## Sunday, November 29, 2009

### A Taxonomy Of The Effects Of Wicksell Effects

Consider a typical circulating capital model, in which commodities are produced from commodities and labor. The technique in use is described by a square Leontief input-output matrix and a vector of labor coefficients. In a long run equilibrium, in which prices are stationary, the technique is selected from a set of techniques to minimize production costs at a given interest rate. That set is known as the technology.

Suppose the composition and quantity of output is taken as given, along with the interest rate and the technology. The difference in the value of the capital goods at two different interest rates is the sum of the price Wicksell and real Wicksell effects. The price Wicksell effect is the sum of the differences in prices among the capital goods for a given technique. But the cost-minimizing technique might not be the same at two interest rates. The real Wicksell effect is the difference in the value of the capital goods for two techniques, given the price system at a one interest rate.

I want to compare the relative magnitude of price and real Wicksell effects at a given interest rate. Thus, I want to consider derivatives at a given interest rate. Therefore, suppose the technology consists of a continuum of techniques that might be eligible along the so-called factor price frontier. Table 1 shows all combinations of price and real Wicksell effects. A Wicksell effect is negative when the equilibrium at the higher interest rate has a lower value of capital, from the effects of price and quantity changes, respectively.

 Technology Property LaborMarketResponse CapitalMarketResponse PriceWicksellEffects RealWicksellEffects HigherWage LowerInterestRate A Negative Negative LessEmployment IncreasedValue ofCapital B Negative Positive MoreEmployment Indeterminate C Positive Negative LessEmployment Indeterminate D Positive Positive MoreEmployment DecreasedValue ofCapital E Zero Negative LessEmployment IncreasedValue ofCapital F? Zero Positive MoreEmployment DecreasedValue ofCapital G Negative Zero UnchangedEmployment IncreasedValue ofCapital H Positive Zero UnchangedEmployment DecreasedValue ofCapital I Zero Zero UnchangedEmployment UnchangedValue ofCapital

Row A in Table 1 conforms to the outdated neoclassical intuition of equilibrium prices as indices of relative scarcity. But, as Edwin Burmeister has noted, nobody knows what special case assumptions need to be imposed on technology to ensure that Wicksell effects happen to fall in any given direction.

I have the response in the capital market shown as indeterminate for rows B and C. The claim is that, for the case of a technology representable by a continuum of techniques, the price Wicksell effect can, but need not, swamp the real Wicksell effect. It is essential for this swamping to occur at a single interest rate that the technology be continuous. Pierangelo Garegnani, Heinz D. Kurz & Neri Salvadori, and Saverio M. Fratini have examples that illustrate some possibilities with a continuum of techniques.

Row E is the case of Samuelson's Surrogate Production Function. Price Wicksell effects are zero when the factor price curve for a given technique is a straight line. The question mark after the label for Row F reflects my belief that this row catalogs an impossibility. If factor price curves are straight lines along their entire length, capital-reversing cannot arise.

Rows G, H, and I are cases in which the real Wicksell effect is zero. The real Wicksell effect is zero in the discrete case when the factor price curves are tangent at a switch point. I'm not sure how this extends to the continuous case, in which all points along the factor price frontier are non-switching points. If the Row I case is possible, the technique is not determined by the location of the corresponding factor price curve. I think this may be so for non-straight line factor price curves, but I'm unsure about this case.

These remarks suggest a research program. First, demonstrate that no possibilities exist that are not listed in Table 1. This would seem to be obvious. But I don't understand Andreu Mas-Colell's paper "Capital Theory Paradoxes: Anything Goes" (in Joan Robinson and Modern Economic Theory (ed. by G. Feiwel) (1989)). He shows some multi-valued relations where I would expect functions. Second, for those rows that are impossible in Table 1, demonstrate this impossibility. Third, for each possible row, construct a numeric example. For rows B and C, one should construct at least two examples, one for each direction of the capital market response. I suppose a third example, in which price and real Wicksell effects are exactly matched in magnitude would be amusing. Much of this research would be non-original; many components are in the literature.