Inverse demand systems and welfare measurement in quantity space.

• Author: Kim, H. Youn

1. Introduction

   Most studies of welfare or cost-benefit analyses are concerned with the welfare effects of price changes [11; 19; 27; 33; 50]. There are, however, many situations in which policy options are directly related to quantity changes. The welfare effects of price changes are analyzed with the traditional demand system in which commodity quantities are determined as functions of their prices. The welfare effects of quantity changes, on the other hand, are associated with the inverse demand system in which commodity prices are dependent on their quantities. In conventional welfare analysis of price change, prices are taken to be exogenous or predetermined, while quantities are endogenous. In contrast, in welfare analysis of quantity changes, quantities are exogenous, while prices are endogenous. Price-based or dual welfare measures are relevant when there are well-functioning competitive markets and quantities are fully adjusted to changes in prices; on the other hand, quantity-based or primal welfare measures are useful in situations where there are constraints on commodity quantities, or when transaction costs impede consumers from fully adjusting to changes in prices.

   The choice between price- and quantity-based welfare measures is empirical, and proper measurement of welfare effects requires the knowledge as to which variable - price or quantity - is the exogenous one. For individual consumers, it may be reasonable to assume that the supply of commodities is perfectly elastic, and therefore prices can be taken as exogenous. But this assumption may not be tenable for consumers in the aggregate or if highly aggregated economy-wide data are used to estimate demand relations. At the aggregate level, quantities are more properly viewed as exogenous than are prices. Although individual consumers make their consumption decisions based on given prices, the quantities of commodities are predetermined by production at the market level and prices must adjust so that the available quantities are consumed [28].(1,2) This implies that although price-based measures are useful for analyzing the welfare of individual consumers, quantity-based measures may be more appropriate at the aggregate level.(3) Given the fact that most of the consumer demand studies based on time-series data involve the estimation of aggregate demand functions, there is a clear need for the inverse demand system and hence welfare analysis of quantity changes in empirical analysis. Moreover, while these results hinge on competitive behavior, quantity-based measures are essential for analyzing the welfare effects for non-competitive firm or industry behavior. For example, a monopoly is a price-maker and the relevant demand is an inverse, rather than direct, demand function, and welfare is analyzed in terms of the quantity of output [10; 48]. Furthermore, many (indivisible) investment projects entail direct changes in quantities (price changes occur indirectly); thus cost-benefit analysis of investment or public projects requires the use of quantity-based welfare measures [29].

   Quantity-based welfare measures are not totally new. Indeed, consumer surplus is often discussed for changes in price or quantity for a single commodity, and the Marshallian surplus measure (together with producer surplus) for quantity changes is used to analyze social welfare (or deadweight loss) or the welfare properties of market equilibrium [29; 48]. There are some limited empirical studies on consumer welfare for quantity changes using the Marshallian surplus. Rucker, Thurman, and Sumner [42] estimate the inverse demand function for tobacco which is subject to quantity restrictions (quotas) and investigate the welfare effect associated with changes in quotas. Bailey and Liu [2] estimate an inverse demand for airline services in which air fares are specified as a function of network scale and examine consumer welfare for changes in network scale. However, the Marshallian surplus is an approximate welfare measure for quantity changes, and there is no formal analysis of exact welfare measures pertinent to the inverse demand system for quantity changes.(4) This is in stark contrast to the literature on price-based welfare measures which provides well-established welfare measures for price changes [11; 13; 19; 27; 48].

   This paper seeks to fill this gap in the literature and presents exact measures of welfare change for the inverse demand system where the changes in welfare arise more reasonably from changes in quantity than in price. Welfare measures are characterized in terms of the distance function where quantities are specified as independent variables with the utility level held constant. The distance function yields the compensated inverse demand system in contrast to the direct utility function which underlies the uncompensated inverse demand system.(5) The distance function is dual to the expenditure function and can be considered a normalized "money metric" utility function; hence it is a natural tool to analyze the welfare effects of quantity changes. The distance function has been used in demand analysis [1; 21; 23; 49], in index numbers [14; 18], and in tax analysis [14; 15; 32; 47]; but it has not been used in welfare analysis. Using the distance function, exact welfare measures for quantity changes are developed by adapting Hicksian compensating and equivalent variations associated with price changes and are related to the compensated inverse demand system. These welfare measures are contrasted to the Marshallian (approximate) surplus measure derived from the uncompensated inverse demand system. The connection between price-based and quantity-based welfare measures is derived, and it is shown that when there are well-functioning competitive markets, quantity-based measures can be used to measure the welfare effects of price changes. Moreover, alternative measures of deadweight loss of monopoly and taxation are presented using the distance function instead of the expenditure function employed in earlier studies [27; 35]. An illustration is given to show the applicability of the proposed welfare measures and the quantitative magnitude of bias arising from the use of the Marshallian surplus instead of exact measures.

2. Uncompensated and Compensated Inverse Demand Systems and Duality Results

   Suppose that there exists a direct utility function u = F(X), which is assumed to be twice-continuously differentiable, increasing, and quasi-concave in X, a vector of commodities whose elements are [X.sub.i] (i = 1, . . ., n). Assuming that consumers are price-takers, consider the following optimization problem:

   [Mathematical Expression Omitted], (1)

   where [Mathematical Expression Omitted] is a vector of normalized prices whose elements are [Mathematical Expression Omitted] ([P.sub.i] is the price of the ith commodity and Y = [[Sigma].sub.i] [P.sub.i][X.sub.i] is income or expenditure on commodities). Its solution, summarized by the Hotelling-Wold identity [6; 13; 49], gives the (normalized) uncompensated inverse demand system [b.sub.i](X) (i = 1, . . ., n):

   [Mathematical Expression Omitted]. (2)

   Inverse demands measure shadow (or virtual) prices, or marginal valuation, or marginal willingness to pay for commodities by consumers. In equilibrium, marginal willingness to pay for a commodity equals its market price.

   Solving (2) for X implicitly gives the uncompensated direct demand system: [Mathematical Expression Omitted]. Equivalently, it can be obtained explicitly from the (normalized) indirect utility function [Mathematical Expression Omitted]:

   [Mathematical Expression Omitted] (3)

   by using Roy's identity [6; 13; 48]:

   [Mathematical Expression Omitted]. (4)

   The indirect utility function is continuous, decreasing, linearly homogeneous, and quasi-convex in [Mathematical Expression Omitted]. Equations (2) and (4) show that the uncompensated inverse and direct demand systems have similar structures. However, while the inverse demand system takes quantities as exogenous, the direct demand system treats prices as exogenous. The duality between the direct and indirect utility functions suggests that the direct utility function can be recovered from the indirect utility function. That is,

   [Mathematical Expression Omitted]. (5)

   Given the direct utility function, the distance function D(u, x) is defined as

   [Mathematical Expression Omitted], (6)

   which gives the maximum amount by which commodity quantities must be deflated or inflated to reach the indifference surface [46]. The utility function exists if and only if D(u, X) = F(X)/u = 1. The distance function is continuous, increasing, linearly homogeneous, and concave with respect to X, and decreasing in u. Given the distance function (6), the expenditure function [Mathematical Expression Omitted]) can be described as

   [Mathematical Expression Omitted] (7)

   if and only if the distance function is expressed as

   [Mathematical Expression Omitted] (8)

   [6; 46; 49]. The expenditure function is continuous, increasing, linearly homogeneous, and concave with respect to [Mathematical Expression Omitted], and increasing in u. These results imply that the distance function can be interpreted as a (normalized) expenditure function and that the two functions are dual to each other.

   Application of Shephard's lemma [6; 13; 31; 46] to the distance function yields the (normalized) compensated inverse demand system [a.sub.i](u, X) (i = 1, . . ., n):

   [Mathematical Expression Omitted]. (9)

   Unlike uncompensated inverse demands, compensated inverse demands are defined with the level of utility held constant. Linear homogeneity of D(u, X) implies that [a.sub.i](u, X) is homogeneous of degree zero in X, and the concavity implies that [a.sub.i](u, X) is negative and symmetric, i.e., [Delta][a.sub.i](u, X)/[Delta][X.sub.i] [less than] 0 and [Delta][a.sub.i](u, X)/[Delta][X.sub.j] = [Delta][a.sub.j](u, X)/[Delta][X.sub.i] (i [not equal to] j). Zero...