Recovering preferences in the household production framework: The case of averting behavior

Transcription

1 Udo Ebert Recovering references in the household roduction framework: The case of averting behavior February 2002 * Address: Deartment of Economics, University of Oldenburg, D-26 Oldenburg, ermany Tel.: (+49) Fax: (+49) ebert@uni-oldenburg.de

2 Abstract The aer considers a simle model relevant for evaluation and welfare measurement in environmental economics: a household consumes one market good directly and emloys another market good and an environmental service as inuts in order to roduce its ersonal environmental quality yielding utility. Starting from the comlete conditional demand system for market goods and the roduction function the aer examines the conditions under which maximization of the utility function imlies the behavior observed and the utility function is consistent with household roduction. Integrability uts restrictions on the Slutsky and Antonelli matrix. The results correct some findings resented in the literature. Keywords: household roduction, integrability, evaluation, Slutsky matrix, Antonelli matrix JEL code: D3, D, Q26

3 - -. Introduction The evaluation of environmental goods is a comlicated task: Since markets do not exist, observing a consumer s behavior is not sufficient for evaluation or welfare measurement. Therefore additional information is required or additional hyotheses have to be maintained. In theory and ractice the latter in general describe the substitutability or comlementarity of environmental goods and market goods (cf. Freeman (985), Kolstad and Braden (99)). Mäler (97) was the first author suggesting and using weak comlementarity in order to evaluate an environmental good. Further hyotheses have been introduced later on. The household roduction framework resents another aroach of giving more structure to the roblem to be solved (see Smith (99)). Assuming that market goods are transformed in the household (or by a consumer) into commodities whose consumtion yields utility, often allows to evaluate environmental goods. When the environment is one of the inuts of roduction, the roduction function imlicitly describes the marginal rate of substitution between the environment and other market goods. Therefore the choice of roduction functions finally determines the (marginal) willingness to ay for environmental goods. Thus the assumtion of household roduction forms another hyothesis which may be sufficient for evaluation and welfare measurement. This aer is concerned with the averting behavior model. It is the simlest model allowing to evaluate one environmental good: There are two market and one nonmarket good. One market good (consumtion) is consumed directly (without being used in household roduction). The other one and the environmental good are emloyed to roduce the consumer s ersonal environmental quality (household roduction). The consumer then consumes the consumtion good and her environmental quality. Her references (or tastes) are reresented by a utility function defined on consumtion and quality. The label of averting behavior is justified if the environmental good is detrimental and household roduction leads to an imrovement in environmental quality. But, of course, this model can also be emloyed if the environmental service is a ositive (and not a negative) externality. The model is comletely determined by choosing aroriate tastes (reference ordering or utility function) and an aroriate technology (roduction function). iven that the consumer maximizes her utility under the assumtions made, a comlete system of demand functions for market goods can be derived. The demand functions are conditional, i.e. they in general deend on the environmental good which is fixed and exogenous for the consumer. In I thank Lutz Mommer for helful comments.

4 - 2 - ractice often a functional form for the utility and roduction function is secified and the demand system is formally derived and then estimated (cf. e.g. Shairo and Smith (98)). If all relevant arameters of the utility and roduction function occur in the demand system, the entire model can be determined. Furthermore, in rincile it can be tested whether the functions estimated ossess the necessary roerties of a utility and roduction function, resectively. Such a test might reject the hyothesis that the household roduction model secified is correct. Otherwise it only roves that the hyothesis and the data are comatible with one another. When the utility and roduction function are known, it is ossible to measure welfare changes and to evaluate the environmental good since the corresonding exenditure and cost function can be derived. The objective of the resent aer is to reverse the roceeding: Starting with observed behavior, i.e. a conditional demand system, the ossibilities of deriving an underlying utility and roduction function are investigated. It is easy to show that they cannot be recovered without adding further information. Therefore it is suosed that the demand functions for market goods and the roduction function are given. The latter might be determined by the technology available for averting damages. It turns out that additional conditions have to be satisfied if the underlying utility function which is then (ordinally) unique is to be recovered. This result is imortant since it demonstrates that in this case observed behavior is not always consistent with an arbitrary technology. In rincile these conditions can be tested. The question osed is comarable to the roblem of integrability of demand functions in demand theory. Indeed, the roblem will be framed in such a way that a mixed demand system is integrated. It consists of the conditional demand functions for market goods and an inverse demand function for the environmental good which is derived from the roduction function and the assumtion of utility maximization. Therefore it turns out that the conditions which are sufficient (and necessary) for integrability imose restrictions on the Slutsky and Antonelli matrix of an aroriately defined utility function. Though these matrices are related to the Hicksian demand functions, the conditions can be checked by emloying the demand functions observed and the roduction function imosed. The alication of the methodology develoed is not limited to the averting behavior model. The model underlying the analysis can be generalized easily. To the best knowledge of the author there is only one aer in the literature dealing with integrability in the household roduction framework: Hori (975) investigates four different scenarios and resents some conditions sufficient for integration. His results are at variance

5 - 3 - with the conditions derived below. A simle examle will demonstrate that his analysis does not rovide a correct solution to the integrability roblem. It turns out that Hori concentrates only on the utility function (and the commodities yielding utility). He neglects the environmental good. Thus it is ossible that a utility function does not exist though his conditions are satisfied. Furthermore he does not check whether the utility function ossesses the usual roerties. Below the roblems inherent in Hori s aroach are revealed and discussed. Afterwards the new, comlete solution will be resented. Smith (979) attacks Hori s aroach for a different reason: he claims that the information requirements necessary are more stringent than direct restrictions to the utility function. Freeman (993) resents this argument in a articular clear form. Our analysis will suort this claim. It is not necessary to know the roduction function or the corresonding cost function itself. Knowledge of the marginal rate of substitution between the environmental and the market good used is already sufficient to recover the utility function given the conditional demand system. The rate of substitution is equivalent to the marginal willingness to ay for the environmental good. On the other hand this fact again demonstrates that the choice of the marginal rate of substitution is crucial for the evaluation of the environmental good and for welfare measurement. As long as only the environmental good is to be evaluated knowledge of the rates of substitution is sufficient. But whenever at least one rice of a market good changes, the entire underlying reference ordering (the utility function or, equivalently, the exenditure function) is required for welfare measurement. Thus solving the integrability roblem is not only a theoretical exercise, but is indisensable for alied work. The aer is organized as follows. Section 2 describes the framework. In section 3 several examles are resented and discussed which investigate the relationshis between utility function, roduction function, and conditional demand system. Section 4 derives the conditions for integrability. Three different aroaches to integrating the system given are resented. Finally section 5 concludes.

6 Framework 2. Basic model The household roduction framework allows to evaluate environmental goods (cf. Freeman (993)). We will consider a simle variant of a model which is dealt with in the literature as a model of averting behavior (see e.g. ourant and Porter (98), Harford (984), Bartik (988)). Using an environmental good and another inut consumers roduce their own ersonal environmental quality. If the environmental good can be interreted as ollution consumers decide on the level of its reduction. The cost incurred are corresondingly defensive exenditures. We assume that there are three goods: Let denote a Hicksian comositum (consumtion) and be a factor, an inut in the household roduction rocess. and are market goods. Their quantity can be chosen by consumers. The nonmarket good, rovided by the environment, is given and exogenous. It is suosed to be a good and not a bad as ollution. Therefore the term averting behavior is a little bit misleading in our model. It would be justified for ollution (change into ). The tyical (or reresentative) consumer is able to combine and in order to roduce the ersonal environmental quality. The technology is described by a roduction function F(, ) = (which is the same for all consumers). The (reresentative) consumer consumes two commodities, namely consumtion (which is identical to the market good ) and environmental quality. Her tastes and references are reresented by a (direct) utility function U(, ) which does not directly deend on the environment. iven market rices and for and and the exogenous income M the consumer is a utility maximizer and solves Problem UF,, max U, st.. = F,, () + = M, (2) fixed, (3) i.e. she observes her budget constraint (2) and the technological restriction (). The quantity cannot be influenced (3).

7 - 5 - This roblem is well defined if the utility and roduction function ossess aroriate roerties. We assume that U is quasi-concave, strictly increasing and twice continuously differentiable in and. The roduction function is suosed to be concave, strictly increasing, and twice continuously differentiable in and in. Then the otimization roblem has a unique solution which is described by = M, (4) = M, (5) (, ) = M = F M, where (, ) =. and are the conditional (derived) demand functions (cf. Pollak (969)) and can in rincile be observed. is unobservable, but can be determined if, and F are known. It should be stressed that there is no market for environmental quality and thus no corresonding market rice. Below we will define the marginal willingness to ay for, an imlicit rice of. 2.2 ost minimization The roerties of the technology are relevant for the roduction rocess. We assume that both inuts are normal inuts; i.e. suosing for a moment that is also a market good cost minimal choices of and are assumed to be increasing in outut. Denoting the inuts by = ( ) and ( ) follows (cf. Bear (965)) = for given rices of and these concets can be characterized as d d normal F 0 is indeendent of F F 0 F inferior and normal d F 0 is indeendent of F F 0 d F inferior where F, F, F etc. denote artial derivatives.

8 - 6 - The roerties of F are also imortant when the corresonding cost function is considered. Here it is taken into account that the environment is exogenous for a consumer. We obtain : = min (, ) F st.. F, where F (, ) denotes the inverse function of F with resect to its first argument. ossesses the usual roerties: It is linearly homogeneous in, convex in and concave in : Lemma: 2 Assume that F concave, F > 0, F 0. Then (i) > 0 F > 0 (ii) > 0 F < 0 (iii) 0 F 0 (iv) 0 0 where FF F F F F + F = F F F inferior (v) 0 indeendent of normal (vi) and normal < 0. Thus marginal cost of are decreasing in the environmental good, given that is normal. The cost function (which corresonds to averting costs or defensive exenditures in a model of averting behavior) allows us to reformulate Problem UF (cf. Pollak and Wachter (975) and Deaton and Muellbauer (980)): Problem U, max U, ( ) st.. +,, = M and (3). 2 All roofs have been delegated to the Aendix.

9 - 7 - The solution is, of course, the same as the solution of UF above. For future use we resent a corresonding first-order condition U U (, ) (, ) = (6) which has to be satisfied in an otimum. By analogy we can introduce the imlicit rice of by defining : = (,,, ) = (,, ) = ( (, ), ) F F. It can be interreted as shadow rice of. Unfortunately the virtual exenditures in general not coincide with costs (,, ) if F f, =. Then = f. do =. Both concets are identical if and only 2.3 Alternative reresentations U to now the consumer s behavior has been described in the household roduction framework. iven the assumtion of concavity of the roduction function it is also ossible to emloy the conventional model of utility maximization. Inserting the roduction function in the utility function and thus eliminating the commodity ersonal environmental quality we obtain Problem U,, ( ) max U, F, st.. (2) and (3). The otimization roblem is comletely defined in the goods sace. Its solution is identical to the corresonding solution of roblem UF and U. This formulation of the underlying roblem is advantageous since now the conventional theory can be alied. Inserting the demand functions (4)-(5) into the direct utility function U gives the conditional indirect utility function V M. It is quasi-convex and decreasing in, increasing and concave in, and homogeneous of degree zero in and M. By inversion we get the conditional exenditure function E (, u, ) which is concave, increasing and linearly homogene-

10 - 8 - ous in, decreasing and convex in, and increasing in u. The exenditure function solves also Problem E, ( ) min +,, st.. U, u Roy s identity yields the demand functions V M,, V,, M (,, M) = and (,, M) = V,, M V,, M M and Shehard s Lemma imlies the comensated demand functions u,, : E = and M,, u : = E. (7) Similarly, we are able to derive (exressions for) the marginal willingness to ay (MWTP) for : M = V M V M (8) M and u = E u. (9) By the enveloe theorem we obtain E = and thus = = (,, (, )) M u F when u = V(,, M) and (,, M) =. Since = F F the MWTP for is essentially determined by the rate of substitution between and. Knowledge of the roduction function is sufficient for an evaluation of. It is well-known (cf. havas (984), ornes (992), Ebert (998)) that it does not matter which concet is considered: the direct utility function U, the conditional indirect utility function V, the conditional exenditure function E, the uncomensated conditional demand/ MWTP system (4)-(5) and (8), and the comensated conditional demand/ MWTP system (7) and (9) imly one another (and are, therefore, equivalent) under the assumtions given. The equivalence will lay a role below.

11 Recoverability: Discussion In the household roduction framework a consumer s behavior can be described by her utility function and roduction function. As shown in section 2 maximization of utility imlies a (conditional) demand system for goods. Then the question arises whether the demand system and knowledge of allow us to recover the underlying tastes and/or technology. In the following we will resent a simle examle which forms the basis for some discussion of the connections between the various concets mentioned. 3. Simle examle Let a consumer s taste be reresented by the utility function α α β U = (0) where 0< α <. The function U is a little bit more general than the utility functions introduced above since it also deends on the environment directly. This assumtion is made deliberately for a moment. It will be droed soon by setting β = 0. The technology is also given by a obb-douglas tye function: = F, = γ δ 0< γ <,0 δ which can be equivalently exressed by the corresonding cost function = γ δγ. Then the solution of the Problem UF can be derived exlicitly and leads to the demand functions M,,, () ( M) = ( ε ) M,, = ε, (2) ( M) M γ M = ε δ α where ε = α γ + α and 0< ε <. (3)

12 - 0 - The examle chosen has the (extreme) roerty that the derived demand for and does not deend on the environment, but it fits well into the framework introduced above (if β = 0 ). hecking the examle carefully we are able to demonstrate a number of results. We obtain Result : If the utility function U deends on the environment directly, U cannot be recovered from the derived demand functions for and and the roduction function. This result is easily roved: The arameter β being related to the marginal utility of does not lay any role in, or F. Therefore it cannot be determined from this information. This outcome is not really surrising when we consider the corresonding roblem U: In this case we know a little bit of the structure of U, namely F(, ) =, but U might vary with directly. On the other hand only two demand functions, and, can be observed. It is wellknown (cf. e.g. Ebert (998, 200)) that then U cannot be recovered comletely. Now set β = 0 for the rest of the subsection, i.e. we are back to our original framework. The conditional demand system ( M,, ) and does it imly? We establish M can in general be observed. What Result 2: If only the demand functions () and (2 ) and are observable there is an infinite variety of combinations of ( FU, ) imlying the observed behavior. iven the demand system () and (2) we can determine the budget share ε, 0 < ε <. Now choose an arbitrary arameter α, 0 < α < and define a corresonding arameter γ ( α ) by ε α γ : = γ ( α) = ε α. γ( α) Then U = α α δ and = imly the given demand system and budget share ε (use (3) to calculate ε!). There might be a roblem with the concavity of F, but as long as α is small enough F is concave for δ <. When α is varied tastes and the technology change. This roves result 2. What are the imlications of the technology? an it be determined from, and the utility function? We obtain Result 3: If the demand system () and (2), the quantity, and the utility function U is observable or known, the technology F cannot be recovered.

13 - - This result even holds if we imose the additional assumtion that the technology is reresentable by a obb-douglas function as it is in our case: The arameter δ cannot be derived from the information given. Then the latter is consistent with an infinite number of different technologies (vary δ!), Finally we turn to the scenario which corresonds to the integrability roblem: What can be inferred from, and the roduction function F? We are able to derive Result 4: If the demand system () and (2), the quantity, and the roduction function F are given and (!) if U is a obb-douglas function, the utility function can be ordinally recovered. The roof is obvious: since ε, γ, and δ are observable α can be determined by equation (3). Thus this result might lead one to exect that knowledge of ( M,, ), ( M,, ), and F(, ) is always sufficient to recover the utility function. Unfortunately this conjecture is wrong as the analysis in section 4 will show. The next subsection will critically reconsider a solution to the integrability roblem roosed by Hori (975). 3.2 Hori s aroach To the best knowledge of the author Hori (975) is the only one aer in the literature dealing with the roblem of integrability in the household roduction framework. He examines four different scenarios. Our model (or averting behavior) can be subsumed under his first scenario. The solution he rovides is based on the following idea: Suose that the utility and roduction functions are given and consider all (, ) satisfying U(, ) = constant. U Total derivation yields U d + U d = 0 or d + d = 0. U For an otimal solution the first-order condition (6) has to be satisfied, i.e. U U =. The right-hand side can be determined from,, and F. Thus we obtain a differential equation. Reversing the argument and solving the differential equation we are able to recover the underlying utility function. Hori resents an additional sufficient condition (cf. Hori (975),

14 ) which does not imly any restriction in our case since there are only two commodities and. Unfortunately we may face some roblems when alying Hori s aroach and solution to the integrability roblem. Let us discuss Problem : The RHS of (6) contains rices. This and the following roblems will be discussed by considering the given demand system M (,, M) = and (,, M) 2 and various technologies. Let = (, ) = or F M = 2 =. Deriving the first-order condition (6) we get,, =. Obviously the rice ratio has to be eliminated since we have to obtain a differential equation in the variables and! Thus we have to invert the conditional demand system in order to get the inverse demand functions M 2 (, ) = and (, ) Inserting them in (6) we obtain d d = =. M =. 2 This condition is not aroriate as the variables and must not occur in the utility function U and therefore they must not be used in the differential equation. We have Problem 2: The differential equation may contain and. In the examle considered this roblem can be solved by using the roduction function: can be relaced by. Then we obtain d = d

15 - 3 - and thus the differential equation d d + = 0, which after integration leads to U, = ln + ln + a, i.e. a obb-douglas utility function. But this trick does not always work: suose that (, ) ( ) ( ) = + for. We then end u with d = = = = d + and uon integration with U = ln + ln ( + ) + a where a is a constant of integration. In other words we have demonstrated roblem 3: = F = and Problem 3: Sometimes cannot be eliminated and the utility function deends on U directly.. Furthermore, the resulting utility function may have unleasant roerties. We have Problem 4: U is ossibly not concave. This roblem can be demonstrated by means of another examle: onsider the demand system M + (,, M) = 2 2 M M = + and 2 2 and the simle technology F(, ) = and (,, ) =. Without exlicit inversion of the demand system we get d d = = = =

16 - 4 - and thus d + d = 0. The corresonding utility function 2 2 U(, ) = + + a 2 2 is convex and the underlying reference ordering concave. Summing u, we recognize that Hori s result and conditions are not (always) sufficient to recover an aroriate utility function. Therefore it is necessary and worthwhile to examine the roblem of integrability anew.

17 Integrability In this section we assume that the conditional demand system ( M,, ) and (,, M ) can be observed and that the roduction function F(, ) is known. The objective is to derive conditions which guarantee the integrability of these functions; i.e. we are interested in obtaining sufficient conditions for the existence of a corresonding utility function. Above we have discussed three variants of the basic model: the roblems UF, U, and U. We will concentrate on roblem U and recover the utility function deending on all goods,, and. iven the structure of our model we are then able to identify the underlying tastes (and reference ordering). Since the analysis will be erformed in terms of,, the roduction function which is art of the utility function to be recovered cannot be used directly. Therefore we utilize which is imlied by F. M, the marginal willingness to ay function for instead 4. Defining the roblem It is reasonable to define the roblem of integrability recisely. We introduce Problem I Suose that the conditional demand system ( M,, ) and ( ) M, the quantity, M,, =, F, M,,, are given. Determine and the MWTP function the utility function U(, F(, )) such that ( M,, ) and M solve the corresonding roblem U and reflects the MWTP for. It should be mentioned that knowledge of a searable utility function (, (, )) U H where H is concave in and does not allow us to recover the roduction function uniquely. The function H(, ) can only be identified u to a monotonic increasing transformation f since ( ) (, ( (, ))): =, ( (, )) U f H U f f H reresents the same reference ordering, as well. Therefore the ersonal environment could be given by H(, ) or f ( H(, )) (or still another transformation of (, ) H ). On the other hand the conditional demand system and the MWTP for, imlied by U, are unique.

18 - 6 - The formulation of the integrability roblem chosen is no longer directly related to the household framework since the roduction function has been relaced by the MWTP function The latter can be interreted as an inverse demand function. The functions ( M,, ), ( M,, ), and usual demand theory can be alied.. M reresent a mixed demand system to which the Ebert (998) resents in Proosition 2 a solution to Problem I. The way it is derived is comlicated: The mixed demand is equivalent to an unconditional demand system. It is integrable if the Slutsky matrix of the unconditional system is symmetric and negative semidefinite. These conditions are translated to the original mixed system. In the next subsections we will consider the given system itself and will determine this and another solution directly. 4.2 Using the exenditure function A direct way to integrate an unconditional demand system is to emloy Shehard s Lemma (cf. Hurwicz (97), Hurwicz and Uzawa (97), Varian (984), and Mas-olell, Whinston, and reen (995)). This idea can also be alied to a mixed demand system if the conditional exenditure function is emloyed. We know that E E =, =, and E =. The comensated demand functions and the comensated willingness to ay function are unobservable. In our case only the Marshallian functions are known. Hicksian functions can be defined by inserting the exenditure function (a well-known identity): ( ),, u =,, E,, u, =... etc. Then we obtain a artial differential equation system: E u E u E u ( ) =,, u =,, E,, u, (4) ( ) =,, u =,, E,, u, (5) ( ) =,, u =,, E,, u. (6)

19 - 7 - This system can be set u since the functions Now two questions have to be osed:,, are given by assumtion. Q) Does there exist a function E(,, u ) satisfying (4)-(6)? [mathematical integrability]? Q2) Has this function E the usual roerties? [economic integrability]? Mathematical integrability requires that the Jacobian matrix of E (the Slutsky matrix S) is symmetric: S = ( sij ) = Symmetry is equivalent to = = (7) (8) = Since the MWTP function. (9) is defined by means of the household cost function, both ingredients the demand system, and the roduction function F have to be taken into account and have to fit to one another. The conditional exenditure function E(,, u ) has to be concave in and convex in. Therefore economic integrability (or the answer to Q2) requires that the submatrix ( sij ), =, i j is negative definite (20) and that is negative. (2) Thus we obtain Proosition The integrability roblem I ossesses a solution if the conditions (7)-(9) and (20)-(2) are satisfied.

20 - 8 - It is easy to see that these conditions are not only sufficient, but also necessary (i.e. they are also satisfied by any solution of Problem U). Furthermore, treating as a market good, the conditional exenditure function E(,, u ) is equivalent to an unconditional one, E u. The conditions listed in Proosition are equivalent to the fact that the Slutsky matrix of E is symmetric and negative semidefinite (cf. also Ebert (998)). 4.3 Using the indirect utility function Similarly we can check whether a conditional indirect utility function V M exists imlying the mixed demand system. Assuming that marginal utility of income is strictly ositive ( 0) V > we again obtain a system of artial differential equations being based on M Roy s identity: V V V V M M (,, M) (,, M) (,, M) (,, M) = = ( M),,, (22) ( M),,, (23) V V M (,, M) (,, M) = ( M),,. (24) This system is more comlicated than the system (4)-(6). In this case an answer to question Q can be given by using a result roved by Frobenius (see Hartman (964) and cf. also Hurwicz and Uzawa (97)). It requires that 3 three integrability conditions are satisfied + = + M M, (25) = + M M = + M M, (26). (27) (see the Aendix) But these conditions are equivalent to (7)-(9) (the symmetry of the Slutsky matrix). 3 A further regularity condition has to be satisfied by,, which is always fulfilled by well-behaved (mixed) demand systems.

21 V M has to be convex in rices and concave in. These roerties of V are then equivalent to (20) and (2) (if signs are taken into account aroriately). Therefore we have rovided another roof of Proosition and shown Proosition 2 The system (22)-(24) ossesses a solution V which is equivalent to a direct utility function solving Problem I if the conditions (7)-(9) and (20)-(2) are satisfied. 4.4 Using the direct utility function Finally we examine the direct utility function and reconsider Hori s aroach. We are looking for (, (, )) U F. Total differentiation of U = const yields U F Ud + UFd + UFd = 0 or d + d + d = 0. U F F Let (,, ), (,, ) and be the inverse demand functions (aroriately normalized). Then the first-order conditions of utility maximization require that (,, ) U = U and (, ) (, ) U F = U F. Therefore the integrability roblem can be reduced to solving the artial differential equations 4 and U,,,, = U,,,, U,,,, F, = = U,,,, F, denotes the utility function to be determined, i.e. the indirect demand where U (,, ) functions,, and describe the sloes of the indifference surfaces. Since in this case 4 Here the conditional demand system ( M,, ) and (,, ) and (,, ). M has to be inverted in order to derive

22 no budget constraint is taken into account and only relative rices are considered, already two artial differential equations determine the solution. Using Frobenius Theorem again we now obtain one necessary condition for mathematical integrability: ( ) ( ) ( ) ( ) = (28) Denoting the LHS and RHS by a and a, resectively, and defining a ( ) ( ) : = and a ( ) ( ) : = we are able to introduce the Antonelli matrix (cf. Katzner (970)) A a a. = a a Thus symmetry of A is required. Since U U is indeendent of, U has to be searable, i.e. it can be reresented in the form (, (, )) U F. As U is to be quasi-concave in,, the matrix A has to be negative definite (cf. Theorem in Katzner (970)). Therefore we have derived Proosition 3 The integrability Problem I ossesses a solution if A is symmetric and negative definite. The analysis demonstrates that at least one condition has to be fulfilled for mathematical integrability. This result is in accord with Samuelson (950) who considered the integrability roblem for the general case. Since Hori concentrated on U(, ), i.e. on two variables, he failed to notice this condition. 4.5 Discussion This section resents three different aroaches to integrate a conditional demand system and roduction function given. Since it does not lay a role whether we know the conditional exenditure function, the indirect utility function or the direct utility function (each of them

23 - 2 - reresents the underlying reference ordering), the solution of roblem I will always be the same (if it exists). On the other hand we have derived two different (sets of) conditions sufficient for integrability. It is worthwhile to relate them. We establish Proosition 4 The conditions (7)-(9) and (20)-(2) are satisfied if and only if the Antonelli matrix is symmetric and negative definite. An exlicit roof of Proosition 4 is relatively comlicated; but one can rovide a simle argument for this result: Both sets of conditions are not only sufficient for integrability, but are also imlied by a given utility function and utility maximization. Furthermore, the Antonelli matrix and the Slutsky matrix (of the unconditional demand system) are directly connected: the former is almost the inverse of the latter (cf. Katzner (970), Stern (986)). (Moreover, the Slutsky matrix of the unconditional demand system and the Slutsky matrix introduced in subsection 4.2 can also be derived from one another.) Thus the conditions resented are consistent. Now we return to Hori s aroach. Subsection 3.2 demonstrates that it is sometimes roblematic. The roblems 2-4 cannot occur in our framework. When the conditional exenditure or indirect utility function is to be derived rices have not to be eliminated (i.e. roblem can be ignored). If the direct utility is to be obtained the conditional demand system has to be inverted, too. The following examle demonstrates that a demand system and household roduction function are not integrable if condition (28) is violated (; in this case no condition has to be satisfied according to Hori s aroach). onsider again the relative rices (inverse demand system for 2 and (,, ) = (cf. Subsection 3.2) and assume that (, ) Then = ( ) and = =. Thus Hori would solve M = ) (,, ) F = +. = 2

24 d + d = 0 or d d 0 + = which leads to U = ln + ln ( ) + a; i.e. U deends on which is not allowed in the framework of this aer. According to Proosition 3 we have to check condition (28). We obtain = = and thus Therefore and = and =. ( ) ( ) = 2 ( ) ( ) = 0 0, i.e. condition (28) is not satisfied; the mixed demand system is not integrable. Thus the examle shows the relevance of the integrability conditions derived. Finally we reconsider the differences between Hori s and our aroach: Hori examines the utility function U(, ), i.e. he takes into account only the commodities and, neglecting the environmental good. Therefore in his aroach only two variables lay a role. In this aer the roblem is formulated in terms of goods (,, and (!) ); i.e. we deal with three variables by transforming the conditional demand system and the roduction function given into a mixed demand system. The discussion and the examles rovided in this aer demonstrate that Hori s aroach does not describe the correct solution whereas the alternative aroach based on market goods is able to coe with the roblems shown.

25 onclusion The aer has investigated the integrability roblem for the averting behavior model within the household roduction framework. It has turned out that a consumer s observable behavior is not consistent with an arbitrary household roduction function but that some integrability conditions have to be satisfied. Thus there are limits for choosing the technology, given a conditional demand system. The conditions derived are equivalent or identical to the conditions one obtains in the conventional framework of demand theory. Their derivation and statement is made comlicated by the fact that in contrast to the conventional analysis the basic demand system is a mixed one since it contains an inverse demand function. The result obtained is relevant for the evaluation of environmental goods. The marginal willingness to ay for the environment can be determined from the roduction function alone. But whenever the rices of market goods change knowledge of the reference ordering (and the technology) is indisensable for welfare measurement. Then it is helful to know a riori whether the demand system observed and the roduction function chosen can be integrated at all. The results of this aer allow one to erform a test. Finally, we would like to stress that the aroaches to solving the integrability roblem which have been resented in this aer can also be used for more general models. They are not limited to the averting behavior model. This analysis will be erformed in a subsequent aer.

26 References Bartik, T.J. (988), Evaluating the benefits of non-marginal reductions in ollution using information on defensive exenditures 5, -27. Bear, D.V.T. (965), Inferior inuts and the theory of the firm, Journal of Political Economy 73, havas, J.P. (984), The theory of mixed demand functions, Euroean Economic Review 24, ornes, R. (992), Duality and modern economics, ambridge University Press, ambridge. ourant, P.N. and R.. Porter (98), Averting exenditure and the cost of ollution, Journal of Environmental Economics and Management 8, Deaton, A. and J. Muellbauer (980), Economics and consumer behavior, ambridge University Press, ambridge. Ebert, U. (998), Evaluation of nonmarket goods: Recovering unconditional references, American Journal of Agricultural Economics 80, Ebert, U. (200), A general aroach to the evaluation of nonmarket goods, Resource and Energy Economics 23, Freeman III., A.M. (985), Methods for assessing the benefits of environmental rograms, in: Allen V. Kneese and James L. Sweeny (eds.), Handbook of Natural Resource and Energy Economics, Vol. I., Elsevier Science Publishers B.V., Amsterdam, hater 6, Freeman III, A.M. (993), The measurement of Environmental and Resource Values, Resources for the Future, Washington D.. Harford, J.D. (984), Averting behavior and the benefits of reduced soiling, Journal of Environmental Economics and Management, Hartman, P. (964), Ordinary Differential Equations, John Wiley, New ork. Hori, H. (975), Revealed reference for ublic goods, American Economic Review 65, Hurwicz, L. (97), On the roblem of integrability of demand functions, in: J.S. himan, L. Hurwicz, M.K. Richter, and H.F. Sonnenschein (Eds.): Preferences, utility, and demand, Harcourt Brace Jovanovich, New ork, hater 9.

27 Hurwicz, L. and H. Uzawa (97), On the integrability of demand functions, in: J.S. himan, L. Hurwicz, M.K. Richter, and H.F. Sonnenschein (Eds.): Preferences, utility, and demand, Harcourt Brace Jovanovich, New ork, hater 6, Katzner, D.W. (970), Static Demand Theory, Macmillan, London. Kolstad,.D. and J.B. Braden (99), Environmental demand theory, in: J.B. Braden and.d. Kolstad (eds), Measuring the demand for environmental quality, hater 2, 7-39, Elsevier Science Publishers, Amsterdam. Mäler, K.-. (97), A method of estimating social benefits from ollution control, Swedish Journal of Economics 73, Mas-olell, A., M.D. Whinston and J.R. reen (995), Microeconomic Theory, Oxford University Press, Oxford. Pollak, R. (969), onditional demand functions and consumtion theory, Quarterly Journal of Economics 83, Pollak, R.A. and M.L. Wachter (975), The relevance of the household roduction function and its imlications for the allocation of time, Journal of Political Economy 83, Samuelson, P.A. (950), The roblem of integrability in utility theory, Economica 7, Shairo, P. and T. Smith (98), Preferences for nonmarket goods revealed through market demands, in: V.K. Smith (Ed.), Advances in alied microeconomics, Vol., JAI Press, reenwich, Smith, V.K. (979), Indirect revelation of the demand for ublic goods: an overview and critique, Scottish Journal of Political Economy 26, Smith, V.K. (99), Household Production Functions and Environmental Benefit Estimation, in: J.B. Braden and.d. Kolstad (eds), Measuring the demand for environmental quality, hater 3, 4-76, Elsevier Science Publishers, Amsterdam. Stern, N. (986), A note on commodity taxation: The choice of variable and the Slutsky, Hessian and Antonelli matrices (SHAM), Review of Economic Studies 53, Varian, H.R. (984), Microeconomic Analysis, Norton, New ork.

28 Aendix Proof of Lemma (, ) = F = Alying the imlicit function theorem to F(, ) = we obtain = F and = F F = F = F F (i) (, ) ( (, ), ) (ii) = F ( F (, ), ) F F ( ) = = F F 2 3 F F =, = F (iii) F (iv) ( (, ), ) (, ), F F N = = : F ( F ) F 2 where F F N = F + F F F F F 2 2 = ( FF 2 F F F + F F ) = : F F (v) F F F + F F F = = 2 2 F F + F

29 onditions yielding = = = = = = F F ( (, ), ) (, ) (, ) F F (, ) (, ) F = F, = f F Derivation of the examle in 3. ( M,, ) and M satisfy the budget constraint. For exogenous the first-order condition is given by U,,,, =. U We obtain U U and thus = ( α ) (,, ) U U = α ( α) ( α) ( ε) α α ( ε ) U M U M = = = γ δ = γ ( α) ( ε) M α ε γ γ δ δ γ γ On the other hand = and =. γ δ γ γ Using the roduction function we get γ δ M δ = F(, ) = = ε. γ

30 Therefore γ γ γ ( ) ( γ M δ δ M δ γ γ ) = ε = ε γ γ δ γ ε ε = = γ γ γ γ γ γ M δ M γ γ δ. γ α ε ε = the first-order condition is satisfied. γ α ε γ Since Derivation of (25)-(27) Total differentiation of V M yields V d + V d + V d + V dm = 0 M This differential equation is equivalent to M M M V = = V M V = = V V V M = = M Integrability is guaranteed by + = + M M = + M M = + M M (See Hartman (964),. 7.)

31 Proof of Proosition 4 The Antonelli matrix A is the inverse of S where S is an aroriately defined 2x2 submatrix of the Slutsky matrix S of the unconditional (!) demand system (see Katzner (970), Theorem 3.2-3). ( S is the Jacobian of the unconditional exenditure function.) S is derived from S by eliminating the row considering the derivatives of and the column containing the derivatives with resect to. S can be constructed from S by using the budget constraint aroriately. S is equivalent to the Slutsky matrix S (see e.g. Ebert (998)). Thus A and S can be derived from one another. Similarly the roerties of symmetry and (semi) definiteness are directly related.