on price characterization and pareto-efficiency of game equilibrium growth

Vol. 46 (1986), No. 4, 347m367 Journol of Economics Z.eltschrift fur Naflonal4~konomle

@ by Springer-Verlag 1986

On Price Characterization and Pareto-Efficiency of Game Equilibrium Growth

By

John Lane, London, U. K., Wolfgang Leininger, Bonn, F .R .G . , and Murray Hill, New Jersey, U. S. A.*

(Received January 27, 1986; revised version received October 17, 1986)

1. Introduction

We consider a well-structured model of capital accumulation with a preference structure that reflects altruism in the specific sense that each generation's welfare is defined not only on its own consumption level but also on that of its immediate descendants. As altruism is limited the interests of distinct generations, who each live for one period only, come into conflict. The solution concept is that of a perfect Nash equilibrium in which each generation chooses a consumption schedule (possibly non-linear). The savings of each generation form the inherited endowment of the next.

Our purpose is to utilize recent results by L e i n i n g e r [6] on the properties of the equilibrium sequence of consumption schedules to provide a simple proof that a// feasible programs that can be generated from this sequence are Pareto-efficient in the modified sense first proposed in L a n e and M i t r a [5].

The restriction is made that the schedules (strategies) be differentiable. By application of L e i n i n g e r ' s "levelling" result [6] and Sard 's theorem on critical values [7] we show, in Section 3 of this paper, that marginal propensities to consume are interior (i. e., belong to the open interval (0, 1)) except on a closed null set i. e.,

* We are grateful to James Foster, Douglas Gale, Martin Hellwig, and Tapan Mitra for many helpful discussions. We also wish to thank a referee for valuable and detailed suggestions. Support for Leininger's research by Deutsche Forschungsgemeinschaft is gratefully acknowledged. An earlier draft of this paper was presented to the 1985 World Congress of the Econometric Society at Cambridge, Mass., August 17--24.

24 Journal of Economics, Vol. 46, No. 4

348 J. Lane and W. Leininger:

only with probability zero can the marginal propensities take the value zero or one. This remarkable result allows us to extend the methods developed in L a n e and M i t r a [5] and obtain our normative results.

Because marginal propensities to consume exist everywhere we can define appropriate prices and so generalize thedefini t ion in [5] of quasi-competitive programs. Programs generated from a perfect Nash equilibrium sequence of consumption schedules must be quasi- competitive (Theorem 2, Section 4). Furthermore, as marginal propensities cannot take on boundary values, it is not difficult to show that any quasi-competitive program that satisfies an obvious transversality condition must be "modified" Pareto-efficient (Proposi- tion 1, Section 5).

On the other hand, we show that with preferences that are separable over time the sequence of capital stocks generated from an equilibrium program is bounded above in the limit by a 'golden rule'-level of capital stock. It is this "turnpike" result that enables us to show, and in contrast to L a n e and M i t r a [5], that all programs generated from the perfect Nash equilibrium sequence of schedules satisfy the transversality condition and so are modified Pareto-efficient.

In L a n e and M i t r a [5] Pareto-efficiency of Nash equilibrium programs was shown, but subject to placing certain restrictions on the possible parameterizations of the model. The result here is stronger, and so it should be. It refers to programs generated [rom a perfect Nash equilibrium sequence of consumption schedules, i. e., programs along which expectations are rational. In [5] each generation chooses a consumption ratio dependent on the histori- cally given initial capital stock before the first period of time; i. e., the Nash solution concept is defined with respect to a set of linear consumption schedules. In a recent contribution to this journal [4] the present authors have shown that this linearity restriction in general prevents equilibria from being perfect. Thus the weaker solution concept yields a weaker result; parameterizations are required to obtain the above turnpike result. L a n e and L e i n i n g e r [4] have also pointed out the very restricted circumstances in which the two solution concepts (if applied relative to the same sets of feasible consumption schedules) coincide and provided a useful re- appraisal of results obtained by D a s g u p t a [2], P h e l p s and Po l - l ak [10] and also L a n e and M i t r a [5] who all drew on the existence results of Pe l eg and Y a a r i [9] for Nasb solutions. The basic insight provided is that if a Nash equilibrium in this intertemporal setting is not perfect then it cannot be based on expectations which

Price Characterization and Pareto-Efficiency 349

are entirely correct off the equilibrium path. Correcting this defect by imposing perfectness, however, can change the entire equilibrium (and hence growth-) path.

In [2] and [10] it is shown that Nash equilibrium programs are not Pareto-efficient, but the unrestricted definition is used, i. e., no restriction is placed on "initial" consumption levels (at t = 1) in the class of comparison programs. An overall improvement in welfare can then always be obtained at the expense of the "preceding" generation (at t=0) , whose interests are no longer considered, by reducing "initial" consumption just enough to "finance" an increased level of consumption for all future generations. In the modified definition of Pareto-efficiency the interests of the "preceding" generation cannot be jeopardized. The appropriate definition is solely dependent on whether one regards t =0 as the beginning of bio- logical time or whether one presumes that at any moment of time there is always a preceding generation; in the latter case, the modified definition is appropriate. Further support to this definition is given by the result that efficiency and modified Pareto-efficiency coincide for programs generated from a perfect Nash equilibrium sequence of consumption schedules (see Section 5).

2. The Model

2.1 P r o d u c t i o n

We consider a one-good economy, with a technology given by a function, [ from R + to itself. The production possibilities consist of inputs x and outputs y = [ (x), for x_>0. The following assumption on [ is maintained throughout:

(F) For x _> 0, f is strictly increasing, concave and twice differentiable. Also

[ (0) = 0, [" (0) > 1, lim [ (x) = o0 and lim [" (x) = O. X---> oo X- -~ oO

Note that (F) allows for constant returns to scale over arbitrary finite ranges of inputs. A feasible production program emanating from some arbitrary input level x is a sequence (x, y )=(x t , yt) satisfying

xo=x, O<xt<yt for t_>l, f (xt)=yt+t for t_>0. (1)

24*


The consumption program <c> = <ce> generated by <x, y> is

c~ = yt - xt ( -> 0) for t _> 1. (2)

The sequence <x, y, c> is called a [easible program, it being under- stood that <x, y> is a feasible production program, and <c> the corresponding consumption program. A feasible program <x, y, c> is called interior if c~ > 0 for t _> 1.

The consumption-ratio sequence <z>=<zD associated with a feasible program <x, y, c> is given, for t_> 1, by

z, = ct/yt if yt > 0, zt =0 if ye = 0. (3)

The corresponding saving ratio sequence is <s> = <st> = <1-zt>.

2.2 P r e f e r e n c e s

Individuals are considered to be identical except for their dates of birth. The group of individuals born at the beginning of period t is called the t-th generation. Each generation lives for precisely one period, and is replaced by an equal number of direct descendents the instant they die.

The preferences of each generation are the same, and are repre- sentable by a welfare function U from R + x R + to R. We consider generation t's welfare, denoted by ut, to be dependent on its own consumption, and on the consumption of its immediate descendants. Thus we can associate with a feasible program <x, y, c> a welfare sequence <u> = <ut>, given by

(w) ue = U (ct, ct+z) for t >_ 1, where U is concave, strictly increasing in its arguments with continuous partial derivatives. More- over, we require

i) lim Ux (ct, Ct+l)= oo for all c,+1>0, Ct--~O

lim U2 (c,, ct+l) = oo for all c, > 0 and r + z-->O

i i) l i ra UI (ct, Ct+l) > d > 0 for al l ct > 0 C~+ i---> oo

lim Uz (ct, Ct+l)=0 for all c ,>0. 65 + 1---> cO

Assumption i) simply means that generations are neither completely selfish nor completely unselfish while ii) says that altruism is limited asymptotically.


2.3 S t r a t e g y S e l e c t i o n

We assume that each generation chooses a consumption schedule c~ (y) from either S or S', which are defined as follows:

S: the space of all continuously differentiable functions c: R + ~ R + ;

S': the restriction of S to all linear functions.

A consumption schedule from S is called feasible if c (y) < y for all y e R +. This implies c (0) = 0.

2.4 D e f i n i t i o n s

A sequence of feasible schedules (c~ (y)) generates a feasible program for any initial input level x0 = x by

Xo=X, y , + l = / ( x t ) for t>_0, x t = y , - c , (yt) for t > l . (4)

The sequence (c~ (y)) may be viewed as a function mapping initial input levels into consumption programs. Alternatively, a consumption program may be regarded as the "realization" of such a function at xo. Different sequences of schedules may generate the same program from x0. Hence, it is not possible in general to infer the generating sequence of schedules from a given program. This is only possible if the schedules are required to be linear.

A feasible program (x, y, c) is efficient if there does not exist a feasible program (x' , y', c'), such that c~" >_ ct, for t >__ 1, and c~' > c~ for some t.

We call a feasible program (x , y, c) Pareto inefficient if there is a feasible program (x', y', c ') such that (i) c1 '=cl , (ii) us'_>ut for t >_ 1, and us '> u, for some t. A feasible program is called Pareto- efficient if it is not Pareto inefficient.

This definition was first proposed in L a n e and M i t r a [5] and departs from the standard definition by requiring that initial consumption be the same for both of the feasible programs under con- sideration and all comparison programs. Without this constraint it is always possible to yield an improvement in welfare for the set of generations born after t = 0 by reducing the welfare of generation zero. It is appropriate to regard the 'state' of the system at t = l as requiring specification of both yl = [ (x0) and cl, for otherwise a sufficient specification of the 'history' at t = 1 has not been provided. Of course, programs should only be compared if they emanate from the same state (for a more complete discussion see [5]).


3. The Solution Concepts and the Role of the Differentiability Assumption

If the strategy space of a player is given by S (resp. S') the Nash equilibrium solution concept requires that for all t, and for all output levels y, 8t (y) maximizes t's utility subject to all other strategies being given by ( ~ (y)). Formally, a sequence of consumption schedules (st (y)) is a perfect Nash equilibrium, if for t_>l

u [et (y), e~+l (? (y-e~ (y)))] _> u [ct (y), et+l (f (y-ct)))] (5)

for all feasible ct (.) ~ S and all y e R +.

Note that the requirement that (5) should hold [or all y e R + is crucial as it ensures that decisions are always optimal and not just for the capital levels realized.

From an individual generation's point of view one may regard 8t+1 (') as generation t's conjecture about generation ( t+ l ) ' s behavior, relative to which t optimizes. Hence generation t faces the following optimization problem

(P) max U (c, et+l (f (y -c ) ) ) c ~ [0, y]

which it has to solve for all y > 0. Since the solution to this problem, for a given y, may not be unique (see L e i n i n g e r [6]) one is lead to consider the correspondence of maximizers, ~ ('18t+1), which is defined by

(yie~+l) = (c'lc' E [0, y] and c" maximizes U (c, gt+~ ([ (y -c ) ) )

over [0, y]}.

Any selection of r i .e . , any function ~(.) satisfying 8 ( y ) ~ ( y l S ~ + t ) for all y, is called an optimal reaction to ~t+t ('). Hence a perfect Nash equilibrium may be described as a sequence of consumption schedules ( ~ (y)) such that 8t (y) is an optimal reaction to 8t+i (y) for all t. Moreover, since (P), for each y, involves the maximization of a real-valued function over a real interval it is immediate that all the properties shown to hold for optimal reactions to unrestricted (and hence possibly non-differentiable) schedules Ct+l (') by L e i n i n g e r [6] must in particular hold for optimal reactions to differentiable schedules.

In L e i n i n g e r [6] existence of a solution to (5), when the strategy space is unrestricted, is demonstrated and the properties of gt (y) are also derived. It will be helpful to briefly restate these


properties. Our immediate concern is then to deduce certain additional properties of these equilibrium consumption schedules when strategy choice is restricted to differentiable functions. Existence of perfect equilibrium relative to our strategy space S (or - - equiva- lently - - existence of perfect equilibria with respect to the general strategy spaces used in [6] that are differentiable) is assumed.

If the strategy space is not restricted then the equilibrium consumption schedules are upper-semicontinuous, almost everywhere differentiable and satisfy the (upper) Lipshitz condition

(L) g (y+A) - g (y) </1 for all y_>0 and A1 >0.

This is the content of Theorem 3 in L e i n i n g e r [6]. Note that (L) is not stated in absolute values. It says that if consumption increases in y then it does so continuously and at most at the same rate as capital, but it can also decrease in y and this can occur in a discontinuous way. At points at which ge(') is differentiable (L) implies that the marginal propensity to consume is less than (or equal to) 1, but it could also be negative. The following diagram illustrates the possibility of such seemingly strange behavior.

Ct§ 1

c~.~ ,,.,. 45 ~ .. ,~5 ~ tP ,e f

./J / ~ St "- 1 / , / opffmol /

, / / cL, reaction'~

Yt.t Yt

Fig. 1

Assume that generation (t+ 1)'s schedule is given by ge+l (ye+l) as in Fig. 1 on the left. We claim that an optimal reaction for generation t to gt+l (') must look like the schedule ge (ye) shown in the diagram on the right. That schedule contains a linear segment of slope 1 followed by a 'downward' jump in reaction to the phe- nomenon that ct+l ( ') is not strictly increasing and hence prescribes over a certain range of capital levels ye+l lower consumption levels


than it does for some values of yt+i that lie below this range. The intuition is the following: If a player, who gets an additional amount of initial capital, realizes that a further increase in his bequest to the next generation leads to a reduction in consumption of that generation (and therefore to a decrease in the utility accruing to him from altruism) he would not make this further bequest, but consume all of his initial capital increment, i. e., the marginal propensity to consume would be unity over an interval. The argument is not affected if the initial decrease in consumption occurs in a continuous way as opposed to a 'jump'. But if the capital level yt has increased by so much that it becomes feasible for t to make a further large bequest that would provide generation (t+ 1) with a capital level yt+i beyond the "sink" in its schedule, then generation t can expect higher consumption by (t + 1) than before and is led to reduce its own consumption drastically. This behavior is caused by the existence of the "sink" in 8~+i ('), but is independent of the shape of 8t+i (') over the "sink". As L e i n i n g e r [6] shows (Prop- osition 5) 8t( ' ) would still be an optimal reaction if we replaced ~t+l (') by the schedule c~+1 (see Fig. 1) which is a 'levelled' version of 8t+i (i. e., identical to it except for the dotted line in Fig. 1). In consequence, the "sink" has reproduced itself in the form of a discontinuity and this discontinuity must reproduce itself in every schedule & (y), z _< t.

What is then the impact of our differentiability assumption? Obviously, it rules out the occurrence of discontinuities. What is more important to us is that it also rules out 'boundary' behavior of the marginal propensity to consume, which cannot be anymore negative or even assume the values 0 or 1. Figure 1 already indicates that boundary behavior of the marginal propensity goes along with the occurrence of discontinuities. To rule out boundary behavior is of importance for the following reason: The positive and in particular normative characterization of interior equilibrium programs (generated from linear schedules) in L a n e and M i t r a [5] makes extensive use of the fact that the marginal propensities of equilibrium schedules always lie (strictly) between 0 and 1. Observe that in the case of linear strategy spaces this (equilibrium) condition on the marginal propensity to consume is implicitly imposed by the feasibility condition 0 < g (y) _< y. In fact it also ensures the lower bound zero for g' (y). Thus, any interior schedule automatically satisfies the condition on marginal propensities. With non-linear strategies, however, this is not true. The restriction on the marginal propensity to consume can become binding even if the feasibility constraint is not (see s t ( ' ) in Fig. 1); the coincidence of the interiority restriction


with that on the marginal propensity to consume is lost. It is shown below that the differentiability assumption restores it.

The requirement that schedules be differentiable is certainly necessary for our purposes, for otherwise the marginal propensity may not be defined and one could not expect to find correct 'shadow' prices. But is it also sufficient? The answer is 'yes' since it also rules out boundary behavior of marginal propensities which implies that the first-order condition of the Nash problem:

U1 (ct (yt), gt+* ([ (xt))) = U2 (ct (yt), Ct+l ([ (xt))) 6t+1" ([ (xt))/" (xt) (6)

for t >_ 1 and for all yt > 0

always would hold with equality at interior points. One could therefore use it as in the linear case. To be precise, we show below that it is 'yes, almost'; i.e., generically the marginal condition is not binding at differentiable equilibria. Thus, whenever an initial capital level, xo, is given and a differentiable equilibrium sequence of schedules is evaluated to generate the corresponding program, it will 'never' (only with probability 0) happen that in this process one evaluates a schedule at a point at which its marginal propensity assumes boundary values (i. e., 0 or 1)1

Moreover, if boundary behavior does not occur for an xo, then it does not occur if x0 is changed a little; and if it should hold at an x0 (the probability of which is 0) then an arbitrarily small change in x0 would make it disappear. Taking into account that 'data' always contain some degree of inaccuracy one is led - - not only by mathematical but also economic reasoning - - to conclude that boundary behavior of the marginal propensity does not occur. It is this sound structural argument which allows us to derive 'shadow prices' from the first-order condition as in the linear case. In particular, these prices are well-defined and positive.

Theorem 1

Let (ce "~ (y)) be a perfect equilibrium sequence of consumption schedules. Then for all t, ce* (y) is stricdy increasing and c~ :~' (y) = 0 or 1 at most on a closed set of y's with measure 0.

eroo/ Let (ct'* (Y))S=0 be a perfect equilibrium. Then it follows from

assumption (W) i) that all schedules are interior schedules, i. e.,

1 That 0<c ~'' (y)_<l must hold in equilibrium was shown by Kohl- berg [3] for stationary equilibria under stronger assumptions.


0<ce* (y)<y for all y > 0 and for all t. We claim that ce* (y) is also strictly increasing for all t=0 , 1, . . . .

If not, there are to and y0 such that ct0*' (y0)<0 and this condition then must hold over an entire interval (yl, y2) around y0.

Let y denote a capital level such that Cto* (9) > ceo* (y) for all y E [0, y0]; clearly, y exists and c~0* (9) > ct0* (y0). We now distin- guish between two cases:

a) y2 is finite and ct0* (y) will increase again to values such that

ce0* (y) > ce0* (9) for y > y2.

This is the case of a "sink" alluded to in Fig. 1. But - - as Fig. 1 illustrates (and L e i n i n g e r [6] proves) - - the occurrence of such a sink is only compatible with a discontinuous optimal reaction to it. But ce-l* (.) is differentiable; hence a) cannot occur.

b) cto* (y)<-ceo* (9) for all y>y2; i .e. , ceo* is bounded by ce,* (9). This case is only compatible with ceo+l*-* oo as y--,oo, since otherwise any sufficiently large bequest by to is wasteful. In addition, Ceo+l* (y) -* oo strictly monotonically because otherwise a "sink" or "levelled" portion of ce0+l* (') would exist that would have to cause a discontinuity in Cto* (.). But ce0* (y) -<ct0* (9) < oo for all y implies that (y-Ceo* (y) )~oo as y-ooo and hence ye+l=[(y -c to*(y ) )~oo as y--~oo because of (F). Thus, ct0+l* (y )~oo as y ~ o o . But a bounded schedule (like c,0* (.)) cannot be optimal against such a ct0+f* (.) as lira U1 (ceo* (~),

ct+l*) >0 while lim U2 (ce0* (9), Ce+l*)=0 by assumption (W) C$ + 1~-->o0

part ii). In conclusion, this case cannot arise either and C,o* (.) must be increasing.

Since ce0* (.) also cannot contain any flat parts the possibility that c,0*' (y)=0 is limited to isolated points and hence ct0* (') must be strictly increasing.

For strictly increasing differentiable functions there is a (differentiable) one-to-one correspondence between critical values and critical points. Since by SaM's Theorem (see e.g. , M i l n o r [7]) the first form a null set (set of Lebesgue measure 0) the latter must likewise form a null set. Moreover, (by the same theorem) the set of regular points (i. e., points y at which ce*' (y) > 0) is everywhere dense in R. Thus, generically ct*' (y)= 0 is ruled out.

But then it is also generically ruled out that ce*'= 1, because if along a program (xt, yt, cD generated by equilibrium schedules,


ce=ct* (yt) is such that ce*" (yt)=1 then ye+l and ce+l* (y), to which ce* (y) is the optimal reaction, must be such that ce+f" (yt+l)=0. Only then can it be optimal for a player to consume all of an in- finitesimal increase in capital, because only then does an infinitesi- mal increase in the bequest not lead to an increase in the next generation's consumption. If ce+l*" (ye+l)>0, it is always better to increase the bequest marginally. []

To sum up, along equilibria boundary behavior of the implied marginal propensities never occurs.

4. The Dual Characterization of Nash Equilibria

In L a n e and M i t r a [5] Nash equilibrium programs were characterized in terms of supporting prices. In particular, Proposition 4 in [5] states that an interior program is a Nash program if and only if it is "quasi-competitive". However, the solution concept employed in [5] is weaker than the one used here, the term "equilibrium program" therefore has to be explained.

In [5] a program (~, ~, g) is called a Nash equilibrium program if its associated consumption ratio sequence (g) satisfies:

u [e,9,, e,+~ f [[1 -e t ] ~,]] > 8 [z 9,, e,+~ f [(1 - z ) 9,]]

for all t >_ 1 and 0 < z -< 1.

(7)

This implies that s is derived under the conjecture by generation t that generation (t + 1) wilt choose the same consumption ratio what- ever its inherited capital stock is. Clearly conjectures are not neces- sarily rational, in which case the equilibrium is not a perfect one as prescribed by (5) (the reason being that (7) is not required to hold for all y).

In L a n e and L e i n i n g e r [4] the relationship between the solution concepts (5) and (7) is discussed in detail. Attention is focused on determining environments for which the solution to (7) is equivalent to the solution to (5) and for which, in addition, the solution to (5) relative to S lies in S', the class of linear schedules. Only in this case solutions in [5] are "correct", i. e., equilibria are perfect.

We now provide a generalization of the price characterization in [5] for these (perfect) equilibria w. r. t. S" to (perfect) equilibria w. r. t. S. This also leads to a critical evaluation of equilibria in [5] and their dual price characterization, in case they are not perfect (i. e., do not satisfy (5)).


Given that the solution concepts (7) and (5) coincide on S', a crucial point in studying the effect on prices of the enlargement of the strategy space from S' to S is whether the sequence of consumption ratios (z,) associated with a program (x, y, c) has to be interpreted as the relevant marginal or average propensities to consume of the underlying schedules (which generate the program).

Definition

An interior program (2, ~, e), generated from the sequence of consumption schedules (g, (.)), is quasi-competitive if there is an associated price sequence (q, l~), such that q ,>0, and ~,>0, for t _ 0 and:

q, u,-i~t a*- [/5,+1/~t+1' (~t+l)] a,+l _> qt U (c, c') -i~* c -

-[lgt+#at+1 ' (~+1)] c', for c, c '>0 , t > l (8)

~+ly,+l-p~2t>_~+l[(x)-~x, for x>_0, t>_0. (9)

Observe that ~t+l/g,+l' (~,+1) is well defined as &~l' (yt+l)r (8) requires some comment:

Let (x, y, c) be any feasible program. Generation t must allocate output yt between c, and xt; these are variables under direct con- trol. To do this optimally it must anticipate a consumption schedule c~+i(.) for the t + l - t h generation since u~=U(c~,ct+l)= = U (c,, c,+1 (f (x,))). That is, the assumed marginal schedule ct+l'(. ) constitutes an externality, which has not been internalized in the preference structure.

If an optimal choice corresponds to that implied by the program (2, ~, ~) it follows that we must take prices as (q, i~) and consider the externality to be fixed at gt+l" (~+~) to have a proper reference point for comparisons. Therefore, the imputed value of generation t's inherited output is /5~ y~ =/3t c~ +i~+i y~+l. Its choice of ct, and so implicitly y~+l, must be over all feasible (ct, y~+l) such that this imputed value remains constant, i .e. , ~tdct+ +~t+l dy~+l = 0 = I~ d ct+ [~5~+~/8~+~' (y~+l)] dc~+~. Because the marginal rate of transformation between c~+1 and y~+~ is 8~+x' (y~+~) this indicates that, along the program (2, y, g), l~+~ is the futures price of a unit of output at t + 1 and/5~+1/&+~' (y~+l) is the futures price of a unit of consumption at t + 1.

It now follows that condition (8) is to be interpreted as "utility maximization subject to a budget constraint" in the presence of an externality which has not been internalized. Condition (9) constitutes "intertemporal profit maximization" and there is no external-


ity present here. For these reasons we call the above conditions quasi-competitive as in L a n e and M i t r a [5].

But note, that in our model a given program alone cannot de- termine futures consumption prices; information (at least locally) about the generating schedules is needed. The same is true in the analysis of Lane and Mitra, but this is somewhat blurred since the information needed is self-evident from the implied linearity of the generating schedules. There, a program determines consumption ratios, ~t, which, if identified with the schedules ct (y)=~ty, determines futures consumption prices as /St+l/~t+l. But then ~t+l has to be interpreted as the marginal propensity to consume rather than the average. The replacement of the budget constraint iOt~t = iOtct +iFt+l yt+l by Pt ~t = ~tct + (~t+x/gt+l) ct+l certainly is a mathematically correct substitution using ct+l = gt+l yt+~. But it is economically only correct to interpret l~t+~/~t+l as the shadow price of future consumption if ~t+~ is identical to the marginal propensity to consume. This is only true if the solution sequence (gt) of (7) is such that (gty) is a solution to (5). If not, the definition of quasi-competitiveness as in [5] implies that players hold a non- rational theory of the formation of future prices.

From (6) we know that the marginal rate of substitution between yt+x and ct, given ct+l (.), is U1 (ct, ct+l)/U2 (ct, ct+l) ct+l" (f (xt) ). The marginal rate of transformation is [" (xt). If (6) is satisfied, i. e., the program is a Nash equilibrium, then these two rates are equal; but this suggests that (8) and (9) are satisfied. The common slope of the indifference curve and the production possibility curve provide the price ratio implicit in (8) and (9). Does the reverse im- plication hold ? One would not expect so. Our definition of a quasi- competitive program requires that conditions (8) and (9) hold, at time t, for the ~t given in (~, Y, ~}. It does not require that it holds for all ~t. Condition (6), however, must remain valid for all st. The obvious way to turn the local supporting hyper-plane condition (8) into a global one is to assume that ut=U(ct , ct+l ([(yt-ct))) be concave in ct for any given yr. A sufficient condition for this is that the consumption schedule ct+l (.) be concave in yt+l. There is no problem with condition (9) since [ (x) is assumed to be concave.

Note that, at any stage t = to, the full game contains a continuum of subgames: each initial capital level at t =to, yt0, gives rise to a new subgame. Hence a strategy for player to specifies an action for each subgame that possibly can arise at t=to (and not just those that may arise given the previous players' strategies). As, in addition, all perfect equilibrium strategies have to be interior schedules because of (W) we can prove:


Theorem 2: Under (F) and (W):

(i) If (ee (y)) is a perfect Nash equilibrium, then all interior programs generated from it (in any subgame) are quasi-competitive.

(ii) If all interior programs (2, Y, c) generated from (& (y)) (in any subgame) are quasi-competitive, and each schedule & (y) is strictly increasing and concave, then (& (y)) is a perfect Nash equilibrium.

Proo[ (i) Since any subgame from t = to onwards is identical to the orig-

inal game, it is enough to consider programs generated by (ce (y)) in the original game. Let (2, Y, ~5 be any program generated from the Nash equilibrium sequence (& (y))~=l. To find a price sequence satisfying (8) and (9), we will simply define (q, if), and check that, under the definition conditions (8) and (9) are indeed satisfied. Define:

t50 = 1, i~+l = pe/[" (~e) for t >__ 0, (10)

qt = pt/U1 (ct, 8~+1) for t > 1. (11)

Note, then, that qe>0 and ~e>O, for t>0 , since (2, Y, e) is interior, and [', U1 > 0. To check that (9) holds, use the concavity of [, and write, for any x>_O, [ ( x ) - [ (~) _<[' (~e) (x-~t) . Multiply through by/5~+1 and use (11) to get ~e+l ([ (x)-[(~2t))<~e ( x - ~ ) . Then (9)is obtained by transposition of the relevant terms. To check that (8) holds, use the concavity of U, and write, for any c, c' > 0:

U (c, 6') - U (e$, et+l) ~-- U1 (el~, e$+l) (c - e$) -~ Ug. (et, e$+l) (c' - et+l) .

Multiply this by qe as defined by (11) to get

qe [U (c, c') - U (at, et+l)] -<l~ (c -a t ) +

+ [U2 (ct, C~+l)/U1 (et, et+l)] fit ( 6 ' - C t + l ) .

Since the given sequence of schedules is a Nash equilibrium we can use (6) to get Uz (&, &+i)/Ui (&, &+I) =l/[&+i" (~+i). �9 [' (2t)] for all t >_ 1. After substitution and use of (11) we finally get for all c, c" and t >_ 1:

q~ [u (c, c') - u (as, e~+~)] _< ~, ( c - e,) + [p~+~/e,+l' (~+1}] ( c ' - ~,+1)

which after re-arrangement of terms yields (8).


(ii) We prove the second statement by showing that if (& (y)) is not a Nash equilibrium then there exists a program generated by it which is not quasi-competitive. If (ct (y)) is not a Nash equilibrium then for some t=to there must exist yt ~ and ct (-) ~ S such that

(*) at = U [et (yt~ et+l (yt+l)] < 8 [c~ (yt~ et+, (yt+i)] = U (c, c')

where y,t+i = f [y t ~ -- c t (yt0)] and yt+l = f [yt ~ - ct (yt~ By assumption (& (y)) generates a quasi-competitive program for every xto-i in the subgame from t = to onwards; in particular, the program generated from 2to-1 = f - i (ytO) is quasi-competitive, i. e., by (8)

pt+l qtt/t-i~t& (yt ~ et+i' (yt+l) &+l (~t+i) >-[ItU (c, c') - ~ t c t (yt ~ -

pg+l e~+~' (y~+~) &+l (yt+l)

which is equivalent to

(**) /St ct (yt ~ -/St & (yt ~ > (/t [U (c, c') -Ut ] +

+ e,+~T~'t+l) [et+l (yt+l)--et+l (yt+i)].

The first square bracketed term in (**) is positive by C:) while the second one is bounded from below by &+i' (yt+i) [yt+i-yt+l] since &+i (-) is strictly increasing and concave by assumption. Hence,

i~tct (yt ~ - ~tSt (yt ~ > i~t+i y t + i - i0t+i yt+i.

Add p t y t ~ to both sides and use ~ t = y t ~ ~ resp. xt = yt 0 - ct (yt ~ to obtain

~t+i yt+l - / ~ t x t < ~t+l yt+l - ~txt.

This contradicts (9). []

If utility is additively separable, as is commonly assumed, then the prices qt would depend only on contemporaneous information. But this tends to obscure the fact that, just as the state variable xt must summarize the past history of the system in so far as generation t needs to know it, so must the price of utility (qt) and futures consumption [pt+x/ct+x" (yt+l)] summarize the future in exactly the same sense. In a model in which altruism extends forward one period these prices should be expected to depend on information


available one period forward; the future is not separable from the present. We note that the prices <q, p> associated with a quasi- competitive program <x, y, c>, in view of the differentiability assumptions on [ and U, are uniquely determined up to a positive scalar multiple by (10) and (11).

5. Pareto Efficiency

The main importance of Theorem 2 is the economic characterization of Nash equilibria that it provides. This is now exploited to demonstrate when any feasible program <x, y, c), generated from the Nash equilibrium sequence <c, (y)), is Pareto efficient (as defined in Section 2.4). The first result provides a sufficient transversality condition for Pareto efficiency.

Proposition 1. If <x, y, c> is any feasible program generated from the perfect equilibrium <ct (y)>, and if

inf ptxt = 0 t>0

then it is Pareto efficient.

Proof Using the results of Sections 3 and 4 and the 'levelling' - - result

from [6] the proof follows the technique used for Proposition 5 in [5].

Suppose (x, y, c> is a program generated by the perfect equilibrium sequence of schedules (ct (y)). Assume it is Pareto inefficient. Then there is a feasible program <x', y', c'> s.t . c1"=cl, ue '>m for t >_ 1 and ut' >ue for some t. Let ~ be the first period for which ut' >ut. Then (xt, ce)=(xe', ce') for 1 _<t_<3, and x,+l"< < xT+~, cr+l' < cr+,. Let ur' - ur = el > 0 by assumption.

Now, note that for t_> 1, we have:

�9 ~t+l p~ (c~ -ct) + c,+1' (y,+1) (ct+l"-c,+l) >_0

as <x, y, c> is quasi-competitive by Theorem 2 (i) and also Pareto inefficient. Therefore (by (1) and (2)):

p, [[ (m- , ' ) - [ (x~-l)- (x,'-xt)] +

p t + i , > + c,+,' (y,+l)[[ (x t ' ) - [ (x , ) - (x~+ , -x ,+ , ) ] _ 0 for t > l .


Define dt=pt (x t -x t ' ) , use the concavity of [ and the definition of pt to confirm that the first term in the above expression is no greater than dt-de-1. Similarly

pe+l [[ ( x ( ) - [ (xe)- (xt+l '-xe+l)] <de+l-de .

From Theorem 1 we know that ct+l" (yt+l) >0 (w. 1. o. g.) so that de - de-1 + [1/ce+l" (yt+l)] (dt+l - de) > O. Consequently,

(*) dt+l >_ de (1 - ce+l" (ye+l) ) + dt- i ct+l" (yt+l).

As noted earlier d r = 0 while d r + l > 0 so d~+z>dr+z(1-cr+2"" �9 (y~+2)) =e2, say. Therefore:

d~+3>_d~+2 ( 1 - c~+3' (y~+3))+d~+l c~+8" (y~+~)

_>e2 (l--Or+3' (yr+3))-]-d~r+l 6"r+3' (yl:+3)

>e2 ( 1 - c~+3' (yr+3)) +e2 c~+~' (yr+3)=e2

(as c~+~" (yr+2) > 0 and therefore dr+l > e2). Repeated application of (*) and the fact that 0 < c((yt) < 1 for

all t now yield dt>ez for all t_>T+3. Consequendy, ptxe>_e2 for t > r +3. But e2 is bounded away from zero because cr+2" (yr+2)< 1. Therefore

inf pt xe > O. t_>O

This completes the proof�9 []

Although the transversality condition in Proposition 1 is a strong one, it will be shown to hold for programs generated from perfect equilibrium schedules if preferences are separable.

In Proposition 2 we study the asymptotic value of the sequence of capital stocks along an equilibrium. To do so we make the assumption that utility is additively separable (with a discount factor b strictly less than one).

Proposition 2

Given ue=v (ct) +by (ct+l), b <1, any feasible program (x, y, c) generated from a perfect equilibrium (ct (y)) satisfies, for all t sufficiently large, xe < x* (where x* is the largest x satisfying [' (x) = 1)

25

i .e . , lim xt<x*. t - + oo

Journal of Economics, Vol. 46, No. 4


Proof Suppose the claim is not true. Then there exists a subsequence

of {x~}, which we can take for the sequence itself, such that xt > x* for all t. Therefore [' (xt) < 1 for all t and the first-order condition (6) yields:

(*) bct+l" ([ (x~)) >_v" (ct)/v' (ct+l ([ (xt)) for all t.

However, b ct+l' ([ (xt)) < 1 as b < 1 and ct+l' ([ (xt)) < 1. Therefore:

v" (c~) <v" (ct+l ([ (xt))) or c~>ct+l ([ (x~)) for all t.

As xt >_ x* > 0 clearly c~ > 0, otherwise the equilibrium condition is violated. Thus {ct) is monotonically decreasing and bounded from below. Therefore {ct) converges to 8, say, from above.

It is easy to see that ~ > 0 is impossible: For then the right-hand side of (*) approaches 1 from below, whereas the left-hand side is less than b (< 1) for all t.

But ~=0 cannot hold either:

0 = lim ct = lim ([ (xt-1)-xe) (by (4)) t---~ 00 S--a- 00

implies that xt-o2 (and therefore yt-*2) for t -00% where 2 > 0 is such that [ ( 2 ) = 2 (that 2 exists is implied by (F)); i .e . , lim c~ (2)= 0 and ~ > 0. t--0- o0

The last statement, however, implies (by the monotonicity of equilibrium schedules proved in Theorem 1), that the schedules c~ (y) converge uni[ormly to the schedule c* (y) =0 on the interval [0, 2] (the convergence behavior on (2, oo) does not concern us here). This means, by Theorem 2 in L e i n i n g e r [6], that c* ( y ) - 0 is the optimal reaction to c* (y)----0 (i. e., itself) at least on the interval [0, [-1 (2)] i. e., the limiting equilibrium sequence (Theo- rem 2 in [6]) contains completely flat parts. But such an equilibrium sequence contradicts Theorem 1. []

Proposition 2 immediately implies

Proposition 3

Arty program (x, y, c) generated from a perfect Nash equilibrium (ct (y)) is efficient.


Proo[ A well-known result by M i t r a [7, Corollary 1] that generalizes

a method proposed by Cass [1] states that if a program (in our model) is inefficient, then inf p tx t>0 . However, Proposition 2 is

t > 0

now shown to imply that inf ptxt =0. t > 0

As x~ <x* for t > T say, {pt} is decreasing for t > T and therefore must converge.

We claim that {pt} converges to zero. If not, it converges to a positive number. But then [' (xt) converges to one, by the definition of prices, and so xt converges to x*. Then yt converges to f (x*). Therefore ct converges to c* = [ (x*) - x* > O, so, by continuity of v" and [', the first-order condition yields lim bc~+l" (yt+l)=l . But

t---~ oo

b < l and ct+l' (yt+l) <1 so bc~+l" (yt+l) _<l -e , e>0.

This contradiction establishes that {pt} converges to zero and, as {xt} is bounded above, (ptxt} also converges to zero. Hence (x, y, c) must be efficient. []

These results are remarkable: By Theorem 2, the perfect Nash equilibrium property of a sequence (ct (y)) is in general stronger than the requirement, that all schedules generated from a sequence (ct (y)) be quasi-competitive. We now see, that this slightly stronger requirement not only provides us with the usual 'competitive' prices for programs generated by (ce (y)), but also with prices whose behavior over time is such that we can infer (long-run) efficiency of the programs generated from them. The well-known fact that competitive prices (as defined by (10)) may lead to long-run inefficiency was attributed to the fact "that there's neither a market signal nor a market adjustment mechanism" associated with a condition like inf pext>0 (Cass [1], p. 220). However, such forces are present t-,~o in a perfect Nash equilibrium as defined by (5) and characterized by Theorem 2. Intuitively, the conflicting interests of successive generations work against the possibility of capital overaccumulation (Proposition 2). Perfectness ensures that interests are correctly perceived and this fact frees us m in contrast to L a n e and M i t r a [5] - - from the need to further parametrize the model to get this result. Since the proof of Proposition 3 reveals that for any program as described in Proposition 2 the transversality condition of Propo- sition 1 holds we can immediately conclude that it is Pareto-efficient (as defined in Section 2.4).

25*


Theorem 3

Given u~=v (ct)+by (ct+l), b <1, a program (x, y, c) generated from a perfect Nash equilibrium (c~ (y)) is Pareto-efficient.

6. Condusions

The analysis in this paper is crucially dependent on the marginal propensities only taking interior values.

In an earlier paper K o h l b e r g [3] showed that the marginal propensities were positive. But his solution concept required not only that schedules be differentiable but also that equilibria are stationary. It was stationarity that provided the result which was then used to study the implications for positive behavior of the economy.

We do not assume stationarity, Also we focus attention on normative results. Similar results to ours have been obtained by R a y and B e r n h e i m [11]. They do not restrict the strategy space to differentiable schedules but only to upper semi-continuous schedules. Therefore existence of their solution is ensured (see L e i n i n g e r [6] and R a y and B e r n h e i m [11]) providing no further ad hoc restrictions are placed on the solution. However, Bernheim-Ray are forced to assume that the programs generated [rom the solution sequence of schedules never require evaluation of a schedule at a point where the marginal propensity to consume is one. The only known case where this happens, apart from that discussed in this paper, is when schedules are linear; but then, as argued earlier, the Lane-Mitra analysis applies. Our approach greatly simplifies the mathematical analysis and also makes the economic content of the result easier to perceive.

References

[1] D. Cass (1972): On Capital Overaccumulation in the Aggregative, Neoclassical Model of Economic Growth: A Complete Characterization, Journal of Economic Theory 4, pp. 200--223.

[2] P. Dasgupta (1974): On Some Alternative Criteria for Justice between Generations, Journal of Public Economics 3, pp. 405---423.

[3] E. Kohlberg (1976): A Model of Economic Growth with Altruism between Generations, Journal of Economic Theory 13, pp. 1--13.

[4] J. Lane and W. Leininger (1984): Differentiable Nash Equilibria in Altruistic Economies, Zeitschrift fiir NationalSkonomie / Journal of Economics 44, pp. 329--347.


[5] J. Lane and T. Mitra (1981): On Nash Equilibrium Programs of Capital Accumulation under Altruistic Preferences, International Eco- nomic Review 22, pp. 309--331.

[6] W. Leininger (1986): The Existence of Nash Equilibria in a Model of Growth with Altruism between Generations, Review of Eco- nomic Studies 55, pp. 349--367.

[7] J. W. Milnor (1965): Topology from the Differentiable Viewpoint, University Press of Virginia.

[8] T. Mitra (1979): Identifying Inefficiency in Smooth Aggregative Models of Economic Growth: A Unifying Criterion, Journal of Mathe- matical Economics 6, pp. 85--111.

[9] B. Peleg and M. Yaari (1973): On the Existence of a Consistent Course of Action when Tastes are Changing, Review of Economic Studies 40, pp. 391 401.

[10] E. Phelps and R. Pol lak (1968): On Second Best National Sav- ings and Game Equilibrium Growth, Review of Economic Studies 35, pp. 185--199.

[11] D. Ray and B.D. Bernheim (1983): Altruistic Growth Eco- nomics, IMSSS (419), Technical Report, Stanford University.

Addresses of authors: John Lane, Ph.D., Lecturer in Economics, London School of Economics, London, U. K.; Ass. Wolfgang Leininger, Ph. D., Department of Economics, University of Bonn, Bonn, Federal Republic of Germany.

on price characterization and pareto-efficiency of game equilibrium growth

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