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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 X1, n,
X2, n, and X3, 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.
l1, 3, l2, 3, and l3, 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