Friday, June 15, 2012

Labor Intensities In Producing Capital And Consumer Goods

Figure 1: Rate of Profits Versus Capital Per Worker

1.0 Introduction

If you study mainstream economics, you get used to economists making arbitrary special-case assumptions. For example, assume that more labor-intensive processes are used in producing capital goods than in producing consumer goods. Less means of production are used, in some sense, in producing means of production than in producing commodities directly for the consumer. Or assume the opposite, that less labor-intensive processes are used in producing capital goods. I do not care; either assumption is ad hoc. One can find textbook authors aware of the arbitrary nature of their assumptions:

"Thus, if the consumer good is more capital intensive, ... If the consumer good is more labour-intensive, i.e. the investment good is more capital intensive ... Rybczynski’s theorem ...

It must be admitted that the condition on relative capital-intensities is not very plausible, not very intuitive, and not really verified or refuted empirically."

-- A. K. Dixit, The Theory of Equilibrium Growth (Oxford, 1976): 127-130.

This post explores, in a simple two-sector model, the consequences of these different assumptions. I here emphasize the direction of price Wicksell effects. I have explored price Wicksell effects before. I find the possibility of positive real Wicksell effects more intriguing. Perhaps, however, variations in the direction of price Wicksell effects leads to the impossibility of being able to impose any well-founded limitation on the direction of real Wicksell effects.

2.0 The Technique In Use

Consider an economy in which two commodities, steel and corn, are produced. Steel is a capital good. It’s only use is as an input in the production of commodities. Corn is a consumer good. The entrepreneurs know only one process for producing each. That is, the technique of production is given in this model. Table 1 defines the coefficients of production, a0,1, a0,2, a1,1, and a1,2. Assume each process exhibits Constant Returns to Scale (CRS). In each process, a certain amount of labor works with steel to produce the output. Doubling, say, the inputs of labor and steel doubles the output.

Table 1: The Technology
Labora0,1 person-yearsa0,2 person-years
Steela1,1 tonsa1,2 tons
Outputs1 ton steel1 bushel corn

The assumptions explored in this post can be stated in terms of relative ratios of coefficients of production (Table 2). One compares the ratio of physical inputs of labor and the capital good in each industry in producing one unit, gross, of the output of that industry. Three special cases thereby arise. For notational convenience below, define the following function of the coefficients of production:

d(r) = a0,2 - (a0,2a1,1 - a0,1a1,2)(1 + r)
Table 2: Arbitrary Special Cases
Case DescriptionAssumptionConsequence
1Corn More
a0,1/a1,1 < a0,2/a1,2Positive Price
Wicksell Effect
2Equal Organic
Compositions of Capital
a0,1/a1,1 = a0,2/a1,2Zero Price
Wicksell Effect
3Steel More
a0,1/a1,1 > a0,2/a1,2Negative Price
Wicksell Effect

3.0 Quantity Flows

The given coefficients of production can be used to calculate the scale at which each industry must be operated to produce a given net output. Table 3 shows the quantity flows needed to produce one bushel of corn, net. For these quantity flows, the steel, a1,2/(1 - a1,1) tons, productivity consumed in producing steel and corn is replaced at the end of the year by the output of the steel industry. The total labor input across industries is d(0)/(1 - a1,1) person-years per bushel corn produced.

Table 3: Quantity Flows to Produce Net Output of One Bushel Corn
Labora0,1a1,2/(1 - a1,1) person-yearsa0,2 person-years
Steela1,1a1,2/(1 - a1,1) tonsa1,2 tons
Outputsa1,2/(1 - a1,1) ton steel1 bushel corn

4.0 Prices

Consider steady states prices. This system of prices is consistent with the smooth reproduction of the economy. The assumption that the same rate of profits is realized in both industries yields the following system of price equations:

a1,1ps(1 + r) + a0,1w = ps
a1,2ps(1 + r) + a0,2w = 1
  • ps is the price of steel (in units of bushels per ton).
  • w is the wage (in units of bushels per person-year).
  • r is the rate of profits

I have chosen a bushel corn, that is, a unit of the consumer good, as the numeraire. It is assumed that wages are paid out of the harvest at the end of the year, not advanced at the beginning of the year.

The price equations comprise a system of two equations in three variables. For a given rate of profits, the system is linear. Consequently, the system can be inverted:

w(r) = [1 - a1,1(1 + r)]/d(r)
ps(r) = a0,1/d(r)

In effect, the price equations have one degree of freedom.

The downward-sloping wage-rate of profits curve (Figure 2) is one manifestation of the class struggle between workers and capitalist. The maximum rate of profits, (1 - a1,1)/a1,1, corresponds to a wage of zero. When the workers receive the entire net output, they get the maximum wage, (1 - a1,1)/d(0). (In drawing the graphs in this post, I have chosen numerical values to fix the intercepts in Figure 2.) The convexities of the wage-rate of profits curves in Figure 2, however, follow from the special-case assumptions. The curve is a straight line for a generic numeraire only in the case of constant organic compositions of capital. It is concave to the origin when the production of consumer goods is more labor-intensive, in some sense, than the production of means of production. It is convex in the third case. (In the general n-good case, the concavity of the wage-rate of profits curve for a technique varies along its extension.)

Figure 2: Wage-Rate of Profits Curves

According to capitalist apologetics, that is, vulgar political economy, capital goods represent deferred consumption ("waiting" or a lengthened "period of production", depending on your taste). At any rate, a physical quantity of capital goods can be evaluated in numeraire units. Capital per worker, k(r), in the production of a unit of the consumer good (Figure 1) is thus:

k(r) = ps(r)a1,2/d(0)

The value of capital per worker is independent of the distribution of income only for constant organic compositions of capital. The slopes, whether increasing or decreasing, follow from the special case assumptions. I gather that the convexities shown in Figure 1 are also not consequences of the specific numeric examples. The variation in the price, with the distribution of income, of a given set of capital goods is known as the price Wicksell effect. A curve sloping upward to the right in Figure 1 is a positive Wicksell effect. A curve sloping downward to the right is a negative Wicksell effect.

5.0 Conclusion

The above analysis can be used to recount various fables. For example, suppose a large part of the workforce spontaneously decides to emigrate. One might expect that labor has become relatively scarcer, with respect to capital, whatever that means. In other words, there is more capital per worker. And that, given diminishing marginal productivity of labor, the real wage, when adjusted to changed conditions, will be higher. In this two-good model, a higher wage is associated with a higher price for the commodity produced by the relatively more labor-intensive industry. Which commodity that is, if any, depends on the special case assumptions.

But does this story make sense? In the first case, more capital per worker is associated with a higher rate of profits and, thus, a lower wage. One requires arbitrary ad-hoc assumptions for price signals to inform entrepreneurs of relative factor scarcities.

Obviously, more than one commodity is consumed in, say, the United States, and some consumer goods may also be used as capital goods. I leave it for others to explore whether National Income and Product Accounts (NIPA) can be used for some sort of notional vertical integration. Can which special case obtains, at a given time in a given country, be determined? If so, what does that tell one?

No comments: