STATUS AND INTERTEMPORAL CHOICE
Debraj Ray
Arthur Robson
New York University
Simon Fraser University
Bianjun Xia
Simon Fraser University
May 1, 2008
1
ABSTRACT
The idea that utility or happiness depends on the comparison of one’s own consumption to that of others can be traced back, at the very least, to Veblen (1899).
Nevertheless, neoclassical economic theory has typically assumed that an agent’s utility depends solely on the absolute level of consumption. This stance has been harder
to maintain in the face of a large body of empirical evidence on an individual’s tendency to evaluate her consumption in comparison with consumption of others, and
there is now a growing literature that includes this relative concern, or status, as
an additional argument in the utility function. This note considers a preference for
status as fundamental. It takes the extreme view that the individual does not care
about the consumption of particular goods per se, but only how she ranks in the distribution of consumption. This view makes it possible to straightforwardly reconcile
an evolutionary basis for preferences with the obvious fact that absolute consumption
levels now vastly exceed any plausible level in hunter-gatherer societies. Despite this
apparently fundamentally distinct basis for choice, this paper establishes a startling
behavioral equivalence between the relative concern model and an absolute concern
model of intertemporal consumption choice. That is, the equilibrium consumption
time path followed by individuals with a pure preference for status is the same as
that followed by individuals with a logarithm utility function in absolute consumption. This result holds for a large class of preferences for status. There may then be
no observable difference at all between a model with a pure concern for status and a
conventional model.
2
“Now here, you see, it takes all the running you can do to keep in the same place.”
Lewis Carroll, Through the Looking-Glass
1
Introduction
The idea that utility or happiness depends in part on comparing one’s own consumption to that of others can be tracked back, at least, to Veblen. Nevertheless,
neoclassical economic theory has typically assumed that an agent’s utility depends
solely on the absolute level of consumption. This stance has been harder to maintain in the face of a large body of empirical evidence of an individual’s tendency to
compare her consumption to that of others (for example, Frank (1985), Brown, Oswald and Qian (2004), and Dynan and Ravina (2007)). There is also now a growing
literature that includes this relative concern as an argument in the utility function
in addition to the absolute consumption level (for example, Robson (2002), Becker,
Murphy and Werning (2005), Hopkins and Kornienko (2004)).
This note takes the view that people’s preference for relative status is fundamental
and arises as a result of long-run evolutionary process. Further, for the sake of
simplicity, it adopts the extreme view that the individual does not care about the
consumption of particular goods per se, but only how this consumption ranks relative
to others. Such a view has the additional merit of reconciling an evolutionary basis
for preferences with the obvious fact that absolute consumption levels now vastly
exceed any plausible level in hunter-gatherer societies.
We construct a theoretical model of consumption over time in which people care
only about status. The agent’s status is her ranking in the social distribution of
consumption in each period. Despite this apparently fundamentally distinct basis
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for choice, this paper establishes a startling behavioral equivalence between the relative concern model and an absolute concern model. Specifically, the equilibrium
consumption time path followed by individuals with pure preference for status is the
same as that followed by individuals with a logarithm utility function in absolute
consumption. This result holds for a large class of preferences for status.
2
Model and Results
Suppose there is a continuum of measure one of agents, where each agent i is endowed
with intial wealth w ≥ 0. The cumulative distribution of the wealth is denoted
by G(w), assumed to be continuously twice differentiable. The associated density
function is denoted by g(w). The agents allocate their wealth over time. Time is
discrete and the horizon is infinite, i.e., t = 1, 2, ....1 The agents can borrow or save
any amount at interest rate r.
The agent cares about her relative standing in the economy ,where this standing
is the agent’s consumption level relative to that of others. Specifically, an agent with
wealth w has status at time t given by Ft (ct (w)), where ct (w) is her consumption
at time t, and Ft (·) is the cumulative density function of consumption at time t.
The associated density function is denoted by ft (·). (We will only need to consider
differentiable Ft (·).) The agent values her status according to some continuously twice
differentiable function v(·) with v 0 > 0. Her objective is to maximize the present
1
The results remain valid for any finite horizon also.
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discounted value of the utility of status over time:
max
{ct }
s.t.
∞
X
v(Ft (ct ))
t=1
∞
X
t=1
(1)
(1 + ρ)t
ct
=w
(1 + r)t
where the discount rate is ρ > 0.
We first make an assumption about the preference for status and the cumulative
distribution function.
Assumption 1. v(G(·)) is strictly concave in w.
The key result is as follows:
Proposition 1. Under Assumption 1, there exists a strict Nash equilibrium in which the equilibrium status for any individual is constant over
time. Therefore, Ft (ct (w)) = G(w), for all t and w, where ct (w) is the
equilibrium consumption level at time t. Indeed, in this equilibrium, every
agent behaves as if she were independently solving the following standard
problem:
max
{ct }
s.t.
∞
X
t=1
∞
X
t=1
ln ct
(1 + ρ)t
(2)
ct
=w
(1 + r)t
Proof: It is straightforward that the optimal time path of consumption in (2) is
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given by
c∗t (w) = P
w
1
(1+ρ)t
(1 + r)t
(1 + ρ)t
∀t
= kt w, say, where kt =
(3)
(1+r)t
(1+ρ)t
P 1
(1+ρ)t
Consider then the intertemporal choice of a particular individual facing (1) in a
population where all other individuals choose c∗t (w) = kt w. It follows that Ft (kt w) =
G(w) so that Ft (ct ) = G(ct /kt ). That is, this particular individual solves the following
problem:
max
{ct }
s.t.
∞
X
v(G(ct /kt ))
t=1
∞
X
t=1
(4)
(1 + ρ)t
ct
=w
(1 + r)t
The first-order conditions for an interior solution to this are then:
v0 (G(ct /kt ))g(ct /kt ) = λ(w)kt
(1 + ρ)t
(1 + r)t
∀t
(5)
for some Lagrange multiplier λ(w). However, given the definition of kt , these firstorder conditions are satisfied at ct (w) = kt w where
λ(w) = v0 (G(w))g(w)
X
1
.
(1 + ρ)t
Given Assumption 1, these first-order conditions are sufficient for the unique global
maximum to the individual’s problem.
Furthermore, the following uniqueness result holds:
Proposition 2. Consider any pure strategy Nash equilibrium, as defined in (1),
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where all consumption levels are positive, Ft (·)) is twice continuously differentiable,
and the support of G contains 0. Suppose individuals have access to any fair bet at
any date, but strictly prefer to decline all such bets. Then this Nash equilibrium can
only be the equilibrium characterized above, and Assumption 1 holds.
Proof: Consider any Nash equilibrium as described above. Since individuals
strictly prefer to decline all fair bets at any time t, v(Ft (·)) is strictly concave for
all t. It follows that (1) has a a solution for consumption ct (w) that is a strictly
increasing and continuously differentiable function of w, since this solution is interior.2
Hence ct (w) = pt (w), say. Furthermore, it must be that F (pt (w)) = G(w) so that
Ft (ct ) = G(p−1
t (ct )).
Indeed, given that all other individuals are choosing in accordance with pt (w), a
particular individual’s problem is
max
{ct }
s.t.
∞
X
v(G(p−1
t (ct )))
t=1
∞
X
t=1
(6)
(1 + ρ)t
ct
=w
(1 + r)t
which has the first-order conditions for an interior solution
−1
v0 (G(p−1
t (ct )))g(pt (ct )) = λ(w)
dpt (1 + ρ)t
dw (1 + r)t
∀t
(7)
for some Lagrange multiplier λ(w). For a Nash equilibrium, it must be that ct = pt (w)
so
v0 (G(w))g(w) = λ(w)
dct (1 + ρ)t
dw (1 + r)t
∀t
(8)
That is
dct
(1 + r)t
= K(w)
dw
(1 + ρ)t
2
Strictly, we make the slightly stronger assumption that
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∀t
d2
dc2t
(9)
[v(Ft (ct ))] < 0 for all ct and all t.
for a suitable constant K(w). Since
X
dct
dw
(1 +
Hence
r)t
P
ct
(1+r)t
= w, it follows that
= 1 = K(w)
X
1
.
(1 + ρ)t
(1+r)t
dct
(1+ρ)t
= P 1 = kt .
dw
(1+ρ)t
Since the support of G contains 0, it must be that ct (0) = 0, so that
ct = kt w ∀t
as before.3 Finally, since v(Ft (ct )) = v(G(ct /kt )), Assumption 1 must hold.
3
Conclusions
This paper considers the behavioral implications of a model of intertemporal choice
in which individuals care only about relative consumption. It may well be that a
relative concern is more fundamental than absolute concern from an evolutionary
perspective. However, there is a startling equivalence of the present model of status
and a conventional model with a logarithmic utility. It could then be very difficult
to distinguish empirically between these two possibilities, despite their apparently
radically different assumptions.
3
If the support of G does
P not contain 0, there are multiple solutions for consumption of the form
ct (w) = αt + kt w, where
αt = 0 and ct (w) ≥ 0 remains true.
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References
[1] Becker, G., K. Murphy and I. Werning (2005), “The Equilibrium Distribution of
Income and the Market for Status,” Journal of Political Economy, Vol. 113(2),
pp. 282-310.
[2] Brown, G. J. Gardner, A. Oswald and J. Qian (2005), “Does Wage Rank Affect
Employees’ Wellbeing?” IZA Discussion Paper No. 1505.
[3] Dynan, Karen and Enrichetta Ravina (2007), “Increasing Income Inequality, External Habits, and Self-Reported Happiness,” American Economic Review, Vol.
97, 226-231
[4] Frank, Robert H. (1985), Choosing the Right Pond: Human Behavior and the
Quest for Status, Oxford University Press
[5] Hopkins, Ed and Tatiana Kornienko (2004), “Running to Keep in the Same Place:
Consumer Choice as a Game of Status,” American Economic Review, September,
1085-1107
[6] Robson, Arthur (1996), “The Evolution of Attitudes to Risk: Lottery Tickets and
Relative Wealth,” Games and Economic Behavior 14 , 190-207
[7] Robson, Arthur (2002), “Status, the Distribution of Wealth, Private and Social
Attitudes to Risk,” Econometrica, Vol. 60, No. 4, 837-857
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