Udo Ebert
Revealed preference and household production
January 2005
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Department of Economics, University of Oldenburg,
D-26111 Oldenburg, Germany
(+49) (0)441-798-4113
(+49) (0)441-798-4116
[email protected]
Abstract
The paper deals with the possibilities of recovering the underlying preference ordering from
observed behavior when an environmental good is employed in household production. This
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 function imposed. Therefore the method
suggested is operational. It is possible that the behavior observed and the household
production function (maintained as hypothesis) are not compatible. This result is important
since the evaluation of the environmental good in this framework crucially depends on the
choice of the production function.
Keywords: household production, integrability, valuation, environmental good
JEL-codes: D13, D11, Q51
-1-
1.
Introduction1
In public policy one has in general to choose among various measures and programs. Therefore a careful assessment of the alternatives available is necessary in order to come to a
rational and well-founded decision. In every case the costs and benefits of a program have to
be taken into account. Here in particular the consumers’ welfare changes implied have to be
determined. The present paper deals with particular aspects of this issue: the evaluation of
environmental goods and welfare measurement in the presence of environmental goods.
Consumer sovereignty is an important cornerstone in cost-benefit analysis. It requires that any
evaluation has to be based on the consumer’s own preference ordering. Therefore we face the
problem of recovering preferences from the consumer’s behavior. It is in principle possible to
observe this behavior in markets for private goods. As long as there are only private goods
one can reveal the underlying preference ordering from a consumer’s demand system (by
integration). Even if it turns out that a utility function cannot be determined in closed form it
is still possible to measure welfare (changes) by using numerical methods (cf. Vartia (1983)).
The task becomes much more complicated when environmental goods have (also) to be taken
into account. Then the (representative) consumer’s behavior in markets is often influenced by
(the level of) environmental goods. Unfortunately, such observations are no longer sufficient:
In this situation neither the complete preference ordering can be recovered nor an evaluation
of an environmental good can be obtained. To solve these problems additional – not observable – information is always required. 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 Mäler (1971) is a
condition which can be used if applicable. The household production framework represents
another possibility of providing more information and giving more structure to the problems
to be solved. In the following we will investigate the problem of revealing preferences in this
framework.
In the household production model 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 input in the
production process and is not consumed directly, the approach allows us to evaluate the
environmental goods and to recover preferences (Hori (1975)). Since the household production function implicitly describes the marginal rate of substitution between the environ1
I am grateful to Hajime Hori for helpful comments on an earlier version of this paper.
-2ment and other market goods, the choice of the production function is crucial for determining
the preference ordering and the (marginal) willingness to pay for environmental goods.
Then two difficulties arise: First, a household production function is often also not directly
observable, i.e. its functional form is based on a hypothesis, as well. As every maintained
hypothesis in this area it cannot be tested econometrically, or as formulated in Smith (1991):
“Most economists … 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.”2 Second, the maintained assumption that the utility function does not directly depend on the environmental good is not necessarily consistent with the behavior observed and the household production function chosen.
This paper is concerned with these problems which are interrelated and provides a partial
solution of the first and a complete solution to the second one. Starting with a complete
system of demand functions for private goods, which may depend on the environmental good,
and the household production function we present an approach for testing their compatibility
(Problem 1) by checking the existence of an appropriate underlying utility function which has
to imply the observed behavior and to possess the usual properties. Furthermore, it must not
depend on the environmental good directly (Problem 2). If these conditions are met it is
possible to recover a unique preference ordering. Then one is able to perform any welfare
analysis one is interested in, and, of course, to evaluate the environmental good simply.
At first we will derive a theoretical result which allows us to reformulate the problems: We
have to investigate the integrability of an (appropriately) defined mixed demand system. Then
the above conditions can be checked explicitly. Sometimes we are able to reject a household
production function. If the observed behavior and the household production function are
compatible, it is still possible that various household production functions are consistent with
the same observed behavior and therefore appropriate.
In order to describe our proceeding more precisely let us suppose for a moment that the
household production function and the utility function are given. In the household production
model our analysis is based on two market and one nonmarket good. One market good
(consumption) is consumed directly (without being used in household production). The other
one and the environmental good are employed to produce the consumer’s personal environmental quality (by household production). The consumer then consumes the consumption
good and her personal environmental quality. Her preferences (or tastes) are represented by a
2
Smith himself is a little bit more optimistic.
-3utility function defined on consumption and quality. The model is similar to the ‘averting
behavior’ model in which the environmental good is detrimental and household production
leads to an improvement in environmental quality. Given that the consumer maximizes her
utility, a complete system of demand functions for market goods can be derived. The demand
functions are conditional, i.e. they in general depend on the environmental good which is
fixed and exogenous for the consumer. When the utility and production function are known, it
is possible to measure welfare changes and to evaluate the environmental good since the
corresponding expenditure and cost function can be derived.
As indicated above the objective of the present paper is to examine the reverse proceeding:
Starting with observed behavior, i.e. a conditional demand system and a given household
production function, we want to investigate the possibilities of deriving the underlying utility
function.
The question posed is comparable to the problem of integrability of demand functions in
demand theory. Indeed, the problem will be framed in such a way that the conditions for the
integrability of a mixed demand system have to be checked. It consists of the conditional
demand functions for market goods and an inverse demand function for the environmental
good which is derived from the household production function and the assumption of utility
maximization. The necessary and sufficient conditions for integrability are derived. They
impose restrictions on the Slutsky matrix of an appropriately defined utility function. Though
this matrix is related to the Hicksian demand functions, the conditions can be checked by
employing the demand functions observed and the production function imposed.
The methodology developed can be generalized easily to more general models.
The paper is organized as follows. Section 2 describes the framework and derives a fundamental result. Section 3 deals with the problem of recovering preferences in the household
production framework. It examines the possibilities and limitations of an approach suggested
by Hori (1975) and presents the new approach. Section 4 derives the conditions for
integrability and discusses the implications for applied work. Finally section 5 concludes.
2.
Basic model and background
Before we are able to go into details we have at first to present the model underlying the
analysis more precisely. Subsection 2.1 introduces the household production model and the
notation. Subsection 2.2 contains a preliminary discussion of the evaluation of the environmental good.
-4-
2.1 Household production
We consider the simplest household production model used in environmental economics (see
e.g. Mäler (1985), Smith (1991), Freeman (1993)). 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 p X 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.3
Household production employs the inputs Y and Q to produce another commodity Z, the
personal environmental quality. The technology is described by a production function
Z = F (Y , Q ) or, equivalently, by the cost function C ( pY , Q, Z ) = pY F −1 ( Z , Q ) where
F −1 ( Z , G ) 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 commodities X and Z. Her tastes and preferences
are represented by a (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
given she maximizes her utility subject to the budget constraint (the exogenous income is
denoted by M). This maximization problem is equivalent to
Problem U ∗
max U ( X , F (Y , Q ) )
X ,Y ,Q
(1)
such that
p X X + pY Y = M
and
(2)
Q fixed.
Given our assumptions the utility function is weakly separable and the functional form of the
part containing Y and Q is known. In the following we suppose that U ( X , F (Y , Q ) ) is
3
If the environmental good is detrimental we obtain the model of ‚averting behavior’. Compare also Courant
and Porter (1981), Harford (1984), and Bartik (1988).
-5concave in X , Y , Q . Then the solution of Problem U ∗ is unique. It is described by the conditional demand system
X = X ( p, Q, M ) and Y = Y ( p, Q, M )
(3)
where p = ( p X , pY ) .
Because of the weak separability of the utility function the marginal rate of substitution
between Q and Y has a simple form
MRSQY ( X , Y , Q ) =
dU dQ FQ (Y , Q )
=
dU dY FY (Y , Q )
(4)
where FQ and FY denote partial derivatives. It equals the marginal rate of transformation
between Q and Y. 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 good wQ is implicitly defined by MRSQY = wQ pY . Therefore
we obtain
(
wQ ( p, Q, M ) = −CQ pY , Q , F (Y ( p, Q, M ) , Q )
)
(5)
since
CQ = − pY FQ FY .
This result is not surprising as an increase in Q lowers the costs of household production CQ .
For completeness we also demonstrate how the personal environmental quality Z can be
evaluated. We reformulate Problem U ∗ to
Problem U ∗ ( Z )
max U ( X , Z )
X ,Z
such that
p X X + C ( pY , Q , Z ) = M
and Q fixed.
Then the solution has to satisfy the first-order condition which can be rearranged to
-6-
MRS ZX ( X , Z ) =
CZ ( pY , Q , Z ) pY
1
=
pX
p X FY (Y , Q )
(6)
where Y = F −1 ( Z , Q ) .
2.2 Informational requirements
Now we want to discuss for the present which information is required to evaluate the
environmental good Q, i.e. to determine wQ ( p, Q, M ) , when the household production model
is given. Considering (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 preference ordering represented by U ( X , Z ) . Knowledge of the household technology and of the consumer’s
behavior (the conditional demand system) is already sufficient to compute wQ .
The situation changes dramatically if we admit a direct effect of Q on the consumer’s
preference ordering, i.e. if U = U ( X , Z , Q ) . In this case we get
MRSQY =
FQ
FY
+
UG
,
U Z FY
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 (5). Then
from an empirical point of view the situation is hopeless. Q cannot be evaluated on the basis
of observations. 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.
We have established
Proposition 1
Assume that the consumer maximizes a utility function V ( X , Z , Q ) for Z = F (Y , G ) under
the 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
Z = F (Y , Q ) .
X ( p, Q , M ) , Y ( p, Q, M ) , and
-7-
(
)
(b) wQ ( p, Q, M ) = −CG pY , Q , F (Y ( p, Q, M ) , Q ) .
(c) V does not depend on Q directly, i.e. ∂V ∂Q ≡ 0 .
Without any doubt the household production framework gives some structure to the evaluation problem. But Proposition 1 clearly demonstrates that an evaluation of the environmental
good Q (and – as we will see below – also the identification of the underlying preference
ordering) requires an additional hypothesis: There must be no direct effect of Q on the
consumer’s preferences. This insight plays an important role in the following section.
3.
The problem of recovering preferences
Above we have introduced the household production model and derived its implications. Now
we reverse our proceeding and we want to discuss the possibilities of recovering the underlying utility function U from observable information. We suppose that the conditional demand
system (3) can be observed and that the household production function Z = F (Y , Q ) is
known. Then the question arises whether there exists a utility function U ( X , Z ) such that –
given the household production function F – the conditional demand system (3) is the solution
to Problem U ∗ . If the answer is in the affirmative the utility function can form the basis for
welfare measurement and the evaluation of the environmental good: We can employ the
corresponding expenditure function or the marginal willingness to pay function wQ ( p, Q, M ) .
In this section we will examine this problem. In subsection 3.1 we will investigate an
approach proposed by Hori (1975). It turns out that there may be some difficulties in applying
it. Therefore in a second step we suggest a new approach which seems to be more promising
and which avoids the problems mentioned (subsection 3.2). Finally an issue which might
complicate the search for a solution is clarified.
3.1 Hori’s approach
Hori (1975) demonstrates that in the case considered in subsection 2.1 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. But even if this condition is not satisfied, it can
be checked by applying the procedure proposed by Hori. Following this method we have to
take some steps. At first we have to invert the demand system in order to express prices in
terms of X and Y. Then we sometimes have to eliminate Y by means of the household
-8production function. Finally we obtain a partial differential equation in terms of X and Z
which must not depend on Q. The integration of this equation is always possible and yields
the utility function.
In order to demonstrate the proceeding and the difficulties which can arise we consider an
example:
Example 14
Let the conditional demand system be given by
12
M  pY 
− 12 (1+α )
X ( p, Q , M ) =
−
 Q
pX  pX 
(7)
12
p 
− 1 (1+α )
Y ( p, Q , M ) =  X  Q 2
 pY 
(8)
and the household production function by Z = Y 1 2Q1 2 for some α .
On the assumption that the utility Uα ( X , Z ) does not depend on Q directly we want to
describe an indifference surface Uα ( X , Z ) = U . By total differentiation we obtain
Uα X ( X , Z ) dX + Uα Z ( X , Z ) dZ = 0 .
(9)
This equation is the starting point of the analysis. It can be rearranged, identified from observable information, and integrated afterwards: Then we recover Uα .
Rewriting (9) we get
dX + Uα Z Uα X dZ = 0 .
Since the consumer maximizes her utility (see subsection 2.1) we know that
U α Z Uα X =
CZ pY
1
in the optimum.
=
p X PX FY (Y , Q )
This optimality condition has to be satisfied and can be used to replace Uα Z Uα X . But the
right hand side still contains the prices p X and pY and the quantities Y and Q. These variables have to be eliminated since we have to express the right hand side in terms of X and Z.
4
The details of all examples are derived in an Appendix.
-9Therefore we have to invert the conditional demand system in a first step. (Here some
regularity conditions are needed.)
We get
pX
1
p
1
=
and Y =
.
2 1+α
1+α
M
M XY Q + Y
X + 1 (YQ )
Furthermore, in a second step, we determine
FY (Y , Q ) =
12
1 Q 
 
2 Y 
12
=
1  QQ 


2  Z2 
=
1Q
2 Z
since Y = F −1 ( Z , Q ) = Z 2 Q .
Then we obtain for the marginal rate of substitution
Uα Z Uα X = pY (1 p X )
 X + 1 (YQ1+α )  1 Z 1 1
1
M
.
=
=


 2 Q 2 Z 3Qα
FY XY 2Q1+α + Y 
M


Using this result we have to integrate dX +
U=X+
(10)
1 1
dZ = 0 which implies
2 Z 3Qα
1
1
( −2 ) 2 α = X − 1 ( Z 2Qα ) = : Uα ( X , Z , Q ) .
2
Z Q
Checking Uα ( X , Z , Q ) we recognize two points. First, the utility function depends on Q
unless α = 0 . This fact cannot be revealed directly by the inspection of the conditional
demand system. But it can already be discovered from an investigation of the marginal rate of
substitution (10) and the partial differential equation. If Q cannot be eliminated from this
expression, it is also still present after integration. That means, we do not have to integrate in
this case. We already know at this stage that for α ≠ 0 no utility function U ( X , Z ) exists
which allows us to derive the conditional demand system as the solution of Problem U ∗ given
the above household production function.
On the other hand, following Hori’s method we have to invert the demand system and solve
the household production function for Y in order to obtain (10). 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 functions. Then it is impossible to decide whether U depends on
Q or not. Another difficulty concerns the properties of Uα . If e.g. α < 0 the marginal utility
- 10 of Q would be negative, i.e. Q would be a bad! This property can only be revealed by an
investigation of Uα ; it cannot be discovered from (7) or (8) directly.5
Nevertheless, in some cases we obtain a solution to our problem: if α = 0 , the corresponding
utility function is given by U 0 ( X , Z ) = X − 1 Z 2 and then
wQ ( p, Q, M ) = ( p X pY ) Q −1 2 .
12
(11)
To sum up, Hori’s method works if the assumption that the utility is independent of Q is satisfied. This is not clear from the beginning. There are some difficulties. First, the inversion of
the demand system (and of the household production function) is required. Second, the
function recovered might have properties which are not welcome. Furthermore, our example
demonstrates that the necessary condition wQ = −CQ 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 subsection will provide a positive answer.
3.2 New Approach
The condition that U must not depend on Q directly is a maintained hypothesis of Hori’s
approach; but as Example 1 demonstrates it is not necessarily satisfied. Proposition 1 tells us
that the condition is equivalent to the fact that wQ = −CQ . Therefore we will now impose this
condition explicitly. We ask the question whether there exists a utility function U ( X , Z ) such
that for Z = F (Y , Q ) the conditional demand system
X = X ( p, Q , M ) , Y = Y ( p, Q , M )
is the solution to Problem U ∗ and such that
(
)
wQ ( p, Q, M ) = −CQ pY , Q , F (Y ( p, Q, M ) , Q ) .
(12)
Then the problem can be reformulated: Is there a utility function U ( X , F (Y , Q ) ) implying
the mixed demand system6
X ( p, Q , M ) , Y ( p, Q, M ) , and wQ ( p, Q, M ) = −CQ .
5
In principle it is even possible that U ( X , Z ) is convex (see Ebert (2002)). See also Samuelson (1950).
6
Cf. e.g. Chavas (1984).
(13)
- 11 We obtain a problem of integrability. The direct demand functions are observed. The marginal
willingness to pay function can be determined from the household production function and the
maintained hypothesis. Thus the latter is taken into account a priori.
Furthermore, the problems mentioned in subsection 3.1 can be avoided: First, the demand
system has not to be inverted. Second, the properties of U ( X , F (Y , Q ) ) can be checked
directly from the inspection of the mixed demand system. In section 4 the details will be
discussed and necessary and sufficient conditions for its integrability will be presented.
For completeness we have a look at Example 1 again. Simple computation (see the Appendix)
yields that the marginal willingness to pay for Q is equal to
wQ = pY MRSQY = (1 + α )( p X pY ) Q
12
− 12 (1+α )
,
– given the utility function Uα ( X , Z , Q ) . On the other hand the condition wQ = −CQ is
equivalent to
wQ ( p, Q, M ) = ( p X pY ) Q
12
− 12
Therefore this condition can be satisfied only if α = 0 , i.e. if there is no direct effect of Q on
utility.
3.3 Supplementary remark
This subsection is not necessary for understanding the rest of the paper and can in general be
skipped. It offers an explanation to those readers who might be worried about a subtle aspect
of the problem. In the following this issue is described and commented on.
Assume that an individual maximizes her utility V ( X , Y , Q ) subject to a budget constraint
p X X + pY Y = M and for given Q. Then a conditional demand system X ( p, Q , M ) and
Y ( p, Q , M ) is implied which can in principle be observed.
Now suppose that the utility function V is transformed by a function f ( v, Q ) which is strictly
increasing in v and depends on Q. If f (V ( X , Y , Q ) , Q ) satisfies the usual assumptions it also
represents a preference ordering. But the ordering is different from the original one represented by V. There are two implications: Firstly, maximization of f (V , Q ) under the above
constraints yields the same conditional demand system, i.e. the preference orderings are
equivalent as far as observations are concerned (since Q is fixed!). Therefore it is impossible
- 12 to distinguish between these preference orderings on the basis of observations. Secondly, the
evaluation of Q depends on the underlying preference ordering. The transformation by
f ( v, Q ) changes wQ . Thus Q cannot be evaluated merely on the basis of observations.
This problem is well-known in the literature (see e.g. Ebert (2001) and Larson (2001)). 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 = −CQ
and since this condition is imposed explicitly in subsection 3.2, the preference ordering which
is recovered is unique, i.e. the utility function is ordinally unique and must not be transformed
by a transformation function depending on Q.
4.
Integrability
In this section we investigate the integrability problem in subsection 4.1. Afterwards the
implications are discussed. Two examples are presented.
4.1 Problem and solution
According to our discussion in section 3 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). 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 u. If the expenditure function exists it satisfies
the following conditions
dE ( p, Q, u ) dp X = X ( p, Q, E ( p, Q, u ) )
(14)
dE ( p, Q , u ) dpY = Y ( p, Q , E ( p, Q, u ) )
(15)
(
(
))
dE ( p, Q, u ) dQ = − wG ( p, Q , E ( p, Q , u ) ) = CQ p, Q , F Y ( p, Q , E ( p, Q , u ) ) , Q .
(16)
(14)-(16) is a system of partial differential equations. For integrability two conditions have to
be fulfilled:
1) A function E ( p, Q, u ) satisfying (14)-(16) has to exist (mathematical integrability).
- 13 2) The function E ( p, Q, u ) has to possess the appropriate properties of a conditional expenditure function (economic integrability).
Mathematical integrability7 requires that the Jacobian matrix of E [the Slutsky matrix
S = ( sij )i , j = p
X
, pY ,Q
] is symmetric, i.e.
∂X
∂X
∂Y
∂Y
+Y
=
+X
∂pY
∂M ∂p X
∂M
(17)
∂w 
 ∂w
∂X
∂X
− wQ
= − Q + X Q 
∂Q
∂M
∂M 
 ∂p X
(18)
∂w 
 ∂w
∂Y
∂Y
− wQ
= − Q +Y Q  .
∂Q
∂M
∂M 
 ∂pY
(19)
Condition (17) is satisfied, since X ( p, Q , M ) and Y ( p, Q , M ) form a (conditional) demand
system by assumption. The other conditions (18)-(19) (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 ( sij )i , j = p
definite
(which
is
again
implied
by
the
conditional
X
, pY
demand
is negatively semisystem)
and
if
sQQ = − dwQ dQ > 0 , i.e. if
dwQ dQ < 0 .
(20)
Finally, the conditional expenditure function is equivalent to a corresponding direct utility
function U which has to be weakly separable, i.e. U = U ( X , H (Y , Q ) ) , because of the form
of the mixed demand system. Since we also get H Q H Y = FQ FY there is U such that
U = U ( X , F (Y , Q ) ) .
Thus we have derived
Proposition 2
The mixed demand system (13) is integrable if and only if the conditions (17)-(20) are
satisfied.
7
Compare also Hartman (1964).
- 14 The result is also implied by Proposition 2 in Ebert (1998) who does not consider household
production. A formal proof can be given in analogy to Hurwicz and Uzawa (1971) or along
the lines sketched in Jehle and Reny (2001), pp. 83-85.
In view of this result we reconsider Example 1: The mixed demand system consists of (7), (8),
and (11). The crucial conditions for mathematical integrability are (18) and (19). They are
satisfied only if α = 0 . Then condition (20) is also fulfilled.
It should be mentioned that one has to add an initial value if the partial differential equations
are to be integrated. Here one can choose e.g. E ( 0, 0, 0 ) = 0 . It normalizes the utility function.
4.2 Implications
Proposition 2 presents necessary and sufficient conditions for the integrability of the mixed
demand system (13). These conditions can be checked directly by inspecting (13). 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
U ( X , Z ) such that the system X ( p, Q , M ) and Y ( p, Q , M ) is the solution to Problem U ∗ .
Then the demand system (the observed behavior) and the household production function are
inconsistent, i.e. do not fit to one another. But it is just possible that the demand system is
compatible with another household production function. 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 both “ingredients” are consistent.
As a consequence the results derived in section 2 are valid. 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 between Q and Y coincides with
the corresponding marginal rate of transformation (see (4)). If 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. The latter 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.
The next example demonstrates both possibilities.
- 15 -
Example 2
We use the utility function U 0 ( X , Z ) = X − 1 Z 2 introduced in Example 1 for α = 0 . It leads
to
12
12
p 
M  pY 
−1 2
X ( p, Q , M ) =
−
and Y ( p, Q , M ) =  X  Q −1 2 .
 Q
pX  pX 
 pY 
Furthermore we consider a family of household production functions
Z = Fε (Y , Q ) : = Y ε Q1−ε for 0 < ε < 1 .
Then Cε ( pY , Q, Z ) = pY Z 1 ε Q1−1 ε .
For integration or a test of integrability we have to take into account the maintained hypothesis
wQ = −Cε Q ( pY , Q , Z ) = (1 ε − 1)( p X pY ) Q −3 2 .
12
Now we have to check (18) and (19) and obtain the condition ε = 1 2 .
Thus whenever ε ≠ 1 2 the respective household production function is not consistent with
the above demand system since then (18) and (19) are violated.
Thus this example shows that both cases can occur. 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 one conditional demand system and a family of household production functions:
At first we introduce the utility function U ε ( X , Z ) = X
ε
ε +1
1
Z
ε +1
for 0 < ε < 1 and employ the
household production functions Fε (Y , Q ) for 0 < ε < 1 again.
Let the conditional demand system be given by
X ( p, Q , M ) =
1 M
1 M
and Y ( p, Q , M ) =
for 0 < ε < 1 .
2 pX
2 pY
It is independent of Q and ε . This demand system is always the solution to Problem U ∗ if
U ε ( X , Z ) and Fε (Y , Q ) are employed. For the marginal willingness to pay for Q we obtain
- 16 -
wQ =
1 1−ε M
2 ε Q
which tends to 0 [to ∞ ] for ε → 1 [for ε → 0 ]. In this case the entire family Fε is consistent
with the demand system observed. But the choice of ε determines wQ . The marginal willingness to pay for Q can attain an arbitrary value depending on ε .
Example 3 reiterates the point raised above: It shows that 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 in this case.
5.
Conclusion
The paper has discussed the possibilities of recovering the underlying preference ordering in
the household production model when an environmental good is employed as input. It turns
out that a hypothesis about the structure of the preference ordering is indispensable: The
environmental good must not influence utility 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 function chosen, and the maintained hypothesis.
Therefore the approach suggested is operational. It turns out that the observed behavior is not
necessarily consistent with an arbitrary technology (household production function). This
result is important since the evaluation of the environmental good (and, of course, the
preference ordering recovered) crucially depends on the choice of the technology.
- 17 -
References
Bartik, T.J. (1988), Evaluating the benefits of non-marginal reductions in pollution using
information on defensive expenditures 15, 111-127.
Chavas, J.P. (1984), The theory of mixed demand functions, European Economic Review 24,
321-344.
Courant, P.N. and R.C. Porter (1981), Averting expenditure and the cost of pollution, Journal
of Environmental Economics and Management 8, 321-329.
Ebert, U. (1998), Evaluation of nonmarket goods: Recovering unconditional preferences,
American Journal of Agricultural Economics 80, 241-254.
Ebert, U. (2001), A general approach to the evaluation of nonmarket goods, Resource and
Energy Economics 23, 373-388.
Ebert, U. (2002), Recovering
Freeman III., A.M. (1985), Methods for assessing the benefits of environmental programs, in:
Allen V. Kneese and James L. Sweeny (eds.), Handbook of Natural Resource and Energy
Economics, Vol. I., Elsevier Science Publishers B.V., Amsterdam, Chapter 6, 223-270.
Freeman III, A.M. (1993), The measurement of Environmental and Resource Values,
Resources for the Future, Washington D.C.
Harford, J.D. (1984), Averting behavior and the benefits of reduced soiling, Journal of
Environmental Economics and Management 11, 296-302.
Hartman, P. (1964), Ordinary Differential Equations, John Wiley, New York.
Hori, H. (1975), Revealed preference for public goods, American Economic Review 65,
978-991.
Hurwicz, L. and H. Uzawa (1971), On the integrability of demand functions, in: J.S.
Chipman, L. Hurwicz, M.K. Richter, and H.F. Sonnenschein (Eds.): Preferences, utility,
and demand, Harcourt Brace Jovanovich, New York, Chapter 6, 114-148.
Jehle, G.A. and P.J. Reny (2001), Advanced microeconomic theory, 2nd edition, Addison
Wesley, Boston.
Larson, Douglas M. (1991), Recovering weakly complementary preferences, Journal of
Environmental Economics and Management 21, 97-108.
- 18 Mäler, K.-G. (1971), A method of estimating social benefits from pollution control, Swedish
Journal of Economics 73, 121-133.
Mäler, K.-G. (1985), Welfare economics and the environment, in: A.V. Kneese and J.L.
Sweeney (eds.), Handbook of Natural Resource and Energy Economics, vol. I., Elsevier,
Science Publishers B.V., Amsterdam, Chapter 1, 3-60.
Samuelson, P.A. (1950), The problem of integrability in utility theory, Economica 17,
355-385.
Smith, V.K. (1991), Household Production Functions and Environmental Benefit Estimation,
in: J.B. Braden and C.D. Kolstad (eds), Measuring the demand for environmental quality,
Chapter 3, 41-76, Elsevier Science Publishers, Amsterdam.
Vartia, Yrjö O. (1983), Efficient Methods of Measuring Welfare Changes and Compensated
Income in Terms of Ordinary Demand Functions, Econometrica 51, 79-98.
- 19 -
Appendix
Example 1
We consider the utility function
Uα ( X , Z , Q ) = X −
1
Z Qα
2
and the household production function
Z = F (Y , Q ) = Y 1 2Q1 2 .
Its cost function is given by
C ( pY , Q, Z ) = pY
Z2
.
Q
a) Derivation of the demand system
max X − 1 (Y Q1+α )
X ,Y
s.t. p X X + pY Y = M
Q fixed
Since X =
M
p
− Y Y we can replace X and obtain the first-order condition
pX pX
 pY  
1
−
 −  − 2 1+α
 pX   Y Q

 = 0.

Thus
12
p 
 p  − 1+α
− 1 (1+α )
.
Y =  X  Q ( ) or Y =  X  Q 2
 pY 
 pY 
2
Furthermore
12
M  pY 
− 1 (1+α )
.
−  Q 2
X=
p X  PX 
Since
wQ
pY
= MRSQY we get
- 20 -
wQ = pY
dUα dQ
Y 2Q1+α
= (1 + α ) pY
= (1 + α ) pY Y Q .
dUα dY
YQ 2+α
On the other hand
−CQ ( pY , G, F (Y , Q ) ) = pY Z 2 Q 2 = pY YQ Q 2 = pY Y Q .
Therefore
wQ = −CQ ⇔ α = 0 .
b) Integration à la Hori
We have to integrate
dX + Uα Z Uα X dZ = 0
since Hori assumes that UαQ = 0 .
By the maximization of utility we know that
U α Z Uα X =
CZ pY
1
in the optimum.
=
p X PX FY (Y , Q )
This condition can be used to replace Uα Z Uα X . But the right hand side contains the variables
Y and Q. These variables have to be eliminated since we have to express the right hand side in
terms of X and Z. Therefore we have to invert the conditional demand system in a first step.
(Here some regularity conditions are needed.)
We get
Y2 =
p X −(1+α )
p
1
and X = 2 1+α .
Q
pY Y Q
pY
Then
12
M  pY 
M
1
− 12 (1+α )
−
=
−
X=
 Q
pX  pX 
p X YQ1+α
and thus
pX
1
p
1
.
=
and Y =
2 1+α
1+α
M
M XY Q + Y
X + 1 (YQ )
Furthermore, in a second step, we determine
- 21 12
1 Q
FY (Y , Q ) =  
2 Y 
12
1  QQ 
=  2 
2 Z 
=
1Q
2 Z
since Y = Z 2 Q .
Then we obtain for the marginal rate of substitution
Uα Z Uα X = pY (1 p X )
=
 X + 1 (YQ1+α )  1 Z 1
M
Z
1
1
=
=


2 1+α
2 1+α
2Q 2Y Q Q
FY XY Q + Y 
M


1
1
Z 1 1
.
=
2  ZY  1+α Q 2 Z 3Qα
 Q2 Q


Using this result we have to integrate
dX +
1 1
dZ = 0
2 Z 3Qα
which implies
U=X+
1
1
( −2 ) 2 α = U α ( X , Z , Q ) .
2
Z Q
Example 2
We define
Z = Fε (Y , Q ) : = Y ε Q1−ε for 0 < ε < 1 .
Then
Y ε = Z Q1−ε = ZQ ε −1
and the corresponding cost function is given by
Cε ( pY , Q, Z ) = pY Y = pY Z 1 ε Q1−1 ε .
Moreover
Cε Q = (1 − 1 ε ) pY ( Z Q ) .
1ε
Then
- 22 1ε
wQ = −Cε Q
Z
= (1 ε − 1) pY  
Q
1ε
 Y ε Q1−ε 
= (1 ε − 1) pY 

 Q 
12
p p 
= (1 ε − 1) pY Y Q = (1 ε − 1) Y  X  Q −1 2
Q  pY 
= (1 ε − 1)( p X pY ) Q −3 2 .
12
We want to check the condition (18) (and (19) similarly)
dw 
 dw
dX
dX
− wQ
= − Q + X Q 
dQ
dM
∂M 
 ∂p X
(*)
for Uα with α = 0 . In that case
12
12
p 
M  pY 
−1 2
X=
−
and Y =  X  Q −1 2 .
 Q
pX  pX 
 pY 
Then
12
1 p 
1
12
LHS of (*) =  Y  Q −3 2 − (1 ε − 1)( p X pY ) Q −3 2
2  pX 
pX
12
1
 p 
=  − 1 ε + 1   Y  Q −3 2
2
  pX 
12
1
 p 
RHS of (*) = −  (1 ε − 1)   Y  Q −3 2 + 0.
2
  pX 
The condition (*) is satisfied if and only if
11 
1 1 
 − + 1 = −  − 1 or ε = 1 2 .
2ε
2 ε


Example 3
We consider again
Z = Fε (Y , Q ) : = Y ε Q1−ε for 0 < ε < 1
and define
Uε ( X , Z ) : = X
ε
ε +1
1
Z
ε +1
Then we have to maximize
.
- 23 ε
ε
1−ε
U ε ( X , Fε (Y , Q ) ) = X ε +1 Y ε +1Q ε +1
subject to the budget constraint and fixed Q. The solution is given by
X=
1 M
1 M
and Y =
.
2 pX
2 pY
It is independent of Q and of ε . For wQ we obtain
wY = pY
dU ε dQ
=
dU ε dY
ε
ε
1−ε
 1 − ε  ε +1 ε +1 ε +1−1
X Y Q

1−ε Y 1 1−ε M
ε + 1 

.
= pY
=
pY
ε
ε
1−ε
ε Q 2 ε Q
 ε  ε +1 ε +1−1 ε +1
Q

X Y
 ε +1
In this case wQ (and U ε ) depend on the choice of ε :
wQ → ∞ for ε → 0
wQ → 0 for ε → 1.
But the mixed demand system is integrable for every ε , 0 < ε < 1 .