Markov Perfect Equilibria in Altruistic Growth Economies with Production Uncertainty*
B . Do u g l a s Be r n h e i m
Department o f Economics, Stanford University, Stanford
,
California 94305A N D
De b r a j Ra y
Indian Statistical Institute, Delhi Center, 7, S.J.S. Sansanwaal Marg, New Delhi 110 016, India
Received O ctober 21, 1986; revised April 13, 1988
This paper concerns the existence of M arkov perfect equilibria in altruistic growth economies. Previous work on deterministic models has established existence only under extremely restrictive conditions. We show that the introduction of production uncertainly yields an existence theorem for aggregative infinite horizon models with very general forms of altruism. Journal o f Economic Literature Classification Num bers: 022, 026, 111.
1. In t r o d u c t i o n
An altruistic grow th econom y consists of a sequence (possibly finite) of generations and p ro d u ctio n technologies. Each generation derives utility fro m its ow n consum ption and the consum ptions of some or all of its d esc e n d a n ts.1 As we have discussed elsewhere (Bernheim an d Ray [4 ] ) , th is fram ew ork is of wide applicability.
* We are grateful for helpful conversations with Peter H am mond, C hristopher Harris, and Ja m es Mirrlees, and to an anonymous referee for useful suggestions. An earlier version of this p a p e r was circulated as Technical Report N o. 467, Institute for M athem atical Studies in the S ocial Sciences, Stanford University (June 1985). Ray thanks the D epartm ent of Economics, S tanford University, his affiliation when the first draft of this paper was written. This research w a s supported by N ational Science Foundation G ran t SES-84-04164 at the Institute for M athem atical Studies in the Social Sciences, Stanford University, Stanford, Ca..
1 Alternatively, each generation might derive utility from its own consum ption and the utilities of its descendants. This “non-paternalistic” formulation raises different issues, but we d o not consider them here (see, e.g., Pearce [1 7 ], Ray [1 9 ], Streufert [20]).
A central concept describing intertem poral behaviour for such a n econom y is th a t of M arkov perfect equilibrium .2 In such an equilibrium , each generation chooses consum ption optim ally, given know ledge of its ow n endow m ent and the endow m ent-dependent behaviour of its des
cendants. This is true of all possible endow m ents, and for every generation.
Since M arkov equilibria are so simple, they m ay be m ore likely th an com plex equilibria to arise in practice, an d their p roperties are certainly m o re am enable to study (see, e.g., Bernheim an d R ay [ 6 ] ) . In addition, M a rk o v equilibria will u ndoubtably tu rn o u t to be very useful in studying th e properties of m ore com plex equilibria.3
The n atural and basic question is: d o M ark o v equilibria exist in a reasonably wide class of altruistic grow th econom ies? This issue rem ained unresolved (see, e.g., Peleg and Y aari [1 8 ]; K ohlberg [ 1 3 ] ) u n til Bernheim an d Ray [ 4 ] an d Leininger [1 5 ] obtain ed independent affirm ative results for an aggregative (one-com m odity) m odel displaying lim ited altruism . Altruism is lim ited in their m odels in the sense th a t each genera
tion derives utility only from its ow n consum ption and the consum ption o f its immediate successor.4
This existence result is useful b u t restrictive. In particular, it is im p o rta n t to study whether the result can be extended to (a ) a disaggregated m u lti
com m odity m odel, and (b) m ore general and far reaching form s o f altruism . Regarding (a), recent interesting w ork by H arris [1 2 ] em ploys techniques similar to th a t in Bernheim an d Ray [ 4 ] to prove a M a rk o v existence result in a m any-com m odity fram ew ork.5 B ut (b) is a to u g h e r n u t to crack. In fact, Peleg and Y aari [1 8 ] co n stru ct a finite h o riz o n counterexam ple, showing the difficulty of ob tain in g a general result.6
In Bernheim and Ray [ 6 ] , we showed th a t the presence of u n c e rta in ty (em bodied naturally in the production technology) paves the way for a very general M arkov existence theorem in finite horizon models. U n fo r-
2 See, e.g., D asgupta [ 8 ] , Kohlberg [1 3 ], Leininger [1 5 ], and Bernheim and Ray [ 5 ] , a n d in the non-patemalistic context, Loury [1 6 ], Streufert [2 0 ], and Ray [1 9 ].
3 There is an analogy here with repeated games, where history dependent strategies incorporate one-shot “punishments” in order to sustain “collusive” outcomes.
4 Such limited altruism models have been explored in a variety o f contexts. See, e.g., A rrow [ 2 ] , D asgupta [8 ], Barro [ 3 ] , K ohlberg [1 3 ], Loury [1 6 ], and Lane and M itra [1 4 ].
5 However, even in a stationary model, Harris [1 2 ] fails to establish the existence o f a stationary equilibrium. This remains an interesting (and difficult) open question.
6 Two points are relevant here. First, m ore general history dependent equilibria will still exist, as G oldm an [9 ] shows for the finite horizon case and H arris [1 1 ] dem onstrates for th e infinite horizon model. But M arkov equilibria still dem and o u r attention, as we have argued elsewhere (Bernheim and Ray [5 ]). Second, it is of some interest th at similar problem s do n o t arise in a non-paternalistic framework and M arkov equilibria can be shown to exist (Ray [19]).
tunately, the techniques used in th a t p ap er are n o t well suited for the infinite h o rizo n problem .
The p u rp o se of this note is to d em o n strate th a t the introduction of uncertainty also yields an existence theorem for statio n ary M arkov equi
libria in aggregative infinite horizon m odels w ith very general forms of altruism . The u n certainty is used to show th a t the best response of a generation, given its descendants’ strategies, exists. O u r p ro o f depends critically o n the fact th a t each g en eratio n ’s equilibrium investm ent is a non-decreasing function o f its endow m ent. As in R ay [1 9 ], this allows m onotone savings functions to be identified w th distributions of probability m easures, an d endow ed with the topology o f weak convergence. In Bernheim an d R ay [ 6 ] , we have show n th a t this “m onotonicity” property has stro n g im plications for the positive an d norm ative features of equilibrium pro g ram s for a related m odel. U nfortunately, m onotonicity of policy functions depends bo th u p o n the existence o f an aggregate good, and on a separability assum pton for preferences. Therefore, the infinite horizon result is m ore lim ited th a n its finite horizon counterpart.
We discuss the m odel an d its assum ptions in Section 2. Section 3 states and proves the m ain theorem .
2. T h e M o d e l
C onsider a n infinite sequence o f generations labelled / = 0 ,1 ,2 , etc.
There is o n e com m odity, w hich m ay be consum ed o r invested. In each time period, decisions concerning p ro d u ctio n an d consum ption are m ade by a fresh generation. T hus, generation t is endow ed w ith som e initial o u tp u t, y , k 0, w hich it divides betw een consum ption, c, ^ 0, and investm ent, x , § 0 (yt = c, + x t). The retu rn to this investm ent form s the endow m ent of the succeeding generation.
The well-being o f each generation will be determ ined by the sequence of consum ption choices. Specifically, we assum e th a t generation f’s preferences can be represented by a utility function £ /,:R + -> IR , satisfying the following assum ptions.
( U .l) t / , ( < O f =0) = «(<•,) + ”(<•,+ !, Ct + 2, •••)•
(U .2 ) v is con tin u o u s in the p ro d u c t topology on real valued sequences.
(U .3) u is strictly concave in c,.
Rem arks, (i) N o te th ro u g h o u t th a t the m odel considered here is stationary. T he techniques used can be ad a p te d to d em o n strate the exist
ence of non-stationary M arkov-perfect equilibrium for n on-stationary environm ents, at the expense of additional notation.
(ii) Implicitly, we assum e th a t each g eneration’s well-being is independent of its ancestors’ choices. Trivially, this assum ption could be weakened to require separability between ancestors’ choices, current choice, and descendants’ choices. F u rth e r w eakening o f the assum ption is clearly impossible: if ancestors’ choices affect the cu rren t generation’s ordinal preferences over descendants’ choices, the use of M ark o v policy functions will, in general, be suboptim al.
The investment chosen by each generation determ ines the endow m ent of its successor up to a random disturbance, a>,, which is realized from the state space [0, 1], Specifically, the p ro d u ctio n function, / : R + x [0, 1] -►
R + , and disturbances co, satisfy the following assum ptions.
(F .l) / i s strictly increasing and continuous in b o th x an d co,.
(F.2) There exists y such th a t for all co,e [ 0 , 1 ] an d x > y , f ( x , c o , ) < x .
(F.3) co=<a>() “ is an i.i.d. sequence of ran d o m variables. The distribution of to is given by an atom less probability m easure r\ on the class of Borel sets in [0, 1]. Let /i denote the p ro d u ct m easure t j ' (see H alm os [10, p. 157]).
Remarks, (i) U n d er (F.2), if y 0 ^ y, then for all feasible program s y , ^ y . O n this basis, we restrict atten tio n to endow m ents in [0, j ] . 7 It is possible to relax assum ption F.2 by using a tru n catio n argum ent (see Bernheim and Ray [4 ]).
(ii) It is relatively straightforw ard to relax the assum ption th a t the co, are i.i.d. However, some subset of p ast realizations will then affect expecta
tions concerning future realizations. Thus, one w ould have to allow strategies to depend on the history of past innovations, as well as current endowm ents. Strictly speaking, the equilibrium strategies would th en not be M arkov. It would not, however, be necessary to allow conditioning of strategies on past actions, independent of their effects on current endow m ents, in order to obtain an existence result.
A M arkov strategy (for any generation) is a function s: [0, y~\ -> [0, v]
such th a t for all j e [ 0 , j ] , 0 ^ s ( y ) ^ y . Let S ° denote the set of conceivable M arkov strategies.
W e will focus atten tio n on stationary equilibria. T hus, we wish to describe the evolution of decisions when all generations select the same M arkov strategy, s. The following recursion determ ines the evolution
7 Or, if y in general, to all feasible program s with v, ^ m a x { y , y } .
of capital stocks, given a>, s, an som e initial investm ent choice x for generation 0
c70(x, co; s) = x
a ,(x ,c o ;s ) = s ( f( ( T ,_ i( x ,c o ; s ) ,a ) ,) ), t = 1 ,2 ,.... (1) This, in tu rn , determ ines the evolution o f co n su m p tio n decisions
V,(x, «>; s) =f(<rr^ i ( x , co; s), co,) - a,(x, co; s), t = 1, 2,....
Let a = (<Tj, a2,...) and y — (y ,, y2, ...), and define V(x, co; s) = v(y(x, co; s)).
The strategy s e S ° constitutes a (stationary M arkov perfect) equilibrium if for each y e [0, y ] , solves
m ax u (y — x ) + E m V(x, a>;s). (2)
O^. x < y
R em ark. N o te th a t by o u r con tin u ity assum ption (U .2) and the com pactness of feasible program s in the p ro d u ct topology, the expectation in (2) is alw ays well defined provided V is m easurable.
3. Ex i s t e n c e
We now state o u r central result.
Th e o r e m. Under the sta ted assumptions, there exists a stationary
M arkov-perfect equilibrium. It is always the case that the equilibrium policy function, s, is non-decreasing, and m ay be chosen to be upper semicontinuous.
The general line o f p ro o f used below is sim ilar to th a t of Ray [1 9 ], and some specific steps are closely related to argum ents therein. We have noted these steps th ro u g h o u t, generally leaving them to the reader, who m ay wish to consult B ernheim and R ay [ 7 ] for com plete details.
Proof. O u r first key lem m a establishes th a t best response policy functions are always non-decreasing. T he p ro o f is identical to th a t of Theorem B in R ay [1 9 ]; we therefore om it it.
Le m m a 1. F ix s e S ° . Suppose that fo r y e { y 1, y 2}, y ' e [0,_p] ( i= 1, 2), y ] > y 2, problem (2) is well defined. Further, suppose x l and x 2 are corre
sponding solutions. Then x l 2: x 2.
H enceforth, we will restrict attention to non-decreasing, upper semicon- tin u o u s (use) functions. Let S ^ S ° denote the set of such functions. O u r next tw o lem m as establish th a t w hen future generations select s e S, then problem (2) is well defined.
L e m m a 2. Suppose s e S . For each x e [ 0 , y~\, and 12 : 0 , o ,(x ,c o ;s ) is continuous in co almost everywhere.
Proof. By induction. Suppose th a t a,_ j does n o t depend up o n (cot , a>l+ l, ...), and th a t <x,_, is continuous in co alm ost everywhere (this holds for / = 1). N o t then th a t a, does n o t depend upon (co, + i ,co, + 2, ...) (inspect (1)). D enote the set of discontinuities of a , _ x by x. Since s is non-decreasing, it has a t m ost a countable num ber o f discontinuities on [ 0 ,y ] ; call them ( d u d2, ...>. Let D ',= { c o \f( o ,_ l (x ,c o ;s),co ,) = d i}. a, is discontinuous at co only if c o e D ,_ u or coeD ', for som e i. S in c e /is strictly increasing in co,, and since a , _ l does n o t depend on co,, every (coT)T#,- section of D \ consists of a single point, and therefore has m easure zero.
Thus, D', has measue zero (see H alm os [10, p. 147]). Since D , is contained in the union of a countable num ber of sets of m easure zero, it has m easure
zero. This completes the induction step. Q .E.D .
Tw o corollaries follow immediately:
Co r o l l a r y 2 .1 . Suppose s e S . For each x e [ 0 , j ] , a (x ,c o ;s), y(x, co; 5), and V(x, co; .s) are continuous in co alm ost everywhere.
Co r o l l a r y 2 .2 . Suppose s e S. For each x ° e [ 0 , j ] , define D (x ° ) = { c o ° e [ 0 , 1 ] ° ° | V(x, c o ° ;s) is discontinuous in x at x °}. For all * °e [ 0 ,y~\, D (x °) has measure zero.
Le m m a 3. Suppose s e S. Then fo r all x, y, 0 ^ x < y ^ y,
is well defined, and continuous in (y , x).
Proof. The first term is continuous in (y , x). V is simply the com posi
tion of m easurable functions, and is therefore m easurable. Since v is b o u n ded on the space of all feasible program s (see (U .2)), the expectation is well defined. T o show continuity for the second term , take some sequence
u ( y - x ) + E w V(x, co; s)
x n -> x. Then
By C orollary 2.2, V(x", to; s) ->■ V (x, co; .v) alm ost everywhere. F urther, since [ 0, y ] K, which is co m p act in the p ro d u ct topology, and since v is continuous, V is bounded. A pplying Lebesgue’s dom inated convergence
theorem establishes continuity. Q.E.D.
Due to difficulties involving the b eh av io u r o f policy functions a t y, it is convenient to w ork w ith quasi-equilibria, defined as follows. An s e S is a quasi-equilibrium if for each y e [0, y ), s ( j') solves (2). Let 5 c S consist of the functions s e S such th a t s ( y ) = y. As in R ay [1 9 ], S can be th o u g h t of as the set o f d istrib u tio n functions o n [ 0, j ] (w here probability is rescaled).
O ur next lem m a indicates th a t if we sta rt w ith som e elem ent o f S, maxi
mization for each y e [ 0, _p] generates a unique “quasi-best” response in S.
Lemma 4. For each s e S, there is a unique fu n ctio n s' = H( s) such that s' e S, and f o r all y e [ 0, _p), s'(j>) solves problem (2 ).
Proof. Let h ( y ) be the correspondence which m aps to solutions of (2).
By Lem m a 4 an d the m axim um theorem , h is u p p er hem icontinuous. Let .v'fy) = m ax {/;(>>)} for y e [0, _p), an d s ' ( y ) = y. Clearly, s' e S. N ow suppose there is an o th e r u.s.c. selection from h(y), s" e S . F o r som e y , s'(,y)>.s"(j>).
Since s" is u.s.c. there is some y > y w ith 5'( j> ) > 5"(^). B ut this contradicts
Lemma 1. Q.E.D.
L em m a 4 defines a m apping, H: S -» S. A fixed p o in t o f this m apping is a quasi-equilibrium . W e need to establish continuity of H . T he key step is to prove th a t E w V is continuous in s.
Lemma 5. Suppose som e sequence <s">o° 5 converges to s e S. Then fo r each x e [0, y ] , E a V(x, co; s n) -> E m V (x, co; s).
P ro o f C hoose any co a t w hich a is continuous in a>. Suppose a ,_ ,(x, co; s") -* a,__, (jc, co; 5) (this holds for t = 1). By a s s u m p tio n ,/is con
tinuous. F u rth e r, since a ,(x , co; s) is con tin u o u s in co, a t co, and since / i s increasing in co,, s m ust be co n tin u o u s a t / (<r,_ ,(x, d>;s), co(). T hus, using (1), ct,(x, co; s") - »a ,(x , co;5). By induction, this holds for all t. Since / i s continuous, y(x, co; j ”) ^ y(x, co; j ) in the p ro d u ct topology. By (U.2),
V(x, d>; s ”) -» V (x, <b; .?).
By C o ro llary 2.1, a is con tin u o u s in co alm ost everywhere. Thus, by the peceding arg u m en t, V (x, co; s n) -> V (x, co; s) alm ost everywhere. C om bining this w ith th e boundedness o f V (see the p ro o f o f L em m a 3) an d Lebesgue’s d o m inated convergence theorem yields the desired result. Q.E.D.
G iven L em m a 5, one proves con tin u ity of the m apping H in a m anner com pletely an alag o u s to the p ro o f o f L em m a 6 in R ay [ 9 ] , F ro m Lem m a 3 of R ay [ 1 9 ] , if 5 is endow ed w ith th e topology of w eak convergence, every co n tin u o u s function from 5 to itself has a fixed point. T hus, a quasi-equi
librium exists, w ith some policy function s e S. Let x ° solve (2) for y = y (by L em m a 3, jc0 exists). Define s(y ) = .y(y) for }> e[0, y ), and 5(^) = x° for y = y. Since / i s increasing in co, (so th a t y , = y iff co, = 1), and since rj is atom less (so th at n l { c o e [0, l ] 00 | cot = 1 for som e ?}] = 0), it follows th at s is an equilibrium . L em m a 1 assures us th a t s m ust be non-decreasing. By
construction, s is use. Q .E.D .
Re f e r e n c e s
1. R . A n d e r s o n a n d H. S o n n e n s c h e i n , O n the existence of rational expectations equilibrium, J. Econ. Theory26 (1 9 8 2 ), 2 6 1 -2 7 8 .
2. K. Ar r o w, Rawls’s principle of just saving, Scand. J. Econ. (1 9 7 3 ), 3 2 3 -3 3 5 . 3. R. Ba r r o, Are government bonds net wealth? J. Polit. Econ. 82 (1 9 7 4 ), 1 0 9 5 -1 1 1 8 . 4. B. D . B e r n h e im a n d D. R a y , Altruistic growth economies. I, Existence of bequest
equilibria, IMSSS Technical Report No. 4 19, Stanford University, 1983.
5. B . D. Be r n h e im a n d D. Ra y, O n the existence of M arkov-consistent plans under production uncertainty, Rev. Econ. Stud. 53 (1986), 877-882.
6. B. D. Be r n h e im a n d D. Ra y, Economic growth with intergenerational altruism, Rev.
Econ. Stud. 54 (1 9 8 7 ), 2 2 7 -2 4 2 .
7. B. D . Be r n h e im a n d D . Ra y, M arkov-perfect equilibria in altruistic growth economies with production uncertainty, mimeo., D epartm ent of Economics, Stanford University, 1986.
8. P. Da s g u p t a, O n s o m e p ro b le m s a ris in g fro m P r o fe s s o r R a w ls ’ c o n c e p tio n o f d i s tr ib u tiv e ju s tic e , Theory and Decision 4 (1974), 325-344.
9. S. Go l d m a n, Consistent plans, Rev. Econ. Stud. 48 (1980), 533-537.
10. P . Ha l m o s, “M easure Theory,” Springer International Student Edition, 1978.
11. C. Ha r r is, Existence and characterization of perfect equilibrium in games o f perfect information, Econometrica 53 (1985), 613-628.
12. C. Ha r r is, Consistent M arkov plans, mimeo., Nuffield College, Oxford University, 1986.
13. E. Ko h l b e r g, A model o f economic growth with altruism between generations, J. Econ.
Theory 13 (1976), 1-13.
14. J. La n e a n d T. Mi t r a, O n N ash equilibrium program s of capital accumulation under altruistic preferences, Int. Econ. Rev.22 (1981), 309-331.
15. W. Le in in g e r, The existence of perfect equilibria in a model of growth with altruism between generations, Rev. Econ. Stud.53 (1986), 349-367.
16. G . Lo u r y, Intergenerational transfers and the distribution of earnings, Econometrica 49 (1981), 843-867.
17. D. Pe a r c e, N onpaternalistic sympathy and the inefficiency of consistent tem poral plans, mimeo., Departm ent of Economics, Princeton University, 1983.
18. B. Pe l e ga n d M. Ya a r i, O n th e e x iste n c e o f a c o n s is te n t c o u rs e o f a c tio n w h e n t a s t e s a r e c h a n g in g , Rev. Econ. Stud. 40 (1973), 391-401.
19. D. R ay , Nonpaternalistic intergenerational altruism, J. Econ. Theory 40 (1987), 112-132.
20. P. St r e u f e r t, O n dynamic allocation with intergenerational benevolence, mimeo., D epartm ent of Economics, Stanford University, 1985.
