Labor Demand In Corn Industry With The Recurrence Of Truncation

Figure 1: Labor Demand In Corn Industry

1.0 Introduction

This post continues my example of the recurrence of truncation without reswitching. Here I present graphs of the economy-wide demand for capital and for labor, as well as a sectorial demand for labor.

2.0 Technology and Techniques

Of the two industries in the model, one produces machines, and the other produces corn. Machines are fixed capital. Their physical life is two production cycles - that is, two years - in each industry. Corn is circulating capital in each industry and also a consumption good. A bushel corn is also the numeraire. Old machines cannot be transferred between industries. Constant returns to scale (CRS) and the free disposal of machines are assumed . Tables 1 and 2 define specific numeric values for a technology that meets these specifications, with a direct labor input in each process.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	1/16	3/20	1/8	53/200
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.

Table 3: Specification of Techniques

Technique	Processes	Notes
Alpha	I, III	Machines truncated in both industries.
Beta	I, II, III	Machines truncated in machine-production.
Gamma	I, III, IV	Machines operated at full physical life in both industries.
Delta	I, II, III, IV	Machines truncated in corn-production.

3.0 Price Systems and the Choice of Technique

The choice of technique can be analyzed in a model of pure fixed capital by constructing the wage frontier as the outer envelope of the wage curves for each technique. I take a bushel of corn as the numeraire. I assume that wages are paid out of the surplus at the end of the year, not advanced at the beginning. Figure 2 shows the wage curves for the example. Figure 2 shows an enlargement.

Figure 2: Wage Curves

Figure 3: Wage Curves (Enlarged)

The order of the cost-minimizing techniques, with an increasing rate of profits or a decreasing wage, is Alpha, Gamma, Delta, and Beta. This is also in order of increasing labor intensity. I measure labor-intensity by employment, economy-wide, needed to produce a net product of a bushel corn.

This is not a reswitching example. Techniques do not reswitch even off the frontier. The economic life od the machine is truncated in the corn industry for Alpha and Beta. Alpha and Beta are cost-minimizing at the furtherest extremes of the rate of profits, with Gamma and Delta cost-minimizing for middling rates of profits. So this is an example of the recurrence of truncation in the corn industry.

4.0 Economy-Wide Demand Curves for Capital and Labor

The above supports the drawing of economy-wide demand curves for capital and labor. Suppose net output for the economy as a whole is one bushel corn. Figure 4 shows the demand curve for capital. The value of inputs of corn and machines needed as inputs for the cost-minimizing technique are aggregated to obtain the quantity of capital at each point on the demand curve. Switch points are horizontal line segments in the graph. Around each switch point, a lower rate of profits is associated with a demand for a greater quantity of capital. This property is consistent with outdated marginalist theory. It is not a general property.

Figure 4: Economy-Wide Demand Function for Capital

The demand function for capital is not vertical between switch prices. These curves reflect the variation of prices with the rate of profits, given the technique. This variation is known as the price Wicksell effect. Edwin Burmeister champions David Champernowne's chain index measure of capital. This chain index eliminates price Wicksell effects. Given no capital-reversing, this chain index can be used to show that the rate of profits equals the marginal product of capital. This equality plays no role in solving the price system or in analyzing the choice of technique.

You can also draw an economy-wide demand curve for labor (Figure 5). Switch points are again shown as horizontal line segments. Here, a more labor-intensive technique is adopted at a lower wage. This, too, is a general property. But to emphasize the effects of the recurrence of truncation, I want an example that does not contradict obsolete marginalist theory at an aggregate level.

Figure 5: Economy-Wide Demand Function for Labor

5.0 Sectorial Demand Curves for Labor

I now consider the demand for labor in each industry. Figure 6 plots a sectorial demand curve in the machine industry. Prices of production are assumed to prevail at each level of the wage. One new machine is produced gross by the machine industry. Alpha and Gamma both operate the machine for a single year in producing new machines. Thus, the amount of labor employed with the given gross output in the machine industry is the same for Alpha and Gamma, as shown by the single vertical line to the left. Beta and Delta both operate the machine for its full physical life of two years in the machine industry, also, as shown in the graph. The sectorial labor demand function in the machine industry is a downward-sloping step function approximation to the traditional, non-justified story.

Figure 6: Labor Demand In Machine Industry

Figure 1, at the top of this post is the demand function for labor in the corn industry. Alpha and Beta both operate the machine for one year in the corn industry, while Gamma and Delta operate the machine in this industry for the full two years. The switch point between Beta and Delta exhibits the reverse substitution of labor. A higher wage around this switch point is associated with firms wanting to ultimately employ more labor per bushel corn produced gross in the corn industry.

Conclusion

So much for explaining wages and employment by well-behaved supply and demand functions for labor.

Even without reswitching or capital-reversing, the marginalist textbook stories do not work.