INCENTIVE EFFICIENT PRICE SYSTEMS IN LARGE INSURANCE ECONOMIES WITH ADVERSE SELECTION

• DOI: http://doi.org/10.1111/iere.12184
• Author: Paolo Siconolfi,Alessandro Citanna
• Published date: 01 August 2016
• Date: 01 August 2016

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 3, August 2016

INCENTIVE EFFICIENT PRICE SYSTEMS IN LARGE INSURANCE ECONOMIES

WITH ADVERSE SELECTION∗

BYALESSANDRO CITANNA AND PAOLO SICONOLFI1

Yeshiva University,U.S.A.; Columbia Business School,U.S.A.

We decentralize incentive efficient allocations in large adverse selection economies by introducing a competitive

market for mechanisms, that is, for menus of contracts. Facing a budget constraint, informed individuals purchase

(lottery) tickets to enter mechanisms, whereas firms sell tickets and supply slots at mechanisms at given prices. Beyond

optimization, market clearing, and rational expectations, an equilibrium requires that firms cannot favorably change,

or cut, prices. An equilibrium exists and is incentive efficient. An equilibrium can be computed as the solution to a

programming problem that selects the incentive efficient outcome preferred by the highest type within an appropriately

defined set. For two-types economies, this is the only equilibrium outcome.

1. INTRODUCTION

In a seminal paper, Prescott and Townsend (1984a, 1984b; hereafter PT) first laid out the ques-

tion of whether constrained optimal allocations could be decentralized through linear prices,

linking explicitly the problem of mechanism design to the description of a competitive market.

This article continues this line of investigation, with a focus on large insurance economies of

adverse selection. As the baseline story, we use a Rothschild and Stiglitz (1976; hereafter RS)

economy.

We introduce a competitive market for mechanisms. A mechanism is a menu of contracts,

one for each type of individual in the population and crucially may allow randomizations and

cross-subsidization across types participating in the mechanism. Contracts are policies specifying

insurance premium and benefit levels and are exclusive; that is, individuals can participate at

most in one mechanism and then choose one among the contracts in the menu.

A competitive market is an exchange system based on prices, where agents’ choice is con-

strained by a budget, and prices are linear in the objects of trade. We take mechanisms as

commodities, measures over mechanisms as commodity vectors, and lotteries over mechanisms

as consumption bundles. The choice of lotteries as consumption bundles is natural in this con-

text where contracts are bilateral and exclusive (see also PT). Prices are linear functional over

the commodity space; thus, the price of a lottery is the average price of mechanisms in its sup-

port. Individuals purchase budget-feasible lottery tickets. Once lotteries are purchased, their

mechanism outcome realizes, and agents choose which contract to commit to.2

Competitive equilibria are usually defined via three standard requirements: optimization,

market clearing, and rational expectations. With adverse selection, the set of such equilibria

can be quite large, as we demonstrate later. To restrict the equilibrium set, we embed in the

∗Manuscript received September 2014; revised February 2015.

1This is a revised version of the working paper presented first at the 6th annual Cowles Conference on General

Equilibrium and Its Applications, 2010. We thank seminar participants at various institutions, the editor of this journal,

and two referees for their comments. The first author is also grateful to Columbia Business School for the hospitality

throughout the development of this project.

Please address correspondence to: Alessandro Citanna, Yeshiva University, 245 Lexington Ave., New York, NY

10016 U.S.A. E-mail: citanna@yu.edu.

2Thus, lotteries are not introduced to provide randomizations in outcomes, as these can be already achieved partici-

pating in a single random mechanism. They are introduced to make the market structure competitive.

1027

C

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

and Economic Research Association

1028 CITANNA AND SICONOLFI

equilibrium definition a no-price-cuts criterion: At equilibrium, firms do not have an incentive

to change the price of any mechanism, expecting that other firms may exit only if it is in

their interest to do so, and stay otherwise. This criterion echoes Wilson’s (1977) notion of

“anticipatory equilibrium.”

We insist that our equilibrium notion is perfectly competitive even though, and unlike for pure

Walrasian models, there is no passive price-taking behavior on the firms’ side. In our equilibrium,

agents, in particular firms, take prices as given only because they do not have an incentive to

change them. This is close in spirit to Makowski and Ostroy (2001, and references therein),

who disentangle passive price-taking behavior from competition by proposing a competitive

equilibrium notion where Walrasian prices must satisfy a no-surplus criterion: Agents take

prices as given only because they do not have monopoly power; that is, they cannot favorably

change prices.

With this market setting and equilibrium notion, we show that the price system achieves

incentive efficiency at equilibrium: Prices are such that all agents participate in a constrained

efficient mechanism, so that a first welfare theorem holds. Moreover, an equilibrium always

exists. With two types, the equilibrium is also payoff unique. It is the separating mechanism

most preferred by the high-quality types among the constrained optimal allocations giving at

least the RS contract utility to the low-quality types. With more than two types, an equilibrium

exists under monotonicity and sorting, mild generalizations of conditions used in the related

literature. In fact, we show that a generalization, which we call the ¯

T-CPO, of the outcome

obtained with two types is always an equilibrium outcome. The ¯

T-CPO generally asks for

cross-subsidies and, with type-dependent utilities, for randomizations.

Our result shows that there always exists an equilibrium outcome with degenerate lotteries,

where the equilibrium mechanism provides the randomizations, if necessary, for incentive effi-

ciency. However, lotteries already provide all possible randomizations, questioning the need for

randomizations to enter at the commodity set level. In different, but equivalent, words, one may

wonder why we cannot reduce the number of commodities, thereby increasing price linearity.

We show that decentralization may fail if commodities are only deterministic mechanisms.

1.1. Related Literature. There is no agreed-upon notion of competition or decentralization

in economies with adverse selection. RS introduced a notion of competition among contract

designers and obtained the disconcerting result that equilibrium may not exist and may not

be constrained efficient. Later, Wilson (1977), Riley (1979), and Hellwig (1987) did propose

versions of the notion of competition that regained existence. Most closely related to this article,

Miyazaki (1977; see also Crocker and Snow, 1985), using C. Wilson’s notion of equilibrium,

studied a labor market economy with adverse selection (with only two types) where firms offer

mechanisms and obtained incentive efficiency of equilibrium.3Our equilibrium notion has much

in common with Wilson–Miyazaki’s. Both notions rule out the existence of changes that, after

loss-making firms have exited the markets, generate positive profits. Those changes are in the

set of offered mechanisms in Wilson–Miyazaki and in market clearing prices in our setting.

However, in Wilson–Miyazaki and all other game-theoretic analyses, trade occurs always only

exclusively, bilaterally between firms and individuals, who choose only one out of the set of

mechanisms offered by the firms. Our allocation system instead is price based, as it is the price

system that coordinates agents’ behavior. Thus, part of our contribution consists in providing a

price-based version of Wilson’s equilibrium. At the very least, we extend the existing analysis

to economies with multiple types and states and type-dependent utilities.

Gale (1992, 1996), Dubey and Geanakoplos (2002), Zame (2007), and Guerrieri et al. (2010)

also explored a notion of competitive market for contracts and proposed some stability-based

refinement of beliefs to pin down the equilibrium, delivering coherence of incentives and

3More recently, several papers (see, e.g., Diasakos and Koufopoulos, 2011; Netzer and Scheuer, 2014) give game-

theoretic foundations to the Wilson–Miyazaki equilibrium by providing extensive form games whose unique equilibrium

outcome is Wilson–Miyazaki.