Echenique, Federico

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CORRESPONDENCE PRINCIPLE
FEDERICO ECHENIQUE
California Institute of Technology
The Correspondence Principle is the relation, which exists in certain
economic models, between comparative statics of equilibria and the
properties of out-of-equilibrium dynamics.
The Correspondence Principle (CP) implies that one obtains unam-
biguous comparative statics by selecting equilibria with desirable dy-
namic properties. Generally, the CP determines comparative statics in
models with a one-dimensional endogenous variable, and in monotone
multidimensional models. It does not determine comparative statics
in general multidimensional models, such as Walrasian general equilib-
rium models with more than two goods.
One-dimensional models.
The CP holds quite generally in one-dimensional models. Consider,
for example, a two-good economy with excess-demand function for good
1 given by z1, shown in the figure below. We fix the price of good 2;
by Walras’s Law the equilibrium prices are the zeroes of z1: there are
three equilibria, p11, p
2
1 and p
3
1.
z1(p1)
zˆ1(p1)
p1p11 p
2
1 p
3
1
Figure 1. Two-good economy.
Date: October 2, 2006.
I thank Kim Border and Matias Iaryczower for comments.
1
2 FEDERICO ECHENIQUE
Now consider the comparative-statics exercise of shifting excess de-
mand up to zˆ1. What is the effect on equilibrium price? Locally, the
price increases if the equilibrium is p11 or p
3
1, but it decreases if it is p
2
1.
The different comparative statics at p11 and p
2
1 corresponds exactly to
the different behavior of taˆtonnement dynamics after a small pertur-
bation: p11 is stable while p
2
1 is unstable.
The difference between comparative statics at p11 and at p
2
1 is easy
to explain. The comparative statics at p11 says: slightly larger prices
than p11 are reached by increasing excess demand, and smaller prices
are reached by decreasing excess demand. Since excess demand is zero
at p11, there must be positive excess demand at slightly larger prices and
negative excess demand at slightly smaller prices. Hence, taˆtonnement
dynamics, which respond to the sign of excess demand, converges to
p11 after a small perturbation from p
1
1. On the other hand, at p
2
1, larger
prices result from a decrease in excess demand; hence excess demand
is positive at larger prices. Similarly, excess demand is negative at
smaller prices. As a result, taˆtonnement dynamics will not approach
p21 after a small perturbation from p
2
1.
If the economy is subject to sporadic shocks, one should not observe
p21, the unstable equilibrium. Hence, as a consequence of the correspon-
dence between comparative statics and dynamics, one should expect an
increase in excess demand to produce an increase in equilibrium price.
I shall give a general statement of the Correspondence Principle for
the one-dimensional case. Consider a model where the endogenous
variable takes values in [0, 1] and equilibria are determined as the fixed
points of f(·, t) : [0, 1] → [0, 1]; t ∈ T ⊆ R is an exogenous parameter.
Assume that T is convex and that f is C1.
A selection of equilibria is a function e : T → [0, 1] such that e(t) =
f (e(t), t) for all t ∈ T . Say that a fixed point x ∈ [0, 1] is stable if
there is a neighborhood V of x such that any sequence {xn} satisfying
x0 ∈ V and xn+1 = f(xn) for n ≥ 1, converges to x. Say that x ∈ [0, 1]
is unstable if, for any neighborhood V of x, there is a neighborhood
W of x such that all sequences defined as above eventually lie in the
complement of W .
Proposition 1. Let f be monotone increasing in t. If e is a continuous
selection of equilibria that is strictly decreasing over some interval
[
t, t
]
,
then for all t ∈ (t, t), e(t) is unstable.
Multidimensional models.
The one-dimensional CP is a relation between the sign of the comparative-
statics change in prices, and the sign of excess demand for smaller and
larger prices. When more than one price is determined, this relation
REFERENCES 3
does not need to exist. Still, the CP holds for monotone models; mod-
els where the different dimensions of the endogenous variables are in
some sense complements. Monotone economic models stem mainly
from game theoretic models with strategic complementarities.
I proceed to give a statement of the CP. Consider a model where
the endogenous variable takes values in a compact rectangle X ⊆ Rn,
and equilibria are determined as the fixed points of f(·, t) : X → X;
t ∈ T ⊆ R is a parameter and T is convex.
Proposition 2. Let f be monotone increasing in (x, t) and let e be a
continuous selection of equilibria.
• If e is strictly decreasing over
[
t, t
]
⊆ T , then for all t ∈ (t, t),
e(t) is unstable.
• If e is strictly increasing over
[
t, t
]
, then for all t ∈ (t, t), if e(t)
is locally isolated, it is stable.
Literature
The CP was formulated by Paul Samuelson (1941; 1942; 1947), who
also coined the term (though Hicks (1939) stated the CP informally).
Samuelson formulated the one-dimensional CP. The version in Propo-
sition 1 is taken from Echenique (2000). Bassett, Maybee, and Quirk
(1968) study the scope of the CP. Arrow and Hahn (1971) present a
critical discussion of the CP, and, because it fails in economies with
more than two goods, conclude that “very few useful propositions are
derivable from this principle.” The monotone multidimensional CP
is from Echenique (2002), who presents a general version of Proposi-
tion 2. Echenique (2004) presents a CP that does not rely on con-
tinuous selections of equilibria. The CP is also effective in dynamic
optimization models (Brock, 1983; Burmeister and Long, 1977; Magill
and Sheinkman, 1979) and in models of international trade (Bhagwati,
Brecher, and Hatta, 1987).
References
Arrow, K. J., and F. H. Hahn (1971): General Competitive Anal-
ysis. San Francisco, Holden-Day.
Bassett, L., J. Maybee, and J. Quirk (1968): “Qualitative Eco-
nomics and the Scope of the Correspondence Principle,” Economet-
rica, 36(3/4), 544–563.
Bhagwati, J. N., R. A. Brecher, and T. Hatta (1987): “The
Global Correspondence Principle: A Generalization,” American Eco-
nomic Review, 77(1), 124–132.
4 REFERENCES
Brock, W. A. (1983): “A revised version of Samuelson’s Correspon-
dence Principle: Applications of recent results on the asymptotic
stability of optimal control to the problem of comparing long run
equilibria,” in Models of Economic Dynamics, ed. by H. F. Sonnen-
schein, vol. 264 of Lecture Notes in Economics and Mathematical
Systems, pp. 86–116. Springer-Verlag.
Burmeister, E., and N. V. Long (1977): “On Some Unresolved
Questions in Capital Theory: An Application of Samuelson’s Corre-
spondence Principle,” Quarterly Journal of Economics, 91(2), 289–
314.
Echenique, F. (2000): “Comparative Statics by Adaptive Dynamics
and The Correspondence Principle,” Working Paper No. E00-273,
Department of Economics, UC Berkeley.
(2002): “Comparative Statics by Adaptive Dynamics and The
Correspondence Principle,” Econometrica, 70(2), 833–844.
(2004): “A Weak Correspondence Principle for Models with
Complementarities,” Journal of Mathematical Economics, 40(1–2),
145–152.
Hicks, J. R. (1939): Value and Capital. Oxford: Clarendon Press.
Magill, M. J., and J. A. Sheinkman (1979): “Stability of Regular
Equilibria and the Correspondence Principle for symmetric varia-
tional problems,” International Economic Review, 20(2), 297–315.
Samuelson, P. A. (1941): “The Stability of Equilibrium: Compara-
tive Statics and Dynamics,” Econometrica, 9(2), 97–120.
(1942): “The Stability of Equilibrium: Linear and Nonlinear
Systems,” Econometrica, 10(1), 1–25.
(1947): Foundations of Economic Analysis. Harvard Univer-
sity Press.
Division of the Humanities and Social Sciences, MC 228-77, Califor-
nia Institute of Technology, Pasadena CA 91125, USA.
E-mail address: fede@hss.caltech.edu
URL: http://www.hss.caltech.edu/~fede/


Correspondence Principle

A revised version of Samuelson's Correspondence Principle: Applications of recent results on the asymptotic stability of optimal control to the problem of comparing long run equilibria,&quot;

A revised version of Samuelson’s Correspondence Principle: Applications of recent results on the asymptotic stability of optimal control to the problem of comparing long run equilibria,”

A Weak Correspondence Principle for Models with

A Weak Correspondence Principle for Models with Complementarities,&quot;

Comparative Statics by Adaptive Dynamics and The Correspondence

Comparative Statics by Adaptive Dynamics and The Correspondence Principle,”

Comparative Statics by Adaptive Dynamics and The Correspondence Principle,&quot;

Foundations of Economic Analysis. Harvard University Press. Division of the Humanities and Social Sciences,

On Some Unresolved Questions in Capital Theory: An Application of Samuelson's Correspondence Principle,&quot;

On Some Unresolved Questions in Capital Theory: An Application of Samuelson’s Correspondence Principle,”

Stability of Regular Equilibria and the Correspondence Principle for symmetric variational problems,”

Stability of Regular Equilibria and the Correspondence Principle for symmetric variational problems,&quot;

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