## Saturday, December 15, 2018

### Gain or Loss from Trade with Multiple Equilibria

 Figure 1: Production Possibility Frontiers
1.0 Introduction

Suppose foreign trade is possible in consumption goods, but not in capital goods. In this example, whether or not England achieves gains from trade depends on relative international prices. If foreign trade were possible in both consumption and capital goods, both England and Portugal would obtain gains from trade. The numeric example in this post is a modification of one in a previous post.

As I understand it, most students of economics are taught this numeric example cannot exist. And it raises questions on, for example, tariffs and the distribution of income that you will be hard-pressed to find discussed.

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.

 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

 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 25 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.

3.0 One of Two Equilibria

One can analyze each country under autarky, that is, under the assumption that foreign trade is not possible. One can find, given the rate of profits in each country, relative prices of corn and linen in each country. Suppose foreign trade is possible in corn and linen, but not in steel. And suppose the ratio of the international price of linen to the international price of corn is between the corresponding ratio of autarkic prices in England and Portugal. (I have chosen the rates of profits so this ratio is lower in England than in Portugal under autarky.) Then the English specialize in producing linen, and the Portuguese specialize in producing corn. I consider international prices at the two extreme ends of this range. This section presents the first extreme.

3.1 Trade in Corn and Linen

Table 3 present prices and costs when trade is only possible in corn and linen. I follow the notation in a previous post. The rows show the international price of corn, the international price of linen, wages in each country, the domestic price of steel, the cost of producing corn, and the cost of producing linen. If anybody wants to work it out, wages and the price of steel are such that the given rate of profits is made in producing steel in each country.

 Variable England Portugal P2 \$15 per Bushel P3 \$49/17 per Sq. Meter w(n) \$45/17 Person-Yr. \$155/99 per Person-Yr. p1(n) \$96/17 per Ton \$320/99 per Ton p1(n)a1,2(n)(1 + r(n)) + a0, 2(n) w(n) \$15 per Bushel \$15 per Bushel p1(n)a1,3(n)(1 + r(n)) + a0, 3(n) w(n) \$49/17 per Sq. Meter \$314/99 per Sq. Meter

Firms in a country will only produce a commodity if its cost of production does not exceed its price. With the prices in the above table, the English are willing to produce both corn and linen, while the Portuguese produce only corn. I want to ignore that the English might want to produce corn. If the price of linen on international markets was just an infinitesimal higher, the English would not be willing to produce corn.

The upper half of the figure at the top of this post illustrates this case. When, at these prices, England specializes in linen, they obtain a loss from trade. Portugal obtains gains from trade throughout.

3.2 Trade in Steel, Corn, and Linen

I now consider this case with foreign trade in steel also. Table 4 shows prices and costs. The first row is for the price of steel on international markets. I also introduce a row for the cost of producing steel. With the same logic as above, I ignore that England can produce steel, as well as corn and linen, with these prices. I take the international prices of corn and linen as unchanged from the previous subsection.

 Variable England Portugal P1 \$96/17 per Bushel P2 \$15 per Bushel P3 \$49/17 per Sq. Meter w(n) \$45/17 per Person-Yr. \$93/34 per Person-Yr. P1 a1,1(n)(1 + r(n)) + a0, 1(n) w(n) \$96/17 per Ton \$96/17 per Ton P1a1,2(n)(1 + r(n)) + a0, 2(n) w(n) \$15 per Bushel \$891/34 per Bushel P1a1,3(n)(1 + r(n)) + a0, 3(n) w(n) \$49/17 per Sq. Meter \$471/85 per Sq. Meter

In this case, both England and Portugal gain from trade. England specializes in corn and linen, and Portugal specializes in steel. The possible consumption baskets for both England and Portugal, under trade in all commodities, is also shown in the upper half of the figure at the top of this page. Even if you click through, it is hard to see that the maximum amount of linen that can be consumed in England is strictly greater than autarky in this case. Samuelson calls the additional gains from trade obtained through foreign trade in capital goods as the "Sraffian bonus". I have previously shown that the Sraffian bonus can be negative.

4.0 A Second Equilibrium

Now suppose the international price of linen is at the opposite extreme, with the same specializations. Again, this is the endpoint of what should be an open interval.

4.1 Trade in Corn and Linen

Table 5 shows prices and costs when foreign trade is possible only in consumer goods. English firms make the going rate of profits in producing steel and linen, but would incur extra costs if they produced corn domestically. Portuguese firms make the going rate of profits in producing any of steel, corn, and linen. But I treat them here as specializing in producing corn for foreign trade and obtaining linen only through foreign trade.

 Variable England Portugal P2 \$15 per Bushel P3 \$314/99 per Sq. Meter w(n) \$4710/1617 Person-Yr. \$155/99 per Person-Yr. p1(n) \$10048/1617 per Ton \$320/99 per Ton p1(n)a1,2(n)(1 + r(n)) + a0, 2(n) w(n) \$26690/1617 per Bushel \$15 per Bushel p1(n)a1,3(n)(1 + r(n)) + a0, 3(n) w(n) \$314/99 per Sq. Meter \$314/99 per Sq. Meter

The bottom half of the figure above shows Production Possibility Frontiers for this case. Both England and Portugal obtain gains from trade. (The PPF for England, under trade in consumption goods, is not easy to visually distinguish from the PPF under autarky.) A given technology and given rates of profits is compatible with a country both obtaining gains and suffering losses from foreign trade in consumption goods, depending on international prices.

4.2 Trade in Steel, Corn, and Linen

International prices of corn and linen are the same in Table 6 below and Table 5 above. Table 6 is drawn up for the possibility of foreign trade in steel, corn, and linen. England specializes in corn and linen, and Portugal specializes in steel. As seen in the bottom half of the figure at the top of this post, both England and Portugal have gains in trade, as compared to autarky and to foreign trade in consumer goods, when trade is possible in all produced commodities.

 Variable England Portugal P1 \$1448/297 per Bushel P2 \$15 per Bushel P3 \$314/99 per Sq. Meter w(n) \$2645/891 per Person-Yr. \$5611/2376 per Person-Yr. P1 a1,1(n)(1 + r(n)) + a0, 1(n) w(n) \$11123/1782 per Ton \$1448/297 per Ton P1a1,2(n)(1 + r(n)) + a0, 2(n) w(n) \$15 per Bushel \$181/8 per Bushel P1a1,3(n)(1 + r(n)) + a0, 3(n) w(n) \$314/99 per Sq. Meter \$28417/5940 per Sq. Meter

5.0 Conclusion

In this example, only one process is known in each country for producing each commodity domestically. The possibility of foreign trade creates a choice of technique. I wonder if more processes existed for each country's technology, would the range of international prices for consumer goods consistent with certain national specializations be narrowed? Would the introduction of consumer demand in the model remove the indeterminism? I suppose, for exploring the last question, I should see what has been done with J. S. Mill's approach to analyzing foreign trade.