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We begin our study of international trade with the classic Ricardian
model, which has two goods and one factor (labor). The Ricardian model
introduces us to the idea that technological differences across countries
matter. In comparison, the Heckscher-Ohlin model dispenses with the
notion of technological differences and instead shows how *factor endowments*
form the basis for trade. While this may be fine in theory, the model
performs very poorly in practice: as we show in the next chapter, the
Heckscher-Ohlin model is hopelessly inadequate as an explanation for historical
or modern trade patterns unless we allow for technological differences
across countries. For this reason, the Ricardian model is as relevant
today as it has always been. Our treatment of it in this chapter is a simple
review of undergraduate material, but we will have the opportunity to refer
to this model again at various places throughout the book.

After reviewing the Ricardian model, we turn to the two-good, two-factor
model that occupies most of this chapter and forms the basis of the
Heckscher-Ohlin model. We shall suppose that the two goods are traded
on international markets, but we do not allow for any movements of factors
across borders. This reflects the fact that the movement of labor and
capital across countries is often subject to controls at the border and generally
much less free than the movement of goods. Our goal in the next
chapter will be to determine the pattern of international trade between
countries. In this chapter, we simplify things by focusing primarily on *one*
country, treating world prices as given, and examine the properties of this
two-by-two model. The student who understands all the properties of this
model has already come a long way in his or her study of international
trade.

**Ricardian Model**

Indexing goods by the subscript *i*, let *[a.sub.i]* denote the labor needed per unit of
production of each good at home, while *[a.sup.*.sub.i]* is the labor need per unit of
production in the foreign country, *i* = 1, 2. The total labor force at home is
*L* and abroad is [*L*.sup.*]. Labor is perfectly mobile between the industries in each
country, but immobile across countries. This means that both goods are
produced in the home country only if the wages earned in the two industries
are the same. Since the marginal product of labor in each industry is
1/*[a.sub.i]*, and workers are paid the value of their marginal products, wages are
equalized across industries if and only if [*p*.sub.1]/[*a*.sub.1] = [*p*.sub.2]/[*a*.sub.2],
where *[p.sub.i]* is the price in each industry. Letting *p* = [*p*.sub.1]/[*p*.sub.2] denote
the *relative* price of good 1 (using good 2 as the numeraire), this condition is *p* =
[*a*.sub.1]/[*a*.sub.2].

These results are illustrated in Figure 1.1(a) and (b), where we graph
the production possibility frontiers (PPFs) for the home and foreign
countries. With all labor devoted to good *i* at home, it can produce *L*/*[a.sub.i]*
units, *i* = 1, 2, so this establishes the intercepts of the PPF, and similarly
for the foreign country. The slope of the PPF in each country (ignoring
the negative sign) is then [*a*.sub.1]/[*a*.sub.2] and [*a*.sup.*.sub.1]/[*a*.sup.*.sub.2].
Under autarky (i.e., no international trade), the equilibrium relative prices *[p.sup.a]* and *[p.sup.a*]*
must equal these slopes in order to have both goods produced in both countries, as argued
above. Thus, the autarky equilibrium at home and abroad might occur
at points A and [A.sup.*]. Suppose that the home country has a *comparative
advantage* in producing good 1, meaning that [*a*.sub.1]/[*a*.sub.2] <
*a*.sup.*.sub.1]/[*a*.sup.*.sub.2]. This implies
that the home autarky relative price of good 1 is *lower* than that
abroad.

Now letting the two countries engage in international trade, what is
the equilibrium price *p* at which world demand equals world supply? To
answer this, it is helpful to graph the world relative supply and demand
curves, as illustrated in Figure 1.2. For the relative price satisfying
*p* < *[p.sup.a]* = [*a*.sub.1]/[*a*.sub.2] and *p* < [*p.sup.a**] =
[*a*.sup.*.sub.1]/[*a*.sup.*.sub.2] both countries are fully specialized
in good 2 (since wages earned in that sector are higher), so the world relative
supply of good 1 is zero. For *[p.sup.a]* < *p* < [*p.sup.a**], the home country is fully
specialized in good 1, whereas the foreign country is still specialized in
good 2, so that the world relative supply is (*L*/[*a*.sub.1])/([*L*.sup.*]/[*a*.sup.*.sub.2]),
as labeled in Figure 1.2. Finally, for *p* > *[p.sup.a]* and *p* > *[p.sup.a*]*, both countries
are specialized in good 1. So we see that the world relative supply curve has a "stair-step"
shape, which reflects the linearity of the PPFs.

To obtain world relative demand, let us make the simplifying assumption
that tastes are identical and homothetic across the countries. Then
demand will be independent of the distribution of income *across* the countries.
Demand being homothetic means that *relative* demand [*d*.sub.1]/[*d*.sub.2] in
either country is a downward-sloping function of the *relative* price *p*, as
illustrated in Figure 1.2. In the case we have shown, relative demand intersects
relative supply at the world price *p* that lies *between [p.sup.a]* and *[p.sup.a*]*, but
this does not need to occur: instead, we can have relative demand intersect
one of the flat segments of relative supply, so that the equilibrium price
with trade *equals* the autarky price in one country.

Focusing on the case where **[p.sup.a]** < *p* < *[p.sup.a*]*, we can go back to the PPF of
each country and graph the production and consumption points with free
trade. Since *p* > *[p.sup.a]*, the home country is fully specialized in good 1 at
point B, as illustrated in Figure 1.1(a), and then trades at the relative price
*p* to obtain consumption at point C. Conversely, since *p* < *[p.sup.a*]*, the foreign
country is fully specialized in the production of good 2 at point [B.sup.*] in
Figure 1.1(b), and then trades at the relative price p to obtain consumption
at point [C.sup.*]. Clearly, *both* countries are better off under free trade
than they were in autarky: trade has allowed them to obtain a consumption
point that is above the PPF.

Notice that the home country exports good 1, which is in keeping with
its comparative advantage in the production of that good, [*a*.sub.1]/*[a.sub.2]* <
[*a*.sup.*.sub.1]/[*a*.sup.*.sub.2].
Thus, *trade patterns are determined by comparative advantage*, which is a
deep insight from the Ricardian model. This occurs even if one country has
an *absolute disadvantage* in both goods, such as [*a*.sub.1] > [*a*.sup.*.sub.1] and
[*a*.sub.2] > [*a*.sup.*.sub.2], so
that more labor is needed per unit of production of *either* good at home
than abroad. The reason that it is still possible for the home country to export
is that its *wages* will adjust to reflect its productivities: under free trade,
its wages are lower than those abroad. Thus, while trade patterns in the
Ricardian model are determined by *comparative advantage*, the level of
wages across countries is determined by *absolute advantage*.

**Two-Good, Two-Factor Model**

While the Ricardian model focuses on technology, the Heckscher-Ohlin
model, which we study in the next chapter, focuses on factors of production.
So we now assume that there are two factor inputs-labor and capital.
Restricting our attention to a single country, we will suppose that it produces
two goods with the production functions *[y.sub.i]* = [f.sub.i]([*L.sub.i*], [*K.sub.i*]), *i*
= 1, 2, where [*y.sub.i*] is the output produced using labor *[L.sub.i]* and capital *[K.sub.i]*.
These production functions are assumed to be increasing, concave, and homogeneous
of degree one in the inputs (*[L.sub.i]*, *[K.sub.i]*). The last assumption means that there
are *constant returns to scale* in the production of each good. This will be a
maintained assumption for the next several chapters, but we should point
out that it is rather restrictive. It has long been thought that *increasing returns
to scale* might be an important reason to have trade between countries:
if a firm with increasing returns is able to sell in a foreign market, this
expansion of output will bring a reduction in its average costs of production,
which is an indication of greater efficiency. Indeed, this was a principal
reason that Canada entered into a free-trade agreement with the
United States in 1989: to give its firms free access to the large American
market. We will return to these interesting issues in chapter 5, but for
now, ignore increasing returns to scale.

We will assume that labor and capital are *fully mobile* between the two
industries, so we are taking a "long run" point of view. Of course, the
amount of factors employed in each industry is constrained by the endowments
found in the economy. These resource constraints are stated as

[*L*.sub.1] + [*L*.sub.2] [less than or equal to] *L*, (1.1)

[*K*.sub.1] + [*K*.sub.2] [less than or equal to] *K*,

where the endowments *L* and *K* are fixed. Maximizing the amount of good
2, [*y*.sub.2] = [*f*.sub.2] ([*L*.sub.2], * K.sub.2*), subject to a given amount of
good 1,

The production possibility frontier summarizes the technology of the
economy, but in order to determine where the economy produces on the
PPF we need to add some assumptions about the market structure. We
will assume perfect competition in the product markets and factor markets.
Furthermore, we will suppose that product prices are given *exogenously:*
we can think of these prices as established on world markets, and outside
the control of the "small" country being considered.

**GDP Function**

With the assumption of perfect competition, the amounts produced in each industry will maximize gross domestic product (GDP) for the economy: this is Adam Smith's "invisible hand" in action. That is, the industry outputs of the competitive economy will be chosen to maximize GDP:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)

To solve this problem, we can substitute the constraint into the objective
function and write it as choosing [*y*.sub.1] to maximize [*p*.sub.1][*y*.sub.1] +
[*p*.sub.2]*h*([*y*.sub.1], *L, K*).
The first-order condition for this problem is [*p*.sub.1] + [*p*.sub.2]([partial
derivative]*h*/[partial derivative][*y*.sub.1]) = 0, or

*p* = [*p*.sub.1]/[*p*.sub.2] = [partial derivative]*h*/[partial derivative][*y*.sub.1] =
[partial derivative][*y*.sub.2]/[partial derivative][*y*.sub.1] (1.3)

Thus, the economy will produce where the relative price of good 1,
*p* = [*p*.sub.1]/[*p*.sub.2], is equal to the slope of the production possibility frontier. This
is illustrated by the point A in Figure 1.4, where the line tangent through
point A has the slope of (negative) *p*. An increase in this price will *raise* the
slope of this line, leading to a new tangency at point B. As illustrated, then,
the economy will produce more of good 1 and less of good 2.

The GDP function introduced in (1.2) has many convenient properties,
and we will make use of it throughout this book. To show just one property,
suppose that we differentiate the GDP function with respect to the
price of good *i*, obtaining

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.4)

It turns out that the terms in parentheses on the right-hand side of (1.4)
sum to zero, so that [partial derivative]*G*/[partial derivative]*[p.sub.i]* = *[y.sub.i]*. In other
words, the derivative of the GDP function with respect to *prices* equals the *outputs* of the economy.

The fact that the terms in parentheses sum to zero is an application of
the "envelope theorem," which states that when we differentiate a function
that has been maximized (such as GDP) with respect to an exogenous
variable (such as *[p.sub.i]*), then we can *ignore* the changes in the
endogenous variables ([*y*.sub.1] and [*y*.sub.2]) in this derivative. To prove that these
terms sum to zero, totally differentiate [*y*.sub.2] = *h*([*y*.sub.1], *L, K*) with respect to
[*y*.sub.1] and [*y*.sub.2] and use (1.3) to obtain [*p*.sub.1]*d]y.sub.1] =
-[p.sub.2]d]y.sub.2], or [p.sub.1]d]y.sub.1] +
[p.sub.2]d]y.sub.2] = 0. This equality must hold for any small movement in [y.sub.1]
and [y.sub.2] around the PPF, and in particular, for the small movement in outputs induced by
the change in [p.sub.i]. In other words, [p.sub.1]([partial derivative][y.sub.1]/[partial
derivative][p.sub.i]) + [p.sub.2]([partial derivative][y.sub.2]/[partial
derivative][p.sub.i]) = 0, so the terms in parentheses on the right of (1.4) vanish and it follows that
[partial derivative]G/[partial derivative][p.sub.i] = [y.sub.i].
*

*
*

*
Equilibrium Conditions
*

*
We now want to state succinctly the equilibrium conditions to determine
factor prices and outputs. It will be convenient to work with the unit-cost
functions that are dual to the production functions [f.sub.i] ([L.sub.i], [K.sub.i]). These
are defined by
*

*
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.5)
*

*
In words, [c.sub.i](w, r) is the minimum cost to produce one unit of output. Becauseof our assumption
of constant returns to scale, these unit-costs are
equal to both marginal costs and average costs. It is easily demonstrated
that the unit-cost functions [c.sub.i](w, r) are nondecreasing and concave in
(w, r). We will write the solution to the minimization in (1.5) as [c.sub.i](w, r) =
[wa.sub.iL] + [ra.sub.iK], where [a.sub.iL] is optimal choice for [L.sub.i], and
[a.sub.iK] is optimal choice for K.sub.i. It should be stressed that these optimal choices for labor
and capital depend on the factor prices, so that they should be written in full as
[a.sub.iL](w, r) and [a.sub.iK](w, r). However, we will usually not make these
arguments explicit.
*

*
(Continues...)
*

*
*

Excerpted fromAdvanced International TradebyRobert C. FeenstraCopyright © 2003 by Princeton University Press. Excerpted by permission.

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