EQUILIBRIUM INDETERMINACY IN A MODEL OF CONSTRAINED FINANCIAL MARKETS

• Author: Luciana C. Fiorini
• Date: 01 August 2016
• Published date: 01 August 2016
• DOI: http://doi.org/10.1111/iere.12178

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 3, August 2016

EQUILIBRIUM INDETERMINACY IN A MODEL OF CONSTRAINED FINANCIAL

MARKETS∗

BYLUCIANA C. FIORINI1

University of Western Australia,Australia

I present a general equilibrium model with incomplete markets in which assets pay in units of a num´

eraire good. In

this economy, agents are constrained to negotiate the same amount of assets in different states of the world. Different

from the standard result of economies with real assets, equilibrium indeterminacy can arise, depending on the structure

of the financial markets. Equilibrium fails to be unique when it is not possible to transfer wealth between states in which

consumers trade a pair of assets that face the same restriction.

1. INTRODUCTION

General equilibrium models with incomplete markets (GEI) generalize the Arrow–Debreu

model by allowing limitations on the structure of financial markets that could represent trans-

action costs, asymmetry of information, or coordination failure. In standard GEI models, an

individual’s demand for assets is restricted only by budget constraints. In some markets, how-

ever, consumers are allowed to sign contracts in advance and precommit on the trade of some

assets. What would be the implications of precommitment in a GEI framework? In this article,

I address this question by modeling situations in which all agents commonly agree on con-

ditioning their demand for assets. More specifically, an agent’s choice regarding the quantity

of an asset determines her choice for the quantity of another asset that is sold in a different

state of the world. I provide necessary and sufficient conditions under which the equilibrium

becomes indeterminate. In economies where indeterminacy prevails, it is possible to affect the

equilibrium allocation by choosing some asset prices. As an alternative to lump-sum transfers,

redistribution of income can be performed by mechanisms that control asset prices.

I use a standard GEI in a finite-horizon pure exchange economy with at least three periods

of time. Consumers have perfect information about prices of commodities and assets, which

pay exclusively in units of the num´

eraire. Financial markets have a very special characteristic;

each individual’s demand for a particular asset, which I refer to as a restricted asset, may be

linearly dependent on her demand for another asset that is traded at the same period of time

but in a different state of the world. I analyze the effects of these restrictions on the equilibrium

outcome. I show that the presence of restricted assets increases the number of variables to be

arbitrarily chosen in the model. In addition to the standard normalization of one spot price in

each state of the world, there are also asset prices to be exogenously determined. Whether the

choice of asset prices affects the equilibrium outcome depends on the structure of the financial

markets. I define the concept of incomplete past history (IPH) to show that when at least one

restricted asset is traded in a state with IPH, the choice of its price affects the equilibrium

allocation. Intuitively, a state exhibits IPH when markets are incomplete on the path that leads

∗Manuscript received November 2012; revised February 2015.

1My special thanks go to Herakles Polemarchakis for his extremely valuable suggestions. I also thank Mario Tirelli,

Sergio Turner, Chris Jones, Jos´

e Alvaro Rodrigues Neto, and Ronald Stauber, the seminar participants at Brown

University, and two anonymous referees for their helpful remarks. All errors are my own responsibility.

Please address correspondence to: Luciana C. Fiorini, University of Western Australia, 35 Stirling Hwy, Crawley

WA 6009, Australia. E-mail: Luciana.Fiorini@uwa.edu.au.

857

C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

858 FIORINI

to that state. Insufficient financial instruments in the relevant path create indeterminacy because

there are not enough assets to compensate for changes in the exogenous prices.

There are several situations in which consumers decide to set a precommitment contract. A

typical example is life insurance with indeterminate premium. The issuer of the insurance agrees

on the delivery of a certain amount of death benefit, which is defined according to the amount of

insurance that has been sold. The insured should make regular premium payments that depend

on contingencies such as changes in her own mortality risk. The amount of insurance is then

preagreed, whereas its premium is state-contingent.

Some savings programs also make use of precommitments, as proposed by Benartzi and

Thaler (2004). In an experiment, they invited workers of three different companies to join an

alternative retirement program, Save More Tomorrow (SMarT). Enrolling at SMarT implied

that employees committed themselves to allocating a portion of their salary increases to their

retirement accounts. Ashraf et al. (2006) designed a similar product for a Philippine bank to be

offered to its clients, who would commit to restrict access to their own savings accounts. In both

cases, significant increases in savings rates were observed. As Benartzi and Thaler (2004) argue,

precommitment works well when agents tend to procrastinate in performing some actions, like

the decision to increase their savings.

My results show that when precommitment is undertaken and financial markets are not

sophisticated enough to prevent indeterminacy, price and allocation fluctuations can occur in

the absence of shocks on incomes or preferences. Offering the same type of precommitment

program will then have different consequences, depending on the type of economy to which it

is introduced.

1.1. Related Literature. In the GEI literature, indeterminacy has been mainly related to

how asset returns are specified. Simultaneous research on indeterminacy was carried out in

the 1980s; Cass (1985) shows that the existence of more households than assets gives room for

indeterminacy when assets prices and returns are fixed. In his acknowledgments, Cass states

that “... after this research was completed, Mas-Colell and Geanakoplos kindly let me see a

preliminary draft of their paper reporting a closely related result.” The model presented by

Geanakoplos and Mas-Colell (1989) is an intermediate case of Cass’s model, which has both

prices and yields either variable or fixed. In Geanakoplos and Mas-Colell (1989), only prices can

vary, whereas returns are specified in units of account. After the work of Cass, Geanakoplos and

Polemarchakis (1986) show that in economies with num´

eraire assets, the equilibrium outcome

is determinate. In an independent study, Balasko and Cass (1989) analyze indeterminacy by

allowing the economy to have more than two periods. Indeterminacy occurs even if only one

group of agents does not have access to complete markets, as in Balasko et al. (1990). The

existing literature has defined a clear separation between economies with real or nominal

assets. Because in my model assets deliver in units of the num´

eraire good, one would expect

that indeterminacy would not arise. However, if assets facing the same restriction are interpreted

as one single asset, then this single asset becomes a nominal asset in which indeterminate prices

represent nominal returns.

My main result requires the existence of at least three periods of time. In the case of overlap-

ping generations that live for three periods, Henriksen and Spear (2012) show that perfect risk

sharing implies a strongly stationary equilibrium, which is not achievable under the presence of

a productive asset. Similar to Henriksen and Spear’s (2012) model, the introduction of Arrow

securities in my model can eliminate wealth effects due to uncertainty.

Section 2 presents the model and the main result. Section 3 illustrates the model with two

examples. Section 4 concludes, with proofs left to the Appendix.

2. THE MODEL

In this economy, there are Tperiods of time, indexed by t=1,2,...,Twith T≥3 and

Iconsumers. The total number of states of the world since the first and the last period of