A Theorem for Capital-Reversing

Figure 1: The Wage Frontier for a Numeric Example of a Real Wicksell Effect of Zero

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 coefficients for Alpha decrease at the rate σ₁ or σ₂, while the labor coefficient for the nth industry in Beta decreases at the rate σ₂. Then the wage curves for Alpha and Beta intersect at a rate of profits of zero at time t₁ if

σ₂ t₁ = σ₁ t₁ - ln[ -z₁/z₂]

where z₁ is a linear combination of the values of the labor coefficients at time zero in the Alpha technique that decrease at rate σ₁, and z₂ is a linear combination of the remaining labor coefficients at time zero in the Alpha technique and of the labor coefficient at time zero in the Beta technique for the process producing the nth commodity.

Proof: Left as an exercise for the reader.

I consider my proof to be inelegant. This theorem is related to my previous theorem. (I've updated that post.)

Thee theorem gives an explicit condition for the wage curves for the Alpha and Beta techniques to intersect at a rate of profits of zero percent at time t₁. 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.

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 net output economy as a whole.

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 negative real Wicksell effect. Less labor is employed, per unit output of net product, at a higher wage around the switch point. After this time, it is associated with a positive real Wicksell effect. More labor is employed in the economy as a whole, given net output, at a higher wage around the switch point. The theorem gives conditions for capital-reversing to emerge, given another switch point on the frontier for the mentioned techniques.