Nonpaternalistic Sympathy and the
Inefficiency of Consistent Intertemporal Plans
David G. Pearce∗
Princeton University, 1983
Abstract
Perfect equilibria of intertemporal “cake-eating” models in which
generations care about one another’s utilities, are shown to be inefficient. Systematic consumption bias is generated without the temporally asymmetric assumptions used in earlier papers where the same
phenomenon arises (4, 16). The definition of subgame perfect equilibrium (13, 14) is substantially generalized to apply to the well-defined
strategic situation analyzed, which is formally not a game.
1
Introduction
Strotz’ path-breaking paper (15) concerns the dilemma of an individual who
realizes that he cannot trust himself to carry out (over time) the course
of action that he currently views as optimal. The literature spawned by
that work discusses a set of interrelated problems including taste formation,
the desire for precommitment, and the existence and efficiency of consistent
intertemporal plans (see, for example, Pollak (10), Phelps and Pollak (9),
Blackorby, Primont, Nissen and Russell (1), Peleg and Yaari (8), Hammond
(6), Yaari (16), and Goldman (4, 5)). As these authors recognize, identical
issues arise when economic decisions are made by successive generations having interdependent preferences. In this context Goldman (4) gives conditions
∗
I wish to thank Dilip Abreu, Bob Anderson and Hugo Sonnenschein for their searching
questions and invaluable technical advice. I enjoyed helpful conversation with Ed Green,
Greg Mankiw, Barry Nalebuff, Evan Porteus, Richard Quandt and Dan Usher.
1
guaranteeing the inefficiency of an interior subgame perfect equilibrium (Selten (13, 14)) of an intergenerational “cake-eating” game and the existence
of a “slower” consumption programme that Pareto dominates the equilibrium outcome. The latter result suggests a “bias” in the decision process,
favouring overly rapid consumption.
The model presented here is almost identical to Goldman’s, except that
instead of being interested in one another’s consumption levels, generations
care about one another’s utilities. Such preferences seem rather natural: if
generation i is sympathetic to j, and j’s utility rises, this should please i,
regardless of whether the increase in j’s utility results from increased consumption, or from an improvement in the welfare of some other generation k
toward whom j is favourably disposed. This kind of interdependence might
be called liberal, or nonpaternalistic, sympathy. In Section 2 I give conditions under which a system of utility functions displaying nonpaternalistic
sympathy generates a unique vector of utilities, and behaves well in a sense
to be made precise. Section 3 then describes the strategies of each of the T
generations, and defines a subgame perfect outcome of the model. This is of
special interest to students of strategic behaviour, because the strategic situation involved is actually not a game, and the definition of subgame perfect
Nash equilibrium must be extended substantially to cover it.
Section 4 establishes that under the conditions of Section 2, nonpaternalistic sympathy leads to the same results (inefficiency and “bias”) obtained
in (4). Analyzing what drives his results, Goldman notes that the “ ‘overconsumption’ arises as a consequence principally of the future bias (A3) and
monotonicity (A4).” ((4), page 624). Our respective axioms are not nested.
However, my assumptions have the advantage of being symmetric in all players, thereby making it clear that the results emerge naturally from the strategic situation, without assistance from asymmetric restrictions on the utility
functions.
The theoretical structure developed in Section 2 has many interesting applications; Section 5 concentrates on one of these. I show that even when all
members of a society exhibit nonpaternalistic sympathy, a “socially optimal”
interior allocation (evaluated by any strictly increasing social welfare function) can never be achieved by appointing a dictator to distribute resources.
The Conclusion, Section 6, includes a brief discussion of some extensions of
the results of Section 4.
2
2
Nonpaternalistic Sympathy and Coherence
A number of difficulties arise when utilities are generated by a system of
interdependent utility functions.1 These difficulties and their resolutions are
discussed here without reference to the intergenerational strategic situation
of ultimate interest; this should make it clear that the issues involved, such
as the existence and uniqueness of mutually consistent utility numbers, are
entirely distinct from questions of strategic equilibrium.
Consider T individuals (or generations) having utilities defined by
ut = Ut (ct , u1 , . . . , ut−1 , ut+1 , . . . , uT ),
t = 1, . . . , T
(∗)
where ct is the quantity of some commodity consumed by t. Thus player t’s
utility is given by his utility function Ut (mapping R+ × RT −1 into R) and
depends upon the level of his own consumption, and the utilities of other
individuals.
The most basic question that presents itself is the following: what assumptions guarantee that for any particular vector (c1 , . . . , cT ) there will
exist utilities u1 , . . . , uT satisfying the system (∗)? That this question has
some substance is illustrated by the following two person example.
u1 = U1 (c1 , u2 ) = u2 + 1
u2 = U2 (c2 , u1 ) = u1 + 1
Clearly no utilities exist that can solve both equations; for any c = (c1 , c2 )
the map
U c (u1 , u2 ) ≡ (U1 (c1 , u2 ), U2 (c1 , u2 ))
= (u2 + 1, u1 + 1)
has no fixed point.
If mutually compatible utility numbers do exist, when will they be unique?
A system such as
U1 (c1 , u2 ) = u2
U2 (c2 , u1 ) = u1
1
Ed Green first alerted me to the problems discussed in this section.
3
certainly cannot predict definitively the utilities that result from a particular
consumption vector, and the derivative of u1 with respect to c1 , for example,
is consequently not well-defined.
Finally, the system may or may not be “stable” in a certain sense. Suppose that
U1 (c1 , u2 ) = −10 + c1 + 2u2
U2 (c2 , u1 ) = −10 + c2 + 2u1
Solving these equations for u1 and u2 yields
c1 2c2
−
3
3
c2 2c1
u2 = 10 − −
3
3
u1 = 10 −
Notice that although the functions U1 and U2 seem to indicate that the
individuals enjoy consumption and are mutually sympathetic, the reduced
form indicates that both u1 and u2 are increased when c1 or c2 is decreased!
This does not correspond to any plausible dynamic adjustment story, such
as the following. If c1 is reduced by 1 unit, this should “initially” lower
u1 by 1, which would then cause u2 to fall 2 units (via the function U2 ),
diminishing u1 by a further 4 units. This downward spiral does not converge;
the only finite solution of the equations entails a perverse increase in utilities
for both individuals. Such a counter-intuitive state of affairs might be called
instability, just as we consider unstable a bare-bones Keynesian model with
a marginal propensity to consume (M.P.C.) of 2, giving an apparent (but
1
= −1.
misleading) government expenditure multipler of 1−M.P.C.
I shall establish that assumptions (A1)–(A3) below guarantee the existence of a unique set of utilities satisfying (∗): the resulting “reduced form”
utility functions are continuously differentiable and respond positively to
increases in consumption levels. Hence I refer to (A1)–(A3) as coherence
conditions for (∗).
Let Utt denote the partial derivative of Ut with respect to ct (holding
all other utilities constant) and Uij denote the partial derivative of Ui with
respect to Uj , i 6= j, holding all other utilities and ci constant. The functional
arguments are suppressed below: (A1)–(A3) hold at all points in the domain
of each function.
(A1) Ut : R+ × RT −1 → R is continuously differentiable, t = 1, . . . , T .
4
(A2) Uij > 0,
i = 1, . . . , T ; j = 1, . . . , T .
(A3) There exist positive real numbers eji such that Uij ≤ eji , and


0 e21 . . . eT1
 e1 0
eT2 

 2
E ≡  ..
.. 
.
.
1
2
eT eT . . . 0
satisfies the Hawkins-Simon condition (6), that is, all the principal
minors of I − E are positive (or equivalently, since E is a nonnegative
square matrix, the largest characteristic root r of E is less than 1 (see
Debreu and Herstein (2), Theorem IV)).
Proposition 1. If equations (∗) satisfy (A1)–(A3), for any given vector
(c1 , . . . , cT ), there exists a unique set of utilities satisfying (∗).
Proof. Existence. For all c = (c1 , . . . , cT ) ∈ RT+ , and u = (u1 , . . . , uT ) ∈ RT ,
define


U1 (c1 , u2 , . . . , uT )


..
U c (u) = 

.
UT (cT , u1 , . . . , uT −1 )
Uic is the ith component of the vector U c . Let


|U1 (c1 , 0, . . . , 0)|


..
bc = 

.
|UT (cT , 0, . . . , 0)|
and Gc (u) = bc + Eu, where E is the matrix of uniform upper bounds on
derivatives,
in (A3). Notice that for u > 0, Utc (u) exceeds Utc (0) by
P defined
i
at most i6=t et (ui − 0). Hence, for all u ≥ 0,
U c (u) ≤ U (c, 0) + Eu
∴ U c (u) ≤ bc + Eu = Gc (u)
(1)
Since by (A3), the determinant of E is positive, I − E is invertible. Let
∴
B c = (I − E)−1 bc
B c = bc + EB c
= Gc (B c ), by the definition of Gc .
5
(2)
Thus B c is a fixed point of Gc . Letting subscripts indicate components of
the vector
B c , let Stc be the closed interval [−Btc , Btc ], t = 1, . . . , T , and
Q
S c = Tt=1 Stc .
Recall that by (A2), U is strictly monotonic. For all u ∈ S c ,
|(c1 , u2 , . . . , uT )| .
c
c
.
U (u) ≤ U .
|(cT , u1 , . . . , uT −1 )|
≤ U c (B c )
≤ Gc (B c ) (by (1))
= B c (by (2))
A symmetric argument establishes that for all u ∈ S c ,
U c (u) ≥ −B c .
Combining these results,
U c (u) ∈ S c
for all u ∈ S c .
Thus, letting Ũ c be the restriction of U c to S c , we have
Ũ c : sc → sc .
Since Ũ c is continuous (by (A1)) and sc is nonempty, compact, and convex,
Brouwer’s Fixed Point Theorem guarantees the existence of a vector v c ∈ S c ,
such that
Ũ c (v c ) = v c
and hence
c
c
Ut (ct , v1c , . . . , vt−1
, vt+1
, . . . , vTc ) = vtc ,
t = 1, . . . , T.
Thus (cc1 , . . . , vTc ) satisfies (∗).
Uniqueness. Suppose that distinct vectors (v1 , . . . , vT ) and (w1 , . . . , wT ) satisfy (∗). Without loss of generality, suppose that for some t, wt > vt . Let
a, b, . . . , f be the components in which w strictly exceeds v, and F denote
6
the matrix obtained by deleting all rows and columns of E except those
corresponding to a, . . . , f . F inherits (A3) from E. Now




wa − va
wa − va




..
..

≤F

.
.
w f − vf
wf − vf


wa − va


..
But this is impossible, because 
 > 0 and F satisfies the Hawkins.
wf − vf
Simon condition (A3) (see Debreu and Herstein (2), Lemma*, with s = 1).
Consequently, it is impossible for two distinct utility vectors to satisfy (∗).
Definition. For the system (∗) satisfying (A1)–(A3), define for each c =
(c1 , . . . , cT )
Vt (c1 , . . . , cT ) = Ut (ct , v1 , . . . , vt−1 , vt+1 , . . . , vT ),
t = 1, . . . , T,
where v1 , . . . , vT are the unique utilities satisfying (∗).
Proposition 2. Vt : RT+ → R is continuously differentiable, and Vij , the
partial derivative of Vi with respect to cj is positive, for all i and j.
Proof. By (A1), Ut is continuously differentiable, and (A3) guarantees that
(I − A) is nonsingular, where


0 U12 . . . U1T
U 1 0 . . . U T 
2 
 2
A ≡  ..
..
.. 
 .
.
. 
1
2
UT UT . . . 0
Then by the Implicit Function Theorem, Vt is continuously differentiable,
t = 1, . . . , T . Totally differentiating (∗), we have


U11 dc1


(I − A)dU =  ... 
UTT dcT
7
Let det(I − A) be the determinant of I − A, and Ci,j denote the cofactor of
the i − j th element of I − A. Then Cramer’s rule gives
PT
U i Ci,t dci
dvt = i=1 i
det(I − A)
and in particular,
Vij
Ujj Cj,i
.
=
det(I − A)
Now Ujj > 0 for all j, by (A2), and det(I − A) > 0 by (A3). Furthermore,
Cj,i > 0 because it can be shown that if not, A has a characteristic root p ≥ 1;
since A and E are nonnegative, A ≤ E implies that E has a characteristic
root r ≥ p ≥ 1 ((2), Theorem 1*), contradicting (A3). Therefore it cannot
be the case that Ci,j ≤ 0 for some i and j. Thus Vij > 0, i = 1, . . . , T ; j =
1, . . . , T .
If the utilities of the first t − 1 individuals are fixed artificially at ū1 , . . . ,
ūt−1 , the system of equations
ui = Ui (ci , ūi , . . . , ūt−1 , ut , . . . , ui−1 , ui+1 , . . . , uT ),
i = t, . . . , T
(**)
generates unique utilities ut , . . . , uT , because the system (**) inherits properties (A1)–(A3) from (∗). This means that the following “partially reduced
form” equations are well-defined, strictly monotonic, and continuously differentiable, as a corollary of Propositions 1 and 2.
Definition. Let the functions U1 , . . . , UT be those of (∗), satisfying (A1)–
(A3). For any (ū1 , . . . , ūt−1 ) ∈ Rt−1 , and (c1 , . . . , cT ) ∈ RT+ , define
vi,t (ū1 , . . . , ūt−1 ; c1 , . . . , cT )
= Ui (ci , ū1 , . . . , ūt−1 , wt , . . . , wi−1 , wi+1 , . . . , wT ),
i = 1, . . . , T
where (w1 , . . . , wT ) is the unique utility vector satisfying the equations (**).
Vi,tj denotes the partial derivative of Vi,t with respect to cj , j > t. Notice
that Vi,t has the redundant arguments c1 , . . . , ct−1 ; this is simply a matter of
what is notationally convenient in the definitions employing these functions.
8
3
The Intergenerational Model
The model studied here is a T generation, one commodity storage economy.
Generation 1 is endowed with 1 unit of a perfectly divisible resource (called
“cake”) that is valued by all generations. 1 decides on an amount of cake
to consume, and passes on the remainder to the second generation, which
eats some portion of what has been left to it, leaving the rest for posterity.
Each generation eats, bequeathes, and dies; no two generations are alive
contemporaneously. No stipulations can be attached to any bequest. Thus,
1 cannot control the way in which the resource left to 2 will be distributed
among generations 2, . . . , T . The pattern of consumption (c1 , . . . , ct−1 ) is
known to t as a matter of historical record; the model is one of perfect
information.
The motivation for a generation’s considering passing on some cake is that
all generations care about one another’s utilities. Specifically, the welfare of
the T generations is determined by the system (∗) of the previous section, and
(A1)–(A3) are assumed to hold. The fact that generations 1, . . . , t care about
Ut+1 , . . . , UT , but do not live to see ct+1 , . . . , cT (on which the T -person utility
system functionally depends) introduces a new complication: generation t’s
welfare is determined by the resource allocation it expects will arise, not the
distribution that actually occurs. Moreover, since Ut+1 has an argument ut ,
person t + 1 needs to deduce what generation t’s welfare was. Attempting
to model this as an extensive form game, one finds that it is not a game:
utilities depend upon more than the endpoint of the extensive form that is
reached, as explained above. Can this complexity be finessed by using the
reduced form functions V1 , . . . , VT , thereby disposing of the need to worry
about “i’s belief about j’s utility”? Surprisingly, the answer is no. Simply
evaluating the endpoints of the intergenerational model by the reduced form
functions and solving for the subgame perfect outcome(s) will produce an
incorrect solution, as the following example illustrates.
Consider 3 generations with utilities given by
U1 (c1 , u2 , u3 ) = c1 + 10u3
U2 (c2 , u1 , u3 ) = c2 + u1
U3 (c3 , u1 , u2 ) = c3 .
9
The associated reduced form is
V1 (c1 , c2 , c3 ) = c1 + 10c3
V2 (c1 , c2 , c3 ) = c1 + c2 + 10c3
V3 (c1 , c2 , c3 ) = c3 .
The functions U1 , U2 , U3 do not satisfy (A2), but could easily be altered
to do so, with some loss of simplicity, but without changing the point of
the example. Solving the “artificial” game (whose outcomes are evaluated
by the functions V1 , V2 , V3 ) by backward induction, one notes that since
generation 3 eats anything it is given, 2’s optimal strategy is to pass on to 3
everything 2 has (because, in the notation of Section 2, V23 = 10 > 1 = V22 ).
Knowing that anything passed on to 2 will ultimately be consumed by 3, 1
eats nothing, bequeathing the entire cake to his successors (because V13 =
10 > 1 = V11 ). The unique subgame perfect outcome of this artificial game,
then, is (c1 , c2 , c3 ) = (0, 0, 1), yielding utilities (10, 10, 1).
But this solution bears no resemblance to the result of individual utility
maximizing behaviour in the intergenerational model. When generation 2
makes its consumption decision, 1 is dead; utility ū1 is fixed; a thing of the
past. 2 cares about ū1 , but cannot affect it. Thus 2’s objective function is
U2 (c2 , ū1 , u3 ) = V2,2 (ū1 ; c2 , c3 )
= ū1 + c2
which generation 2 maximizes (treating ū1 as a parameter) by eating everything it is bequeathed. Understanding this, and deriving no satisfaction from 2’s consumption, 1 eats the entire cake. This “consistent plan”
(c∗1 , c∗2 , c∗3 ) = (1, 0, 0) yields utility (1, 1, 0), and is not remotely similar to the
solution of the artificial game analyzed above.
What is needed is a formalization of the intuition that lets us solve the
preceding example. The required generalization of subgame perfect equilibrium must also confront the problem (not encountered in the example) of a
generation’s optimal decision depending upon its perception of earlier generations’ utility levels. Before attacking these problems, I define strategies for
the generations, and allocation functions that determine consumption vectors
as a function of strategy profiles. The notation used for these two definitions
follows closely that of Goldman (4) who cannot, however, be blamed for the
labyrinthine complexity of the subsequent definitions.
10
Definition. A strategy for generation t is a function ht such that
(
)
"
#
t−1
t−1
X
X
ht : (c1 , . . . , ct−1 ) :
ci ≤ 1 → 0, 1 −
ci .
i=1
i=1
Ht denotes the set of all strategies of t.
Definition. The allocation functions xt : H1 × · · · × Ht → Rt+ are given
inductively by
x1 (h1 ) = h1
xt (h1 , . . . , ht ) = (xt−1 (h1 , . . . , ht−1 ), ht (xt−1 (h1 , . . . , ht−1 ))),
t = 2, . . . , T.
Definition. An allocation (c1 , . . . , cT ) is interior if ct > 0 for all t.
P
Definition. An allocation (c1 , . . . , cT ) is feasible if Ti=1 ci = 1.
In a subgame perfect equilibrium, expectations at the beginning of any
subgame (whether on or off the equilibrium path) about other players’ strategies are determined by the strategy profile induced on that subgame by the
equilibrium strategy profile of the original game. In the intergenerational
model with nonpaternalistically sympathetic preferences, players need to
form beliefs not only about others’ strategies, but also about the utility levels
other have attained, or will attain. The belief formation functions I shall introduce, inductively generate expectations about the utilities of others on the
same principle as strategic expectations are implicitly generated in subgame
perfect equilibrium. Given an equilibrium profile (h∗1 , . . . , h∗T ), an arbitrary
history generated by (possibly disequilibrium) strategies h1 , . . . , ht−1 , and
utilities ū1 , . . . , ūt−1 determined by the first t − 1 belief formation functions,
t should expect the outcome xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T ) to occur if t employs the strategy ht . Then t’s utility from playing ht (given the history
xt−1 (h1 , . . . , ht−1 )) should be
Vt,t (ū1 , . . . , ūt−1 ; xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T )),
so this is the number that the tth belief formation function associates with
the history xt (h1 , . . . , ht ) under equilibrium (h∗1 , . . . , h∗T ). In the complete
formal definition that follows, the profile (h∗1 , . . . , h∗T ) can be thought of as
an equilibrium, while the second profile (h1 , . . . , hT ) simply serves to generate
an arbitrary history for each t.
11
Definition. For all pairs of strategy profiles (h∗1 , . . . , h∗T ) and (h1 , . . . , hT ),
the belief formation functions f 1 (h1 ; h∗ ), . . . , f T (h1 , . . . , hT ; h∗ ) are defined
inductively by
f 1 (h1 ; h∗ ) = V1 (xT (h1 , h∗2 , . . . , h∗T )
f t (h1 , . . . , ht−1 ; h∗ ) = (f t−1 (h1 , . . . , ht−1 ; h∗ ), Vt,t (f t−1 (h1 , . . . , ht−1 ; h∗ );
xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T ))),
t = 2, . . . , T.
Definition. A strategy profile h∗ = (h∗1 , . . . , h∗T ) is a pure strategy subgame
perfect equilibrium (S.P.E.) if for each profile (h1 , . . . , hT ) and each t,
Vt,t (f t−1 (h1 , . . . , ht−1 ; h∗ ); xT (h1 , . . . , ht−1 , h∗t , . . . , h∗T ))
≥ Vt,t (f t−1 (h1 , . . . , ht−1 ; h∗ ); xT (h1 , . . . , ht , h∗t+1 , . . . , h∗T )).
Definition. An allocation (c1 , . . . , cT ) is an S.P.E. outcome if xT (h∗1 , . . . , h∗T )
= (c1 , . . . , cT ) for some S.P.E. (h∗1 , . . . , h∗T ).
The existence of such an equilibrium is a complex issue, and is not discussed here. I hope to study existence questions for “generalized games” (of
which the present model is an example) in a subsequent paper.
4
Intertemporal Inefficiency
Proposition 3. Suppose that T ≥ 3, and that utilities in the T generation model are given by (∗), and (∗) satisfies (A1)–(A3) (see Section 2). If
c = (c1 , . . . , cT ) is an interior S.P.E. outcome, there exists ε > 0 such that
(c1 , . . . , cT −2 , cT −1 − ε, cT + ε) is feasible, and Pareto dominates c; that is, for
all t,
VT (c1 , . . . , cT ) < VT (c1 , . . . , cT −2 , cT −1 − ε, cT + ε).
Proof. Let ū = (ū1 , . . . , ūT ) ≡ (V1 (c), . . . , VT (c)). Since c is an interior S.P.E.
outcome, generation T − 1 has divided its inheritance between cT −1 and
cT to maximize VT −1,T −1 (ū1 , . . . , ūT −2 ; xT −1 , xT −2 ) subject to xT −1 + xT =
P −2
1 − Tt=1
xt ; xT −1 , xT ∈ [0, 1].
A necessary condition for this interior solution is
−1
T
VTT−1,T
−1 (ū1 , . . . , ūT −2 ; cT −1 , cT ) = VT −1,T −1 (ū1 , . . . , ūT −2 ; cT −1 , cZ )
12
(3)
By totally differentiating (∗) holding the first T −2 consumptions and utilities
constant at c1 , . . . , cT −2 and ū1 , . . . , ūT −2 respectively, one gets (suppressing
the functional arguments)
−1
UTT−1
1 − UTT−1 UTT −1
UTT−1 UTT
=
1 − UTT−1 UTT −1
−1
VTT−1,T
−1 =
VTT−1,T −1
which are well-defined because the denominators are non-zero (by (A3)), as
the denominators are a principal minor of I −A, and A satisfies the HawkinsSimon condition (see the proof of Proposition 2). Then (3) implies that at
the S.P.E. outcome,
−1
UTT−1
= UTT−1 UTT
(4)
For each t let
dVt
ds
be the derivative of
Vt (c1 , . . . , cT −2 , cT −1 − s, cT + s)
t
>0
with respect to s, evaluated at s = 0. It will be sufficient to show dV
ds
for all t; because then for ε sufficiently small, (c1 , . . . , cT −2 , cT −1 − ε, cT + ε)
is feasible and Pareto dominates (c1 , . . . , cT ). Differentiating the system (∗)
with respect to s, holding c1 , . . . , cT −2 constant, one obtains


0


..


.


(I − A)dV = 
,
0


T
−1
−UT −1 ds
UTT ds
that is

1
−U21
..
.
−U12
1
. . . −U1T −1
−U2T −1
−U1T
−U2T
..
.

dV1


0
..
.



 



 


  ..  


 .  = 

0


 

T
−1
1
2
T
−UT −1 −UT −1
 −UT −1 ds
1
−UT −1  
dVT
UTT ds
−UT1
−UT2 . . . −UTT −1
1
13
and by Cramer’s rule, for t 6= T ,
1
−U12 . . .
−U21
1
.
.
dVt
1
.
=
ds
det(I − A) −UT1 −1 −UT2 −1
−UT1
−UT2 . . .
0
0
..
.
−1
−UTT−1
UTT
|{z}
tth column
T −UT −1
...
1 ...
−U1T
−U2T
..
.
Subtracting UTT times the last column, from the tth column (which leaves the
determinant unaltered), and using (4), yields (for t 6= T )
1
−U12 . . .
U1T
. . . −U1T −U21
1
U2T
−U2T .
..
.. ..
T
.
. UT
dVt
T
=
U
T −2
ds
det(I − A) 0
−UTT−1 0
...
1 −UT1 −UT2 . . .
|{z}
th
t
column
Expanding by cofactors down the tth column gives
T −2
dVt X Uit Utt Ci,t
=
,
ds
det(I
−
A)
i=1
t 6= T
where ci,j is the i − j th cofactor of I − A. But since A satisfies the HawkinsSimon conditions, det(I − A) > 0, and ci,t > 0 for all i, t. Moreover by (A2),
Uit > 0 for all i, t. Thus
dV
> 0 for all t 6= T .
ds
(5)
Finally, differentiate UT (cT + s, u1 , . . . , uT −1 ) with respect to s (evaluating at
s = 0).
T −1
X dVt
duT
dVT
=
= UTT +
UTt
> 0 (by (5) and (A2)).
ds
ds
ds
t=1
Therefore
dVt
ds
> 0 for all t.
14
The intuition behind this result is quite straightforward. When T − 1
makes its consumption decision, the utilities of 1 to T − 2 are treated parametrically. If all players were assured that the allocation (c1 , . . . , cT −2 , cT −1 −
ε, cT + ε) would be enforced, the “initial” impact on T − 1’s utility (through
UT −1 ) would be an almost negligible negative change, per unit increase in ε,
because his derivative with respect to this transfer is 0 at the equilibrium. T is
made better off because final period consumption rises, and there is no significant compensating decline in UT −1 (the only conceivable objection T might
have had to the transfer). Since UT has risen substantially per unit transferred, and UT −1 has fallen inappreciably per unit, the utilities u1 , . . . , uT −2
are increased, since U1 , . . . , UT −2 have positive derivatives in UT −1 and UT .
A further feedback occurs: UT −1 is favourably disposed toward 1, . . . , T − 2,
and hence is pleased by the increases in u1 , . . . , uT −2 . The matrix algebra of
the proof confirms that this pleasure more than compensates T − 1 for its
decreased share of the consumption, for sufficiently small ε.
Thus, the proof works because the utilities of T − 2 generations are immutable when the second last generation’s decision is taken. It would work
equally well if only one generation’s utility were fixed at that time; in other
words, Proposition 3 could easily be extended to cover overlapping generations models, as long as generation 1 is dead before generation T − 1 eats.
(But defining the belief formation functions would be a torturous business.)
Some readers may feel that it would be more natural for each generation
to care only about future generations’ utilities (and its own consumption)
rather than past and future utilities. In this case a slightly altered version
of Proposition 3 applies, and is proved below. There is no need for (A3),
because the utility system is recursive rather than fully simultaneous.
Proposition 4. Suppose that in the intergenerational model with T ≥ 3,
utilities are generated not by (∗), but by the equations
ut = Ut (ct , ut+1 , . . . , uT ),
t = 1, . . . , T,
where for all t, Ut is differentiable and Uij > 0 for i ≤ j. If c̄ = (c̄1 , . . . , c̄T ) is
an interior S.P.E. outcome, there exist s, a > 0 such that (c̄1 −as, c̄2 , . . . , c̄T −2 ,
c̄T −1 − (1 − a)s, c̄T + s) is feasible, and Vt (c̄1 − as, c̄2 , . . . , c̄T −2 , c̄T −1 − (1 −
a)s, cT + s) > V (c̄) for all t.
Proof. Since c̄ is an interior S.P.E. allocation,
−1
UTT−1
= UTT−1 UTT
15
(6)
(as in the proof of Proposition 3). For some a > 0, define
c1 (s) = c̄1 − as
ct (s) = c̄t , t = 2, . . . , T − 2
cT −1 (s) = c̄T −1 − (1 − a)s
cT (s) = c̄T + s
t
It is sufficient to show that for some a, dU
> 0 for all t. Successively
ds
differentiating UT , UT −1 , . . . , U1 with respect to s, and evaluating at s = 0
(i.e., at the S.P.E. outcome)
dUT
(c̄T + s) = UTT > 0
ds
(7)
dUT −1
dUT
−1
(c̄T −1 − (1 − a)s, UT ) = (a − 1)UTT−1
+ UTT−1
ds
ds
−1
−1
= aUTT−1
− UTT−1
+ UTT−1 UTT
−1
= aUTT−1
>0
(from (7))
(from (6))
For t = 2, . . . , T − 2,
T
X
dUi
dUt
(c̄t , ut+1 , . . . , uT ) =
Uti
>0
ds
ds
i=t+1
by induction.
T
X dUi
dU1
(c̄1 − as, u2 , . . . , uT ) = −aU11 +
U1i
.
ds
ds
i=2
Setting a = 0 in the above derivatives, one gets
dUt
> 0,
ds
t 6= T − 1
dUT −1
= 0.
ds
Thus by the continuous differentiability of V1 , . . . , VT , there exists an interval
(0, ā) such that a ∈ (0, ā) implies
dUt
> 0,
ds
t = 1, . . . , T.
16
5
Sympathetic Dictators
Gibbard (3) and Satterthwaite (12) have shown that only dictatorial social
choice functions can be implemented via dominant strategies (as long as the
social choice function must have at least three elements in its range), and
work of Roberts (11) implies the same conclusion for Nash implementation.
One might ask: “If people are sufficiently sympathetic to one another, could
the appointment of a dictator lead to a socially attractive outcome?” I
show that if a society of nonpaternalistically sympathetic persons satisfies
the “coherence” conditions of Section 2, no interior allocation chosen by
a dictator can maximize any strictly increasing differentiable social welfare
function.
In the proposition that follows, the functions Vt are the reduced form
utility functions of Section 2.
Proposition 5. For T ≥ 2, consider a T -person society (possessing 1 unit
of a single commodity) in which everyone lives contemporaneously, and utilities are generated by (∗), (Section 2) satisfying (A1)–(A3). Suppose the
interior allocation c̄ = (c̄1 , . . . , c̄T ) maximizes t’s utility over all feasible consumption vectors. For every strictly increasing differentiable social welfare
function W (V1 , . . . , VT ) there exists some s > 0 such that c(s) ≡ (c1 − s, c2 +
s
s
, . . . , cT + T −1
) is feasible and
T −1
W (V1 (c(s)), . . . , VT (c(s))) > W (V (c̄), . . . , VT (c̄)).
Proof. Without loss of generality let i = 1, and suppose c̄1 , . . . , c̄T is a feasible
interior allocation maximizing 1’s utility. Define
c1 (s) = c̄1 − s
ct (s) = c̄t +
s
,
T −1
t = 2, . . . , T.
Notice that (c1 (s), . . . , cT (s)) is feasible for |s| sufficiently small. Evaluating
at s = 0,
du1
=0
(8)
ds
because at s = 0, 1’s utility is at a maximum. It is sufficient to show
that dW
(V1 (c(s)), . . . , VT (c(s))) > 0. Totally differentiating the last T − 1
ds
17
equations of (∗) with respect to s yields
T
dut
Utt
1 X i dUi
=
+
,
U
ds
T − 1 T − 1 i6=t t ds
t = 2, . . . , T
T
Utt
1 X i dUi
=
+
U
T − 1 T − 1 i6=1,t t ds
(from (8)).
Arranging this as a T − 1 person matrix system, one obtains
  du   U 2 

2
2
1
−U23 . . . −U2T

.
T
−
1




..   ds

1
−U32
.
..  = 

.
 .  
 .
.


T 
 ..

 UT 
−UT −1
duT
T
−UT2 −UT3 . . .
1
ds
T −1
Let Ci,j (1) denote the cofactor of the left-hand side matrix, involving the
deletion of the ith and j th persons’ row and column respectively (the i − 1th
row and j−1th column of the (T −1)×(T −1) matrix itself). All such cofactors
can be shown to be positive as a result of the Hawkins-Simon condition (A3),
as is the determinant C1,1 (a cofactor of I − A). Cramer’s rule yields
T
1
dut X i
=
Ui Ci,t (1)
,
1
ds
(T
−
1)C
1
i=1
t = 1, . . . , T
>0
Thus,
du1
=0
ds
dut
> 0,
ds
t = 2, . . . , T.
Consequently
T
X ∂W dui
dW
(V1 (c(s)), . . . , VT (c(s))) =
ds
∂Vi ds
i=1
> 0.
18
6
Conclusion
Proposition 3 established the inefficiency of interior outcomes of a multiperiod cake-eating model, with nonpaternalistic sympathy amongst generations. Although I explained in Section 4 that the proposition extends to easily
cover overlapping generations models, and proved that inefficiency arises even
if sympathies are restricted to caring about future utilities, the reader may
wonder about the generality of the results. Are they restricted to one commodity storage economies? Notice that Propositions 3 and 4 did not use in
any essential way the fact that the resource is transferred linearly from period
to period; the results apply equally to any situation in which the “storage
functions” are strictly increasing and differentiable. These functions could
encompass possibilities such as production, inventory holding costs, spoilage
and so on.
Generalization of the results to a k-commodity world is so immediate that
the proof can be sketched in a few sentences. Starting at an interior S.P.E.
outcome, fix the allocation of all commodities except the first, and regard the
reduced form utility functions (generated by a system satisfying (A3), and
(A1), (A2) in each argument) as functions of the first commodity only. These
functions satisfy the conditions of Proposition 3, and hence there exists some
small shift of commodity 1 from T −1 to T , that improves everyone’s welfare.
Proposition 4 generalizes in the same way.
A trivial implication of the Pareto inefficiency of outcomes in these models, is that relative to any interior S.P.E., the first generation would always
strictly increase its utility if it were able to dictate a “slower” consumption
stream (in the sense of Proposition 3 or 4). Interpreting the generations as
one person living for T periods, one finds that under (A1)–(A3), an individual always wishes to precommit his future decisions, and invariably feels now
that he could benefit from being more patient or self-disciplined in the future
(going on a diet tomorrow, stopping smoking next Monday, and so on).
An incidental benefit of investigating nonpaternalistic sympathy in an intergenerational model has been encountering a well-specified strategic problem that is not a game. This presented the opportunity to study the complications involved in extending the subgame perfect equilibrium concept to
cover a situation in which all players observe all earlier players’ moves before
moving themselves (suggesting “perfect information”) and yet most players
do not observe which endpoint of the extensive form is reached.
19
References
(1) Blackorby, C., D. Primont, D. Nissen, and R. R. Russell (1973) “Consistent Intertemporal Decision Making”, Review of Economic Studies,
40, 239–248.
(2) Debreu, G. and I. N. Herstein (1953) “Nonnegative Square Matrices”,
Econometrica, 21, 597–607.
(3) Gibbard, A. (1973) “Manipulation of Voting Schemes: A General Result”, Econometrica, 41, 587–602.
(4) Goldman, S. (1979) “Intertemporally Inconsistent Preferences and the
Rate of Consumption”, Econometrica, 47, 621–626.
(5) Goldman, S. (1980) “Consistent Plans”, Review of Economic Studies,
47, 533–537.
(6) Hammond, P. (1976) “Changing Tastes and Coherent Dynamic
Choice”, Review of Economic Studies, 43 (1), 159–173.
(7) Hawkins, D. and H. A. Simon (1949) “Note: Some Conditions of Marcoeconomic Stability”, Econometrica, 17, 245–248.
(8) Peleg, B. and M. E. Yaari (1973) “On the Existence of a Consistent
Course of Action when Tastes are Changing”, Review of Economic
Studies, 40 (3), 391–401.
(9) Phelps, E. S. and R. A. Pollak (1968) “On Second-Best National Savings and Game Equilibrium Growth”, Review of Economic Studies, 35
(2), 185–199.
(10) Pollak, R. A. (1968) “Consistent Planning”, Review of Economic Studies, 35 (2), 201–208.
(11) Roberts, K. W. S. (1979) “The Characterization of Implementable
Choice Rules”, in J. Laffont (ed.) Aggregation and Revelation of Preferences: New York, North Holland, 321–348.
(12) Satterthwaite, M. A. (1975) “Strategy-Proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures
20
and Social Welfare Functions”, Journal of Economic Theory, 10, 187–
217.
(13) Selten, R. (1965) “Spieltheoretische Behandlung eines Oligopolmodells mit Nachfrageträgheit”, Zeitschrift für die Gesamte Staatswissenschaft, 121 (2), 301–324.
(14) Selten, R. (1975) “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games”, International Journal of Game
Theory, 4, 25–55.
(15) Strotz, R. H. (1956) “Myopia and Inconsistency in Dynamic Utility
Maximization”, Review of Economic Studies, 23 (3), 165–180.
(16) Yaari, M. E. (1977) “Consistent Utilization of an Exhaustible Resource
or How to Eat an Appetite-Arousing Cake”, Walras-Bowley Lecture,
North American Meeting of the Econometric Society, Atlanta, June
1977.
21