Download Weak Complementarity, Path Independence, and the Intuition of the

Weak Complementarity, Path Independence, and the Intuition
of the Willig Condition
by
Raymond B. Palmquist
Department of Economics
North Carolina State University
Raleigh, NC 27695-8110
USA
[email protected]
January 2004
The author is grateful to Dan Phaneuf, Kerry Smith, Wally Thurman, Spencer Banzhaf, and
Roger von Haefen, as well as the editor and two referees, for comments on an earlier version.
Weak Complementarity, Path Independence, and the Intuition
of the Willig Condition
Valuing public goods is important, but typically markets for those goods do not exist.
Using information revealed in the markets for private goods to infer information about the
demand for a public good is often the best option. However, this is not possible unless certain
conditions are met. One of the most popular of these maintained hypotheses is weak
complementarity, but even with this condition it is complex to derive useful results. The purpose
of this paper is to provide a new interpretation of the conditions under which information on the
value of an improvement in a public good can be derived from estimates of the uncompensated
demand for a related private good and to consider the implications for welfare measurement.
The concept of weak complementarity was introduced by Karl-Göran Mäler in 1974.
Since that time it has been used widely in valuing non-market goods from observed behavior,
particularly in recreation studies. The assumption of weak complementarity requires that there is
a private good that is consumed with the non-market environmental good. The environmental
good can be thought of as a public good or equivalently as the unpriced quality of the private
good. It is assumed that the private good is nonessential, which implies that there is a choke
price sufficiently high that the consumption of the private good falls to zero. When consumption
of the weakly complementary private good falls to zero, changes in the environmental good will
be of no value to the consumer.
Under these circumstances, it is possible to estimate the willingness to pay for the
improvement in the environmental good by estimating the change in the compensating variation
the consumer receives from the consumption of the private good. Mäler emphasized that this
would necessitate the use of Hicksian demands or the expenditure function, neither of which is
1
directly observable. Thus, there are two main options. Mäler (1974) and Larson (1991) describe
methods for integrating back from Marshallian demands to obtain the expenditure function.1 The
more straight-forward alternative is to estimate Marshallian demands and to use them to
approximate the true welfare measures. As Bockstael and McConnell (1993) show, this is more
complex than was originally thought. In the case where the demand for quality is known,
Randall and Stoll (1980) and Hanemann (1991) have established that the compensating and
equivalent measures bound the uncompensated welfare measure. However, the uncompensated
measure in terms of the weakly complementary private good will not necessarily equal the
uncompensated measure in terms of the environmental good. This is where the Willig condition
is relevant. In 1978 in an article on dealing with quality change in price indexes, Willig
established three equivalent conditions under which quality changes could be converted into
price changes. However, the link between the Willig condition and more traditional welfare
measurement techniques was unclear until an important series of papers by Bockstael and
McConnell (see, for example, Bockstael and McConnell, 1993) established the importance of the
Willig condition for valuing quality changes.2 However, the source of the importance of the
Willig condition can still be clarified. It should be emphasized that the two conditions, weak
complementarity and the Willig condition are completely independent. Either can hold with or
without the other.
This paper shows that the Willig condition can be derived easily from the condition for
path independence of a line integral. The importance of path independence in measuring the
welfare effects of multiple price changes was established by Silberberg (1972).3 Path
independence with quality changes has received considerably less attention. When there is path
independence with an environmental good and its weakly complementary private good, the
2
uncompensated welfare measure in the environmental good space is equivalent to the
uncompensated welfare measure in the private good space. The uncompensated measure derived
in private good space can provide a bound for the compensated measure of the value of a quality
improvement. In addition to deriving the three versions of the Willig condition from the
condition for path independence, this note derives a fourth version that is equivalent and links the
Willig condition in quality space to the Silberberg conditions in price space. Examples of the
application of these conditions to representative empirical specifications are given, and the
implications for implementation are discussed
Path Independence and Weak Complementarity
In a typical application of weak complementarity there is a change in one environmental
good or environmental quality. If the uncompensated demand for environmental quality, q, were
observable, the path of integration would not be an issue. If the demand were continuous, the
welfare measure would simply be a Riemann integral, since the direct path of integration moves
between the two levels of quality without allowing the price of the private good to vary.
However, such demands for environmental quality usually are not revealed directly. As soon as
we use changes in the private good, x, whose price is p, to reveal information about the
environmental good, we must consider changes in both the quality of the environmental good
and the price of the private good. This indirect path first holds q at its initial value, increases p to
its choke price with quality at q0, and then increases the price further to its choke price with
quality at q1. Then quality can be increased from q0 to q1 (without affecting the level of
satisfaction because of weak complementarity), and then p can be returned to its initial
equilibrium level. We now have a line integral, and the value of that integral may vary with the
path of integration that is chosen. We are really only interested in two paths between (p0,q0) and
3
(p0,q1), this indirect path and the direct path with only q changing. Path independence would
insure that the latter measure (which is want we want) would equal the former measure (which is
what we can observe). 4
The intuition of this integration can be seen using two diagrams similar to those in
Bockstael and McConnell (1993). Figure 1 shows the quality change directly in quality space
using inverse demands and virtual prices. The welfare measures when quality changes from q0
to q1 can be obtained by integrating under the various inverse demands between q0 and q1. The
true uncompensated welfare measure would be the area q 0 abq1 , while the compensated
measures would be the areas q 0 acq1 or q 0 dbq1 depending on whether utility was at its initial
level or its new level. Figure 2 is the typical weak complementarity diagram showing the
demands for the weakly complementary private good. The compensated demands are only
shown for the initial level of utility. H and M superscripts on the demand functions represent
Hicksian (compensated) and Marshallian (uncompensated) demands. The Hicksian choke prices
are labeled p̂ j and the Marshallian choke prices are labeled ~
p j with j being 0 or 1 depending on
the level of q. In the diagram, the market price is p*. These various prices are also used as
labels for points on the price axis. The uncompensated welfare measure in this diagram would
be the area p * ~
p1 g − p * ~
p 0 e , whereas the compensated measure would be the area
p * pˆ 1 f − p * pˆ 0 e . Because of path independence, the compensated welfare measures in the two
diagrams are identically equal. The uncompensated measures will differ unless a further path
independence condition holds.
For simplicity assume that in addition to the weakly complementary private good and
environmental quality, there is a composite numeraire good, z. The two goods and quality
provide utility, u = U(x,q,z) with x, q and z all greater than or equal to zero. Since q is a non-
4
market good, the budget constraint is M = px + z. Solving this constrained maximization
problem yields conditional Marshallian demands, x = xm(p,q,M) and z = z(p,q,M), which can be
substituted back into the direct utility function to yield the conditional indirect utility function, u
= V(p,q,M). Changes in the parameters p, q, and/or M will result in changes in utility.
The change in utility can be expressed in terms of the conditional indirect utility function,
∆u = V(p1,q1,M1) - V(p0,q0,M0)
= ∫ dV
L
=
∫ [V
p
dp + Vq dq + VM dM ]
L
=
 Vp
∫ λ ( p, q, M )V
L
dp +
M

dq + dM  .
VM

Vq
(1)
where L is the path of integration, and λ is the Lagrange multiplier, which, by the Envelope
Theorem, is the marginal utility of income, VM. As in the case of pure price changes, if λ were
constant, it could be taken outside the line integral and used to convert the change in utility to an
equivalent change in income. However, λ cannot be constant with respect to all three of the
parameters5. Fortunately, the uncompensated weak complementarity argument does not require
income to change, so λ may be constant with respect to p and q. Even without assuming λ to be
constant with respect to p and q, it is possible to define an uncompensated surplus measure as
Vq
 Vp

S = ∫
dp +
dq 
V
VM 
L M
(2)
(see Silberberg, 1972). This is a line integral that is no longer an exact differential and thus not
necessarily path independent. Thus, integrating along the two paths described above need not
yield the same answers. Path independence requires
5
 Vp
∂
 VM
∂q


≡
 Vq
∂
 VM
∂p


.
(3)
Taking the partial derivatives
V pqVM − VMqV p
(VM )
2
=
VqpVM − VMpVq
(VM )2
.
Using Young's Theorem6, the first terms in the numerators on each side of the equation are
equal. This implies that
VqM V p = V pM Vq
if and only if the line integral is path independent. Since
 Vq
∂
V
 p
∂M


 V V −V V
pM q
 = qM p
,
2
(V p )
 Vq
∂
V
 p
∂M



 =0
then
(4)
if and only if the line integral is path independent. This is one version of the Willig condition.
When income changes, the relative change in Vq must be equal to the relative change in Vp. This
is the counterpart of quasi-homotheticity of the utility function when there are only private
goods.7
Additional insights are available by further consideration of the conditional indirect
utility function, V(p,q,M). The familiar Envelope Theorem results, Vp = -λxm(p,q,M) and VM =
λ imply that
6
Vq
Vp
=
Vq
− λx ( p, q, M )
m
=−
 Vq
1

x ( p, q, M )  VM
m

 .

(5)
The term in parentheses is the marginal willingness to pay for environmental quality. With path
independence, the identity in equation (3) holds. Integrating (3) over p from p0 to ∞,

 Vq

∂
∞ 
 dp =  VM
∫ ∂p
p0
 Vp
∂
 VM
∫p ∂q
0
∞


 dp
(6)
The right-hand integral in (6) can be evaluated as
Vq
VM
−
p =∞
Vq
VM
=−
p = p0
Vq
VM
p = p0
because weak complementarity and nonessentiality implies that Vq = 0 at p = ∞. The left-hand
integral in (6) is
∞
∂x
∫ − ∂q dp
p0
by Roy’s Identity.
Thus,
Vq
VM
∞
= ∫ x q dp .
(7)
p0
Combining (5) and (7) yields
Vq
∞
1
−
= m
x q dp .
V p x ( p, q, M ) p∫0
(8)
This is the second version of the Willig condition.8 The fact that Vq/Vp is independent of
income, as implied by equation (4), implies that
7
∞
1
x q dp
m
x ( p, q, M ) p∫0
(9)
is independent of income,9 which is the third version of the Willig condition. The marginal
willingness to pay for environmental quality per unit of the weakly complementary private good
must be independent of income.
Virtual Prices
Introducing the virtual price of quality, as done by Bockstael and McConnell (1993),
provides another view of the path independence condition. Let πM(p,q,M) be the Marshallian
virtual price of quality. It can be derived by solving the hypothetical problem
~
max U ( x, q, z ) subject to M ≡ M + π M q = px + π M q + z .
x,q , z
The first order conditions include Uq=λπM, so πM = Uq/λ=Vq/VM. Substituting these results and
Roy's Identity in equation (2) gives
[
]
S = ∫ − x M ( p, q, M ) dp + π M ( p, q, M ) dq .
L
In this case, the path independence condition is
∂x M ∂π M 10
−
=
.
∂q
∂p
(10)
Let compensated (Hicksian) and uncompensated (Marshallian) functions be denoted with H and
M superscripts respectively and let E(p,q,u) be the conditional expenditure function. The identity
π H ( p, q, u ) ≡ π M ( p, q, E ( p, q, u ))
implies that
∂π M
∂π H ∂π M
,
+x
=
∂M
∂p
∂p
(11)
8
while the identity
x H ( p, q, u ) ≡ x M ( p, q, E ( p, q, u ))
implies that
∂x H ∂x M
∂x M
,
=
−π
∂q
∂q
∂M
(12)
using the fact that Eq = -πH.11 Since
∂x H
∂π H
,
= − E qp = − E pq = −
∂q
∂p
(13)
combining equations (10), (11), (12), and (13) and rearranging yields
M  ∂π M
π  ∂M
 M  ∂x 
=
.
 x  ∂M 

(14)
The income flexibility of quality must be equal to the income elasticity of the private good in
order for the uncompensated welfare measure to be path independent. This is analogous to the
Silberberg result that the income elasticities must be equal for private goods whose prices
change.
Specific Preferences
The implications of these results can be illustrated by considering some common
specifications of preferences and demand functions. In these examples with known preferences,
the true welfare measures in quantity space can be calculated and compared with those derived in
price space. This will highlight the importance of the path independence condition.
The linear expenditure system (LES) can be modified to incorporate quality and impose
weak complementarity (see, for example, Larson, 1991). The LES is generated by a StoneGeary utility function, which is a Cobb-Douglas utility function that has been translated by the
9
subsistence levels of the goods. Quality affects the exponent of x. With this utility function
weak complementarity is guaranteed only if the subsistence level of x is -1.12 To keep the
example simple, the subsistence level of z is taken to be zero and the exponents other than q are
set to one.13 Finally, the log transform of the utility function is used. Maximizing u = q ln(x+1)
+ ln z subject to the budget constraint, M = px +z yields the Marshallian demand
x=
qM − p
.
(1 + q ) p
(15)
Substituting (15) and the Marshallian demand for z in the direct utility function yields the
indirect utility function
V ( p, q, M ) = q ln(q ) + (1 + q ) ln( M + p ) − (1 + q ) ln(1 + q ) − q ln( p ) .
(16)
and by inverting (16), the expenditure function is
 (u − q ln(q ) + q ln( p ) 
M = − p + (1 + q ) exp 

1+ q


(17)
Vq = ln(q ) + ln(M + p ) − ln(1 + q ) − ln( p )
(18)
Since
and
Vp =
(1 + q )
q
− ,
( M + p) p
(19)
it is apparent that the Willig condition (equation 4) cannot hold. How much of a problem does
this pose?
For example, assume M =10, p = 1, and quality increases from an initial level of 5 to 6.
At the initial level of quality, equation (15) can be solved for the Marshallian choke price equal
to 50, whereas at q = 6, the choke price is 60. The two Marshallian consumer surpluses are
10
 50 1 
∫1  6 p − 6 dp = 24.4335 and
50
60
 60
1
∫  7 p − 7 dp = 26.6658 , so the change in consumer surplus from
1
x because of the change in q is 2.2323.
Using equation (16), one can calculate that the initial utility, u0, is 11.684 and after the
quality improvement u1 is 13.914. These values for u and the parameter values can be used in
the expenditure function (equation 17) to calculate the expenditure necessary to get to u0 with
q=6 and u1 with q=5. The expenditures necessary to get to u0 with q=5 and u1 with q=6 are both
10, the actual income. Thus, the compensating surplus or quantity compensating variation,
QCV, is 3.0015, while the equivalent surplus or quantity equivalent variation, QEV, is 4.9528.
The Marshallian consumer surplus measure, CS, should be between QCV and QEV, yet
estimating it from the uncompensated demands for x comes up with a value that is significantly
lower than either value. The absence of path independence can lead to significant
miscalculations.
The correct uncompensated welfare measure can be calculated by integrating the
Marshallian virtual price, π M =
Vq
VM
, as q changes from 5 to 6.
 ( M + p )[ln(q ) + ln( M + p ) − ln(1 + q ) − ln( p )]
 dq
∫ 
+
1
q

5
6
(20)
While this integral cannot be solved analytically, numerical integration at the parameters above
yields 3.7815. This true uncompensated measure is between QCV and QEV, but it is roughly 70
percent larger than the incorrect measure that is often invoked.14
This example also highlights another point. While x and q are weak complements, this
does not necessarily imply anything about their complementarity at the actual price. When q
increases from 5 to 6, the uncompensated quantity of x demanded does increase from 8.167 to
11
8.429, but the compensated quantity demanded is reduced to 5.856. The compensated quantity
goes down at the market price, but the integral of the Hicksian demand increases as expected
because the two Hicksian demands cross above the equilibrium price. While the demands with
this utility function are nonlinear, this type of shift can be visualized using Figure 2. Suppose
DH(p,q1,u0) were steeper. Point h would be further to the left, and, more surprisingly, point f
could be to the left of point e. This is the case in this example. The typical weak
complementarity demand diagram is a useful expository device, but actual cases can be more
complex.
A second example, one where the Willig condition holds, is a variant on the simple
repackaging model introduced by Fisher and Shell (1971) and generalized by Willig (1978).
Suppose the indirect utility function is
q

V ( p, q, M ) = ln( M ) + ln + 1)  .
p

The partial derivatives of V are V p = −
(21)
1
q
, and VM = 1 / M . Roy’s Identity
, Vq =
(q + p) p
q+ p
gives
x=
qM
,
(q + p ) p
(22)
and inversion of equation (23) gives the expenditure function
M =
exp(u )
.
q

 + 1
p 
(23)
The Marshallian and Hicksian choke prices are now infinity, but these are valid choke prices
because the integrals to the left of the demands are finite. The utility function is weakly
12
complementary because as p → ∞ , changes in q have no effect on utility. Since neither Vq nor
Vp depends on M, neither does their ratio. The Willig condition holds.
Using the same parameter values as in the first example, one can calculate the initial and
final utility levels as u0 = 4.0943 and u1 = 4.2485. Substituting these values in the expenditure
function yields the welfare values QCV = 1.4286 and QEV = 1.6667. Integrating the
Marshallian demands for x over p from the actual price to the choke price at the two quality
levels yields
∞
∞
 p 
∫ 5 p + p 2 dp = 10 ln p + 5  = 17.9176
1
1
∞
50
∞
 p 
∫ 6 p + p 2 dp = 10 ln p + 6  = 19.4591
1
1
60
so the change in consumer surplus is CS = 1.5415. Now the uncompensated measure is between
the two compensated measures as expected. This can be confirmed using the uncompensated
virtual price πM = M/(q+p) and integrating to get the same value.15 Also, the income flexibility
of this uncompensated inverse demand for quality is unitary and therefore equal to the income
elasticity of x.
A final example is the common linear demand for x,
x = α + βp + γq + δM .
Larson (1991) shows that weak complementarity requires that the indirect utility function be

β
 1 
V ( p, q, M ) =  M +  α + βp + γq +
δ
 δ 

This means that
13
 δ

 exp − (γδ + βp ) .

 β
(24)
− δγ 
β  γ
 1 
M +  α + βp + γq +  +

Vq
δ  δ
β 
 δ 
=
,
Vp

β  β
 1 
− δ  M +  α + βp + γq +  +
δ  δ
 δ 

(25)
which obviously varies with M unless δ = 0 so that equation (25) equals γ/β. If income does not
affect x, then the Marshallian and Hicksian demands are identical and path independence holds
by default. This special case is not very interesting. Otherwise, the Willig condition does not
hold, and the consumer surplus cannot be based on the Marshallian demands for x.
Implications
The estimation of the willingness to pay for environmental quality using related markets
often invokes weak complementarity. If compensated demands are used, weak complementarity
guarantees that the welfare measure in the private good space is identical to the welfare measure
in the public good space. However, when uncompensated demands are used, the two measures
need not be equivalent. This note shows that the conditions under which the uncompensated
measures are equivalent correspond to the condition for path independence of a line integral
generating the welfare measure. This path independence provides intuition for the three
expressions that have become known as the Willig condition. The implications of path
independence for the virtual price of environmental quality also imply the equivalence of the
income elasticity of the private good and the income flexibility of the public good. This is a
strong assumption that is unlikely to be satisfied in most cases, just as most private goods do not
have identical income elasticities. If it is not satisfied, there can be major errors in the
uncompensated welfare measure in terms of the weakly complementary private good.
The Willig condition has played a major role in the evolution of our understanding of
ways to recover welfare measures for environmental changes using related market goods (e.g.,
14
Bockstael and McConnell, 1993, and Smith and Banzhaf, forthcoming) and therefore has
substantial pedagogical value. However, it is only possible to confidently derive welfare
measures for quality changes directly from estimates of the demand for the weakly
complementary private good if the path independence conditions hold. This requires conditions
on the indirect utility function or the uncompensated virtual price function. Yet if one knows
either of those functions, it is unnecessary to derive the welfare measures from the estimated
uncompensated demands for the private good. It is preferable to specify an indirect utility
function that satisfies nonessentiality and weak complementarity but not the Willig condition and
derive compensated and uncompensated welfare measures directly as environmental quality
changes. By not requiring the Willig condition, greater flexibility in specifying functional forms
is possible. If a Willig form were nested within the estimated form, one could test the implied
restrictions.
15
1
Larson (1991) analytically solves the differential equation implied by the Marshallian demand. Numerical
solutions as in Vartia (1983) might seem an attractive alternative. However, as Bockstael and McConnell (1993)
show, determining the bounds for the numerical integration requires additional information such as that provided by
the Willig condition discussed here.
2
Smith and Banzhaf (forthcoming) present a graphical interpretation of the Willig conditions.
3
See Just, Hueth, and Schmitz (1982) for a good summary of this literature.
4
The issue of path independence does not arise when Hicksian demands are used because the expression being
integrated is an exact differential of the conditional expenditure function introduced below. Line integrals of exact
differentials are path independent.
5
See, for example, Just, et al. (1982).
6
Young’s Theorem proves that as long as V is twice continuously differentiable, Vpq=Vqp.
7
Silberberg (1972) showed that when there were only private goods and income did not change, consumer surplus
was path independent only when the direct utility function was homothetic. Homotheticity implies that the marginal
rate of substitution between two goods is invariant to income and all income elasticities are equal to 1. Lau (1970)
showed that if the direct utility function was homothetic, the indirect utility function was also homothetic. If not all
prices change, the income elasticities of those goods must be equal but not necessarily equal to 1. This is quasihomotheticity. With the conditional system here, Vq>0 while VP<0, and these concepts of homotheticity have not
been used. Nonetheless, the condition for path independence here is in the same spirit. When virtual prices are
used later, this link will be further clarified.
8
The minus sign was inadvertently omitted in Bockstael and McConnell (1993).
9
The change of sign does not influence the independence.
10
Bockstael and McConnell (1993) derive a similar equation from Willig rather than path independence.
However, a minus sign is omitted. They start with their equation (12), which, in the notation here,
is
Vq
VM
∞
( p0 ) = ∫ x q dp ,
p0
which is equivalent to equation (7) here. The left-hand side is the marginal willingness to pay for q when p=p0, so it
is equal to the uncompensated virtual price of quality when p=p0. They suggest differentiating both sides by p,
perhaps using the Fundamental Theorem of Calculus (integration and differentiation are inverse operations), to get
∂π M ∂x M
.
=
∂p
∂q
However, this is a definite integral, and differentiating with respect to the equilibrium price is differentiating with
respect to the lower limit of integration. This means that (e.g., see Taylor and Mann (1972))
∞
∂ ∫ x q dp
p0
∂p 0
∂x M ( p 0 )
=−
,
∂q
which yields the equation (10) here.
16
11
See, for example, Hanemann (1991).
12
Herriges, et al. (2004) introduce a modified form of this utility function where the ‘subsistence” level can take on
any values while maintaining weak complementarity. This generalization is empirically important, but for the
current example the simpler form is used.
13
In empirical work, these parameters would be estimated.
14
The Marshallian virtual price also shows that the Willig condition does not hold. The income elasticity of
demand for x is equal to 1, while the income flexibility of the inverse demand for quality is equal to M/(M+p).
15
Even though the Willig condition holds, the compensated changes in x at the equilibrium price show that an
increase in quality leads to a reduction in the quantity demanded of x. The Hicksian demands again cross when q
changes.
17
References
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18
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19
p
d
p
H
(P*,q,u1)
a
p
b
c
q0
p
q1
Figure 1
H
M
(P*,q,M)
(P*,q,u0)
q
P
P1
P1
P0
P0
P*
h
DM(p,q0,M)
g
f
e
DH(p,q0,u0)
DM(p,q1,M)
DH(p,q1,u1)
x
Figure 2