Liviatan Macro Article

8/3/2019 Liviatan Macro Article

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American Economic Association

A Diagrammatic Exposition of Optimal GrowthAuthor(s): Nissan LiviatanReviewed work(s):Source: The American Economic Review, Vol. 60, No. 3 (Jun., 1970), pp. 302-309Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/1817980 .

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A DiagrammaticExposition o fOpt ima l G r o w t h

By NISSAN LIVIATAN*The theory of optimal growth can beanalyzed, as is well known, by using opti-mal-control methods or by applying thetechnique of dynamic programming. Itseems, however, that the graphical exposi-tions for simple growth models have been

developed only for the former approach.Thus it is customary to present the opti-mal growth path in terms of a trajectory inthe phase-space.' However, as far as Iknow, there has been no attempt to formu-late a graphical analysis which will bringout the essence of the dynamic program-ming approach2 to optimal growth.The purpose of this paper is to fill thismethodological gap. Moreover, it will beseen that the graphical presentation of thedynamic programming approach is closelyrelated to the traditional Fisherine dia-gram of intertemporal analysis and to thestandard Hicksian tools of demand anal-ysis. Indeed, we intend to show that theexposition of the optimal time path in asimple growth model does not requiremuch more than a dynamic version of thewell-known two-period Fisherine diagram,with current consurmLptionn one axis andnext year's capital on the other.

I. A Restatement of SomeElementary ResultsIn our analysis, which is based on adiscrete time model, we shall deal with a

one-sector model with no technologicalprogress. The production function of theeconomy is assumed to be of the followingform:(1) St+, + Ct+1= F(Sty LJ)where St denotes the stock of capital whichenters as an input in the production func-tion in period t, Lt is labor input in periodt, and Ct+1is consumption in period t+ 1.(We may assume that (1) incorporates anexponential depreciation factor.) An alter-native way of writing (1), which is moreuseful for our purposes, is as follows. De-fine K, as the capital stock available (forproduction or consumption) in the begin-ning of period t. We then have(2) Kt = St + CtSubstituting (2) into (1) we obtain(3) Kt+1 = F(Kt - Ct, Lt)It should be noted that F1=aF/a(Kt-Ct)is the marginal product of capital in periodt in producing the capital stock of periodt+ 1. Hence (F1- 1) is the net own rate ofreturn on capital, which we shall assumeto be nonnegative.We shall assume as usual that F is sub-ject to constant returns to scale withrespect to the inputs (Kt- Ct) and Lt andthat Lt grows according to(4) Lt+1=nLt n > 1, for all t* The author is professor of economics at the HebrewUniversity of Jerusalem and visiting professor at Uni-versity of California, Berkeley. This study was doneduring the tenure of a Ford Foundation Fellowship.I See, for example, the presentation in David Cass(pp. 236-38) which is based on control theory, or thegraphical exposition by Tjalling C. Koopmans based onan equivalent calculus-of-variations formulation.

2 For an analytic exposition of optimal gowth basedon dynamic programmingsee, for example, Roy Radner.302


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LIVIATAN: OPTIMAL GROWTH 303where n is given exogenously. Using theproperty of constant returns to scale, wemay divide all variables n (3) by Lt, whichyields

Lt Lt LtDividing both sides of (5) by n and using(4), we have(6) K -F (---,1) .Lt+l n Lt LtUsing lower case letters to denote percapita variables, we may write (6) as1(7) kt+l =-F(kt -ct, 1).n

As is usual in growth theory, we makethe following assumptions aboutF(kt-ct, 1):(8) F(0, 1) = 0(9) F, > 1, Fl, < O for all kt -ct > O(10) F1(0, 1) = oo, Fi(oo, 1) = 1whereFn denotes a2F/a(Kt- Ct)2. Since nis a constant throughout our analysis, it isconvenientto define a new function

1(11) f(kt - ct) _-F(kt - Ct, 1)nwheref incorporates mplicitly the growthfactor n. Applying assumptions (8)-(10)to f, we may write(8') f(O) = 0;(9') f'> - f"


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304 THE AMERICAN ECONOMIC REVIEWunique positive maximum3 at some k*O u" < 0 for ct > 0andin addition,to excludecornersolutions

for ct (where ct =0), we shall assume(15) lim U'(Ct) = ?.

The problem of determining the optimalprogram can then be stated as follows:00Maximize U3 E atu(ct) with respecttGo

(16) to co,cl, . . . ,subject to kt+l =f(kt - Ct),O


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LIVIATAN: OPTIMAL GROWTH 305to a positive stationary limit). Moreover,the program satisfying the foregoing con-ditions is unique. The proof of these state-ments' is of no direct concern to our dis-cussion and we shall therefore take themas given, or treat them as assumptions.An important feature of the solution of(16) is that it will remain optimal (as faras present and future consumptionis con-cerned) when the economy reexamines itas it moves actually into the future. This"dynamic consistency" property of theoptimal program6 ollows directly from theform of the utility function in (13). Arelated aspect of (16) is that the solutionis independent of the calendar time. Theindex t should be interpretedas indicatingthe "number of periods ahead" at anygiven calendar date. Thus the process is astationaryone.We have formulated the maximizationproblem subject to a given arbitrary levelof initial capital. As a preliminary inquirywe may, however, disregard nitial capitaland examine whether our system is at allcapable of a stationary optimal solutionwhere t= c and kt= k for all t. Substitutingthe constant values of c and k into (17) weobtain(18) 1/a = f'(k - c).We know that there must exist a positivesolutionto (18) iff'(0) > 1 a >f'(cc). Since1/a > 1 this is guaranteed by our foregoingassumption (10'). It also follows from thefact that f' is monotonically decreasingthat (k-c) is uniquelydeterminedby (18).In a stationary solution we must also have(19) k =-f(k - c).Wemay thereforedeterminefrom (18) and(19) a unique solution for c and k individ-ually. The foregoing values, to be denoted

by e and k, constitute an optimal sta-tionary solution. Using the fact that 1/a=f'(k -e) >1, we may infer that k and care smaller than the correspondinggoldenrule values, i.e., k


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306 THE AMERICAN ECONOMIC REVIEWmizing E -o atu(ct) for a given value ofko.We may then use the foregoingv func-tion to write

00(21) max E atu(ct+i)= v(kl).Ct+1 t=OSubstituting in (20), we have(22) max U = u(c0) + av(k1).

C+1This completesthe first stage of maximiza-tion, from which we have obtained thereducedutility function (22). (It is under-stood, of course, that the reduced utilityfunction depends not only on the originalutility function but also on the productionfunction.)In the second stage we treat c0and k1asendogenous variables and maximize thereduced utility function (22) subject to thepresent period's production constraintk1=f(ko-c0). This determinesthe currentperiod's optimal values of coand ki. Thesecond-stageproblem9can then be writtenas max [u(c0)+ av(ki)]

Co t1(23)subject to ki = f(ko - c.), and given ko.

The recursivenature of this system is clear.Thus the value of ki determined by (23)becomesthe next period's kowhich is usedto determine by means of the same func-tions the value of the new ki and c0and soon. Note also that by the definition of v,(23) satisfies(24) v(k0)=max {u(co)+ avjf(ko -c.)I

CO

which is the fundamental functional equa-tion of dynamic programming.10

As a necessary condition for a maximumof the right-handside of (24), we have(25) u'(c0)= af'(k0 - c)]f'(k0 - c0).Denote the maximizing value of coby cwhere the latter can be considered as afunction of ko. Substituting in (24), wehave(26) v(ko)= u(eo) + av f(k. -c)Differentiating (26) with respect to koandusing (25), we obtain

v'(ko)= u'(o)deo/dko+ av'Vf(k -co)](27) *f'(ko - o)(1 - deo/dko)

u' (eo)i.e. the marginal utility of current capitalequals the marginal utility of currentconsumption (and consequently v'> 0).This is only natural, since one of the alter-natives of using an increase in kois con-sumingit in the currentperiod.It follows from (27) and (14) that v isconcave if deo/dko>O. Consider the hy-pothesis that deo/dko


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LIVIATAN: OPTIMAL GROWTH 307deo/dko 1. Thus differentiating(25) withrespect to koand rearrangingterms, weobtain

dco ~~~~1(28) dko 1 _ __ _

a vf" + f'2v")which is between zero and one. It alsofollows from these results that dkl/dko=f * 1-deo/dko) > 0.

III. TheDiagrammaticAnalysisof OptimalGrowthThe short-runequilibriumat somegivendate is illustrated in Figure 2 by means ofa Fisherine diagram, where on the hori-zontal axis we have co and on the verticalaxis k1.The productionpossibility frontierfor initial capital ko s given by AB whichrepresents he function k1=f(k,,- co) in the(co, ki) plane. The slope of this curve atany point is given by df/dco0=J'


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308 THE AMERICAN ECONOMIC REVIEW

A0~~~~~0

80 81~~~R

ko N k, K) kCoFiGuE 3

from which it follows that c' is the sta-tionary value of consumptionwhich corre-sponds to ki. Using our earlier notation,we have c'=s(ki) 14Let us now introduce in Figure 3 the(stationary state) function c=s(ki). Thisis represented by the SS' curve whoseproperties are derived from Figure 1. Wecan then see that if we start at Eo we mayfind a point (QO) n the next period'spro-duction frontierby the intersection of thehorizontalline passing throughEo and theSS' curve. If in the next period theeconomy chooses E1 as its equilibriumpoint, then in the following period theproductionfrontiermust pass through Ql.A similaranalysis appliesto the possibilitywherethe equilibriumpoint is to the rightof SS', as in the case when the economy is

initially at R0.The productionfrontier ofthe next period will then pass through QO.Some additional features of Figure 3should be noted. We know fromour earlierdiscussionthat the point G corresponds othe golden rule levels of c and k. At thispoint the slope of the production frontierABis unitary, i.e.,f'(k*-c*) = 1.Similarly,for any point (c, k) on SS' below G theslope of the productionfrontier is f'(k-c)>1, and above G we havef'(k-c) 1/n>O.16In the logarithmic model, where e(c) =A cy. wehave e'(O)= + oo


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LIVIATAN: OPTIMAL GROWTH 309we must have an intersection of EE' andSS' for some positive k and c. Let M be anintersectionpoint of the two curves. Sincethis point is both a stationary solution anda short-runequilibrium, t must representa stationary optimum. We have seenearlier that in a stationary optimumi'(k-C) = 1/a wherekand j aredeterminedby (18) and (19). It follows therefore thatthe intersection of EE' and SS' is uniqueand that the intersection point must bebelow G17as drawn in Figure 4. Supposealternatively that at the origin e'(O) k, thesystem would converge to M from above.Consider the golden rule point G. Sincethis point is necessarily above EE', itfollows that at G the marginalrate of sub-stitution of k1for c0 on the consumptionside is greater than on the productionside,i.e., cois more valuable (at the margin) inconsumption than in production. Thus ifthe economy is given an initial capital

k, St

A4-

k E~~

E~~~~EE0

VS B,, B, |B2 IB3 |B4koC'O ~~k k' CO

FIGuE 4

equal to k*, it will not stay at G but willrather increase consumption immediatelyto the point R, and then, in subsequentperiods, will reduce consumption andcapitalgraduallyto the point M.

17 Sincel/a=fJ'(k-e)>1.18The possibility of tangency can be ruled out bysimple considerations of continuity. Thus suppose thatEE' is below SS' except at M, where the two curves aretangent. Then for initial ko>k the sequence ct will con-verge to e, while for ko

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Liviatan Macro Article