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UNIVERSITY OF
FLORIDA
Institute of Food ad Agricultural Sciences
THE PRODUCTION THEORY APPROACH TO IMPORT
DEMAND ANALYSIS: A COMPARISON OF THE ROTTERDAM
MODEL AND THE DIFFERENTIAL PRODUCTION APPROACH
By
Andrew A. Washington and Richard L. Kilmer
INTERNATIONAL AGRICULTURAL TRADE
AND POLICY CENTER
MISSION AND SCOPE: The International Agricultural Trade and Policy Center (IATPC) was
established in 1990 in the Food and Resource Economics Department (FRED) of the Institute of
Food and Agricultural Sciences (IFAS) at the University of Florida. Its mission is to provide
information, education, and research directed to immediate and longterm enhancement and
sustainability of international trade and natural resource use. Its scope includes not only trade
and related policy issues, but also agricultural, rural, resource, environmental, food, state,
national and international policies, regulations, and issues that influence trade and development.
OBJECTIVES:
The Center's objectives are to:
Serve as a universitywide focal point and resource base for research on international
agricultural trade and trade policy issues
Facilitate dissemination of agricultural trade related research results and publications
Encourage interaction between researchers, business and industry groups, state and
federal agencies, and policymakers in the examination and discussion of agricultural
trade policy questions
Provide support to initiatives that enable a better understanding of trade and policy
issues that impact the competitiveness of Florida and southeastern agriculture
specialty crops and livestock in the U.S. and international markets
The Production Theory Approach to Import Demand Analysis: A Comparison of the
Rotterdam Model and the Differential Production Approach
by
Andrew A. Washington
(Assistant Professor, Department of Economics, Southern University)
and
Richard L. Kilmer
(Professor, Food and Resource Economics Department, University of Florida and a member of
the International Agricultural Trade and Policy Center (IATPC) at the University of Florida)
Citation
This article was published in the Journal ofAgricultural andApplied Economics,
34(3)(December 2002), Washington, Andrew A. and Richard L. Kilmer, "The Production
Theory Approach to Import Demand Analysis: A Comparison of the Rotterdam Model and the
Differential Production Approach", pages 431443, Copyright 2002, and is posted with the
permission of the Southern Agricultural Economics Association,
http://www.agecon.uga.edu/jaae/. Copies of the article can be downloaded and printed only
for the reader's personal research and study.
Abstract
Results indicate that when comparing the unconditional derived demand elasticities to the
unconditional consumer demand elasticities, significant differences emerge due to the
differences in the first stage estimation procedure between the differential production approach
and the Rotterdam model. In comparing the consumer demand price/cross price elasticities to the
derived demand price/cross price elasticities it is clear that use of the Rotterdam model when a
production approach should be used can lead to overestimation, underestimation and incorrect
signs in deriving unconditional price effects.
Key Words: Dairy, demand, imports, international, production, Rotterdam, trade
JEL Classifications: D12, D24, F10, F14, Q17
The Production Theory Approach to Import Demand Analysis: A Comparison of the
Rotterdam Model and the Differential Production Approach
The Rotterdam model application to import demand has been accomplished by a number of
studies (Lee, Seale, and Jierwiriyapant; Seale, Sparks, and Buxton; Zhang, Fletcher, and Carley).
In past studies, imports are considered to be final goods that enter directly into the consumer's
utility function and the resulting demand equations for imports are derived from utility
maximization theory. However, given the nature of international trade, where traded goods are
either used in other production processes or go through a number of domestic channels before
reaching the consumer, it is more appropriate to view imported goods as intermediate products
than as final consumption goods even if no transformation takes place (Davis and Jensen). The
primary objective of this paper is to compare and contrast the use of the differential production
approach with the Rotterdam model. Both approaches are applied to Japan's derived demand for
imported whey differentiated by source country of production. Unconditional elasticities from
both approaches are then compared.
The application of production theory to international trade is by no means a new concept.
Past research that used a production theory approach to international trade include Burgess
(1974a) and (1974b), Kohli (1978) and (1991), Diewert and Morrison, and Truett and Truett.
Each of these studies acknowledged that most goods entering into international trade require
further processing before final demand delivery. They further acknowledged that even when a
traded product is not physically altered, activities such as handling, insurance, transportation,
storing, repackaging, and retailing still occur. This results in a significant amount of domestic
value added when the final product reaches the consumer. Therefore it is more appropriate to
view imported products as inputs rather than as final goods even if goods are not transformed.
5
Davis and Jensen (pp. 41012) meticulously discuss the advantages of the production
theory approach over the utility approach to import demand estimation. Their first point is that
most imported agricultural commodities are inputs and not final goods. Second, specifying the
first stage aggregates is more intuitive when using the production theory approach. Third, it is
easier and more intuitive to estimate unconditional elasticities using production theory. Their last
point is that the estimated parameters using production theory will be structural parameters.1
Kohli (1991) notes that viewing imports as intermediate goods not only has its merits in
correctness, but it also leads to substantial simplifications theoretically. One simplification is that
the demand for imports can be derived from production theory and there is no need to model
final demand. Second this approach allows for the avoidance of the difficulties that arise when
we aggregate over individual consumers. To expound on this point, data is typically reported in
aggregate terms. Therefore, if we are estimating demand, we are estimating aggregate demand,
and if we are estimating derived demand it is aggregate or industry derived demand. The
differences between aggregate demand and aggregate derived demand is that one is an
aggregation over consumers and the latter is an aggregation over firms. When we consider
optimizing behavior by both consumers and firms, do the properties derived from consumer and
producermaximizing behavior hold in the aggregate? MasColell, Whinston, and Green indicate
that when consumer preferences and wealth effects are identical across consumers, the aggregate
demand function satisfies all of the properties of an individual demand function.2 However, if
there is the slightest difference in preferences and if these differences are independent across
consumers (as one would expect), the property of symmetry, which is a common property tested
in most empirical demand studies, will almost certainly not hold.3
When we aggregate across firms, there are no such conditions required for the properties
of optimal firm behavior to hold in aggregation. This is because the aggregate profit obtained
when each production unit maximizing profit separately, taking prices as given, is the same as
that which would be obtained if they were to coordinate their actions in a joint profit maximizing
decision (MasColell, Whinston, and Green).4 This result implies that the profit maximizing
output arrived at if all firms coordinated their actions is the same as the sum of the individual
output of each profitmaximizing firm. It further implies that the total cost of production for the
coordinated output is the same as the sum of total cost for each individual firm if firms are price
takers in the input market (MasColell, Whinston, and Green). Therefore, if we estimate input
demand functions and output supply functions using aggregate data, the properties of the demand
and supply functions for each individual firm will theoretically hold in aggregation.5
Overview of Theory
The differential approach to the theory of the firm is comparable to the differential
approach to consumer theory proposed by Barten (1964) and Theil (1965). The empirical
application of the differential approach to consumer demand resulted in the Rotterdam model,
which has been used extensively in demand studies and to a lesser extent in import demand
studies. The majority of import demand studies that used the Rotterdam model assumed that
imported goods entered directly into the consumer's utility function and strong assumptions were
made about how consumers view imported and domestic goods and how they grouped
commodities. Furthermore, it was often assumed that these commodity groups were to some
degree independent in terms of the consumer's utility function (For example, see Lee, Seale, and
Jierwiriyapant; Seale, Sparks, and Buxton; and Zhang, Fletcher, and Carley). In these studies, the
intermediate nature of imports was not considered.
The Rotterdam Model
The estimation of import demand using the Rotterdam model is accomplished in two
stages. First consumers allocate total expenditures between product groups (first stage) and
second, consumers allocate total group expenditures among goods within the product group
(second stage).6 It is also assumed that product groups are blockwise dependent, that is the
utility interaction among goods are a matter of the groups and not the individual goods.
The first stage of the consumer budgeting process results in a system of composite
demand equations where each equation is expressed as
G
(1) Wgd(logQ,)= ,d(logQ )+ f ighd(logPh),
h=1
where d(logQg) and d(logPh)are the group Divisia volume and Frisch price indexes
respectively; Wg, g, and ng are the budget share, marginal share, and absolute price
coefficient respectively; d(logQ ) is the percentage change in real income (Theil, 1980, p. 101).
Equation (1) states that the composite demand for the product group depends on real income and
the Frisch price indexes for each group. The size of the system represented by equation 1 is equal
to the total number of groups specified in the consumer's utility function. When estimating
import demand, the total number of equations in the system can be as large as the total number of
goods imported which makes estimating equation (1) problematic.
The demand for individual goods within a group conditional on total group expenditures
(second stage) results in a system of demand equations where each equation is expressed as
n
(2) wd(log q,) = d(log Qg) + ,, d(logp,),
j=1
where w, represents the share of group expenditures allocated to good i and 0, is the conditional
marginal share; q, and p, are the quantities and prices, respectively; z, 's are the conditional
Slutsky price coefficients; and n is the number of goods within the product group (Theil, p. 103).
Dividing equation (1) by Wg and substituting into equation (2) yields the unconditional
demand equation
0 GH n
(3) w d(log q,)=0, [ d(log Q)+ d(logPh)]+ YZ d(logp,).
g h=1 g j=1
From equation (3) we get the unconditional income elasticity
d(log q,) O 0,g
(4) r =
Sd(log Q) w, Wg
which is the product of the conditional expenditure elasticity 0, /w, and the expenditure
elasticity for the group O, /W We also get the unconditional price elasticity
d(logq,) 0, gh n, j
(5) 7, +
d(logp,) w, W, w,
where Hgh /Wg is the ownprice elasticity for the group and ;i,,rw is the conditional price
elasticity for the ith good.
The Differential Production Approach
Using the methodology of Laitinen and Theil, Laitinen, and Theil (1980), the differential
production model will also be used to estimate the import demand. The differential production
model is derived from the differential approach to the theory of the firm where firms maximize
profit in a twostage procedure. In the first stage, firms determine the profit maximizing level of
output to produce and in the second stage firms minimize the cost of producing the profit
maximizing level of output. According to Laitinen and Theil, and Davis and Jensen, this
procedure is consistent with a onestep or direct profit maximization procedure. In the first stage
9
the output supply equation is obtained and the conditional factor demand system is obtained in
the second stage. Using the results of both stages, a system of unconditional derived demand
equations is derived.
In the first stage a competitive firm seeks to identify the profitmaximizing level of
output by equating marginal cost with marginal revenue. This procedure yields the differential
output supply equation
(6) d(logQ*)= od(logp*)+ ,Z d(logw,),
j=1
where Q*, p* and w, represent the output, output price and the price of inputs respectively; (p and
r are the price elasticity of supply and the elasticity of supply with respect to input prices
respectively. Nis the total number of inputs used in production.
In the second stage, the differential factor demand model is derived, which will be used
to estimate the system of source specific derived demand equations. This model is specified as
(7) f/ d(logx,)= 0d(log X)+ Y ; d(log w),
j=1
where f, is the factor share of imported good x from source country i in total input cost; x, and w,
represent the quantity and price of inputs which include the price of each imported good from
source country i; d(logX) = ft,d(logx,)whered(logX) is the Divisia volume input
1=1
index; 0 is the mean share of the ith input in the marginal cost of the firm; r, is the conditional
price coefficient between the ith andjth importing sources or inputs; n is the number of inputs in
the system, n e N.7
The differential factor demand model requires that the following parameter restrictions be
met in order for the model to conform to theoretical considerations: ;V = 0 (homogeneity),
and n K, = ,, (symmetry). The second stage procedure results in the conditional own price/cross
price elasticity
(8) d(log x,)
(8) E
S d(logw,) f
and the conditional Divisia volume input elasticity,
d(log x, ) 0
(9) ex 
d(log X) f,
Using the relationship between the Divisia volume input index and output,
d(logX) = d(logQ*)8, equation (6) can be substituted into equation (7) to yield the
unconditional derived demand system
n n
(10) f d(logx,)= 0/7[pd(logp*)+ + d(logw,)]+ Y ,; d(logw,).
J=1 7=1
Dividing through equation (10) by f and using equations (8) and (9) we get the unconditional
derived demand elasticities. The unconditional elasticity of input demand with respect to output
price is
d(log x,)
(11) Exp = & xX Y P,
S d(logp*)
and the unconditional own price/cross price elasticity of input demand is
d(log x, )
(12) E = 7 2 + Exw +
X d(log w,)
Lastly we get the unconditional elasticity of derived demand with respect to the price of an input
contained in N but not in n
d(log x, )
(13) = = =YS7 .
d) (log w, )
Inputs contained in N but not in n include labor and other inputs that are not part of the imported
whey group.
The second stage procedures in the consumer and production approaches yield
empirically identical demand systems, equation (2) and equation (7), resulting in identical
conditional elasticities. Davis and Jensen note that this similarity explains the empirical success
of consumer based conditional demand systems even though they may be conceptually flawed.
However given the differences in the first stage, equation (1) and equation (6), unconditional
elasticities differ between the two approaches. Also, the production approach results in the
unconditional elasticity of derived demand with respect to output price whereas the Rotterdam
model results in the unconditional income elasticity. This suggests that the use of the Rotterdam
model, when a production approach is more appropriate, not only leads to biased unconditional
own price/cross price elasticity estimates but also leads to the reporting of unconditional income
elasticities when the concern should the unconditional elasticity of derived demand with respect to
output price.
Application to the Derived Demand for Imported Whey in Japan
This study assesses the competitiveness of whey imports into Japan from the U.S. compared to
whey imported from other countries such as the EU, Australia, and New Zealand. Following
Armington, similar imported dairy products such as EU whey and US whey are both individual
goods that are part of the product group whey, but different based on their source country of
production. There are a number of reasons why similar products are viewed as different based on
12
their source country of origin. Dairy products from different sources may actually be physically
different. Physical differences include quality, protein, fat content, and taste. There may also be
perceived differences, such as a country's reputation for a quality product, trade history,
reliability and consistency, and political issues tied to trade (Zhou and Novakovic). The crux of
this assumption is that within an importing country, a particular dairy product imported from a
given source is considered a substitute for that same product from another source. However,
because of the physical and perceived differences attributed to the product due to its origin, these
products are imperfect substitutes.
In this paper it is assumed that dairy products are imported through firms that exclusively
import. Although, there are firms within Japan that import whey as well as transform whey into
other products, it is assume that there is a separate entity within the firm that deals primarily with
the procurement of imported dairy products. Also, dairy imports through this type of firm make
up a smaller percentage of imports in Japan. In addition to providing imported products to other
firms, these firms also provide the services that are associated with importing. These services
include, search and acquisition, transportation, logistics, and storing. A major characteristic of
this firm type is that it deals primarily in imported goods. This suggests that the procurement of
imported goods by firms is a unique process separate from the procurement of similar products
produced domestically. Even if the firm is a subsidiary or branch of a larger firm that purchases
domestic and foreign produced inputs, it is not unlikely that the subsidiary that is responsible for
imported inputs deals primarily in this activity. This is because the acquisition of foreign
produced goods is more involved than purchasing domestically produced goods.
If we assume a production function for these firms, then the output of these firms is the imported
goods that are sold to other firms and the inputs are the imported goods from the various
exporting countries. If we minimize cost subject to this production function, the system of input
demand equations resulting from the optimization procedure will be a system of import demand
equations. If we assume product differentiation across source countries, then each import
demand equation represents the demand for a product from a particular source.
In the firststage, the importing firm seeks to maximize profit by equating marginal cost
with marginal revenue. This procedure yields the differential output supply equation (expressed
in finite log changes)
N
(14) AQ = pAP + _,jAwjr + c, ,
j=1
where AQt = log(Qt / Qt_ ), Apt = log(p, / Pt) and Aw, = log(w,, /w, ), where q, p and w,'s
represent the output, output price and input prices; (p and r are the parameters to be estimated
which are also the ownprice elasticity of supply and the elasticity of supply with respect to input
prices respectively; c, is the disturbance term. Q* represents Japan's total imports of whey that
is to be supplied, p is the price at which firms in Japan sell whey, and the w, 's are the prices paid
for whey imports from each of the exporting countries, the price of labor (wages), and the price
of other inputs used. Nis the total number of inputs used in production.
In the second stage, the differential factor demand model is derived, which is used to
estimate the system of derived demand equations where each equation is the derived demand for
imported whey from a particular source. This model is specified as follows (expressed in finite
log changes)
(15) f, Ax, = AX,+ ZrAwj, +E,
j=1
where f, = (f, + f _)/2; Ax,, = log(x,, / x, ) and Aw, = log(w,, / w,_ ), where x, and w,
represent the quantity and price of imported whey from source country i;
n
AX, = f Ax, where AX, is the finite version Divisia volume input index; 6 and z*c
1=1
parameters to be estimated; n is the number of inputs in the system; E't is the disturbance term.
In addition to the imports from each individual source country, labor and other inputs are
used in the production process. The labor demand and demand for other inputs are expressed in
general terms as
(16) Labor = f (output, wages, input price index)
(17) Other Inputs =f (output, wages, input price index).
Equations (16) and (17) represent the system of derived demand equations for labor and other
inputs where these inputs are a function of the total amount to be supplied, wages, and an input
price index which represents the price of all inputs except labor and whey imports. Here we
assume that labor and other inputs are independent of the source specific whey imports. This is
to say that although labor and other inputs affect the total to be imported, these inputs do not
directly affect the amount imported from an individual source country.
Empirical Results
Using United Nations Commodity Trade Statistics, the derived demand for imported whey into
Japan was estimated. The exporting countries considered were the United States, European
Union, Oceania (aggregation of Australia and New Zealand), and rest of the world (ROW),
which is an aggregation of all other countries. The time period for the data set was 1976 to 1998.
During this period, the United States on average accounted for 35% of all whey exports to Japan,
while Oceania, EU, and ROW accounted for 17, 19 and 27%, respectively. All values and
quantities where reported through Japanese customs. Values were on a cost, insurance, and
freight basis. According to FAO statistics, Japan primarily imports dry whey, which is used as
both cattle feed and an ingredient in infant formula. In the last decade, imports of dry whey have
accounted for 100% of all whey imports.
First stage estimation required the domestic wholesale price of whey in Japan. This price
series was not available. However, a proxy was used which was the per unit wholesale price of
all milk powders which is reported by the Statistic Bureau Management and Coordination
Agency for the Government of Japan. To account for the labor requirement in the importation of
whey, an index of Japan's hourly wages was included in the estimation (U.S. Department of
Labor). To account for other inputs, an industry input price index was also included
(Economagic.com).
SecondStage Estimation and Conditional Elasticities
Table 1 presents the loglikelihood values, the likelihood ratio (LR) statistics, and the critical
value for the LR test for autocorrelation. A Likelihood ratio test indicated that firstorder
autocorrelation could not be rejected at the .05 significance level; thus, all results presented have
the AR(1) error structure imposed.9
[Place Table 1 approximately here]
LR tests were also used to test if the data satisfied the economic properties of
homogeneity and symmetry. The results of these tests are summarized in Table 2. LR tests
indicate that the property of homogeneity could be rejected. However, Laitinen's test for
homogeneity, which is a more precise test, indicated that homogeneity could not be rejected.
Given the homogeneity constraint, symmetry could not be rejected. The property of negative
semidefiniteness was verified by inspection of the eigen values of the price coefficient matrix.
This property is validated when all of the eigen values are less than or equal to zero. All eigen
values were nonpositive in the Japanwhey system.
[Place Table 2 approximately here]
Table 3 presents the conditional parameter estimates for the derived demand and
consumer demand for imports of whey into Japan. With the exception of the ROW, all ownprice
coefficients are negative and all are significant by at least the .05 significance level. The
condition marginal factor share estimates indicate a positive relationship between the Divisia
volume index of all imports and the imports from the individual sources except for the ROW.10
In the consumer demand (Rotterdam) model the conditional marginal factor shares are
interpreted as the conditional marginal expenditure share. Crossprice parameter estimates
indicate that the U.S and Oceania whey imports, Oceania and EU imports, and EU and ROW
imports are substitutes.
[Place Table 3 approximately here]
Table 4 presents the conditional elasticties for the derived demand and consumer demand
of imported whey." The Divisia index elasticities for imports of whey into Japan are .914, 2.295,
2.336 and .500 for the U.S., Oceania, EU and the ROW, respectively. These indicate that as the
Divisia volume index increases, imports from the US will increase proportionately while imports
from Oceania and the EU will increase by more than twice as much. In the consumer demand
model, these are interpreted as conditional expenditure elasticities. The ownprice elasticities are
1.031, 2.930, 1.574, and .296 for the U.S., Oceania, EU, ROW, respectively. With the
exception of the ROW, all are significant at the .10 significance level. Conditional crossprice
elasticities of derived demand for whey in Japan indicate significant substitutional relationships
between whey imports from the exporting sources. The U.S./Oceania crossprice elasticity is
1.003, while the Oceania/U.S. elasticity is 2.106, reflecting the higher value placed on U.S.
whey. The Oceania/EU and the EU/Oceania elasticities are .441 and .401, respectively,
indicating fairly equal substitutability between the two sources. EU whey imports are the only
imports that were substitutes for whey from the ROW.
[Place Table 4 approximately here]
FirstStage Estimation and Unconditional Elasticities
Firststage estimation required the estimation of equation (14), which is the output supply
equation. Results are presented in Table 5. The output price parameter estimate (1.2963) is
positive as expected and significant at the .01 significance level. This estimate is also the price
elasticity of supply, which indicates that the supply of whey in Japan is price elastic. Parameter
estimates for all import prices are insignificant. The parameter estimate for the price of labor and
the price of other inputs (1.4888 and 3.3351, respectively) are negative and significant
indicating that wages and other input prices are inversely related to the output supplied, which is
to be expected. These are also the elasticity of output supply with respect to the price labor and
with respect to the input price index. These indicate that the supply of imported whey in Japan is
relatively sensitive to wages and other input prices. Firststage estimation in the differential
production model is possible and correct estimates could be used to derive unconditional derived
demand elasticities.
[Place Table 5 approximately here]
Unconditional elasticities for the Rotterdam model and the unconditional derived demand
elasticities are presented in Tables 6 and 7, respectively. To derive the unconditional income
elasticities for the consumer demand (Rotterdam) model (equation (4)), the income elasticity for
the product group whey was estimated to be one.12 For the unconditional own price/cross price
elasticities (equation (5)), it is assumed that the price elasticity of the demand for the product
group is .40 (Zhu, Cox, and Chavas). Unconditional derived demand elasticities were derived
using equations (11), (12), and (13).
[Place Table 6 approximately here]
[Place Table 7 approximately here]
In comparing the unconditional Rotterdam elasticity estimates in Table 6 to the
unconditional derived demand elasticities in Table 7, the biasness due to the inappropriate
application of consumer theory to import demand analysis becomes clear. First, the elasticity of
derived demand with respect to output prices, the elasticity of derived demand with respect
wages, and the elasticity of derived demand with respect to other input prices would not be
considered if the consumer demand model were applied. These derived demand elasticities
suggest that the derived demand for whey is highly responsive to these factors.
In addition to not reporting some of the derived demand elasticities, the Rotterdam model
leads to substantial differences in the unconditional own price/cross price elasticities. In the case
of the own price elasticities, the Oceania and EU elasticities derived using the Rotterdam model
are substantially larger in absolute terms than the derived demand elasticities. In the case of the
own price elasticity of demand for Oceania whey, the Rotterdam model overstates the ownprice
effect by 1.6 percentage points.
Unconditional cross price elasticities differ between the approaches as well. Of the 12
unconditional crossprice elasticities, 11 are significant in the derived demand model while 8 are
significant when using the Rotterdam model. Five crossprice elasticities actually change signs
(U.S./EU, Oceania/ROW, EU/Oceania, EU/ROW, and EU/U.S). The largest difference occurred
with EU/Oceania elasticity, which was estimated to be .534 in the Rotterdam model and 1.114
in the derived demand model. Using the Rotterdam elasticities, one would assess that EU whey
and Oceania whey were complements while the derived demand model indicates a substitutional
relationship.
Summary and Conclusions
The primary objective of this paper was to compare and contrast the use of the differential
production approach with the Rotterdam model. Given the intuitive and conceptual appeal of a
production approach to import demand analysis instead of a consumer approach (the Rotterdam
model), this article investigates the empirical differences due to approach selection. When one
compares the conditional derived demand to the conditional consumer demand system, there is
no empirical difference. However, when comparing the unconditional derived demand
elasticities to the unconditional consumer demand elasticities, significant differences emerge.
This is due to the differences in the first stage estimation procedure between the two approaches.
In fact, first stage estimation using the Rotterdam model is often not accomplished due to
difficulty in defining product groups that make up the first stage. However, in this study, it was
shown that first stage estimation is possible with the production approach and lead to
unconditional elasticity estimates. One empirical difference is that with the production approach,
the derived demand elasticity with respect to output price, wages, and other input prices are
derived. This is not the case with the Rotterdam model. In comparing the consumer demand own
price/cross price elasticities to the derived demand own price/cross price elasticities, it is clear
that use of the Rotterdam model when a production approach should be used can lead to
overestimation, underestimation, incorrect signs, and erroneous insignificance when deriving the
unconditional price effects.
Footnotes
1. For a more indepth discussion of the conceptual and theoretical advantages of the
production approach see Davis and Jensen.
2. The properties of a system of demand equations for a utility maximizing consumer are
adding up, homogeneity, and the symmetry and negative semidefiniteness of the matrix of
price effects.
3. The property of negative semidefiniteness holds in aggregation under less strict conditions.
If each individual demand function satisfies the uncompensated law of demand, then the
aggregate demand function satisfies the week axiom of revealed preference, which implies a
negative semidefinite price effect matrix.
4. Prices are assumed as given even with coordination.
5. The properties of the input demand function are the same as the properties of the consumer
demand function. The properties of the supply function are that the matrix of price effects is
symmetric and positive semidefinite. The authors assumed that firms are still price takers
even with coordination. Production technology can vary over firms.
6. Given a common assumption that imports and domestics goods are independent, there is an
additional stage before the two mentioned where total expenditures are allocated between
imports and domestic goods (Seale, Sparks, and Burton).
7. The derivation of equations (6) and (7) are found in Laitinen and Theil.
8. 7 is the elasticity of cost with respect to a proportionate output increase. According to
Laitinen (p. 113), 7 is also the ratio of revenue to cost. When calculating elasticities, the
average of the geometric mean of y for periods t and t1 is used, where yt = 1 is the
T
two period geometric mean and 7 = =ly is the average of yt across all observations.
9. The AR(1) process is the same for all equations in the system.
10. Homogeneity and symmetry are imposed on the parameters. AR(1) is also imposed.
11. All elasticities are evaluated at the mean.
12. The income elasticity for the group whey was estimated using the Workings Model (Theil
and Clements, p. 14). The income elasticity for the group whey was equal to one.
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Table 1. Likelihood Ratio Test Results for Autocorrelation in the Derived Demand and
Consumer Demand Models
Country/Product Model Log LR* P[2
likelihood
Value
JapanWhey AR(1) 55.125
NoAR(1) 48.729 12.7927 3.84(1)a
a The number of restrictions are in parentheses.
Table 2. Likelihood Ratio Test Results for Economic Constraints and Laitinen's Test For
Homogeneity in the Derived Demand and Consumer Demand Models
Country/Product Model Log LR* P[ 2)
likelihood
Value
Japan Whey Unrestricted 55.541
Homogeneity 51.179 8.726 7.81(3)a
Symmetry 48.998 4.362 7.81(3)
Laitinen's Test
W*b P[T2
Japan Whey Homogeneity 9.217 11.186
a The number of restrictions are in parentheses.
b W* is the Wald statistic for the homogeneity constraint.
c T2 is the Hotelling's T2 statistic.
Table 3. Conditional Derived Demand (Consumer Demand) Parameter Estimates for Japan
Imports of Whey
Marginal
Price Coefficients, z; and(;z ) Factor
Exporting U.S. Oceaniaa EU ROWb Shares,
nO and(O )
Country 0,and(
U.S. .3653*** .3556*** .1032 .0935 .3239**
(.1254)c (.0686) (.0739) (.0884) (.1729)
Oceania .4947*** .0744** .0647 .3874***
(.0973) (.0426) (.0836) (.0948)
EU .2926** .1150* .4341***
(.0628) (.0649) (.1166)
ROW .0862 .1454
(.1286) (.1228)
2
System R2 = .79
a Australia and New Zealand aggregation.
b ROW= rest of the world.
c Asymptotic standard errors are in parentheses.
*** Significance level= .01.
** Significance level = .05.
* Significance level = .10.
Table 4. Conditional Divisia and Price Elasticities of the Derived Demand and Consumer
Demand for Imported Whey
Elasticities
Exporting Divisia Conditional
Country Index OwnPrice Conditional CrossPrice
U.S. Oceaniaa EU ROWb
U.S. .914* 1.031*** 1.003*** .291 .264
(.488)c ( .354) (.193) (.209) (.249)
Oceania 2.295*** 2.930*** 2.106** .441* .383
(.562) ( .577) (.252) (.495)
(.406)
EU 2.336*** 1.574*** .555 .401* .618*
(.627) ( .338) (.397) (.229) (.349)
ROW .500 .296 .321 .222 .395*
(.422) (.442) (.303) (.287) (.222)
a Australia and New Zealand aggregation.
b ROW = rest of the world.
SAsymptotic standard errors are in parentheses.
*** Significance level= .01.
** Significance level= .05.
* Significance level .10.
Note: A Wald statistic was used which has a X2 distribution.
Table 5. Parameter Estimates for the Supply of Whey in Japan
Output Price
Input Price Coefficients, r Coefficient
U.S. Oceaniaa EU ROWb Wage Input Price
Index
.0322 .1638 .0001 .0575 .4888*** 3.3351** 1.2963***
(.0974)c (.1477) (.0670) (.1890) (.4143) (1.6403) (.3709)
R =.57
a Australia and New Zealand aggregation.
b ROW= rest of the world.
Asymptotic standard errors are in parentheses.
*** Significance level= .01.
** Significance level= .05.
* Significance level = .10.
Table 6. Unconditional Elasticities of the Consumer Demand Model (Rotterdam Model)
Elasticities
Exporting Income OwnPrice CrossPrice
Country U.S. Oceaniaa EU ROWb
U.S. .914* 1.396*** .638*** .074 .629***
(.488)c ( .195) (.195) (.195) (.195)
Oceania 2.295*** 3.848*** 1.188*** .477** .535**
(.562) ( .225) (.225) (.225) (.225)
EU 2.336*** 2.509*** .379 .534** .316
(.627) ( .251) (.251) (.251) (.251)
ROW .500 .096 .121 .422** .595***
(.422) ( .169) (.169) (.169) (.222)
a Australia and New Zealand aggregation.
b ROW = rest of the world.
SAsymptotic standard errors are in parentheses.
*** Significance level= .01.
** Significance level= .05.
* Significance level .10.
Note: A Wald statistic was used which has a X2 distribution.
Table 7. Unconditional Elasticities of Derived Demand Model
Elasticities
Exporting Output Wage Input price OwnPrice CrossPrice
Country Price index
U.S. Oceaniaa EU ROWb
U.S. 2.209* 2.537* 5.684* 1.085*** 1.283*** .291*** .165***
(1.179)c (1.355) (3.034) (.029) (.149) (.000) (.052)
Oceania 5.547*** 6.371*** 14.272*** 2.229*** 1.968*** .442*** .629***
(1.358) (1.660) ( 3.494) (.172) (.034) (.000) (.060)
EU 5.646*** 6.484*** 14.526*** 1.574*** .415*** 1.114*** .869***
(1.516) (1.742) (3.901) (.000) (.038) (.192) (.067)
ROW 1.208 1.387 3.109 .349*** .291*** .070 .395**
(1.021) (1.172) (2.626) (.045) (.025) (.129) (.000)
a Australia and New Zealand aggregation.
b ROW = rest of the world.
c Asymptotic standard errors are in parentheses.
*** Significance level= .01.
** Significance level= .05.
* Significance level .10.
Note: A Wald statistic was used which has a X2 distribution.
