Elsevier

Economics Letters

Volume 125, Issue 2, November 2014, Pages 233-235
Economics Letters

On the optimality of not allocating

https://doi.org/10.1016/j.econlet.2014.09.016Get rights and content

Highlights

  • We study an environment where the first best efficient allocation is not feasible.

  • We characterize the second best allocation.

  • The seller can increase expected social surplus by sometimes not allocating.

  • No standard auction achieves the second best allocation.

Abstract

We show that the commitment to not allocate may be exploited by a seller/social planner to increase the expected social surplus that can be achieved in the sale of an indivisible unit.

Introduction

This note illustrates a novel strategic use of the option of not allocating. It has been well known since Myerson (1981) that in order to maximize revenues, the optimal mechanism may require the seller to retain the object. In a setting with externalities, Jehiel et al. (1996) have shown that the seller may be better off not selling at all. In the bargaining literature, it is known that the option of value destruction can be strategically exploited to improve the buyer’s bargaining position; see for instance Dasgupta and Maskin (2007). A common feature of the above papers is that not allocating, or voluntary destroying value, are instruments used by one of the participants in the mechanism to increase his/her own surplus at the expense of that of some other party. Instead, we point out that not allocating can be a tool to increase expected social surplus. This work is part of our research agenda on second best efficiency; see our companion papers Hernando-Veciana and Michelucci, 2011, Hernando-Veciana and Michelucci, 2013. Our approach differs from most of the literature on efficient auctions, which focuses on environments where the first best allocation is feasible; see Maskin (2003) for a review. From a technical point of view, we adapt the ironing techniques introduced by Myerson (1981) to characterize the second best allocation.

Section snippets

The model

One unit of an indivisible good is put up for sale to a set of 2 potential buyers. The seller’s value is assumed to be zero. Let s=(s1,s2)R2 be a vector where si corresponds to the realization of an independent random variable with distribution Fi and with a strictly positive density in a bounded support SiR. Buyer i1,2 privately observes si and gains a von Neumann–Morgenstern utility vi(s)p if she gets the good for sale at price p, and utility p if she does not get the good and pays a

Feasible allocations and first best efficiency

We are interested in the set of allocations that can be implemented. According to the revelation principle, there is no loss of generality when restricting to direct mechanisms. A direct mechanism is a pair of measurable functions (p,x), where p is an allocation and x:SR2 a payment function. Let an allocation be a measurable function p:S[0,1]2, where Si{1,2}Si, i{1,2}pi(s)1, and where pi(s) denotes the probability that the good is allocated to i when the vector of types is sS. We say

Second best efficiency and the optimality of not allocating

In our environment the unique symmetric equilibrium of standard auctions (e.g. FPA, SPA, EA) allocates the good to the buyer with highest type, who is the buyer with lowest value. Consequently, standard auctions implement the allocation that induces the lowest expected surplus among the allocations that always allocate the good to one of the buyers.

Definition

We say that an allocation p is second best efficient if it is feasible and it maximizes Si=12(α(s1+s2)+hi(si))pi(s)ds.

The first best allocation is

Conclusions

We provide a novel rationale for a seller/social planner to credibly commit to retain the object. Interestingly, reserve price and entry fees are not helpful in implementing the most efficient allocation because they are not conditional on the type (or bid) of all the buyers.

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There are more references available in the full text version of this article.

Cited by (2)

  • Inefficient rushes in auctions

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We would like to thank Jacob Goeree and Philippe Jehiel for helpful discussions. Angel Hernando-Veciana also acknowledges the financial support of the Spanish Ministry of Economics and Competitiveness through grant ECO2012-38863.

1

CERGE-EI, a joint workplace of Charles University and the Economics Institute of the Academy of Sciences of the Czech Republic, Politickych veznu 7, 111 21 Prague, Czech Republic.

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