Optimal (Under-)Pricing and Allocation of Publicly Provided Goods

David Scrogin

doi:10.4236/tel.2017.74049

Abstract

This paper exploits the links between a private value distribution’s hazard rate, mean residual value, and eta functions in order to characterize posted-price rules for a public agency to allocate scarce units of an indivisible good under the utilitarian distributional objective of maximizing expected consumer surplus. Sufficient conditions on the monotonic and non-monotonic classes of the functions are established that identify either market assignment at the clearing price or lottery assignment with partial or complete under-pricing as the optimal allocation mechanism. The results are summarized across a wide range of parametric value distributions, and selected non-monotonic cases are evaluated numerically to determine the relative scarcity or abundance of the good necessary for market or non-market assignment to dominate.

Highlights

Public agencies are commonly commissioned with allocating scarce units of indivisible goods and services over large numbers of individuals or households
In the mechanism design literature, [6] considered the case of two individuals with independent private valuations competing for a single unit of a good and demonstrated that expected net efficiency is maximized by random assignment if individuals incur sufficient signaling costs and the hazard rate of the distribution of individual private valuations is increasing. [7] found random assignment to be optimal given an arbitrary number of ex ante identical individuals and a value distribution with an increasing hazard rate, whereas market assignment is optimal if sufficient screening costs are incurred by the agency and the hazard rate is decreasing
In settings in which the agency has a limited number of units to be allocated over a large population of individuals, if private values are beta distributed with α < 1, the expected consumer surplus from screening low-valued individuals by posting a non-zero price and assigning units by lottery will exceed that resulting from selling the units outright

Summary

Introduction

Public agencies are commonly commissioned with allocating scarce units of indivisible goods and services over large numbers of individuals or households. In the mechanism design literature, [6] considered the case of two individuals with independent private valuations competing for a single unit of a good and demonstrated that expected net efficiency is maximized by random assignment if individuals incur sufficient signaling costs and the hazard rate of the distribution of individual private valuations is increasing. The analysis extends results from [10]’s study of the effects of price controls on consumer surplus in competitive markets and demonstrates that maximizing ECS may be consistent with the agency achieving alternative objectives in allocating units of indivisible goods and services, such as maximizing revenues, net efficiency, or distributional equity. The results are summarized across a wide range of parametric distributions, and selected non-monotonic cases are evaluated numerically to determine the relative scarcity or abundance of the good necessary for market or non-market assignment to dominate

The Setting and Allocation Problem

Results

Numerical Examples

Log-Logistic Distribution

Beta Distribution

Conclusions