Optimal rationing within a heterogeneous population

• Author: Stéphane Gauthier,Philippe Choné
• Date: 01 June 2017
• DOI: http://doi.org/10.1111/jpet.12220
• Published date: 01 June 2017

Received: 10 June 2016 Accepted: 13 June 2016

DOI: 10.1111/jpet.12220

ARTICLE

Optimal rationing within a heterogeneous

population

Philippe Choné1Stéphane Gauthier2

1CREST-ENSAE

2PSEand University of Paris 1

PhilippeChoné, ENSAE, 3 avenue Pierre

Larousse,92245 Malakoff, France

(philippe.chone@ensae.fr).Stéphane Gauthier,

MSE,106–112 bd de l’Hôpital, 75013 Paris,

France(stephane.gauthier@univ-paris1.fr).

Agovernment agency delegates to a provider (hospital, medical gate-

keeper, school, social worker) the decision to supply a service or

treatment to individual recipients. The agency does not perfectly

know the distribution of individual treatment costs in the population.

The single-crossing property is not satisfied when the uncertainty

pertains to the dispersion of the distribution. We find that the pro-

vision of service should be distorted upward when the first-best effi-

cient number of recipients is sufficiently high.

1INTRODUCTION

We consider an agency in charge of supplying a service or a treatment to a population of potential recipients. Examples

include medical procedures for patients, after-school programs for low-income children, social care for the disabled,

or training programs for the unemployed.When the cost and the benefit of the treatment vary across individuals, effi-

ciency recommends to supply the service only to those with a low enough cost benefit ratio. When these variables are

well observed by the agency,rationing by denial can be used to implement the efficient policy.

In many instances, however,there remains substantial unobserved heterogeneity in cost and benefit conditional on

observable variables. The agency may then observe the number of treated recipients, but not their individual charac-

teristics. In this circumstance, it can rely on rationingby selection, i.e., leave the selection of recipients to the discretion

of a better informed service provider (Klein and Mayblin, 2012) and fund the system on the basis of the number of

treated recipients. However,because the provider’s preferences are in general not perfectly aligned with those of the

agency, the population of selected recipients typically departs from the first-best efficient recommendation. The lit-

erature generally assumes that the agency,when designing the provider’s compensation scheme, perfectly knows the

underlying distribution of cost and benefit in the population of potential recipients. For instance, the assumption is

made in Makris and Siciliani (2013) and Malcomson (2005), with the former article investigating provideraltruism and

the latter considering many treatment varieties—two issues not addressed here.

Perfect knowledge of the cost distribution is a strong assumption. Indeed, because of data availability, only few

studies have estimated the distribution of individual treatment costs in specific contexts. It is obviously difficult for

researchers and agencies to obtain information about all the individual characteristics that may affect treatment

costs. One first difficulty comes from a possible selection bias when only the characteristics of the treated recipients

are observed. Even assuming that a researcher observes all relevant variables for a particular sample of recipients,

econometric analysis only provides agencies with statistical estimates whose precision depends, among other

things, on the size of the considered sample.

Journal of Public Economic Theory.2017;732–738. wileyonlinelibrary.com/journal/jpet c

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