optimal growth with scale economies in the creation of ... · considering an entire economy on the...

Optimal Growth with Scale

Economies in the Creation of

Overhead Capital1

1. SUMMARY

Following closely the approach to optimal economic growth taken in the work of Frank Ramsey [11], a highly simplified two-sector model is presented in which the " overhead capital " sector exhibits increasing returns to scale. Basic properties of the optimal growth path are discussed. From an economic standpoint, the model might be relevant in bearing on some basic issues of development programming. Mathematically, this kind of a model has an interesting structure because it is a combination of convex and concave sub-problems.

2. INTRODUCTION

In the context of development economics it is useful to distinguish two types of capital according to how round-about a role each plays in producing output. One type, the quantity of which is denoted K, is the ordinary directly productive quick-yielding capital which, when it is combined with labour, creates output according to classical laws of production. A second kind of capital, Kl, is the indirectly productive infrastructure which lays down the basic framework within which directly productive economic activities can function. Capital of this variety has come in for increased scrutiny by development economists. At least in part this is due to the growing suspicion that , capital, comprising those essential services without which ordinary production cannot operate, plays an especially important role in the early stages of economic growth.

For the purposes of this paper the total capital stock of the economy is thought of as being partitioned between two sectors-K. belonging to the ac sector and Kl to the , sector. This being the case, it becomes a fair question to ask for operational criteria which can be used to distinguish ac from , capital. Unfortunately it is difficult to be precise about this issue. For one thing it depends upon how aggregative a view one is prepared to take.

Considering an entire economy on the most general level, ,B might consist of all social overhead capital including public service facilities for education, scientific research, sanita- tion engineering, public health, and law enforcement, agricultural overhead such as drainage and irrigation systems, and hard public utilities like transportation, communications, power and water supply installations. A somewhat more satisfactory interpretation might limit / to the hard public utilities. There is even an interesting way of looking at this model which restricts the economic scenario to manufacturing and treats , as structures, cx as producers' durable equipment.

For the purposes of this paper probably the most useful formulation is the middle one which treats , as overhead capital for producers' services. In any case, the basic features are taken to be the following.

1 For their helpful comments I would like to thank D. Cass, T. C. Koopmans, and A. S. Manne. The research described in this paper was carried out under grants from the National Science Foundation and from the Ford Foundation.

555

556 REVIEW OF ECONOMIC STUDIES

(i) Capital of the f type is strongly complementary with cx. Investment in cx capital will be productive only if it has been preceded by sufficient investment in ,B capital.

(ii) The ,B sector is highly capital intensive and usually consists primarily of structures and installations. It is typically characterized by a significantly higher capital-labour ratio than the cx sector.

(iii) There are substantial economies of scale in creating , capacity. The main reason is that due to indivisibilities there is obvious cost lumpiness involved in creating a transportation, communications, or power and water supply system as a whole. Geometric- engineering considerations are also important in the case of many structures because the cost of an item is frequently related to its surface area while the capacity increases according to its volume.1

(iv) Both ,B and cx capitals are specific to the role for which they have been created and cannot be shifted.

3. THE BASIC MODEL

The highly stylized economy under consideration is centralized and closed. A single homogeneous output, denoted Y, is produced which is perfectly general before it has been committed, and can be used for any purpose. The planners seek to maximize welfare by appropriately manipulating the available instruments-in this case the destination of final output. For simplification the following are assumed: stationary labour force and population, tastes independent of time, constant technology, no capital deterioration.

As an abstraction of proposition (ii), it is postulated that ,B capital has only negligible manpower requirements. This makes the labour allocation problem trivial because all available workers will be assigned to work with cx capital.

If , capital were abundant at time t, Y(t) would depend only on the stock of Ka(t) and the labour force. Since the latter is treated as constant, the production function in this case could be written as

Y(t) = F(Ka(t)). Decreasing returns to a single factor implies that F(K.) is concave. Purely for convenience, we assume that a first derivative exists and that F'(K,) >0 for all K. > 0

With K,(t) plentiful, the production function would simply be Y(t) = Kp(t).

Note the implied asymmetry in capital measurement; K. is gauged by the usual criterion of real production cost, whereas it will prove useful to quantify K, in capacity units. Of course strict identification of K, with " capacity " would be possible only if there were a negligible elasticity of substitution between K, and F(K.), a condition which we readily assume following (i).

In the general case, Y(t) = min {F(K,(t)), Kp(t)}.

With cx capital all investment goes into capital formation in the usual direct form

Ka = ial

where I. denotes investment in cx capital.2 However, with , capital there is a meaningful distinction between capital accumulation

and investment. Because of the presumed increasing returns to scale described in (iii) it will typically be better not to invest directly in , capital. Rather it will pay to first accumu- late what could be thought of as either a generalized inventory of materials or as projects

1 In addition the usual internal economies of specialization and information handling may be present. Note that the increasing returns relates only to the design stage when the amount of installed capacity is treated as variable. Ex post the size of an installation is considered to be fixed and unalterable.

2 A dot over a variable denotes differentiation with respect to time. Variables may not be explicitly specified as functions of time if this interpretation is otherwise clear.

OPTIMAL GROWTH WITH SCALE ECONOMIES 557

in progress, denoted X. Only after a while should some generalized inventory be trans- formed into new K; available for operation.'

Let AX represent a portion of generalized inventory earmarked for conversion into operating ,B capital. Naturally 0 _ AX < X. Suppose that AK, units of K, are created, where

AK, = G(AX).

Reflecting economies of scale, G is taken to be a convex, monotonically increasing, continuous function of AX defined for all AX > 0. It is assumed that lim G(AX) = so, G(O) = 0,

Ax Xoo and

lim G(AX) = 0. ... (1) AX-O+ AX

Something like the latter condition is necessary to ensure that economies of scale are taken advantage of and that in fact generalized inventories must be accumulated for this purpose.2

We will also find it useful to work with the function H, defined as the inverse of G. H can be interpreted as an investment cost function relating the cost in cumulated output units of a given , capital increase according to the schedule

AX = H(AKp) = G-1(AKp).

Displaying decreasing unit costs, the continuous, monotonically increasing, concave cost function H is defined for all AKp > 0 and possesses the properties lim H(AK.) = co,

AK-co H(AK ~ ~ ~ ~

K

H(O) = 0, and lim H(AK) = AKP-O+ AKp

Before turning to the main problem, we digress in the next two sections to consider a pair of related problems whose solution will prove useful in characterizing an optimal path for the general case.

4. OPTIMAL GROWTH IN A MACROECONOMIC MODEL

The social utility of consuming amount C(t) at time t is taken to be U(C(t)). The instantaneous utility function U is monotonic increasing, concave and differentiable. For simplification the condition

lim U'(C)= oo

is imposed, guaranteeing non-zero consumption for all time. Finally, it is necessary to make a boundedness qualification of the form

sup U(F(K,)) = B < oo. KC 2 0

Ramsey called the least upper bound B the bliss level. A state of bliss would be attained (in the limit) as consumers became sated with goods or as the effects of capital saturation were so pronounced as to make it impossible to increase production past a certain output no matter how much investment were undertaken.

1 This reasoning is easy to spell out by a simple example. If a certain amount of material can be moulded into a pipeline to carry a given volume per unit time, twice as much material would result in a pipeline able to transport four times the previous volume (assuming a given thickness of pipe). Under these circumstances it may be better to wait a while for a bigger pipeline even though larger inventory stocks would be standing idly by in the interim.

2 Generalizing to the case where (1) does not hold is not difficult. Continuous adjustments are then possible, as well as discrete jumps.


For a given consumption path {C(t)}, Ramsey defined his social welfare criterion V[{C(t)}] as a sum of the difference between instantaneous utility and the bliss level:

V[{C(t)}]- [U(C(t))-B]dt. 0

There is no a priori reason why this evaluation integral ought to be finite for any given {C(t)}. Should V be equal to - oo for each of two consumption paths, there would be nothing to recommend one over the other; however a path yielding a finite value of V would be preferable to both.1

Temporarily forgetting all about ,B capital, Ramsey's problem is to

max [U(C)-B]dt ... (2) 0

subject to C+I =F(K), ... (3)

K =I ... (4)

C, I O, 0,.(5)

K(O) = Ka(O) >0, given. ... (6)

Any path {K(t), C(t)} satisfying (3)-(6) is called feasible. It is presumed that U and F are such that (2)-(6) is a meaningful problem in the sense

that an optimum exists.2 An optimal path {K(t), 2(t)} must be feasible, satisfy

-soo < V--=V[{C(t)}]

and possess the property that for any feasible path {K(t), C(t)},

V[{C(t)}] ? V.

Using the calculus of variations, Ramsey was able to characterize3 an optimal path {K(t), C(t)} as the unique solution to the differential equations

UI(C) K= B- U(C), ... (7)

C F(K= .(8)

with the given initial condition K(O) = Ka(). .. (9)

Equation (7) is the famous Keynes-Ramsey rule of optimal allocation. Along an optimal path both capital and consumption grow monotonically until one or the other goes to or asymptotically approaches its saturation level. The other variable goes to or asymptotically approaches a corresponding level which is determined from the production function. Thus the bliss level is reached at least asymptotically. An optimal solution is shown in Fig. 1.

The overtaking criterion makes {C1(t)} preferable to {C2(t)} if there exists a T such that

U(C1 (t))dt > U(C2(t))dt

for all T > T. Were {C1(t)} preferable to {C2(t)} by Ramsey's evaluation integral it would also be preferable by the overtaking criterion. The converse is not necessarily true.

2 It is easy to see that (2)-(6) would have to be a well-defined problem if U(F(K.)) reached a maximum for finite K,. The various sufficiency conditions of Gale and Sutherland [3], McFadden [8] and von Weizsacker [14] are stated in terms of the overtaking criterion but many of them can be routinely modified to treat the Ramsey evaluation integral case.

3 A rigorous argument is contained in Koopmans [7].


Define Y()_F(K(t) . (10)

q(t) --U'(C(t)), ... (11) I()_K(t ... .(12)

The dual price q(t) is interpretable as the value of an extra unit of output at time t imputed in terms of the evaluation integral. Differentiating (7) and (8) with respect to time, substituting from (11) and rearranging yields

q - - =- F(K), q

a standard relation of optimal growth theory. Note that q<0, Y_I' 0.

Y~(t)

Time t FIGURE 1

5. OPTIMAL CAPACITY EXPANSION

Now temporarily neglecting oc capital entirely, treat { Y(t)} and {q(t)} as if they were prescribed data. { Yf(t)} is considered in the present context to be a fixed final demand schedule which must always be fulfilled. Thinking of q(t) as the cost in current terms of investment funds at time t, the present discounted cost of creating extra capacity AK at time t is q(t)H(AK).

The least cost capacity expansion problem is to schedule capacity {K(t)} to meet final demands { Y(t)} at minimum total present discounted cost. Mathematically, the problem is to find times t1, t2, ... and capacity increments AK(t1), AK(t2), ... which'

00

min f[{tj}, {AK(ti)}] _ q(tj)H(AK(t1)) ...(13) i = 1

1 At times other than ti, t2, ..., AK(t) is thought of as being zero. Manne [9] discusses several practical examples of optimal capacity expansion problems.


subject to K(t)_ Y(t),..(4

K(t) = K(ti - 1) + AK(ti - 1) for ti_ - < t _ ti, . ..(15) K(O) = Kp(O) > Y(O), given, .. .(16)

to - 0.

Let {?i, A1Q(?X)} represent an optimal policy.' Define ;- [{Ji, {Ak(?i)}]. Using (15), an optimal path can alternatively be expressed by {k(t)}.

Demand, capacity

Capacity k,(t)

| f IL/ Demand ?(t)

-kP(3) _

kp(72) - r

K(O) = fl(7j)

Y(0 r . I I I,

t, t2 t 3 Time t FIGURE 2

1 Although the existence of an optimal policy is not proved, we indicate why it has been assumed. Consider a feasible policy of installing 8 units of extra capacity whenever K Y. The cost of such a policy would be

a(S)-- H() z q(i'1(K(O)+j8))8.

Passing to the limit,

lim E q(5-1(K(O)+jS))8= q(5-1(F))df= f q(t)Ydt= f q(t)F'(K)Kdt a6-O + j = O K(O) tj tj

= F'(K)[B- U(C)]dt < F'((O)) ( [B- U(C)]dt < oo, where ti _ Yl(K(O)). () is well defined for

all S>O. It follows that for some value S'>0 and some M< co, 0(8') = M. M provides,a finite upper bound on I. Since zero is a lower bound, a finite greatest lower bound on 0 exists, denoted 0. This means that feasible policies exist with costs arbitrarily close to ?,. In practice such a result is as good as an existence theorem, given the uncertainties of the data.

Strict existence of an optimal capacity schedule in the mathematical sense could be rigorously proved if we had chosen to formulate the model in terms of period analysis instead of continuous time. Let Q be


It is easy to see that along an optimal path, R(fi) = if(fi). No extra capacity will be installed while some excess capacity already exists. With q < 0, it pays to postpone intended construction until the day when some must be undertaken because full capacity will have been reached.' A typical minimum cost policy is illustrated in Fig. 2.

6. FORMULATION AND SOLUTION OF THE BASIC MODEL

Having treated separately the Ramsey and Capacity Expansion problems, we are in a position to tackle the main problem introduced in Section 3. Properties of the present model are vaguely recognizable as some sort of a rough combination of features belonging to the two simpler problems. As we shall see, the optimal solution will combine, in a well defined sense, the Ramsey and capacity expansion optimal trajectories.

We use the same objective as Ramsey. However, in the context in which it is presently

employed, [U(C)-B]dt is denoted W[{C(t)}] to avoid confusion.

The problem is to select times t t, ... and to choose values for the instruments Y(t) C(t) s I(t) Ia(t) , IXt,X( n SK(1i) to2

max W[{C(t)}] [U(C(t))-B]dt ...(17) 0

subject to Y(t)

_ F(Ka(t))9..(8

Y(t) _ Kf(t) . . . (19) C(t) + I(t) = Y(t), ...(20) Ia(t) +IX(t) =I(t) . ..(21)

Ka(t) = Ia(t), . .. (22) X(t) = IX(t) for t #0 li, ... (23)

lim X(li + e) = X(i) - AX(is), .. .(24) C-0 +

AKpQi) = G(AX(1i)), ... (25)

Kp(t) = Kp(li- 1) + &Kp(1i_ 1) for li- 1 < t ? li, .. .(26) Y(t) C(t) I(t) Ia(09 IX(09 X(t) AX(ii) > 09 ... (27)

Ka(O)q Kpy(O)q X(O) given. ... (28) Where not otherwise noted, t is any non-negative time and i is any positive integer. By convention o ? 0.

the Cartesian product set of all feasible x /AK(n). Clearly Q is closed. With M an upper bound on , n =O

we can restrict AK(n) to values 0 < AK _G C ( for n = O, 1, 2,.... A standard application of the

Tychnoff theorem (cf. Kelley [6], p. 143) shows the Cartesian product set Q' = x G, G to be

compact. It follows that the set Q" _ QnQ' is compact. Define the function 0 min {0, M) for all 00 x AK(n) E Q". Being real valued and continuous under the product topology on the compact set Q", 0

n =O

must attain a minimum for some value x AK(n) E Q", concluding the proof. n =O

1 This simple conclusion is an example of a " regeneration point theorem ". Such a result greatly simplifies computation because the search for an optimum can be limited to full capacity regeneration points and these can typically be efficiently examined via the appropriate dynamic programming algorithm.

2 For economy of notation, positive values of AX and positive changes in Ka have been restricted a priori to discrete times {f1}. In fact this is a vacuous restriction because condition (1) implies that even with the possibility of continuous adjustments available, an optimal policy would always call for jump adjustments in Ka at certain distinct times. For times t # li, it is useful to interpret AX(t) and AK0(t) as being zero. Note that AX(fi) is defined as minus the algebraic change in X at time li, so that AX 2 0.

2N


With only insignificant loss of generality, we restrict attention to initial configurations of the state variables Ka, K;, X obeying'

Kp(O) _ F(Ka(?))s X(O) = 0.

The following theorem is the main result.

Define {1} and {!*} by the recursive equations2 i-1

= z ~ _ j = I

t- *+ H(Ak(li))

i = 1,2 ....

Under the assumptions of the model, an optimal solution of (17)-(28), whose variables are denoted with asterisks, can be described as follows:

AKp*(t) = AX*(t) = 0, t =A7 ,

AK*(ts*) = Kt)

AX*(!*) =H(Ak(ii))

K*(t) = K t- ? (J*-t*)

K*(t) = 0(I),

C*(t) = c (t 5 (tj*-tY))

X*(t) = i .

Y*(t) = (n I

C*(t) = C ) t

Fortunately a relatively simple interpretation can be placed on this formidable looking prescription.

The optimal policy is depicted in Fig. 3. Suppose that all the [t*, i*] sections of that diagram were compressed into points and the gaps removed by pushing together and

1 Later it should become clear what to do if these initial conditions should not be met. In fact an optimal policy will always call for K0(t) ? F(Kit))-for all t ? 0 if K0(O) _ F(K~(O)), and for all sufficiently large t if K0(O) < F(Ka(O)). The latter condition must therefore have arisen, in some sense, out of mismanage- ment prior to time zero. The case of X(O) positive is not difficult to handle; we normalize to zero just for notational convenience.

2 The reader is reminded that variables capped with a circumflex are defined in Section 5, whereas those capped with a tilde are defined in Section 4.


connecting the {Y*(t)}, {I*(t)} and {C*(t)} curves. The resulting trajectories would be exactly the { {1(t)}, {C(t)} graphs of Fig. 1. In other words, the sole effect of introducing the [t*, t] regions into Fig. 3 is to stretch out into intervals what would otherwise be isolated single points of the original Ramsey optimal trajectory.

An analogous interpretation is available for fl capital. Except for the [tn, i*] sections, the time profile of Ki in Fig. 3 would look identical to the Fig. 2 portrayal of an optimal capacity schedule {?i, AR (?I)}. It is easily seen that the {Ii} of Fig. 2 are exactly those isolated points of the Ramsey optimal policy which in Fig. 3 are stretched out into time segments of positive length.

While time seems to stand still during the period [t, t*], all investment is going into accumulating an inventory of X(t!) starting from an initial level of zero at time t*. The

, C*, K*

C*(t)

Y*(t)

--)Time t t* -* * ' i*t -i-1 ti1 -- l i -L+1 till

FIGURE 3

interval [I, ti] lasts precisely long enough to build up that amount of X which will coalesce to form the AR( i) units of P capital dictated by the solution to the capacity expansion problem. Throughout the period [t*, !i], Y*(t), I*(t), C*(t) are maintained at the constant levels Yi'Q), 1Qi), C(Q). At time ti the entire generalized inventory X(t*) is formed into fi capital and the economy picks up again at that point of the Ramsey trajectory where it left off at time t* because the Kl ceiling had become binding.

In terms of economic development, [t*, t*] represents a big push period.' During this time all investment is being funneled at a constant rate into the as yet unproductive overhead capital project. As will presently be demonstrated, big push stages are likely to be more predominant in the earlier stages of development. From the viewpoint of social policy, the big push is probably a critical time because no real growth occurs and consumption is stagnant.

1 Cf. Rosenstein-Rodan [12], Hirschman [5], Scitovsky (13].


Let p(t) represent the value of an extra unit of output at time t imputed in terms of the evaluation integral. Obviously p(t) = U'(C*(t)), so that

p(t) = q (t- L t) _ 1 <t<t*.

p(t) = q(ii), t,* < t _ t*, i = 1, 2, ....

For t belonging to the Ramsey growth phase (71, t, the dual price declines over time; an extra unit of output is worth more if it is received early because it could be pro- ductively invested in a capital to yield increased future returns. However, in the big push phase [t,, it] the social output price is stationary; whether received early or late in a big push stage an extra unit of output cannot be used to increase returns but could only be invested in non-directly-productive generalized inventory.

7. PROOF OF THE MAIN THEOREM

The strategy of proof can be easily outlined. A consumption path {Ce(t)} is efficient in the usual sense if it is feasible and if, for any other feasible path {Cf(t)} with the property

00 Cf(t) ? Ce(t) for all t > 0, [Cf(t)- Ce(t)]dt = 0. We frst exhibit three obvious

efficiency conditions (a)-(c). Any candidate for an optimal path must satisfy these three criteria. Next, considering only comparisons among paths so restricted, we show that an optimal solution must belong to an even more exclusive family of paths with special additional features. Finally, the proposed solution is shown to be an optimal member of this family of special paths.

The three efficiency conditions are:

(a) If F(K.(t)) < Kp(t), no investment in X occurs at time t. This condition specifies that all investment must go into building a capital if there is

excess ,B capacity. Let T be the first time later than t when F(K,-(T)) = Kp(t) (if this never happens, T_ oo). With F(Ka(t'))<Kp(t'), there is no loss of generality in restricting AX(t') to be zero for t ? t'<T. It certainly won't depress consumption at any future or present time to put off until z setting AX> 0. It may even help to wait and take advantage of increasing returns by later converting a bigger chunk of AX into more AKP. Now

suppose contrary to the hypothesis that f X(t')dt' = y >0. Since nothing useful is going Jt

to get done with X until time T at the earliest, we can consider a different policy. Maintain the same consumption levels as in the original policy. Now, however, first devote all investment to building up Ka to level K{(T) = F- 1(Kf(t)) and then put all investment into creating amount y of X. Let this alternative policy take (T' - t) time units to complete. It is easy to see that ' <T because the second alternative takes advantage of the direct productivity of a capital for all times when F(Ka) < K,. Thus, the identical stocks of Ka, K., and X are achieved at time T and consumption is the same for all times belonging to [t, T'], but the second alternative permits attainment of a strictly higher amount of consumption at level Kp(t) = F(K(-(T)) for all times of (T', T]. It follows that the original path must have been inefficient.

(b) If F(Kx(t)) _ Kp(t), no investment in K. occurs at time t. This condition prohibits investment in a capital so long as full capacity already exists

in that sector. Let T be the first time after t when AK; >0. Suppose that F(K,(t')) > K,,(t')

for I E (t, T] and that { K,,(t')dt' = y>0. It is easy to see that a superior policy is first to


invest only in X and then to coalesce H(AKp(T)) units of X into AK,(T) as soon as X(T) - X(t) units of X have been accumulated, say at time '< T. Only then should the y units of a capital be built up. All the while the same consumption levels of the original policy are maintained. This alternative ends up with the same values of Ka, K,, and X immediately after time z, maintains an identical consumption level for all times belonging to [t, T'], and still allows a splash of extra consumption to occur in the interval (z', T] due to taking advantage of the direct productivity of a capital.

(c) If F(Ka(t)) > K,,(t), if X(t) = 0, and if z is the first time after t when AX> 0, then AX(T) = X(T).

This condition requires that all X built up from zero for the purpose of increasing K, must be coalesced into AK,, all at once. Suppose to the contrary that AX(T) = y < X(T).

r' Let T' < be such that J (t')dt' = y. If F(Kx(t)) = K1(t), it will be better to set AK(T') = y

t and then, duplicating the reasoning of (a), to restrict further immediate investments to a capital. The advantage of setting AK(-c') = y for the case F(K,(t)) > K,(t) is obvious. In either event, the original path is inefficient.

The following comments apply to efficient paths. It follows from (b) and the initial condition F(K,(O)) < Kp(O) that F(K,(t)) < Kp(t)

for all t > 0. Let {Iij represent the times (in order) at which AX>0; that is, AX(t)>O if and only if t = ti for some positive integer i. Immediately after 1i-1, F(Ka) <Ks, and all investment goes into a capital. Let ti be the first time after ti- I when F(K.) = K.. Define

00

R U (1i -1, ti), with to 0. If t E R, F(Ka(t))<K ,(t), and, from (a), I(t) = I"(t). Let i = 1

S- U [, tiJ. For t'eS, F(K,(t'))=Kp(t'), and (b) stipulates thatI(t') = IX(t'). Thus i = 1

any efficient path can be divided into two distinct investment specialization phases. Property (c) combined with the initial condition X(0) = 0 implies that AX(b) = X(71), or that X(t) = 0 for t E R.

Let qi be the value of an optimal policy starting at time ti. Note that qi is a function of KQ(ti), KQ(ti) and X(ti) but does not depend on t, explicitly because the objective function and the constraints of the main problem are time autonomous. We know that all investment during the period (ti_1, ti) is restricted to a capital alone. Consider ti-1, Ka(t.i_1) K,(Qi), as fixed. To be part of an optimal path, ti, treated as a variable, must be chosen to

I'ti max [U(C)-B]dt+ i ... (29)

subject to C+ = F(K)(30)

K=-Ia ... (31)

C,I> _ 0, ... (32)

ii- 1, KJ! i - 1), Kx(tij) given. .. (33)

Using the calculus of variations, the solution to this free-time fixed-endpoint problem is

U'(C)Kc = B- U(C). ... (34)

c + Ka F(Kx), ... (35)

i - 1 KJ! i - ), Kx(t j) given. ... .(36)

his, of course, is nothing but a section of the Ramsey optimal trajectory (7)-(9). Now consider an analogous problem over the time interval [ti, 7]. Let Ob be the value


of an optimal policy starting at time bi with capital stocks Ka(Ii), K,(ii), X(Ii). All investment during the period [ti, i] goes into building up an inventory of X(Ii) starting from zero. Also Y(t) = Y(t) = F(K=(ti))=Kp(ti) = Y(Ii) is constant for t e [ti, Ji].

Look upon Y(t), ti, XQ() and X(i) as constants. An optimal policy must have the property that bi, treated here as a variable, is selected to

rZi max [U(C)- B]dt+0k ... (37)

subject to C+IX= Y(i), ...(38)

X-=I~X, ... (39)

C, >x 0 0, ... (40)

ti, X(tQ) = 0, X(Ii), Y(tQ) given. ... (41)

The calculus of variations solution to this free-time fixed-endpoint problem is

U'(C)X = B- U(C), ... (42)

C+X = Y(i)-F(Ka()))=.. (43)

ti, X(ti) = 0, X(QI), Y(tQ) given. ... (44)

Optimal values of IX(t) and C(t) are constants for t E [ti, ti]. They are equal to the optimal values, respectively, of I,(t) and C(t) in the solution of problem (29)-(33) at t = ti, since at that time equations (34)-(36) and (42)-(44) are identical. Using the same reasoning, IX(t) and C(t) from (42)-(44) are also equal to the optimal values of I,(t) and C(t) at time t = ii for the free-time fixed-endpoint problem which is identical to (29)-(33) except for taking place over the interval (ii, ti+ 1) instead of (i- 1, ti).

These results justify the basic features of an optimal policy as they have been depicted in Fig. 3. Y*(t), I*(t), and C*(t) are continuous. Each big push stage of S is just a single Ramsey optimal point stretched out into an interval of positive length. If these isolated points were reassembled back into the appropriate niches of R and the resulting conglomer- ation treated as if it were a set connected under continuous time, the complete Ramsey trajectory (7)-(9) would emerge. A Ramsey growth stage of R ends whenever full fi capacity is encountered and it becomes necessary to devote investment to building generalized inventories. Ramsey growth continues as soon as all inventories have been converted into excess , capacity.

In our search for an optimal policy, without loss of generality further attention is restricted to the family of paths described above. The social objective W[{C(t)}] can then be split up as follows:

W[{C(t)}]= f [U(C)-B]dt= f [U(C)-B]dt+ [U(C)-B]dt. O JR JS

As we have seen, the value of the former integral along an optimal path must be RI the optimal Ramsey objective. As for the latter integral,

[U(C)-B]dt = [U(C(i))-B](i-ti) S ~~~~i=l1

= E [-HUA(C(Q))I(O)1 H (-t))

= - Ej U'(C(ti))H(AKf(li)). ... (45)


i-i Think of ti = t i- E (tj-t!) as the single Ramsey optimal point which has been

j = 1 magnified into the interval [ti, 1J. In terms of the capacity expansion problem (13)-(16), (45) is equal to

00 00

- Z _ U'(c(t))H(AK(ti))=- E q(ti)H(AK(ti)) i= i=l

= -+[{ti}, {AK(ti)}], where ti+1 =

Thus { [U(C)-B]dt will be maximized whenever f is minimized subject to the s

constraints (14)-(16). We can translate from {1i} to {tfl and to {t*} by using the relations

= i + E (t -t ) and t*-t* = __/i___) j = 1 0i

This concludes the proof.' An interesting side result is that

W* =V-.

The maximum social objective is less than the optimal value of the Ramsey problem by the social cost of implementing the cheapest feasible capacity schedule.

8. CAPACITY EXPANSION AT A CONSTANT GEOMETRIC RATE

The features of an optimal capacity schedule can be particularized by restricting the general functions H(AK), Y(t), and q(t) to specific parameterizations. Here we consider a constant elasticity investment cost function of the form2

H(AK) = A(AK)a, ... (46)

where A is just a positive constant of proportionality. The exponent a measures the (constant) ratio of incremental to average costs of installed capacity. Consistent with the presumed existence of economies of scale,

O<a< 1. We also assume that

Yt= q(t) -

..(7

for some constants 1 and s obeying 0<1,

O<s< a

1 Note the critical dependence of our results on the time autonomy of the system under consideration. In the original Ramsey problem the introduction of discounting, population growth, and time dependent depreciation or technical change does very little to change the basic properties of a solution (so long as the time dependence is exponential !). This is not the case for the present model. If time dependent features were introduced, the qualitative properties of a welfare maximizing path would be roughly similar, but there would be no possibility of cleanly decomposing the optimal trajectory into two distinct sub-problems. On the other hand, incorporating into the model such time autonomous features as Arrow's " leaming by doing " [I] would not significantly change the nature of an optimal trajectory. It is not difficult to extend the model to handle the case of two or more overhead capital goods which must operate in fixed proportions. Introducing a positive elasticity of substitution between Kx and Ka would considerably complicate the picture, although some features of an optimal policy would remain similar.

2 This parameterization is popular as a pre-design approximation to investment costs of installed capacity for such continuous process industries as chemicals, petroleum, cement, electricity generation, and primary metals. It should be emphasized that in the cost engineering literature AXK would refer to the extra capacity created from the establishment of a typical complete process balanced plant and not to the creation of semi-mythical " social overhead capital " for the economy as a whole. Cost engineers usually treat the exponent a as being equal to about two-thirds. Cf. Chilton [2], Haldi and Whitcomb [4], and Moore [10].


An equivalent way of writing (47) is

g(t) = sr(t), ...(48) where g(t) _ 5Y is the growth rate of final demand and r(t) - -q/q is the discount rate, both evaluated at time t.

In the framework of the isolated capacity expansion model of section 5, expression (48) can be looked upon purely as a convenient parameterization of the underlying data. The parameterization (48) generalizes the case sometimes analyzed where fixed demand grows exponentially and the discount rate is constant.

However, within the context of the two-sector model we have been analyzing, relation (48) has an interesting additional interpretation. It could come about as a result of society choosing to save at the constant rate s:

g(t) = t = F'(K)K sF'(K) s sr(t) y y ~~~~~q

In turn, a proportional savings function is the optimal behaviour befitting any solvable Ramsey problem with a constant elasticity of marginal utility. Such a utility function is of the form

U(C) = B-DC'-,

Ul, where B > 0, D > 0 and 7 =- C > 1 are given constants. U,

With the specific parameterization (46), (47) the minimum cost capacity schedule has a particularly simple characterization. When capacity must be increased (because no more slack exists), it is always incremented by the same constant percentage of existing capacity.'

We prove this interesting result by considering a schedule {ti, AK(tj)} which is a candidate for minimizing present discounted cost f. Without loss of generality it can be presumed that K(ti) = Y(ti), so that no extra capacity is installed while excess capacity is already in place.

*= Z

q(tj)H(AK(tj)) i = 1

00 i-1

- Zi l[K(O)+ E AK(tj)]- s A [AK(ti)]a i=l j=

=1AK(O)'a-; 1+ 3j AKt)~FA(~1 ... (49) j i = i K(O) L K(O) ]

Because 0 can be written in the form (49), it is apparent that the cost minimizing values of AK(tj)/K(0) are independent of K(0). Now consider the problem of finding a least cost capacity schedule which begins at time tn with capacity K(tn) instead of at time 0 with capacity K(0). This is a sub-problem of the original. The cost function for the new problem can be written in a form identical to (49) except for obvious index renumbering and the interchanged roles of K(tj) and K(0). But the optimal incremental capacity sequence expressed in units of initial capacity is independent of the initial capacity level. Hence, for an optimal path {1j, A K(?j)},

AkRt) _ Ak(Q1) 1, 2.

KQi) K(0)

1 Note that if g(t) (and hence r(t) also) is constant, an optimal policy would call for scheduling extra

capacity increments at equally spaced time intervals of length l In I + I. This constant cycle time g w pK(Oe result was proved by Srinivasin in Manne [9] for the special case mentioned above.


Let

_AQ) i_= 1,2, ...,

be the constant fraction by which capacity is always incremented. It is not difficult to demonstrate that 8y48a<0 and 04u/as>0.

9. CONCLUDING REMARKS

The time spent in a typical big push period will be H(AKfl)/i. A Ramsey growth phase will last approximately AK/Ytime units.' The fraction of time spent in big push stages will be approximately

H(AKp) +K AKFK H(AK O) +1

H(AIK) + ~Kpi H(AKO) A KP_ F( H(AKP)+1 I Y 1~I F'Kzk AK#

The earlier the stage of development, the higher the anticipated values of F'(KD) and H(AKp)/AK .2 Thus, it is to be expected that the percentage of time spent in big push stages should decline over time.

This quantifies the generally accepted notion that infrastructure is somehow a much more important ingredient in the growth of an underdeveloped than of a mature economy. The increased significance of big push stages during the early years of development means more time spent in no-growth stagnant consumption phases awaiting the completion of overhead facilities. Of course the present model over-emphasizes certain structural rigidi- ties, but the conclusions accord well with the customary feeling that the creation of social overhead capital is a more formidable barrier to growth in a less developed economy.

Cowles Foundation M. L. WEITZMAN Yale University

First version received May 1969; final version received December 1969

REFERENCES

[1] Arrow, K. J. "The Economic Implications of Learning by Doing ", Review of Economic Studies (June 1962).

[2] Chilton, C. H. (ed.). Cost Engineering in the Process Industries (New York, McGraw- Hill, 1960).

[3] Gale, D. and Sutherland, W. R. "Analysis of a One Good Model of Economic Development ", in Dantzig and Veinott (eds.) Mathematics of the Decision Sciences, Part 2 (Providence, American Mathematical Society, 1968).

1 This is just a first approximation which will be increasingly accurate as Y is close to being constant over the Ramsey growth phase. We are comparing a big push period (corresponding to a single Ramsey time point) with the growth phase which directly precedes or follows it. Both I and Y are evaluated at this single Ramsey time point.

2 It has already been noted that d- F'(K) = q < 0. Under a variety of conditions it can be proved that

the average cost of capacity H(AKft)/AKft will decline with time (due to taking greater advantage of economies of scale in later periods). For example, the condition Y >0 would be sufficient to demonstrate that AKa

increases over time. Note that in Section 8 H(AK0)/1AK,l declines over time irrespective of the sign of ?.


[4] Haldi, J. and Whitcomb, D. " Economies of Scale in Industrial Plants ", The Journal of Political Economy (August 1967), 373-385.

[5] Hirschman, A. 0. The Strategy of Economic Development (New Haven, Yale University Press, 1958).

[6] Kelley, J. M. General Topology (New York, Van Nostrand, 1955).

[7] Koopmans, T. C. " On the Concept of Optimal Economic Growth ", Pontificae Academiae Scientiarum Scripta Varia (1965), 225-300.

[8] McFadden, D. " The Evaluation of Development Programmes ", The Review of Economic Studies, January 1967, 25-50.

[9] Manne, A. S. Investments for Capacity Expansion (Cambridge, M.I.T. Press, 1967).

[10] Moore, F. T. " Economies of Scale: Some Statistical Evidence ", Quarterly Journal of Economics (May 1959), 232-245.

[11] Ramsey, F. P. " A Mathematical Theory of Savings ", Economic Journal (December 1928), 219-226.

[12] Rosenstein-Rodan, P. N. "Notes on the Theory of the 'Big Push' ", Ch. 3 in H. S. Ellis (ed.) Economic Development for Latin America (I.E.A. Conference) (London, Macmillan, 1961).

[13] Scitovsky, T. " Growth-Balanced or Unbalanced? " in Abramovitz (ed.) The Allocation of Economic Resources (Stanford, Stanford University Press, 1959), 207- 217.

[14] von Weizsacker, C. C. "Existence of Optimal Programs of Accumulation for an Infinite Time Horizon ", Review of Economic Studies (April 1965), 85-104.

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