When Borch's Theorem Does Not Apply: Some Key Implications of Market Incompleteness, with Policy Relevance Today

• Date: 01 October 2016
• DOI: http://doi.org/10.1111/sjoe.12162
• Author: Jacques Drèze
• Published date: 01 October 2016

Scand. J. of Economics 118(4), 755–784, 2016

DOI: 10.1111/sjoe.12162

When Borch’s Theorem Does Not Apply:

Some Key Implications of Market

Incompleteness, with Policy Relevance

Today ∗

Jacques Dr`

eze

CORE, Catholic University of Louvain, Louvain-la-Neuve, B-1348, Belgium

jacques.dreze@uclouvain.be

Abstract

Markets are incomplete when the assets available to the agents do not span the space of

future contingencies. In that case, competitive equilibria on the markets for assets and com-

modities fail (generically) to be constrained efficient. Pareto-superior allocations can be im-

plemented through price/wage rigidities and quantity constraints. However, nominal rigidities

are conducive to multiple equilibria, implying endogenous macroeconomic uncertainties that

compound the primitive (exogenous) uncertainties. This paper defines a temporary general

equilibrium for which there exists a set of equilibria defining an inflation – unemployment

locus. Various policy implications are drawn, with relevance to the current crisis.

Keywords: Constrained efficiency; coordination failures; multiple equilibria; price rigidities;

temporary equilibrium

JEL classification:D50; D52; D82

I. Introduction

On previous occasions, I have suggested that “the macroeconomic impli-

cations of microeconomics may be a more fruitful research topic than

the microeconomic foundations of macroeconomics” (quoted from Dr`

eze,

1987, Section 3.4). The present paper provides an original validation of

that claim. Indeed, Theorem 1 establishes the multiplicity of equilibrium

pairs of macro variables (underemployment and inflation) on the basis of

a model (temporary general equilibrium) rightly labeled micro. The multi-

plicity, a form of endogenous uncertainties, has theoretical as well as policy

relevance – as spelled out in Sections VIII and IX.

Theorem 1 is not original to this paper; it is proven in Dr`

eze (2013). The

originality comes from spelling out and integrating, within a basic standard

∗This is a revised version of the text distributed in Bergen in 2009 on the occasion of a

Borch Lecture and available as CORE DP 2013/09. I am grateful to Pierre Dehez, P. Jean-

Jacques Herings, Michael Magill, and two anonymous referees, as well as lecture attendees,

for helpful comments.

CThe editors of The Scandinavian Journal of Economics 2015.

756 When Borch’s theorem does not apply

model (Sections III and VII), some nested implications of market incom-

pleteness (of exceptions to Borch’s theorem; see Section II): constrained

suboptimality of market equilibria (Section IV); second-best price/wage

rigidities (Section V); multiplicity of equilibria (Section VI). These impli-

cations are insufficiently stressed by macro-theorists, and have not, to my

knowledge, been integrated in published work. One reason is their technical

complexity. In this paper, I aim to provide a readable account of the main

elements.

The emphasis is placed on the chain of implications from a specific

market imperfection (incompleteness), which is so patent that it does not

require further empirical documentation.1Of course, it is not claimed that

this is the single path towards spelling out the macro-implications of mi-

croeconomics. However, it is important enough to be privileged here – at

the cost of limiting to brief references the mention of complementary paths.

II. From Borch’s Theorem to Incomplete Market Economies

The theorem of Borch (1960) is the beautiful, transparent statement of a

basic property of efficient risk-sharing among a set of agents. Let Nagents

each be endowed with a random wealth prospect xiof known probability

distribution, and with preferences representable by the expectation of a

concave function of wealth; then, every efficient risk-sharing ar rangement

calls for pooling all risks and sharing the aggregate wealth X=ixi

among all the agents. The share of the aggregate risk borne by each agent

is allowed to depend upon the level of that risk, reflecting individual risk-

tolerances (a property of the utility functions).

Borch (1960) makes reference to the equally general result in Arrow

(1953), where random prospects are defined with reference to an underly-

ing set of Sexogenous states of the world. A wealth prospect is then de-

fined by an S-vector of state-dependent wealth levels. The state-dependent

preferences of an agent are assumed representable by a concave function

of such vectors. Arrow’s theorem states that every efficient risk-sharing

arrangement corresponds to a competitive equilibrium on the Smarkets for

claims contingent on the states (for Arrow securities). It is readily verified

that Arrow’s efficient arrangements satisfy Borch’s theorem.

Borch’s research was motivated by reinsurance problems. He noted mod-

estly that these provide a unique application of Arrow’s model, which

otherwise would miss an empirical counterpart: markets for contingent

claims are not common, and opportunities for risk-sharing remain limited

in today’s world.

1See Magill and Quinzii (1996) for further discussion of that theme.

CThe editors of The Scandinavian Journal of Economics 2015.