Proof Of The Falsity Of The Factor Price Equalization "Theorem" (Part 1)

1.0 Introduction
This sequence of posts explores a numeric example. The example was created by L. Mainwaring (1976). Mainwaring used the example to prove the factor price equalization theorem to be false. Before explaining this implication of the example, I want to consider a single country.

2.0 Technology
Consider a simple economy in which three commodities, iron, steel, and corn, are produced from inputs of labor, iron, steel, and corn. All production processes in in this example require a year to complete and exhibit Constant Returns to Scale. One process is known for producing iron, and two processes each are known for producing steel and corn. These processes are shown in Table 2-1. The inputs for each process are purchased at the beginning of year. The inputs provide their services over the course of the year, and the iron, steel, and corn inputs are totally used up each year in the production processes. The outputs become available at the end of the year.

Table 2-1: Technology

INPUTS HIRED AT START OF YEAR	IRON- PRODUCING PROCESS (A)	FIRST STEEL- PRODUCING PROCESS (B)	SECOND STEEL- PRODUCING PROCESS (C)	FIRST CORN- PRODUCING PROCESS (D)	SECOND CORN- PRODUCING PROCESS (E)
Labor	1/20 Person-Year	1/5 Person-Year	3/50 Person-Year	9/20 Person-Year	19/100 Person-Year
Iron	0 Ton Iron	1/5 Ton Iron	0 Ton Iron	1/10 Ton Iron	1 18/25 Tons Iron
Steel	1/5 Ton Steel	0 Ton Steel	0 Ton Steel	1/2 Ton Steel	0 Ton Steel
Corn	0 Bushel Corn	1/10 Bushel Corn	8/25 Bushel Corn	0 Bushel Corn	0 Bushel Corn
OUTPUT	1 Ton Iron	1 Ton Steel	1 Ton Steel	1 Bushel Corn	1 Bushel Corn

A technique consists of three processes, where one process is used to produce iron, a second process produces steel, and the third process produces corn. As shown in Table 2-2, three techniques are available in this economy. Note that for each technique, no commodity can be produced without inputs either directly or indirectly of each other commodity For example, consider the Delta technique. Since corn is produced in this technique with process E, iron is required to be reproduced for production to continue year after year. Steel is required to reproduce iron by process A in this technique. And corn is required to produce steel by process C. In the jargon, iron, steel, and corn are all said to be “basic commodities”. Sraffa invented this terminology.

Table 2-2: Techniques And Processes

Technique	Processes
Alpha	A, B, D
Beta	A, B, E
Gamma	A, C, D
Delta	A, C, E

I leave as an exercise for the reader to check that, for each technique, some level of operation of the processes comprising the technique results in a positive net output. That is, after replacing the commodities used up in production, some output is left over to be used for consumption or accumulation. One method of checking this condition is to confirm the Hawkins-Simon conditions are met for each technique.

Consider cost-minimizing (or profit-maximizing) competitive firms facing this technology. Suppose that, although labor is hired at the start of the year, the workers are paid their wages at the end of the year. Iron, steel, and corn inputs are paid for at the beginning of the year, and interest (or profit) is charged on these payments. For each level of the rate of profits, which technique is cost minimizing? Is one technique not cost-minimizing at any rate of profits? I answer these questions in the next part by presenting the results of some tedious algebra.

References

• Mainwaring, L. (1976). "Relative Prices and 'Factor Price' Equalisation in a Heterogeneous Capital Goods Model", Australian Economic Papers, Republished in Fundamental Issues in Trade Theory (Ed. by Ian Steedman), Macmillan, 1979.