A Theorem For The Reverse Substitution Of Labor

Figure 1: The Wage Frontier for a Numeric Example

Theorem: Consider a model of an economy in which n commodities are produced by means of commodities. Let Alpha be a technique in which each of the n commodities is produced by a fixed-coefficients, constant-returns-to-scale process. Suppose the Beta technique differs from Alpha only in the process operated in the nth industry. For simplicity, assume all n commodities are Sraffian basics in both techniques. Let both techniques undergo technical change, with only labor coefficients varying through time. The labor coefficient for the nth industry declines at the rate ρ for the Alpha technique:

a^α_{0, n}(t) = a^α_{0, n}(0) e^{-ρ t}

The corresponding labor coefficient for the Beta technique declines at the rate σ:

a^β_{0, n}(t) = a^β_{0, n}(0) e^{-σ t}

Then the wage curves for the Alpha and Beta techniques intersect at time t₀ at a rate of profits of -100 percent if the following condition holds:

σ t₀ = ρ t₀ - ln[ a^α_{0, n}(0)/a^β_{0, n} (0)]

Proof: Left as an exercise for the reader.

I arrived at this theorem from a somewhat more general setting. Assume that in each industry, M processes are available to produce the corresponding commodity, at each instant in time, and that each of these processes has constant returns to scale. Each of these processes requires a positive input of labor. Consider the M techniques (out of Mⁿ techniques) in which each commodity is produced, and the mth process in each industry is operated for the mth technique. Suppose Harrod-neutral technical change occurs for each one of these techniques, with the rate of increase of labor productivity varying among the techniques.

The theorem gives an explicit condition for the wage curves for the Alpha and Beta techniques to intersect at a rate of profits of -100 percent at time t₀. Suppose a switch point also exists at this time at a positive rate of profits that is less than the minimum of the maximum rate of profits for the Alpha and Beta techniques. At that time, one has:

a^α_{0, n}(t₀) = a^β_{0, n}(t₀)

Around the switch point, a variation in the rate of profits or the wage is associated with no change in the quantity of labor hired per unit of gross output in the nth industry.

The wage frontier illustrates for a numeric example with three produced commodities and two processes available in each industry. The techniques mentioned in the theorem are labeled "Gamma" and "Delta" in this example. Before the illustrated time in the example, this switch point is associated with a forward substitution of labor, in which less labor is employed in the nth industry per unit output of gross product of that industry. After this time, it is associated with a negative substitution of labor, in which increased employment per unit of gross product is associated with an increased wage around the switch point.

The ability to explicitly state mathematical theorems is a step forward for my approach of using fluke cases to partition parameter spaces associated with models of prices of production.