The uniform impatience hypothesis, a joint requirement on endowments and preferences, was imposed in the literature to prove equilibrium existence in infinite horizon sequential economies. In this note, we characterize this assumption in terms of asymptotic properties on inter-temporal discount factors.Uniform impatience, inter-temporal discounting. JEL Codes: D50, D52.

Juan Pablo Torres Martinez

MArio Páscoa

Myrian Petrassi

English

Research Papers in Economics

 
 
No. 555 
 
 
Inter-temporal discounting and uniform 
impatience 
 
Mário R. Pascoa 
Myrian Petrassi 
Juan Pablo Torres-Martínez 
 
 
TEXTO PARA DISCUSSÃO
DEPARTAMENTO DE ECONOMIA 
www.econ.puc-rio.brINTER-TEMPORAL DISCOUNTING AND UNIFORM IMPATIENCE
M´ ARIO R. P´ ASCOA, MYRIAN PETRASSI AND JUAN PABLO TORRES-MART´ INEZ
Abstract. The uniform impatience hypothesis, a joint requirement on endowments and prefer-
ences, was imposed in the literature to prove equilibrium existence in inﬁnite horizon sequential
economies. In this note, we characterize this assumption in terms of asymptotic properties on
inter-temporal discount factors.
Keywords: Uniform impatience, inter-temporal discounting.
JEL classiﬁcation: D50, D52.
The uniform impatience assumption—see Magill and Quinzii (1994, Hypotheses A2 and A4), Her-
nandez and Santos (1996, Assumption C.3) or Magill and Quinzii (1996, Assumptions B2 and B4)—is
a usual requirement for existence of equilibrium in economies with inﬁnite-lived debt-constrained
agents. In this note, we characterize uniform impatience in terms of inter-temporal discount factors.
As a consequence, we show that the uniform impatience assumption does not hold for agents with
hyperbolic inter-temporal discounting (see Laibson (1998)).
We follow the notation of Magill and Quinzzi (1994). Consider a framework where an inﬁnite-lived
price-taker agent demands L diﬀerent commodities at any node ξ of an inﬁnite countable event-tree
D. This agent may trade ﬁnancial assets to implement inter-temporal transfers. She receives at
any ξ ∈ D a physical endowment w(ξ) ∈ RL
+ and she makes contingent consumption plans, x(ξ), to
maximize her preferences, which are represented by a function U : R
L×D
+ → R+ ∪ {+∞}. Aggre-
gated physical endowments in the economy at node ξ are given by Wξ ∈ RL
++. The date associate
with a node ξ ∈ D is denoted by t(ξ).
Assumption A. Let U(x) :=
P
ξ∈D βt(ξ)ρ(ξ)u(x(ξ)), where ρ(ξ0) = 1 and, for each ξ ∈ D,
ρ(ξ) =
P
µ∈ξ+ ρ(µ). For any ξ ∈ D, βt(ξ) is a strictly positive number and the function u : RL
+ → R+
is concave, continuous and strictly increasing. Also,
P
ξ∈D βt(ξ)ρ(ξ)u(Wξ) is ﬁnite.
Uniform impatience. There are π ∈ [0,1) and (v(µ);µ ∈ D) ∈ R
D×L
+ such that, given a consump-
tion plan (x(µ);µ ∈ D), with 0 ≤ x(µ) ≤ Wµ, we have that,
u(ξ,x(ξ) + v(ξ)) +
X
µ>ξ
u(µ, π0 x(µ)) >
X
µ≥ξ
u(µ,x(µ)), ∀ξ ∈ D, ∀π0 ≥ π.
Moreover, there is δ > 0 such that, w(ξ) ≥ δv(ξ), ∀ξ ∈ D.
M. Petrassi wants to disclaim that the views expressed herein are not necessarily those of the Central Bank of
Brazil. J.P.Torres-Mart´ ınez acknowledges support from CNPq through project 307554/2004-0.
12 M´ ARIO R. P´ ASCOA, MYRIAN PETRASSI AND JUAN PABLO TORRES-MART´ INEZ
The requirements of impatience above depend on both preferences and physical endowments. As
particular cases we obtain the assumptions imposed by Hernandez and Santos (1996) and Magill
and Quinzii (1994, 1996). Indeed, in Hernandez and Santos (1996), for any µ ∈ D, v(µ) = Wµ.
Also, since in Magill and Quinzii (1994, 1996) initial endowments are uniformly bounded away from
zero by an interior bundle w ∈ RL
+, they suppose that v(µ) = (1,0,...,0), ∀µ ∈ D.
Our main result is,
Proposition 1. Suppose that Assumption A holds, (Wξ;ξ ∈ D) is a bounded plan and that there
is w ∈ RL
+ \ {0} such that, w(ξ) ≥ w, ∀ξ ∈ D. For each t ≥ 0, let st = 1
βt
P+∞
r=t+1 βr.
Then, the function U satisﬁes uniform impatience if and only if (st)t≥0 is bounded.
Proof. Assume that (Wξ)ξ∈D is a bounded plan. That is, there is W ∈ RL
+ such that, Wξ ≤
W, ∀ξ ∈ D. If (st)t≥0 is bounded, then there exists s > 0 such that, st ≤ s, for each t ≥ 0. Also,
since F := {x ∈ RL
+ : x ≤ W} is compact, the continuity of u assures that there is π ∈ (0,1)
such that u(x) − u(π0 x) ≤
u(W+w)−u(W)
2s , ∀x ∈ F, ∀π0 ≥ π. Thus, uniform impatience follows by
choosing δ = 1 and v(ξ) = w, ∀ξ ∈ D. Indeed, given a plan (x(µ);µ ∈ D) ∈ R
L×D
+ such that,
x(µ) ≤ Wµ ∀µ ∈ D, the concavity of u assures that, for any ξ ∈ D and π0 ≥ π,
X
µ>ξ
βt(µ)ρ(µ)u(x(µ)) −
X
µ>ξ
βt(µ)ρ(µ)u(π0x(µ)) ≤
βt(ξ)st
2s
ρ(ξ)
 
u(W + w) − u(W)

< βt(ξ)ρ(ξ)u(x(ξ) + v(ξ)) − βt(ξ)ρ(ξ)u(x(ξ)).
Reciprocally, suppose that uniform impatience property holds. Then, given (x(µ);µ ∈ D) ∈
R
L×D
+ such that, x(µ) ≤ Wµ, for all µ ∈ D, there are (π,δ) ∈ [0,1) × R++ and (v(µ);µ ∈ D) R
D×L
+
satisfying, for any ξ ∈ D, wh(ξ) ≥ δv(ξ), such that
1
βt(ξ)ρ(ξ)


X
µ>ξ
βt(µ)ρ(µ)u(x(µ)) −
X
µ>ξ
βt(µ)ρ(µ)u(πx(µ))

 < u(x(ξ) + v(ξ)) − u(x(ξ)), ∀ξ ∈ D.
It follows that, for any node ξ,
1
βt(ξ)ρ(ξ)


X
µ>ξ
βt(µ)ρ(µ)u(w) −
X
µ>ξ
βt(µ)ρ(µ)u(πw)

 < u

1 +
1
δ

W

.
Therefore, we conclude that, for any ξ ∈ D,
1
βt(ξ)
(u(w) − u(πw))
+∞ X
t=t(ξ)+1
βt < u

1 +
1
δ

W

,
which implies that the sequence (st)t≥0 is bounded. 
Suppose that Assumption A holds and initial endowments are bounded away from zero. If inter-
temporal discount factors are constant, i.e. ∃c ∈ R++ :
βt(ξ)+1
βt(ξ) = c, ∀ξ ∈ D, then c < 1 and
st = c
1−c, for each t ≥ 0. Therefore, in this case, if aggregated physical endowments are uniformly
bounded along the event-tree then U satisﬁes uniform impatience condition.INTER-TEMPORAL DISCOUNTING AND UNIFORM IMPATIENCE 3
However, even with bounded plans of endowments, uniform impatience is a restrictive condition
when inter-temporal discount factors are time varying. For instance, if we consider hyperbolic inter-
temporal discount factors, that is, βt = (1+at)− b
a, where b > a > 0, then the function U, as deﬁned
in the statement of Proposition 1, satisﬁes Assumption A and the sequence st goes to inﬁnity as t
increases. Therefore, in this case, uniform impatience does not hold.
References
[1] Hernandez, A., and M. Santos (1996): “Competitive Equilibria for Inﬁnite-Horizon Economies with Incomplete
Markets,” Journal of Economic Theory, 71, 102-130.
[2] Magill, M., and M. Quinzii (1994): “Inﬁnite Horizon Incomplete Markets,” Econometrica, 62, 853-880.
[3] Magill, M., and M. Quinzii (1996): “Incomplete Markets over an Inﬁnite Horizon: Long-lived Securities and
Speculative Bubbles,” Journal of Mathematical Economics, 26, 133-170.
[4] Laibson, D. (1998): “Life-cicle consumption and hyperbolic discount functions,” European Economic Review,
42, 861-871.
Faculdade de Economia, Universidade Nova de Lisboa
Travessa Estev˜ ao Pinto, 1099-032 Lisbon, Portugal.
E-mail address: pascoa@fe.unl.pt
Central Bank of Brazil and Department of Economics, PUC-Rio
Rua Marquˆ es de S˜ ao Vicente 225, G´ avea, 22453-900 Rio de Janeiro, Brazil.
E-mail address: myrian@econ.puc-rio.br
Department of Economics, Pontifical Catholic University of Rio de Janeiro (PUC-Rio)
Rua Marquˆ es de S˜ ao Vicente 225, G´ avea, 22453-900 Rio de Janeiro, Brazil.
E-mail address: jptorres martinez@econ.puc-rio.b r 
 
Departamento de Economia    PUC-Rio 
Pontifícia Universidade Católica do Rio de Janeiro 
Rua Marques de Sâo Vicente 225  - Rio de Janeiro 22453-900, RJ 
    Tel.(21) 35271078     Fax (21) 35271084 
www.econ.puc-rio.br 
flavia@econ.puc-rio.br 

Competitive Equilibria for Inﬁnite-Horizon Economies with Incomplete Markets,”

Incomplete Markets over an Inﬁnite Horizon: Long-lived Securities and Speculative Bubbles,”

Life-cicle consumption and hyperbolic discount functions,” European Economic Review, 42, 861-871. Faculdade de Economia, Universidade Nova de Lisboa Travessa Estev˜ ao Pinto, 1099-032 Lisbon, Portugal. E-mail address: pascoa@fe.unl.pt Central Bank

Inter-temporal discounting and uniform impatience

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