A Theory of the Rate of Change in Exchange Rates

1.0 Introduction
I created this model by thinking about what would happen if no basic commodity existed, and yet no commodity could be produced with unassisted labor alone. That is, suppose (seed) corn can be used to produce corn, and rice can be used to produce rice. But corn does not enter either directly or indirectly into the production of rice. Nor does rice enter either directly or indirectly into the production of corn.

The mathematical problem posed by these thoughts can be set out in a model of two countries trading. I end up with an explanation of relative rates of currency appreciation across countries by the interaction of technology and the distribution of income - class struggle, if you will - in each country. I think this model illustrates that non-mainstream ways of thinking about economics can suggest new models and new insights.

I don't claim originality for this model. I was re-reading Samuelson (1957) to confirm my impression that that is where Samuelson describes Marx as "a minor Post-Ricardian". (I'm fairly sure Samuelson sets out his eraser algorithm in his 1971 JEL article.) I did not recall that Samuelson had set out a Marxist scheme of reproduction in his 1957 paper, albeit with a poor supply and demand interpretation. Anyways, I stopped at this passage:

"Without going into the social relations of the past or future, any economist... can evisage a case where Industry III [luxuries] alone, by virtue of having a₃ = 0 and b₃ < 1 will determine its own-rate of profit by itself, and he will realize that if this new r differs from that of (11) what must give is not bourgeois economic theory or the capitalistic institutional economy but rather the assumption of stationary relative prices." -- Paul A. Samuelson (1957)

This quote jostled my memory of a Joan Robinson review of Sraffa's book. I also dimly recall Keynes' 1923 analysis of arbitrage in forward trades in international currency markets and expected rates of inflation. I'd have to review whether one of the papers collected in Steedman (1979) sets out something like this model. I don't recall any conclusion as simple as the one I obtain, but I think there must be something like this in older Marxist models.

2.0 The Model
Consider two countries, each producing one of two commodities, corn and rice. The commodity produced in each country is a basic good in that country's economy. Assume no migration of labor is possible between the countries. Hence, wages can vary across economies. Assume, however, that no barriers to international flows of (financial) capital have been erected. Thus, a tendency exists for the same rate of profits to arise across countries, where financial outlays and revenues are calculated in some common abstract unit of account.

Without loss of generality, assume the price of corn is always one dollar per unit. The price of rice is one yen per unit. And the exchange rate at the start of the year is one yen per dollar.

These assumptions allow one to formulate equations for the prices of production in a common unit of account, for example, dollars. In the corn-producing country, prices of production satisfy the following equation:

a_c,c (1 + r) + a_0,c w_c = 1

where

• a_c,c is the amount of corn needed as input per unit corn produced
• a_0,c is the person-years labor needed as input per unit corn produced
• w_c is the wage in units corn in the corn-producing country
• r is the rate of profits

I here follow Sraffa's approach of regarding the wage as being paid at the end of the cycle of production. Given the assumptions, both sides of the above equation can be considered as being expressed in terms of dollars per unit corn produced. The rate of profits shows how many dollars are returned for each dollar invested.

The remaining equation specifying the model relates the revenues, in terms of dollars per unit rice produced, to costs in the rice-producing country. That equation is:

a_r,r (1 + r) + a_0,r w_r/p = 1/p

where

• a_r,r is the amount of rice needed as input per unit rice produced
• a_0,r is the person-years labor needed as input per unit rice produced
• w_r is the wage in units rice in the rice-producing country
• p is the exchange rate of yens per dollar at the end of the year

The above equations express the idea that the capitalists are indifferent between investing their dollars in producing corn in the corn-producing country or producing rice in the rice-producing country.

One can easily solve the above equations for the exchange rate at the end of the year:

p = [(1 - a_0,r w_r)/a_r,r]/[(1 - a_0,c w_c)/a_c,c]

The left-hand side of the above equation is ratio of the exchange rate at the end of the year to the exchange rate at the start of the year. The right-hand side is the quotient of two ratios, each ratio characterizing one of the two countries. These are the ratios of the net product remaining after compensating the workers for their labor power to the outlay needed to produce that surplus.

3.0 Conclusion
This model suggests that the smaller the rate of surplus value the capitalists are able to extract from the workers in a given country, the stronger their currency tends to become.

Update: I originally had the conclusion incorrect.

References

• John Maynard Keynes (1923) A Tract on Monetary Reform
• Paul A. Samuelson (1957) "Wages and Interest: A Modern Dissection of Marxian Economic Models", American Economic Review, V. 47, N. 6 (December): pp. 884-912
• Paul A. Samuelson (1971) "Understanding the Marxian Notion of Exploitation: A Summary of the So-Called 'Transformation Problem' Between Marxian Values and Competitive Prices", Journal of Economic Literature, V. 9, N. 2: pp. 399-431.
• Ian Steedman (editor) (1979) Fundamental Issues in Trade Theory, Macmillan