The Loss From Trade: A Numeric Example With Three Countries And Three Produced Commodities

Figure 1: The Production Possibility Frontier, With And Without Trade, In "Germany"

1.0 Introduction

This post presents a numeric example of a Ricardian model of small, open economies engaged in trade. Each of three countries specializes in producing one of three commodities. Technology is modeled following an Austrian approach. Each commodity can be produced in each country from inputs of labor and "capital". Endowments of labor are taken as given parameters. It makes no sense to take the endowment of capital as a given parameter.

The with-trade Production Possibilities Frontier (PPF) can be compared to the autarkic PPF in each of the three countries. And it is unambiguously rotated inwards in one country for the set of international prices and rates of profits I consider. One cannot correctly conclude, in the traditional textbook approach, that free trade makes the inhabitants of a country better off in the aggregate.

I have previously written up an analogous argument in the two-commodity, two-country case. I should have cited Steedman and Metcalfe (1973). If I had, I would have known that my originality was less than I suggested. Shiozawa argues convincingly that the sort of model illustrated in this post can only be offered as an intellectual exercise. A more empirically applicable model would have trade in intermediate products, and the number of traded commodities would exceed the number of countries. On the other hand, I do not know of any comparable write-up of an example with three or more commodities and countries, as here.

2.0 Assumptions

Consider a model of three countries - England, Portugal, and Germany - in which three commodities are potentially traded on international markets by each country.

• Each country can produce any of the three commodities.
• The managers of firms in each country know a given flow input-point output technology with the structure described in Section 3. The technology differs among countries.
• Each country has a given endowment of labor, the only non-produced factor of production in each country. The endowment of labor may vary between countries.
• Only commodities produced for consumption can be traded internationally. Workers can neither immigrate nor emigrate. Intermediate goods, also known as capital goods, cannot be traded internationally.
• Financial capital is only invested domestically. Consequently, the rate of profits may vary across countries.
• Free competition obtains in all domestic markets; transport costs are negligible; and free trade exists in all commodities produced for consumption.

3.0 Technology

In each country, each commodity can be produced by a uniform application of labor for a specified number of years (Table 1). In each country, less total labor is required to produce a unit wine than is required to produce a unit of corn, with the labor uniformly distributed over a longer period of time in producing wine. Wine is a less labor-intensive and more capital-intensive commodity than corn, in some sense. In the same sense, linen is a less labor-intensive and more capital-intensive commodity, as compared with wine. England has an absolute advantage over Portugal, and Portugal has an absolute advantage over Germany, in producing each commodity. Nevertheless, a set of prices on international markets can exist, for corn, wine, and linen, such that firms in each country will want to specialize in producing a single commodity.

Table 1: Example Technology

Produced Commodity	Years of Labor	Labor Input per Year (Person-Yrs)
		England	Portugal	Germany
Corn	1	100	220	320
Wine	2	40	75	120
Linen	3	25	45	200/3 ≈ 66.6

The data on technology, along with the endowment of labor in each country, is enough to draw the Production Possibility Frontier (PPF) for each country in an autarky (without trade). Let X_{1, n}, X_{2, n}, and X_{3, n} be the quantity of corn, wine, and linen consumed in the nth country (n = 1, 2, 3). The plane outlined by the heavy lines in Figure 1, above, is the autarkic PPF for Germany. L₃ is the endowment of labor in Germany. l_{1, 3}, l_{2, 3}, and l_{3, 3} is the labor embodied in each commodity, when produced in Germany. These quantities are 320, 240, and 200 labor-years for this technology.

4.0 Prices and Specialization With Trade at a Rate of Profits of Zero

Next, consider an equilibrium with trade. Suppose the prices of a unit of corn, wine, and linen are as in Table 2. The question arises of whether there is a pattern of specialization among countries and a distribution of income in each country consistent with the given international prices. At this point, I take the rate of profits as zero. And I have calculated the wages shown for each country.

Table 2: Prices with Trade with Zero Rate of Profits

Commodity	England	Portugal	Germany
Corn	p1 = $300
Wine	p2 = $225
Linen	p3 = $200
Rate of Profits	r1 = 0	r2 = 0	r3 = 0
Wage	w1 = $3	w2 = $3/2	w3 = $1

Table 3 shows relative prices in each country, where the numeraire varies among countries. That is, a person-year of domestic labor is taken as the numeraire. (In Table 2, I have implicitly taken German labor - the lowest paid, as numeraire.) Notice that, in Table 3, the prices of corn in England, wine in Portugal, and linen in Germany are all equal to the labor values in the respective countries. And the prices of all other commodities falls below their labor values. Thus, since the rate of profits is zero, English firms will specialize in producing corn, Portuguese firms will produce only wine, and German firms will produce only linen. To obtain a domestic consumption basket, in any country, that contains all three commodities, each country must engage in international trade. It turns out that the PPF is unambiguously rotated outward for each country, for a pattern of specialization with a rate of profits of zero.

Table 3: Renormalized Prices with Zero Rate of Profits

Commodity	England	Portugal	Germany
Corn	p1/w1 = 100	p1/w2 = 200	p1/w3 = 300
Wine	p2/w1 = 75	p2/w2 = 150	p2/w3 = 225
Linen	p3/w1 = 200/3	p3/w2 = 400/3	p3/w3 = 200

5.0 Prices and Specialization With Trade at Positive Rates of Profits

It is well-known that, in general, prices deviate from labor values when the rate of profits is positive. Suppose the prices of corn, wine, and linen are as shown in Table 4. I also take the rates of profits as given in each country. These data yield the wages shown in the Table.

Table 4: Prices with Trade

Commodity	England	Portugal	Germany
Corn	p1 = 4275/26
Wine	p2 = 9063/52
Linen	p3 = 855/4
Rate of Profits	r1 = 3/5	r2 = 1/2	r3 = 9/10
Wage	w1 = 9063/5408	w2 = 1	w3 = 855/1664

I suppose the fractions are somewhat less awkward when normalized, as in Table 3. Unlike the case with a zero rate of profits, one should not compare these prices with labor values, in order to figure out the pattern of specialization. Rather, one should compare these prices with dated labor inputs costed up at the going rate of profits. For example, consider England. Since corn is produced with only one year of labor, its labor cost is still 100 person-years. (I am assuming wages are paid at the end of the year, not advanced.) Since p₁/w₁ is less than 100, England will import corn and not produce any. The cost of wine is 40(1 + 3/5) + 40 = 104. England will produce wine, and no super-normal profits are earned in its production. Linen is costed up as 25(1 + 3/5)² +25(1 + 3/5) + 25 = 129. p₃/w₁ is, approximately, 127.55. It is more costly to produce linen domestically, and, so, England will not do that. As a result of similar calculations, one can see that Portugal will specialize in producing linen and Germany in corn. The prices permit a consistent pattern of specialization, with all commodities being produced in some country and no firm earning more than the going rate of profits. And every country specializes in producing a different commodity shown above for the pattern with a zero rate of profits.

Table 3: Renormalized Prices

Commodity	England	Portugal	Germany
Corn	p1/w1 = 5200/53	p1/w2 = 4275/26	p1/w3 = 320
Wine	p2/w1 = 104	p2/w2 = 9063/52	p2/w3 = 1696/5
Linen	p3/w1 = 128440/1007	p3/w2 = 855/4	p3/w3 = 416

One can draw the PPFs, for each country, with this pattern of specialization and prices on international markets. The PPFs for England and Portugal are rotated out. For any consumption basket that contains some commodity not produced domestically, more is available to the country as a whole in England and Portugal. But the PPF is rotated inwards in Germany, as illustrated in Figure 1. The possibility of trade has diminished the commodities available for consumption in Germany.

6.0 Conclusion

I like that, in the above example, the pattern of specialization has each country producing a different commodity in the case with a positive rates of profits, as compared to the case with a rate of profits of zero. I'd like to convince myself that no other pattern of specialization is possible when the rate of profits is zero. I'd also like to find an example where the with-trade PPF is rotated outwards on one dimension and inwards on another. So whether every commodity in a nation's consumption basket is improved or decreased by trade would depend on its composition. I can show in the above model how a country's endowment of capital varies in value with the domestic rate of profits. And the model can be set out, in general, with any number of produced commodities and countries, with the number of commodities not exceeding the number of countries. In such a general setting, I think I will retain the severe restrictions of an Austrian model so as to exhibit that my point does not depend on, for example, capital-reversing.

It has been known for decades that the argument from comparative advantage is not a valid justification for a lack of tariffs (also known as free trade). Even setting aside such matters as, for example, increasing returns to scale or Keynesian failures of aggregate demand preventing a country from being on its PPF, the argument fails on its own terrain. This post is one more demonstration. Of course, this does not imply that any random, ill-natured, and ill-considered imposition of tariffs is likely to be a good idea in any specific case.

References

• Kurose, Kazuhiro and Naoki Yoshihara (2016). The Heckscher-Ohlin-Samuelson Model and the Cambridge Capital Controversies. Working paper.
• Metcalfe, J. S. and Ian Steedman. 1974. A Note on the Gain from Trade, Economic Record. Reprinted in Fundamental Issues in Trade Theory (Ed. by I. Steedman). Aldershot: Greg Revivals (1979, 1991).
• Shiozawa, Yoshinori. 2018. An Origin of the Neoclassical Revolution: Mill’s ‘Reversion’ and its Consequences.
• Steedman, Ian and J. S. Metcalfe. 1973. ’On Foreign Trade,’ Economia Internazionale. Reprinted in Fundamental Issues in Trade Theory (Ed. by I. Steedman). Aldershot: Greg Revivals (1979, 1991).
• Vienneau, Robert (2014). On the Loss from Trade