A Country Worse Off With Trade In Capital Goods

Figure 1: PPFs in Portugal

1.0 Introduction

This post continues these two posts. I change the model here to have wages advanced, not paid out of the surplus at the end of the year.

I here consider an example of a model of stationary states in which two countries can trade in produced commodities to be used for consumption. The countries face given prices on international markets for traded commodities. (They are small open economies, in the jargon.) I take the rate of profits as given in each country. They may differ, since I assume that finance capital cannot be traded internationally. I also assume labor is immobile between countries.

I contrast the model with and without foreign trade being possible in capital goods. Paul Samuelson calls the supposed gains from trade in capital goods the Sraffian bonus. This post demonstrates that the Sraffian bonus can be negative. The inhabitants of a country might be better off, in the sense that the total bundle of consumption goods is larger, if international markets do not exist in capital goods.

2.0 Parameter Values

I assign the numeric values in Table 1 to coefficients of production. For those who do not want to click, a unit of steel can be made in England from a direct and unassisted labor input of l_{2, C, E} person-years. A unit of corn can be made from one unit of steel and l_{1, C, E}. A unit of linen is made from l_{1, L, E} person-years of direct and unassisted labor.

Table 1: Technology for the Example

Parameter	England	Portugal
l1, C, n	3	7
l2, C, n	2	2
l1, L, n	1	2
Ln	1	1
rn	300%	400%

I take the endowments of labor - L_E and L_P person-years, respectively - as given. Production Possibility Frontiers (PPFs) are constructed per worker.

I also take rates of profits, r_E and r_P, as given. I do not strive for realism. But, if you are concerned by the sizes of the rates of profits, pretend that my "year" is actually a decade or so.

3.0 Stationary States with and without Trade in Capital Goods

I now consider what prices could be consistent with stationary states.

First, suppose foreign trade is not possible in capital goods. Only corn and linen can be traded on international markets. Suppose prices are as in Table 2. The cost of producing linen in England:

l_{1, L, E} w_E (1 + r_E) = 1 (1/6) (1 + 3) = 2/3

If firms in England manufacture linen for both domestic consumption and for foreign trade, they make the going rate of profits. The cost of producing corn in England is:

[l_{2, L, E} (1 + r_E) + l_{1, L, E}] w_E (1 + r_E) = [2 (4) + 3](1/6)(4) = 22/3

Firms in England will not want to produce corn. They would be undercut by foreign competition. You can do the analogous calculations for Portugal. Portuguese firms will produce corn and the needed steel to continue production. They will not produce linen. With these prices and this specialization, consumers in both countries can consume baskets of commodities containing both corn and linen. And firms will be minimizing costs.

Table 2: Example with Foreign Trade in Corn and Linen

Variable	England	Portugal
PC	6
PL	2/3
Cost of producing corn	22/3	6
Cost of producing linen	2/3	12/7
Specialization	Linen	Corn and Steel
wn	1/6	6/85

Suppose now that international markets exist in corn, linen, and steel. Table 3 shows prices for consideration in this case. One tabulates the cost of producing corn with the steel input evaluated at the international price. For example, the cost of producing corn in England is:

(P_S + l_{1, C, E} w_E) (1 + r_E) = [1 + 3 (1/6)](4) = 6

Going through these tabulations, one will find that the firms in England specialize in producing corn and linen. The cost of producing steel in England exceeds its price. Likewise, firms in Portugal specialize in producing steel. Cost-minimizing firms in Portugal are unwilling to produce either corn or linen.

Table 3: Example with Foreign Trade in Corn, Linen, and Steel

Variable	England	Portugal
PC	6
PL	2/3
PS	1
Cost of producing corn	6	17/2
Cost of producing linen	2/3	1
Cost of producing steel	4/3	1
Specialization	Corn and Linen	Steel
wn	1/6	1/10

Notice that the international prices of corn and linen are unchanged between Tables 2 and 3. Steady states are here shown as resulting in an increased wage in Portugal when foreign trade is possible in steel. But rates of profits and the wage in England are shown as constant.

3.0 Production Possibility Frontiers

What about physical quantities of commodities? I restrict myself to stationary states. Figure 2 shows PPFs in England. The PPF under autarky is constructed from technical data on coefficients of production and the endowment of labor in England. When only linen is produced in England, whether foreign trade is possible or not, the same amount of linen is produced as under autarky. Consequently, all three PPFs are shown as rotated around the same intercept in Figure 2.

Figure 2: PPFs in England

Consider England when foreign trade is only possible in corn and linen. Since England specializes in linen, the maximum amount of corn consumed by the English is (L_E/l_{1, L, E})(P_L/P_C). In this example, that quantity is less than L_E/(l_{1, C, E} + l_{2, C, E}), the maximum quantity of corn consumed in England without foreign trade.

When foreign trade is also possible in steel, the maximum quantity of corn manufactured in England is (L_E/l_{1, C, E}) units. But not all of this corn can be consumed. Steel must be purchased from Portugal to continue production on the same scale. That is, (L_E/l_{1, C, E})(P_C - P_S) numéraire units are available for consumption. So the maximum corn consumption in England is (L_E/l_{1, C, E})(P_C - P_S)/P_C units of corn. For England, the Sraffian bonus is positive. The possibility of foreign trade in steel has left the PPF for domestic consumption rotated outwards, even beyond the maximum consumption under autarky.

The situation in Portugal, as illustrated by Figure 1 at the top of this post, is quite different, however. Foreign trade in consumer, but not capital goods, results in a PPF rotated outwards from the autarkic PPF. This conforms to the nonsense long-suffering students in economics taught out of mainstream textbooks must endure. At the prices considered above, Portugal produces only steel when foreign trade is possible in all produced goods. The maximum amount of linen that can be consumed in Portugal is (L_P/l_{2, C, E})(P_S/P_L) units. Neither intercept with an axis for this PPF is equal to the corresponding intercept for autarky. Furthermore, the PPF with foreign trade in all goods is strictly inside the PPF with foreign trade only in consumer goods. The Sraffian bonus is negative. Suppose one compares the PPF with foreign trade in all goods to the autarkic PPF for Portugal. Whether or not there are gains from trade is ambiguous. It depends on the consumption basket.

4.0 Conclusion

So this post has extended long-ignored proofs that the theory of comparative advantage does not provide a valid a-priori argument for so-called free trade. Opening up markets in capital goods may not provide a country with more goods, setting aside problems of adjustment.

I know of some empirical work purporting to demonstrate gains from trade. I do not know of any that addresses the issues brought forth in this post.

References

• Paul A. Samuelson (2001). A Ricardo-Sraffa Paradigm Comparing Gains from Trade in Inputs and Finished Goods. Journal of Economic Literature 39, 4: 1204-1214.
• Ian Steedman (1980). Trade amongst Growing Economics. Cambridge University Press.