Advanced International Trade

Theory and Evidence
By Robert C. Feenstra

Princeton University Press

Copyright © 2003 Princeton University Press
All right reserved.

ISBN: 978-0-691-11410-1

Chapter One

Preliminaries: Two-Sector Models

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, [y.sub.1] =_[f.sub.1] ([L.sub.1], [K.sub.1]), and the resource constraints in (1.1) give us [y.sub.2] = h([y.sub.1], L, K). The graph of [y.sub.2] as a function of [y.sub.1] is shown as the PPF in Figure 1.3. As drawn, [y.sub.2] is a concave function of [y.sub.1], [[partial derivative].sup.2] h([y.sub.1], L, K) / [partial derivative][y.sup.2.sub.1] < 0. This familiar result follows from the fact that the production functions [f.sub.i]([L.sub.i], [K.sub.i]) are assumed to be concave. Another way to express this is to consider all points S = ([y].sub.1], [y].sub.2]) that are feasible to produce given the resource constraints in (1.1). This production possibilities set S is convex, meaning that if [y.sup.a] = ([[y.sup.a].sub.1], [y.sup.a.sub.2] and [y.sup.b] = ([[y.sup.b].sub.1] , [[y.sup.b].sub.2]) are both elements of S, then any point between them [lambda][[y.sup.a] + (1 - [lambda]) [y.sup.b] is also in S, for 0 [less than or equal to] [lambda] [less than or equal to] 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:


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


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


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.


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