Friday, August 08, 2014

Labor Demand In A Fog

Figure 1: Labor Demanded Per Unit Output in a Stationary State
1.0 Introduction

As a Sraffian, I have no problem with open models in which room exists for exogenous political forces to determine distribution. The example here, though, has more indeterminancy than I expect.

2.0 Technology

Consider a simple capitalist economy, composed of workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. The workers are paid a wage, w (in units of bushels corn per person year) out of the harvest. Capitalists obtain the rate of profits, r. The technology1 consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques. In each technique, a bushel of corn is produced from inputs of:

  • l0 person-years of labor performed in the year of the harvest.
  • l1 person-years of labor performed one year before the harvest-year.
  • l2 person-years of (unassisted) labor performed two years before the harvest-year.

Each technique is determined, given the values of the two index variables s and t. s is a non-negative real number less than or equal to the parameter c. t is a non-negative real number.

l0(t, s) = A - B + (t + 1)(B - s)/2
l1(t, s) = s
l2(t, s) = (B - s)/[2 (t + 1)]

where A, B, and c are positive constants and


In effect, the above has traced out isoquants for a production function, where the quantity of output is a function of dated labor inputs2. For a given value of the index variable s, labor inputs in the harvest year and two years before can be traded off. That is, if the amount of labor two years before is lower, then more labor must be expended in the harvest year. Likewise, for a given value of the index variable t, more labor being expended one year before the harvest mandates less labor being expended in the harvest year and two years before. So this specification of technology allows for substitution among inputs, at least in comparing steady states3, 4.

3.0 Choice of Technique

As usual, I consider a competitive, steady state economy in which capitalists have chosen the cost-minimizing technique, at an exogenously specified wage or rate of profits. Consider the function v(r, t, s):

v(r, t, s) = (1 + r)2 l2(t, s) + (1 + r) l1(t, s) + l0(t, s)

Take a bushel of corn as numeraire. The condition that all income be paid out to workers and capitalists leads to a wage-rate of profits curve, as a function of the rate of profits and the technique (specified by the values of the two index variables):

w(r, t, s) = 1/v(r, t, s)

A wage-rate of profits curve can be drawn for each technique. The wage-rate of profits frontier, consistent with a competitive steady-state, is the outer envelope (Figure 2) of all these curves. That is, for a given wage, one finds the values of the index variables that maximizes the wage among all techniques. This maximization does not fix s. But, for each value of s, the maximum is found by setting the index variable t equal to the rate of profits r. The equation for the frontier is:

w(r) = 1/(A + Br)

Notice the frontier is independent of the labor input, s, in the first year before the harvest. In this case, each point on the frontier is consistent with a continuum of profit-maximizing techniques. And these techniques vary continuously along the frontier. None of this indeterminancy is apparent by looking at the frontier5.

Figure 2: The Wage-Rate of Profits Frontier
4.0 Labor Inputs

The analysis of the choice of technique allows one to plot labor inputs versus selected variables from the price system. In any year in a stationary state, some workers will be gathering the harvest, some will be working on preparing for the harvest one year out, and some will be working on preparing for the harvest two years out. So employment, per the unvarying net output, is the sum of l0, l1, and l2. And these labor inputs can be found from a given rate of profits and a choice of s. From the wage-rate of profits frontier, one can calculate the wage for any given rate of profits. Thus, one has the two dimensions needed to draw the curves in Figure 1. One sees that, for any given wage in an interval from zero to a maximum, the quantity of labor demanded by the firms per unit output is a relation, not a function of the wage. If the relation shown were considered to be a labor demand curve, the curve would have a certain (varying) thickness.

5.0 Capital Inputs

The analysis of the choice of technique also allows one to plot the value of capital goods6 versus selected variables from the price system. I define the value of capital per unit output, given the rates of profit and the technique like so:

k(r, t, s) = (1 + r) l2(t, s) w + l1(t, s) w

This definition is such that the value of capital advanced, discounted to harvest time, and the wages paid out of the harvest add up to unity:

k(r, t, s) (1 + r) + l0(t, s) w = 1

Impose the condition here, too, that only cost-minimizing techniques are considered for a given rate of profits. Then one obtains the curves shown in Figure 3. Here, too, the analysis yields an obvious indeterminancy.

Figure 3: Capital Demanded Per Unit Output in a Stationary State
6.0 Conclusion

Does this example undermine Sraffian analysis, as well as introductory textbook labor economics?

  1. Notation and numerical values are chosen to be consistent with a past post.
  2. I am unsure how to explicitly represent such a production function.
  3. With three or more inputs, some complementarity among inputs is possible. I am not sure how to express this formally.
  4. I suppose the production function consistent with the data exhibits non-negative marginal returns. I am not sure it would exhibit non-increasing marginal returns. If not, I would like to see either a proof, in the general case with n dated labor inputs, that the shaded violet regions cannot arise, given such conventional properties for a production function. Or, I would like to see a concrete numerical illustration like mine, but with such conventional properties shown to hold.
  5. Also, notice the analysis of the choice of technique leads to simpler equations than those in the specification of the technology. This is not an accident.
  6. I gather that, for any given value of s, unassisted labor two years before the harvest can be used to produce one of a continuum of capital goods, depending on the value of t. And once one of these capital goods is selected, the minimum dated labor inputs in each of the three years are fixed. Maybe this way of thinking about capital goods makes issues of convexity raised in Footnote 4 of little interest.
  • Enrico Bellino (1993). Continuous Switching in Linear Production Models, Manchester School, V. 61, Iss. 2 (June): pp. 185-201.
  • Christian Bidard (2014). The Wage Curve in Austrian Models, Centro Sraffa Working Papers n. 3 (June).

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