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JOURNAL OF ECONOMIC THEORY 49, 36t%375 (1989)

Existence, Uniqueness, and Stability of Equilibriumin an Overlapping-Generations Modelwith Productive Capital

ODED GALOR AND HARL E. RYDER*Department of Economics, Brown University,

Providence. Rhode Island 02912

Received March 15, 1988; rev ised November 7, 1988

This paper analyzes the existence, uniqueness, and stabi l i ty of a steady -state equi-l ibrium in an overlapping-generations model with productive capital. It is shownthat for any feasible set of well-behaved preferences there e xists a production func-tion that satisties the Inada con ditions under which the econom y experiences globalcontraction and the steady-s tate equilibrium is characterized by the absence ofproduction and consum ption. The study establ ishes a strengthened Inada conditionwhich is necessa ry to preclude global con traction. and sul l icient conditions forthe existence of a unique and globally stable non-trivial steady-s tate equil ibrium.Journal of Economic Literature Classit ication Num bers: 020, 111. 0 1989 AcademicPress, Inc.

1. INTRODUCTIONIn the last decade overlapping-generations models have become astandard tool of modelling dynamic economic behavior over an infinite

time horizon. The evolving literature has been based on the consumption-loans version, presented in the seminal contribution by Samuelson [ 131, aswell as on the production economy along the lines of Diamond [3].Despite the intensive use of the original setting of the production economy,the analysis of existence (as well as uniqueness and stability) of equilibriumin this widely applied setting has largely been neglected.

* We w ould l ike to thank an anonymous referee, an associate editor, and J . Zeira for usefulcomments .

The existence, uniqueness, and stabi l i ty of equi libria in the pure-exchange conte xt havebeen addressed by Gale [5 ], Balasko, C ass , and Shell [ l ] , Balasko and Shell [2], Kehoe andLevine [9], as well as others.0022-0531/89 $3.00Copyright 0 1989 by Academic Press, Inc.All rights of reproduction in any form reserved.

360

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OVERLAPPING-GENERATIONS MODEL 361

Consequently, analysis based on this simple overlap~ing~generatio~smodel with production have been conducted, to a large extent, either (i)under the false presumption that the Inada conditions (and well-behavedpreferences) are suflkient to guarantee the existence of non-trivial steady-state equilibrium, or (ii) under the implicit or explicit assumption thatequilibrium exists and is unique. Clearly, since conditions under which thelatter holds have not been placed on the fundamental features of theeconomy (i.e., technology and preferences), results based on comparativestatics may be internally inconsistent.The properties of stationary equilibria in different types of overlap generations models with production have been studied previously.[4] has analyzed the convergence to a steady-state equilibrium in aspecialized two-sector model. IvIuller and Woodford [ll, 12] haveanalyzed the existence and determinacy of any stationary equilibria in amodel characterized by the existence of both infinitely lived and finitelylived individuals and by the presence of a non-depreciating asset-land.This paper, in contrast, analyzes the existence, uniqueness, and stabilityof a non-trivial steady-state equilibrium in the widely used over~a~p~~g~generations model with productive capital, in which individuals live for twoperiods and a single good is produced using capital and labor2 Theanalysis indicates that an existing presumption according to which a non-trivial state equilibrium exists in the economy if the Inada conditions aresatislied (and preferences are well-behaved), is unfounded.3 It is shown thatfor any set of well-behaved preferences there exists a pro ion functionthat satisfies the Inada conditions under which the only s y-state equblibrium is characterized by the absence of production and co~sum~tio~~

The study establishes a strengthened Inada condition which is necessaryin order to preclude this contraction phenomenon. Nevertbeless,existence of non-trivial steady-state equilibrium is not guaranteed.shown that if the extended Inada condition is satisfied, the nature of theinteraction between preferences and technology is sti ll critical in &X-mining the nature of a steady-state equilibrium.The paper derives a set of sufficient conditions for the existence of a non-trivial steady-state equilibrium as well as for the existence of a unique andglobally stable non-trivial equilibrium. It is shown that the widely usedassumption concerning the existence of unique (God-trivial) steadi-state

* In a seque l to this paper Galor and Ryder [6] analyze the dynamic efkiency of steady-state equilibr ia in this mode l. The study establishes sutkient conditions under which over-investment relative to the Golden Rule is ruled out and the economys steady-state equilibriaare therefore dynamically effkient. The implications of these conditions fcx the feasible rangesof the technological parameters of the CobbDouglas and CES production functions areanaiyzed as well.

3 E.g., McCallum [lo, p. 121.

642:49 2-1 I

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362 GALOR AND RYDERequilibrium is rather restrictive.4 It requires conditions on the thirdderivatives of the utility function as well as the production function,derivatives which could otherwise fluctuate wildly without violatingconcavity.

2. THE MODELConsider a perfectly competitive world where economic activity isperformed over intinite discrete time. At any period of time a single good

is produced using two factors, capital and labor, in the production process.The endowment of labor at time t, L(, is exogenously given by

where n 2 -1 is the rate of population growth. The endowment of capital attime t, Kr, is equal to the resources not consumed in the preceding period,

where 6 is the rate of capital depreciation, 0 6 8 < 1, and Yf- i and Ctp iare the aggregate production and consumption, respectively, at time t - 1.Production occurs within a period according to a constant returns toscale production function which is invariant through time. The outputproduced at time t, Yt, is governed by a neoclassical production function,

(3)The production function f is twice continuously differentiable, positive,increasing, and strictly concave:

Inada conditions are satislied at the origin,lim f(k) = 0, lim f(k) = cm,

k-0 k-0 (5)and there is an upper bound to the attainable capital E, such that

E.g., Ihori [8], Tirole [14], and Weil [15].s Alternative ly, we may assume that lim k+ mf(k) =0, which, in conjun ction with (5),

s&ices to assure (6).

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364 GALORANDRYDERwhere tt+ I is the anticipated return on next periods capital-a price thatis not observable at the time this decision is under consideration. Given(+v~, r+ i), it follows from the postulated properties of the util ity functionthat the optimal saving, ~(vv~, f+,), exists and is uniquely determined. Wemay consider agents to be endowed with perfect foresight,

or alternatively with static expectations,

Under the assumptions given above, utility-maximizing agents choosestrictly positive consumption in both periods so long as positive capital hasbeen inherited from the previous period. Once the current capital stock kZand the anticipated return to capital tl+ I are specilied, the consumptionchoice is unique. Both of these variables are unambiguously determined ifagents form expectations that depend only on past and current variables(e.g., static expectations, adaptive expectations, extrapolative expectations).In any such case there is a well-defined path following from any init ial con-liguration of capital endowments and relevant history. Hence the futurecourse of the economy can be predicted accurately. These expectations,however, do not coincide with the actual course of the economy unless theeconomy is in long-run steady state.For this reason, it is customary to assume that the agents are capable ofpredicting the future course of the economy and that they adopt thesepredictions as their expectations. Such rational or perfect foresightexpectations are independent of past observations and must be self-fullilling. The subsequent section establishes conditions under which aunique self-fulfilling expectation exists and is interior for every positive levelof initial capital. In the absence of this condition, perfect foresight may leadto indeterminate dynamic equilibria, since some initial levels of capitalstock may have more than one self-fulfilling expectation.

3. CHARACTERIZATION OF EQUIL IBRIUMThe dynamics of the system is characterized by the set of equations

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OVERLAPPING-GENERATIONSMODEL 365

Thus given k*, a level of kt+ l that is a self-fulfiiling expectation satisfies

LEMMA 1. Given kl > 0, there exists a unique lc+ l > 0 that is a sel$fulfilling expectation if savings are a Bon-decreasing f~~etio~ of the interestrate, i.e., if

ProojI Consider Fig. 1, where each side of Eq. (18 ) is plotted as a fun&oQfkt+1. Since s(w~, r r+ll 0 (and therefore ~2~ O)>there exists kt+ I >O which satisfies (18) if lirn~~+~-~~~w~,~(k~+~)~~~.Noting (1 I), a sufficient condition for the existence of kr+ I > 0 is therefore

Umqueness is satisfied if in addition the derivative of the right hand sideof (18) with respect to kt+ l is everywhere smaller than one, i.e., if

Ii t+1FIG. 1, Existence of a uniq ue k,+ , > 0 which f&i& expectations.

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366 GALOR AND RYDERThus, since y(kf+ I) c 0, uniqueness is guaranteed by the lemmas condi-tion. 1

Remark 1. Note that

iff the substitution effect created by an increase in the interest rate is notsmaller (in absolute value) than the income effect, i.e., iff

Remark 2. Preferences which are represented by a strictly concaveutility function and satisfy the properties (9)-( 11) are not sufficient topreclude limkf+, ~ 0 ,s(wz, rf + I) = 0 and consequently cannot guarantee theexistence of a kt + l > 0 that is a self-fulfilling expectation. For instance theutility function

satisfies the required properties but fails to ensure positive savings when theinterest rate approaches infinity.Thus, if savings are a non-decreasing function of the interest rate,k t+ l = d&h where 4 is a single-valued function, b(O) = 0,

andd2k+ 1- = #(kJdk;

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QVERLAPPING-GENERATIQNSMODEL 367

DEFINIT ION 1. A dynamic equilibrium is a sequence {kz] FL0 underwhich

where k. is exogenously given.DEFINIT ION 2. A steady-state equilibrium is a stationary ca~itaI-Gabonratio, k, under which

Given the specilications of the production and util ity functions, there isa bounded range into which every path will enter and within which it willremain. In particular

so O

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OVERLAPPING-GENERATIQNS M0DEL 369

We see that this production function satisfies all the standard assumptions,but the behavior specified by the overlapping-generations model leads toglobal contraction, as

5 A STRENGTHENED INADA CONDIT IONCan some strengthening of the Inada condition rule out the l&nof technology that would force contraction to the trivial steady-stateequilibrium?PROPO~ITIQN 2. Consider the overlapping-geflerations economy. Tkereexists k > 0 such that

only iflim kl 215 for all kO>t-m

lim [ -kp(k)] > I i-n.k-0

Booj: If limt+ ~ kt3k>0 for all kG>Q, then k,+I>ki fm aI1kr E (Q2k). IIence

Rearranging,

n the limit, using lHospitals rule,lim .W9 -W(k) = lim [ -kj(k)] > I+ n.k-0 k k-.0

LEMMA 2. The conditionlim [ -kr(k)] > I + n

k+Ois stronger than the Znada condition

lim f(k) = CD.k+O

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370 GALORANDRyDERProoj It is sufficient to show that the Iirst inequality implies the Inadacondition but the converse implication does not hold. For 0 < k-~ k,,, we

have f(k) < - (1 + n)/k. Integrating both sides, we obtainf(kO) -f(k) = 1; f(k) dk < - 1; (I+;

= (1 +n)[ln k-ln k,,]. (39)Rearranging,

f(k)> -(I +n) ln k+ [(1 +a) ln kO+y(kO)]. (40)In the limit,

lim~(k)>(l+~)~mO(-lnk)+[(l+n)lnkO+f(kO)]=co. (41)k+OThus limk +o[ - kp(k)] > 1 + n implies limk+o f(k) = CX. The converseimplication does not hold, since the example of Section 3 satislieslim k+Oy(k)=a, but has -kf(k)=a 1 +n is stronger than limk+oy(k)= co. 1

6. SUFFICIENTCONDITIONS FOR THE NON-EXISTENCEOFNON-TRIVIAL STEADY-STATEEQUILIBRIUMPROPOSITION . For any given set of preferences, lj+ he production func-tion satisfies the Inada conditions, the on/y steady-state equilibrium in theoverlapping-generations economy is characterized by zero production andconsumption lj-

(a) limk-0 [--kY(k)] -C 1 +n,(b) -kf(k) 6 I + n, for all k > 0.

ProoJ Suppose that sz= HJ( (i.e., there is no utility from lirst periodconsumption). Clearly, if global contraction is established under theabove conditions for .Y~ PV , it can be established for all other feasiblesets of preferences under which sz< wt. Thus, modifying (lS),k r+ r = We= f(kt) - kJ(kf) and dk t + JdkZ = - kJ(kl). Consider Fig. 2.Noting that b(O) = 0, the only steady-state equilibrium is the trivial one ifthe curve k $+ r =~#(k~) intersects the 45 line only at the origin. Theproposition therefore follows. 1

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OVERLAPPING-GENERAT~ONSMODEL 371

FIG. 2. G loba l contraction

7. EXISTENCE, UNIQUENESS, AND STABILITY OF NON-TRIVIAL STEADY-STATEEQUILIBRIUM

The existence of a non-trivial steady-state equilibrium, however, is notguaranteed even with the strengthened Inada condition. Restrictions on thenature of the interaction between preferences and technology are requireas well.PROPOSITION 4. The overlapping-generations economy experiences mm-trivial steady-state equilibrium &f k0 > 0 and

(a) sr(w, r) > Ofor all (w, r) 2 0.

(c) Firn% (k) = 0.

ProoJ Noting (21), the proposition follows immediately from Figure 3~Condition (a) guarantees the existence of the single valued function 4,condition (b) guarantees that the function 4 is steeper than the 45 line atthe origin, and condition (c) guarantees that there exists a k > 0 su&&kJ=k. 1

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372 GALORANDRYDER

FIG. 3. Existence of equi libria.

PROPOSITION 5. The overlapping-generations economy experiences aunique and gIobally stable (non-trivial) steady-state equilibrium zyk,, > 0 and

(a) lim - vV-~~~ , lk-o 1 +n-srf(k) (b) jima f(k) = 0.(c) d(k) 20 for all k>O.(d) d(k) < 0 for all k > 0.(e) sJ+v, r)20 for all (w, r)>O.ProojI Consider Figure 4. Uniqueness and global stability of non-trivialstationary equilibrium is satisfied if (i) a single valued function 4 exists, (ii )

the curve d(k) is strictly concave, (i ii) limk+o 4(k) > 1, and (iv) the curveintersects the 45 line at k> 0. Condition (e) is suffkient for (i). Notingthat &k) 1 + n,k-0

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374 GALORANDRYDERIt is easily verified that r =y(k) is a continuous, positive, decreasingfunction and that w =f(k) -ky(k) is a continuous, positive, increasingfunction for k E (0, lo]:

14-ink for Ock

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