Variations In Switch Points With Markups In The 'Corn' Industry

Figure 1: Variation of Switch Points with the Markup in the Corn Industry

1.0 Introduction

I have been re-creating some of my past analyses. The graphs in this post look a bit different because I impose a requirement that the relative markups sum to unity.

2.0 Technology

Consider an economy which produces three commodities, iron, steel, and corn, with the technology specified in Table 1. Two processes are available for producing each commodity. The coefficients of production in a column specify the person-years of labor, tons of iron, tons of steel, and bushels of corn required to produce a unit of output of the given industry.

Table 1: The Technology

Input	Iron Industry		Steel Industry		Corn Industry
	a	b	c	d	e	f
Labor	1/3	1/10	5/2	7/20	1	3/2
Iron	1/6	2/5	1/200	1/100	1	0
Steel	1/200	1/400	1/4	3/10	0	1/4
Corn	1/300	1/300	1/300	0	0	0

Eight techniques (Table 2) are defined for this technology. Each technique is defined by the operation of one process in each of the three industries. All three commodities are Sraffian basics in all techniques. That is, each commodity is a direct or indirect input in the production of all commodities. For example, iron is used directly as an input in the first corn-producing process, and steel is used indirectly in producing corn with this process since steel is an input in either iron-producing process

Table 2: Techniques

Technique	Processes
Alpha	a, c, e
Beta	a, c, f
Gamma	a, d, e
Delta	a, d, f
Epsilon	b, c, e
Zeta	b, c, f
Eta	b, d, e
Theta	b, d, f

3.0 Prices of Production

Prices of production are defined here for given ratios of markups among industries. The ratios of rates of profits among industries are assumed stable, but rates of profits are not necessarily uniform. Lack of uniformity in rates of profits can result from variations in evaluations of profits among industries due to idiosyncratic properties of investment; from barriers to entry arising from, for example, secrets in manufacture; and from legal monopolies (D’Agata 2018). Let s₁ r, s₃ r, and s₃ r be the rate of profits in the iron, steel, and corn industries respectively. I call r the scale factor for the rate of profits. The usual system of equations, with labor advanced, must be satisfied for prices of production for a given technique.

As a matter of scaling, suppose the markups lie on a simplex:

s₁ + s₂ + s₃ = 1

Suppose that a bushel of corn is the numeraire. In drawing various graphs, I consider only variations in the markup in the corn industry, with markups in producing iron and steel assumed identical:

s₁ = s₂

The solution to this system, for each technique, has a single degree of freedom, which can be expressed with the wage as a function of the scale factor for the rate of profits

4.0 The Choice of Technique with Competitive Markets

Figure 1 graphs the wage curves for four techniques, given competitive markets. The same relative markups are obtained in all industries. The cost-minimizing technique at a given wage maximizes the scale factor for the rate of profits. The cost-minimizing technique at a given scale factor maximizes the wage. The outer frontier of all wage curves shows the variation of the cost-minimizing technique with distribution. Wage curves are graphed in Figure 1 only for the techniques on the outer frontier. This type of figure, usually for competitive markets, is the most well-known graph in post-Sraffian price theory

Figure 2: Capital-Reversing with Competitive Markets

Around the so-called perverse switch point, the firms in the corn industry switch from the second corn-producing process to the first at a lower wage. That is, they adopt a process that requires less labor to be hired per bushel of corn produced gross. This is known as the reverse substitution of labor (Han and Schefold 2006). For the economy as a whole, the technique adopted at a lower wage requires less labor per unit of net output. This is a consequence of capital-reversing as manifested in a comparison of stationary states (Harris 1973).

5.0 Fluke Cases

Five fluke cases can be found by perturbing the relative markup in the corn industry (Table 3). Figure 3 depicts the wage frontier for the first fluke case. This markup occurs when reswitching is just emerging.

Table 3: Fluke Switch Points

Markup for Corn	Fluke Case
s3 ≈ 0.211996	Reswitching pattern for Gamma vs. Delta.
s3 ≈ 0.249246	Four technique pattern for Gamma, Delta, Eta, and Theta.
s3 ≈ 0.8232415	Alpha vs Beta switch point at wage of zero.
s3 ≈ 0.8696757	Four technique pattern for Alpha, Beta, Gamma, and Delta.
s3 ≈ 0.9307414	Beta vs Delta pattern over r axis

Figure 3: Wage Curves for Gamma and Delta Tangent at Switch Point

6.0 The Choice of Technique with the Full Range of the Markup in the Corn Industry

Figure 1, at the top of this post, is my new type of diagram illustrated for depicting the analysis of the choice of technique. The abscissa is the markup in the corn industry, with given markups of unity in the iron and steel industry. The maximum wage and the wage at switch points along the frontier are plotted. The number and sequence of switch points along the wage frontier are invariant in each numbered region. Fluke switch points partition the numbered regions. Figure 4 enlarges Figure 1 on the right for low wages

Figure 4: Variation of Switch Points with the Markup (Detail)

The qualitative properties of the wage frontier are invariant in each numbered region in Figures 1 and 4. Table 4 describes each numbered region. The cost-minimizing technique along the wage frontier is listed, from a wage of zero to the maximum wage. Some salient properties of switch points and the cost-minimizing technique are summarized in Table 5. Figure 2 depicts the wage frontier for a markup in the corn-industry in region 3, while Figure 3 depicts the wage frontier on the boundary between regions 1 and 2.

Table 4: Variations in the Cost-Minimizing Technique

Region	Range	Technique
1	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ wmax,η	Eta
2	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ w3	Delta
	w3 ≤ w ≤ w4	Gamma
	w4 ≤ w ≤ wmax,η	Eta
3	0 ≤ w ≤ w1	Alpha
	w1 ≤ w ≤ w2	Gamma
	w2 ≤ w ≤ w3	Delta
	w3 ≤ w ≤ w4	Theta
	w4 ≤ w ≤ wmax,η	Eta
4	0 ≤ w ≤ w1	Beta
	w1 ≤ w ≤ w2	Alpha
	w2 ≤ w ≤ w3	Gamma
	w3 ≤ w ≤ w4	Delta
	w4 ≤ w ≤ w5	Theta
	w5 ≤ w ≤ wmax,η	Eta
5	0 ≤ w ≤ w1	Beta
	w1 ≤ w ≤ w2	Delta
	w2 ≤ w ≤ w3	Theta
	w3 ≤ w ≤ wmax,η	Eta
6	0 ≤ w ≤ w1	Delta
	w1 ≤ w ≤ w2	Theta
	w2 ≤ w ≤ wmax,η	Eta

Table 5: Notes on Regions

Region	Summary
1	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.
2	Reswitching of techniques between Gamma and Delta. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta at the lower wage. Process recurrence of the first process in the corn industry.
3	No reswitching. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta. Process recurrence of the first process in the corn industry.
4	No reswitching. Capital-reversing and the reverse substitution of labor at the switch point between Gamma and Delta. Process recurrence of both processes in the corn industry.
5	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.
6	No reswitching, no capital-reversing, no reverse substitution of labor, no process recurrence.

This example allows for a graphical display showing that reswitching arises with an increased markup in corn-production, starting from a markup much less than in other industries. The ‘perverse’ switch point between Gamma and Delta remains on the wage frontier after the other switch point between these techniques falls off the frontier at a higher markup. Eventually, the ‘perverse’ switch point is no longer on the frontier when corn-production has a much higher markup than other industries.

7.0 Conclusion

The properties of the wage frontier might be thought to have some impact on the struggle between capitalists and workers. These properties can be altered both by technical change and by variations in relative market power among capitalists.