microeconomics ii lecture 3 constrained envelope...

Leonardo Felli 23 October, 2002

Microeconomics II Lecture 3

Constrained Envelope Theorem

Consider the problem:

maxx

f (x)

s.t. g(x, a) = 0

The Lagrangian is:

L(x, λ, a) = f (x)− λ g(x, a)

Necessary FOC are:

f ′(x∗)− λ∗∂g(x∗, a)

∂x= 0

g(x∗(a), a) = 0

1

Microeconomics II 2

Substituting x∗(a) and λ∗(a) in the Lagrangian we

get:

L(a) = f (x∗(a))− λ∗(a) g(x∗(a), a)

Differentiating we get:

dL(a)

d a=

[f ′(x∗)− λ∗

∂g(x∗, a)

∂x

]d x∗(a)

d a−

−g(x∗(a), a)dλ∗(a)

da− λ∗(a)

∂g(x∗, a)

∂a

= −λ∗(a)∂g(x∗, a)

∂a

by the necessary FOC.

In other words — to the first order — only the direct

effect of a on the Lagrangian function matters.

Microeconomics II 3

3. Roy’s identity:

xi(p, m) = −∂V/∂pi

∂V/∂m

By the constrained envelope theorem and the obser-

vation that:

V (p, m) = u(x(p, m))− λ(p, m) [p x(p, m)−m]

we shall obtain:

∂V/∂pi = −λ(p, m) xi(p, m) ≤ 0

and

∂V/∂m = λ(p, m) ≥ 0

which is the marginal utility of income.

Microeconomics II 4

(Notice that the sign of the two inequalities above

prove property 1 of the indirect utility function V (p, m).)

We conclude the proof substituting

∂V/∂m = λ(p, m)

into

∂V/∂pi = −λ(p, m) xi(p, m)

and solving for xi(p, m).

4. Adding up results. From the identity:

p x(p, m) = m ∀p, ∀m

Microeconomics II 5

Differentiation with respect to pj gives:

xj(p, m) +

L∑i=1

pi∂xi

∂pj= 0

or, more interestingly, with respect to m gives:

L∑i=1

pi∂xi

∂m= 1

There does not exist a clear cut comparative-static

property with the exception of:

0 ≥L∑

i=1

pi∂xi

∂ph= −xh(p, m)

which means that at least one of the Marshallian

demand function has to be downward sloping in ph

Microeconomics II 6

Effect of a change in income on the level of the Mar-

shallian demand:∂xl

∂m

In the two commodities graph the set of tangency

points for different values of m is known as the in-

come expansion path.

In the commodity income graph the set of optimal

choices of the quantity of the commodity is known as

Engel curve.

Microeconomics II 7

We shall classify commodities with respect to the ef-

fect of changes in income in:

• normal goods:∂xl

∂m> 0

• neutral goods:∂xl

∂m= 0

• inferior goods:∂xl

∂m< 0

Notice that for every level of income m at least one

of the L commodities is normal:

L∑l=1

pl∂xl

∂m= 1

Microeconomics II 8

If the Engel curve is convex we are facing a luxury

good in other case a necessity.

-

6x(p, m)

m

luxury

necessity

Microeconomics II 9

Expenditure Minimization Problem

The dual problem of the consumer’s utility maxi-

mization problem is the expenditure minimization

problem:

min{x}

p x

s.t. u(x) ≥ U

Define the solution as:

x = h(p, U) =

h1(p1, . . . , pL, U)

...

hL(p1, . . . , pL, U)

the Hicksian (compensated) demand functions.

We shall also define:

e(p, U) = p h(p, U)

as the expenditure function.

Microeconomics II 10

Properties of the expenditure function:

1. Continuous in p and U .

2. ∂e∂U > 0 (2.1) and ∂e

∂pl≥ 0 (2.2) for every l =

1, . . . , L.

Proof: (2.1): Suppose not: there exist U ′ < U ′′

(denote x′ and x′′ the corresponding solution to the

e.m.p.) such that p x′ ≥ p x′′ > 0.

If the latter inequality is strict we have an immediate

contradiction of x′ solving e.m.p.;

if on the other hand p x′ = p x′′ > 0 then by con-

tinuity and strict monotonicity of u(·) there exists

α ∈ (0, 1) close enough to 1 such that u(α x′′) > U ′

and p x′ > p αx′′ which contradicts x′ solving e.m.p..


(2.2): consider p′ and p′′ such that p′′l ≥ p′l but p′′k =

p′k for every k 6= l.

Let x′′ and x′ be the solutions to the e.m.p. with p′′

and p′ respectively.

Then by definition of e(p, U)

e(p′′, U) = p′′ x′′ ≥ p′ x′′ ≥ p′ x′ = e(p′, U).

3. Homogeneous of degree 1 in p.

Proof: The feasible set of the e.m.p. does not change

when prices are multiplied by the factor k > 0.

Hence ∀k > 0, minimizing (k p) x on this set leads

to the same answer. Let x∗ be the solution, then:

e(k p, U) = (k p) x∗ = k e(p, U).


4. Concave (graphic intuition) in p.

Proof: let p′′ = t p + (1− t) p′ for t ∈ [0, 1]. Let x′′

be the solution to e.m.p. for p′′. Then

e(p′′, U) = p′′ x′′ = t p x′′ + (1− t) p′ x′′

≥ t e(p, U) + (1− t) e(p′, U)

since u(x′′) ≥ U and by definition of e(p, U).

Properties of the Hicksian demand func-

tions:

h(p, U)

1. Shephard’s Lemma.

∂e(p, U)

∂pl= hl(p, U)

Proof: by constrained envelope theorem.


2. Homogeneity of degree 0 in p.

Proof: by Shephard’s lemma and the fact that the

following theorem.

Theorem. If a function F (x) is homogeneous of

degree r in x then (∂F/∂xl) is homogeneous of

degree (r − 1) in x for every l = 1, . . . , L.

Proof: Differentiating the identity that defines hom-

geneity of degree r:

F (k x) = kr F (x) ∀k > 0

with respect to xl we obtain:

k∂F (k x)

∂xl= kr∂F (x)

∂xl


The latter equation corresponds to the definition of

homogeneity of degree (r − 1):

∂F (k x)

∂xl= k(r−1) ∂F (x)

∂xl.

Euler Theorem. If a function F (x) is homoge-

neous of degree r in x then:

r F (x) = ∇F (x) x

Proof: Differentiating with respect to k the identity:

F (k x) = kr F (x) ∀k > 0


we obtain:

∇F (kx) x = rk(r−1) F (x)

for k = 1 we obtain:

∇F (x) x = r F (x).

3. The matrix of cross-partial derivatives (Substitu-

tion matrix) with respect to p

S =

∂h1∂p1

· · · ∂h1∂pL

... . . . ...∂hL∂p1

· · · ∂hL∂pL

is negative semi-definite and symmetric. (Main di-

agonal non-positive).


Proof: Simmetry follows from Shephard’s lemma

and Young Theorem.

Indeed:

∂hl

∂pi=

∂

∂pi

(∂e(p, U)

∂pl

)=

∂

∂pl

(∂e(p, U)

∂pi

)=

∂hi

∂pl

While negative semi-definiteness follows from the con-

cavity of e(p, U) and the observation that S is the

Hessian of the function e(p, U).


Identities:

V [p, e(p, U)] ≡ U

xl[p, e(p, U)] ≡ hl(p, U) ∀l

e[p, V (p, m)] ≡ m

hl[p, V (p, m)] ≡ xl(p, m) ∀l

Slutsky decomposition:

start from the identity

hl(p, U) ≡ xl[p, e(p, U)]

if the price pi changes the effect is:

∂hl

∂pi=

∂xl

∂pi+

∂xl

∂m

∂e

∂pi


Notice that by Shephard’s lemma:

∂e

∂pi= hi(p, U) = xi[p, e(p, U)]

then∂hl

∂pi=

∂xl

∂pi+

∂xl

∂mxi.

or∂xl

∂pi=

∂hl

∂pi− ∂xl

∂mxi.

Own price effect gives Slutsky equation:

∂xl

∂pl=

∂hl

∂pl− ∂xl

∂mxl.


Slutsky decomposition:

∂xl

∂pi=

∂hl

∂pi− ∂xl

∂mxi.

Slutsky equation:

∂xl

∂pl=

∂hl

∂pl− ∂xl

∂mxl.

This latter equation corresponds to the distinction

between substitution and income effect:


Substitution effect:

∂hl

∂pl

Income effect:∂xl

∂mxl

x2

6x1

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x1

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We know the sign of the substitution effect it is non-

positive.

The sign of the income effect depends on whether the

good is normal or inferior.

In the case that:

∂xl

∂pl> 0

we conclude that the good is Giffen.

This is not a realistic feature, inferior good with a

big income effect.

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