p/

Xfbf .p/

D@ˇ.p/

@piC

�@˛.p/=@pi

˛.p/

��Xfyf � ˇ.p/

�i D 1; � � � ; N

(6.46)

where ˇ.p/ �Pf b

f .p/, ˛.p/ �QFfD1 a

f .p/�f

. Thus there exist aggregateMarshallian demand functions of Gorman Polar Form. A GPF implies that aggregatedemands are identical to those that could be generated by a single consumer withexpenditure

Pf y

f , and by assumption this consumer maximizes utility. Thus theaggregate demand relations (6.46) inherit the properties of utility maximization. ut

We have the following corollary to the above result.

Corollary 6.1. Suppose that the utility function of individual consumers are homo-thetic, so that the expenditure function of individual consumers have the form

Ef .p; uf / D af .p/uf f D 1; � � � ; F (6.47)

and also suppose that each consumer’s share �f in total expenditurePf y

f isfixed over the data set. Then aggregate Marshallian demand relations

Pf x

f D

X.p;Pf y

f / exist, have the form

Xfxfi D

�@˛.p/=@pi

˛.p/

�Xfyf i D 1; � � � ; N (6.48)

(using the notation of (6.46)) and inherit the properties of utility maximization.In order to see that this is a special case of the result proves above, simply notethat (6.47) is a special case of (6.41) where bf .p/ � 0, and that bf .p/ � 0

.f D 1; � � � ; F / reduce (6.45) to

6.4 Condition for exact nonlinear aggregation over agents 75

yf D �fFXfD1

yf > 0 f D 1; � � � ; F

where �f > 0,FXfD1

�f D 1

(6.49)

In one respect the assumption of homotheticity in (6.47) is more restrictive thanthe quasi-homothetic Gorman Polar form (6.23) at constant price p the ratio of costminimizing demands is independent of the level of expenditure, i.e. increases inexpenditure yf lead to increases in consumption of all commodities along a rayfrom the origin rather than from an arbitrary point. On the other hand, (6.47) ismore general than Gorman Polar Form (6.23) in the sense that the function af .p/can vary over agent, so that (linear) Engel curves can vary over agents.

6.4 Condition for exact nonlinear aggregation over agents

The previous sections assumed that aggregate output or expenditure Y is to be con-structed as a simple linear sum Y D

Pf y

f of the outputs of expenditure yf ofindividual agents. More generally, the aggregation relation can be written as

Y D Y.y1; � � � ; yF /: (6.50)

Then the conditions for existence of an aggregate cost function for producers (forexample) can be written as

C�w; Y.y1; � � � ; yF /

�D

Xf

cf .w; yf / (6.51)

for all .w; y1; � � � ; yF /. Differentiating (6.51) w.r.t. yf ,

@C.w; Y /

@Y

@Y

@yfD@cf .w; yf /

@yff D 1; � � � ; F (6.52)

which implies

@Y

@yf

�@y

@ygD@cf .w; yf /

@yf

,@cg.w; yg/

@ygf; g D 1; � � � ; F: (6.53)

Condition (6.53) implies that Y D Y.y1; � � � ; yF / is strongly separable (see Deatonand Muellbauer 1980, pp. 137–142), so that

Y D Y�X

fhf .yf /

�: (6.54)


The corresponding aggregate cost function and aggregate demands are

C�w; Y.y1; � � � ; yf /

�D ˛.w/ Y

�Xfhf .yf /

�C ˇ.w/

Xi

�w; Y.y1; � � � ; yF /

�D@˛.w/

@wiY�X

fhf .yf /

�C@ˇ.w/

@wii D 1; � � � ; N:

(6.55)

The above cost function is more general than the Gorman Polar Form since the ag-gregate output Y in not restricted to the linear case

Pf y

f . The aggregate cost func-tion inherits the propertied of cost minimization, so the aggregate factor demandsX�w; Y.y1; � � � ; yF /

�can be derived from the aggregate cost function using Shep-

ard’s Lemma. these conditions for exact nonlinear aggregation are less restrictivethan Gorman Polar Form, but it is not easy make this distinction operational (seeDeaton and Muellbauer 1980, pp. 154–158, for one attempt).

References

1. Deaton A. & Muellbauer, J. (1980). Economics and Consumer Behavior: pp. 144–145, 151–153

2. Debreu, G. (1974). Excess demand functions, Journal of Mathematical Economics 1: pp. 15–22

3. Diewert, W. E. (1977). Generalized slutsky conditions for aggregate consumer demand func-tions, Journal of Economic Theory, Elsevier, vol. 15(2), pages 353-362, August.

4. Diewert, W. E. (1980). Symmetry Conditions for Market Demand Functions, Review of Eco-nomic Studies: pp. 595–601

5. Mantel, R.(1977). Implications of Microeconomic Theory for Community Excess DemandFunction, pages 111–126 in M. D. Intriligator, ed., Frontiers of Quantitative Economics,Vol. III A, North-Hollard

6. Sonnenschein, H. (1973). Do Walras’ identity and continuity characterize the class of com-munity excess demand functions?. Journal of Economic Theory 6: pp. 345–354

Chapter 7Aggregation Across Commodities: Non-indexNumber Approaches

Consumers and also producers generally use a wide variety of commodities, so sub-stantial aggregation (grouping) of commodities is necessary to make econometricsstudies manageable. This is particularly the case with flexible functional forms,where the number of parameters to be estimated increases exponentially with thenumber of commodities that are modeled explicitly. Presumably aggregation overcommodities generally misrepresent the choices faced by consumers and produc-ers, so the microeconomic theory that applies to the true behavioral model withdisaggregated commodities may not generalize to a model with highly aggregatedcommodities.

This leads to the following questions: when does aggregation over commodi-ties not misrepresent the agent’s choices or behavior, and how is this aggregationprocedure defined? This lecture summaries two types of results related to this ques-tion: the composite commodity theorem and conditions for two stage budgeting. Thecomposite commodity theorem of Hicks demonstrates that certain restrictions on thecovariation of prices permit consistent aggregation, and the discussion of two stagebudgeting shows that separability restrictions (plus other restrictions) on the struc-ture of utility functions, production functions etc. also permit consistent aggregationover commodities.1

The next lecture provides a more satisfactory answer to the above question. Cer-tain index number formulas for aggregation over commodities can be rationalizedin terms of certain functional forms for production functions, cost functions, etc.,including popular second order flexible functional forms. Thus certain index num-ber formulas for aggregation over commodities inherit the desirable properties ofapproximation that characterize the corresponding second order flexible functionalforms.

1 For simplicity we will assume a single agent, i.e. we will abstract from problems in aggregatingover agents.

77

78 7 Aggregation Across Commodities: Non-index Number Approaches

7.1 Composite commodity theorem

If the prices of several commodities are in fixed propositions over a data set, thenthese commodities can be correctly treated as a single composite commodity withone price. For example, suppose that a consumer maximizes utility over three com-modities x1; x2; x3 and that prices p2 and p3 remain in fixed proportion over thedata set, i.e.

p2;t D �tp2;0 p3;t D �tp3;0 for all times t (7.1)

where p2;0 and p3;0 are base period prices of commodities 2 and 3, and �t is a(positive) scalar that varies over time t . Equivalently the consumer can be viewed assolving a cost minimization problem

E .p1; p2; p3; u/ D minx

3XiD1

pixi

s.t. u.x/ D u�

(7.2)

Combining (7.1) and (7.2),

E .p1;t ; p2;t ; p3;t ; ut / D E .p1;t ; �tp2;0; �tp3;0; ut /

D QE .p1;t ; �t ; ut / for all t(7.3)

Cost minimizing behavior (7.2) implies that QE .p1; �; u/ inherits all the propertiesof a cost function, including Shephard’s Lemma:

@ QE.p1; �; u/

@p1D x1

@ QE.p1; �; u/

@�D

@

@�.p1x1 C �p2;0x2 C �p3;0x3/

D p2;0x2 C p3;0x3

(7.4)

by the envelope theorem.Thus p2;0x2;t C p3;0x3;t can be interpreted as the quantity xc of a composite

commodity at time t , and �t is the price of the corresponding composite commodityat time t . The resulting system of Hicksian demands

x1 D xh1 .p1; �; u/

xc D p2;0x2 C p3;0x3 D xch .p1; �; u/

(7.5)

inherit the properties of cost minimizing demands. Therefore the corresponding sys-tem of demands

x1 D x1 .p1; �; y/

xc D p2;0x2 C p3;0x3 D xc.p1; �; y/

(7.6)

7.2 Homothetic weak separability and two-stage budgeting 79

inherit the properties of utility maximizing demands.However, this composite commodity theorem may be of limited value in justify-

ing and defining aggregation over commodities. Even if two commodities are per-ceived as relatively close substitutes in consumption, their relative prices may varysignificantly due to differences in the supply schedules of the two commodities.

7.2 Homothetic weak separability and two-stage budgeting

The assumption of “two-stage budgeting” has often been used to simplify studies ofconsumer behavior. The general utility maximization problem (3.1) can be writtenas

maxx

u.xA1 ; � � � ; xANA; xB1 ; � � � ; x

BNB; � � � ; xZ1 ; � � � ; x

ZNZ/ � V.p; y/

s.t.NAXiD1

pAi xAi C

NBXiD1

pBi xBi C � � � C

NZXiD1

pZi xZi D y

(7.7)

where x D .xA1 ; � � � ; xANA; xB1 ; � � � ; x

BNB; � � � ; xZ1 ; � � � ; x

ZNZ/, i.e. there are NA C

NB C � � � CNZ commodities. Two-stage budgeting can then be outlined as follows.In the first stage total expenditures y are allocated among broad groups of com-modities x. For example the consumer decides to allocate the expenditures yA tocommodities xA D .xA1 ; � � � ; x

ANA/, yB to commodities xB D .xB1 ; � � � ; x

BNB/, � � � ,

yZ to commodities xZ D .xZ1 ; � � � ; xZNZ/, where yA C yB C � � � C yZ D y. This

allocation of expenditures among broad groups requires knowledge of total expen-ditures y and of an aggregate price QpA; QpB ; � � � ; QpZ for each group of commodities.

Thus in the first stage a consumer is viewed as solving a problem of the form

maxQxu. QxA; QxB ; � � � ; QxZ/

s.t. QpA QxA C QpB QxB C � � � C QpZ QxZ D y(7.8)

where Qx D . QxA; � � � ; QxZ/ and Qp D . QpA; � � � ; QpZ/ denote aggregate commoditiesand prices for the broad groups A; � � � ; Z. Alternatively, utilizing the equivalencebetween utility maximization and cost minimization (see pages 27 to 28), in the firststage a consumer can be viewed as solving a cost minimization problem of the form

minQxQpA QxA C QpB QxB C � � � C QpZ QxZ

s.t. u. Qx/ D u�(7.9)

In the second stage the group expenditure yA; yB ; � � � ; yZ are allocated amongthe commodities within each group. Here the consumer solves maximization prob-lems of the form


maxxA

uA.xA1 ; � � � ; xANA/ � � � max

xZuZ.xZ1 ; � � � ; x

ZNZ/

s.t.NAXiD1

pAi xAi D y

A� � � s.t.

NZXiD1

pZi xZi D y

Z(7.10)

where uA.xA1 ; � � � ; xANA/; � � � ; uZ.xZ1 ; � � � ; x

ZNZ/ are interpreted as “sub-utility func-

tion” for the commodities within each group A; � � � ; Z.What restrictions on the consumer’s utility function u.x/ imply that two-stage

budgeting is realistic, i.e. under what restrictions do the general utility maximiza-tion problem (7.7) and the two-stage procedure (7.8) and (7.10) yield the same so-lutions x�? First consider the second stage of two-stage budgeting. Define a weaklyseparable utility function as follows:

Definition 7.1. A utility function u.x/ is defined as “weakly separable” in commod-ity groups xA D .xA1 ; � � � ; x

ANA/; � � � ; xZ D .xZ1 ; � � � ; x

ZNZ/ if and only if u.x/ can

be written as

u.x/ D QuhuA.xA1 ; � � � ; x

ANA/; � � � ; uZ.xZ1 ; � � � ; x

ZNZ/i

over all x for some macro-utility function u D Qu.uA; � � � ; uZ/ and sub-utility func-tion uA D uA.xA1 ; � � � ; x

ANA/; � � � ; uZ D uZ.xZ1 ; � � � ; x

ZNZ/.

The second stage of two-stage budgeting is equivalent to the restriction that theconsumer’s utility function u.x/ is weakly separable. To be more precise,

Property 7.1. The general utility maximization problem (7.7) and a series of“second stage” maximization problems (7.10) (conditional on group expen-diture yA; � � � ; yZ) yield the same solution x� if and only if u.x/ is weaklyseparable in the above manner in Definition 7.1.

Proof. First, suppose that u.x/ is weakly separable as in Definition 7.1 and that@ Qu=@uA > 0; � � � ; @ Qu=@uZ > 0. Then utility maximization (7.7) requires thateach subutility uA; � � � ; uZ be maximized conditional on its group expenditureyA; � � � ; yZ . For example if xA� solving Definition 7.1 does not solve uA.xA/s.t.

PNAiD1 p

Ai x

Ai D yA, then yA could be reallocated among commodities xA so

as to increase uA without decreasing uB ; � � � ; uZ , i.e. so as to increase the total util-ity level u without violating the budget constraint px D y. Thus the second stageof two-stage budgeting is satisfied if u.x/ is weakly separable. Second, supposethat two-stage budgeting is satisfied. two-stage budgeting implies xA� D xA.p; y/

solving (7.7) can be written as

xA�i D xAi .p

A; yA/ i D 1; � � � ; NA (7.11)

and similarly for subgroups B; � � � ; Z. Without loss of generality xA� solves

7.2 Homothetic weak separability and two-stage budgeting 81

maxxA

u�xA; xB�; � � � ; xZ�

�s.t.

NAXiD1

pAi xAi D y

A

(7.12)

where xB ; � � � ; xZ are fixed at their equilibrium levels solving (7.7), (7.11) im-plies that (given pA; yA) xA� is independent of pB ; � � � ; pZ and hence indepen-dent of xB�; � � � ; xZ�. Thus (7.12) reduces to (7.10), i.e. there exists a subutil-ity function uA.xA/ for commodity group A that is independent of the levels ofother commodities xB ; � � � ; xZ . Thus two-stage budgeting implies weak separabil-ity u

�uA.xA/; � � � ; uZ.xZ/

�. ut

Second, consider the first stage of two-stage budgeting. The critical point hereis to be able to construct a price QpA; � � � ; QpZ for each commodity group A; � � � ; Zsuch that a first stage utility maximization or cost minimization problem yields theoptimal allocation of expenditure across subgroups A; � � � ; Z.

Given weak separability of u.x/, a sufficient condition for the first stage is ho-motheticity of each subutility function uA.xA/; � � � ; uZ.xZ/. To be more precise,

Property 7.2. If u.x/ is weakly separable and each subutility functionuA.xA/; � � � ; uZ.xZ/ is homothetic, then there exists a first stage maximizationproblem that obtains the same distribution of group expenditures yA; � � � ; yZ

as in the general case (7.7).

Proof. Weak separability of u.x/ implies second stage maximization (7.10) andequivalently a series of cost minimization problems.

minxA

NAXiD1

pAi xAi D E

A.pA; uA�/

s.t. uA.xA/ D uA�

:::

minxZ

NZXiD1

pZi xZi D E

Z.pZ ; uZ�/

s.t. uZ.xZ/ D uZ�

(7.13)

where uA� D uA.xA�/; � � � ; uZ� D uZ.xZ�/ for xA�; � � � ; xZ� solving (7.7). Un-der weak separability the consumer’s general cost minimization problemE.p; u�/ D

minxpx s.t. u.x/ D u� can be restated as


E�p; u�

�D minuA;�� ;uZ

EA�pA; uA

�C � � � CEZ

�pZ ; uZ

�s.t. Qu

�uA; � � � ; uZ

�D u�

(7.14)

uA.xA/ homothetic implies EA.pA; uA/ D �.uA/EA.pA/ (see Property 1.3.a onpage 6) where without loss of generality we can define �.uA/ D uA (constantreturns to scale), since the indexing of the indifference curves representing the con-sumer’s preferences is arbitrary. Therefore under homotheticity (7.14) reduces to

E�p; u�

�D minuA;�� ;uZ

uAeA�pA�C � � � C uZeZ

�pZ�

s.t. Qu�uA; � � � ; uZ

�D u�

(7.15)

This can be interpreted as a first stage cost minimization problem with aggregateprices cA.pA/; � � � ; cZ.pZ/ and leading to the following optimal allocation of ex-penditures y over subgroups:

yA D uA�eA.pA/; � � � ; yZ D uZ�eZ.pZ/

Weak separability places substantial restrictions on the degree of substitution be-tween commodities in different groups. Since weak separability implies that utilitymaximizing demands for a group of commodities xA depends only on prices pA

and group expenditure yA (7.11), it follows that other prices pB influence demandsxA only through changes in the optimal level of expenditure yA for the group. Thisimplies the following restriction on the Hicksian substitution effects across groups:

@xAi .p; u�/

@pBjD 'AB.p; y/

@xAi .pA; yA/

@yA

@xBj .pB ; yB/

@yB

i D 1; � � � ; NA j D 1; � � � ; NB

(7.16)

where the function 'AB .p; y/ is independent of the choice of commodities i; jfrom group A;B (see Deaton and Muellbauer, pp. 128–129 for a sketch of a proof),moreover, this relationship (7.16) is both necessary and sufficient for weak separa-bility of groups A and B , i.e. restrictions (7.16) exhaust the implications of weakseparability.

The most obvious application of this discussion of two-stage budgeting is in theestimation of, e.g., consumer demand for food. In general the demand for food goodsxA D .xA1 ; � � � ; x

ANA/ will depend on the prices of all food and non-food final goods

and on total expenditure y. On the other hand suppose that the consumer’s utilityfunction u.x/ is weakly separable in food commodities xA and all other commodi-ties xB , i.e. u.x/ D Qu

�uA.xA/; uB.xB/

�. Then the demand equations for food

xAi D xAi�pA; pB ; y

�.i D 1; � � � ; NA/ can be simplified to xAi D xAi

�pA; yA

�.i D 1; � � � ; NA/ (7.11) which can be derived from a subutility maximization prob-lem

7.3 Implicit separability and two-stage budgeting 83

maxxA

uA.xA/ D V A.pA; yA/

s.t.NAXiD1

pAi xAi D y

A(7.17)

Here V A.pA; yA/ has the properties of an indirect utility function and the corre-sponding Marshallian demands xA D xA.pA; yA/ inherit the properties of utilitymaximization.

One complication in estimating the above model (7.17) for food demand is thatfood expenditure yA is not exogenous to the consumer, and ignoring this fact gen-erally leads to biases in estimation. In the absence of any further assumptions (be-yond weak separability) yA generally depends on all prices and total expenditure,i.e. yA D yA.pA; pB ; y/.

However if the subutility functions uA.xA/ and uB.xB/ are homothetic (im-plying two-stage budgeting), then the corresponding expenditure functions can bewritten as uAeA.pA/ and uBeB.pB/ and the relation yA D yA.pA; pB ; y/ can berewritten as

yA D yAheA.pA/; eB.pB/; y

i(7.18)

Thus we can posit homothetic functional forms EA D uAeA.pA/ and EB DuBeB.pB/, derive the corresponding indirect utility function V A D yA=eA.pA/

and differentiate this indirect utility function to obtain the estimating equationsxA D xA.pA; yA/, and use these functional forms eA.pA/; eB.pB/ as the aggre-gate prices in (7.18). Unfortunately the expenditure equation (7.18) still depends onprices pB as well as pA, but at least the estimating equation (7.18) is more restrictedthan the general equation yA D yA.pA; pB ; y/.2

7.3 Implicit separability and two-stage budgeting

An alternative two-stage budgeting procedure can be obtained if certain separabilityrestrictions hold for the consumer’s cost function E.p; u/ rather than for his utilityfunction u.x/. Preferences are defined as “implicitly separable” in broad groupsA; � � � ; Z if the cost function can be written in the form

E.p; u/ D QEheA.pA; u/; � � � ; eZ.pZ ; u/; u

i(7.19)

where pA D .pA1 ; � � � ; pANA/; � � � ; pZ D .pZ1 ; � � � ; p

ZNZ/. Note that total util-

ity u (rather than subutilities uA; � � � ; uZ) appear in each of the cost functioneA.pA; u/; � � � ; eZ.pZ ; u/, so there are no group subutilities in contrast to the casewhere u.x/ is weakly separable.

2 Alternatively we can eliminate yA from the demand equations xA D xA.pA; yA/ using theidentity yA �

PNAiD1 p

AixAi

.


The following two-stage budgeting procedure is defined by simple differentiationof the macrofunction QE.eA; � � � ; eZ/ and then the group cost function eA.pA; u/; � � � ; eZ.pZ ; u/:

(first stage) sA �

PNAiD1 p

Ai x

Ai

yD@ log QE.eA; � � � ; eZ ; u/

@ log eA

:::

sZ �

PNZiD1 p

Zi x

Zi

yD@ log QE.eA; � � � ; eZ ; u/

@ log eZ

(7.20a)

(second stage) sAi �pAi x

Ai

yAD@ log eA.pA; u/

@ logpAii D 1; � � � ; NA

:::

sZi �pZi x

Zi

yZD@ log eZ.pZ ; u/

@ logpZii D 1; � � � ; NZ

(7.20b)

where y � yAC � � � C yZ(i.e. total expenditure) and yA �PNAiD1 p

Ai x

Ai (expendi-

ture of group A), etc. Thus the budget shares sA; � � � ; sZ of groups A; � � � ; Z are ob-tained by simple logarithmic differentiation of the macrofunction QE.eA; � � � ; eZ ; u/,and the share sAi .i D 1; � � � ; NA/, � � � , sZi .i D 1; � � � ; NZ/ of each commodity ingroup expenditure is obtained by simple logarithmic differentiation of the groupcost functions eA.pA; u/; � � � ; eZ.pZ ; u/.

Proof. Differentiating (7.19) w.r.t. pAi and applying Shephard’s Lemma,

@E.p; u/

@pAiD@ QE.�/

@eA@eA.�/

@pAi

D xAi .p; u/ i D 1; � � � ; NA;

(7.21)

Thus

yA �

NAXiD1

xAi pAi

D@ QE.�/

@eA

NAXiD1

@eA.�/

@pAipAi

D@ QE.�/

@eAeA.�/ by Euler’s Theorem

(7.22)

References 85

since eA.pA; u/ is linear homogeneous in pA.3

sA �yA

y

D@ QE.�/=@eA

y=eA.�/

D@ log QE.�/@ log eA

(7.23)

which is (7.20a). Substituting @ QE.�/

@eAD

yA

eA.�/(7.22) into @eA.�/

@pAi

@ QE.�/

@eAD xAi (7.21)

yields (7.20b).

Note that implicit separability is sufficient for the two-stage budgeting procedureoutlined in (7.20). In contrast weak separability of u.x/ was sufficient only for thesecond stage of the budgeting procedure discussed in the previous section. Alsonote that implicit separability implies that the ratio of commodities within a groupis independent of prices of commodities outside the group, i.e.

@hxAi .p; u/=x

Aj .p; u/

i@pBk

D 0 i; j D 1; � � � ; NA k D 1; � � � ; NB (7.24)

(see (7.21)). In contrast, weak separability of u.x/ implied that the ratio of marginalrates of substitution between commodities within a group is independent of levelsof commodities outside the group.

References

1. Deaton A. & Muellbauer, J. (1980). Economics and Consumer Behavior

3 E.p;u/ is linear homogeneous in p only if eA.pA; u/; � � � ; eZ.pZ ; u/ are also linearhomogeneous in prices and QE.eA; � � � ; eZ ; u/ is linear homogeneous in eA; � � � ; eZ . How-ever note that eA.pA; u/ does not equal total expenditure yA on group A (see (7.22)). ThuscA.pA; u/; � � � ; cZ.pZ ; u/ are to be interpreted as group price indexes that depend on utilitylevel u.

Chapter 8Index Numbers and Flexible Functional Forms

In this lecture we show that particular index number formulas for aggregating overcommodities can be rationalized in terms of particular functional forms for produc-tion functions or dual cost or profit functions. Approximately correct procedures foraggregating over commodities are presented for cases of Translog and GeneralizedLeontief functional forms. Since these functional forms provide a second order ap-proximation to any true form, the corresponding index number formulas can oftenbe interpreted as approximately correct.

The results obtained here should be contrasted with the previous lecture. Thereconsistent aggregation over commodities was rationalized essentially in terms ofassumptions of separability between groups of commodities. Here specific aggre-gation procedures are rationalized essentially in terms of specific functional formsfor production functions or dual cost functions. Since assumptions of Translog orGeneralized Leontief functional forms are usually considered less restrictive thanassumptions of (homothetic) weak separability, this lecture presents a more usefulbasis for a theory of approximate aggregation over commodities.

8.1 Laspeyres index numbers and linear functional forms

Until recently most index number computations have used simple base periodweighting schemes, and the most common of these are Laspeyres quantity and priceindexes. The Laspeyres quantity index can be written as

X1

X0D

PNiD1 pi;0xi;1PNiD1 pi;0xi;0

(8.1)

where p0 D .p1;0; � � � ; pN;0/ denotes the prices for the N commodities in the baseperiod .t D 0/, x0 D .x1;0; � � � ; xN;0/ denotes the quantities of the N commoditiesin the base period .t D 0/, and x1 D.x1;1; � � � ; xN;1/ denotes the quantities of theN commodities in any other period t D 1.

87

88 8 Index Numbers and Flexible Functional Forms

This aggregation procedure (8.1) can be defined as correct if the ratio of aggre-gates X1, X0 provides an accurate measure of the contributions of inputs 1; � � � ; Nto the producer’s output in different time periods. Thus in the case where aggrega-tion is defined over all inputs (1; � � � ; N is to be interpreted as all inputs) and there isa single output, the quantity indexX1=X0 in (8.1) is correct ifX1=X0 is equal to theratio of outputs f .x1;1; � � � ; xN;1/=f .x1;0; � � � ; xN;0/ for any time periods t D 0; 1.

Similarly a Laspeyres price index can be written as

P1

P0D

PNiD1 xi;0pi;1PNiD1 xi;0pi;0

(8.2)

where the prices p D .p1; � � � ; pN / for any period are weighted by the base periodquantities x0 D .x1;0; � � � ; xN;0/. Interpreting commodities 1; � � � ; N as inputs inproduction and assuming cost minimizing behavior, this aggregation procedure canbe defined as correct if the ratio of the aggregates P1, P0 provides an accuratemeasure of the contribution of inputs 1; � � � ; N to the cost of attaining a given levelof output y. In the case where aggregation is defined over all inputs (1; � � � ; N is tobe interpreted as all inputs), the price index P1=P0 is correct if P1=P0 is equal tothe ratio of minimum costs C.p1;1; � � � ; pN;1; y/=C.p1;0; � � � ; pN;0; y/ for any twotime periods and a common output level y.

Are the above aggregation procedures (8.1)–(8.2) correct for some cases of pro-duction functions? The answer is yes: assuming static competitive profit maximizingor cost minimizing behavior for a firm, an aggregation procedure (8.1) or (8.2) overinputs x D .x1; � � � ; xN / can be rationalized in terms of a linear production functionwith a family of parallel straight line isoquants

x1

x2

0 (8.3)

and also in terms of a linear production function with fixed coefficients, i.e. rightangle isoquants

8.2 Exact indexes for Translog functional forms 89

x1

x2

0 (8.4)

The first case (8.3) assumes perfect substitution between inputs and the second case(8.4) assumes zero substitution between inputs.

More formally, given Laspeyres quantity and price indexes X1=X0 (8.1), a linearproduction function y D f .x/ satisfying either (8.3) or (8.4), and static competitiveprofit maximizing or cost minimizing behavior, then

X1

X0Df .x1;1; � � � ; xN;1/

f .x1;0; � � � ; xN;0/

P1

P0Dc.p1;1; � � � ; pN;1/

c.p1;0; � � � ; pN;0/for all t D 1; 0

(8.5)

where c.p/ denotes a unite cost function (the minimum cost of producing one uniteof output y given prices p) (see Diewert 1976 pp. 182–183 for a proof of (8.5) ).Under the above assumptions X1=X0 is the ratio of output and P1=P0 is the ratioof unite cost of output for the two periods t D 1; 0. In this sense Laspeyres quantityand price indexes are exact for both a linear production function satisfying (8.3) anda linear production function satisfying (8.4).

In either case linear production functions (with either zero or perfect substitu-tion between inputs are very unrealistic and highly restrictive. Linear productionfunctions can provide only a first order approximation to an arbitrary productionfunction.1 This suggests that Laspeyres indexes may often lead to substantial er-rors in aggregation over commodities, and it is unlikely that aggregate data col-lected in this manner inherits properties of profits maximization or cost minimiza-tion from disaggregate data. In the next two sections we obtain more positive resultsby demonstrating that particular quantity and price indexes are correct for Translogand Generalized Leontief functional forms.

8.2 Exact indexes for Translog functional forms

Economists have frequently advocated the use of Divisia indexes rather than Laspeyresindexes for aggregating over commodities (e.g. Hulten 1973, pp. 1017–1026). A

1 It can easily be shown that Laspeyres indexes also provide a first order approximation to anarbitrary (true) index (see Deaton1980, pp.173–174).


Divisia quantity index for (e.g.) inputs can be defined in continues time by the line

integralXt

X0D exp

�Z �XN

iD1si .t/

@xi .t/=@t

xi .t/

��where si .t/ �

wi .t/xi .t/PNjD1wj .t/xj .t/

:

The following Tornqvist index is the most commonly used discrete approximationto the Divisia quantity index for inputs:

log�X1

X0

�D

NXiD1

si log�xi;1

xi;0

�

where si �1

2

wi;1xi;1PNjD1wj;1xj;1

Cwi;0xi;0PNjD1wj;0xj;0

!i D 1; � � � ; N:

(8.6)

The Tornqvist quantity index (8.6) is exact for a Translog production functionwith constant returns to scale. In other words, assuming static competitive cost min-imizing behavior,

log�Xt

Xs

�D log

�f .xt /f .xs/

�(8.7)

for all periods s, t when f .x/ is Translog constant returns to scale and the quantityindex is calculated as in (8.6). Such an index, which is exact for a constant returns(or variable returns) to scale flexible form for f .x/, is termed superlative.

Proof. First we establish an algebraic result for a quadratic function

g.´/ D a0 CX

iai´i C

1

2

Xi

Xjaij´i j .aij D aj i for all i; j /

or, in matrix notation,

g.z/ D a0 C aT zC1

2zTA z .A symmetric/ (8.8)

Then

g.z1/ � g.z0/ D aT .z1 � z0/C1

2zT1 A z1 �

1

2zT0 A z0

D aT .z1 � z0/C1

2zT1 A.z1 � z0/C

1

2zT0 A.z1 � z0/

since A is symmetric

D1

2.aC Az1 C aC Az0/T .z1 � z0/

(8.9)

which implies

g.z1/ � g.z0/ D1

2

�@g.z1/@z

[email protected]/@z

�T.z1 � z0/ (8.10)


Moreover (8.10) implies (8.8), i.e. (8.8) is correct if and only if (8.10) iscorrect. Since a Translog production function logf D a0 C

PNiD1 ai log xi CPN

iD1

PNjD1 aij log xi log xj is quadratic in logs, (8.10) and @ logf .x/

@ logxiD

@f .x/=@xif .x/=xi

.i D 1; � � � ; N / imply

logf .x1/ � logf .x0/ D1

2

NXiD1

�@f .x1/@xi;1

xi;1

f .x1/C@f .x0/@xi;0

xi;0

f .x0/

�log

�xi;1

xi;0

�:

(8.11)Static competitive cost minimization implies @f .x/

@xiD

[email protected];y/=@y .i D 1; � � � ; N /,

and constant returns to scale implies f .x/ DPNiD1

@f .x/@xi� xi (Euler’s theorem).

Substituting these to (8.11),

log�f .x1/f .x0/

�D1

2

NXiD1

wi;1xi;1PNjD1wj;1xj;1

Cwi;0xi;0PNjD1wj;0xj;0

!log

�xi;1

xi;0

�(8.12)

where right hand side is the Tornqvist input quantity index (8.6).

Moreover, the Tornqvist quantity index (8.6) is exact only for a Translog con-stant returns to scale production function (this follows from the equivalence between(8.10) and a quadratic function g.z/ (8.8)).

The assumption of a constant returns to scale production function appears to becrucial to the interpretation of the particular Tornqvist quantity index (8.6) as exact.This assumption is not crucial only in the case of pair-wise comparisons of aggregateinputs, i.e. if the aggregate index (8.6) is to be calculated only for two time periodst D 0; 1 (see Diewert 1976, Op. cit.).

Nevertheless, a quantity index closely related to (8.6) can be interpreted as exactfor a general Translog production function. Define the following quantity index:

log�X1

X0

�D1

2

NXiD1

�wi;1xi;1

py;1f .x1/C

wi;0xi;0

py;0f .x0/

�log

�xi;1

xi;0

�(8.13)

where py;t � price of the (single) output in period t . This deviates from theTornqvist quantity index (8.6) only in that wi;txi;t is divided by total revenuepy;tf .xt / rather than by total cost

PNiD1wi;txi;t . In the case of constant returns to

scale f .x/ DPNiD1

@f .x/@xi� xi (Euler’s theorem) and in turn pyf .x/ D

PNiD1wixi .

Thus (8.11) reduces to the Tornqvist index (8.6) in the case of constant returns toscale and competitive profit maximization. Unlike the index (8.6), the above quan-tity index (8.13) is exact for a general (variable returns to scale) Translog productionfunction. This result requires the assumption of profit maximization rather than sim-ple cost minimization, in contrast to (8.6).

Proof. Substituting the first order conditions for static competitive profit maximiza-tion @f .xt /

@xiD

wi;tpy;t

.i D 1; � � � ; N / into (8.11),


log�f .x1/f .x0/

�D1

2

NXiD1

�wi;1xi;1

py;1f .x1/C

wi;0xi;0

py;0f .x0/

�log

�xi;1

xi;0

�(8.14)

i.e. (8.13) is exact for the general Translog f .x/.

This index (8.13) can be interpreted as an alternative Tornqvist index approximatingthe continuous Divisia quantity index.

Given an input quantity index X1=X0 such as (8.6) or (8.13), we can easily con-struct a corresponding implicit input price index using the following factor reversalequation:

B�W1W0

��X1

X0

�D

PNiD1wi;1xi;1PNiD1wi;0xi;0

(8.15)

i.e. the product of the input price and quantity indexes is equal to the ratio of totalexpenditure on the N disaggregate inputs for the corresponding time periods t D0; 1. Assuming X1=X0 D f .x1/=f .x0/ (8.7), (8.15) implies

B�W1W0

�D

PNiD1wi;1xi;1=f .x1/PNiD1wi;0xi;0=f .x0/

(8.16)

i.e. C.W1=W0/ can be interpreted as the index of average costs of production. Obvi-ously if a quantity index is superlative then the corresponding implicit price indexis superlative in an analogous manner.

Alternatively an input price index can be calculated directly rather than by using(8.15). Now assume that the production function is constant returns to scale and thatthe cost function C.w; y/ D y c.w/ is Translog: log c.w/ D a0C

PNiD1 ai logwiCPN

iD1

PNjD1 aij logwi logwj . Define the following Tornqvist price index for in-

puts:

log�W1

W0

�D

NXiD1

si log�wi;1

wi;0

�(8.17)

where s D .s1; � � � ; sN / are calculated as in (8.6).2 Proceeding as in the proof of(8.7), we obtain the result

log�W1

W0

�D log

�c.w1;1; � � � ; wN;1/

c.w1;0; � � � ; wN;0/

�(8.18)

which is analogous to (8.16). Thus the Tornqvist price index (8.17) is exact for aTranslog unit cost function. In the absence of constant returns to scale in production,the assumption of a Translog cost function C.w; y/ implies

2 Since flexible functional forms are not self-dual (e.g. a Translog production function does notimply a Translog cost function, or vice-versa), the price index (8.17) for a Translog unit cost func-tion is not equivalent to the implicit price index (8.16) for a Translog constant returns to scaleproduction function.


log�W1

W0

�D

NXiD1

si log�wi;1

wi;0

�D log

�C.w1;1; � � � ; wN;1; y1/

C.w1;0; � � � ; wN;0; y0/

� (8.19)

i.e. the Tornqvist index (8.17) is equal to the ratio of total costs in different time pe-riods t D 0; 1: Thus, in the absence of constant returns to scale, the direct Tornqvistindex (8.17) cannot strictly be interpreted as a price index for inputs since it dependsupon a measure y1; y0 of input quantities x1; x0 as well as upon input prices w1;w0.

In contrast, consider the index C.W1=W0/ calculated implicitly from (8.15) usingthe quantity index (8.13), science this index satisfies (8.16) for a general Translogproduction function and profit maximizing behavior, it can be interpreted as the ratioof average costs of production

logB�W1W0

�D log

�C.w1;1; � � � ; wN;1; y1/=y1

C.w1;0; � � � ; wN;0; y0/=y0

�(8.20)

even in the absence of constant returns to scale.The derivations of exact index number equations are complicated somewhat in

the case of multiple outputs y D .y1; � � � ; yM /. Generalizations of the Translogproduction function do not appear to provide an entirely satisfactory basis for indexnumber formulas. For example, assume a Translog transformation function

logy1 D a0 CTXiD1

ai log ´i CTXiD1

TXjD1

aij log ´i log j (8.21)

where z � .y2; � � � ; yM ;�x1; � � � ;�xN / .T DM � 1CN/ and static competitiveprofit maximization

maxy;x

MXiD1

piyi �

NXiD1

wixi

s.t. y1 D y1.z/

(8.22)

Then the quadratic identity (8.11) implies

log�yi;1

yi;0

�D

NXiD1

sxi log�xi;1

xi;0

��

MXiD2

syi log

�yi;1

yi;0

�where sxi �

1

2

�wi;1xi;1

pi;1yi;1Cwi;0xi;0

pi;0yi;0

�syi �

1

2

�pi;1yi;1

pi;1yi;1Cpi;0yi;0

pi;0yi;0

� (8.23)


(8.23) defines quantity indexes that are exact for the Translog transformation func-tion (8.21). However the output quantity index

PMiD2 s

yi log .yi;1=yi;0/ incorporates

the level of the first output y0 only indirectly via the definitions of the weights syifor output 2; � � � ;M .

A more satisfactory approach to the derivation of index numbers in the case ofmultiple outputs may be in terms of the cost function C.w; y/. Assume a Translogjoint cost function

logC D a0 CNXiD1

ai logwi CMXiD1

bi logyi

C

NXiD1

NXiD1

aij logwi logwj CMXiD1

MXiD1

bij logyi logyj

C

TXiD1

TXiD1

cij logwi logyj

(8.24)

and static competitive profit maximizing behavior. Then the quadratic identity(8.11), Shephard’s Lemma and @C.w; y/=@yi D pi .i D 1; � � � ;M/ imply

log�C1

C0

�D

NXiD1

sxi log�wi;1

wi;0

�C

MXiD1

syi log

�yi;1

yi;0

�

where sxi �1

2

wi;1xi;1PNiD1wi;1xi;1

Cwi;0xi;0PNiD1wi;0xi;0

!

syi �

1

2

pi;1yi;1PNiD1wi;1xi;1

Cpi;0yi;0PNiD1wi;0xi;0

!:

(8.25)

The first sumPNiD1 s

xi log .wi;1=wi;0/ can be interpreted as a Tornqvist price

index log .W1=W0/ for all inputs, and the second sumPMiD1 s

yi log .yi;1=yi;0/ can

be interpreted as a Tornqvist quantity index log .Y1=Y0/ for all outputs. In order tosee this, rewrite equation (8.25) as

C1

C0D

�W1

W0

��

�Y1

Y0

�where log

�W1

W0

�D

XN

iD1sxi log

�wi;1

wi;0

�log

�Y1

Y0

�D

XN

iD1syi log

�yi;1

yi;0

� (8.26)

i.e. the index of total costs is always equal to the product of the input price index andthe output quantity index. This confirms that Y1=Y0 can be interpreted as an output


quantity index and W1=W0 can be interpreted as an index of the contributions ofinputs to the average cost of aggregate output.

The corresponding implicit Tornqvist input quantity and output price indexes canthen be calculated from the equations�

W1

W0

�B�X1X0

�D

PNiD1wi;1xi;1PNiD1wi;0xi;0

A�P1P0

��Y1

Y0

�D

PMiD1 pi;1yi;1PMiD1 pi;0yi;0

(8.27)

These implicit indexes .BX1=X0/ and .BP1=P0/ are also superlative.3 Finally, con-sider the problem of deriving index numbers for aggregation of commodities in thecase of consumer behavior. Assuming that the utility function u.x/ is homothetic (orequivalently constant returns to scale) and that the unit cost function e.p/ D min

xpx

s.t. u.x/ D 1 is Translog, then the Tornqvist price index

log�P1

P0

�D1

2

NXiD1

pi;1xi;1PNjD1 pj;1xj;1

Cpi;0xi;0PNjD1 pj;0xj;0

!log

�pi;1

pi;0

�(8.28)

and the corresponding implicit quantity index

logB�X1X0

�D log

PNiD1 pi;1xi;1PNjD1 pj;0xj;0

!� log

�P1

P0

�(8.29)

are exact (see the discussion of (8.17)–(8.18)).Alternatively suppose that the consumer’s utility function u.x/ is Translog as

well as homothetic:

logu D a0 CNXiD1

ai log xi CNXiD1

NXjD1

aij log xi log xj (8.30)

u.x/ constant returns to scale implies that the indirect utility function V.p; y/ isconstant returns to scale in expenditure y, so that (using Euler’s theorem) V.p; y/ [email protected];y/@y�y. Applying the quadratic identity (8.10) to (8.30), and using the first order

conditions @u.x/@xiD

@V.p;y/@y

� pi .i D 1; � � � ; N / and in turn @V.p; y/=@y D u=y,we obtain the result

3 In the case of constant returns to scale in production, W1=W0 can also be interpreted as a priceindex of the contributions of inputs to the marginal cost of aggregate output.


log�u.x1/u.x0/

�D

NXiD1

si log�xi;1

xi;0

�

where si �1

2

pi;1xi;1PNjD1 pj;1xj;1

Cpi;0xi;0PNjD1 pj;0xj;0

! (8.31)

The right hand side of (8.31) defines a Tornqvist quantity index log.X1=X0/ whichis exact for a homothetic Translog utility function u.x/, and the corresponding im-plicity price index can be calculated using (8.29).

8.3 Exact indexes for Generalized Leontief functional forms

Results in the previous section demonstrated that many index numbers closelyrelated to popular Tornqvist indexes can be rationalized in terms of underlyingTranslog functional forms. This section briefly illustrates that different index num-ber formulas are implied by Generalized Leontief functional forms. Nevertheless,since both classes of functional forms provide a second order approximation to atrue form, the various index number formulas should lead to similar results at leastfor small changes in quantities and prices.

First, assume a constant returns to scale generalized Leontief production function

f .x/ DNXiD1

NXjD1

aijpxipxj (8.32)

and static competitive profit maximization. This implies

f .x1/f .x0/

D

PNiD1 si;0.xi;1=xi;0/

1=2PNjD1 sj;1.xj;0=xj;1/

1=2

where si;t �wi;txi;tPNkD1wk;txk;t

(8.33)

i.e. the quantity index number X1=X0 corresponding to the right hand side of (8.33)is exact for the production function (8.32).

Proof. Competitive profit maximization and constant returns to scale imply p @f .x/@xiD

wi and pf .x/ DPNiD1wixi , so that

vi;t �wi;tPN

jD1wj;txj;t

D@f .xt /=@xi;tf .xt /

.i D 1; � � � ; N /

(8.34)

8.4 Two-stage aggregation with superlative index numbers 97

for all time periods t . Substituting the derivatives of (8.32) into (8.34),

vi;t D.xi;t /

�1=2PNjD1 aij

pxj;t

f .xt /(8.35)

multiplying vi;0 bypxi;1pxi;0 and summing over i D 1; � � � ; N ,

NXiD1

pxi;1 � vi;0 �

pxi;0 D

PNiD1

PNjD1

pxi;1 � aij �

pxi;0

f .x0/(8.36)

and similarly,

NXiD1

pxi;0 � vi;1 �

pxi;1 D

PNiD1

PNjD1

pxi;0 � aij �

pxj;1

f .x1/(8.37)

Dividing (8.36) by (8.37) (noting ai;j D aj;i andpxi;1pxi;0 D

pxi;1=xi;0 � xi;0),P

i .xi;1=xi;0/1=2� vi;0 � xi;0P

j

�xj;0=xj;1

�1=2� vj;1 � xj;1

Df .x1/f .x0/

: (8.38)

Alternatively , assume constant returns to scale in production and a GeneralizedLeontief unit cost function c.w/ D C.w; y/=y:

c.w/ DNXiD1

aijpwipwj (8.39)

These assumptions and competitive profit maximization imply

c.w1/c.w0/

D

PNiD1 si;0 .wi;1=wi;0/

1=2PNjD1 sj;1

�wj;0=wj;1

�1=2 (8.40)

where s0, s1 are defined as in (8.33) (the proof is analogous to the above proof of(8.33)). Thus the above price index for inputs is exact for a Generalized Leontiefunit cost function (8.39).

8.4 Two-stage aggregation with superlative index numbers

All of the above index number formulas and their relations to Translog and General-ized Leontief functional forms were essentially calculated as a one stage aggregationprocedure. For example all inputs 1; � � � ; N were aggregated directly into a singleinput quantity index and a single input price index rather than into several quantity


and price subindexes. On the other hand, commodities typically are aggregated intovarious subindexes for use in econometric models.

This leads to the following important question: are two stage aggregation pro-duces (using index numbers formulas) exact or approximately exact? For example,suppose that the Tornqvist quantity index number formula (8.6) and the correspond-ing implicit price index formula are applied separately to two subsets of inputs (NAand NB inputs, respectively, where NA C NB D N , the total number of inputs),resulting in two quantity indexes QA;QB

QA� log

�XA1

XA0

�

D1

2�

NAXiD1

wAi;1x

Ai;1PNA

jD1wAj;1x

Aj;1

CwAi;0x

Ai;0PNA

jD1wAj;0x

Aj;0

!� log

xAi;1

xAi;0

!

QB� log

�XB1

XB0

�

D1

2�

NBXiD1

wBi;1x

Bi;1PNB

jD1wBj;1x

Bj;1

CwBi;0x

Bi;0PNB

jD1wBj;0x

Bj;0

!� log

xBi;1

xBi;0

!(8.41)

and corresponding implicit prices indexes QpA, QpB . Then the same quantity indexnumber formula (8.6) is applied to the subindexes QA, QB , QpA, QpB , resulting in aquantity index yQ:

yQ � log2�X1X0

�D1

2

�QpA1Q

A1

QpA1QA1 C Qp

B1 Q

B1

CQpA0Q

A0

QpA0QA0 C Qp

B0 Q

B0

�log

�QA1

QA0

�C1

2

�QpB1 Q

B1

QpA1QA1 C Qp

B1 Q

B1

CQpB0 Q

B0

QpA0QA0 C Qp

B0 Q

B0

�log

�QB1

QB0

� (8.42)

Are the results of one stage and two stage aggregation identical? For particular,is the quantity index yQ calculated in (8.41)–(8.42) exact for a constant returnsto scale Translog production function y D f .x1; : : : ; xN /, i.e. does yQ equallog.f .x1/=f .x0//?

In general the two stage aggregation procedure (8.41)–(8.42) is not exact, i.e. yQ ¤log.f .x1/=f .x0//. This result is not surprising, since results in the previous chapterindicated that two stage budgeting is correct only under strong restrictions on thestructure of production or utility functions (e.g. homothetic weak separability).

On the other hand the two stage aggregation procedure (8.41)–(8.42) is approx-imately exact, i.e. yQ approximates log .f .x1/=f .x0//. Moreover, two stage aggre-gation procedures based on known superlative index number formulas are approxi-mately exact. This can be explained very briefly as follows (see Diewert 1978).

8.4 Two-stage aggregation with superlative index numbers 99

The Vartia I price index P V and quantity index QV , where

logP V �NXiD1

si log�pi;1

pi;0

�

logQV�

NXiD1

si log�xi;1

xi;0

�

where si �

24 pi;1xi;1 � pi;0xi;0

log .pi;1xi;1=pi;0xi;0/

, PNjD1 pj;1xj;1 �

PNjD1 pj;0xj;0

log�PN

jD1 pj;1xj;1

.PNjD1 pj;0xj;0

�35 ;

(8.43)are know to be consistent in aggregation, i.e. one stage and two stage aggregationprocedures based on (8.43) lead to identical index numbers. Unfortunately theseVartia indexes are exact for a constant returns to scale Cobb-Douglas productionfunction y D f .x1; : : : ; xN / rather then for a second order flexible functionalform.4

Nevertheless it can be shown that the vartia quantity index QV differentiallyapproximates the Tornqvist quantity. index QT � .X1=X0/ (8.6) to the secondorder at any point where the prices and quantities are the same for the two timeperiods comparison .p1 D p0; x1 D x0/, i.e.

QV .z/ D QT .z/

@QV .z/@í

D@QT .z/@í

@2QV .z/@í@ j

D@2QT .z/@í@ j

for all i; j

(8.44)

where z � .p1; p0; x1; x2/ and p1 D p0, x1 D x0. Identical results hold forthe Vartia price index P V and the Tornqvist price index pT � .W1=W0/ (8.18).These results do not require any assumptions about optimizing behavior log agents.Moreover, these results can be extended to other superlative indexes, e.g. indexescorresponding to Generalized Leontief production functions or cost functions.

Therefore two stage aggregation procedures using superlative index number for-mulas are approximately correct (exact) for relatively small changes in prices andquantities between the comparison time periods t D 0; 1. These changes in pricesand quantities are usually smaller when indexes are constructed by chaining obser-vations in successive period rather than using a constant base period. On this basisit is recommended that a Tornqvist quantity macroindex or subindex (8.6), e.g., beconstructed as

4 Laspeyres and paasche index numbers are also consistent in aggregation, but the correspondingproduction are much more restrictive than the Cobb-Douglas (see Section 8.1).


QAt D

XNA

iD1sAi;t log

xAi;t

xAi;t�1

!t D 1; � � � ; T (8.45)

where sAi;t D1

2

wAi;tx

Ai;tPNA

jD1wAj;tx

Aj;t

CwAi;t�1x

Ai;t�1PNA

jD1wAj;t�1x

Aj;t�1

!(chaining observations in successive periods) rather than as

QAt D

XNA

iD1sAi;t log

xAi;t

xAi;0

!t D 1; � � � ; T (8.46)

where sAi;t D1

2

wAi;tx

Ai;tPNA

jD1wAj;tx

Aj;t

CwAi;0x

Ai;0PNA

jD1wAj;0x

Aj;0

!(using a constant base period t D 0).

8.5 Conclusion

The above results indicate that, at least in the case of simple static maximizationmodels, various index number procedures can be used to obtain approximately con-sistent aggregation of commodities provided that the variation in prices and quan-tities between comparison periods is small. For time series data this condition canusually be satisfied by chaining observations in successive periods.

However in the case of cross-section data there may be substantial variation inquantities or prices between successive periods, and here different superlative indexnumber formulas may lead to significantly different results. In this case it may beuseful to compare the variation in the N quantity ratios xi;1=xi;0 to the variation inthe N price ratios pi;1=pi;0.

Usually there is much less variation in the price ratios than in the quantityratios. Then a directly defined price index PAt D

PNAiD1 s

Ai;t log.pAi;t=p

Ai;t�1/ is

less sensitive to the variation in data than is a directly defined quantity indexQAt D

PNAiD1 s

Ai;t log.xAi;t=x

Ai;t�1/ (since both equations use the same shares sAt ).

This suggests that the best strategy in this case is to calculate the price indexes di-rectly and to employ the corresponding implicit quantity indexes (see Allen 1981,pp. 430-435).

References

1. Allen, R. C. and Diewert, W. E. (1981). Direct versus implicit superlative index number for-mulae. The Review of Economics and Statistics, 63(3):430–435.

2. Deaton, A. and Muellbauer, J. (1980). Economics and Consumer Behavior. Cambridge Uni-versity Press.

3. Diewert, W. E. (1976). The Economic Theory of Index Number: A Survey, volume 1. ElsevierScience Publishers.

References 101

4. Diewert, W. E. (1978). Superlative index numbers and consistency in aggregation. Economet-rica, 46(4):883–900.

5. Fuss, M., McFadden, D., and Mundlak, Y. (1978). A Survey of Functional Forms in the Eco-nomic Analysis of Production, volume 1. McMaster University Archive for the History ofEconomic Thought.

6. Hulten, C. R. (1973). Divisia index numbers. Econometrica, 41(6):1017–1025.


8.6 Laspeyres and Paasche cost of living indexes

minx

px

s.t. u.x/ D u�

)! E.p; u/

u.x/ homothetic E.p; u/ D uE.p; 1/„ ƒ‚ …e.p/

.

True cost of living (COL) index conditional on u�:

P1

P2DE.p1; u�/E.p0; u�/

:

Homotheticity) COL index is independent of any u�:

P1

P0Du� � e.p1/u� � e.p0/

De.p1/e.p0/

:

Laspeyres COL index is > true COL (at u�0):�P1

P0

�L�

x0p1x0p0„ ƒ‚ …

Laspeyres COL index

>

D

E.p1; u�0/E.p0; u�0/„ ƒ‚ …true COL index

:

Paasche COL index is < true COL(at u�1):�P1

P0

�P�

x1p1x1p0„ ƒ‚ …

Paasche COL index

D

>

E.p1; u�1/E.p0; u�1/„ ƒ‚ …true COL index

:

So (assuming homotheticity)�P1

P0

�P< true COL <

�P1

P0

�L(8.47)

Note: (8.47) marks no assumptions about form of preferences (such as Translog),except for homotheticity.

If (8.47) does no place close enough bounds on true COL, then we should specify(e.g.) a Tornqvist consumer price index (assuming a Translog CRTS cost functionE.p; u/).

8.7 Fisher indexes (for inputs) 103

8.7 Fisher indexes (for inputs)

CRTS ! C.w; y/ D yC.w; 1/ where C.w; 1/ � c.w/: unit cost function.Assume the following quadratic c.w/ function:

c.w/ DhX

i

Xjaijwiwj

i1=2(8.48)

Define the following price index:

�W1

W0

�FD

"�W1

W0

�L�W1

W0

�P# 12(8.49)

This is called a Fisher price index (it is a geometric mean of a Laspeyres andPaasche index).

We can prove the following result:

Theorem 8.1. Assume CRTS and the quadratic cost function (8.48) (all inputs areat static cost minimizing equilibrium). Then�

W1

W0

�FDc.w1/c.w0/

�AC1

AC0:

Alternatively define a Fisher input quantity index as�X1

X0

�F��.X1=X0/

L„ ƒ‚ …w0x1=w0x0

.X1=X0/P„ ƒ‚ …

w1x1=w1x0

� 12 (8.50)

and assume a quadratic CRTS production function

y DhX

i

Xjbijxixj

i1=2(8.51)

Then, assuming all inputs are at static cost minimizing equilibrium, we can show�X1

X0

�FDy1

y0:

Note: A quadratic cost function (8.48) is essentially equivalent to a quadratic pro-duction function (8.51). (In contrast, Translog and G.L. forms are not “self-dual”).


Proof. Assuming CRTS, define the unit cost function c.w/ D C.w; y/=y. Assume

c.w/ DhX

i

Xjaijwiwj

i1=2(8.52)

So

@c.w/@wi

D1

2

hXi

Xjaijwiwj

i�1=22X

jaijwj .aij D aj i /

D

Pj aijwj

c.w/by (8.52)

(8.53)

[email protected]/[email protected]/

D

Pj aijwj

c.w/2(8.54)

Cost minimization for all inputs implies Shephard’s Lemma

[email protected]/@wi

D xi : (8.55)

Dividing (8.55) by yc.w/ �Pi wixi ,

xiPi wixi

[email protected]/[email protected]/

: (8.56)

The Fisher input price index is�W1

W0

�FD

h.W1=W0/

L.W1=W0/Pi1=2

D Œ.w1x0=w0x0/ � .w1x1=w0x1/�1=2

D Œ.w1x0=w0x0/=.w0x1=w1x1/�1=2

D

��[email protected]/[email protected]/

��[email protected]/[email protected]/

��1=2by (8.56)

D

�Xi

Xjwi;1

aijwj;0

c.w0/2

�Xi

Xjwi;0

aijwj;1

c.w1/2

�1=2by (8.54)

D

h1=c.w0/2

.1=c.w1/2

i1=2D c.w1/=c.w0/

Chapter 9Measuring Technical Change

In previous chapters we assumed the existence of an unchanged production functiony D f .x/ as transformation function g.y; x/ D 0, where output levels y are deter-mined by input levels x. In contrast here we allow for shifts in the technology of thefirm or industry that lead to changes in output levels y that cannot be accounted forsolely by changes in input levels x.

In principle studies of productivity as in this chapter control for changes in thequality of measured inputs. Consequently application of these methods to a partic-ular sector of the economy (e.g. agriculture) will not (in principle) quantify the ef-fect of improved quality of measured inputs on the sector’s output. Improvement inquality of inputs should be analyzed as technical changes in the industries producingthose factors. In order to study the effect of improvements in quality of measured in-puts as well as changes in levels of unmeasured inputs on agricultural output, it hasbeen suggested that agriculture and manufacturing industries supplying measuredinputs to agriculture should be combined into1 a single sector where these inputsare treated as intermediate goods (Kisler and Peterson 1981).

This chapter introduces several approaches to the measurement of technicalchange. Section 9.1 considers econometric production models incorporating a timetread as a proxy for technical change. Section 9.2 summarizes index number pro-cedures for calculating changes in productivity without recourse to econometricsor specification of specific functional forms for the production function, cost func-tion or profit function (in this sense the procedures are non-parametric). Section 9.3summarizes parametric index number procedures for calculating changes in produc-tivity without recourse to econometrics. These sections 9.2 and 9.3 both assume thatthere is no variation in utilization rates of capital over time; i.e. the industry or firmis assumed to be in long-run equilibrium. Section 9.4 demonstrates that (under re-

1 In principle it is possible to incorporate technical change into the production function by expand-ing the list of inputs x so as to include all variables that contribute to technical change, e.g. researchresults of agricultural experiment stations and agricultural extension activities. However these ad-ditional inputs are not easily quantified. This is the primary rationalization for treating technicalchanges as a residual of output changes that is unexplained by changes in levels of observableinputs.

105

106 9 Measuring Technical Change

strictive assumptions) these non-econometric procedures can be modified to allowfor variable utilization rates in capital, and that econometric procedures can (undercertain restrictive assumptions) allow for variable utilization rates in capital. Section9.5 conclude the chapter.

9.1 Dual cost and profit functions with technical change

In both primal and dual econometric models of the industry or firm, technicalchanges usually is proxied simply by a time trend variable t D 1; 2; 3; � � � ; T whereT is the number of time periods. By allowing for interactions between this time trendand other variables in the model, this specification is designed to proxy more than aregular secular time trend for technical changes over historical time. Of course thetime trend may also proxy secular trends in any relevant variables that have been ex-cluded from the model. In addition it is assumed that technical change is exogenousto the industry or firm.

A multiple output Translog cost function C D C.w; y/ can be generalized toincorporate a time trend t D 1; 2; 3; � � � ; as follows:

logC D ˛0 CNXiD1

˛i logwi CMXiD1

ˇi logyi C '1t

C1

2

NXiD1

NXjD1

˛ij logwi logwj

C1

2

MXiD1

MXjD1

ˇij logyi logyj

C1

2

NXiD1

MXjD1

ij logwi logyj

C

NXiD1

�i t logwi CMXiD1

ıi t logyi C '2t2

(9.1)

Shephard’s lemma implies the following factor share equations:

si �wixi

CD ˛i C

NXjD1

˛ij logwj CMXjD1

ij logyj C �i t i D 1; � � � ; N (9.2)

Assuming competitive profit maximization, the first order conditions pj D @C.w; y; t /[email protected] D 1; � � � ;M/ and (9.1) imply

9.1 Dual cost and profit functions with technical change 107

pjyj

CD@ logC@ logyj

D j C

NXiD1

ij logwi CMXiD1

j i logyi C ıj t j D 1; � � � ;M

(9.3)

Differentiating (9.1) with respect to t ,

@ logC@t

�@C.w; y; t /=C.w; y; t /

@t

D '1 C

NXiD1

�i logwi CMXjD1

ıj logyj C 2'2t(9.4)

Assuming technical progress (not regress), @ logC=@t < 0. Note that equation (9.4)cannot be estimated directly because the percentage reduction in cost due to techni-cal change (@ logC=@t ) is unobserved, and also note that the coefficients '1, '2 of(9.4) cannot be inferred from estimates of factor demand and output supply equation(9.2)–(9.3). In order to obtain estimates of all coefficients of (9.4), it is necessary toestimate directly equation (9.1) defining the Translog cost function. Unfortunatelythe data will often permit estimation of equation (9.2)–(9.3), but not direct estima-tion of the cost function equation (9.1).

Nevertheless we can easily derive a measure of technical change @C.w; y; t /=@tdirectly from estimates of the factor share equation (9.2). Since C.w; y; t / �PNiD1 xi .w; y; t /wi ,

@C.w; y; t /@t

�

NXiD1

wi@xi .w; y; t /

@t(9.5)

Differentiating si .w; y; t / � wixi .w; y; t /=C.w; y; t / with respect to t and substi-tuting in (9.5) yields @s

@tD A @x

@twhere matrix A has full rank.2 Then @x

@tcan be

calculated as @x@tD A�1� using the estimates of @s

@tfrom the share equations (9.2),

and in turn @C@t

can be calculated using (9.5).A single output generalized Leontief cost function with technical change can be

specified as

C D yXi

Xj

aijpwipwj C y

2Xi

aiwi C tyXi

biwi (9.6)

2

A D

26664w1C

�1� w1

C

��w1C

w2C

� � � �w1C

wNC

�w1C

w2C

w2C

�1� w2

C

��

w2C

wNC

::::::

::::::

�w1C

wNC

�w2C

wNC

� � �wNC.1� wN

C/

37775which generally has full rank. Deleting the equation for share sN from the estimation of (9.2),�N � 1�

PN�1iD1 �i ,


Here @C.w; y; t/=@t D yPNiD1 biwi , and all coefficients of this equation can be

recovered directly from the estimated factor demand equations (in contrast to theTranslog).

The shift in the underlying production function or transformation function due totechnical change can easily be calculated from the reduction in cost @C.w; y; t/[email protected] competitive cost minimization and a production function y D f .x; t / fora single output y, the increase in output @f .x�; t /=@t due to technical change can becalculated as follows:

@f .x�; t /@t

D �@C.w; y; t/

@[email protected]; y; t/

@y(9.7)

or, assuming competitive profit maximization,

@f .x�; t /

@tD �

@C.w; y; t /@t

=p (9.8)

In the case of constant returns to scale in production C.w; y; t/ D Cy.w; y; t/y(Euler’s theorem), and in turn (9.7) implies

@f .x�; t /@t

=y D �@C.w; y; t/

@t=C (9.9)

Thus under constant returns to scale the rate of growth of output (at equilibrium x�)is equal to the rate of reduction in cost due to technical change. Similarly in the caseof a multiple output transformation function y1 D g.y2; � � � ; yM ; x; t / � y1.Qy; x; t /and assuming competitive profit maximization, the shift in transformation functiondue to technical change @g.Qy; x; t /=@t can be calculated as

@g.Qy; x; t /@t

D �@C.w; y; t /

@t=p1 (9.10)

These relations (9.7)-(9.10) are proved in the next section 9.2.Biases as well as magnitudes of technical change can be calculated from cost

functions. For this purpose it is convenient to define Hicks-neutral technical changeas technical change that does not alter the firm or industry’s input expansion path.Then the change in cost shares @si .w; y; t /=@t .i D 1; � � � ; N / provides a measureof bias if the production function is homothetic, i.e. the expansion in input space islinear from the origin for a given state of technology.

However the change in shares @s.w; y; t /=@t does not provide an accurate mea-sure of biases in technical change when the production function is non-homothetic.For example suppose that Hicks-neutral technical change occurs, i.e. technicalchange does not alter the input expansion path. This technical change simply leadsto a re-numbering of the output levels for given isoquants (@f .x�; t /=@t > 0 andoutput y constant implies the firm moves to a lower isoquant). If the expansion pathis not linear and the firm’s output level must be held constant as this reindexing oc-curs, then the firm moves to a different isoquant with a different cost-minimizing

9.1 Dual cost and profit functions with technical change 109

ratio of inputs. Thus the changes in shares @si .w; y; t /=@t .i D 1; � � � ; N / are notequal to zero even though technical change is Hicks-neutral.

In order to correct for non-homotheticity in calculating biases in technicalchange, note that the change in shares can be decomposed into a scale effect dueto movement along the initial expansion path and a bias effect due to the shift in theexpansion path. The share equation si D si .w; y; t / can be defined equivalently assi D Osi .w; i

�.t/; t/ where i�.t/ indexes the isoquant yielding output level y giventechnology t . Differentiation of this composite function with respect to t yields

@si .w; y; t /@t

D@Osi .w; i�; t /

@tC@si .w; y; t /

@y

@f .x�; t /@t

(9.11)

where @Osi .w; i�; t /=@t provides a true measure of bias in technical change. Combin-ing (9.7) and (9.11), a true measure of bias in technical change for a non-homotheticproduction function can be defined in terms of a cost function as

@si .w; y; t /@t

C@si .w; y; t /

@y

@C.w; y; t /@t

[email protected]; y; t /

@yi D 1; � � � ; N (9.12)

or in elasticity terms as

@ log si .w; y; t /@t

C@ log si .w; y; t /

@ logy@ logC.w; y; t /

@t=@ logC.w; y; t /

@ logyi D 1; � � � ; N

(9.13)Homotheticity implies @si .w;y;t/

@yD

@ log si .w;y;t/@ logy D 0 .i D 1; � � � ; N /. Similarly

in the case of multiple outputs, a true measure of bias in technical change can becalculated as

@ log si .w; y; t /@t

C

MXjD1

@ log s.w; y; t /@ logyj

@ logC.w; y; t /@t

=@ logC.w; y; t /

@ logyj(9.14)

Technical change can be incorporated into dual profit functions in a manner anal-ogous to the above treatment of cost functions. Here we simply note the followingrelation between changes in profits and costs due to technical change

@�.w; p; t/=@t D �@C.w; y�; t /=@t (9.15)

This relation is proved in the next section 9.2.Finally it is important to note that most of the above discussion applies equally

well to short-run and long-run equilibrium models. The above models assumed thatall inputs are freely variable and attain static equilibrium levels. In contrast we couldpostulate, e.g. , a Translog cost function C.w; y; k; t/ conditional on the stocks Kof quasi-fixed inputs.

In place of (9.7) we have the following relation between the shift @f .x�; K; t/=@tin the production function and the shift @C.w; y; k; t/=@t in cost:


@f .x�; K; t/

@tD �

@C.w; y;K; t/

@[email protected]; y;K; t/

@y(9.16)

where Cy.w; y;K; t/ D p assuming competitive profit maximization. On the otherhand the measurement of biases is complicated somewhat when the cost function isshort-run equilibrium in nature.

9.2 Non-Parametric index number calculations of changes inproductivity

In this section we consider methods for calculating technical change that require nei-ther econometrics nor the specification of particular functional forms for a produc-tion function as cost function. However this index number approach to productivityis defined in terms of continuous time, and errors in approximation result from theuse of discrete time data. Moreover these calculations usually assume that all inputsare freely variable and are at static long-run equilibrium levels.

Initially assume a single output production function y D f .x; t/. Then outputat time t is related to inputs at time t and technology as indicated by the relationy.t/ D f .x.t/; t/. Differentiating with respect to t yields

Vy D

NXiD1

@f .x; t/

@xiVxi C

@f .x; t/

@t(9.17)

where Vy � @y.t/@t

, Vx � @x.t/@t

denotes the change in output and input levels withrespect to time. The standard first order conditions for static competitive cost min-imization are wi � Cy.w; y; t / @f .x

�;t/

@xiD 0 .i D 1; � � � ; N /, and substituting these

into (9.17) yields

Vy D

NXiD1

�wi

Cy.w; y; t /

�Vxi C

@f .x; t/

@t(9.18)

Competitive profit maximization implies Cy.w; y; t / D p, and substituting thisinto (9.18) yields

Vy D

NXiD1

�wi

p

�Vxi C

@f .x; t/

@t(9.19)

or equivalently

Vy=y D

NXiD1

�wi

p

xi

y

�Vxi

xiC@f .x; t/

@t=y (9.20)

The weightingswixi=py for input changes Vxi=xi define a Divisia quantity indexfor inputs in continuous time. Thus, assuming continuous time and all inputs are at

9.2 Non-Parametric index number calculations of changes in productivity 111

static long-run equilibrium levels, technical change @f .x; t/=@t could be calculatedas the residual in (9.18) or (9.19)-(9.20).

However data is measured at discrete time intervals rather than continuously, andequation (9.19) or (9.20) can only be approximated using discrete data. For exampleintegrating (9.19) over an interval t D 0; 1 yields

y.1/ � y.0/ D

NXiD1

Z 1

tD0

wi

p.t/@xi .t/

@tdt C

Z 1

tD0

@f .x.t/; t/

@tdt (9.21)

However, unless all price ratios wi=p are independent of t .i D 1; � � � ; N /,closed form solutions for the integrals

R 1tD0

wi

p.t/ @x

i .t/@t

dt generally cannot be de-fined. (9.20) is most commonly approximated using Tornqvist approximations tothe Divisia index:

TF �y.t/ � y.t � 1/

y.t/�

NXiD1

1

2

�wixi

py.t/C

wixi

py.t � 1/

�xi .t/ � xi .t � 1/

xi .t/

(9.22)where Tf �

R tvDt�1

@f .x.v/;v/@v

=y.v/dv denotes the integral of the residual in (9.19)attributed to technical change.

There is a further complication in interpreting the calculated shifts in the pro-duction function @f .x�; t /=@t . The equilibrium point x� where the continuous ondiscrete calculations are made varies over time. Therefore, in order to interpret theseresults as simple comparable indicators of shifts in the production function overtime, it is necessary to assume Hicks-neutral technical change and constant returnsto scale. Similar comments apply to the interpretation of other index number calcu-lations of productivity presented in this section and the following section C.

In the case of multiple outputs y D .y1; � � � ; yM /, we can define a transfor-mation function normalized on y1: y1 D g. Qy; x; t/ where Qy � .y2; � � � ; yM /.Differentiating y1.t/ D g. Qy.t/; x.t/; t/ with respect to t yields

Vy1 D

MXiD2

g Qy. Qy; x; t/ VQyiC

NXiD1

gxi . Qy; x; t/ VxiC @g. Qy; x; t/=@t (9.23)

The competitive maximization problem maxy;x;�L � py�wx��.g. Qy; x; t/�y1/ has first order conditions piCp1gyi . Qy; x; t/ D 0 .i D 2; � � � ;M/,�wipigxi . Qy; x; t/ D0 .i D 1; � � � ; N /, and substituting these into (9.23) yields

MXiD1

.pi=p1/ Vyi D

NXiD1

.wi=p1/ Vxi C @g. Qy; x; t/=@t (9.24)

However the numerical measure of technical change @g. Qy; x; t/=@t varies withthe choice of output i , i.e. the measure of technical change is not symmetric withrespect to the normalization of the transformation function.


In the case of multiple outputs a move useful measure of technical change maybe obtained in terms of the cost function C.w; y; t /. At time t C.w.t/; y.t/; t/ �PNiD1w

i .t/xi .t/, and differentiating with respect to t yields

NXiD1

Cwi .w; y; t / VwiC

Xj

Cyi .w; y; t / [email protected]; y; t /

@tD

NXiD1

Vwixi C

NXiD1

wi Vxi

(9.25)Competitive cost minimization impliesCwi .w; y; t / D xi .w; y; t / .i D 2; � � � ; N /

(Shephard’s lemma), and substituting these conditions into (9.25) yields

@C.w; y; t /@t

D

NXiD1

wi Vxi �

MXjD1

Cyj .w; y; t / Vyj (9.26)

Competitive profit maximization further implies Cyj .w; y; t / D pj .j D

1; � � � ;M/, and substituting into (9.26) yields

@C.w; y; t /=@t DNXiD1

wi Vxi �

MXjD1

pi Vyj (9.27)

or equivalently

@C.w; y; t /@t

=C D

NXiD1

�wixi

C

�Vxi

xi�

MXjD1

�pjyj

C

�Vyj

yj(9.28)

How the simple relations between primal measures and dual cost and profit mea-sures of technical change can easily be established as follows. Comparing (9.18)and (9.26) for the case of a single output and cost minimization establishes

@C.w; y; t /@t

D �@f .x�; t /

@t

@C.w; y; t /@y

(9.29)

or in the case of competitive profit maximization

@C.w; y; t /@t

D �p@f .x�; t /

@t(9.30)

Similarly comparing (9.24) and (9.27) in the case of multiple outputs and profitmaximization establishes

@C.w; y; t /@t

D �p1@g. Qy�; x�; t /

@t(9.31)

In term of the profit function �.w; p; t/ we have the identity �.w.t/; p.t/; t/ �p.t/y.t/ � w.t/x.t/ for profits at time t , and differentiating with respect to t andapplying Hotelling’s lemma yields

9.3 Parametric index number calculations of changes in productivity 113

@�.w; p; t/

@tD

MXiD1

pi Vyi �

NXiD1

wi Vxi (9.32)

Comparing (9.27) and (9.32) establishes

@C.w; y�; t /

@tD �

@�.w; p; t/

@t(9.33)

9.3 Parametric index number calculations of changes inproductivity

The Divisia approaches of the previous section required the approximation of con-tinuous time derivatives by discrete differences but did not require the specificationof a particular functional form for a production function or cost function. In thissection we discuss an alternative approach to calculating changes in productivity: aflexible functional form is assumed for a production function or cost function, andthen an index number formula is derived that is consistent with the functional formand with discrete time data. Here there are no errors in approximation due to the useof data for discrete time intervals, but there are errors in approximation to the truefunctional form for the production on cost function. Here we derive index numbersformulas for technical change corresponding to thus different functional forms: aTranslog cost function, a Translog transformation function, and a Translog profitfunction.

First assume a multiple output Translog cost function C.w; y; t / as in (9.1). Theunderlying transformation function need not be constant returns to scale. Since thiscost function is a quadratic form, the quadratic identity 8.11 of chapter 8 implies


log .C1=C0/ D1

2

NXiD1

�@ logC1@ logwi

C@ logC0@ logwi

�log

�wi1=w

i0

�C1

2

MXiD1

�@ logC1@ logyi

C@ logC0@ logyi

�log

�yi1=y

i0

�C1

2

�@ logC1@t

C@ logC0@t

�.t1 � t0/

D1

2

NXiD1

�@C1

@wiwi1C1C@C0

@wiwi0C0

�log

�wi1=w

i0

�C1

2

MXiD1

�@C1

@yi

yi1C1C@C0

@yi

yi0C0

�log

�yi1=y

i0

�C1

2

�@C1

@t=C1 C

@C0

@t=C0

�since t1 D t0 C 1

D1

2

NXiD1

�wi1x

i1

C1Cwi0x

i0

C0

�log

�wi1=w

i0

�C1

2

MXiD1

�pi1y

i1

C1Cpi0y

i0

C0

�log

�yi1=y

i0

�C1

2

�@C1

@t=C1 C

@C0

@t=C0

�

(9.34)

by Shephard’s lemma and @C=@yi D pi .i D 1; � � � ;M/. This index number equa-tion is identical to 8.26 discussed on pages 8.10-8.11 of chapter 8, except that a timetrend variable t D 1; 2; 3; � � � is incorporated into the Translog cost function. Thefirm is assumed to be a competitive profit maximizer with all inputs variable andattaining static long-run equilibrium levels.

Rearranging (9.34)

1

2

�T C1 C T

C0

�D

(log .C1=C0/ �

1

2

NXiD1

�wi1x

i1

C1Cwi0x

i0

C0

�log

�wi1

wi0

�)

�1

2

MXiD1

�pi1y

i1

C1Cpi0y

i0

C0

�log

�yi1

yi0

� (9.35)

where T CV �@C.wV ;yV ;tV /

@t=C.wV ; yV ; tV /, CV �

PNiD1w

iV x

iV .V D 0; 1/V .

12.T C1 C T

C0 / is the average percentage reduction in cost due to technical change at

time t D 0; 1.The second set of terms within brackets f� � � g on the right hand side of (9.35) is

the logarithm of a Tornqvist price index .W1=W0/ for inputs (see equation 8.18), sothe brackets f� � � g enclose an implicit Tornqvist quantity index

�Ax1=x0� for input

9.3 Parametric index number calculations of changes in productivity 115�Ax1=x0� D .C1=C0/ = .w1=w0/ by the factor reversal equation analogous to 8.16.The term on the right hand side of (9.35) that is outside the brackets can be inter-preted as the logarithm of a quantity index .Y1=Y0/ for outputs (constant returns toscale in production would imply C D py and this index would be equivalent toa Tornqvist quantity index for outputs). Hence the average percentage reduction incost is the logarithm of the ratio of input and output quantity indexes:

1

2

�T C1 C T

C0

�D log

h.Ax1=x0/=.Y1=Y0/

i(9.36)

This calculated change in cost can easily be related to shifts in the production ortransformation function at equilibrium levels of commodities x�, y� using equations(9.7)-(9.10).

Second, assume a constant returns to scale Translog production function y Df .x; t/ and competitive cost minimization with all inputs variable at long-run equi-librium levels. Proceeding as in the proof of 8.8, we obtain

1

2

�T t1 C T

t0

�D log

�y1

y0

��1

2

NXiD1

�wi1x

i1

w1x1Cwi0x

i0

w0x0

�log

�xi1

xi0

�(9.37)

where T tV �@f .xV ;tV /

@t=yV .v D 0; 1/. Here the average percentage change in pro-

ductivity is equal to the logarithm of the ratio of output and input quantity indexes:

1

2

�T t1 C T

t0

�D log

h.y1=y0/=.x1=x0/

Ti

(9.38)

where .x1=x0/T is the Tornqvist quantity index for inputs. Alternatively if we as-sume a Translog production function (which is not necessarily constant returns toscale) and competitive profit maximization, then we obtain the closely related indexnumber formula

1

2

�T t1 C T

t0

�D log

�y1

y0

��1

2

NXiD1

�wi1x

i1

p1y1Cwi0x

i0

p0y0

�log

�xi1

xi0

�(9.39)

Similarly in the case of multiple outputs assume a Translog transformation func-tion y1 D g. Qy; x; t/ . Qy � .y2; � � � ; yM // and competitive profit maximizing be-havior with all inputs variable. Then

1

2

�Tg1 C T

g0

�D log

�y1

y0

��1

2

MXiD1

�pi1y

i1

p1y1Cpi0y

i0

p0y0

�log

�yi1

yi0

�

�1

2

NXiD1

�wi1x

i1

p1y1Cwi0x

i0

p0y0

�log

�xi1

xi0

� (9.40)


where T gV �@g. Qy�

V;x�V;tV /

@t=yV .V D 0; 1/. This measure of technical change varies

in a simple manner with the commodity chosen as numeraire in the transformationfunction. For example, given the alternative normalization y1 D g.y2; � � � ; yM ; x; t/and y2 D h.y1; y3; y4; � � � ; yM ; x; t/,

@g.y2�; yM�; x�; t /=@t D .p2=p1/@h.y1�; y3�; � � � ; yM�; x�; t /=@t (9.41)

Third, assume a Translog profit function and competitive profit maximizationwith all inputs variable at long-run equilibrium levels. The quadratic identity andHotelling’s lemma establish

1

2

�T �1 C T

�0

�D log

��1

�0

��1

2

MXiD1

�pi1y

i1

�1Cpi0y

i0

�0

�log

�pi1

pi0

�

C1

2

NXiD1

�wi1x

i1

�1Cwi0x

i0

�0

�log

�wi1

wi0

� (9.42)

where T �V �@�.wV ;pV ;tV /

@t=�.wV ; pV ; tV /. This measure of the effect of technical

change on profits �.w; p; t/ is easily related to changes in costs and shifts in theproduction function or transformation function using equations (9.7)-(9.10). Inter-preting the second and third terms of the right hand side of (9.42) as price indexesfor outputs and inputs, respectively, and noting that � � py � wx, it follows that12

�T �1 C T

�0

�is the logarithm of the ratio of implicit quantity indexes for outputs

and for inputs.The behavioral models of the firm employed in sections B and C have assumed

static long-run equilibrium and lead to index number formulas such as (9.20) thatare defined in terms of the flows xt of all inputs used in production. Since the flowof capital services is not generally observable, it is usually assumed that the flow ofcapital services is proportional to the stock of capital assets (this assumption maybe reasonable at long-run equilibrium). Then the growth rate of the capital serviceflow Vxkt =x

kt is equal to the growth rate of the capital stock VKt=Kt .

The capital stock Kt is often approximated by the perpetual inventory method:Kt D It�1C.1�ı/Kt�1 where It�1 denotes gross investment at time t�1 and ı is aconstant rate of depreciation, and substituting backwards forKt�1; Kt�2; � � � ; Kt�Syields the approximation

Kt � It�1 C

SXSD2

.1 � ı/t�SC1It�S (9.43)

Thus time series data on capital stockK and in turn the growth rate of the capitalservice flow is approximated from data on gross investment I and an assumed rateof depreciation ı. Then the rate of technical change is calculated using index numberformulas such as (9.20), (9.28), (9.35), (9.37) (e.g. Ball 1985).

9.4 Incorporating variable utilization rates for capital 117

9.4 Incorporating variable utilization rates for capital

There is considerable evidence that utilization rates of capital vary significantly overtime. This implies that firms generally are not in long-run equilibrium, and in turnthat (a) flows of capital services are not in fixed proportion to the levels of capitalstocks and (b) the marginal value product of capital services is not equal to a marketwage or rental rate. Nevertheless the index number procedures of sections B-C andmany econometric models incorporate these assumptions.

One approach to avoiding or reducing these problems in econometric modelsstems from the assumption that the rate of depreciation for capital varies with theutilization rate of capital. Then the firm’s short-run profit maximization problem canbe written as

max.xt ;K

ft /

pf .x;Kt ;Kft / � wxt C Qp

KKft � �.w; p;

Qpk ; Kt / (9.44)

whereKt � capital stock predetermined at beginning of period t ,Kft � position ofKt remaining at end of period t , and QpK � pK=.1C r/ is the asset price of capitalat the end of period t discounted back to the beginning of period t . The derivativesof the production function are fx.P/ > 0, fK.P/ > 0 and ftf .P/ < 0 (an increasein Kft for a given initial stock Kt implies a lower depreciation rate and utilizationrate of the initial stock Kt ). However there is a serious difficulty in implement-ing models such as (9.44) where the depreciation rate as well as utilization rate ofcapital is treated as endogenous: time series data on capital stocks have invariablybeen constructed assuming a constant rate of depreciation, and there are substantialeconometric problems in the estimation of (9.44) where Kt is unobserved (see Ep-stein and Denny, 1980 for an illustration). Hereafter we shall assume for simplicitythat variations in capital utilization do not influence the rate of depreciation.

Next consider the effects of short-run equilibrium on non-parametric Divisia in-dex number calculations of technical change, as discussed in section B. Let the pro-duction function be y D f .x;K; t/ where x denotes flows of variable inputs andKdenotes stocks of predetermined (quasi-fixed) capital inputs, and assume short-runcompetitive profit maximization

maxxpf .x;K; t/ � wx � �.w; p;K; t/ (9.45)

Differentiating the identity y.t/ � f .x.t/;K.t/; t/ and applying the first orderconditions pfxi .x

�; K; t/�wi D 0 .i D 1; � � � ; N / for a solution to (9.45) yields

@f .x�; K; t/

@t=y D Vy=y �

NXiD1

�wixi

py

�Vxi

xi�

Xj

@f .x�; K; t/

@KjVKj =y (9.46)

However the marginal products of capital @f .x�; K; t/=@Kj are not observeddirectly are equal to price ratios wk

j=p only in a static long-run equilibrium.


Also note that the productivity index (9.46) employs the change in capital stocksVK rather than a change in flow of capital services from the stocks. Equation (9.46)

provides an exact measure of change in productivity in continuous time. Thus errorsin calculations of changes in productivity @f .x�;K;t/

@tunder the erroneous assumption

of long-run equilibrium are due to mismeasurement of weights for changes in capitalstocks Vk rather than to mismeasurement of the flow of capital services (the correctweights for VK are the unobserved marginal products of capital). In other words,in productivity studies there is no need to derive capital service flows from capitalstocks irrespective of the industry being in long-run or short-run equilibrium. Thisis an important result since the concept of capital service flows, which has beenwidely employed in earlier productivity studies, is an artificial construct with littleempirical basis.

In the special case of constant returns to scale�f .�x; �K; t/ D �f .x;K; t/

�and

one capital good K, the marginal product of capital can be calculated directly as

@f .x�; K; t/

@KD

"y �

NXiD1

�wi

p

�xi

#=K (9.47)

Using Euler’s theorem and first order conditions for (9.45). Then changes in pro-ductivity can be calculated directly from a discrete approximation to (9.46).

Similarly the variable cost function C D C.w; y;K; t/ � wx and short-runcompetitive behavior (9.45) imply

@C.w; y�; K; t/

@t=C D

NXiD1

�wixi

C

�Vxi

xi�

MXjD1

�piyi

C

�Vyi

yi�

Xi

@C.w; y�; K; t/

@KiVKi=C

(9.48)In the case of constant returns to scale in production and a single capital goodK,

the unobserved shadow price of capital can be calculated as

@C.w; y;K; t/

@KD

0@C � MXjD1

pjyj

1A =K < 0 (9.49)

(using C.w; �y; �K; t/ D �C.w; y;K; t/ and Euler’s theorem).Substituting (9.49) into (9.48), the resulting index number formula for technical

change can be approximated using data for discrete time intervals. This calculationof technical change under constant returns to scale and a single capital good Krequires neither econometrics nor the assumption of long-run equilibrium.

Next consider the effects of short-run equilibrium on parametric Divisia indexnumber calculations of technical change, as discussed in section C. Assuming aTranslog variable cost function C.w; y;K; t/ and competitive short-run profit max-imization ((9.45)), equation (9.35) must be rewritten as

9.4 Incorporating variable utilization rates for capital 119

1

2

�T C1 C T

C0

�D

(log .C1=C0/ �

1

2

NXiD1

�wi1x

i1

C1Cwi0x

i0

C0

�log

�wi1

wi0

�)

�1

2

MXiD1

�pi1y

i1

C1Cpi0y

i0

C0

�log

�yi1

yi0

�

�1

2

Xj

�@ logC1@ logKj

C@ logC0@ logKj

�log

Kj1

Kj0

! (9.50)

In general @ logC@ logKj

�@C

@Kj= C

Kjcannot be measured without recourse to econo-

metric estimation of the factor demand equations. However in the case of a singlequasi-fixed capital good K and constant returns to scale in production, @ logC

@ logK canbe calculated from (9.49). Thus (as in non-parametric Divisia index number for-mulas) given the assumptions of constant returns to scale and one quasi-fixed input,parametric measures of technical change can be obtained without recourse to econo-metrics or the assumption of long-run equilibrium.

Thus, if there is more than one quasi-fixed input or returns to scale are not con-stant, econometric methods are required for the measurement of technical change.For example we could postulate a short-run Translog cost function C.w; y;K; t/analogous to (9.1) and estimate factor share equations for the variable inputs. Thenthe change in technology @C.w;y;K;t/

@tcan be calculated directly from the estimates

of the share equations as discussed in section A (page 9.4). Alternatively the shadowprices can be calculated as

@C.w; y;K; t/

@Kj�

NXiD1

wi@xi .w; y;K; t/

@Kj(9.51)

From the estimates of the share equations, and then the change in technology12

�T C1 C T

C0

�can be calculated from the index number equation (9.50). The disad-

vantage of this second approach is that equation (9.50) requires the assumption ofshort-run competitive profit maximization, whereas estimation of the cost functiononly requires the weaker assumption of short-run competitive cost minimization.

We conclude this section with a brief discussion of empirical measures of capac-ity utilization based on microeconomic theory. Unexpected changes in output pricesor input prices are likely to lead to short-run combinations of variable and quasi-fixed inputs that are inappropriate for the long-run, i.e. under or over-utilization ofcapacity is likely in the short-run. A fruitful approach to the measurement of capac-ity utilization is by comparing the observed level of output and “capacity output”.One definition of capacity output is the output level corresponding to the minimumpoint on the firm’s long-run average cost curve, but this definition is not useful dueto difficulties in identifying the long-run average cost curve.

A more useful definition of capacity output is the output level yC at which theshort-run average total cost curve (with quasi-fixed inputs fixed at their short-runequilibrium levels) is tangent to the long-run average cost curve. If there is constant


returns to scale in the long-run, then capacity output corresponds to the minimumpoint on the short-run average total cost curve (see diagram on following page). Therate of capacity utilization CU is then defined as the ratio of actual output y tocapacity output yC :

CU � y=yC (9.52)

Econometric estimates of a short-run cost function C.w; y;K; t/ can be em-ployed in calculations of a capacity utilization index CU (Berrdt and Herse 1986).

9.5 Conclusion

In this chapter we have discussed several non-econometric and econometric ap-proaches to the measurement of technical change. The non-econometric approachesdo not require a long time series of data or the use of highly aggregated data. Onthe other hand these approaches cannot test hypothesis about technical change, andthese approaches can disentangle shifts in productivity and variations in capital uti-lization rates only in the case of constant returns to scale and a single quasi-fixedinput. In contrast econometric approaches require a substantial number of obser-vations and substantial aggregation of commodities, but these approaches can testhypothesis and can calculate changes in productivity given multiple quasi-fixed in-puts.

In practice non-econometric and econometric approaches to the measurement oftechnical change can often be viewed as complementary. Most of the effort requiredto construct non-econometric.

Measures of (full) Capacity OutputA.

yC

SRAC ≡ AC(w,wK , y,K) LRAC ≡ AC(w,wK , y)A

C

Fig. 9.1 figure 1

9.5 Conclusion 121

B. Assuming constant returns to scale in the production function y D f .x;K/,�f .�x; �K/ D �f .x; k/

�:

yC0

SRAC1 ≡ AC(w,wK , y,K1)A

CyC1

SRAC0 ≡ AC(w,wK , y,K0)

Fig. 9.2 figure 2

Short-run average cost function (SRAC)

AC.w;wK ; y;K/ D minx

wx C wkK

@y

s.t. f .x;K/ D y

Long-run average cost function (LRAC):

AC.w;wK ; y/ D minx;K

wx C wkK

y

s.t. f .x;K/ D y

Index number calculations of technical change will also be required to constructeconometric models, and data problems may be indicated more clearly by non-econometric measures of technical change. Thus the following sequence may beappropriate: first construct non-econometric index number measures of technicalchange, and then (data permitting) estimate a short-run equilibrium econometricmodel.

However at least two serious problems remain for the econometric models andDivisia index number formulas outlined here. First technical change is proxied bya simple time trend t D 1; 2; 3; � � � (which may interact with other variables) oris calculated as a simple residual. Thus there is no theory endogenizing technical


change to the model. In principle a well constructed theory of demand and supplyfor changes in technology should improve both econometric and non-econometricmeasures of technical change. Second, there has been little attempt to incorporateexplicitly into the analysis changes in quality of inputs supplied to the industry (secBerrdt 1983 and Tarr 1982).

4. It should be noted that cross-sectional differences in productivity (e.g. dif-ferences in productivity between countries, regions or firms) can be measured byparametric index number procedures somewhat similar to the procedures discussedin section C (Caves, Christensen and Diewert 1982; Denny and Fuss 1983).

Index

C

composite commodity 78cost function

short-runGeneralized Leontief 59normalized quadratic 59Translog 59

cost of living indexes 102

D

Divisia index 90dual cost function 1dual expenditure function 28dual indirect utility function 27dual profit function 11dual profit function, restricted 18

E

equations of motion 24Euler’s theorem 4

F

factor reversal equation 92Fisher indexes 103flexible functional forms 47

second order 49Frobenius theorem 4

G

Generalized Leontief functional form 7Generalized Leontief short-run cost function

59

Gorman polar form 65

H

Hicksian consumer demands 28Hotelling’s Lemma 12, 14

I

implicitly separable 83integrability problem 4

in economics 32

L

Laspeyres index numbers 87Laspeyres indexes 102Le Chatelier principle 17

M

marginal frim 21Marshallian consumer demands 28

N

Normalized Quadratic indirect utility function56

normalized quadratic cost function 55normalized quadratic profit function 52normalized quadratic short-run cost function

59

P

Paasche indexes 102primal-dual relations 39

Q

quadratic dual profit function 51

123

124 Index

quasi-homothetic 65

R

ratio of commodities 85ratio of marginal rates of substitution 85Roy’s theorem 29

S

second order flexible functional forms 49separable

implicitly 83weakly 80

Shephard’s Lemma 2Slutsky equation 31sub-utility function 80superlative 90

T

Tornqvist index 90Translog cost function 56Translog dual profit function 53Translog indirect utility function 57Translog short-run cost function 59two-stage budgeting 79

V

Vartia I price index 99

W

Walras Law 63weakly separable 80

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