Measuring the Value of a Nonmarket Good Using Market Data: Theory and Practice

Transcription

1 Measuring the Value of a Nonmarket Good Using Market Data: Theory and Practice Proposed running title: Measuring Nonmarket Good Value David S. Bullock * Assistant Professor Department of Agricultural and Consumer Economics University of Illinois Nicholas Minot International Food Policy Research Institute June 25, 1996 All correspondence should be addressed to: David S. Bullock Department of Agricultural and Consumer Economics University of Illinois 305 Mumford Hall 1301 W. Gregory Drive Urbana, IL dsbulloc@uiuc.edu phone: (217) fax: (217)

2 Abstract Weak complementarity and weak neutrality have been suggested as sometimes reasonable impositions on preferences that permit measurement of the welfare effects of changes in a nonmarket good. We present a general theoretical context for the measurement of the value of a nonmarket good using market data, and use this general theoretical context to provide general definitions of weak complementarity and weak neutrality, and we illustrate and analyze the relationship between these preference restrictions. We present a new method of using either weak complementarity or weak neutrality to measure the value of nonmarket goods using market data. Our numerical method is more flexible than the existing analytical approach because it can be used with any Marshallian demand function. Our approach uses minimal human and computer effort and can measure the value of nonmarket goods with any desired degree of accuracy. 2

3 List of symbols p 0 = (p 1 0,..., p n-1 0 ) vector observed prices of market goods i = 1,..., n -1 x = (x 1,..., x n-1 ) vector of quantities demanded of market goods i = 1,..., n - 1 x n : a numeraire (composite) good y 0 : observed income level z: a parameter describing (the quantity or quality of) some nonmarket good z 0 : initial level of nonmarket good parameter z 1 : subsequent level of nonmarket good parameter u(x, z) individual s utility function u 0 utility level at the observed prices, income, and nonmarket good parameter level m(p, z, u): expenditure function x i (p, z, y): Marshallian demand function for good i x i c (p, z, u): Hicksian demand function for good I p*, p**, p***: vectors of constant prices. L: an arbitrary path of integration L straight : a straight-line path of integration S 1 : a subpath of integration with endpoints (p 0, z 1 ) and (p**, z 1 ) S 2 : a subpath of integration with endpoints (p**, z 1 ) and (p***, z 0 ) S 3 : a subpath of integration with endpoints (p***, z 0 ) and (p 0, z 0 ) ˆ p (z, u): Hicksian choke price function S choke : the choke-price subpath of integration ε: an arbitrarily small constant S ε-choke : subpath of integration along which Hicksian quantities demanded equal a small number, ε p (z, u): function used to define S ε-choke parametrically D 1, D 2, D 3, B 1, B 2, B 3 : line integrals (see equation (4)) α: constant in Marshallian demand function α, β, γ, δ: constant, price, quality, and income coefficients in Marshallian demand function θ(z, u): constant of integration for expenditure function TV(z 0, z 1 ): total value of the change in the nonmarket good parameter z NUV: nonuse value of the change in the nonmarket good parameter z UV: use value of the change in the nonmarket good parameter z m (p,z,θ(z,u)): quasi-expenditure function p: incremental price change in numerical algorithm S j : approximation of willingness to pay for p in the jth iteration of the numerical algorithm z: incremental quality change in the numerical algorithm 3

4 Measuring the Value of a Nonmarket Good Using Market Data: Theory and Practice INTRODUCTION The limitations of the contingent valuation method being well known, economists sometimes desire to use market data to measure the effects of a change in a nonmarket good (perhaps environmental quality) on the welfare of individuals. The theoretical method is to use information available about the expenditure function or its accompanying (Hicksian) demand curves for market goods to obtain compensating or equivalent variation measures of a change in a nonmarket good. As is made clear in Larson [7, 8] and LaFrance and Hanneman [6, p. 272], such exact welfare measures cannot be obtained unless further restrictions are imposed on preferences. So far in the literature, weak complementarity and weak neutrality have been suggested as sometimes reasonable impositions on preferences that permit the use of exact welfare measures. However, because the theory of measurement of the welfare effects of nonmarket goods has not been laid out in a sufficiently general context, the relationship between weak complementarity and weak neutrality remains poorly understood. Also, different definitions of these important concepts have appeared. An analytical method for using weak complementarity or weak neutrality assumptions to recover Hicksian demands and their associated quasi-expenditure function is present in the literature. But this analytical method is limited because only a few demand functions can be integrated back to obtain a quasiexpenditure function, and fewer still allow the derivation of the constant of integration under weak neutrality or weak complementarity. A numerical method using weak neutrality has been proposed in the literature, but we show that this method is not actually applicable. In this paper, we use line integral theory to first lay out in a general theoretical context what is needed to measure the welfare effects of a change in a nonmarket good. We use this general theoretical context to provide general definitions of weak complementarity and weak neutrality, we explore the relationship between these preference restrictions, and we offer brief 4

5 explanations of why these preference restrictions permit measurement of the welfare effects of changes in a nonmarket good. We then present a new method of measuring the value of nonmarket goods using market data, under either weak neutrality or weak complementarity. Our numerical method is applicable to real-world data. It also is more flexible than the existing analytical method because it can be used with almost any Marshallian demand function. Our approach uses minimal human and computer effort and can measure the value of nonmarket goods with any desired degree of accuracy. Furthermore, we show that this method can be adapted for use when the Hicksian choke price is infinite. Finally, we explore the possibility of extending this approach to the case of more than one (non-numeraire) market good. I. MEASURING THE VALUE OF A NONMARKET GOOD IN THEORY Since Mäler [9], a number of papers have appeared suggesting theoretical methods of measuring the value of a nonmarket good using market data. As discussed in LaFrance and Hanneman [6], restrictions must be placed on preferences before the value of a nonmarket good can be measured using market data. The principal preference restrictions referred to in the literature have been weak complementarity, Hicks neutrality, and weak neutrality. Most often in the literature it has been assumed that information is available about the Marshallian demand function for one market good. Also, it has been most often assumed that preferences are weakly complementary or weakly neutral at the choke price. Considerable confusion exists in the literature because of inconsistent definitions of key concepts. What is lacking in the literature is a theoretical synthesis of main results and definitions, and a broad presentation of how such preference restrictions allow the measurement of nonmarket good value using market data. The goal of this section of our paper is to use line integral theory to provide a general theoretical context in which to view methods of measuring the value of nonmarket goods using market data. This general theoretical context allows us to provide general, explicit definitions of weak complementarity and weak neutrality, which encompass and synthesize earlier definitions 5

6 appearing in the literature. The general theoretical context serves as a basis for a new method of calculating the value of nonmarket goods. A general theoretical context Basically following notation in Larson [8], let p 0 = (p 1 0,..., p n-1 0, 1) be a vector of observed price levels for goods x = (x 1,..., x n-1, x n ) in terms of x n, a numeraire. Let y 0 be the observed income level of an individual. Let z be a parameter describing (the quantity or quality of) some nonmarket good, and let z 0 denote the observed level of z. Let u(x, z) be the individual s differentiable utility function. The individual s utility at the observed prices, income, and nonmarket good availability is then u 0 = max x expenditure function may be defined as m(p, z, u) min x u(x, z 0 ) s. t. p 0 x y 0. The individual s px s. t. u u(x, z). By Shephard s lemma, the partial derivatives of the expenditure function with respect to prices are identical to the Hicksian demand functions: m(p, z, u)/p i x i c (p, z, u), i = 1,..., n - 1. Let the level of the nonmarket good parameter change from z 0 to z 1. Given no changes in prices, the total value of this change (in terms of compensating variation) is (1) TV( z 0,z 1 )= m( p 0,z 0,u 0 ) m( p 0,z 1,u 0 ). Since z characterizes a nonmarket good, the challenge to the applied economist is to measure (1) somehow from data which reflect observed behavior in markets. Much of the theory of how to measure the welfare effects of changes in nonmarket goods has focused on measuring geometric areas behind Hicksian demand curves for market goods. To see the basic procedure in a general theoretical context, assume that m(p, z, u 0 ) has continuous first partial derivatives of p, and z. Then the following can be established from line integral theory (Kaplan [5, p. 293]) and Shephard s lemma: (2) TV( z 0,z 1 )= m p,z,u 0 0 n 1 m p,z,u dp p i + dz i=1 i z = n 1 x c ( p,z,u m 0 p,z,u0 )dp i i + dz, i=1 z L L 6

7 where L is a path of integration in (p, z)-space from point (p 0, z 1 ) to point (p 0, z 0 ). Furthermore, the line integrals in (2) are path independent (Kaplan [5, Theorem I, p. 292]), so L may be any (piecewise smooth) path between endpoints (p 0, z 1 ) and (p 0, z 0 ). The path independence of the line integral on the far right-hand side of (2) causes it to be general measure which explains and encompasses methods previously presented of measuring TV(z 0, z 1 ). We therefore call equation (2) a general theoretical context for the measurement of the total value of a change in a nonmarket good. Concepts used in (2) are illustrated in figure 1, for the simple case of n = 2 market goods, with p being the price of x 1 in terms of units of x 2, a numeraire. Figure 1 shows the domain of m(p, z, u 0 ). From (1), the total value of the change in z is the change in height of the m(p, z, u 0 ) function (not shown) when we move from (p 0, z 1 ) at point g to (p 0, z 0 ) at point b. Path independence in (2) implies that we could take the line integral along any path between points b and g in figure 1 and still find TV(z 0, z 1 ). In figure 1, the depicted paths g-b, g-e-d-b, and g-h-a-b (and any other piecewise smooth path between g and b) would all result in the same calculation of TV(z 0, z 1 ) from (2). The simplest path is a straight line from g to b, labeled L straight, which changes z from z 1 to z 0 while holding p constant at p 0. For the general case of n 2, such a straight-line path of integration causes the line integral in (2) to convert into the definite integral (proof follows easily from Kaplan [5, p. 275], and is available upon request): (3) TV( z 0,z 1 )= m ( z p,z,u0 )dz = m ( z p0,z,u 0 )dz. L straight Comparing (2) and (3), it may seem that taking the straight-line path of integration provides the simplest method of measuring TV(z 0, z 1 ). For to employ the straight-line path all that is needed is knowledge of the function revealing marginal willingness to pay for the nonmarket good given the observed prices of market goods, m(p 0, z, u 0 )/z. But in the general theoretical context of (2), it is necessary not only to identify m(p, z, u 0 )/z along the chosen path of integration L, but also the Hicksian demand functions x i c (p, z, u 0 ) along L. In z 0 z 1 7

8 any case, equation (2) makes it clear that even if Hicksian demand functions can be identified from market data, having sufficient knowledge about the marginal willingness to pay function m(p, z, u 0 )/z is still the essence of the whole problem of measuring total willingness to pay TV(z 0, z 1 ). While the general theoretical context of (2) does not entirely remove the requirement of identifying the marginal willingness to pay function m(p, z, u 0 )/z, it provides considerable additional flexibility to the researcher to meet that requirement. This flexibility is useful because the function revealing marginal willingness to pay for the nonmarket good given the observed prices of market goods, m(p 0, z, u 0 )z, in general is not derivable from market data; thus the straight-line path of integration from (p 0, z 1 ) to (p 0, z 0 ) cannot be used to find the total value of the change in the nonmarket good z. But equation (2) reveals that, given knowledge of the Hicksian demand functions, it is not necessary to identify m(p, z, u 0 )/z along its entire domain, nor even along the straight line between (p 0, z 1 ) and (p 0, z 0 ). Rather it is at most necessary to identify m(p, z, u 0 )/z along some arbitrary path of integration L which runs between points (p 0, z 1 ) and (p 0, z 0 ). The advantage of the general theoretical context of (2) is that any path of integration between (p 0, z 1 ) and (p 0, z 0 ) may be chosen. That is, (2) gives the researcher the increased flexibility of choosing a convenient path of integration, which has a subpath along which the marginal willingness to pay function m(p, z, u 0 )/z may be identified. This convenience provided by the arbitrariness of the path of integration L is a key component to the application of the weak complementarity and weak neutrality assumptions commonly employed in the literature on measuring willingness to pay for nonmarket goods. In what follows, the general theoretical context of (2) and the role of the arbitrariness of L in the weak complementarity and weak neutrality assumptions is further explained and highlighted. Weak complementarity: definition and use 8

9 A number of slightly different definitions of weak complementarity have appeared in the literature. We can use the general theoretical context provided by (2) to define weak complementarity as follows: Definition. Preferences are said to exhibit weak complementarity along a path of integration if m(p, z, u 0 )/z = 0 for all (p, z) on that path of integration. 1 Let p** and p*** be arbitrary vectors of prices. We can break up any path L between (p 0, z 1 ) and (p 0, z 0 ) into three subpaths. The first of these subpaths is called S 1 and travels between (p 0, z 1 ) and (p**, z 1 ) maintaining z constant at z 1. The second subpath is called S 2 and is an arbitrary path traveling between points (p**, z 1 ) and (p***, z 0 ). The third subpath is called S 3 and travels between (p***, z 0 ) and (p 0, z 0 ) maintaining z constant at z 0. (In figure 1 an example of such a path L is g-h-a-b, where S 1 is subpath g-h, S 2 is the subpath shown between h and a, and S 3 is subpath a-b.) For any such path L broken into three such subpaths S 1, S 2, S 3, we have from (2), z (4) TV( z 0,z 1 n 1 m )= x c i ( p,z,u 0 p,z,u0 )dp i + i=1 L n 1 dz = x c i ( p,z,u 0 c )dp i + S x i ( p,z, u 0 )dp i + i=1 S i= x c i ( p,z,u 0 )dp i S i= D 1 n 1 D 2 n 1 m p,z,u0 m p,z,u0 m p,z,u0 + dz + z S 1 dz + z S dz. z S B 1 D 3 B 2 Equation (4) implies that if preferences exhibit weak complementarity along subpath S 2, TV(z 0, z 1 ) can be calculated provided that the Hicksian demand curves can be identified. For since z is a constant along subpaths S 1 and S 3, the integrals labeled B 1 and B 3 in (4) are equal to zero; furthermore, since preferences exhibit weak complementarity along subpath S 2, the integrand m(p, z, u 0 )/z is zero everywhere along S 2, so B 2 equals zero. Thus, under weak B 3 9

10 complementarity along S 2, TV(z 0, z 1 ) equals the sum of the line integrals labeled D 1, D 2, and D 3 in (4), which can be calculated if the functional forms of the Hicksian demands are known. A convenient path of integration frequently proposed in the literature is illustrated by path g-e-d-b in figure 1. Path g-e-d-b can also be broken into three subpaths, where now we let S 1 be g-e, we let S 3 be d-b, and we let S 2 be the subpath labeled S choke, where S choke is a locus of zs and their corresponding choke prices, which are just high enough for the Hicksian demand for good 1 to be zero. (We follow the custom of defining the choke price function as ˆ p (z, u 0 ) = min { p: x c 1 ( p,z,u 0 ) = 0 }. S choke in figure 1 is then the locus of ( ˆ p (z, u 0 ), z) points defined parametrically by substituting every z [z 0, z 1 ] into ˆ p (z, u 0 ).) Mäler [9] (implicitly) used a path of integration like g-e-d-b to analyze the simple case of n = 2 market goods and one relative price, and he assumed weak complementarity along S choke. Bockstael and Kling [1] generalized Mäler s [9] analysis to deal with the case of n non-numeraire market goods, and assumed ˆ p (z, u 0 ) was a vector of choke price functions. 2 Their chosen path of integration moved along a subpath (again, we can call it S 1 ) from (p 0, z 1 ) to ( ˆ p (z 1, u 0 ), z 1 ) 3, then it moved along a subpath S 2 from ( ˆ p (z 1, u 0 ), z 1 ) to ( ˆ p (z 0, u 0 ), z 0 ) all along which prices are maintained at the choke levels, then it moved along a subpath S 3 from ( ˆ p (z 0, u 0 ), z 0 ) to (p 0, z 0 ). Bockstael and Kling [1] also assumed weak complementary along the choke-price subpath S choke. By assuming S 2 to be the choke-price subpath, Mäler [9] and Bockstael and Kling [1] not only caused B 1, B 2, and B 3 to equal zero in (4), but also caused D 2 to equal zero, since all Hicksian demands x i c (p, z, u 0 ) are zero everywhere along the choke-price subpath S choke, making the integrand in D 2 to be zero everywhere along S choke. The line integrals D 1 and D 3 in (4) can be shown to equal the following definite integrals (this is easily proved from the description of S 1 and S 3 in footnote 3): (5a) D 1 = ˆ p 2 ( z 1,u 0 ) x c 1 t,p 0 0 ( 2,...,p n 1,1,z 1,u 0 c )dt + x 2 ( p ˆ 1 ( z 1,u 0 ),t,p 0 3,...,p 0 n 1,1,,z 1,u 0 ) dt + ˆ p 1 z 1,u 0 p 1 0 p

11 p ˆ n 1 ( z 1, u 0 ) c... + x n 1 ( p ˆ 1 ( z 1, u 0 ),..., p ˆ n 2 ( z 1, u 0 ), t,1,z 1,u 0 )dt 0 p n 1 (5b) D 3 = ˆ p 2 ( z 0,u 0 ) ˆ p 1 z 0,u 0 c x 1 ( t, p 0 2,..., p n 1,1,,z 0,u 0 )dt + c x 2 ˆ 0 p 1 0 p 2 p 1 ( z 0, u 0 ),t, p 0 0 3,...,p n 1,1,,z 0,u 0 dt + p ˆ n 1 ( z 0,u 0 ) c... + x n 1 ( p ˆ 1 ( z 0,u 0 ),..., p ˆ n 2 ( z 0,u 0 ), t,1,z 0,u 0 )dt 0 p n 1. Thus, under the assumption that preferences are weakly complementarity along the choke-price subpath, the problem of not having information about the marginal willingness to pay function m(p, z, u 0 )/z is overcome. The total value of a change in the nonmarket good can be assessed by taking definite integrals behind Hicksian demand curves: TV(z 0, z 1 ) = D 1 + D 3. This is the essence of Bockstael and Kling s [1] and Mäler s [9] procedure. If we define subpath S 2 in (4) to be the choke-price subpath, then Hicksian demands x 1 c (p, z, u 0 ),..., x n-1 c (p, z, u 0 ) are all zero everywhere along S 2, and D 2 in (4) must be zero. Choosing S 2 = S choke, the value of B 2 in (4) is called the nonuse value of the change in z, NUV(z 0, z 1 ), because along S choke no utility can be derived from using z together with Hicksian goods x 1 c,..., x n-1 c, since along S choke these goods are not used. An immediate implication of weak complementarity along the choke-price subpath, also pointed out by Larson [7, p. 110] (though not in terms of line integral theory), is that NUV(z 0, z 1 ) = 0. Use value of the change in z, called UV(z 0, z 1 ) is defined as the difference between total value and nonuse value: UV(z 0, z 1 ) = TV(z 0, z 1 ) - NUV(z 0, z 1 ). Under weak complementarity along S choke, use value is equal to total value: UV(z 0, z 1 ) = TV(z 0, z 1 ) = D 1 + D 3, as defined in (4) and (5). (This last result uses line integral theory to generalize equation (7) of Larson [8] for the case of multiple market goods.) For a nonmarket good with no nonuse value, letting S 2 = S choke lends intuitive appeal to the assumption that m(p, z, u 0 )/z = 0 everywhere along S 2. For it seems reasonable that if utility from z only is attained when z is consumed with (x 1,..., x n-1 ), then if none of (x 1,..., 11

12 x n-1 ) is consumed, a change in z should not alter the level of utility, nor should it alter the minimum expenditures necessary to obtain that level of utility. Thus, using the arbitrariness of the path of integration L and the general theoretical context of (2) to assume S 2 = S choke provides an intuitively appealing method of measuring TV(z 0, z 1 ) when the nonmarket good z has no nonuse value. Weak neutrality: definition and use We use the general theoretical context of (2) to define weak neutrality as follows: Definition. Preferences are said to exhibit weak neutrality along a path of integration if for at least one i {1,..., n}, x c i (p, z, u 0 )/z = 0 for all (p, z) on that path of integration. 4 Hicksian and Marshallian demands are related by the identity x c i (p, z, u) x i (p, z, m(p, z, u)). Weak neutrality of good i along a path of integration S allows the researcher to use a Slutsky-Hicks decomposition of this identity to capture of the marginal willingness to pay function in terms of Marshallian demand derivatives 5 : (6) z m p,z, u0 z m p,z, u 0 x i p,z,m p,z,u 0 = z x i p,z,m p,z,u 0 ( y ( n 1 x j p,z,m p,z,u 0 p j j=1 z = n 1 x j p,z,m p,z, u 0 1 p j j=1 ( y for all (p, z) on S, if i {1,..., n - 1}, for all (p, z) on S, if i = n. Breaking any path L into three subpaths S 1, S 2, and S 3, defined as before, B 1 and B 3 in (4) are again zero. Then if we assume preferences are weakly neutral along S 2, then substituting (6) into (4) we see that B 2 can be written, 12

13 (7) ( x i p,z,m p,z,u 0 m ( p, z,u 0 ) dz = z z S1 2 x i ( p,z,m( p,z,u 0 ) dz, if i {1,..., n - 1}, S 2 B 2 y n 1 x j p m ( p, z,u 0 ) j ( p,z, m( p,z,u 0 ) j=1 z dz = z n 1 S 1 2 x j ( p,z,m( p,z,u 0 dz, if i = n. S 2 ) 1 p B j 2 j=1 y Substituting (7) into (4), we see that under weak neutrality of good i along S 2, TV(z 0, z 1 ) = D 1 + D 2 + D 3 + B 2, where D 1,D 2, D 3, and B 2 are all expressed in terms of Hicksian and Marshallian demand functions. Thus if the Hicks and Marshallian demand functions are known, then under weak neutrality along any S 2, TV(z 0, z 1 ) can be calculated. As will be explained in detail in section II, a number of technical difficulties present themselves in the calculation of the righthand side of (7). Also, as will be discussed, even in the general context of (2), there may not be an obvious subpath S 2 for which weak neutrality is a justifiable assumption. Nevertheless, under the assumption of weak neutrality along some subpath S 2, TV(z 0, z 1 ) can be calculated. If we let S 2 be the choke-price subpath S choke, then under weak neutrality along S choke, D 2 in (4) is zero, TV(z 0, z 1 ) = D 1 + D 3 + B 2, NUV(z 0, z 1 ) = B 2, and UV(z 0, z 1 ) = D 1 + D 3. The relationship between weak complementarity and weak neutrality Does weak neutrality of the numeraire along S choke imply weak complementarity along S choke? Larson [8] and Flores [2] have disagreed about a fundamental characteristic of weak neutrality. Stated in terms used in our paper, Larson argued that an assumption of weak neutrality of a numeraire (composite) good along the choke-price subpath S choke could be used to calculate nonuse value of a change in the nonmarket good. Larson also claimed that his assumption has intuitive appeal. Flores [2], on the other hand, argued that assuming all non-numeraire goods to 13

14 be Hicksian complements to z implies that imposing weak neutrality at the choke price will result in a nonuse value of zero (i.e., B 2 = 0 in (4)), concluding that "weak neutrality... at the choke price is a special case of weak complementarity." Unfortunately, some technical problems are present in Flores proof 6, and Flores intuitive argument is not completely convincing 7, so the issue remains unresolved in the literature We argue that, strictly speaking, Flores criticism of Larson s method is valid assuming (as Larson and Flores do) that the non-numeraire goods are Hicks complements to z. We offer a proof of Flores argument in Appendix B to resolve this issue. If however, at least one numeraire good is a Hicksian substitute to z 8 (and there seems to be no general reason why one should not be), then weak neutrality along S choke need not imply weak complementarity along S choke, and need not imply nonuse value of zero. Thus, there is no technical reason why Larson s method cannot be used as long as at least one non-numeraire good is a Hicksian substitute for z. Graphic illustration Weak complementarity along a subpath S 2 is illustrated in figure 2 for the case of two market goods, x 1 and x 2, where p is the price of x 1 and x 2 is a numeraire. Three slices of the expenditure function m(p, z, u 0 ) are shown in (p, z)-space: m(p**, z, u 0 ), m(p, z, u 0 ), and m(p***, z, u 0 ). It is assumed that z is never a bad, and thus m(p, z, u 0 ) is nonincreasing in z. But for all (p, z) on S 2, the marginal willingness to pay for quality is zero; particularly shown are m(p**, z, u 0 )/z = 0, m(p, z, u 0 )/z = 0, and m(p***, z, u 0 )/z = 0. An illustration of weak neutrality of a good i along subpath S 2 would look similar to figure 2, except that the verticle axis would represent the Hicksian demand for i. Weak neutrality would require that for all (p, z) along S 2, the slope x i c (p, z, u 0 )/z = 0. If we were to replace the arbitrary subpath S 2 with the choke-price subpath S choke, the x i c (p, z, u 0 ) function would be nonincreasing in p i until reaching the choke price ˆ p i, after which it would remain at zero for all p i > ˆ p i. 14

15 How weak complementarity and weak neutrality restrict preferences Under our general definition, the assumption that there exists a subpath of integration S along which preferences are weakly complementary or weakly neutral is not a particularly restrictive imposition on preferences. The challenge in empirical work, however, lies not in proving the existence of, nor even in finding such subpath along which a known expenditure function m(p, z, u 0 ) displays weak complementarity or weak neutrality. Rather, the challenge lies in recovering the unknown expenditure function from known Marshallian demand functions (which are compatible with many expenditure functions). It is by combining those Marshallian demand functions with the assumption that preferences are weakly complementary or weakly neutral along a particular subpath of integration S that preferences are restricted--we narrow down which of the many expenditure functions compatible with our Marshallian demand functions is the one implied by the data and our assumptions. II. MEASURING THE VALUE OF A NONMARKET GOODS IN PRACTICE In this section, we discuss and compare two methods for applying the theory described in the previous section to estimate the value 9 of changes in a nonmarket good ("quality") using ordinary Marshallian demand parameters and standard restrictions on preferences (weak complementarity or weak neutrality). The first is Larson's [8] method of analytically deriving the quasi-expenditure function and its accompanying Hicksian demand functions from Marshallian demand parameters and restrictions on preferences. To calculate the total value of a change in quality, the quasi-expenditure function is used as the expenditure function is used in (1), and the Hicksian demand functions are used in (2). 15

16 We present a second method of applying the theory, a new approach involving numerical integration of the expressions in (4) and (5). Both Larson [8] and Flores [2] suggested that numerical methods could be used to measure the value of nonmarket goods, but neither described how this approach would be implemented under the standard restrictions on preferences, weak neutrality along the choke-price subpath or weak complementarity along the choke-price subpath. The numerical method proposed here can be implemented with any Marshallian demand function; in contrast, the analytical method described by Larson [8] is limited to the relatively small set of demand functions which can be integrated back to an explicit expenditure function. As we show in this section, our approach can be used to measure the value of quality changes with any desired degree of accuracy. Preliminary results suggest that the numerical approaches may be useful even when the Hicksian choke price is infinite. The computer programming needed for our approach is relatively simple, and the computational burden is light. (We present a detailed example of our approach, a sample program of about fifty lines of computer code.) Analytical derivation of willingness to pay for nonmarket goods In this section, we describe Larson's [8] method of analytically deriving the value of nonmarket goods. The goal is to provide a basis of comparison between Larson's analytical method and our numerical approach and to illustrate how restrictions on preferences allow calculation of the value of quality changes using (2). We follow Larson [8] in illustrating his analytical approach with a linear model of the Marshallian demand for fishing trips (x 1 ) as a function of price of a trip (p), income (y), and the quality of the fishing (z), measured by the average number of fish caught per trip: 16

17 (8) x 1 = α + βp+ γz + δy. This demand equation can be integrated back to obtain the following quasi-expenditure function, where θ(z, u) is the constant of integration: (9) m (p,z,θ(z,u)) = θ(z,u)e δ p 1 [ α +βp + γz + β/δ δ ]. For any level of utility u, the choke-price subpath may be defined parametrically by z [z 0, z 1 ] and the pair of functions ( ˆ p (z, u), z), where Larson showed the Hicksian choke price function associated with (9) to be (10) ˆ p (z,u) = 1 δ ln [ β /[δ2 θ(z,u)]. When S choke is used for subpath S 2 in (4), D 1 and -D 3 from (4) and (5) represent the value of access to x 1 at p = p 0 and u = u 0 when quality is z 1 and z 0, respectively. Thus, D 1 - (-D 3 ) = D 1 + D 3 is the increase in the value of access as quality changes from z 0 to z 1, known as the use value of the quality change. B 2 from (4) represents nonuse value since it is the value of a change in z, holding u = u 0 and x 1 c ( ˆ p (z, u 0 ),z, u 0 ) = 0. Combining (9) with expressions for the two constants of integration, θ(z 0, u 0 ) and θ(z 1, u 0 ), would allow exact calculation of the value of quality changes to be performed directly using (1), given a linear demand equation. θ(z 0, u 0 ) is easily obtained by substituting y 0 = m(p 0, z 0, u 0 ), p 0, and z 0 into (9) and solving for θ. Derivation of θ(z 1, u 0 ) can only be found by imposing restrictions on preferences. Larson [7] showed that if we assume weak complementarity along the choke-price subpath, the constant of integration associated with the linear demand equation takes the form 17

18 θ(z, u) = φ(u)e (zγδ/β), where φ is a utility index. The value of φ(u 0 ) can be found by setting z = z 0 and θ(z, u) = θ(z 0, u 0 ) and solving for φ. With φ(u 0 ), we can obtain θ(z 1, u 0 ), which allows us to calculate the exact value of a change in quality by using θ(z 0, u 0 ) and θ(z 1, u 0 ) in (9) and substituting into (1). From (4), we know that under weak complementarity at the choke price, nonuse value (B 2 ) is zero. If we assume weak neutrality at the choke price, Larson's [8] approach is more complex and less exact. He demonstrated that, under weak neutrality of the numeraire good 2 along S choke, the nonuse value (NUV) can be expressed in terms of the Marshallian demand function for good 1 (provided that good 1 is a Hicksian substitute for z, otherwise NUV is zero- -see section I). If S 2 is the choke-price subpath, the term B 2 in our equations (4) and (7) may be converted into Larson s [8] equation (13) (Kaplan [5, p. 227, equation (5.6)]): m p,z,u 0 z 1 (11) NUV(z 0,z 1 ) = dz = m p ˆ ( z,u 0 ),z,u 0 z S 1 choke z z B 2 = z 1 z 0 ˆ p z,u 0 1 p ˆ z,u 0 x 1 ˆ x 1 ˆ dz p ( z,u 0 ),z,m( p ˆ ( z, u 0 ),z,u 0 z ( p ( z, u 0 ),z,m ( p ˆ ( z,u 0 ),z, u 0 dz. ) y According to Larson [8, p. 116], this expression "allows one to express nonuse value in terms of observables." 10 Although (11) is useful, applying it is more problematic than first appears since ˆ p is neither constant nor, strictly speaking, observable. Rather, ˆ p is a function of z and θ(z, u), and we do not (yet) have a method of identifying θ(z i, u 0 ) for z i z 0 under weak neutrality. Larson [8, footnote 11] suggested numerically approximating (11). But such a 18

19 numerical approximation would require an expression for θ(z 1, u 0 ), and if we had an expression for θ(z 1, u 0 ) we could calculate the exact nonuse value directly with (1), substituting the quasiexpenditure function in for the expenditure function, and not bother with (11) in the first place. In his bass fishing example, Larson [8] circumvented these problems of identifying θ under weak neutrality with some simplifying assumptions. He calculated (11) assuming that ˆ p (z 1, u 0 ) = ˆ p (z 0, u 0 ). He then approximated θ(z 1, u 0 ) by substituting m = y 0 + B 2 + (-D 3 ), z 1, and ˆ p into (9) and solving for θ. 11 Presumably, Larson used ˆ p (z 0, u 0 ) or ˆ p (z 1, u 1 ) as an approximation of ˆ p (z 1, u 0 ) in this second step. In summary, Larson's method provides an exact measure of total value of quality changes under weak complementarity with certain demand functions. This approach, however, is not exact when measuring the value of quality changes under weak neutrality. Furthermore, the method can only be applied with demand functions that can be analytically integrated back to their quasi-expenditure functions. Numerical calculation of willingness to pay: previously suggested methods applying Vartia-type algorithms of Vartia s numerical method of calculating willingness to pay for a price change This section describes Vartia's [12] method of numerically approximating exact measures of the welfare impact of price changes. It also reviews previous studies that have suggested applying this method to the measurement of the value of nonmarket goods. This discussion provides useful background and justification for the next section, which describes our approach for numerically calculating the value of nonmarket goods under weak neutrality and weak complementarity. 19

20 Vartia [12] showed that the compensating variation and equivalent variation associated with a price change could be approximated numerically using Marshallian demand parameters. In his method, the price change is divided into a many small increments, here called p. After each incremental price change, income is adjusted to maintain utility (approximately) constant. For the case of one relative price, the value of demand (x) after the j th incremental price change is: j (12) x j = x p 0 + j p,y 0 + S i, where S j is an income adjustment to compensate for the j th incremental price increase. Vartia [12, p. 88] suggested three algorithms for the income adjustment: x j-1 p, x j p, and 0.5(x j-1 + x j )p. The first algorithm makes (12) recursive, while the second and third make it an implicit expression for x j which must itself be calculated iteratively. He showed that the third algorithm converges quickest and calls this the "main algorithm." Whichever way S j is calculated, it will be a good approximation of "exact" measures i=1 (compensating or equivalent variation) if p is small. Thus, x j (p 0 + p, y 0 + S j ) x c (p 0 + p, u 0 ) and so lies (approximately) on the Hicksian demand curve x c (p, u 0 ). If we start at x(p 0, y 0 ), the sum of the adjustments is (approximately) equal to the compensating variation of the price change from p 0 to p 0 + jp. The equivalent variation can be approximated by starting at x(p 0 + jp, y 0 ) and working "back" toward p 0. The advantage of Vartia's approach is that it provides flexibility in the specification of the demand equation; unlike the method suggested by Hausman [3], it is not limited to the set of 20

21 demand functions associated with explicit expenditure functions. Unlike the approach of McKenzie [10], based on Taylor-series expansions, Vartia's methods allows accuracy to be increased to any desired level with little (human) effort (see Minot [11]). Flores suggestions for applying Vartia s method for a change in a nonmarket good Both Larson [8] and Flores [2] suggested applying Vartia's numerical method to approximate the value of nonmarket goods using market data. Larson [8, p ] developed an expression (his equation (6)) for the value of quality changes in terms of Marshallian demand parameters for a good that is Hicks neutral to the nonmarket good 12. Larson briefly suggested that numerical techniques similar to those of Vartia [12] could be used to approximate his equation (6). Flores [2] criticized Larson for paying insufficient attention to detail, and pursued Larson s idea by providing three iterative equations based on Vartia's three algorithms to approximate Larson's equation (6). The third and most accurate approximation is given by Flores equation (25): x i p,z 0 + ( k 1) z,m k 1 (13) m k = m k 1 + z x i p,z 0 + ( k 1) z,m k 1 y x i p,z 0 + k z,m k 1 + z x i p,z 0 + k z,m k 1 y where z is a small change in quality z and m k is an approximation of m(p 0, z 0 + kz, u 0 ). Following Vartia's idea, the quality change (z 1 - z 0 ) is divided into small increments (z); after each increment, the marginal value of quality is calculated and used to "compensate" the consumer to maintain utility approximately constant at the original level. z 2, 21

22 Unfortunately, (13) suffers from some important limitations. First, as Flores recognized, the assumption of Hicks neutrality is a very strong restriction on preferences. As we show below, the value of quality changes can be numerically calculated using the less restrictive assumptions of weak complementarity and weak neutrality along a subpath. Second, (13) is an implicit equation (m k is on the left and right side), and Flores does not explain the iterative procedure for obtaining m k. This is not an insurmountable problem, but the omission is unfortunate given Flores' comments about the "lack of attention to important implementation details" in Larson s work. Flores [2] also discusses the case where weak neutrality holds along a straight-line subpath between (p*, z 1 ) and (p*, z 0 ), where p* is a vector of constant prices. He suggests using the following equation (his equation (27)) for numerical calculation: (14) TV( z 0,z 1 n )= [ x c i ( s,z 1, u 0 ) x c i ( s,z 0,u 0 )] ds + [ m( p *,z 0,u 0 ) m( p *,z 1,u 0 )]. p * p i=1 This equation attempts to restate a well-known decomposition of total value (see Larson [8, equation (7)]) and is expressed in terms of Hicksian demand curves and expenditure functions. Equation (14) can be derived from our equations (4), where S 2 is the straight-line subpath between (p*, z 1 ) and (p*, z 0 ). (Though Flores equation presents several notational difficulties for the case of n - 1 > 1.) Flores' brief discussion of (14) does not resolve or even acknowledge the main problem (and several smaller ones) in implementing the equation. The key obstacle in implementing (14) is that, contrary to Flores' [2] assertion, the integral in (14) cannot be approximated "by direct application of Vartia's algorithm." In particular, we lack a starting point x c i (p 0, z 1, u 0 ) from which to calculate the area behind x c i (p, 22

23 z 1, u 0 ) over p 0 to p*. (This area corresponds to (-D 3 ) in our equations (4) and (5b).) We cannot obtain the starting point x c i (p 0, z 1, u 0 ) without knowing the amount of money which would compensate the consumer for the change in z, which is, of course, the objective of the whole exercise. Another problem is that since Flores intended (14) to describe the multi-good case, the integral in (14) is a line integral. Flores ignored important details of how (14) should be evaluated following a specific path of integration in which prices are raised sequentially to their choke price levels, similar to what is shown in our equations (5). Finally, (14) does not take into account the additional complexities that occur when S choke is used for S 2 in (4), instead of using Flores straight-line subpath for S 2, as it is in the standard applications of weak neutrality and weak complementarity. The most difficult problem occurs in the multi-good case, since the choke price for each varies depending on the prices of other goods (this issue is discussed further below). Thus, Flores [2] follows an interesting line of inquiry, but is not an applicable method for calculating the value of quality changes under normal restrictions on preferences. Equation (13) can be implemented but applies only under Hicks neutrality, a very strong restriction on preferences, while (14) applies under weaker assumptions about preferences (since any value of p* can be chosen to define the straight-line subpath), but cannot be implemented as presented. Numerical calculation of willingness to pay: Our method Our approach extends Vartia to the problem of measuring the value of nonmarket goods using Marshallian demand parameters. Unlike Larson's analytical method, our approach can be used 23

24 with any well-defined Marshallian demand function, whether or not it can be integrated back to obtain an explicit expenditure function. Unlike Flores' numerical method, our procedure measures the value of quality changes under the conventional restrictions on preferences, weak complementarity and weak neutrality along the choke-price subpath. Our approach is based on the general theoretical context of (2) and its ramifications shown in (4) and (5). Our approach illustrated Our approach can be implemented with a relatively simple computer program (Appendix A provides an example, written for GAUSS. 13 ) The researcher chooses the restriction on assumptions, weak neutrality or weak complementarity along S choke (line 3 in Appendix A). 14 The integrals approximated are D 1 and D 3 in (5) and B 2 in (11). These integrals allow us to approximate TV(z 0, z 1 ), UV(z 0, z 1 ), and NUV(z 0, z 1 ) under either weak complementarity or weak neutrality along S choke. Any desired level of accuracy can be reached by setting the size of the increments p and the number of increments for z (lines 4 and 5). The researcher also specifies the demand function, the initial levels of income (y 0 ), price (p 0 ), and quality (z 0 ), and the final level of nonmarket good (z 1 ) (lines 7-11). The procedure involves two loops, one nested within the other. In the first ("outer") loop, the value of z is raised in small increments z from the initial value, z 0, to the final value, z 1 (the loop starts on line 21). For each z i in the outer loop, the second ("inner") loop uses Vartia's method to trace the Hicksian demand curve from p 0 to the Hicksian choke price, ˆ p (z i, u 0 ) (lines 27-32). Figure 3 illustrates this inner loop process for z = z 0. The inner loop measures the value of access to the nonmarket good, which for z = z 0 is the integral (-D 3 ) in (4) and (5b). In 24

25 the first price iteration, the price rises from p 0 to p 0 +p while demand falls from x(p 0, z 0, y 0 ) = x c (p 0, z 0, u 0 ) to x(p 0 + p, z 0, y 0 ) along the Marshallian demand curve. The (approximate) change in consumer surplus associated with this price increase, represented by area S 1, is used to adjust income upward, raising demand to x(p 0 +p, z 0, y 0 + S 1 ). If the price increment is small, this point is close to x c (p 0 + p, z 0, u 0 ). Note that the size of the price increments has been exaggerated in Figure 3 to demonstrate the method: in tests described below, we use p = 0.1, resulting in hundreds of increments between p 0 and ˆ p. In the second price iteration, income is further adjusted by S 2, so y = y 0 + S 1 + S 2. The process continues until the Hicksian choke price is reached (Hicksian demand is zero). The sum of the S i is (approximately) the area behind the Hicksian demand curve x c (p, z 0,u 0 ) above p 0, and the approximation can be made as accurate as desired by shrinking the size of the price increment p. This area is a measure of the value of access to the nonmarket good (-D 3 from (5b)). The iterations of the outer loop, which increase the value of z by an increment z, are illustrated in figure 4. (In figure 4, it is assumed that there are R such increments: z 1 - z 0 = Rz.) In the second iteration of z, the inner loop process of tracing the Hicksian demand curve is repeated starting from x(p 0, z 0 + z, y 0 ). The difference between the area behind the original Hicksian demand curve and the new one (area UV 1 in Figure 4) is a measure of the use value associated with the increase in z from z 0 to z 0 +z (see line 35 in Appendix A). Under weak neutrality, the updated nonuse value is approximated using a trapezoidal approximation of (11) over z 0 to z 0 + z (lines 36-48). If weak complementarity is assumed, these calculations are skipped and nonuse value remains at zero. 25

26 The estimates of use value and nonuse value are used to adjust the income level for the third and subsequent iterations of z. As shown in Figure 4, the third Hicksian demand curve is traced upward using the inner loop process from the point x(p 0, z 0 + 2z, y 0 -UV 1 ). 15 For small changes in z, this is close to x c (p 0, z 0 + 2z, u 0 ). In general, at the end of the i th iteration in z, income is adjusted using the interim estimates of use values (sum of area UV(1) to UV(i - 2)) and nonuse values (NUV(1) to NUV(i - 2)) if the assumption is weak neutrality (line 50). This adjusted income is used in iteration i + 1 so that utility is maintained (approximately) constant as z varies. The final estimate of use value is the difference in area behind the Hicksian demand curve x c (p 0, z 1, u 0 ), representing D 1 from (5a), and the area behind Hicksian demand curve x c (p 0, z 0, u 0 ), representing -D 3 from (5b). The intermediate steps are needed to estimate the income adjustments needed to keep utility constant as z increases. The GAUSS program in Appendix A is written to follow Larson s [8] bass fishing example under weak neutrality along the choke-price subpath, with R = 5 iterations in z and price increments of 0.5 (see lines 3-11). (The assumption of weak neutrality can be changed to weak complementarity simply by assigning assum = 2 in line 3.) Using our numerical approach, this program calculates the use value at -$7.69 and the nonuse value at $10.11, virtually identical to Larson s results of -$7.69 and $10.10 (the small difference in nonuse value is probably due to Larson s simplifying assumption of a constant choke price as z changes since our result does not change with more iterations). Evaluating the accuracy of our approach Assuming linear demand in the form of (8), we evaluate the accuracy of numerical methods of measuring the value of changes in the nonmarket good by comparing our results with the exact 26

27 value under weak complementarity. The exact value is measured using Larson's analytical approach. We cannot evaluate the numerical method under weak neutrality because, as noted above, the methods proposed by Larson [8] yield only approximations. Furthermore, since Larson suggests using numerical methods to approximate (11), the difference between our approach and Larson's is minimal in the calculation of nonuse value (the only difference is that under our approach, the terms in (11) are evaluated numerically rather than analytically). We calculate the error associated with our numerical approach using 64 combinations of demand parameters, three sizes of total increase in quality z, and four levels of precision, determined by the number of iterations in z. Following Larson s bass fishing example, we set p 0 = 30, z 0 = 5, and y 0 = Four price parameters (β) are selected to generate own-price elasticities of -4.0, -2.0, -1.0, and -0.5 at the original point. We use four income parameters (δ) yielding income elasticities of -1.0, 0.5, 1.0 and 2.0, and four quality parameters (γ) producing quality elasticities of -1.0, 0.5, 1.0, and 2.0, evaluating all elasticities at the original point. This results in 64 parameter combinations. The constant (α) is set in each case so that x(p 0, z 0, y 0 ) = 10. These parameters are chosen to represent a somewhat wider range than would normally be found in empirical work. The purpose of including somewhat extreme parameter combinations is to examine the full range of possible errors stemming from our numerical approach to measuring the value of quality changes. We vary the number of iterations R in z and the size of the total change (z 1 - z 0 ) in our test. We allow R to take on values of 5, 10, 100, and 1000, and allow the overall increase in z to range from 10 percent to 100 percent. The price increment p is kept constant at 0.1. Table 1 summarizes the results of the 64 parameter combinations for different changes in quality and different numbers of iterations in z. Three conclusions can be drawn from this table. First, the errors are quite small. Even if only five iterations are used to measure a 100 percent increase in z, the mean error across the parameter combinations is less than 0.5 percent, while the median error is less than 0.25 percent. With a thousand iterations, the mean error falls to percent. Second, the degree of precision is roughly proportional to the number of 27