1.0 IntroductionI have previously suggested a
taxonomy of Wicksell effects. This post presents an example with:
 The costminimizing technique varying continuously along the socalled factorprice frontier
 Negative price Wicksell effects
 Positive real Wicksell effects
 Price Wicksell effects greater in magnitude than real Wicksell effects.
This example is due to Saverio M. Fratini (
"Reswitching and Decreasing Demand for Capital").
2.0 TechnologySuppose technology consists of a continuum of techniques indexed by the parameter θ, where θ is a real number restricted to the interval [0, 1]:
0 ≤ θ ≤ 1
Each technique consists of the three ConstantReturnstoScale processes in Table 1. No commodity is basic, in Sraffa's sense, in any technique in this technology. In the first process in a technique, θgrade iron is produced directly from unassisted labor. In the second process, labor transforms the θgrade iron into θgrade steel. Finally, in the third process, labor transforms θgrade steel into corn, the consumption good in the model. All processes take a year to complete, and all processes totally use up their input.
Table 1: The Technique Indexed by θInputs  Industry Sector 
θGrade Iron  θGrade Steel  Corn 
Labor (PersonYrs)  1/(1 + θ)  θ  3/(1 + θ) 
Iron (Tons)  0  1  0 
Steel (Tons)  0  0  1 
Corn (Bushels)  0  0  0 
Output  1 Ton  1 Ton  1 Bushel 
Capital goods are specific in their uses in this example. θ
_{1}grade steel cannot be made out of θ
_{2}grade iron when θ
_{1} ≠ θ
_{2}.
3.0 StationaryState Quantity FlowsSuppose in Table 1 that:
 The first process is used to produce (1 + θ)/(4 + θ + θ^{2}) tons of θgrade iron
 The second process is used to produce (1 + θ)/(4 + θ + θ^{2}) tons of θgrade steel
 The third process is used to produce (1 + θ)/(4 + θ + θ^{2}) bushels corn
Then one personyear would be employed over these three processes. Capital goods would consist of (1 + θ)/(4 + θ + θ
^{2}) tons of θgrade iron and (1 + θ)/(4 + θ + θ
^{2}) tons of θgrade steel. The capital goods would be used up throughout the latter two sectors, but reproduced at the end of the year. Net output would consist of (1 + θ)/(4 + θ + θ
^{2}) bushels corn per personyear.
4.0 PricesGiven the technique, stationary state prices must satisfy the following three equations:
[1/(1 + θ)] w = p_{1}
p_{1}(1 + r) + θ w = p_{2}
p_{2}(1 + r) + [3/(1 + θ)] w = 1
where:
 p_{1} is the price of a ton θgrade iron;
 p_{2} is the price of a ton θgrade steel;
 w is the wage;
 r is the rate of profits.
A bushel corn is the numeraire. The above equations embody the assumption that labor is paid at the end of the year.
The above is a system of three equations in four unknowns, given the technique. It is a linear system, given the rate of profits. The solution in terms of the rate of profits is easily found. The socalled factorprice curve for a technique is:
w(r, θ) = (1 + θ)/[3 + θ(1 + θ)(1 + r) + (1 + r)^{2}]
The price of a ton θgrade iron is:
p_{1}(r, θ) = 1/[3 + θ(1 + θ)(1 + r) + (1 + r)^{2}]
The price of a ton θgrade steel is:
p_{2}(r, θ) = [(1 + r) + θ(1 + θ)]/[3 + θ(1 + θ)(1 + r) + (1 + r)^{2}]
Given the technique and the rate of profits, these prices can be used to evaluate the value of the capital goods used in a stationary state.
5.0 The CostMinimizing TechniqueThe optimal technique to use at any given rate of profits maximizes the wage. The firstorder condition for such maximization is found by equating the derivative of the factorprice curve to zero:
dw/dθ = 0
Or:
3 + θ(1 + θ)(1 + r) + (1 + r)^{2}  (1 + θ)(1 + 2θ)(1 + r) = 0
For 0 ≤
r ≤ 2, the costminimizing technique is then:
θ(r) = {[3 + (1 + r)^{2}]/(1 + r)}^{1/2}  1
For
r > 2, a corner solution is found:
θ(r) = 1
Figure 1 illustrates the costminimizing technique.

Figure 1: The Choice of Technique 
The graph in Figure 1 reaches a minimum at a rate of profits of (3
^{1/2}  1). For (12
^{1/4}  1) < θ < 1, two rate of profits have the corresponding costminimizing technique indexed by the given value of θ. In other words, this is an example of reswitching.
The index for the costminimizing technique can be plugged into the factor price curve for the technique to which it corresponds at a given rate of profits. Figure 2 displays the resulting socalled factor price frontier. The index θ varies continuously for 0 ≤
r ≤ 200% in Figure 2. As the rate of profits increases without bound, the frontier approaches a wage of zero.

Figure 2: The FactorPrice Frontier 
6.0 Capital and Labor "Markets"Fratini’s notes that this is a reswitching example in which the capital market initially appears to be in accord with outdated neoclassical intuition. The above analysis has shown how to find physical quantities of capital goods per worker, how to evaluate them at equilibrium prices, and how to find net output per worker. Figure 3 shows the resulting plot of the value of capital per unit output. Fratini looks at the value of capital per worker instead. Either curve is continuous and downwardsloping. The regions above and below the rate of profits of (3
^{1/2}  1) appear qualitatively similar and visually indistinguishable. This curve might be said to be a downwardsloping demand function for capital.

Figure 3: The Capital Market 
The analogous curve looks very different for the labor market (Figure 4). The region with a positive Wicksell effect is a region with a high rate of profits and thus a low real wage. The demand function for labor might be said to be upwardsloping in the region with a positive real Wicksell effect.

Figure 4: The Labor Market 
7.0 ConclusionThe example makes Fratini’s point. The shape of the relationship between the value of capital, either per worker or per unit output, and the rate of profits is not necessarily a good indicator of the presence of reswitching or reverse capitaldeepening.