Recurrence Of Capital-Output Ratio Without Reswitching

Figure 1: Recurrence of Capital-Output Ratio

1.0 Introduction

This example is from Arrigo Opocher and Ian Steedman. It illustrates the analysis of an isolated industry in equilibrium. This analysis is therefore more akin to partial equilibrium than to general equilibrium. Sometimes (mainstream?) economists say that the Cambridge Capital Controversies were only about aggregate neoclassical theory, that is, macroeconomics. Or that the CCC has been subsumed by General Equilibrium Theory. The example illustrates that such economists are, as has long been apparent, spouting poppycock.

2.0 Indirect Average Cost Function

Consider a firm that produces widgets from inputs of widgets, unskilled labor, and skilled labor. Let the indirect average cost function be:

c(p, w₁, w₂) = sp + w₁ + w₂

+ 2(pw₁)^1/2 + 2(pw₂)^1/2 + 2γ(w₁w₂)^1/2

where

0 < s < 1

0 < γ

γ ≠ 1

and

• p is the price of a widget. Widgets used as inputs are assumed to be totally consumed in one production period.
• w₁ is the wage for unskilled labor.
• w₂ is the wage for skilled labor.

The indirect average cost function shows the average cost of producing each widget (net), when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm.

This indirect average cost function is homogeneous of the first degree:

c(a p, a w₁, a w₂) = a c(p, w₁, w₂)

This is a conventional assumption for cost functions.

Suppose the firm faces a given price of widgets and given wages for skilled and unskilled labor. By Shephard's lemma, the quantity of each input the firm wants to hire per unit output, given the price of each input, is the derivative of the indirect average cost function with respect to the price of that input. Hence, the capital-output ratio, k(p, w₁, w₂), is:

k(p, w₁, w₂) = ∂c/∂p = s + (w₁/p)^1/2 + (w₂/p)^1/2

Notice that the capital-output ratio is a pure number, unambiguously defined in this example, and independent of prices.

By the same logic, the amount of unskilled labor the managers of the firm desire to hire per widget produced is:

l₁(p, w₁, w₂) = ∂c/∂w₁ = 1 + (p/w₁)^1/2 + γ(w₂/w₁)^1/2

The amount of skilled labor the managers of the firm desire to hire per widget produced is:

l₂(p, w₁, w₂) = ∂c/∂w₂ = 1 + (p/w₂)^1/2 + γ(w₁/w₂)^1/2

The matrix of second derivatives of the indirect average cost function is:

(I am not sure whether it is more common to define the above matrix as the transpose of what I have above.) Anyway, for a positive price of widgets and positive wages, the signs of the second derivatives are as follows:

The signs along the principal diagonal show that the slopes of the per-unit input demand functions slope down. That is, given prices for all but one input, a lower price of that input is associated with a willingness of the firm to employ more of that input per unit produced. The positivity of the off-diagonal elements of the above matrix show that widgets, considered as inputs; unskilled labor; and skilled labor are all substitutes, not complements, in some sense. These signs for the matrix of second derivatives of the indirect average cost function are also conventional properties for cost functions.

3.0 Full Industry Equilibrium

Suppose the industry in which widgets are produced has no barriers to entry or exit. Thus, in the long run, economic profits will have been competed away. For firms to be earning no economic profits, the price of widgets must be equal to the average cost of manufacturing them:

p = c(p, w₁, w₂)

So far, no numeraire has been specified. Let widgets themselves be numeraire. Then:

1 = c(w₁, w₂)

where the argument in the indirect average cost function for widgets has been dropped as otiose.

Consider various levels of w₁, the wage of unskilled labor. For the industry to continue to be in long run equilibrium, the wage of skilled labor, w₂, must vary as well, thereby leaving the average cost of producing a widget as unity. Figure 2 illustrates the resulting wage-wage frontier. (Figures are drawn for s = 1/10 and γ = 2/3.) The highest wage for unskilled labor (when the wage for skilled labor is zero) is ((2 - s)^1/2 - 1)². Since this model is symmetric in skilled and unskilled labor, the highest wage for unskilled labor is likewise ((2 - s)^1/2 - 1)². As long as the rate of accounting profits is zero and technology is given, the wage of unskilled labor can only be higher if the wage of skilled labor is lower.

Figure 2: Wage-Wage Frontier

The wage-wage frontier can be used to find the wage of skilled labor for a given wage of unskilled labor between zero and the maximum. In other words, the frontier is helpful in calculating the ratio of the wage of skilled labor to the wage of unskilled labor, given the wage of unskilled labor. This ratio of wages is independent of the choice of the numeraire.

4.0 Capital and Labor

With the chosen numeraire, the capital-output ratio is:

k(w₁, w₂) = s + (w₁)^1/2 + (w₂)^1/2

Given the wage of unskilled labor, one can find the wage of skilled labor and, consequently, both the ratio of wages of the two types of labor and the capital-output ratio. Figure 1, at the start of this post, graphs the capital-output ratio as the derived function of the ratio of wages.

The capital-output ratio is the same when either skilled or unskilled labor is earning their maximum wage, with the other type of labor being paid a wage of zero. In these two extreme cases, the capital-output ratio is (2 - s)^1/2 - (1 - s). Likewise for any ratio but one of the wage of skilled labor to the wage of unskilled labor between these extremes of zero and infinity, the capital-labor ratio is non-unique. The exception is the ratio of wages at which the function in Figure 1 peaks.

One can see that recurrence of the capital-output ratio is not reswitching. Figures 3 and 4 show, respectively, unskilled labor and skilled labor per unit output as a function of the ratio of wages. As shown in Figure 3, a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ more unskilled labor per unit output. Likewise, a a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ less skilled labor per unit output. As far as unproduced inputs go, this example of the isolated firm in long run equilibrium does not contradict outdated and exploded neoclassical intuitions about substitution and the mistaken notion of equilibrium prices as scarcity indices. But, since the functions in Figures 3 and 4 are monotonic, no reswitching of techniques arises in this example.

Figure 3: Unskilled Labor Employed per Unit Output

Figure 4: Skilled Labor Employed per Unit Output

5.0 Conclusion

This post has presented an example of an isolated firm in a long period equilibrium. The indirect average cost function, which includes the cost of the use of an input which itself is produced by the firm's industry, otherwise has utterly conventional properties. The analysis of the firm in a long run equilibrium demonstrates that it is an incoherent thought experiment to consider the equilibrium response of the firm to the variation of one price at a time. Only the variation of more than one price at a time can yield an equilibrium analysis that could be at all empirically relevant.

A result of this analysis is to reveal a non-monotonic response of the capital-output ratio to variations in the relative prices of the two unproduced inputs used by this firm in production. In fact, every possible capital-output ratio, except for one, recurs in the example. This is a step in an argument leading to the conclusion that economic theory is consistent with competitive firms wanting to employ more input per unit output for higher prices of that input, a finding that seems consistent with empirical results.