Gains And Losses From Foreign Trade: A Numeric Example

Figure 1: Production Possibility Frontiers

1.0 Introduction

This post presents a numeric example of foreign trade in a model of the production of commodities by means of commodities. This is a modification of the model here, which considers a flow-input, point output technology. As usual, I show neoclassical economics is mistaken. Frictions, increasing returns, information asymmetries, principal agent problems, and so on do not need to be introduced to explain why the outcomes of free markets are not always ideal. Even under ideal conditions, problems can arise.

2.0 Technology, Endowments, And The Rate Of Profits

I assume each of two countries (Tables 1 and 2) have a fixed-coefficients technology for producing three commodities. The technology varies between countries, although it has the same structure in both. Steel is the only capital good. Each commodity can be produced, in a year, from inputs of labor and steel. A coefficient of production shows the quantity of an input needed per unit output. For example, in England, one person-year and 1/30 tons of steel must be purchased per square meter of produced linen. Steel is totally used up in production, and constant returns to scale obtains.

Table 1: Coefficients of Production in England

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(E) = 2	a0, 2(E) = 3	a0, 3(E) = 1
Steel	a1, 1(E) = 1/20	a1, 2(E) = 1	a1, 3(E) = 1/30

Table 2: Coefficients of Production in Portugal

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(P) = 2	a0, 2(P) = 7	a0, 3(P) = 2
Steel	a1, 1(P) = 1/40	a1, 2(P) = 1	a1, 3(P) = 1/100

I take endowments of labor as given, as in the Ricardian model of foreign trade. Let England and Portugal both have available a labor force consisting of one person-year. So Production Possibilities Frontiers (PPFs) are found per person-year. By assumption, workers neither immigrate nor emigrate. In this model, full employment is assumed.

I also take the rate of profits as given, at 300 per cent, in both countries. I originally intended to assume that financial capital cannot flow between countries. So the rate of profits need not be the same across countries. (If you find the rate of profits unacceptably high, read "year" as "decade" throughout this post.)

3.0 Aspects of Autarky

Suppose all three commodities are each produced in each country. Foreign trade is not possible. The technology allows one to calculate the labor embodied in each commodity. For steel, the number of person-years embodied in each ton of steel is:

v₁(n) = a_{0, 1}(n)/(1 - a_{1, 1}(n))

The labor embodied in corn and linen is:

v_j(n) = a_{0, 1}(n) a_{1, j}(n)/(1 - a_{1, 1}(n)) + a_{0, j}(n), j = 2, 3.

Labor values are useful in drawing the PPF for each country, under autarky. Consumers in England can consume 1/v₂(E) bushels of corn per person-year of labor hired, if they consume no linen. Or they can consume 1/v₃(E) square meters per person-year, with no corn. Any linear combination of these two consumption baskets, with positive quantities of both corn and linen, can also be consumed.

Each PPF embodies a rate of transformation between corn and linen, in a comparison of stationary states. For example, in England 61 bushels of corn can be traded off, in some sense, for 291 square meters of line. England has a comparative advantage in linen, as compared to corn, at a rate of profits of zero.

v₃(E)/v₂(E) < v₃(P)/v₂(P)

Portugal has a comparative advantage in steel, as compared to both corn and linen, at a rate of profits of zero.

4.0 Trade in Corn and Linen

In this section, I assume that foreign trade is possible in the consumer commodities, corn and linen. But international markets do not exist in the capital good, steel. The introducition of the possibility of foreign trade creates a choice of technique.

In the small country model, firms take prices, including on international markets, as given. I introduce the following notation:

• P₂: The price of a bushel corn on international markets.
• P₃: The price of a square meter of linen on international markets.
• w(n), n = E, P: The wage.
• r(n), n = E, P: The rate of profits.
• p₁(n), n = E, P: The domestic price of steel.
• p₁(n)a_1,2(n)(1 + r(n)) + a_{0, 2}(n) w(n), n = E, P: The cost of producing a bushel corn domestically.
• p₁(n)a_1,3(n)(1 + r(n)) + a_{0, 3}(n) w(n), n = E, P: The cost of producing a square meter of linen domestically.

Table 3 shows the value of each of these variables. Firms will not produce commodities when their cost of producing it exceeds what it can be purchased for on international markets. Accordingly, England specializes in producing linen and the necessary steel at these prices. Portugal produces corn and linen. The domestic price of steel and wages are such that firms cannot make extra profits in steel, corn, or iron production.

Table 3: Trade in Consumer Goods

Variable	England	Portugal
P2	$6 per Bushel
P3	$2/3 per Sq. Meter
w(n)	$19/32 Person-Yr.	$117/286 per Person-Yr.
p1(n)	$35/64 per Ton	$69/88 per Ton
p1(n)a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$127/32 per Bushel	$6 per Bushel
p1(n)a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$2/3 per Sq. Meter	$1869/2200 per Sq. Meter

The ratio of the prices of linen and corn are a key variable here. Countries specialize in the consumer commodity in which relative international prices exceeds the domestic relative price, as calculated under autarky. Since the rate of profits is positive, relative domestic prices differ from the slope of the autarkic PPF. That is, the relative autarky price is what determines comparative advantage, in some sense. But the slope of the autarkic PPF is important in analyzing whether gains from trade are positive or negative.

The possibility of foreign trade in consumer goods has made consumers in England worse off, in a comparison of stationary states. The English PPF is rotated inwards, when firms specialize as induced by these prices. Consumers in Portugal, on the other hand, are better off. Their PPF is rotated outwards.

4.0 Trade in Steel, Corn, and Linen

I now assume trade is possible in all goods, including the capital good. Let P₁ be the price of steel on international markets. Cost of domestic production are modified in the obvious way in Table 4.

Table 4: Trade in Capital and Consumer Goods

Variable	England	Portugal
P1	$10/9 per Bushel
P2	$6 per Bushel
P3	$2/3 per Sq. Meter
w(n)	$14/27 Person-Yr.	$1/2 per Person-Yr.
P1 a1,1(n)(1 + r(n)) + a0, 1(n) w(n)	$34/27 per Ton	$10/9 per Ton
P1a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$6 per Bushel	$143/18 per Bushel
P1a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$2/3 per Sq. Meter	$47/45 per Sq. Meter

England specializes in the production of corn and linen, and Portugal specializes in the production of steel. Firms in England obtain the steel they need to continue production by trading corn and linen in foreign trade. Likewise, consumers in Portugal obtain corn and linen from firms selling the surplus steel product in foreign trade. No firm incurs extra costs or obtains extra profits in any process which they operate. And operated process would incur extra costs.

The opening up of foreign markets in steel has made England better off, both in comparison to autarky and in comparison with foreign trade only being possible in consumer goods. (Although it is difficult to see in Figure 1, the intercept of the PPF, for trade in all commodities, with the ordinate strictly exceeds the intercept for the other PPFs). The opening up of foreign trade in steel has made Portugal worse off, as compared to trade only in consumer goods. Whether the Portuguese are better off as compared to autarky is ambiguous. It depends on the consumption basket.

5.0 Conclusion

Why, oh why, do mainstream economists teach untruths about the theory of trade?