Rationed price controls and “prices versus quantities”

• Published date: 01 December 2022
• Author: John Bennett
• Date: 01 December 2022
• DOI: http://doi.org/10.1111/jpet.12619

Received: 18 June 2021

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Accepted: 21 August 2022

DOI: 10.1111/jpet.12619

ORIGINAL ARTICLE

Rationed price controls and “prices versus

quantities”

John Bennett

Department of Economics, Royal

Holloway, University of London,

London, UK

Correspondence

John Bennett, Department of Economics,

Royal Holloway, University of London,

London, UK.

Email: j.bennett@rhul.ac.uk

Abstract

We interpret Weitzman's prices‐versus‐quantities

model as applying to a private‐good market, and we

formulate a “rationed price control,”with quantity

determined endogenously as the minimum of supply

and demand. With efficient rationing and no

externalities, the optimal rationed price control

exceeds the price at the expected demand–supply

intersection if (the numerical value of) the demand

slope curve exceeds the supply slope, and is less than

this level for the reverse ordering of slopes. If

uncertain demand is steeper than deterministic

supply, or if uncertain supply is steeper than

deterministic demand, expected welfare is greater

with the optimal rationed price control than with the

optimal quantity control. We obtain, for any speci-

fied price level, a partial ranking of the rationed price

control and alternative simple price controls. Ex-

tending the model to incorporate externalities, we

show that, for demand steeper than supply, with a

negative externality on either benefit or cost, the

optimal rationed price control exceeds the price at

the expected intersection of marginal social benefit

and marginal social cost; a converse result also

obtains. For an example with inefficient rationing,

we show that the optimal pure price control always

exceeds the price at the expected intersection of

demand and supply.

J Public Econ Theory. 2022;24:1364–1385.wileyonlinelibrary.com/journal/jpet1364

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© 2022 Wiley Periodicals LLC.

1|INTRODUCTION

In a seminal paper Weitzman (1974) compares a price control with a quantity control when a

regulator has incomplete information on the costs and benefits of a good whereas producers

possess full information. He develops his formulation with reference to a large organization or

planned economy, but notes that the model can been interpreted in numerous scenarios,

including the choice between emissions standards and pollution taxes, and in the context of

supply and demand in a market economy. An extensive theoretical literature has developed

using this framework, including with regard to pollution control (Karp & Traeger, 2019),

monopoly regulation (Basso et al., 2017; Chen, 1990; Freixas, 1980), cartel behavior

(Yohe, 1977), and bank regulation (Weitzman, 2018). Hepburn (2006) and Basso et al. discuss,

with examples, the practical operation of price and quantity controls.

Weitzman assumes that if a price control is announced it is transmitted to producers, who

are then assumed to choose the output at which price equals marginal cost. This formulation

(which we call “supply‐side”price control) has been applied extensively in the context of public

goods or externalities. However, in the case of a private good, there may be excess supply at the

price set. In development of Weitzman's framework, Laffont (1977) argues that the price

control could instead be announced to consumers, who determine the level of output that

maximizes their benefit, net of price (see also Schlee, 2013). The information is transmitted to

producers, who are assumed to make this level of output (we call this demand‐side price

control). Although, in the case of a private good, this could in principle leave the market with

excess demand, it is often assumed in the regulation literature that the regulated firms must

meet consumer demand, and regulation in practice may include this stipulation (see, e.g.,

Armstrong & Sappington, 2007; Lewis & Sappington, 1988).

The present paper follows Weitzman's model directly, but interpreted specifically for a

private‐good market. We consider an alternative policy whereby, when a price control is set,

quantity is left to be determined by the smaller of the realized quantities demanded and

supplied. Except in a knife‐edge case, the market will then be in disequilibrium and one side of

the market will be rationed. The regulator chooses the level of the price control consciously

exploiting the potential market disequilibria that may follow. With this “rationed price

control,”marginal revenue is constant up to the quantity at which demand is fully satisfied.

Thus, a firm's market power is neutralized in this quantity range and the model applies as well

to a monopoly as to an oligopoly—though, for simplicity, we refer throughout to the “supply

curve.”

1

We keep our analysis to the basic formulation of the prices‐versus‐quantities problem,

not taking into account potential generalizations such as correlated uncertainty of the demand

and supply curves (see Stavins, 1996) or incomplete enforcement (Montero, 2002).

Our initial formulation, in Section 2, is based on the assumptions that there are no

externalities on either the cost or the benefit side and that any rationing is efficient. We assume

that demand is downward‐sloping and supply upward‐sloping. We show that, unlike a supply‐

side or a demand‐side price control, the optimal (expected welfare‐maximizing) rationed price

control is not located at the intersection of the regulator's expected demand and supply curves.

If the supply curve is steeper than the demand curve, the optimal rationed price control is

located below this intersection and exceeds expected marginal cost.

2

If the demand curve is the

1

The model may also apply in the context of a large organization or a planned economy.

2

Throughout we refer to the numerical value of the slope of demand as its steepness.

BENNETT

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