Skilled Labor, Labor Values, Prices Of Production

1.0 Introduction

Suppose not all labor in a capitalist economy is 'unskilled'. Some jobs require specific skills that somehow command a premium. How can heterogeneous labor be handled in modern treatments of classical and Marxian political economy?

My inclination is just to assume that relative wages for different labor categories are stable. I read Ricardo and others as doing the same:

"I must not be supposed to be inattentive to the different qualities of labour, and the difficulty of comparing an hour's or a day's labour, in one employment, with the same duration of labour in another. The estimation4 in which different qualities of labour are held, comes soon to be adjusted in the market with sufficient precision for all practical purposes, and depends much on the comparative skill of the labourer, and intensity of the labour performed. The scale, when once formed, is liable to little variation." -- David Ricardo, Principles

But in this post, I want to consider a different approach. I assume that skilled labor is produced by prior training. This post is only a start.

2.0 Technology

I consider the simplest model of the production of commodities for making my point. Table 1 specifies the technology for this example. Each column shows the bushels corn, person-years of skilled labor, and the person-years of unskilled labor paid by the capitalist operating that process to produce one unit of output. I assume constant returns to scale.

Table 1: Technology

Input	Sector
	Training	Corn
Corn	a1,1 Bushels	a1,2 Bushels
Skilled Labor	c1,1 Person-Years	c1,2 Person-Years
Unkilled Labor	a0,1 Person-Years	a0,2 Person-Years
OUTPUT	One Person-Year	One Bushel

I assume the skilled labor produced by the training sector only has skills for the next year. After working for one year, a skilled worker needs to be retrained. In this way, the model resembles a model with only circulating capital and no fixed capital. The inputs to the training sector do not include the workers who emerge as skilled workers. Rather, those workers are customers, paying the capitalist, who obtains returns from operating the process in the training seector.

All coefficients of production are assumed to be positive. I assume the Hawkins-Simon conditions:

a_{1, 2} < 1

c_{1, 1} < 1

a_{1, 1} c_{1, 2} < (1 - a_{1, 2})(1 - c_{1, 1})

These asumptions ensure that the economy can produce a surplus product.

3.0 Labor Values

I first want to calculate labor values. Labor values are defined in terms of unskilled labor. Introduce the following two variables:

• v₁ is the (unskilled) labor value of corn, in unskilled person-years per bushel.
• v₂ is the (unskilled) labor value of skilled labor, in unskilled person-years per skilled person-year.

The labor value of a bushel corn is the sum of the labor values of the inputs needed to produce it.

v₁ = a_{0, 2} + a_{1, 2} v₁ + c_{1, 2} v₂

The specification of a process for training skilled labor allows for a parallel definition of the labor value of skilled labor:

v₂ = a_{0, 1} + a_{1, 1} v₁ + c_{1, 1} v₂

The above is a system of two equations in two unknowns. The following abbreviation is useful in setting out the solution:

denom = (1 - a_{1, 2})(1 - c_{1, 1}) - a_{1, 1} c_{1, 2}

The solution is:

v₁ = [a_{0, 1} c_{1, 2} + a_{0, 2}(1 - c_{1, 1})]/denom

v₂ = [a_{0, 1}(1 - a_{1, 2}) + a_{0, 2} a_{1, 1}]/denom

The Hawkins-Simon conditiones guarantee that labor values are positive. Despite the existence of heterogeneous labor, the labor value of corn, in terms of one type of labor is well-defined.

4.0 Prices of Production

I skip exploring various ratios and other aspects of the system of labor values. I introduce the following variables for prices of production:

• p₁ is price of corn, in numeraire-uints per bushel.
• w₁ is the wage of unskilled labor, in numeraire units per person-year.
• w₂ is the wage of skilled labor, in numeraire units per person-year.
• r is the rate of profits, a pure number.

The system of equations for prices of production includes an equation for corn:

(p₁ a_1,2)(1 + r) + c_1,2 w₂ + a_0,2 w₁ = p₁

In this equation, wages of skilled and unskilled labor are paid out of the surplus, at the end of the year. The following equation applies to the training sector:

(p₁ a_1,1)(1 + r) + c_1,1 w₂ + a_0,1 w₁ = (w₂ - w₁)/(1 + r)

The Right Hand Side is the present value of the skill premium obtained by a trained worker.

Specifying the numeraire removes one degree of freedom for the system of prices of production. One degree of freedom remains.If the wage of unskilled labor is taken as given, the system is closed.

5.0 Conclusion

Obviously, this analysis can go in many directions. What would it mean for the organic composition of of capital not to vary between the training sector and the corn-producing sector? How does Marx's account of exploitation work here? Can skilled workers exploit unskilled worker, as in Ian Steedman's book? How can I draw on the theory of fixed capital to account for skills lasting for multiple periods? Can I introduce risk and distinguish between the rate of profits and the interest rate used for time-discounting? How does the the theory of rent apply to inate talents? (Bootlickers will insist this is the dominant case.)

If you want to apply this approach empirically, you must answer some of these questions. The approach can be extended to more sectors and more types of labor. I have seen some input-output tables broken down to have two types of labor.