ON THE CONSUMER VALUE OF COMPLEMENTARITY:
A BENEFIT FUNCTION APPROACH
MICHELE BAGGIO AND JEAN-PAUL CHAVAS
[Corrections added after Online Publication September 9, 2008.]
The article develops a conceptual model of the consumer value of complementarity and illustrates its
usefulness in an application to fisheries. Complementarity arises when some goods have a positive effect
on the marginal value of other goods. We propose a measurement of the value of complementarity
based on the benefit function. Our econometric analysis of fish consumption in Italy examines these
issues, with special attention given to dynamics. Our results show that, while short-run fish demand
is characterized by substitution relationships, complementarity does develop in the intermediate run
and in the long run.
Key words: benefit function, complementarity, duality, inverse demand.
The concept of complementarity has been
a standard part of economic analysis (e.g.,
Hicks 1956; Samuelson 1950, 1974). Intuitively,
two activities are complementary if increasing
one has a positive effect on the value of the
other. Hicks (1956) was the first to provide
a systematic treatment of complementarity in
the context of consumer theory. He evaluated complementarity in terms of quantity effects (called q-complementarity) as well as
price effects (called p-complementarity).1 Our
analysis focuses on q-complementarity, where
two commodities are said to be complements
(substitutes) when increasing the quantity of
one tends to increase (decrease) the marginal
value of the other. While there has been interest in the economics of complementarity
(e.g., Samuelson 1950, 1974), the linkages with
Michele Baggio is a graduate student at Department of Agricultural and Resource Economics, University of Maryland. He was
also research assistant at Department of Economics, University
of Verona, Italy, when this article was written. Jean-Paul Chavas
is professor in the Department of Agricultural and Applied Economics, University of Wisconsin-Madison.
The authors thank Ted McConnell, Federico Perali, and the journal reviewers for their useful comments and suggestions on an earlier draft of the article. Michele Baggio thanks Paolo Accadia and
Nando Cingolani for providing information on Italian fishery industry. He also thanks Shinsuke “Yagi” Uchida and his classmates
for the helpful discussions.
1
P-complementarity (or p-substitution) is defined from the
effects of prices on compensated Hicksian demands, holding
utility constant. In this context, two commodities are said to
be p-complements (p-substitutes) if their Hicksian compensated
cross-price effect is negative (positive) (Hicks 1956; Samuelson
1950,1974). Alternatively, q-complementarity (or q-substitution)
is defined from the effects of quantities on compensated inverse
demands, holding utility constant (Samuelson 1950; Deaton 1979).
Two commodities are said to be q-complements (q-substitutes)
if their compensated cross-quantity effect is positive (negative)
(Hicks 1956; Samuelson 1950, 1974; Deaton 1979).
consumer valuation and welfare analysis remain somewhat unexplored.
The objective of this article is to develop a
conceptual model of the consumer value of
complementarity and to illustrate its usefulness in an application to fisheries. Measuring
the value of resources based on consumers’
willingness to pay is standard in economic
and welfare analysis. What is not clear is how
to apply such measurements to the valuation
of complementarity. This article develops a
methodology for such a measurement. Complementarity arises when the willingness to pay
for some goods increases with the consumption of other goods. In this article, we develop
a general approach to the consumer valuation
of complementarity, and we illustrate its usefulness with an application to the analysis of
fish consumption.
As a starting point, the methodology we propose is based on a standard welfare investigation of marginal willingness to pay. The main
challenge to value complementarity is that we
need to know more than just the value of particular goods. Indeed, to evaluate complementarity relationships, we need to know how the
marginal value of a good is influenced by other
goods. Obtaining this information requires a
joint evaluation of willingness to pay across
goods. This creates three specific challenges.
First, the scope of the analysis requires a system approach to consumer valuation. Second,
evaluating possible complementarity among
goods in a given bundle requires assessing how
marginal willingness to pay varies depending
on the composition of the bundle. This suggests relying on welfare measures that can be
Amer. J. Agr. Econ. 91(2) (May 2009): 489–502
Copyright 2008 Agricultural and Applied Economics Association
DOI: 10.1111/j.1467-8276.2008.01188.x
490
May 2009
easily aggregated across goods within a bundle. Third, the approach must be empirically
tractable.
This article addresses these three challenges
to the consumer value of complementarity.
The methodology we propose infers the economic value of complementarity through the
analysis of prices. The approach applies when
goods are heterogeneous and their prices depend on their characteristics. It also applies to
nonstorable goods where short-run supply is
inelastic and prices adjust to clear the market (Barten and Bettendorf 1989). The determination of market-clearing prices induced
by consumer behavior suggests a system of
inverse demands. Note that using a system
approach to value consumer’s marginal willingness to pay is not new. Previous literature
has typically relied on the distance function
first proposed by Shephard and analyzed by
Deaton (1979). Following the early work of
Barten and Bettendorf (1989), a distance function approach has been used in the analysis
of consumer behavior by Eales and Unnevehr
(1994), Holt and Goodwin (1997), Beach and
Holt (2001), Holt and Bishop (2002), Moro
and Sckokai (2002), and Wong and McLaren
(2005). It provides a framework to analyze consumer’s marginal willingness to pay for particular commodities in a way consistent with
the consumer theory (Deaton 1979). However, Shephard’s distance function has one
significant drawback: being based on a proportional rescaling of quantities consumed, it
does not have “nice” aggregation properties
(because proportional measurements cannot
be easily added across heterogeneous consumers). Yet, the evaluation of complementarity would be simpler when relying on welfare measures that can be easily aggregated
across goods within a bundle. This suggests
that the Shephard distance function is not
well suited to this task. What is needed is a
welfare measure that has “nice” aggregation
properties.
A welfare measure with this property is
Luenberger’s benefit function (Luenberger
1992). The benefit function provides a measure
of willingness to pay for goods expressed in
number of units of a reference bundle, holding
utility constant. While the benefit function differs from the distance function, they are closely
related. Indeed, Chambers, Chung, and Färe
(1996) showed that when the reference bundle is chosen to be current consumption then
the benefit function b(·) and the distance function D(·) satisfy 1 – b(·) = 1/D(·). As argued
Amer. J. Agr. Econ.
by Luenberger (1995, 1996), as long as the reference bundle is kept constant and includes
only private goods, aggregate benefits can be
obtained simply by summing individual benefit across consumers.2 This provides the main
motivation for using the benefit function in our
analysis.3
On that basis, we develop our analysis of
the consumer value of complementarity using Luenberger’s benefit function. We show
that the benefit function provides a conceptual framework to develop a welfare measure
of the value of complementarity. The proposed framework involves the specification
and estimation of a system of “inverse demands.” It is an alternative to the distancebased inverse demand systems widely used in
the literature (e.g., the inverse almost ideal demand system, IAIDS). The empirical tractability of the approach and its usefulness are illustrated in an econometric application to fish
consumption.
The article is organized as follows. The next
section presents a brief introduction to the
benefit function, and its linkages with welfare
analysis in a system framework. The evaluation
of complementarity is then discussed based on
Luenberger’s benefit function. A flexible specification of the benefit function is proposed
that provides a basis for our empirical analysis.
An econometric application to the Italian fisheries is then presented, with special attention
given to dynamics. The investigation illustrates
the importance of dynamics. Our results show
that, while short-run fish demand is characterized by substitution relationships, complementarity does develop in the intermediate run and
in the long run. Finally, the last section presents
concluding remarks.
The Model
In this section, we present the basic consumer model. Consider a household choosing consumer goods x = (x1 , . . . , xM ) T ∈ RM
+,
where the superscript “T” denotes transpose.
2
This convenient aggregation property of the benefit function
does not apply to the distance function. Indeed, when consumption
varies across consumers, the distance function involves the rescaling of different consumption bundles, implying that the distance
function cannot be meaningfully added across consumers. On that
basis, applications of the distance function have been limited to a
single representative consumer (e.g., Deaton 1979; Anderson 1980;
Barten and Bettendorf 1989). In addition, aggregate behavior can
represent individual behavior only when strong restrictions are
imposed on consumer preferences (Varian 1992; Stocker 1993).
3
See Chambers (2001) for additional discussion of the use of the
benefit function in welfare analysis.
Baggio and Chavas
Consumer Value of Complementarity
The household preferences are represented
by the utility function u(x), assumed to be
quasi-concave and continuous on RM
+ . Let g =
(g1 , . . . , gM )T ∈ RM
be
some
reference
bundle
+
satisfying g ≥ 0 and g = 0. Following Luenberger (1992), define the benefit function as
(1)
b(x, U)= max{ : u(x − g)≥U, (x −g) ∈ R+M ,
if there is a satisfying u(x − g) ≥ U,
(x − g) ∈ R+M ,
= −∞ otherwise.
Throughout, we treat the reference bundle g
as fixed. To simplify the notation, we omit g as
an argument of the benefit function. The benefit function b(x, U) in (1) measures the largest
number of units of the bundle g the household
is willing to give up to move from the reference
utility level U to the point x. In the case where
the bundle g has a unit price, this provides a
measure of household willingness to pay for x.
And when b(x, U) is differentiable in x, it follows that the marginal benefit ∂b(x, U)/∂ x is
a measure of the marginal willingness to pay
for x.
The properties of the benefit function have
been investigated by Luenberger (1992) and
Chambers, Chung, and Färe (1996). They are
briefly summarized. First, if preferences satisfy
u(x + g) > u(x) for all x ∈ RM
+ and all > 0,
then u(x) = U implies b(x, U) = 0. Second, if
(x − g) ∈ RM
+ for some > 0, then b(x, U) =
0 implies u(x) = U.This shows that b(x, U) = 0
is an implicit representation of household preferences u(x). Third, b(x, U) is nonincreasing
in U. Fourth, b(x + g, U) = + b(x, U).
When b(x, U) is differentiable in x, this implies
(∂b/∂x)g = 1. Finally, if u(x) is quasi-concave
in x, then b(x, U) is concave in x. In the case
where b(x, U) is twice-continuously differentiable in x, this implies that (∂ 2 b/∂x ∂xT ) is a
symmetric, negative semi-definite matrix that
satisfies (∂ 2 b/∂x ∂xT )g = 0.
Luenberger (1992, 1996) has shown that the
benefit function is dual to the standard expenditure function E(p, U) = pT xc (p, U) =
minx {pT x: u(x) ≥ U, x ∈ RM
+ }, where p =
(p1 , . . . , pM ) T ∈ RM
is
the
vector
of prices for
++
x, and xc (p, U) is the quantity-dependent Hicksian demand. While the expenditure function
is commonly used in welfare analysis involving price changes, the benefit function provides a convenient counterpart for conduct-
491
ing welfare analysis involving quantity changes
(Luenberger 1996). In addition, as argued by
Luenberger (1992, 1995, 1996), in situations
where the reference bundle g remains constant
and includes only private goods, then aggregate benefits can be obtained simply by summing individual benefit across consumers.
From consumer theory, it is well known that
Marshallian and Hicksian demands are closely
related: they satisfy the standard Slutsky equation relating Marshallian and Hicksian price
effects (Deaton and Muellbauer 1980). Similar relationships also apply to inverse demands.
To see that, let V( p/m) = max x {u(x): p T x ≤
m, x ∈ RM
+ } be the indirect utility function,
where m > 0 denotes household income.
Assuming that u(x) is quasi-concave and
continuous, define the inverse Marshallian demand p∗ (x) as the solution of the minimization problem u(x) = min p {V( p/1): p T x ≤ 1,
p ∈ RM
+ }, where prices have been normalized
to satisfy pT x = 1 (Anderson 1980). Following Luenberger (1992, 1996), define the Luenberger price equation pL (x, U) as the solution
of the minimization problem minp { p T x − E( p,
U): p T g = 1, p ∈ RM
+ }. The Luenberger price
equation pL (x, U) is a compensated pricedependent (or inverse) demand, expressing
prices as a function of quantities, holding utility U constant. It is subject to the price normalization rule pT g = 1. When the utility function
is quasi-concave and monotonic, Luenberger
(1992, 1996) showed that the benefit function
satisfies b(x, U) = minp { p T x − E( p, U): p T g =
4
1, p ∈ RM
+ }. Under differentiability, applying
the envelope theorem gives
(2)
∂b(x, U )/∂ x = p L (x, U ).
This shows that the Luenberger price equation pL (x, U) measures marginal benefit. Luenberger (1996) showed that the price normalization rule pT g = 1 implies that5
(3)
p L (x, U ) = p/ p T g.
Since U is typically unobservable, inverse
demands pL (x, U) in (2) are not observed.
To make the analysis empirically tractable, we
4
In addition, Luenberger (1992) showed the following dual relationship between E( p, U) = minx { p T x − b(x, U) ( p T g): x ∈ RM
+ }.
5
Note that the Luenberger price equation satisfies p L (x, U) =
p c (x, U)/[ p c (x, U) T g], where p c (x, U) ∈ argminp { p T x: V( p/1) =
U, p ∈ RM
+ } is the price-dependent compensated demand analyzed
by Deaton (1979). Compared to p L (x, U), note that p c (x, U) uses
a different price normalization rule: it satisfies p T x = 1.
492
May 2009
need to establish linkages between the Luenberger price function pL (x, U) and its Marshallian counterpart p∗ (x). Using (3) evaluated at
U = u(x) yields
(4)
p ∗ (x)/[ p ∗ (x)T g] = p L (x, u(x)).
Note that p∗ (x) and pL (x, U) use different
price normalization rules: pT x = 1 in the former, and pT g = 1 in the latter. It follows from
(4) that p∗ (x) and pL (x, u(x)) are in general
proportional to each other. This generates the
following duality result (Chavas and Baggio
2007)
(5)
p ∗ (x) = (x, g) p L (x, u(x))
where (x, g) is a proportionality factor satisfying
(6)
(x, g) = p ∗ (x)T g.
The proportionality factor (x, g) rescales
the marginal benefit pL (x, u(x)) to account
for the different normalization rules. Equations (5) and (6) establish the relationship between normalized (observable) prices p∗ (x)
and the Luenberger price functions pL (x, U).
We will make use of the duality relationship
(5) below.
The Consumer Value of Complementarity
Denote by I = {1, . . . , M} the index set of M
goods. Consider the partitions I = {I A , I B }
where I A is the index set of goods of interest
to the investigator, and I B is the set of other
goods. The household faces x = (xA , xB ), where
xA = {xj : j ∈ I A } denotes the goods in I A and
xB = {xj : j ∈ I B } are the other goods. We are
interested in measuring the value of the goods
xA . Given some reference utility U, the household total value for the goods xA is
(7)
V A (x, U ) = b(x A , x B , U ) − b(0, x B , U ).
It will be useful to try to decompose this
total value into some of its components. One
of its components is the value of complementarity associated with the goods in xA . To investigate this issue, consider partitioning the
set I A into K mutually exclusive subsets: I A =
{I A1 , . . . , I AK }, where I Ak denotes the kth subset of goods in I A , k = 1, . . . , K. Let xAk = {xj :
j ∈ I Ak } denote the goods in the subset I Ak ,
k = 1, . . . , K. And let xA\Ak = {xj : j ∈ I A \I Ak },
Amer. J. Agr. Econ.
where I A \I Ak denotes the subset of commodities that are in I A but not in I Ak Given some
reference utility U, the household incremental
value for the goods xAk is defined as
V Ak (x, U ) = b(x Ak, x A\Ak, x B, U )
(8)
− b(0, x A\Ak, x B, U )
where k = 1, . . . , K. The incremental value
V Ak (x, U) in (8) reflects the willingness to pay
for xAk , holding other goods (xA\Ak , xB ) and
utility U constant. It applies only to the kth
subset of goods in xA . Of special interest is
the sum of the incremental values
K across the K
sets of goods {x A1 , . . . , x AK } : k=1
V Ak (x, U ).
In general, this sum will not be the same as
the total value V A (x, U) in (7). The reason is
that in (8), goods in xA are valued one group
at a time, while they are valued jointly in (7).
This suggests defining the household’s value of
complementarity for goods in xA as
W (x, U ) =
(9)
K
V Ak (x, U ) − V A (x, U ).
k=1
In general, W(x, U) in (9) can be positive,
zero, or negative. It can be zero if the value
of the goods in each group is independent of
goods in other groups. And, we argue below
that it is positive when the consumer values
complementarity. Note that the incremental
value V Ak (x, U) involves the valuation of xk
given xA\Ak ≥ 0, while the total value V A (x,
U) involves the joint valuation of the xk ’s
compared to x = 0. Intuitively, under complementarity, the benefit of xk tends to be larger
evaluated at xA\Ak > 0 compared to xA\Ak =
0 (since a higher complementary xA\Ak is expected to stimulate the value of xk ). It follows
that complementarity is associated with the
sum of incremental values being higher than
the joint value, that is, with W(x, U) ≥ 0. Such
arguments are formalized next.
Consider the simple case where there are
two groups in I A : K = 2. Then, for the household, x = (xA1 , xA2 , xB ), and equations (7)–(9)
take the form
(7 )
(8a )
V A (x, U ) = b(x A1 , x A2 , x B , U )
− b(0, 0,x B , U )
V A1 (x, U ) = b(x A1 , x A2 , x B , U )
− b(0, x A2 , x B , U )
(8b )
V A2 (x, U ) = b(x A1 , x A2 , x B , U )
− b(x A1 , 0, x B , U )
Baggio and Chavas
Consumer Value of Complementarity
and
(9 )
493
Benefit Specification
W (x, U ) = V A1 (x, U )
+ V A2 (x, U ) − V A (x, U )
= b(x A1 , x A2 , x B , U )
− b(0, x A2 , x B , U )
− b(x A1 , 0, x B , U )
+ b(0, 0, x B , U ).
In the case where b(xA , xB , U) is twice differentiable in xA , this gives
x A2 x A1
(9 ) W (x, U ) =
0
0
× [∂ 2 b(a 1 , a 2 , x B , U )/∂a 1 , ∂a 2 ] da 1 da 2 .
This generates the following results.
PROPOSITION 1. Let xA = (xA1 , xA2 ) ≥ 0
where xA = 0 and xB = 0. Assume that the household benefit function b(xA , xB , U) is twice continuously differentiable in xA . The household
value of complementarity W(x, U) satisfies
(a) W (x, U ) = 0 if ∂ 2 b(x A1 , x A2 , x B , U )
/∂ x A1 ∂ x A2 = 0 for all (x A , x B ) ≥ 0,
(b) W (x, U ) < 0 if ∂ 2 b(x A1 , x A2 , x B , U )
/∂ x A1 ∂ x A2 < 0 for all (x A , x B ) ≥ 0,
(c) W (x, U ) > 0 if ∂ 2 b(x A1 , x A2 , x B , U )
/∂ x A1 ∂ x A2 > 0 for all (x A , x B) ≥ 0.
Proposition 1 shows how the sign of the
value of complementarity W(x, U) is determined. From (a), a sufficient condition for a
zero value of W is that the marginal benefit of
xA1 is independent of xA2 . From (b), a sufficient condition for W < 0 is that the marginal
benefit of xA1 decreases with xA2 . In this case,
the incremental benefit of xA1 is smaller when
xA2 increases. This means that the goods in
xA behave as substitutes across groups. Finally,
from (c), a sufficient condition for W > 0 is that
the marginal benefit of xA1 increases with xA2 .
Then, the incremental benefit of xA1 is higher
when xA2 increases. This associates a positive
value of complementarity for the goods in xA ,
W(x, U) > 0, with the presence of synergy or
complementarity across groups in I A .
Proposition 1 shows that equations (8) and
(9) provide a basis for evaluating the value
of complementarity. But this requires knowing
the benefit function b(x, U). Next, we propose
a methodology to support an empirical estimation of the benefit function.
For the analysis in this article, we consider the
following specification for the benefit function
(10)
b(x, U ) = (x) − [U (x)]/[1 − U (x)]
where (x) > 0, and [1 − U (x)] > 0. In neoclassical consumer theory, the benefit function
is nonincreasing in U, and nondecreasing and
concave in x. This imposes corresponding constraints on (x), (x), and (x). It means that
[(x) – [U(x)]/[1 – U (x)]] is nonincreasing
in U, nondecreasing in x, and that ∂ 2 [(x) –
[U(x)]/[1 – U (x)]]/(∂x∂xT ) is a negative
semi-definite matrix.
When (x) takes a quadratic form, then (10)
provides a flexible representation of quantity
effects (see below). When (x) = 0, specification (10) becomes linear in U (in a way similar
to the specification IAIDS). However, when
(x) = 0, (10) can generate more flexible “utility effects.” As such, (10) provides an attractive
general and flexible specification of consumer
benefits (and the associated consumer preferences).
From (10), equation (2) gives the Luenberger price equation
(11)
p L (x, U )
= (∂/∂ x) − (∂/∂ x)U /[1−U (x)]
− (∂/∂ x)(x)U 2 /[1 − U (x)]2 .
From Luenberger (1992, 1995), when u(x +
g) is strictly increasing in and x > 0, the
benefit function b(x, U) evaluated at 0 is an
implicit representation of the utility function
u(x). Hence, solving the equation b(x, U) = 0
for U yields U = u(x). It follows from (10) that
U/[1 − U (x)] = (x)/(x) when U = u(x).6
Substituting this term in (11) and using the duality relationship in equation (5), we obtain the
inverse Marshallian demand
(12)
p ∗∗ (x) = (∂/∂ x) − (∂/∂ x)(x)/(x)
− (∂/∂ x)(x)2 /(x)
where p∗∗ (x) ≡ p∗ (x)/[(x, g)]. Consider the
following parametric specification for (x),
6
It follows u(x) = (x)/[(x) + (x) (x)]. An alternative way of
specifying Marshallian inverse
demands would then be to use Wold
identity, p ∗j (x) = (∂u/∂ x j )/[ m
k=1 (∂u/∂ x k )x k ], which is obtained
from applying the envelope theorem to u(x) = minp {V( p/1):
p T x ≤ 1, p ∈ RM
+ }.
494
May 2009
Amer. J. Agr. Econ.
(x), and (x):
(15)
(13a) (x) =
M
j x j +
j=1
M
M 1/ x x
2 jk j k
(13b) (x) = exp
M
j x j
j=1
and
M
j x j .
j=1
This specification (12)–(13) has some similarities with the more traditional inverse demand systems (e.g., Eales and Unnevehr, 1994;
Holt and Goodwin, 1997; Wong and McLaren,
2005). The specification in (12)–(13) is flexible. Equation (12) identifies three components
in the price equation: the first term (13a) is a
flexible quadratic form, the second term (13b)
means that ln((x)) is linear in x, and the
third term includes (x) specified as linear in
x in (13c). Note from equation (11) that, when
nonzero, the third term allows for some additional flexibility by capturing quadratic utility
effects. We have seen that (∂b/∂x)g = 1 holds
for all x and U. This implies (∂/∂x)g = 1,
(∂/∂x)g = 0, and (∂ /∂x)g = 0, holding for
all x. This generates the following restrictions:
(14a)
M
jk x k − j (x)
− j (x)2 /(x), j = 1, . . . , M.
j=1 k=1
M
k=1
where jk = kj as symmetry restrictions for all
j = k,
(13c) (x) =
p ∗∗
j (x) = j +
j g j = 1
j=1
Further Considerations on Heterogeneity
and Extension to Panel Data Analysis
In the analysis presented below, we rely on
panel data, involving per capita consumption
patterns across R regions and T time periods. In the rth region at time t, let the observed quantity be xrt = (x1rt , . . . , xMrt )T with
corresponding prices prt = (p1rt , . . . , pMrt )T , r =
1, . . . , R, t = 1, . . . T. In this context, we want to
account for changes over space. Define Dr as a
dummy variable equal to 1 for region r, and 0
otherwise. As an extension of (13a), consider
the specification
(16a) (xr t ) =
M
jr t x jr t +
j=1
+
M
M R M
jr Dr x jr t
r =1 j=1
1/ x
2 jk jr t x kr t .
j=1 k=1
The parameters in (16a) allow for variations in consumer preferences across regions.
Note that, (∂/∂xrt )g = 1 implies the additional
restrictions M
j=1 jr g j = 0, r = 2, . . . , R.
Next we want to account for changes over
time. At time t, this can be done by specifying
jrt in (16a) as follows:
2
(16b) jr t = j0 + j t + j t
M
+
jkL xkr,t−1 + jL p ∗∗
jr,t−1
k=1
(14b)
M
jk g j = 0, k = 1, . . . , M
j=1
(14c)
M
j g j = 0
j=1
and
(14d)
M
j g j = 0.
j=1
Finally, equations (12) and (13) imply that
the household’s inverse Marshallian demand
for the jth good is
j = 1, . . . , M. Note that, in the context
of equation (16a)–(16b), imposing the
restriction (14a) globally (i.e., for all values of t, xkr,t−1 , and pjr,t−1 ) implies that
M
M
M
j g j =
j=1 j0 g j = 1,
j=1 j g j = 0,
M
M j=1
0, j=1 jkL g j = 0, and
j=1 jL g j = 0.
The specification (16a)–(16b) allows the
parameters in to vary over time and across
regions. The parameters jkL and jL in (16b)
introduce dynamics in the analysis. The parameters jkL account for one-period-lagged
quantity effects. This includes lagged ownquantity effects (when j = k) as well as lagged
cross-quantity effects (when j = k). And the
parameters jL capture lagged own-price
Baggio and Chavas
effects. In general, the jkL ’s capture intermediate run dynamics (after one period), while
the jL ’s capture longer-term dynamics.7 This
provides a flexible representation of dynamics
and its effects on consumer welfare. This will
provide a basis for investigating the role of
dynamics in the valuation of complementarity.
One important remaining issue is the choice
of the reference bundle g. As discussed above,
to obtain nice aggregation properties, we
choose g so that it includes private goods that
remain constant for all consumers. In our application, we choose g to be a (M × 1) vector
g = (1, 0, . . . , 0)T . This means that the reference
bundle g is expressed in units of the first good in
the bundle g.8 From equations (14) and (16),
note that choosing g = (1, 0, . . . , 0)T implies
choosing (x, g) = p∗1rt in (5) and (6). This is
equivalent to normalizing prices, all prices being deflated by the price of the first good: p∗∗
jrt ≡
p∗jrt / p∗1rt , j = 1, . . . , M. This reflects the fact that
all valuations are made relative to the value
of the first commodity. With p∗∗
1rt = 1, this implies that there are no parameters to estimate
in the inverse demand for the first commodity.
For that reason, the first equation is dropped
from the empirical analysis presented below.
In other words, our econometric analysis focuses on the estimation of equation (15) for
(M − 1) goods: j = 2, . . . , M.
An Application
The model is applied to estimate a system of
inverse demands for fish landed at Italian regional ports.9 The data start with annual observation of tons of landed fish and average prices
per kilogram of forty-seven marine species registered by port authorities in twelve Italian
regions.10 We further collected data on population, estimates of resident population at the
beginning of each year, and tonnage of the fishing vessels registered in each region. The sample period covers 1974 through 2003, for a
Consumer Value of Complementarity
495
total of 360 observations. The data were obtained from available publications of the Italian Institute of Statistics (ISTAT) and from the
Istituto Ricerche Economiche per la Pesca e
l’Acquacoltura (IREPA onlus).11
Given the large number of species, to make
the analysis manageable, we aggregated the
species into five main categories: (1) small
pelagic species (anchovies, mackerels, and sardines), (2) other fish, (3) cephalopods (squid,
octopus, and cuttlefish), (4) mussels, and (5)
crustaceans. Those broad categories represent
the main groups of species reported in the Italian statistics.12
Consumption data for each category and
each region are obtained as follows: CONS
= RL − NE, where CONS denotes regional
consumption, RL is regional landing (tons of
fish), and NE is net export.13 Finally, quantities
are expressed in per capita terms (kg/person)
by dividing regional consumption by regional
population. And regional prices for each category are calculated for each region by dividing
total value by total quantity. Descriptive statistics are presented in table 1.
Estimation and Empirical Results
Given the availability of aggregate consumption data, the econometric specification is written as a system of equations obtained after
adding an error term to equation (15). For the
jth good in region r at time t, and using (16),
this gives
(17)
2
p ∗∗
jr t = j0 + j t + j t +
R M
js Ds
s=2 j=1
+
M
jkL xkr,t−1 + jL p ∗∗
jr,t−1
k=1
+
M
jk xkr t − j (·)
k=1
− j (·)2 /(·) + e jr t
7
For alternative specifications of dynamics in the context of inverse demands, see Holt and Goodwin (1997).
8
This is a common normalization in the literature (Luenberger
1995, 1996). This choice is convenient: it makes estimation simpler
(by dropping an equation) and it gives a simple interpretation of
welfare measures.
9
The underlying assumption is that fish is weakly separable from
other goods.
10
The data set includes Italian regions facing the sea: 1-Abruzzi,
2-Calabria, 3-Campania, and 4-Emilia Romagna, 5-Friuli V. Giulia,
6-Liguria, 7-Marche, 8-Puglia, 9-Sardegna, 10-Sicilia, 11-Toscana,
12-Veneto. Note that some regions are excluded due either to lack
of price data (Lazio and Basilicata), or to the presence of many
missing values (Molise).
11
We thank Monica Gambino and IREPA for making available
part of the data on tonnage (elaborations on data from the Italian
Ministry for Agricultural and Forestry Policies-Mipaf).
12
Similar but broader categories were analyzed in Moro and
Sckokai (2002).
13
We measure net export as follows: NE = RL(RTSL/TTSL)
(TP – RP)/TP, where RP is regional population, TP is total Italian
population, RTSL is regional tonnage of fishing fleet, and TTSL
is total tonnage of the Italian fleet. This states that net export in
a region is proportional to regional landing RL, to the regional
relative fleet size (RTSL/TTSL), and to the relative population of
“other regions” (TP – RP)/TP.
496
May 2009
Amer. J. Agr. Econ.
Table 1. Descriptive Statistics for Fish Consumption in the Italian Regions
Per Capita Consumption (Kg/Person)
Fish
Category
x1
x2
x3
x4
x5
Prices (Euros/Kg)
Mean
Standard
Deviation
Min
Max
Fish
Price
Mean
Standard
Deviation
Min
Max
2.18
3.55
0.67
2.31
0.51
2.34
3.24
0.71
2.81
0.56
0.07
0.72
0.06
0.02
0.04
15.64
17.51
5.43
21.89
3.33
p1
p2
p3
p4
p5
1.68
5.30
5.74
3.40
10.27
0.78
1.46
1.61
1.90
5.15
0.41
0.99
2.26
0.22
3.27
5.09
11.47
11.64
13.72
31.96
Table 2. Model Diagnostics
Model A
Model B
Model B
Single equation R2
Parameters estimated
Log likelihood
No. of observations
Hypothesis
Restrictions
Wald
P-Value
j = 0
jkL = 0
is = 0
p2
0.662
70
−2,043.811
348
4
20
20
p3
0.702
5.279
34.807
46.569
p4
0.588
0.260
0.021
0.0007
p5
0.652
Note: j = 2, . . . , M; k = 1, . . . , M; s = 2, . . . , R.
j = 2, . . . , M, where (·) and (·) are given by
equations (13a), (13b), and (16), and ejrt is an
error term distributed with mean zero. In equation (16a), xjrt represents per capita consumption of the jth fish in the rth group of regions14
at year t, r = 1, . . . , R(R = 6), t = 1974, . . . ,
2003 (T = 30), with p∗∗
jrt as the corresponding
normalized price.
Before estimation, it is important to consider the stochastic properties of the system
demands. Given g = (1, 0, . . . , 0)T , we proceed
to estimate equation (17) for j = 2, . . . , M (after dropping the first equation, as discussed
above).15 The associated error terms ejrt are assumed to be serially uncorrelated (recall that
the dynamics are captured in (17) through the
inclusion of lagged quantities xkr,t−1 and lagged
prices pjr,t−1 ). However, we allow for correlation across goods, with ert ≡ (e2rt , . . . , eMrt )T
satisfying E[ert ] = 0 and E[ert ert T ] = Σ, where
Σ denotes the contemporaneous covariance
14
To avoid overparameterization, we group regions according
to their geographic position. Our analysis considers six regional
groups: Northern and Southern Adriatic Sea, Northern and Southern Tirrenian Sea, Sardegna, and Sicilia.
15
The estimation is not invariant to the choice of g. The empirical analysis was also conducted under alternative choices for g.
While this affected the quantitative estimates, the main qualitative
findings reported below remained unaffected.
matrix. We can proceed with estimating the set
of (M − 1) equations in (17). Since equation
(17) is nonlinear in the parameters, it requires
using nonlinear estimation methods. The system can be estimated by a nonlinear seemingly
unrelated regression (NLSUR) procedure, allowing for correlation across equations. This
was done using GAUSSX 7.0,16 NLS method,
with GAUSS algorithm and ROBUST option
for heteroskedastic-consistent standard errors.
This yields consistent and asymptotically efficient parameter estimates.
We conducted a series of tests on the model
specification. This was done using Wald tests,
with results presented in table 2. We first estimated the complete model (model A) presented in equation (17). This model includes
the parameters jkL ’s capturing the effects of
lagged quantities xkr,t−1 , as well as jL capturing the effects of lagged prices pjr,t−1 . And in
the context of equation (11), model A also includes the parameters j ’s capturing the linear effect of U/[1 – U (x)], as well as the
parameters s capturing the quadratic effects
of U/[1 – U (x)]. The second model (model
B) is similar to model A, except that it imposes the restrictions: j = 0, j = 2, . . . , M (thus
16
We thank Anna Alberini for making the software available.
Baggio and Chavas
Consumer Value of Complementarity
limiting the effects of U/[1 – U (x)] to be linear in (11)). The Wald test of these restrictions gave a p-value of 0.2598. Thus, we do
not find strong statistical evidence that the j ’s
are nonzero. This suggests that specifying the
effects of the utility term U/[1 – U (x)] as linear appears appropriate in equation (11). Next,
we estimated model B, defined to be model
A after imposing the restriction j = 0, j =
2, . . . , M. In this context, we investigated the
significance of dynamics (as represented by
the null hypothesis jkL = 0, j = 2, . . .M, k =
1, . . . , M) and of regional differences (as represented by the null hypothesis js = 0, s =
2, . . . , R). As reported in table 2, both hypotheses are rejected at the 5% significance level.
Hence, we find strong statistical evidence of dynamics, where lagged quantity effects influence
marginal benefits. And we also find strong statistical evidence against regional homogeneity of preferences. On the basis of these tests,
we proceed with our empirical analysis based
on model B (which allows for lagged quantity
effects and regional heterogeneity). The resulting econometric estimates are presented in
Appendix A. The single-equation R2 reported
in table 2 confirm that the model shows a good
fit.
Testing the Theory
The estimated specification (model B) allows
us to study the dynamic properties of the
model. Our analysis focuses on three time
horizons: short-run effects (SR) that correspond to the current period effects of quantities xt ; intermediate-run effects (IR) that
consider both current and one-period-lagged
effect of quantities, xt, and xt−1 ; long-run effects (LR) corresponding to effects after many
periods. The long-run effects consider the scenario where the marginal benefits are allowed
to adjust to their long-run equilibrium pejr
∗∗
where pejr = p∗∗
jrt = pjr,t−1 for all t. From equations (11) and (16), and given (·) = 0, this
corresponds to p ejr (x, U ) = (1 − jL )−1 [ j0 +
M
k=1 jkL x kr − (∂/∂ x)U ]. The role of dynamics is further discussed below.
In each of these three scenarios (SR, IR,
and LR), the concavity property of the benefit
function is evaluated in the context of model
2
B. The Hessian of the benefit function ∂ x∂ ∂bx T
t
t
and its eigenvalues are calculated. We call
2
the matrix ∂ x∂ ∂bx T the “Luenberger matrix.”17
t
17
t
Note that, in general, the Luenberger matrix differs from the
“Antonelli matrix” (∂ p c /x) where pc (x, U) the price-dependent
497
Table 3. SR Luenberger Matrix Evaluated at
Sample Means
Fish
Category
x1
x2
x3
x4
x5
x1
x2
x3
x4
x5
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
−0.077
(0.003)
0.006
(0.005)
−0.004
(0.001)
−0.113
(0.005)
0
(0)
0.006
(0.005)
−0.608
(0.012)
−0.024
(0.002)
−0.380
(0.010)
0
(0)
−0.004
(0.001)
−0.024
(0.002)
−0.089
(0.001)
−0.065
(0.003)
0
(0)
−0.113
(0.005)
−0.380
(0.010)
−0.065
(0.003)
−0.802
(0.019)
Inference on Eigenvalues
Estimated
Bootstrapped
t-Statistic
P-Value
−1.109
−0.335
−0.084
−0.048
0
−1.108
−0.335
−0.084
−0.048
0
−69.589
−25.499
−55.409
−12.067
−
0
0
0
0
−
Note: Bootstrapped standard errors in parentheses are below the corresponding
estimates.
Under a quasi-concave utility function u(x),
2
the theory implies that the matrix ∂ x∂ ∂bx T is sint
t
gular (since ∂ x∂ ∂bx T g = 0)18 and negative semit
t
definite (from the concavity of b(xt , ·)). It follows that, under a quasi-concave utility function u(x), the eigenvalues of the Luenberger
matrix must all be nonpositive, with at least
one eigenvalue equal to zero.
To investigate the statistical properties of the
Luenberger matrix and other results, we conducted simulations based on the parameter estimates. We evaluated the standard errors of
the simulations by bootstrapping using 5,000
draws from the asymptotic distribution of the
estimated parameters.
The SR Luenberger matrix and its eigenvalues are reported in table 3 (evaluated at
sample means). All the nonzero elements of
the Luenberger matrix are statistically different from zero. And all the nonzero eigenvalues
are negative and statistically different from
zero. This provides statistical evidence that the
Luenberger matrix is negative semi-definite
2
demand defined in footnote 6 and analyzed by Deaton (1979).
The “Antonelli” matrix naturally arises from analyzing compensated behavior using the Shephard distance function D(x, U) =
max { : u(x/) ≥ U, x ∈ RM
+ }. When u(x) is increasing and
quasi-concave, and under differentiability, Deaton (1979) showed
c
that p (x, U) = ∂D(x, U)/∂ x. There is a close relationship between
the Shephard distance function and the benefit function: b(x, U) =
1 − 1/D(x, U) when g = x (see Chambers, Chung, and Färe 1996).
However, the two functions differ when g = x.
∂2b
18
In general, in the short run, the Luenberger matrix ∂ x,∂
is at
xt
t
most of rank (M − 1). In our case, given g = (1, 0, . . . , 0)T , its first
row and column are zeros by construction.
498
May 2009
Amer. J. Agr. Econ.
Table 4. SR Compensated Flexibilities Evaluated
at Sample Means
Fish
Category x1
p1
p2
p3
p4
p5
0
(0)
0
(0)
0
(0)
0
(0)
0
(0)
x2
0
(0)
−0.075
(0.003)
0.001
(0.0009)
−0.003
(0.0008)
−0.016
(0.0008)
x3
0
(0)
0.005
(0.004)
−0.102
(0.002)
−0.014
(0.0009)
−0.048
(0.001)
x4
0
(0)
−0.007
(0.002)
−0.007
(0.0005)
−0.089
(0.001)
−0.014
(0.0006)
x5
0
(0)
−0.059
(0.003)
−0.037
(0.001)
−0.022
(0.001)
−0.060
(0.001)
Note: Bootstrapped standard errors in parentheses are below the corresponding
estimates.
as expected, that is, the benefit function is
concave and well behaved. Note that the
derivation of the Luenberger matrix in the SR
involves only changes of the current quantities xt , keeping the lagged quantities and prices
fixed.
We also evaluated price flexibilities. Capturing the effect of the kth quantity on the
jth price, they are: f ∗jk ≡ [∂ln(pj ∗ (x))/∂ln(xk )],
the uncompensated (Marshallian) price flexibility; and f Ljk ≡ [∂ln(pLj (x, U))/∂ln(xk )], the
compensated (Luenberger) price flexibility, j,
k = 1, . . . , M. (See Appendix B.) The short-run
compensated price flexibilities are presented
in table 4.
Compensated flexibility f Ljk measures the relative effect of a 1% change in the kth good on
the marginal benefit of the jth good, holding
utility constant. Own-quantity compensated
flexibilities are all negative, as expected from
the theory. This means that the inverse demand functions are downward sloping. Their
small magnitude19 (in absolute value) is consistent with previous findings in the literature
(Barten and Bettendorf 1989; Beach and Holt
2001; Holt and Bishop 2002). However, our
estimates are less consistent with those obtained by Moro and Sckokai (2002) using similar groups of species for annual fish landing in
Italy. Category 3 (cephalopods) has the largest
(in absolute value) own-quantity compensated
flexibility of –0.10: a 1% increase in this good
induces a 0.10% decrease of its price. Category
5 (crustaceans) has the smallest quantity
elasticity of –0.06. When statistically significant,20 fish categories are all q-substitutes:
the cross-compensated flexibilities range from
–0.003 to –0.059. The largest (in absolute value)
cross-quantity compensated flexibility is between category 2 (other fish) and category 5
(crustaceans). It appears that, since the species
in the two groups are very different in characteristics (e.g., taste, ways of cooking, etc.), they
behave as strong substitutes as households are
less likely to consume them together. An expected result is that cephalopods are found
to be very small q-substitutes with respect to
mussels. These two categories represent similar species that should (intuitively) be more
likely consumed together. It is less clear why
mussels and other fish exhibit a lower degree
of q-substitution. The empirical results indicate that consumers are more likely to consume mussels and other fish together.
A similar analysis is conducted in the IR,
now considering a marginal change in both
xt and xt−1 . For example, the IR Luenberger
2
matrix involves evaluating ∂ x∂ ∂bx T , where xt =
t
t
xt−1 = x. We found that the IR Luenberger
matrix is no longer negative semi-definite, that
is, the benefit function is no longer concave in
the intermediate run. This shows that dynamics plays a significant role in consumer welfare.
Intermediate run compensated flexibilities, reported on the top panel of table 5, show that
marginal consumption effects are stronger in
the intermediate-run than in the short-run case
Table 5. IR and LR Compensated Flexibilities
Evaluated at Sample Means
Fish
Category
p1
p2
p3
p4
p5
p1
p2
p3
p4
19
Since we are dealing with an inverse demand system, note that
small-quantity elasticities correspond to large price elasticities.
20
The flexibility for categories 2 and 3, other fish and
cephalopods, is positive, but it is not statistically different from
zero.
p5
x1
0
(0)
−0.178
(0.005)
0.002
(0.001)
−0.043
(0.002)
0.026
(0.002)
0
(0)
−0.598
(0.017)
0.007
(0.004)
−0.115
(0.005)
0.062
(0.004)
x2
x3
Intermediate Run
−0.030
0.002
(0.0008) (0.001)
0.057 −0.110
(0.008) (0.013)
−0.023 −0.304
(0.003) (0.006)
−0.018 −0.083
(0.002) (0.005)
0.024 −0.077
(0.002) (0.005)
Long Run
−0.102
0.006
(0.003) (0.003)
0.369 −0.372
(0.022) (0.037)
−0.078 −0.793
(0.008) (0.018)
−0.043 −0.205
(0.006) (0.012)
0.11
−0.074
(0.005) (0.012)
x4
−0.018
(0.0008)
−0.043
(0.006)
−0.042
(0.002)
−0.155
(0.004)
0.017
(0.002)
x5
0.016
(0.001)
0.090
(0.008)
−0.060
(0.004)
0.026
(0.003)
−0.049
(0.004)
−0.047
0.039
(0.002) (0.002)
−0.103
0.411
(0.014) (0.020)
−0.103 −0.058
(0.006) (0.009)
−0.265
0.113
(0.010) (0.007)
0.074 −0.035
(0.004) (0.009)
Note: Bootstrapped standard errors in parentheses are below the corresponding
estimates.
Baggio and Chavas
(SR, reported in table 4). This means that after
one period, the marginal willingness to pay for
a good tends to increase over time. One important result is that some pairs of goods that were
substitutes in the SR are now complements in
the IR, for example, mussels (category 4) and
crustaceans (category 5), and other fish (category 2) and crustaceans (category 5). These results indicate that the possibility of substitution
can deteriorate over time, with the rise of complementarity relationship in the intermediate
run. Now also small pelagic are substitutes with
other fish and mussels, while they are complements with cephalopods and crustaceans.
Next, a similar analysis is presented in the
LR, which considers permanent changes in x
(i.e., over many periods, not just two periods
as in the IR analysis). As discussed above, this
involves letting prices adjust to their long-run
equilibrium pt = pt−1 = p. Again, we found that
2
the LR Luenberger matrix ∂ x∂ ∂bx T is not negat
t
tive semi-definite, that is, the benefit function
is not concave in the long run. The LR compensated price flexibilities are reported in the
bottom panel of table 5. In general, the LR
results reinforce the findings for the IR scenario. For example, the evidence of complementarity becomes stronger for small pelagic
(category 1) and cephalopods (category 3), as
well as crustaceans (category 5), and for other
fish (category 2), and crustaceans (category
5) (as their positive cross-flexibility becomes
larger). Interestingly, substitutability becomes
weaker between cephalopods and crustaceans,
suggesting a more complicated dynamic
behavior.
Substitutability and Complementarity
The curvature property of the benefit function is important not just for the consistency
with the theory, but also for the derivation
of welfare measures, and therefore for the
applicability of measures of the consumer
value of complementarity developed here. The
benefit function estimated through the inverse demand system measures the household
willingness to pay to reach consumption level x
starting from utility level U. Using our empirical estimates we now analyze the complementarity relationships between fish categories. In
particular, we derive the consumer willingness
to pay and value of complementarity given in
equations (7 )–(9 ).
As shown in proposition 1, the sign of the
off-diagonal terms of the Luenberger matrix
Consumer Value of Complementarity
499
Table
6. Value
of
Complementarity
Evaluated at Sample Means
(x1 , x2 )
(x1 , x3 )
(x1 , x4 )
(x1 , x5 )
(x2 , x3 )
(x2 , x4 )
(x2 , x5 )
(x3 , x4 )
(x3 , x5 )
(x4 , x5 )
SR
IR
LR
0
(0)
0
(0)
0
(0)
0
(0)
0.014
(0.011)
−0.035
(0.010)
−0.203
(0.001)
−0.037
(0.003)
−0.130
(0.003)
−0.077
(0.003)
−0.389
(0.010)
0.005
(0.003)
−0.094
(0.004)
0.056
(0.004)
−0.297
(0.035)
−0.229
(0.031)
0.314
(0.028)
−0.223
(0.012)
−0.209
(0.013)
0.091
(0.010)
−1.304
(0.037)
0.016
(0.009)
−0.251
(0.012)
0.135
(0.009)
−1.003
(0.100)
−0.549
(0.077)
1.428
(0.070)
−0.552
(0.032)
−0.200
(0.032)
0.394
(0.029)
Note: Bootstrapped standard errors in parentheses are below the
corresponding estimates. Annual per capita value in euros.
is sufficient to establish the sign of the value
that consumers assign to complementarity.21
As noted above, the nature of complementarity varies with the time horizon: SR, IR,
and LR. Table 6 presents the estimate of the
value of complementarity from equation (9 )
for each pair of fish. As expected the value
varies across the SR, IR, and LR scenarios. In a
way consistent with our SR compensated flexibility estimates (reported in table 4), table 6
shows that the SR value of complementarity
is negative for all pairs. This means that all
goods are substitutes in the short run. However, this changes with the planning horizon.
As shown in the third and fourth column of
table 6, in the IR and LR scenarios, small
pelagic-cephalopods (categories 1 and 3), small
pelagic-crustaceans (categories 1 and 5), other
fish-crustaceans (categories 2 and 5), as well
as mussels-crustaceans (categories 4 and 5) are
complements. In general, all the relationships
strengthen in the LR. The corresponding values of complementarity (expressed in euros
per capita) are 0.005, 0.056, 0.314, and 0.091 in
the intermediate run, and 0.016, 0.135, 1.428,
and 0.394 in the long run. This shows that their
complementarity value becomes stronger in
21
The Luenberger matrix is also found to be negative semidefinite (in the short run) when evaluated at x = 0. This is relevant
for our simulations on the value of complementarity.
500
May 2009
Amer. J. Agr. Econ.
Table 7. Relative Value of Complementarity
Evaluated at Sample Means
(x1 , x2 )
(x1 , x3 )
(x1 , x4 )
(x1 , x5 )
(x2 , x3 )
(x2 , x4 )
(x2 , x5 )
(x3 , x4 )
(x3 , x5 )
(x4 , x5 )
SR
IR
LR
0
(0)
0
(0)
0
(0)
0
(0)
0.0008
(0.0006)
−0.002
(0.0005)
−0.011
(0.0005)
−0.004
(0.0003)
−0.018
(0.0005)
−0.008
(0.0003)
−0.015
(0.0004)
0.0008
(0.0004)
−0.008
(0.0004)
0.006
(0.0004)
−0.010
(0.001)
−0.007
(0.0009)
0.010
(0.0009)
−0.016
(0.0009)
−0.018
(0.001)
0.005
(0.0006)
−0.065
(0.002)
0.009
(0.005)
−0.032
(0.002)
0.018
(0.001)
−0.050
(0.005)
−0.022
(0.003)
0.061
(0.003)
−0.065
(0.004)
−0.025
(0.004)
0.030
(0.002)
Note: Bootstrapped standard errors in parentheses are below the
corresponding estimates.
the long run (compared to the intermediate
run). This illustrates a situation where complementarity relationships that are absent in the
short run can develop under longer planning
horizons.
Finally, table 7 presents relative value of
complementarity, that is, the values reported in
table 6 as a proportion of the value of each pairs
of goods, obtained from equation (7 ) using the
benefit function evaluated with and without
those goods. One noticeable result is that the
values are decreasing for substitutes, and increasing for complements, as the time horizon
becomes longer. This means that substitution
and complementarity relationships become
stronger with a longer planning horizon. In addition, the time horizon effects are particularly
strong moving from the intermediate run to
the long run: often, the relative values more
than double (in absolute value) from such a
move. The pairs other fish-crustaceans (categories 2 and 5) and mussels-crustaceans (categories 4 and 5) move from SR substitutes (with
small negative values) to complements in the
intermediate run and long run, with LR relative values of complementarity equal to 6.10%
and 3%, respectively.
Concluding Remarks
This article has addressed the question of how
consumers value complementarity. This was
done by specifying and estimating consumer
benefits from fish consumption. The analysis is
illustrated by an application to the Italian fishery using a panel data for twelve regions over
thirty years.
Our proposed approach is based on the benefit function developed by Luenberger (1992).
The benefit function provides a powerful way
of conducting welfare analysis, and provides
a convenient framework to investigate the
value of complementarity as perceived by consumers. The model presented in this article
is proposed as an alternative to the inverse
AIDS models that have been commonly used
in previous literature. Our analysis of fish
consumption in Italy illustrates the empirical
tractability of the approach, with special attention given to dynamics. Our results show that,
while short-run fish demand is characterized
by substitution relationships, complementarity
does develop in the intermediate run and in the
long run.
While our application focused on fish consumption, it can be extended in a number of
directions. First, our approach can be used by
policy makers and economic analysts to simulate the effects of policy instruments (such
as quotas) used in the management of natural resources. In this context, our findings on
longer-run complementarity can help evaluate
optimal harvest policies. For example, when
the fish stock is negatively affected by pollution, the value of complementarity may be
considered in a social welfare problem so to
provide incentives for controlling pollution. Finally, showing the empirical tractability of our
approach should prove useful in general applied welfare analysis, with a focus on the estimation of consumer benefits and their use in
welfare evaluation.
[Received March 2007;
accepted May 2008.]
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Appendix A
Table A1. Parameter Estimates and Standard Errors (SE) for Model B
Parameter
2
3
4
5
2,2
2,3
2,4
2,5
3,3
3,4
3,5
4,4
4,5
Coeff.
∗∗∗
1.153
1.112∗∗∗
1.106∗∗∗
3.070∗∗∗
−0.075
0.006
−0.004
−0.115
−0.608∗∗
−0.024
−0.381
−0.089∗∗∗
−0.066
SE
0.391
0.411
0.334
0.876
0.062
0.104
0.029
0.127
0.279
0.037
0.235
0.033
0.066
Parameter
5,5
2
3
4
5
21
22
23
24
25
31
32
33
Coeff.
∗
−0.799
0.007
0.004
0.003
−0.009
0.967∗∗
0.247
0.430
0.236
0.240
1.254∗∗∗
0.409
0.618
SE
Parameter
Coeff.
SE
0.428
0.010
0.014
0.006
0.025
0.389
0.279
0.324
0.288
0.313
0.466
0.293
0.397
34
35
41
42
43
44
45
51
52
53
54
55
0.539
0.349
0.585∗∗
0.121
0.557∗∗
0.210
−0.001
0.908
0.480
1.969∗∗
0.072
1.078
0.345
0.378
0.245
0.262
0.277
0.232
0.220
0.842
0.531
0.777
0.603
0.687
502
May 2009
Amer. J. Agr. Econ.
Table A1. Continued
Parameter
Coeff.
∗∗∗
2L
3L
4L
5L
2,1L
2,2L
2,3L
2,4L
2,5L
3,1L
3,2L
0.701
0.708∗∗∗
0.624∗∗∗
0.586∗∗∗
−0.050
0.121∗∗
−0.257
0.003
0.306∗
0.003
0.119
SE
Parameter
Coeff.
SE
Parameter
Coeff.
SE
0.060
0.059
0.085
0.068
0.031
0.055
0.192
0.030
0.170
0.041
0.099
3,3L
3,4L
3,5L
4,1L
4,2L
4,3L
4,4L
4,5L
5,1L
5,2L
5,3L
−0.169
−0.0005
0.617∗∗
−0.019
−0.021
−0.087
0.022
0.266∗∗
0.050
0.125
−0.358
0.225
0.024
0.313
0.020
0.046
0.157
0.027
0.116
0.073
0.139
0.596
5,4L
5,5L
2
3
4
5
2
3
4
5
−0.045
0.537
−0.038
−0.086∗∗∗
−0.031
−0.041
−0.0007
0.003∗∗∗
0.0007
−0.00002
0.052
0.435
0.029
0.031
0.027
0.068
0.0008
0.0009
0.0008
0.002
Note: Single asterisk (∗ ), double asterisks (∗∗ ), and triple asterisks (∗∗∗ ) denote significance at 10%, 5%,and 1% levels.
Appendix B
xk /p∗∗
j , (B1) gives
Price Flexibilities
(B2)
Differentiating equation (5) with respect to x
yields
(B1)
∂ p ∗∗
j (x)/∂ x k
≡ ∂ p Lj (x,U )/∂ x k + [∂ p cj (x, U )/∂U ]
× [∂u(x)/∂ x k ]
j = 1, . . . , K, k = 1, . . . , K, evaluated at U = u(x),
where p∗∗ (x) ≡ p∗ (x)/[(x, g)]. The term ∂u(x)/∂xk in
(B1) can be obtained by applying the implicit function theorem to b(x, u(x)) = 0, yielding ∂u(x)/∂xk =
– [∂b(x, U)/∂xk ]/[∂b(x, U)/∂U], evaluated at where
U = u(x). Using equation (2), and multiplying by
f jk∗∗ = f jkL − ∂ p Lj (x,U )/∂U / p ∗∗
j
× [ pk∗∗ (x, U )x k ] / [∂b(x, U )/∂U ]
∗∗
evaluated at U = u(x), where f ∗∗
jk ≡ [∂ln(pj (x))/
∂ln(xk )] is the uncompensated (Marshallian) price
flexibility and f Ljk ≡ [∂ln(pLj (x, U))/∂ln(xk )] in the
compensated (Luenberger) price flexibility. Equation (B2) gives the relationship between the uncom∗∗
pensated price flexibility f ∗∗
jk ≡ [∂ln(pj (x))/∂ln(xk )]
and the compensated price flexibility f Ljk ≡
[∂ln(pLj (x, U))/∂ln(xk )] capturing the effect of the
kth quantity on the jth price. It shows that f ∗∗
jk and
f Ljk differ from each other by the income/utility effect
∗∗
[∂pLj (x, U)/∂U]/p∗∗
j · [pk (x, U) xk ]/[∂b(x, U)/∂U].