specific equations avoid the generalities of
broad-based indexes.
This article first presents the conditions for
simple linear aggregation of demand equations,
then tests for evidence of simultaneous equation
bias (problem [1]) and aggregation bias (problem
[2]), and finally presents a multiple-equation
estimation procedure to reduce the effects of
both types of bias. The market-specific multiple-
equation estimation procedure presented herein
provides estimates of specific exchange rate
effects on U.S. soybean exports to individual
major markets, thus addressing problem [3].
CONDITIONS FOR AGGREGATION
There are several ways of demonstrating the
conditions for aggregation. Deaton and
Muellbauer (pp. 148-53) demonstrate the condi-
tions for aggregating individual consumer de-
mand functions whose arguments are prices
and total expenditures. They point out that
linearly aggregated demand functions are
subject to aggregation bias if aggregate demand
is a function of the distribution of expenditures
across consumers as well as the level of aggre-
gate expenditures. We provide a simple demon-
stration of sufficient conditions required for ag-
gregating any two demand functions whose ar-
guments are prices and any other variables.
These demand functions may represent the
import demand of two different countries as
well as being input or consumer demand
functions.
Suppose the demand functions for two coun-
tries are linear in price and another variable
such as income. The demand for country i is
(1) di = biP + aiYi,
where di is the quantity purchased in the ith
country as a function of world price (P) and that
country’s income (Yi). The demand for country
j is
(2) dj = bjP + ajYj,
where d. is the quantity purchased in the jth
country as a function of price (P) and the jth
country’s income (Yj). Aggregate demand (D),
expressed as
(3) D = BP + AY,
is a function of price (P) and aggregate income
(Y), and, by definition, equals the sum of the
individual country demand functions,
(4)D = di + dj.
Substituting terms from equations (1), (2), and
(3) into equation (4) yields
(5) BP + AY = (biP + aiYi) + (bjP + ajYj).
Assume that the price effects (BP) in the aggre-
gate demand function, equation (3), equal the
sum of the price effects in the individual de-
mand functions,
(6) BP = biP + bjP.
Subtracting equation (6) from (5) shows that
the income effects in D equal the sum of the
income effects in di and dj,
(7) AY = aiYi + ajY..
Dividing both sides of equation (6) by P simply
shows that the sum of the parameters on P
equals the aggregate parameter on price, or
that
(8) B = bi + bj.
By definition, aggregate income equals the sum
of income in the two countries,
(9) Y = Y + Yj.
Substituting terms from equation (9) into (7)
gives
(10) AY = A(Yi + Yj) = AYi + AYj = aiYi + ajYj,
which is true when a. = aj. Furthermore, Deaton
and Muellbauer state that for exact linear ag-
gregation, the parameters on the Y term must
be equal in each equation (p. 150). Zellner af-
firms (without the simplifying assumption of
equation (6)) that there will be no aggregation
bias involved in simple linear aggregation if the
parameters on income are equal across individ-
ual demand functions. However, this argument
applies not only to income aggregated across
individuals but to any variable summed across
equations or individual countries.
Applying these conditions for linear aggre-
gation of demand functions to linear aggrega-
tion of import demand functions, we derive the
null hypothesis to test for evidence of aggrega-
tion bias: parameters on all the linearly aggre-
gated exogenous variables are the same across
market-specific import demand equations.
130