Revealed preference and household production

ARTICLE IN PRESS

Journal of Environmental Economics and Management 53 (2007) 276–289 www.elsevier.com/locate/jeem

Revealed preference and household production Udo Ebert Department of Economics, University of Oldenburg, D-26111 Oldenburg, Germany Received 25 February 2005 Available online 26 January 2007

Abstract The paper deals with the possibilities of recovering the underlying preference ordering from observed behavior when nonmarket goods are employed in household production. The problem is relevant for the evaluation of environmental goods and for the measurement of welfare in environmental policy. It is shown that preferences can be recovered if and only if a corresponding (mixed) demand system can be integrated. This system can be derived from observable behavior and the household production functions imposed. The conditions for its integrability are presented and can be checked. Therefore the approach suggested is operational and allows to decide whether the behavior observed and the household production functions chosen (as maintained hypothesis) are consistent. This result is important since the evaluation of nonmarket goods in this framework crucially depends on the choice of the household production functions. r 2007 Elsevier Inc. All rights reserved. JEL classification: D13; D11; Q51 Keywords: Household production; Integrability; Valuation; Nonmarket good

1. Introduction The task of valuing changes to environmental amenities (or any nonmarket good1 for that matter) is complicated by the fact that observations on consumer behavior (i.e., revealed preferences) alone are insufficient for the characterization of the consumer’s complete preference ordering. Additional information is always required.2 The additional hypotheses imposed in general describe the relationship between environmental goods and market goods, i.e., their substitutability or complementarity. For instance, weak complementarity introduced by Ma¨ler [22] is a condition which can be used if applicable (see also [4]). The household production framework presents another possibility of providing more information and of giving more structure to the problems to be solved. In the following we will investigate the revelation of preferences in this framework.

Fax: +49 441 798 4116.

E-mail address: [email protected]. The analysis is also valid for public goods. 2 Instead of using an indirect method of revealing preferences one could also employ stated preference methods. 1

0095-0696/$ - see front matter r 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.jeem.2006.09.003

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In the household production model one has to distinguish between goods and commodities: it is assumed that the (utility maximizing) consumer employs market and environmental goods in order to produce commodities whose consumption yields utility. Indeed, on the assumption that the environment is used only as an input in the production process and is not consumed directly, the framework allows us to evaluate the environmental goods and to recover preferences [18]. Since the household production functions implicitly describe the marginal rate of substitution between the environment and other market goods, the choice of the production functions is crucial for determining the preference ordering and the (marginal) willingness to pay for environmental goods. Two difficulties arise: first, a household production function is often also not observable, i.e., its functional form cannot be tested econometrically, or as formulated in Smith [26]: ‘‘Most economists y would characterize the net result of models describing ‘consumers as producers’ as providing a good vehicle for the ‘story-telling’ component of model development, but offering a paucity of new testable hypotheses.’’3 Second, the maintained hypothesis that the utility function does not directly depend on the environmental goods is not necessarily consistent with the behavior observed and the household production functions chosen. The objective of the paper is to present a new (direct) approach which enables us to check the consistency of all ‘‘ingredients’’ and to recover the underlying preferences. The method is based on observations and on a maintained hypothesis. It is assumed that (i) the complete (conditional) demand system for market goods, which may depend on environmental goods, can be observed and that (ii) the household production functions are known. Furthermore, it is supposed that (iii) the environmental goods do not enter the underlying utility function directly (hypothesis). On this basis it is possible to reformulate the problem in an appropriate way: Using (i)–(iii) one can derive the marginal willingness to pay functions (inverse demand functions) for the nonmarket goods which augment the conditional demand system. Then we have to investigate the integrability of a mixed demand system (consisting of the demand functions for market goods and the inverse demand functions for the nonmarket goods). Proceeding in this way we obtain a direct approach since the problem is formulated in terms of market and nonmarket goods (and not indirectly in terms of commodities). The demand system allows us to check whether the available information and the maintained hypothesis are consistent. If the test fails it is clear that the demand functions observed, the household production functions chosen and the maintained hypothesis are incompatible. On the other hand, consistency does not prove that the household production functions and the maintained hypothesis are the correct ones. It is in particular possible that there are many other production functions that could be consistent with the behavior observed and the maintained hypothesis. If consistency is satisfied, the (necessary) properties of the underlying utility function can also be checked. If these conditions are additionally fulfilled, a unique preference ordering can be recovered and the corresponding (conditional) expenditure function can be derived by integration. Then one is able to perform any welfare analysis one is interested in and, of course, to evaluate the environmental goods in a simple manner. It has to be emphasized that the approach does not require further assumptions. It provides a useful and operational basis for recovering preferences. The article contributes to two strands of the literature: one is concerned with the integrability problem, the other one with the measurement of welfare when environmental goods have to be taken into account. The integration of a (complete) demand system for private goods is nowadays well understood. Samuelson [25] deals with the derivation of the direct utility function. Hurwicz and Uzawa [19] consider the conditions for integrability and show how to derive the expenditure function. Hausman [16] suggests a method of (direct) integration in the case of two goods. He presents an explicit solution for two particular functional forms of the demand system which are relevant for empirical work. Finally, Vartia [27] proposes a numerical method of integrating any demand system. It can be applied even if the solution of the integration problem does not exist in closed form. Every representation of a preference ordering (direct or indirect utility function or expenditure function) can also be employed for welfare measurement. Conversely, the Hicksian welfare measures also allow us to define corresponding utility functions (cf. e.g., [7]). The (general) integrability problem in the presence of environmental goods is investigated in Ebert [9]. Hori [18] suggests a method of integration when there is household production. He derives the utility function. His approach is different from ours since he investigates the integration problem in the commodity space. Then the 3

Smith himself is a little bit more optimistic.

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observable information about market goods has to be transformed appropriately. This process may lead to complications. Hori’s indirect approach and its difficulties will be discussed in more detail below. There are only two papers explicitly dealing with welfare measurement in the household production model (when there are environmental goods): Bockstael and McConnell [2,3]. The first one derives (Hicksian) welfare measures from a preference ordering which is known. The approach is based on compensated demand functions for market goods and requires weak complementarity and the essentiality of the relevant input (to household production). The later paper deals with an approximation of these measures by means of Marshallian demand functions. It reveals the importance of the Willig condition (see [28] for a definition and [24] for an interpretation). The condition guarantees that the consumer surplus measure of a change in the environmental good is an acceptable approximation of the Hicksian measures. The focus of the present paper is different: its starting points are the observed demand functions for market goods and the household production functions. Conditions like weak complementarity or the essentiality of inputs are not required. The imposition of the Willig condition is also not necessary since here the entire preference ordering is recovered and therefore exact welfare measures can be derived. The paper is organized as follows. Section 2 describes the framework and introduces a simple model. Section 3 deals with the problem of recovering preferences in the household production framework and examines the limitations of Hori’s [18] indirect approach. Section 4 presents the new direct approach and the conditions for integrability. Section 5 discusses the implications for applied work, the assumptions and the application of the method to more general models. Finally Section 6 concludes. 2. Household production In the core of the paper we consider the simplest household production model used in environmental economics (see e.g., [13,22,26]). There are three goods. A Hicksian composite commodity X (consumption), which is consumed directly, and a good Y being an input in the household production process. Both are assumed to be market goods. Their prices are denoted by pX and pY, respectively. The quantities of X and Y can be chosen by the (representative) consumer. Furthermore, the nonmarket good Q, provided by the environment, is given and exogenous for the consumer. It is supposed to be a ‘‘good’’ and not a ‘‘bad’’ like pollution.4 Household production employs the inputs Y and Q to produce another commodity Z, the (consumer’s) personal environmental quality. The technology is described by a twice continuously differentiable production function Z ¼ F(Y,Q) or, equivalently, by its cost function C(pY,Q,Z) ¼ pYF1(Z,Q) where F1(Z,Q) denotes the inverse of F with respect to the first argument. It is assumed that F is strictly increasing and concave in Y and Q. The (representative) consumer consumes the commodities5 X and Z. Her tastes and preferences are represented by a twice continuously differentiable (direct) utility function U(X,Z) which does not depend on the environment Q directly. Taking into account household production and the fact that the level of Q is determined exogenously she maximizes her utility subject to the budget constraint (the exogenous income is denoted by M). This maximization problem can be stated as Problem U*: max UðX ; F ðY ; QÞÞ X ;Y

such that pX X þ pY Y ¼ M

(1)

Q fixed:

(2)

and

4

If the environmental good is detrimental we obtain the model of averting behavior. Compare also [1,6,14]. In this simple model X is a market good and—since it is not employed for household production—simultaneously a commodity yielding utility. 5

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The utility function is weakly separable and the functional form of F(Y,Q) is known. In the following we suppose that U(X,F(Y,Q)) is strictly quasi-concave6 in X, Y, Q. Then the solution of Problem U* is unique. It is described by the conditional demand system X ¼ X ð p; Q; M Þ and

Y ¼ Y ð p; Q; M Þ,

(3)

where p ¼ (pX,pY). Inserting the demand functions in the direct utility function we obtain the conditional indirect utility function V(p,Q,M). Inversion with respect to income yields the conditional expenditure function E(p,Q,u) where u denotes the utility level. Both concepts can be used to define the Hicksian equivalent and compensating variation (cf. [8]). Because of the weak separability of the utility function the marginal rate of substitution between Q and Y has a simple form MRS QY ðY ; QÞ ¼

dU=dQ F Q ðY ; QÞ ¼ , dU=dY F Y ðY ; QÞ

(4)

where FQ and FY denote partial derivatives. It equals the marginal rate of technical substitution of Y for Q. Furthermore, it is independent of X and already completely determined by the household production function. In the optimum of Problem U* the marginal willingness to pay for the environmental good7 wQ is implicitly defined by MRSQY ¼ wQ/pY. Therefore we obtain   wQ ð p; Q; M Þ ¼ pY MRS QY ¼ C Q pY ; Q; F ðY ð p; Q; M Þ; QÞ

(5)

since  C Q ¼ pY F Q F Y . This result is not surprising as an increase in Q by one unit lowers the costs of household production by CQ. Since we want to discuss Hori’s [18] approach below we now switch to the commodity space and reformulate Problem U* to Problem U*(Z): max UðX ; ZÞ X ;Z

such that pXX+C(pY,Q,Z) ¼ M and Q is fixed. Its solution is given by X(p,Q,M) and Z(p,Q,M) where Z(p,Q,M) is not a demand function in the usual sense (see [2]). But it allows us to derive Y(p,Q,M) since Y ¼ F1(Z,Q). The first-order conditions can be rearranged to MRS ZX ðX ; Z Þ ¼

  C Z pY ; Q; Z p 1 . ¼ Y pX F Y ð Y ; Q Þ pX

(6)

The marginal willingness to pay for commodity Z, i.e., wZ(p,Q,M), equals its marginal costs. Strictly speaking we have two different preference orderings: one is defined in the commodity space and represented by U(X,Z), the other one is defined in the goods space and given by U(X,F(Y,Q)). Since wQ ¼ wZ@Z/@Q ¼ wZFQ, the solutions of Problem U* and U*(Z) are identical. Therefore we will often not distinguish between both orderings and use the general term ‘‘preferences’’. 6

Strictly speaking concavity with respect to Q is not necessary to solve Problem U* since Q is fixed, but it is required for recovering the preference ordering. 7 The (usual) marginal willingness to pay for Q can also be derived by means of Roy’s identity wQ ¼ VQ/VM and the compensated one by Shephard’s Lemma: wcQ ¼ E Q .

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3. Recovering preferences 3.1. Problem Above we have introduced the household production model and derived its implications. Now we want to reverse the proceeding and discuss the possibilities of recovering the underlying preferences from observable information. There are two possibilities of framing this problem: one can represent the preferences by UðX ; ZÞ in the commodity space (indirect approach). Then we have to translate observations (3) appropriately. Or one can consider the problem in the goods space and recover UðX ; F ðY ; QÞÞ (direct approach). In this case the commodity Z ‘‘disappears’’ since the household production function is inserted in the utility function. Below we will consider both approaches. In both cases we suppose that the conditional demand system (3) and the quantity Q can be observed and that the household production function Z ¼ F ðY ; QÞ is known. Then the question arises whether there exists a preference ordering such that the conditional demand system (3) can be derived. If the answer is in the affirmative, the preferences are well defined and can form the basis for welfare measurement and an evaluation of the environmental good. In Section 3.2 we will present and investigate the indirect approach proposed by Hori [18]. It turns out that there may be some difficulties in applying it. Therefore in Section 4 we suggest a direct approach which avoids the complications. In a preliminary step we derive a fundamental result for recovering a preference ordering in our framework. We have to discuss which information is required to evaluate the environmental good Q, i.e., to determine wQ ðp; Q; MÞ, when the household production model is given. Considering (4) and (5) we recognize that the Problem U* describes an ‘‘ideal’’ situation: In order to evaluate the environmental good Q it is not necessary to know the entire preference ordering. Knowledge of the household technology and of the consumer’s behavior (the conditional demand system) is already sufficient to compute wQ. The situation changes entirely if we admit an additional direct effect of Q on the consumer’s preference ordering, i.e., if U ¼ UðX ; Z; Q). In this case we get MRS QY ¼

FQ UQ þ , F Y U ZF Y

i.e., we have to take into account an additional term on the right-hand side. It depends on the unobservable utility function and it also changes the marginal willingness to pay for Q described in (5). Then from an empirical point of view the situation is hopeless (unless we invoke further conditions like e.g., weak complementarity (see also [13]): Q cannot be evaluated on the basis of observations and U cannot be recovered. Thus, we recognize that the marginal willingness to pay for Q can be derived from observable behavior if and only if the underlying preference ordering does not depend on Q directly.8 We have established Proposition 1. Assume that the consumer maximizes a utility function U(X,Z,Q) for Z ¼ F(Y,Q) under conditions (1) and (2) and that the solution is described by (3) and wQ(p,Q,M). Then the following statements are equivalent: (a) wQ can be determined uniquely by means of X(p,Q,M), Y(p,Q,M) and Z ¼ F(Y,Q). (b) wQ(p,Q,M) ¼ CQ(pY,Q,F(Y(p,Q,M),Q)). (c) U does not depend on Q directly, i.e., @U/@Q ¼ 0. Thus the household production framework gives some structure to the evaluation problem. Proposition 1 clearly demonstrates that an evaluation of the environmental good Q and therefore the identification of the 8

U(X,Z) ¼ U(X,F(Y,Q)) satisfies this property.

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underlying preference ordering requires an additional hypothesis: There must be no direct effect of Q on the consumer’s preferences. This insight has to be taken into account in the following. 3.2. Hori’s (indirect) approach Hori demonstrates that in the case considered9 in Section 2 the utility function U(X,Z) can be reconstructed by integration if it is a priori guaranteed that U does not depend on the environmental good Q directly. He does not impose this (necessary) condition explicitly, but introduces it as maintained hypothesis (in the background). In order to demonstrate his proceeding we consider an example: 10

Let the conditional demand system be given by   pY M þ pY Q1=2   , X ð p; Q; M Þ ¼ pX pX Qa þ pY

Example 1.

Y ð p; Q; M Þ ¼

pX Qa M  p2Y Q1=2   pY pX Q a þ pY

(7)

(8)

for some a 2 ð12; 12Þ and the household production function by Z ¼ F(Y,Q) ¼ Y+Q1/2. Obviously Y is a that YX0 and to confine oneself to the domain nonessential good.11 Then one has to guarantee n o  a 1=2 2  D ¼ ð p; Q; M Þ p Q MXp Q and Q40 . For ð p; Q; M Þ 2 D we have X40, YX0 and Z40 and (7) X

Y

and (8) is an inner12 solution of the utility maximization problem. On the assumption that the (underlying) utility function Ua(X,Z) does not depend on Q directly one wants ¯ By total differentiation we obtain to describe an indifference surface U a ðX ; ZÞ ¼ U. U aX ðX ; Z Þ dX þ U aZ ðX ; Z Þ dZ ¼ 0.

(9) 13

This equation is the starting point of the analysis. It can be rearranged, identified from observable information and integrated afterwards in order to recover Ua. Rewriting (9) yields dX þ U aZ ðX ; Z Þ=U aX ðX ; Z Þ dZ ¼ 0.

(10)

Since by assumption the consumer maximizes her utility (see Section 2) one knows that U aZ ðX ; ZÞ=U aX ðX ; Z Þ ¼

C Z pY 1 in the optimum. ¼ pX PX F Y ðY ; QÞ

This optimality condition has to be satisfied and can be used to replace UaZ/UaX in (10). But the right-hand side still contains the prices pX and pY and the quantities Y and Q. These variables have to be eliminated since the right-hand side has to be expressed in terms of X and Z. Therefore the conditional demand system has to be inverted in a first step. (Here some regularity conditions are needed, see e.g., [25].) In our example one obtains  1=2 Y þ Q1=2 X 1=2 pX ¼  1=2 M X 1=2 Y þ Q1=2 þ YQa=2 9

Hori [18] examines several scenarios. The computations of all examples are suppressed, but are available from the author on request. 11 Smith [26], p. 50 provides an interpretation for this kind of household production function: Given a fish population Q one can enjoy fishing (Z) using a boat (Y). But a boat is not necessarily required (Y ¼ 0) since one can also fish from a pier or the bank. 12 The first-order conditions are satisfied. 13 In the two-variable case no additional conditions have to be fulfilled for integrability (cf. [18,25]). 10

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and pY ¼ M

Qa=2 .  1=2 X 1=2 Y þ Q1=2 þ YQa=2

Furthermore, in a second step, one gets14 F Y ðY ; QÞ ¼ 1 and then for the marginal rate of substitution U aZ =U aX ¼

pY M 1 X 1=2 ¼ 1=2 Qa=2 . M pX F Y ð Y ; Q Þ Z

(11)

Using this result one has to integrate X 1=2 dX þ Z1=2 Qa=2 dZ ¼ 0. The integration of this equation is always possible. It yields U ¼ 2X 1=2 þ 2Z 1=2 Qa=2 ¼: U a ðX ; Z; QÞ. Thus the basic idea of Hori’s proceeding is simple. It (in principle) allows us to recover the preference ordering and to evaluate the environmental good Q: Simple computation yields that the marginal willingness to pay for Q is equal to  1=2 1=2 2ap Y þ Q Y p , (12) wQ ¼ pY MRS QY ¼ Y1=2 þ Q 2Q given the utility function Ua(X,Z,Q). But we may face some difficulties with the indirect approach which are discussed in the following. Checking Ua(X,Z,Q) we recognize three points. First, the utility function depends on Q unless a ¼ 0. This fact cannot be revealed by an inspection of the conditional demand system (7) and (8). But it can already be discovered from an investigation of the marginal rate of substitution (11) and the partial differential equation. If Q cannot be eliminated from this expression, it is still present after integration. That means, we do not have to integrate in this case. We already know at this stage that for a6¼0 no utility function U(X,Z) exists which allows us to derive the conditional demand system as the solution of Problem U*(Z) given the above household production function. Second, following Hori’s method we have to invert the demand system (and in general to solve the household production function for Y) in order to obtain (11). Both steps can be laborious. It is even possible that these steps cannot be performed explicitly: There might be no closed analytical form of the inverse demand functions or of the inverse of the household production function. Then U cannot be recovered and it is impossible to decide whether U depends on Q or not. A third difficulty concerns the properties of Ua. If e.g., ao0 the marginal utility of Q can be negative, i.e., Q can be a bad! This property can be revealed only by an investigation of the utility function Ua; it cannot be seen from (7), (8) or (11) directly.15 To sum up, Hori’s method works if the maintained hypothesis that the utility does not directly depend on Q is satisfied. It is not clear from the beginning whether this condition is fulfilled. Furthermore, there are some complications. The inversion of the conditional demand system (and of the household production function) is required. The function recovered might have undesired properties. Furthermore, our example demonstrates that the necessary condition (formulated in Proposition 1) is not taken into account in the process of integration! Therefore one can ask oneself whether there is an alternative approach avoiding these difficulties. The next section will provide a positive answer. 14 In our case Fg(Y,Q) is a constant. In general it depends on Y and Q. Then Y has to be replaced by Y ¼ F1(Z,Q) in order to get an expression depending only an X and Z. 15 It is even possible that U(X,Z) is convex (cf. [11]). See also Samuelson [25].

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4. Direct approach 4.1. Mixed demand system Now we suggest a direct approach. We know that U must not depend on Q directly. But as Example 1 demonstrates this requirement is not necessarily satisfied. Proposition 1 tells us that the condition is equivalent to the requirement that wQ ¼ CQ. Therefore we will now impose this condition explicitly. We ask the question whether there exists a (weakly separable) utility function U(X,F(Y,Q)) such that the conditional demand system X ¼ X ðp; Q; MÞ;

Y ¼ Y ðp; Q; MÞ

is the solution to Problem U* and such that   wQ ð p; Q; M Þ ¼ C Q pY ; Q; F ðY ð p; Q; M Þ; QÞ .

(13)

(14)

Then the problem is formulated in the goods space. We have to integrate the mixed demand system16 (13) and (14) for ð p; Q; M Þ 2 D where D denotes the set of feasible arguments. We obtain a problem of integrability. The demand functions are observed. The marginal willingness to pay function can be determined from the demand functions, the household production function and the maintained hypothesis. Thus the latter is a priori taken into account. Furthermore, it will turn out below that the other difficulties mentioned in Section 3.2 can also be avoided: the demand system and the household production function do not have to be inverted. The properties of U(X,F(Y,Q)) can be checked directly by the inspection of the mixed demand system. In the next subsection the details will be discussed and necessary and sufficient conditions for integrability will be presented. 4.2. Integrability According to our discussion we have to check whether the mixed demand system is integrable, i.e., whether there is a weakly separable direct utility function U(X,F(Y,Q)) generating (13) and (14). Existence of U is equivalent to the existence of a conditional expenditure function E(p,Q,u) which is concave, increasing and linearly homogeneous in prices p, decreasing and convex in Q and increasing in the utility level u. If the expenditure function exists it satisfies the following conditions17: @E ð p; Q; uÞ=@pX ¼ X ð p; Q; E ð p; Q; uÞÞ,

(15)

@E ð p; Q; uÞ=@pY ¼ Y ð p; Q; E ð p; Q; uÞÞ,

(16)

@E ð p; Q; uÞ=@Q ¼  wQ ð p; Q; E ð p; Q; uÞÞ ¼ C Q ð p; Q; F ðY ð p; Q; E ð p; Q; uÞÞ; QÞÞ.

ð17Þ

Eqs. (15)–(17) is a system of partial differential equations. For integrability two conditions have to be fulfilled: (1) A function E(p,Q,u) satisfying (15)–(17) has to exist (mathematical integrability). (2) The function E(p,Q,u) has to possess the usual properties of a conditional expenditure function (economic integrability). Mathematical integrability18 requires that the Jacobian matrix of E, i.e., ðE ij ðpX ; pY ; Q; uÞÞi;j¼pX ;pY ;Q is symmetric. If we replace u by the conditional indirect utility function V(p,Q,M), this condition is equivalent to some conditions on the Slutsky matrix S ¼ ðsij Þi;j¼X ;Y ;Q of the mixed demand 16

Cf. e.g., Chavas [5] for a discussion of mixed demand systems. We assume that the demand functions given are once continuously differentiable. 17 Shephard’s lemma is applicable. 18 Compare also Hartman [15].

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system (13) and (14): qX qX qY qY ¼ ¼ sYX , þY þX qpY qM qpX qM  qwQ qwQ qX qX  wQ ¼ ¼ þX ¼ sQX , qQ qM qpX qM  qwQ qwQ qY qY  wQ ¼ ¼ þY ¼ sQY . qQ qM qpY qM

sXY ¼

(18)

sXQ

(19)

sYQ

(20)

These conditions have to be fulfilled for all ðp; Q; MÞ 2 D. Condition (18) is a priori satisfied, since X(p,Q,M) and Y(p,Q,M) form a (conditional) demand system by assumption. The other conditions (19), (20) (which are not independent) are not satisfied automatically and postulate that the demand system and the household production function fit to one another. Economic integrability is guaranteed if the Slutsky submatrix ðsij Þi;j¼X ;Y is negatively semidefinite (which is again implied by the conditional demand system) and if sQQ ¼

qwQ qwQ  wQ o0 for ð p; Q; M Þ 2 D. qQ qM

(21)

Finally, if the integrability conditions are satisfied the system (15)–(17) can be integrated. Many demand systems used in practice allow us to derive an explicit solution E(p,Q,u). But there are also cases in which the integration does not yield a closed-form solution. Then numerical methods can be applied (see [12]). If the conditional expenditure function E(p,Q,u) exists, it is equivalent to a corresponding direct utility function U which has to be weakly separable, i.e., U ¼ U(X,H(Y,Q)), because of the form of the mixed demand system. Since we also get HQ/HY ¼ FQ/FY there is U~ such that U ¼ U~ ðX ; F ðY ; QÞÞ. It has to be stressed that the utility functions U and U~ are ordinal. They also represent the underlying preference ordering. Thus we have derived Proposition 2. Given our assumptions the mixed demand system (13) and (14) is integrable on D if and only if conditions (18)–(21) are satisfied for ð p; Q; M Þ 2 D. The result is also implied by Proposition 2 in Ebert [10] who does not consider household production. A formal proof can be given in analogy to Hurwicz and Uzawa [19] or along the lines sketched in Jehle and Reny [20, pp. 83–85]. Two remarks have to be made: first, one has to add an initial value if the system of partial differential equations is to be integrated. Here one can choose E(p0,Q0,u0) ¼ 1 e.g., for p0 ¼ (1,1) and Q0 ¼ 1. The condition normalizes the (ordinal) utility function. Second, the discussion in this subsection demonstrates how weak separability is used to restrict the Slutsky matrix to recover preference information. Separability is implied by the household production framework and the maintained hypothesis. They enable us to derive the marginal willingness to pay for Q (see Proposition 1) which in turn allows us to identify the missing parts of the Slutsky matrix. In view of this result we reconsider Example 1: The mixed demand system consists of (7), (8) and condition (12). The crucial conditions for mathematical integrability are (19) and (20). They are satisfied only if a ¼ 0. Then condition (21) is also fulfilled. 5. Discussion 5.1. Implications Proposition 2 presents necessary and sufficient conditions for the integrability of the mixed demand system (13) and (14). These conditions can be checked directly. There are two possibilities: either one of the conditions is violated or all conditions are satisfied. In the first case it is clear that—given the household production function—there is no utility function U(X,F(Y,Q)) implying the system X(p,Q,M), Y(p,Q,M) and

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wQ(p,Q,M) as the solution to Problem U*. Then the demand system (the observed behavior), the household production function and the maintained hypothesis are inconsistent, i.e., they do not fit to one another. But, nevertheless, it is possible that the demand system is compatible with another household production function.19 This outcome is interesting. It proves that it is not possible to augment an observed conditional demand system by an arbitrary household production function (if we maintain the hypothesis that the underlying preference ordering must not depend on Q directly). In the second case all ‘‘ingredients’’ are consistent. Then the marginal willingness to pay for Q can be derived directly from the household production function without solving the integration problem since the marginal rate of substitution of Y for Q coincides with the corresponding marginal rate of technical substitution (see (4)). When a general welfare analysis is to be performed, then, of course, the mixed demand system has to be integrated in order to obtain the conditional expenditure function. It is unique and allows us to compute the Hicksian measures of welfare change. One point should be stressed (see also below): this kind of consistency does not prove that the household production function is the ‘‘correct’’ one, but several others can be ruled out. We present two examples demonstrating possible outcomes. Example 2. We introduce the conditional demand system X ð p; Q; M Þ ¼

p M  Y , pX pY þ pX Q

(22)

Y ð p; Q; M Þ ¼

p Q  X M pY pY þ pX Q

(23)

and the household production functions   

Z ¼ F  ðY ; QÞ ¼ Y 1=2 þ 1 Q for  2 0; 1=2 which imply the cost functions ( 0    2 C  pY ; Q; Z ¼ pY Z=Q  1

for ZpQ ; for Z4Q :

For a test of integrability or for integration we have to take into account the maintained hypothesis (14) which is given by     wQ ð p; Q; M Þ ¼ C Q pY ; Q; Z ¼ 2pY Y 1=2 þ 1 Y 1=2 =Q, where Y ¼ Y ð p; Q; M Þ. Now we have to check (19) and (20) and obtain the condition  ¼ 12. Thus whenever a12 the respective household production function is not consistent with the above demand system since (19) and (20) are violated. One could have the impression that there is always at most one household production function being consistent with a given conditional demand system. This conjecture is not correct as the following example demonstrates: Example 3. We consider the conditional demand  system  (22), (23) again and use a slightly different family of household production functions: F g ðY ; QÞ ¼ Y 1=2 þ g Q1=2 for g40. 19

Then the marginal willingness to pay function changes too!

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For the marginal willingness to pay for Q we then obtain  1=2  Y þ g Y 1=2 , wQ ð p; Q; M Þ ¼ pY Q where Y ¼ Y ð p; Q; M Þ. In this case it turns out that the entire family Fg is consistent with the demand system observed. Obviously the choice of g determines the marginal willingness to pay for Q which can therefore attain many different values depending on g. Example 3 reiterates the point raised above: an infinite variety of household production functions can be consistent with given behavior. The fact of consistency does not prove anything: There is no possibility of testing the value of g. In such a situation we obtain different preference orderings and welfare measures depending on the parameter g. The researcher has at least two possibilities of dealing with this problem. First, one can make an attempt at choosing exactly one of the feasible technologies on the basis of additional information (if available). This proceeding is the simplest way of dealing with the problem of multiple solutions. Additional data can help to restrict the choice set. Second, it is sometimes possible to leave the household production framework and to augment the conditional demand system for market goods directly by a marginal willingness to pay function for the environmental good (cf. [9]). Then revealed and stated preference methods are combined. A priori it is not clear which alternative is viable and which one is better. The decision depends on the problem to be investigated, the data available, and the effort required. But in most cases it will probably be easier to determine the technology than to elicit the marginal willingness to pay function for a nonmarket good by stated preference methods. Therefore the setup suggested in this paper seems to be preferable whenever applicable. 5.2. Assumptions For clarity we discuss the general assumptions which have to be made when the direct approach is to be employed. The conditional demand system and the household production function must be known. X ðp; Q; MÞ and Y ðp; Q; MÞ have to be homogeneous of degree zero in prices and income. They have to satisfy the adding-up condition. The corresponding Slutsky matrix has to be symmetric (see Eq. (18)) and to be negative semidefinite. F has to be strictly increasing and concave. These properties have to be fulfilled for the feasible levels of the environmental good Q. On this basis and on the assumption that U is weakly separable one can introduce the maintained hypothesis by wQ ¼ C Q . Then the conditions for integrability listed in Proposition 2 can be checked explicitly. Furthermore, we want to emphasize that some assumptions, which are often made in this field of environmental economics, are not required. At first we reconsider Example 1. It demonstrates that for a ¼ 0 the demand system (8), (9) and (12) can be integrated. The solution can be derived for X 40, Y X0, Q40 and Z40 since the analysis in this paper is based on inner solutions of the utility maximization problem (see the derivation in Section 2). Using the direct approach we obtain the conditional expenditure function and can also determine the corresponding utility function UðX ; ZÞ for all X 40 and Z40. The utility function allows us to consider the corner solutions for Y ¼ 0. Inspection of (8) shows that they are implied if pX Mop2Y Q1=2 . Then we obtain X ðp; Q; MÞ ¼ M=pX and Y ðp; Q; MÞ ¼ 0. Furthermore we get wQ ð p; Q; M Þ ¼

1 M 1=2 2 p1=2 Q3=4

(24)

X

in this case (Y ¼ 0). It is easy to see that because of continuity the solutions in (12) (for a ¼ 0) and (24) are equal if pX M ¼ p2Y Q1=2 . Therefore this example can be used to further clarify the assumptions necessary for the direct approach. Since the approach allows us to recover the underlying preference ordering, the example demonstrates three points: first, the essentiality of the input Y is not required. Second, the input Y and the environmental good Q

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do not have to be weak complements.20 Third, the Willig condition which often plays a role in the measurement of welfare is not involved.21 A referee has pointed out that Example 1 assumes perfect substitution between the private good Y and an affine transformation of Q in the household production function and that this case has already been dealt with by Ma¨ler [23]: With perfect substitution assumed the marginal willingness to pay for Q can be derived directly from the conditional demand system. Indeed, Ma¨ler’s argument allows to determine the marginal (!) willingness to pay for the environmental good under the condition that the price of Y is fixed. But the direct approach is more comprehensive: it allows us to recover the entire preference ordering and the corresponding expenditure function. The latter is needed if e.g., the price of the input Y and the quantity Q are changed simultaneously. In such a case Ma¨ler’s approach is silent. The household production functions used in Examples 2 and 3 are different. In these cases the input Y is essential and Y and Q are not weak complements. Furthermore the Willig condition is also violated. These examples demonstrate that the direct approach does not require perfect substitutability. Finally, we have to deal with a subtle aspect of the revelation problem and to offer an explanation to those readers who might be worried about the uniqueness of the preference ordering. Assume that an individual maximizes her utility U^ ðX ; Y ; QÞ subject to a budget constraint pX X þ pY Y ¼ M and for given Q. Then a conditional demand system is implied which can in principle be observed. Now suppose that the utility ^ function by a function f ðv; QÞ which is strictly increasing in the utility level v and depends on  U is transformed  ^ Q. If f U ðX ; Y ; QÞ; Q satisfies the usual properties it also represents a preference ordering. But the ordering is  ^ There are two implications: firstly, maximization of f U; ^ Q different from the original one represented by U. under the above constraints yields the same conditional demand system (since Q is fixed!). Therefore it is impossible to distinguish between these preference orderings on the basis of observations. Secondly, the transformation by f ðv; QÞ changes wQ. Thus Q cannot be evaluated merely on the basis of observations. This problem is well known in the literature (see e.g., [10,17,21]). Therefore one has to think about its consequences for our investigation. It turns out that the indeterminacy shown cannot occur in our framework. The reason is that the kind of transformation mentioned would contradict the maintained hypothesis that the utility function does not depend on Q directly. Since this property is equivalent to wQ ¼ C Q and since this condition is imposed explicitly, the preference ordering which is recovered is unique. 5.3. Extension Up to now we have considered a very particular model of household production. The direct approach suggested above can easily be generalized and also be applied to a larger class of models. Assume now that there are nX2 commodities Zi, n environmental goods Qi for i ¼ 1; . . . ; n and m private goods Y ij for j ¼ 1; . . . ; mi and i ¼ 1; . . . ; n where m ¼ Smi . Production of  commodity i can be described by a production

function Fi: i.e., Z i ¼ F i ðY i ; Qi Þ where Y i ¼ Y i1 ; . . . ; Y imi or by the corresponding cost function C i ðpi ; Qi ; Z i Þ

where pi denotes the price vector for Yi. The private goods Y ij , for j ¼ 1; . . . ; mi , and the environmental good Qi are used exclusively for producing Zi. Let the utility function UðZ 1 ; . . . ; Z n Þ depend only on commodities. Using the price vector p ¼ ðp1 ; . . . ; pn Þ and defining Q ¼ ðQ1 ; . . . ; Qn Þ we suppose that the consumer solves      max U ðZ 1 ; . . . ; Zn Þ ¼ U F 1 Y 1 ; Q1 ; . . . ; F n Y n ; Qn Y 11 ;:::;Y nmn

s:t:

mi n X X

pij Y ij ¼ M

i¼1 j¼1

and Q1 ; . . . ; Qn fixed. Then we obtain the conditional demand system Y ij ð p; Q; M Þ for j ¼ 1; :::; mi and i ¼ 1; :::; n. 20

For Y ¼ 0 we obtain dU=dQ ¼ 1=ð4Q3=4 Þ40. It is violated since qY =qQ ¼ pY =ð2Q1=2 ðpX þ pY ÞÞ and qwQ =qpY ¼ 1=ð2Q1=2 Þ for Y 40 (cf. equation (10) in [24]).

21

(25)

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The approach presented in this paper can be applied to this framework: assuming that the conditional demand functions Y ij ð p; Q; M Þ and the household production functions can be observed and supposing that the underlying utility function UðZ 1 ; . . . ; Z n Þ does not depend on the environmental goods directly we can derive the marginal willingness to pay functions for Qi:  i  i F iQi Y ; Qi  wQi ð p; Q; M Þ ¼ p1 F iY i1 Y i ; Qi      for i ¼ 1; . . . ; n. ð26Þ ¼  C iQi pi ; Qi ; F i Y i p; Qi ; M ; Qi We again obtain a complete mixed demand system. For the derivation of the conditional expenditure function Eðp; Q; uÞ we have to consider a system of partial differential equations (the analogue to (15)–(17)): qE ð p; Q; uÞ ¼ Y ij ð p; Q; E ð p; Q; uÞÞ for j ¼ 1; . . . ; mi and i ¼ 1; :::; n, qpij qE ð p; Q; uÞ ¼ wQi ð p; Q; E ð p; Q; uÞÞ @Qi

for i ¼ 1; . . . ; n.

The corresponding conditions for (mathematical and economic) integrability can be expressed by the Slutsky matrix of the mixed demand system S. Suppose that ! S YY S YQ S¼ S QY SQQ where the submatrices are defined as above: SYY is the Slutsky matrix of the demand system Y ðp; Q; MÞ, SQQ the Slutsky matrix of the system of inverse demand functions wðp; Q; MÞ, etc. Then we get a generalization of Proposition 2 (see [9] for a proof): Proposition 3. Given our assumptions the mixed demand system (25) and (26) is integrable if and only if SYQ ¼ S0QY and SQQ is negative definite. These conditions can be checked and—if they are satisfied—the preference ordering can be recovered. The case considered here is one of the scenarios examined by Hori [18]. The basic idea of the direct approach—investigating the problem of revealing preferences in the goods space—can, of course, also be applied to more complicated household production models. 6. Conclusion The paper has discussed the possibilities of recovering the underlying preference ordering in the household production model when environmental goods are employed as input. We have obtained a number of results. It turns out that a hypothesis about the structure of the preference ordering is indispensable: The environmental goods must not enter the utility function directly. Given this assumption preferences can be recovered if the existence of a corresponding utility function is guaranteed. The existence depends on the integrability of a well-defined mixed demand system. The corresponding conditions can be checked since the demand system is based on observations (in markets for private goods), the household production functions chosen and the maintained hypothesis. Observed behavior is not necessarily consistent with an arbitrary technology (household production function). Thus in some cases the approach allows us to reject household production functions. This result is in particular important since the evaluation of the environmental goods (and, of course, the preference ordering recovered) crucially depends on the choice of the technology. Therefore the direct approach is operational and can be used in practice (if the necessary information is available). It provides a basis for the measurement of welfare when environmental goods have to be taken into account.

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