Labor Reversing Without Capital: An Example

Figure 1: Skilled Labor Hired by Firms per Unit Output

1.0 Introduction

This example is from Opocher and Steedman (2015). They present many examples in which the reader is expected to work them out, as illustrated in this post.

This is an example in which cost-minimizing firms desire to hire more labor (of a specific type) for an increased wage, around a specific wage. This example is of a firm producing a single commodity from inputs of specific types of land and specific types of labor. No produced capital goods exist in this example, and the interest rate is assumed to be zero. Yet perverse behavior arises on the demand side of markets for factors of production anyway - where results are called perverse merely if they violate neoclassical intuitions shown to be mistaken half a century ago. The most complicated aspect of this example is that some techniques of production are specific to specific types of land.

2.0 Indirect Average Cost Functions

Consider a firm that produces widgets from inputs of skilled labor, unskilled labor, and land of one of two types. Suppose the price of widgets is unity. Define:

• p_α is the rent for alpha-type land.
• p_β is the rent for beta-type land.
• w₁ is the wage for unskilled labor.
• w₂ is the wage for skilled labor.

The indirect average cost function for widgets produced on land of type alpha is:

c_α(p_α, w₁, w₂) = (1/2)[(w₁ p_α)^1/2 + (w₁ w₂)^1/2

+ (w₂ p_α)^1/2]

The indirect average cost function for widgets produced on land of type beta is:

c_β(p_β, w₁, w₂) = (3/5)(w₁ p_α)^1/2 + (3/10)(w₁ w₂)^1/2

+ (11/20)(w₂ p_α)^1/2

The indirect average cost function shows the average cost of producing each widget, 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.

These indirect average cost functions are homogeneous of the first degree. For the indirect average cost function for land of type alpha, this property is expressed as:

c_α(ap_α, aw₁, aw₂) = a c_α(p_α, w₁, w₂)

This a traditional assumption for cost functions.

Consider the indirect average cost function for a specific type of land. That type of land, unskilled labor, and skilled labor are substitutes. No inputs are complements in this example. In other words, the off-diagonal elements of the Hessian matrices formed from each indirect average cost function are all positive. The elements along the principal diagonal of each Hessian matrix are negative.

3.0 The Wage-Wage Frontier

Consider a long run equilibrium of the firms in which pure economic profits have been competed away and no firm is making a loss. Perhaps, the prospect of firms entering or exiting the industry has caused this situation to arise. Furthermore, suppose rents for both types of land happen to be unity. (Without this assumption, this example would have two more degrees of freedom.) If firms are producing on a given type of land, the indirect average cost function for that type of land will be equal to unity. For alpha type land, one has:

1 = c_α(1, w₁, w₂)

Or:

w_{1, α} = [(2 - w₂^1/2)/(1 + w₂^1/2)]²

As shown in Figure 2, given the type of land employed, the wage for unskilled labor is a declining function of the wage for skilled labor. The maximum wage for unskilled labor, 4 widgets per person-year, corresponds to skilled labor working for free. Symmetrically, the maximum wage for skilled labor likewise corresponds to unskilled labor working for free.

Equating the indirect average cost function for production on land of type beta yields another trade-off in long run equilibrium between the wages of unskilled and skilled labor.

w_{1, β} = [(20 - 11 w₂^1/2)/(12 + 6 w₂^1/2)]²

When land of type beta is used, the maximum wage for unskilled labor is 2 7/9. The maximum wage for skilled labor is 3 37/121.

Figure 2: Wage-Wage Curves and the Frontier

For some combination of wages of skilled and unskilled labor, firms will be indifferent between producing widgets with land of type alpha and type beta. The cost-minimizing technique at these wages, on each type of land, is equally cheap. These combinations can be found by equating the wages of unskilled labor for the expressions above. After some manipulation, one obtains the equation:

5 w₂ - 9 w₂^1/2 + 4 = 0

This equation can be factored:

(w₂^1/2 - 1)(5 w₂^1/2 - 4) = 0

Firms will thus be indifferent to the type of land used in production for ordered pairs of wages of unskilled and skilled labor, (w₁, w₂), of (1/4, 1) and (4/9, 16/25).

Firms produce widgets on land of type alpha for wages for skilled labor between zero and 16/25, and for wages of skilled labor between one and four. For wages for skilled labor between 16/25 and four, firms produce widgets on land of type beta. The outer frontier allows one to determine the wage of unskilled labor for any feasible wage for skilled labor, given the model assumptions. As well soon be apparent, this is not an example of reswitching. The overall indirect average cost function is almost always differentiable. It is not differentiable only at the two points found by the construction of the outer frontier.

4.0 Land and Labor

We have seen that when rents are unity, long run equilibrium of the firm necessitates that the wages of unskilled labor is a declining function of the wages of skilled labor. Shepherd's lemma can be used to find the coefficients of production for each feasible combination of wages of unskilled and skilled labor. The quantity of each input the firm wants to hire per unit output is the derivative of the indirect average cost function with respect to the price of that input. Thus, when land of type alpha is used, the number of acres of land employed per unit output of widgets is:

t_α(w₁, w₂) = (1/4)(w₁^1/2 + w₂^1/2)

The number of acres of land of type beta per unit output of widgets, when land of that type is used, is:

t_β(w₁, w₂) = (1/40)(12 w₁^1/2 + 11 w₂^1/2)

In what I hope is obvious notation, person-years of unskilled labor employed per unit output of widgets is, depending on the type of land used:

l_{1, α}(w₁, w₂) = (1/4)(1 + w₂^1/2)/(w₁^1/2)

l_{1, β}(w₁, w₂) = (3/20)(2 + w₂^1/2)/(w₁^1/2)

Finally, person-years of skilled labor employed per unit output of widgets is given by one of the following two functions of wages:

l_{2, α}(w₁, w₂) = (1/4)(1 + w₁^1/2)/(w₂^1/2)

l_{2, β}(w₁, w₂) = (1/40)(6 w₁^1/2 + 11)/(w₂^1/2)

5.0 Bringing it all Together

The above algebra can be used to generate various graphs. Figure 1 shows person-years of skilled labor firms desire to hire per unit output. As one moves to the right in the figure, the wage of skilled labor rises and the wage of unskilled labor falls. But at every point in the figure, the wages of the two types of labor are such as to maintain wages as on the outer frontier in Figure 2. That is, firms are minimizing costs, and the output price and input prices are such as to enforce the equilibrium condition that no pure economic profits are available in this industry.

Figure 3 shows the analogous graph for unskilled labor. The point for wages of 4/9 widgets per person-years and 16/25 widgets per person-year for unskilled and skilled labor, respectively, is emphasized. At any point to the left, wages for unskilled labor are higher, and wages for skilled labor are lower. And an infinitesimal variation around this point is associated with firms wanting to employ unskilled labor more intensively when their wage is relatively higher.

Figure 3: Unskilled Labor Hired by Firms per Unit Output

Reswitching of techniques arises when one technique of production is cost-minimizing at, say, a high and low wage but not at an intermediate wage. A technique of production is specified by four coefficients of production in this example. The amount of skilled labor and unskilled labor hired per unit output are two of these coefficients. The acres of land of each type rented per unit output are the other two. The latter two coefficients of production obviously vary, depending on which type of land can be used in a cost-minimizing technique. In fact, the coefficients of production for the type of land not employed is zero. As can be seen in Figures 1 and 3, the coefficient of productions for the two types of labor vary monotonically with relative wages, given the type of land employed.

At one of the two switch points highlighted in Figure 2, two techniques of production are cost-minimizing. (This is the definition of a switch point.) In one technique, one type of land is used. And in the other, the other type of land is used. But a different pair of techniques of production is cost-minimizing at the other switch point. The coefficients of production vary among, for example, the cost-minimizing techniques in which alpha-type land is used at a switch point. Hence, as noted, no reswitching of techniques exists in this example.

6.0 Conclusion

This example has cost-minimizing firms in equilibrium in a single industry. Price and quantity relationships among factors of production have been analyzed, where factors of production consist of land of two types and labor of two types. Quantity relationships have been presented in terms of inputs per unit output for a firm. For simplicity, only the case in which the interest rate is zero and rents of land per acre are unity has been considered. When beta type land is adopted, more acres are cultivated for alpha-type land, for the same level of output. Thus, land has a higher proportion of total unit cost when beta-type land is used. Both skilled and unskilled labor are a lower proportion of total unit cost (as seen in Figures 1 and 3) than they would be if alpha type land was employed. A wage has been found for unskilled labor in which a higher relative wage for unskilled labor is associated with firms desiring to hire more unskilled labor per unit output. And a different relative wage for skilled labor has been found with the analogous property.

I wonder whether an example can be found with a continuum of types of land in which the analog of Figures 1 and 2 come out as continuous U-shaped curves.

So much for explaining wages and employment by well-behaved supply and demand curves in competitive labor markets.

Reference

• Opocher, Arrigo and Ian Steedman (2015). Full Industry Equilibrium: A Theory of the Industrial Long Run, Cambridge University Press