Chapter 14 Introduction to Non-Market Valuation

Transcription

1 1 Chapter 14 Introduction to Non-Market Valuation Throughout Parts I and II of this book we have referred to damage, abatement cost, profit, and other economic functions in our discussion of environmental policy. As we have seen, these functions arise from household preferences and firm production technologies, and are therefore usually not directly observable. Nonetheless, to this point we have assumed regulators and other agents possess knowledge of these functions or estimates thereof, and conditional on this we have proceeded with our mostly theoretical discussion of policy design, usually at the industry or economy level. In Part III we turn attention to methods that have been developed to estimate the damage and cost functions needed to implement policy. With this our emphasis switches to examining in detail models of individual household or firm behavior, since the measurement concepts of interest arise out of individuals interactions with the environment. Furthermore, Part III of the book has a greater emphasis on econometrics, since our discussion will also examine ways that these models can be brought to data to produce empirical results. The themes we present in Part III are usually referred to collectively as the sub-field of nonmarket valuation. The general problem of non-market valuation is to compute a monetary value for environmental resources or services that, in the general setup of market economies, are not subect to exchange and therefore do not have an observable market price. For example, we might be interested in knowing the monetary value of preserving a wooded area in an urban landscape, relative to its value in a residential use. Likewise large scale biodiversity preserves usually preclude extractive use, such as mining and timber harvest, within the boundaries of the preserve. What is the monetary value of the biodiversity protection relative to the market value of the forgone minerals and lumber? To answer this and similar questions, conceptual models that connect the environmental resource to people s preferences (or in some cases firm profits), and econometric approaches that connect data to the conceptual models, are needed. In this chapter we begin by our study of non-market valuation by examining the former. In particular, we develop a general model of household behavior that allows us to formally define useful concepts of value, which arise from neoclassical welfare theory. It is not by accident that neoclassical welfare theory provides the starting point for study in Part

2 2 III, as it did in Part I of the book. Indeed, it helps us stress that the valuation problem should not be viewed as separate from the policy design problem, since they are two sides of the same coin. Policy design involves using neoclassical welfare theory to derive optimal policy rules, while valuation uses the theory to suggest measurement methods needed to implement the policy rule. The externality example from chapter 1 illustrates this nicely: when emissions result from production of a good, a Pareto optimal outcome can be obtained using an emission tax (equal to the marginal damage from pollution), but the socially optimal tax rate depends on the structure of individuals preferences. Actual implementation therefore requires empirically examining people s behavior, in order to learn something about their preferences, from which the correct tax rate can be computed. More generally, much of non-market valuation is concerned with estimating behavioral functions that are sufficiently revealing about preferences to allow calculation of the monetary value of marginal or discrete changes in environmental conditions. It is useful at the outset to define carefully what we mean by monetary value, since it plays such a large role in the discussion that follows. As described at the book s outset, the concept of value that we rely on is individualistic and based on consumer sovereignty, so that a person s preferences determine how outcomes (e.g. consumption levels, states of the environment, or his own health) translate to value. Outcomes can, of course, change through any number of channels. Prices can adust, indirectly shifting consumption levels, or policy might directly alter environmental quality, thereby impacting the level of well-being a person obtains. Establishing a monetary value for any action that directly or indirectly changes outcomes involves (a) defining a baseline state and an ending state, and (b) computing the person s willingness to pay (WTP) to secure the ending state, or his willingness to accept (WTA) to forgo it. The key points here are twofold. First, reference to monetary value is for something specific. We cannot define a person s monetary value for a generality like clean drinking water, but we can define the health effects value of a specific (small or large) reduction in carcinogen contaminants, relative to current conditions. Second, WTP and WTA are income equivalents that link the starting and ending states to preferences. Consider, for example, a reduction in the price of a private good. The price reduction expands the range of consumption outcomes the person can obtain, and thereby potentially increases well being. The starting point is the original price and the ending point is the new price. WTP is the amount of money the person would give up to have the new

3 3 price, and hence the higher level of well being. WTA is the amount of money the person would need to be given in lieu of the price decrease, and therefore reflects an equivalent way of obtaining the higher level of well being. WTP and WTA are useful concepts because they convert information on preferences, which is inherently unquantifiable, into money equivalents. As we will see, substitutability among elements in an individual s preference function is a key determinant of WTP and WTA. For the price decrease example, if abundant and inexpensive alternatives to the now-cheaper good are already available, the reduction in its price is less valuable to the person than if the good were relatively unique. In establishing the monetary value of environmental goods, the notion of substitutability between market goods and environmental goods turns out to be critical. Non-market valuation is largely about measuring a person s WTP or WTA (often referred to collectively as welfare measures or welfare effects ) for changes in environmental variables. As such, references to WTP and WTA are ubiquitous throughout Part III of the book. We begin this chapter with a review of consumer welfare theory, and how economists have used the theory to develop standard empirical techniques for measuring the welfare effects of changes in private good prices. We present the well established duality results linking estimable demand functions to preferences, and ultimately to individual willingness to pay or accept measures. Since most of non-market valuation involves examining the welfare impacts of changes in quasifixed goods (such as the level of an environmental indicator), we then discuss how the standard price change techniques must be modified when we consider quantity changes. We show that while duality can still be used to link demand for the quasi-fixed good to the preference function, the absence of market exchange rules out the use of observed behavior to directly estimate a demand function. Instead, extra-market information is needed to infer individuals values for these goods, the sources of which we consider in subsequent chapters. We close this chapter by discussing generalizations needed to define welfare measures that are appropriate for use under state of the world uncertainty, and then by examining the conceptual relationship between willingness to pay and willingness to accept measures General setup The traditional goal of applied welfare analysis is to use observed behavior to characterize the

4 4 components of preferences needed to calculate welfare measures. A necessary first step for this is to describe a model that gives rise to the observed behavior. Consider a generalization of the consumer problems given in earlier chapters. The utility function is now U(x,z,q) where x is a J dimensional vector of private market goods, z is a numeraire good with price normalized to one, and q denotes a vector or scalar of environmental goods and services. In this generalization q can represent a wide range of human interactions with the environment. This can include health effects, residential and recreational amenities, aesthetic impacts, ecosystem services, and public goods in general. In actual applications the specific definition of q will depend on the nature of the problem and pragmatic considerations such as the availability of data. For purposes of exposition here, we assume that q is a scalar quantity measured and expressed in known units, and that it generates positive marginal utility (i.e. q is a good ). The consumer s problem is to choose amounts of x and z to maximize utility given the fixed level of q and market prices p, subect to an exogenous income level y. The formal problem is given by xz, max U( x, z, q) y z px, (14.1) where is the Lagrange multiplier. The first order conditions for this problem are U x p, 1,..., J, y z px, (14.2) and if we assume there is an interior solution to the problem, we can solve for the ordinary demand functions x (p,y,q), the LaGrange multiplier (p,y,q), and the level of the numeraire. Note that the ordinary demand functions are directly estimable, since they depend on the observable quantities p, y, and q. If we substitute the demand functions into the direct utility function, we obtain the indirect utility function V(p,y,q). It is worth emphasizing that q appears in both the direct and indirect utility functions, due to the fact that its level is not chosen by the consumer. Finally, via the Envelope Theorem, (p,y,q) can be interpreted as the marginal increase in realized utility due to a small change in the income constraint. This is often referred to as the marginal utility of income, and it will become important in later discussions. The consumer s behavior can also be represented by the expenditure minimization problem

5 xz, min px z u U( x, z, q), (14.3) where ū is a reference level of utility. The first order conditions for this problem are U x p, 1,..., J, U z 1 (14.4) U ( x, z, q) u, and assuming an interior solution we obtain the compensated demand functions h (p,ū,q) for each x and h z (p,ū,q) for z. Note that, in contrast to the ordinary demand functions, the compensated demand functions are not directly estimable, since they depend on a particular level of utility rather than income. By substituting the compensated demand functions into (14.3) we obtain the expenditure function E(p,ū,q), which is the smallest amount of income a person would need to reach utility level ū. 5 Duality relationships are critical in applied welfare analysis, and none is more so than the identity that connects the ordinary and compensated demands at the point of consumption. Suppose that u 0 is the utility level that is obtained in the utility maximization problem in (14.1), and that E(p,u 0,q) is the expenditure needed to obtain u 0. By construction y=e(p,u 0,q), and the solutions to the utility maximization and expenditure minimization problems are linked at the observed point of consumption by x p E p u q q h p u q (14.5) 0 0, (,, ), (,, ). The identity allows us to state the relationship between the price responses for the ordinary and compensated demand functions. Differentiating both sides with respect to p leads to the Slutsky equation x p y q h p u q x p y q E p u q p p y p 0 0 (,, ) (,, ) (,, ) (,, ) 0 h( p, u, q) x ( p, y, q) x p y ( p, y, q), (14.6) where the second equality follows from Sheppard s Lemma and the identity in (14.5). Equation (14.6) is important for two related reasons. It shows that the price effect for ordinary and compensated demand functions differ by an income gradient, so that if there is no income effect i.e. x ( )/ p =0 then the two demand functions are equivalent. It also makes clear that utility

6 6 is held constant for movements in price along the compensated demand curve, while movements in price along the ordinary demand curve confound two effects: the pure price effect, and an implicit income effect (the so-called substitution and income effects). Because of the latter, movements along the ordinary demand curve do not hold utility constant. This difference is important in distinguishing between the types of measures typically employed in applied welfare analysis Price change welfare measures Equations (14.1) through (14.6) provide the building blocks for defining measures that reflect the value consumers have for changes in prices, income, or the level of quasi-fixed goods. As noted above, these have come to be known as welfare measures, although in general we are not able to measure individual utility changes. Rather, we are interested in measuring a person s willingness to pay for or accept for the proposed change. In this section we review techniques that have been developed specifically for the case of price changes Compensating and equivalent variation Consider a change in the price of a private consumption good, and suppose we are interested in assessing how the price change may impact the well-being of a person or group of people. For example, we may be interested in knowing how subsidized tuition for vocational training (an effective tuition price decrease) affects low income households. There are two related concepts that we can use to measure the effect, which differ by their choice of reference point. The first of these is compensating variation, which is summarized in the following definition: Definition 14.1 Following a price decrease, the compensating variation (CV) is the amount of money that would need to be taken from a person to restore the original utility level. Following a price increase, CV is the amount of money that would need to be given to the person to allow them to once again reach the original utility level. As is clear from the definition, CV uses the pre-change level of utility as a reference point, and is the income offset that would, following the change, restore the original utility level. Analytically it is easiest to see the definition of CV using the indirect utility function. For a change in prices

7 7 from p 0 to p 1, the compensating variation is implicitly defined by the equation 0 1 V( p, y, q) V( p, y CV, q). (14.7) The left hand side of the equation is the utility level obtained at baseline price conditions, denoted by p 0. The right hand side summarizes the optimal behavioral outcome under the new conditions, with price p 1, but with an income adustment to maintain the original utility level. As written here CV>0 for a price decrease (money is taken away) and CV<0 for a price increase (income is augmented). Importantly, the equation also shows that CV can be interpreted as a willingness to pay or accept measure. For a price decrease CV is the maximum amount of money a person would pay to have the lower price, since any smaller payment would provide a well-being improvement and the actual amount CV leaves him indifferent. For a price increase CV is the minimum amount a person would be willing to accept in compensation for agreeing to the increase; anything less would leave him worse off. The second welfare measure, equivalent variation, is defined based on the ex post utility level, and it is summarized in the following definition: Definition 14.2: For a price decrease that increases utility, the equivalent variation (EV) is the payment that allows the person to reach the new utility level, absent the price change. For a price increase lowering utility, EV is the amount that would need to be taken from the person to move him to the new (lower) level of utility, absent the price change. EV differs from CV in that it uses the post-change level of utility as its reference point, and it is the income adustment that moves the person to the new level of well-being, without the actual change. The indirect utility function is again useful for providing an analytical definition: 1 0 V( p, y, q) V( p, y EV, q). (14.8) The left hand side shows the utility level that is reached given the new price vector. The right hand side summarizes behavior at the original price vector, but with an income adustment to maintain utility at the reference level. As written, EV>0 for a price decrease since more income is needed to obtain the new (higher) utility level. In contrast, EV<0 for a price increase, since money needs to be taken away to move utility to its new (lower) level.

8 8 Like compensating variation, EV has a clear WTP/WTA interpretation. For a price decrease equivalent variation is the willingness to accept compensation to forego the decrease. For a price increase equivalent variation is the willingness to pay to prevent the increase. The link between EV and CV and WTP and WTA is central to applied welfare analysis. To begin the chapter we argued that consumer sovereignty implies that the appropriate measure of value for an exogenous change is a consumer s willingness to pay to have the change or his willingness to accept to forego the change. This suggests that the operational goal of applied welfare analysis is to estimate a sufficient component of preferences to allow calculation of compensating or equivalent variation. Two additional analytical expressions for CV and EV help move us in the direction of an operational strategy. The measures can be expressed using the expenditure function by CV E p u q E p u q (,, ) (,, ) y E p u q 1 0 (,, ), EV E p u q E p u q (,, ) (,, ) E p u q y 0 1 (,, ), (14.9) where u 0 and u 1 are the original and new levels of utility, respectively. Recalling that Shepard s Lemma implies E( p, u, q) h ( p, u, q), 1,..., J, p (14.10) the expressions in equation (14.9) for the specific case of a change in p can also be expressed by the definite integrals 0 p CV h p p u q dp E p u q E p u q 1 p 0 p 1 p (,,, ) (,, ) (,, ) EV h p p u q dp E p u q E p u q (,,, ) (,, ) (,, ), (14.11) where p denotes the vector of prices without p. From this we see the familiar result that CV or EV for a price change can be computed by measuring the area under the appropriate compensated demand curve and between the two price levels. If we know or can estimate the compensated demand curves, this calculation is straightforward. Normally, however, we estimate ordinary demand curves and the integrals in (14.11) cannot be directly computed. We discuss this apparent dilemma in detail below.

9 9 Figure 14.1 summarizes graphically the three equivalent ways of expressing EV and CV for the case of a single good and a numeraire, when the price decreases and the good is normal. Panel A illustrates behavior and the welfare measures in utility space. Income is shown as y 0 along the vertical access showing consumption of z, the budget constraint for the initial price is labeled p 0, and the person chooses initial consumption amount x 0 and obtains utility level u 0. The price decrease results in the new budget constraint labeled p 1, which expands the feasible choice set. With the new price the person chooses consumption level x 1 and reaches a new, higher level of utility at u 1. Compensating variation is calculated by finding the expenditure needed to reach u 0, given the new price. In the figure the dashed line p 1 is a hypothetical budget line drawn for price p 1, but with income that is only sufficient to reach u 0. This income level is E(p 1,u 0,q), and the graph shows that CV=y 0 E(p 1,u 0,q). Likewise, equivalent variation is found using the hypothetical budget line p 0, which is drawn for the original price p 0 but with the income needed to reach u 1. This is the expenditure level E(p 0,u 1,q), from which we can see that EV= E(p 0,u 1,q) y 0. Panel B illustrates the same measures in indirect utility space, where income is on the vertical access and price is on the horizontal access. Recall that indirect utility is non-decreasing in income and non-increasing in price. Point a shows the initial outcome, where the person obtains utility V 0 =u 0 given y 0 and p 0. The price decrease leads to the new outcome at point b, in which V 1 =u 1 is reached given y 0 and p 1. The willingness to pay for the price decrease (CV) is the vertical distance between point b and hypothetical point c, which sits on the original utility curve V 0, but at the new price. The willingness to accept to forego the price decrease (EV) is the vertical distance from point a to the hypothetical point d, which sits on the new utility curve V 1, but at the old price. The hypothetical outcome points c and d are also shown on Panel A, where they mark the unobserved compensated consumption levels. Panel C connects utility space to demand space. The price change from p 0 to p 1 traces out the ordinary demand curve x(p,y,q), with points a and b once again representing the initial and final observable outcomes, respectively. Reference to Panels A and B reinforce the notion that the utility level changes as we move along the ordinary demand curve. The two compensated

10 10 demand curves, in contrast, show fixed-utility responses to the price change. The budget lines p 0 and p 1 trace out the response to the price change along u 0, shown as the movement from a to c, resulting in the compensated demand curve h(p,u 0,q). Likewise the budget lines p 1 and p 0 trace out h(p,u 1,q) via the movement from b to d. The fact that the compensated demand curves are based on hypothetical outcome points reinforces the notion that they are not directly observable. In Panel C, compensating and equivalent variation are given by the change in the areas behind the appropriate compensated demand curve. In particular, CV is the area traced out by the points p 0 acp 1, and EV is the area given by p 0 dbp Consumer surplus Equation (14.11) implies that EV and CV can be computed using the compensated demand curve, and as such they can be linked to a behavioral function for the good that has a price change. Typically, of course, it is ordinary rather than compensated demands that are estimated, and the integrals as given cannot be directly computed. A consequence is that consumer surplus has often been used in place of EV or CV, which is defined as follows: Definition 14.3: Consumer surplus is the integral under the ordinary demand curve for a good, computed between a reference price and the point where the demand curve intersects the price axis (i.e. the choke price). The change in consumer surplus (C) for a good is the integral under the ordinary demand curve between an initial and new price level, analytically given by 1 p C x (,, ). 0 p y q dp (14.12) p Since the integral in (14.12) is based on the ordinary demand curve, it can be readily computed using an estimate of consumer demand. Consumer surplus and its relation to EV and CV are shown graphically in panel C of Figure 15.1, where C=p 0 abp 1. Although it can be readily calculated, using C in place of CV or EV begs the questions of what is being measured and how it relates to willingness to pay or willing to accept. Hundreds of pages in dozens of ournals have been devoted to this question since Marshall introduced the concept of consumer surplus early in the last century. Here we summarize the differences that are most important for applied welfare analysis, focusing on the welfare theoretic interpretation of C, its uniqueness, and its

11 11 performance as an approximation. Interpretation of Consumer Surplus In the most general case consumer surplus has no meaningful WTP or WTA interpretation. Unlike a movement along a compensated demand curve, a price change along an ordinary demand curve implies a utility change. This was shown in Figure 15.1, where the change from p 0 to p 1 leads to an increase in utility from u 0 to u 1. Since utility is changing, the area behind the ordinary demand curve cannot be a utility-constant measure of willingness to pay or accept in the sense that EV and CV are. At best we can hope that C is something else perhaps a measure that transforms a change in utility into a money-based (and hence quantitatively meaningful) reflection of the well-being change. To examine this possibility it is useful to rewrite the expression for consumer surplus using Roy s identity as C 1 p 0 p 1 p 0 p V V p y ( p, y, q) dp ( p, y, q) Vp ( p, y, q) dp, ( p, y, q) (14.13) where subscripts on V denote partial derivatives. By the Envelope Theorem we know that V y ( )=p,y,q and hence the second equality follows, where ( ) is the marginal utility of income that arose from the solution to (14.1). Note as well that the numerator reflects the marginal (dis)utility of price. Finally, if p,y,q is not a function of p i.e. / p =0 then the denominator in equation (14.13) can be moved outside the integral and consumer surplus rewritten as 1 1 p C V (,, ) 0 p p y q dp ( p, y, q) p V ( p, y, q) V ( p, y, q) ( p, y, q) (14.14) This is an important result. If the marginal utility of income is constant with respect to p (i.e. is not a function of p ), consumer surplus is indeed a money-metric reflection of the change in utility, and therefore has quantitative meaning. In particular, we can see from the second equality in (14.14) that C consists first of a utility change, which is then monetized by scaling it times a term that reflects the money value of a unit of utility change. This implies the

12 12 measurement unit of C is the change in money that is implied by a change in utility, when / p =0. Consumer theory does not generally imply that / p =0; instead, it is an extra assumption that needs to be maintained if C is to have the desired interpretation. To see how strong or weak this assumption might be, note that Roy s identity for good implies that V p ( p, y, q) x ( p, y, q) V ( p, y, q) x ( p, y, q) V ( p, y, q) V ( p, y, q). y p y (14.15) If we differentiate both sides of the second equality with respect to y we can derive the expression x () x ( ) Vyy ( ) Vy ( ) V p ( ). y y (14.16) If the marginal utility of income V y ( ) is not a function of p, then Vyp ( ) 0. By Young s theorem this implies V ( ) 0, and so it is the case that py and after some manipulation, x ( ) x ( ) Vyy ( ), y V () x ( ) y y Vyy ( ). y x ( ) V ( ) y y (14.17) (14.18) This has ramifications for how the demand for good responds to income changes. Note that, the income elasticity for good, is the same for all in which / p =0. Thus, for consumer surplus to have a meaningful interpretation, the income elasticity of demand must be a priori set equal for all the goods whose price may change as part of the analysis. The restrictiveness of this condition depends on the context. If a subset of the prices is held constant throughout, the income elasticities for the goods with price changes must be equal, but the common elasticity can be any magnitude. For some applications this may be intuitively acceptable, even if statistical tests do not support the restriction. In instances when the analysis involves changing most or all of the prices, the assumption becomes more tenuous. To see why this is, we can differentiate the consumer s budget constraint identify with respect to y and rearrange to obtain the Engle

13 13 aggregation condition: J y p x ( p, y, q) z( p, y, q) 1 x ( ) z( ) x ( ) y x z( ) y z 1 y y y x y y z y J J p 1 p 1 J y p x ( ) z( ). 1 z (14.19) Note that to impose the constraint = for all goods =1,,J, it must be the case that z = which is unlikely to be a supportable restriction econometrically, or intuitively. Uniqueness of Consumer Surplus An additional difficulty with consumer surplus is that, absent restrictions, it is generally not unique for multiple price changes. When more than one price change is part of the analysis, consumer surplus is a line integral defined by Vp k ( p, y, q) C dpk, (14.20) V ( p, y, q) L k y where is the set of goods for which prices changes in the analysis, and L is the path of integration. A line integral consists of a sum of interdependent definite integrals, and a path of integration that indicates the order in which the variables change. For example, if prices p 1 and p 2 in a two-good system change, we need to decide the order in which we calculate C. We could first change p 1, and calculate the consumer surplus change from the movement along x 1 ( ). To this we would add the consumer surplus change from the subsequent change in p 2 based on the movement along the (now shifted) x 2 ( ) curve, to obtain C. Analytically this is p1 1 1 p p1 p2 (14.21) C x ( p, p, y, q ) dp x ( p, p, y, q ) dp. We could, however, perform these calculations by first changing p 2 and then p 1, which is p1 1 2 p p2 p1 (14.22) C x ( p, p, y, q ) dp x ( p, p, y, q ) dp. The problem is that these two calculations will lead to different answers if certain restrictions do not apply, which casts further doubt on the usefulness of consumer surplus as a welfare measure. Formally, a line integral has a unique value only if the integrand is an exact differential of a function, which the expression in (14.20) is not. However, if income and the prices of goods not

14 14 included in remain constant, then the integral in (14.20) can be rewritten as 1 C Vp ( p, y, q) dp, k k V ( p, y, q) (14.23) L k y since V y ( ) does not change within the path of integration L. With this restriction, the integrand in (14.23) is an exact differential of V(p,y,q), and the resulting measure C is unique. From this we see that the restriction needed for uniqueness is equivalent to the restriction that provides C with a meaningful welfare interpretation for price changes. In general, consumer surplus can only be a meaningful welfare measure under potentially restrictive assumptions on how income affects demand. In particular, all income elasticities for the goods being analyzed must be set equal; an equivalent condition is that the ordinary demand cross price effects need to be equal (see exercise x.x). The restrictiveness of this assumption will vary with the application, but is nonetheless unlikely to hold in many empirical applications. Consumer Surplus as an Approximation While we have seen that consumer surplus has disadvantages as a measure of well-being change, it has the notable advantage of being based on a directly estimable behavioral function. Because of this applied welfare economists have long relied on consumer surplus in spite of its theoretical shortcomings. Given this, it is reasonable to ask how large the error will be when C is used in place of CV or EV. Figure 15.1 provides a hint at the answer. Note that for the case of a normal good C is bound by the preferred measures, in that CV<C<EV. This observation suggests consumer surplus may be an acceptable approximation in some circumstances. In an influential paper, Willig (1976) rigorously confirms this intuition for the single good case. Willig shows that the difference between C and the WTP/WTA measures is dependent on the magnitude of the good s income elasticity, and that in many cases the difference will be small. Specifically, if the income elasticity is small, or the change in consumer surplus is small relative to the overall budget (which seems like for many price change analyses), the error from using C will be of second order importance compared with errors in measurement, specification, and estimation. Willig goes so far as to derive analytical bounds on the error that suggest, for income elasticities falling in plausible ranges, only trivial differences between WTP and WTA can be expected to arise.

15 15 Subsequent research has also examined the quality of C as an approximation for multiple price changes, particularly as regards the issue of uniqueness. Just et al. (2004) show that, while a line integral as shown in equation (14.20) is not unique, any path of integration will lead to an estimate of C that lies between CV and EV for multiple price changes. Thus the bounding results derived by Willig carry over to multiple good cases. Based on this the title of Willig s article Consumer surplus without apology seems appropriate, and changes in consumer surplus for price changes can, with confidence, be applied. But need they be? Towards measurement Willig s paper provided legitimacy to applied economists who relied on consumer surplus measures for evaluating the welfare effects of price changes. However, influential work by Hausman (1981) and Hanemann (1980) appearing shortly after Willig s paper introduced the contemporary techniques used to directly calculate measures of compensating and equivalent variation for price changes, and made reliance on approximations unnecessary. The fundamental insight is that ordinary demand curves, if specified to be consistent with a well-defined utility maximization problem, contain sufficient information to recover the portion of the underlying preference function generating the observed behavior. Sufficient conditions for behavioral consistency are summarized by the integrability conditions. These in essence require that the functional form for the ordinary demand equations obey the curvature conditions implied by theory, and that the matrix of Slutsky substitution effects (i.e. the price gradients of the compensated demand curves) is symmetric and negative semi-definite. Two identities from consumer theory are central to showing how the integrability conditions are operationally relevant for applied welfare analysis. The first is shown in (14.5), which says that the observed demand level represents the solution to both the utility maximization problem and the expenditure minimization problems. The second identity is u V p, E( p, u, q), q, (14.24) which connects the indirect utility function to the expenditure function for a reference level of utility ū. Differentiating (14.24) with respect to p leads to V ( ) V ( ) E( ) 0. p y p (14.25)

16 16 We can rewrite this expression as E( p, u, q) V() p p V () y x ( p, y, q), (14.26) where the second equality holds by Roy s identity. This can be interpreted as a differential equation relating income and p, which can in principle be solved to obtain E( ). Specifically, suppose we specify x ( ) and estimate its parameters using consumption and price data. With a known parameterization of x ( ), we can solve the differential equation y( p ) x( p, p, y, q) p (14.27) to obtain y p, k( p, q), (14.28) where p is the price vector absent p and k is the constant if integration that lumps together all terms outside of p. If p and q remain fixed in the analysis (so that k( ) remains fixed) it is possible to interpret (14.28) as a quasi expenditure function, and compute welfare measures using it. Specifically, because utility is ordinal only comparisons, not levels, matter we can without loss of generality normalize the constant of integration to the baseline level of utility, so that u 0 =k(p -,q), and the expenditure function is denoted by Ê(p,u 0 ). This expenditure function is referred to as quasi, since it is only possible to recover the component of preferences related to the demand for good. Anything else appearing in the full expenditure function is, by construction, assumed to be fixed. So long as only changes in p are considered, and the constant levels of p and q can be subsumed into the normalization, it is simple to calculate compensating variation using Ê(p,u 0 ), which is fully characterized by the estimated parameters for the demand curve. An example adapted from Freeman (2003, p.70) helps to show the power of this integrating back technique. Suppose we are analyzing the demand for a single good and a numeraire, and we specify a linear demand equation by x= p+y, where ( are parameters that we estimate econometrically, and the other notation follows from above. Using equation (14.27), we have the differential equation

17 17 dy( p) p y, (14.29) dp which is a common form with a known analytical solution, given by 1 y( p) k exp( p) p. (14.30) If we normalize the constant of integration such that k=u 0 (the baseline level of utility), then the quasi expenditure function is ˆ(, ) exp( ), 1 E p u 0 u 0 p p from which, by substituting y= Ê(p,u 0 ), we can derive the indirect utility function as (14.31) (, ) exp( ). 1 0 u V p y p y p This exercise shows that with estimates of the demand function parameters, the structural functions that are needed to compute welfare measures for a price change are also known. (14.32) There are two generalizations to the Hausman/Hanemann logic that deserve elaboration. First, the techniques described here are based on a demand equation that has a closed form solution for the differential equation, and thus an analytical expression for the quasi expenditure function. This limits options to a handful of functional forms that may not, for reasons of econometric fit and flexibility, be the best choice. Vartia (1983) describes a numerical algorithm that enables computation of welfare effects when there is no closed form for the expenditure function, and so in principle researchers have available a wide range of specifications whose only requirement is consistency with utility maximization conditions. Second, the discussion thus far has focused on a single demand curve. In many cases x is a vector, and we are interested in recovering components of the preference function related to all of the goods under analysis. For this the techniques of integrating back need to be generalized to a system of demand context. LaFrance and Hanemann (1989) rigorously examine how the single equation differential equation in (14.26) generalizes to a system of partial differential equations for the multi-good case. While the notation and mathematical techniques are substantially more involved than for the single good case, the basic intuition carries through. First, the analyst specifies a functional

18 18 form for the system of demand equations in the analysis. As in the single good case, the specification needs to obey certain properties in order to be consistent with a well-defined utility maximization problem; these restrictions are the integrability conditions defined above. As we show in an example below, the restrictions needed in the system context can be quite severe. Second, the parameters of the J dimensional demand system are estimated using data on prices and quantities consumed for the collection of goods. Integrating back the J partial differential equations provides expressions for the quasi expenditure and indirect utility functions, which are then known based on the demand system parameter estimates. Finally, welfare analysis proceeds using the estimated functions. The following example helps show how this method operates in practice. Suppose we specify a system of J demand equations using a log-linear functional form ln x p k p k y, 1,..., J, (14.33) k where (, k, ) are parameters to be estimated. For this system to be consistent with utility maximization i.e. for it to obey the integrability conditions several restrictions are needed. These include k =0 for all k and = for all. Note the severity of these restrictions, which imply that all ordinary demand cross price effects are zero, and the income coefficient is equal for all of the goods. With the integrability conditions imposed, the system in (14.33) can be integrated to obtain a closed form solution for the preference functions. For example, the indirect utility function corresponding to the log-linear system is exp( y) J exp( p ) V ( p, y). (14.34) 1 Von Haefen (2002) discuss how a large variety of functional forms used in systems estimation need to be restricted in order to recover expenditure and indirect utility function parameters from their estimation. While the usual challenges associated with any empirical measurement task apply, the main take away message concerning applied welfare analysis for price changes is that the techniques are well established, understood, and accepted. Whether a single or a set of goods is considered, the procedure for exact welfare analysis is the same. A properly restricted demand curve/system is specified and estimated, and the parameters that are recovered characterize the quasi expenditure

19 19 and indirect utility functions. The ordinal nature of utility allows the terms not related to prices to be normalized away in the constant of integration. In instances when the integrating back strategy is not applied, Willig and his followers have shown that consumer surplus changes can often provide an acceptable approximation to compensating and equivalent variation. The ideas discussed in this section are nicely reviewed by Slesnick (1998), and studied in much greater detail in Just et al. (2004). As we show below, however, the operational techniques that are so effectively applied in price change analysis do not directly transfer to the quantity change case Quantity change welfare measures In environmental economics we are usually more concerned with measuring the well-being effects of changes in quasi-fixed quantities related to the environment, rather than price changes for private goods. This problem requires that we define welfare measures for these changes, and consider operational strategies for quantifying them. In the following two subsections we begin to address these tasks Compensating and equivalent variation The definitions of compensating and equivalent variation for quantity changes follow logically from their price change counterparts. Using the indirect utility function the compensating variation for a change in q from q 0 to q 1 is given by 0 1 V( p, y, q ) V( p, y CV, q ), (14.35) and equivalent variation is analogously defined as 1 0 V( p, y, q ) V( p, y EV, q ). (14.36) For an increase in the quantity of the quasi-fixed good, CV>0 can be interpreted as the willingness to pay to have the improvement, and EV>0 is the willingness to accept to forego the improvement. When analyzing changes in fixed quantities the distinction between WTP and WTA is perhaps of greater importance than for private good price changes. Here, the choice of CV or EV as a welfare measure has associated with an implicit assignment of property rights for the quasi-fixed good. The use of WTP implies ownership of the resource lies elsewhere and the person is not entitled to the improvement. The person may, however, choose to purchase the improvement. In contrast the use of WTA implies the person is entitled to the improvement, and any outcome that does not provide it must be met with appropriate compensation. If in practice

20 20 the difference between WTP and WTA is not large, the choice is of little importance. We return to this point in section 14.6 below. As in the price change case, we can also define CV and EV using the expenditure function by CV E p u q E p u q (,, ) (,, ) y E p u q 0 1 (,, ), EV E p u q E p u q (,, ) (,, ) E p u q 1 0 (,, ) y. (14.37) Finally, we can define the valuation measures using demand curves. Because q is fixed from the perspective of the individual, however, we need to examine inverse demand functions, which in general track the prices a person would pay for different fixed quantities of a good. Define the compensated inverse demand for q function by q E( p, u, q) ( p, u, q), (14.38) q and note that q ( ) is also the marginal willingness to pay for q function, since it is the change in income that holds utility constant following a marginal change in q. We will use the terms inverse demand and willingness to pay function interchangeably. With knowledge of q ( ) we can calculate EV and CV by computing the areas under the appropriate inverse demand curve: 1 q q 0 CV ( p, u, q) dq 0 q EV 1 0 q E( p, u, q) (,, ) (,, ), 0 dq E p u q E p u q q q 1 q 0 q q 1 (,, ) p u q dq 1 1 q E( p, u, q) (,, ) (,, ). 0 dq E p u q E p u q q q (14.39) Figure 14.2 illustrates these concepts graphically. The important difference between the behavior shown here and that in Figure 14.1 is that q is fixed from the perspective of the individual, and it does not have a market price. Total spending on private goods therefore does not change when q changes. Panel A shows the behavior and welfare measures in utility space, where spending on private goods (income) is on the vertical axis, and q is on the horizontal axis.

21 21 At observed income y 0 and initial quantity level q 0, the person obtains utility level u 0. This is shown as point a in the figure. An exogenous increase in quantity to q 1 allows the person to reach a higher utility level u 1 while maintaining private good spending y 0 ; this is shown as point b. The welfare measures are shown on the y axis. Compensating variation is the vertical distance between u 1 and u 0 at q 1. Note that this is the amount of money the person would pay for the increase, since it would move him to the hypothetical point c on the original utility curve. Equivalent variation is the vertical distance between u 1 and u 0 at q 0. To reach the new utility level without the quantity increase (point d), the person would need an increase in income equal to E(p,u 1,q 0 ) y 0. Tracing out the (inverse) demand curves for q is more involved than for the price case, since there is no market price, budget constraint, or choice related to q. However, we can identify virtual prices and income that would lead the person to freely choose a given level of the quasifixed good, if it were exchanged in markets. Several of these thought-experiment price and income level budget constraints are shown via the dashed lines in Panel A, each labeled as l (ỹ l ). Consider for example 0 (ỹ 0 ), which is tangent to the initial outcome a. At point a the person consumes q 0, but does not pay anything to do so. If q were instead exchanged in a market, at price 0 and income ỹ 0 >y 0 the person would, by construction, purchase q 0. Note that virtual income ỹ 0 is greater than actual income in this case, since the budget must allow for expenditures on q. The virtual prices are useful because, like market prices, they identify the marginal willingness to pay for q at different points in the indifference map. These allow us to trace out the compensated inverse demand curves shown in Panel B. For example, the virtual prices 0 and 2 give us two points a and c on q (p,u 0,q); likewise, 1 and 3 provide points d and b on the compensated inverse demand curve q (p,u 1,q). The curves are compensated because as q changes, the amount of private good spending adusts up or down to maintain the reference level of utility. The calculations shown in equation (14.39) are illustrated in Panel B by the area under the appropriate inverse demand curve. Compensating variation is the area q 0 acq 1, and equivalent variation is the area q 0 dbq 1.

22 22 The information in Panel A also allows us to trace out an ordinary inverse demand for q curve, which we denote q (p,y,q). By definition ordinary demands are uncompensated, in that they maintain the initial income level while allowing utility to change. Compensated demands are so named because they adust income to maintain utility. The same idea applies to inverse demands. In Panel A the initial and final outcomes are shown as points a and b, respectively. At these two points, ( 0, ỹ 0 ) and ( 1, ỹ 1 ) show the marginal willingness to pay and virtual income needed to maintain private good spending y 0, while utility changes. Note that ỹ 0 =y 0 +q 0 0, and ỹ 1 =y 0 +q 1 1, which allows the person to purchase q without changing spending on private goods. Connecting points a and b in Panel B allows us to trace out q (p,y,q) the uncompensated marginal willingness to pay for q function. Equations (14.35) through (14.39) are fairly intuitive extensions of the price change definitions given above, and there is an appealing symmetry between the expressions in (14.11) and those in (14.39). In particular, both seem to point to a link between the welfare measures of interest and a well-defined behavioral function. Figures 14.1 and 14.2 further reinforce the symmetry, in that ordinary and compensated demand curves are shown to intersect at the observed consumption outcomes. The similarities, unfortunately, do not extend beyond these conceptual parallels. For the price change case the intersection of compensated and ordinary demand curves was critical, since estimates of x (p,y,q) based on observed choices could be used to recover sufficient information about preferences to allow computation of welfare measures. A parallel idea would be to similarly use estimates of q (p,y,q). However, with no market for q there is no mechanism through which people reveal their marginal values for q, and hence it is not possible to estimate the inverse demand curve by observing individual behavior. This is the crux of the problem, and does much to explain why it is notably more difficult to estimate the welfare effects of changes in the level of quasi-fixed goods than prices for private goods The fundamental challenge of measurement To best understand the challenge of welfare measurement for quantity changes recall the fundamental role that equation (14.26) played in the price change case. Then it was possible to relate observed behavior to the expenditure function, and solve a differential equation to recover the necessary component of the preference function. The analytics are identical for the quantity

23 23 change case. If we differentiate the identity in (14.24) with respect to q and rearrange we recover the relationship E( p, u, q) V ( p, y, q) q q ( p, y, q), (14.40) q V ( p, y, q) y where q ( ) is the ordinary inverse demand (willingness to pay function) for q, shown in Figure If it were possible to estimate the inverse demand for q the techniques described above would be relevant here. Solving the differential equation in (14.40) would allow recovery of the quasi expenditure function for q and, following estimation of q (p,y,q), welfare measurement could proceed as usual. As noted, however, q ( ) cannot be estimated using observed behavior, and so the integrating back strategy is not operationally useful. A comparison of equations (14.27) and (14.40) nicely summarizes the fundamental challenge of welfare measurement for quantity changes. In the former information on the observable demand for private goods allows recovery of the quasi expenditure function, but the normalization needed to eliminate the constant of integration k(p -,U 0,q) requires that q remains fixed in the subsequent analysis. In the latter equation information is not available to estimate the parameters entering the differential equation, and the quasi expenditure cannot be recovered. The main conclusion we draw from this is clear: without additional assumptions on specific aspects of the preference function, or information beyond individuals observed behavior, quantity change welfare analysis is not possible. We return to this theme in detail throughout subsequent chapters in Part III of the book WTP and WTA Thus far in the chapter we have used WTP and WTA more or less interchangeably, having only noted with little elaboration that the choice of one or the other for applied welfare analysis has associated with it an implicit assumption on the distribution of property rights. For many environmental applications this distinction can matter in principle, since it determines (a) whether agents suffering the consequences of pollution are endowed with the right to demand pollution reduction or compensation for losses; or (b) they have no such right, and therefore must implicitly buy a reduction. The WTP versus WTA distinction therefore relates to the perennial ethical debate regarding the rights and responsibilities of polluters versus those suffering