Inverse demand systems and welfare measurement in quantity space.

Kim, H. Youn

I. 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.

II. 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 homogeneity of (9) implies

[Mathematical Expression Omitted], (10)

where [Mathematical Expression Omitted], compensated price flexibility, with [Mathematical Expression Omitted] and sign [Mathematical Expression Omitted]. Two goods i and j are net q-complements if [Mathematical Expression Omitted] and net q-substitutes if [Mathematical Expression Omitted].

Solving (9) for X implicitly gives the compensated direct demand system [Mathematical Expression Omitted], which is equivalently obtained explicitly by applying Shephard lemma [6; 13; 46] to the expenditure function:

[Mathematical Expression Omitted]. (11)

Thus the compensated inverse and direct demand systems have similar structures, the difference being whether prices or quantities are exogenous.

To derive the relationship between compensated and uncompensated inverse demands, equate (2) and (9) and substitute u = F(X) into (9) to obtain

[b.sub.i](X) [equivalent to] [a.sub.i](F(X),X). (12)

Partial differentiation of (12) with respect to [X.sub.j] yields the Antonelli decomposition of the price effect of a quantity change into the substitution and scale effects:

[Delta][b.sub.i](X)/[Delta][X.sub.j] = [Delta][a.sub.i](u, X)/[Delta][X.sub.j] + ([Delta][a.sub.i](u, X)/[Delta]u)([Delta]F(X)/[Delta][X.sub.j]). (13)

In elasticity form, (13) becomes

[Mathematical Expression Omitted], (14)

where [[Eta].sub.ij] [equivalent to] [Delta] ln [b.sub.i] (X)/[Delta] ln [X.sub.j], uncompensated price flexibility, and [[Mu].sub.i] [equivalent to] ([Delta]ln[a.sub.i](u, X)/[Delta]ln u) ([[Sigma].sub.i] [Delta] ln F(X)/[Delta] ln [X.sub.i]), scale flexibility,(6) with [S.sub.i] (expenditure share of the ith commodity) [Mathematical Expression Omitted], derived from (2). Two goods i and j are gross q-complements if [[Eta].sub.ij] [greater than] 0 and gross q-substitutes if [[Eta].sub.ij] [less than] 0.(7) For a normal good, a change in quantities has a negative scale effect, i.e., [[Mu].sub.i] [less than] 0, with [[Mu].sub.i] = -1 for homothetic preferences. This implies that the uncompensated inverse demand is more quantity-elastic than the compensated inverse demand.

Since [Mathematical Expression Omitted], this implies the restriction on [[Eta].sub.ij]:

[summation over i] [[Eta].sub.ij][S.sub.i] = -[S.sub.j]. (15)

Summing (14) over j to satisfy (10) and noting that [[Sigma].sub.j][S.sub.j] = 1, we obtain the restriction on [[Mu].sub.i]:

[[Mu].sub.i] - [summation over j] [[Eta].sub.ij], (16)

which shows that the scale flexibility is obtained as the sum of the uncompensated price flexibilities. Moreover, summing (15) over j, we obtain the restriction on (16):

[summation over i][S.sub.i][[Mu].sub.i] = -1, (17)

which says that the weighted sum of the scale flexibilities (with the weights given by the expenditure shines) is equal to -1.

Equation (14) shows that when the expenditure share of a good is small or when a change in quantities has no scale effects, i.e., [[Mu].sub.i] = 0, the uncompensated and compensated inverse demands coincide. An issue of great concern is under what condition a change in quantities has no scale effects. This occurs when the indirect utility function is quasi-linear. In the case of two goods, the quasi-linear indirect utility function is of the form:

[Mathematical Expression Omitted], (18)

where indirect utility is linear in [Mathematical Expression Omitted] but nonlinear with respect to [Mathematical Expression Omitted], which implies that the (price) indifference curves are vertical translates of each other with respect to the [Mathematical Expression Omitted] axis.(8) Following (5), minimization of (18) with respect to [Mathematical Expression Omitted] and [Mathematical Expression Omitted] subject to [Mathematical Expression Omitted] yields [Mathematical Expression Omitted]. This implies that the inverse demand for [X.sub.1] is independent of the scale of the quantities of [X.sub.1] and [X.sub.2], in which case the uncompensated and compensated inverse demands for [X.sub.1] coincide.

III. Exact versus Approximate Measures of Welfare Change in Quantity Space

Consider parametric changes in commodity quantities to examine how they affect the welfare of consumers. Many circumstances in which quantity changes are relevant in welfare analysis are discussed in the Introduction. The distance function is a natural tool to define welfare measures for quantity changes; it is a normalized money metric utility function because [Mathematical Expression Omitted], which is a monotonic transformation of the direct utility function for fixed quantities X and is itself a utility function. Further, it is dual and symmetric to the expenditure function used to investigate the welfare effects of price changes. As such, it can be used to examine the welfare effects of quantity changes by adapting Hicksian measures of compensating and equivalent variations for price changes [11; 13; 19; 27; 48].

The compensating variation (CV) associated with a change in quantities from [X.sup.0] to [X.sup.1] is defined as

CV [equivalent] D([u.sup.0], [X.sup.1]) - D([u.sup.0], [X.sup.0]), (19)

which, upon using the Fundamental Theorem of Calculus and Shephard's 1emma (9), reduces to

[Mathematical Expression Omitted], (20)

where [u.sup.0] [equivalent to] F([X.sup.0]).(9,10) CV is the amount of additional (normalized) expenditure required for the consumer to reach the utility level [u.sup.0] while facing the quantity vector [X.sup.1]. When [X.sup.1] [less than] ([greater than])[X.sup.0], CV measures willingness to pay (accept). The consumer is clearly worse (better) off while facing quantities [X.sup.1] if CV is greater (less) than 0.

The equivalent variation (EV) of a change in quantities from [X.sup.0] to [X.sup.1] is defined as

EV [equivalent to] D([u.sup.1], [X.sup.1]) - D([u.sup.1], [X.sup.0]), (21)

which reduces to

[Mathematical Expression Omitted], (22)

where [u.sup.1] [equivalent to] F([X.sup.1]).(11,12) EV is the amount of additional (normalized) expenditure that would enable the consumer to maintain the new utility level [u.sup.1] while facing the initial quantities [X.sup.0]. When [X.sup.1] [less than] ([greater than])[X.sup.0], EV measures willingness to accept (pay). As in the case of the CV, a positive (negative) value of EV implies that the consumer is clearly worse (better) off under [X.sup.1] than under [X.sup.0].

These results show that the CV and EV of a quantity change are exact (normalized) measures of welfare change. Figure 1 illustrates the CV and EV associated with an increase in the quantity of one good [X.sub.1]. The indifference curve is defined over price space characterized by the indirect utility function (3). The slope of the budget line is the ratio of the commodity quantities - [X.sub.1]/[X.sub.2]. From Roy's identity (4), in equilibrium the slope of the (price) indifference curve is equal to the ratio of the quantities. The initial equilibrium is at A. With an increase in [X.sub.1], the new equilibrium occurs at B. Note that CV is conditional upon the utility level [u.sup.0], while EV is associated with the utility level [u.sup.1]. In general, the relationship between CV and EV cannot be ascertained.(13) However, when a change in quantities has no scale effects, the two welfare measures coincide. Moreover, (20) and (22) suggest that CV and EV can be measured by the area under the compensated inverse demand curve from [X.sup.0] to [X.sup.1] with the old and new utility levels, respectively. For an increase in the quantity of one good, the compensated inverse demand curve [a.sub.i]([u.sup.1], X) lies below the compensated inverse demand curve [a.sub.i]([u.sup.0], X) because of the negative scale effect when the good in question is a normal good. This implies that EV is smaller than CV for an increase in the quantity of one good. (This is illustrated in Figure 2 and will be discussed later.)

In contrast to CV and EV which can be described by the compensated inverse demand functions (9), the Marshallian surplus is expressed in terms of the uncompensated inverse demand functions (2). Formally, the Marshallian surplus (MS) associated with a change in quantities from [X.sup.0] to [X.sup.1] is defined as

[Mathematical Expression Omitted], (23)

which is the area below the uncompensated demand curve from [X.sup.0] to [X.sup.1]. When the scale flexibility is zero (see (13) or (14)), MS coincides with CV and EV, i.e., MS = CV = EV. When a change in quantity has a scale effect, however, MS will bias the true welfare change. For a normal good, the uncompensated inverse demand curve is steeper than the compensated curve - note that for the direct demand curve, the compensated demand curve is steeper than the uncompensated demand curve - implying that the MS associated with a quantity increase is bounded from below by the EV and from above by the CV(EV [less than] MS [less than] CV).

Figure 2 portrays MS in relation to CV and EV, using the inverse demand curves, for the case of a single quantity increase. The price axis pertains to a range of implicit prices corresponding to the domain of quantities being considered. [a.sub.i]([u.sup.0], X) and [a.sub.i]([u.sup.1], X) are the two compensated inverse demand curves corresponding to initial and new utility levels [u.sup.0] and [u.sup.1], while [b.sub.i](X) is the uncompensated inverse demand curve. The initial situation is at A, given by [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. The final situation is at B, given by [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. CV is shown by the area [Mathematical Expression Omitted] under the compensated inverse demand curve [a.sub.i]([u.sup.0], X). MS is the area [Mathematical Expression Omitted] under the uncompensated inverse demand curve [b.sub.i](X), which is bounded by CV and EV.

There is no previous analysis using the distance function to define CV or EV as a welfare change measure for quantity changes; instead MS is used as the relevant welfare measure. The issue is whether MS is a theoretically valid measure of welfare change. The result is well known for price-based welfare measures, which basically can be extended to quantity-based measures. The MS is a relevant welfare measure for quantity changes when preferences are homothetic or when a quantity change has no scale effects. Homothetic preferences are, however, unrealistic, and commodity demands are found to have pronounced scale effects [4; 21; 30]. Moreover, when many goods are considered, MS is not independent of the path of quantities chosen for integration since the associated uncompensated inverse demands are not symmetric in contrast to the compensated inverse demand functions associated with CV or EV. This implies that MS is an approximate welfare measure for quantity changes relative to CV or EV. Nevertheless, MS is employed as the relevant measure for quantity changes, especially in analysis of social welfare or welfare properties of market equilibrium.(14)

IV. Relationship between Welfare Measures in Price and Quantity Space

Quantity-based welfare measures are relevant when dealing with situations where there are constraints on quantities. Since the distance function is defined without regard to market conditions (see (6)), this implies that associated CV and EV measures do not intrinsically rely on competitive behavior. Price-based welfare measures, in contrast, are useful where there are well-functioning competitive markets such that quantities are fully adjusted to changes in prices. An issue of importance is whether quantity-based welfare measures can be used to investigate the welfare effects associated with price changes when consumers can freely adjust quantities in response to changes in prices. This section examines the relationship between quantity-based and price-based welfare measures.

Price-based welfare measures are well known [11; 19; 27; 50]. Briefly, the compensating variation ([CV.sub.P]) associated with a change in prices from [Mathematical Expression Omitted] to [Mathematical Expression Omitted] is defined as

[Mathematical Expression Omitted], (24)

which, upon using the Fundamental Theorem of Calculus and Shephard's lemma (11), reduces to

[Mathematical Expression Omitted], (25)

where [Mathematical Expression Omitted]. The equivalent variation ([EV.sub.P]) of a change in prices from [Mathematical Expression Omitted] to [Mathematical Expression Omitted] is defined as

[Mathematical Expression Omitted], (26)

which reduces to

[Mathematical Expression Omitted], (27)

where [Mathematical Expression Omitted].

Equations (25) and (27) suggest that the [CV.sub.P] and [EV.sub.P] are the areas to the left of the compensated direct demand curve for a change in prices from [Mathematical Expression Omitted] to [Mathematical Expression Omitted] associated with the utility levels [u.sup.0] and [u.sup.1], respectively. The Marshallian surplus ([MS.sub.P]) of a change in prices from [Mathematical Expression Omitted] to [Mathematical Expression Omitted], in contrast, is defined as

[Mathematical Expression Omitted], (28)

which is the area to the left of the uncompensated demand curve from [Mathematical Expression Omitted] to [Mathematical Expression Omitted].(15)

Consider now the CV of a price change (25) and integrate by parts to obtain

[Mathematical Expression Omitted], (29)

so that

[Mathematical Expression Omitted], (30)

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. This relationship shows that the CV of price changes can be obtained from the CV of quantity changes by allowing for some changes in expenditure. The same result holds for EV and the MS as well. These results suggest that when there are well-functioning competitive markets, the welfare effects of price changes can be estimated from quantity-based measures when an adjustment is made for changes in expenditures.(16)

An important application of this analysis is the measurement of welfare or deadweight loss due to monopoly or taxation. Harberger [26] did pioneering work on measuring the welfare loss of monopoly, and many studies have found that the welfare loss is inconsequential (see references in Bergson [5] and Kay [35]). Bergson [5], however, argues that the compensated demand curve is the appropriate one for welfare loss measurement and that the use of the ordinary demand curve in general biases the welfare loss estimates. In Figure 3, which rests on constant costs, monopoly equilibrium is at [Mathematical Expression Omitted] and [X.sup.M], while competitive equilibrium is at [Mathematical Expression Omitted] and [X.sup.C]. The deadweight loss of monopoly based on the ordinary demand schedule, [Mathematical Expression Omitted] or [b.sub.i](X), is given by the area QST. In contrast, the deadweight loss based on the compensated demand schedule, [Mathematical Expression Omitted] or [a.sub.i](u, X), is given by the area QRU. Clearly the use of the ordinary demand schedule will bias the true welfare loss derived from the compensated demand schedule. Hausman [27] has shown that while the Marshallian approximation by the ordinary demand schedule may be adequate for consumer welfare measurement in certain situations, it is often not accurate for measurement of the deadweight loss.

Following Bergson's [5] idea, Kay [35] proposes the use of the expenditure function to measure the true welfare loss of monopoly DWL. For many monopolized products, Kay's measure is given by

[Mathematical Expression Omitted], (31)

which reduces to

[Mathematical Expression Omitted], (32)

where [Mathematical Expression Omitted].

While Kay's measure is useful and is an improvement over earlier measures, it implicitly rests on the assumption that prices are exogenous, and direct demand functions are used. For a monopoly, inverse demand functions are more appropriate, which naturally requires the use of the distance function. The welfare loss measure based on the distance function is given by

[Mathematical Expression Omitted], (33)

which reduces to

[Mathematical Expression Omitted], (34)

where [Mathematical Expression Omitted].

These results can be directly applied to measure the deadweight loss of taxation. Diamond and McFadden [17] and Hausman [27] propose the use of the expenditure function; however, the distance function can yield an alternative measure of the deadweight loss of taxation.

V. An Illustration

This section discusses the applicability of the exact welfare measures, CV and EV, and illustrates the quantitative magnitude of bias arising from the use of the MS.

Consider an adaptation of Deaton and Muellbauer's [16] AIDS (Almost Ideal Demand System) form for the distance function [21]:

[Mathematical Expression Omitted], (35)

where [[Gamma].sub.ij] = [[Gamma].sub.ji] (i [not equal to] j) due to symmetry. Linear homogeneity implies the parametric restrictions: [[Sigma].sub.i] [[Alpha].sub.i] = 1, [[Sigma].sub.j] [[Gamma].sub.ij] = 0, and [[Sigma].sub.i] [[Beta].sub.i] = 0. Applying Shephard's lemma (9) to (35), we obtain the expenditure share equations:

[Mathematical Expression Omitted]. (36)

These equations are not in estimable form because utility is unobservable. To derive an estimable form, set D(u, X) = 1 in (35) and solve it for u to yield

[Mathematical Expression Omitted]. (37)

Substituting (37) into (36) for u gives

[S.sub.i] = [a.sub.i] + [[Gamma].sub.ij] ln [X.sub.j] + [[Beta].sub.i] ln Q, (38)

where

[Mathematical Expression Omitted]. (39)

In equation (38), the parameter can be estimated using observed quantities and expenditure shares.

With the expenditure share equation (38), the uncompensated price flexibilities are obtained as

[[Eta].sub.ii] = 1 + {[[Gamma].sub.ii] + [[Beta].sub.i]([S.sub.i] - [[Beta].sub.i] ln Q)}/[S.sub.i] (40)

and

[[Eta].sub.ij] = {[[Gamma].sub.ij] + [[Beta].sub.i]([S.sub.j] - [[Beta].sub.j] ln Q)}/[S.sub.i] (i [not equal to] j). (41)

The scale flexibilities are given by

[u.sub.i] = 1 + [[Beta].sub.i]/[S.sub.i]. (42)

From these uncompensated price and scale flexibilities, the compensated price flexibilities can be derived using the relation (14). Thus the AIDS distance function (35) is flexible and does not impose any restriction on price and scale flexibilities.

Once the parameters of the expenditure share equations (38) are estimated, they can be used to recover the distance function (35) and to derive CV and EV to analyze the welfare effects of parametric changes in commodity quantities.

To illustrate how the use of the MS biases the exact welfare measures, CV and EV, consider the consumer's preferences represented by the Cobb-Douglas utility function:

[Mathematical Expression Omitted], (43)

which gives the uncompensated inverse demands of the form:

[Mathematical Expression Omitted], (44)

where [Alpha] [equivalent to] [[Sigma].sub.i] [[Alpha].sub.i], the degree of homogeneity.

The distance function associated with (43) is

[Mathematical Expression Omitted], (45)

with compensated inverse demands:

[Mathematical Expression Omitted], (46)

which, upon substituting for u in (43), yield the uncompensated inverse demands (44). The uncompensated price flexibilities are given by [[Eta].sub.ii] = -1 and [[Eta].sub.ij] = 0 (i [not equal to] j), and the scale flexibility is [[Mu].sub.i] = -1, which implies the unitary (negative) scale flexibility, signifying homothetic preferences. The compensated price flexibilities are [Mathematical Expression Omitted] and [Mathematical Expression Omitted], which implies that all goods are net q-complements.

Table I. Comparison of CV, EV, and MS for Quantity Changes

[X.sub.1] EV MS CV

1 0.00 0.00 0.00
2 6.69 6.93 7.18
3 10.40 10.99 11.61
4 12.94 13.86 14.87
5 14.86 16.09 17.46

The CV and EV associated with a change in [X.sub.1] are given by

[Mathematical Expression Omitted] (47)

and

[Mathematical Expression Omitted]. (48)

On the other hand, the MS of a change in [X.sub.1] is obtained as

[Mathematical Expression Omitted]. (49)

These welfare measures are fairly simple and do not depend on variables other than that which effects a change.

Table I gives the welfare estimates for [[Alpha].sub.1] = 1/10 and [[Alpha].sub.2] = 9/10. Since the welfare measures defined in this study are normalized measures, for ease of understanding they are converted into "non-normalized" estimates by multiplying them by the expenditure level ($100). As is clear from the table, MS is not the same as EV and CV. Further, the change in MS always lies between EV and CV, which is the bounding relationship between the three welfare measures associated with a single quantity increase.

In contrast to the Cobb-Douglas utility function, other utility functions do not yield easily manipulable solutions for welfare measures. For instance, for the CES utility function the distance function and inverse (uncompensated and compensated) demand functions can be easily obtained. However, while CV and EV can be analytically derived, the MS measure is not analytically integrable. This suggests that the use of direct utility functions has a limited value in welfare analysis of quantity changes. Instead, a more appropriate procedure is to specify and estimate the distance function and derive welfare measures, as is shown with the AID distance function in this section. This procedure gives exact welfare measures, and thus no approximation is needed. In fact, it is the procedure exploited in welfare analysis of price changes in which the indirect utility function rather than the direct utility function is specified and exact welfare measures are derived [33; 37].(17)

VI. Summary and Conclusion

This paper has examined the measurement of welfare changes for the inverse demand system and provided exact welfare measures associated with quantity changes. There are many circumstances that warrant the use of quantity-based welfare measures, in contrast to the conventional price-based measures. The distance function is employed to develop compensating and equivalent variations for quantity changes, which are contrasted to the Marshallian surplus. Many results derived for quantity changes are parallel to those of welfare measures for price changes. In view of the increasing use of the inverse demand system and the distance function, welfare measures of quantity changes are of great importance in policy analysis. Moreover, quantity-based welfare measures can also deal with the welfare effects of price changes when there are well-functioning competitive markets.(18)

This research was supported by a Summer Faculty Research Fellowship from Western Kentucky University. The author wishes to thank the referee, John Wassom, and Dennis Hanseman for helpful comments and suggestions.

1. According to Hicks, "When we are studying the behavior of the individual consumer, it is natural to regard the former ('price into quantity,' i.e., direct demand) approach as primary, for the consumer is concerned with given prices on the market, and he chooses how much to purchase at a given price. But when we are studying market demand, the demand from the whole group of consumers of the commodity, the latter ('quantity into price,' i.e., inverse demand) approach becomes at least as important. For we then very commonly begin with a given supply, and what we require to know is the price at which that supply can be sold" [28, 83]. Katzner [34] argues that the inverse demand system may be useful to the economic planner since he may be interested in the prices required to clear the market of planned commodities. See Huang [30], Barten and Betterdoff [4], and Eales and Unnevehr [21] for the rationale of the use of the inverse demand system in food demands.

2. Bronsard and Salvas-Bronsard [9] examine whether a direct or inverse demand system is appropriate in empirical analysis and find that the level of commodity aggregation is important. In particular, their test rejects the exogeneity of prices in three-commodity models, but prices are often considered as exogenous at a more disaggregate level.

3. This is true in a general equilibrium view of the economy where total supply is fixed for the economy, while it is not fixed for individual consumers.

4. There is a growing literature on quantity-based welfare measures for the restricted or partial demand system in which some subset of commodities are subject to quantity restrictions. Hicks [28] originally introduced so-called compensating and equivalent surplus measures for this situation. Maler [40] shows that Hicksian compensating and equivalent variations defined for price changes can be readily adapted to welfare measures of quantity changes for a partial demand system. Randall and Stoll [42] demonstrate that with appropriate modifications, Willig's [50] formulas for bounds on compensating and equivalent variations for price changes carry over to welfare measures of quantity changes (see also Haneman [25]). Several studies have appeared to analyze quantity-constrained welfare effects arising from changes in the availability of nonmarket goods or environmental amenities, or changes in the fixed quantities of rationed goods or quotas [7; 8; 39]. However, the partial and inverse demand systems have different properties and also different welfare measures.

5. There has been an increasing use of the inverse demand system in applied demand analysis [4; 12; 21; 30; 41; 45], in noncompetitive firm analysis [3; 10; 20; 24; 48] and in hedonic price model [22; 43], all of which involve quantity (or quality) changes for welfare analysis.

6. For a good diagrammatical discussion of the scale flexibility and the Antonelli decomposition, see Anderson [1] and Cornes [13]. It may be noted that for the inverse demand system the income flexibility has no significance because it is equal to unity. This is in contrast to the partial demand system in which the income flexibility can take on any value [25; 42].

7. For a detailed discussion of gross and net substitutability or complementarity associated with inverse demand systems, see Kim [38].

8. A quasi-linear indirect utility function does not imply, nor is it implied by, the quasi-linear direct utility function [48, 164] which produces a zero income effect for the direct demand function.

9. CV is related to the Laspeyres-Malmquist quantity index [16; 18]. The Laspeyres-Malmquist quantity index [Q.sub.L]([X.sup.0], [X.sup.1]; [u.sup.0]) is defined as

[Q.sub.L]([X.sup.0], [X.sup.1]; [u.sup.0]) [equivalent to] D([u.sup.0], [X.sup.1])/D([u.sup.0], [X.sup.0]).

The relationship between CV and the Laspeyres-Malmquist quantity index is given by

CV - {D([u.sup.0], [X.sup.1])/D([u.sup.0], [X.sup.0])}D([u.sup.0],[X.sup.0]) - D([u.sup.0], [X.sup.0]) = [[Q.sub.L]([X.sup.0], [X.sup.1]; [u.sup.0]) - 1]D([u.sup.0],[X.sup.0]).

10. All welfare measures in this analysis are expressed in a normalized form. They can be convened into "non-normalized" measures by multiplying them by income or expenditure. For example, a non-normalized CV is given by

CV [equivalent to] Y[D([u.sup.0], [X.sup.1]) - D([u.sup.0], [X.sup.0])].

where [Y.sup.0] is income before a change in quantities.

11. EV is related to the Paasche-Malmquist quantity index [16; 18]. The Paasche-Malmquist quantity index [Q.sub.P]([X.sup.0], [X.sup.1]; [u.sup.1]) is defined as

[Q.sub.P]([X.sup.0], [X.sup.1]; [u.sup.1]) [equivalent to] D([u.sup.1], [X.sup.1])/D([u.sup.1], [X.sup.0]).

The relationship between EV and the Paasche-Malmquist quantity index is given by

EV = D([u.sup.1], [X.sup.1]) - {D([u.sup.1], [X.sup.0])/D([u.sup.1], [X.sup.1])}D([u.sup.1], [X.sup.1]) = [1 - {1/[Q.sub.P]([X.sup.0], [X.sup.1]; [u.sup.1])}]D([u.sup.1], [X.sup.1]).

12. For all welfare measures in this analysis, it is assumed that income remains unchanged when quantities of commodities change. However, when income changes, an adjustment must be made. For example, when income changes, CV and EV are defined by

CV [equivalent to] [Y.sup.0][D([u.sup.0], [X.sup.1]) - D([u.sup.0], [X.sup.0])] - [Y.sup.1] - [Y.sup.0]),

and

EV [equivalent to] [Y.sup.1][D([u.sup.1], [X.sup.1]) - D([u.sup.1], [X.sup.0])] - [Y.sup.1] - [Y.sup.0]),

where [Y.sup.0] and [Y.sup.1] are income before and after a change in quantities.

13. When preferences are homothetic (such that [Q.sub.L]([X.sup.0], [X.sup.1]; [u.sup.0]) = [Q.sub.p]([X.sup.0], [X.sup.1]; [u.sup.1]) = Q([X.sup.0], [X.sup.1])) and D([u.sup.0], [X.sup.0]) = D([u.sup.1], [X.sup.1]), CV and EV are related to each other by

CV = EV x Q([X.sup.0], [X.sup.1]).

14. Hotelling [29], in his pioneering study on welfare, addresses the relevance of total surplus defined as the sum of consumer and producer surpluses as a social welfare measure, and shows that the required condition is that the inverse demand and supply functions be integrable. The inverse supply or marginal cost functions are integrable because they are symmetric. In the case of demand, the integrability conditions hold only for the compensated inverse demand functions because they are symmetric. Hotelling, however, does not consider the compensated inverse demand functions. An implication of this discussion is that the conventional measure of total surplus based on the Marshallian consumer surplus derived from the uncompensated inverse demand function is biased in relation to the exact measure derived from the compensated inverse demand function.

15. While Figure 2 can be used to describe CV and EV for price changes, it cannot be used to describe the MS for price changes because the uncompensated direct demand curve has a steeper slope than the compensated direct demand curve, whereas the uncompensated inverse demand curve has a steeper slope than the compensated inverse demand curve.

16. Equation (30) also suggests that when there are well-functioning markets, the welfare effects of quantity changes can be estimated from price-based measures by allowing for some changes in expenditures.

17. An alternative procedure is to specify and estimate the uncompensated inverse demand system and derive the distance function, which gives the CV and EV. This procedure is employed by Hausman [27] to evaluate the welfare effects of price changes. The problem with this approach is that unless a simple demand function is specified, the distance function cannot be analytically derived.

18. While the analysis in this paper is conducted in a static framework, its extension to a dynamic or intertemporal framework is possible, adapting the line of inquiry for price-based welfare measures [36].

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