2
The Preliminaries
This chapter discusses features of static trade theory that are
important components of the dynamic, multi-sector models de-
veloped in later chapters. Most of the notation used throughout
the text is introduced and the style used to state the model’s
primitives, and to define and characterize equilibrium is pre-
sented.
The first section reviews key concepts and results from indi-
vidual consumer and producer theory relevant to neoclassical
trade theory. The exposition is simplified by assuming produc-
tion technologies and preferences are differentiable and homo-
thetic functions. Throughout the text we draw heavily upon the
so called dual or indirect functions that characterize the con-
strained optimization behavior of individual agents. Readers in-
terested in a more rigorous exposition of consumer theory should
consult Cornes (1992) or Mas-Colell et al. (1995). A more rig-
orous treatment of producer theory can be found in Chambers
(1988), and Fare and Grosskop (2004).
Using the concepts developed in Sections 1 and 2 introduces
the Heckscher-Ohlin-Samuelson (HOS) model of a small open
and competitive economy. The basic features of equilibrium and
comparative statics as provided by the Stopler-Samuelson and
Rybczynski theorems are discussed. Woodland (1982) provides
an excellent characterization of this model. Section 3 considers,
brieﬂy, some further generalizations of the comparative statics
of the HOS model. Section 4 concludes this chapter and presents
a model of two traded goods, a home-good and three factors of
production. A dynamic version of this model follows in later
chapters.
T.L. Roe et al.,
Multisector Growth Models
, DOI 10.1007/978-0-387-77358-2
2,
9
c
Springer Science+Business Media, LLC 2010

10
2.
The Preliminaries
2.1
Microeconomic foundations
Throughout the text, the following notation denotes factor en-
dowments, factor rental rates and output prices. Sectors are
indexed by
j
∈ {
1
, ..., M
}
,
and denote the quantity of sector-
j
’s
output by the scalar
Y
j
.
Corresponding output prices are de-
noted
p
= (
p
1
, ..., p
M
)
∈
R
M
++
, with the scalar
p
j
representing
the per-unit price of sector-
j
output. We index factor endow-
ments by
i
∈ {
1
, ..., N
}
,
and denote the economy’s level of en-
dowment
i
by the scalar
V
i
and the vector of factor endowments
by
V
≡
(
V
1
, ..., V
N
)
∈
R
N
++
.
Corresponding factor rental rates
are denoted
w
= (
w
1
, ..., w
N
)
∈
R
N
++
,
with the scalar
w
i
repre-
senting the rental rate of factor
V
i
.
For simplicity, outputs are
often given a sector specific designation, such as agriculture,
a,
manufacturing,
m,
and the home-good,
s
. Likewise, endowments
are often given designations like labor,
L,
capital,
K,
and land
H
.
2.1.1
Consumer preferences
The economy is composed of a large number of atomistic house-
holds. Each household faces the same vector of prices
p
and the
same vector of factor rental rates
w
. Let
υ
h
=
(
υ
h
1
, ..., υ
h
N
)
∈
R
N
++
denote the level of factor endowments held by household-
h,
with
υ
h
i
representing the household’s endowment of factor
i.

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