Slope Of "Demand Curve" Varying With Numeraire

1.0 Introduction
As I have previously blogged, Ian Steedman has a number of articles explaining price theory. These articles typically explain the implications, for example, for the slopes of certain functions, but often do not contain graphical illustrations. Here then is an opportunity for me to develop blog posts. For instance, this post works through the example in Section 4 of Arrigo Opocher and Ian Steedman's article, "Input price-input quantity relations and the numéraire" (Cambridge Journal of Economics, V. 33, N. 5 (2009): 937-948).

2.0 Technology
Consider a simple economy in which three commodities - iron, steel, and corn - are produced. Corn is the only commodity people consume, and corn is not used as an input in the production of any commodity. The problem considered here is how much of each input into the production of corn will be demanded by the firms in the corn-production industry.

Iron is produced by unassisted labor. The production function for iron is:

q₁ = f₁(l₁, x_1,1, x_2,1, x_3,1) = l₁

where:

• q_i; i = 1, 2, 3; is the quantity of the ith commodity produced in the period under consideration.
• f_i( ); i = 1, 2, 3; is the production function for the ith commodity.
• l_j; j = 1, 2, 3; is the number of person-years hired in the period under consideration to produce the jth commodity.
• x_i,j; i = 1, 2, 3; j = 1, 2, 3; is the quantity of the ith commodity used in the production of the jth commodity in the period under consideration.

Steel is produced by labor from iron. Steel production is modeled by a Cobb-Douglas production function:

q₂ = f₂(l₂, x_1,2, x_2,2, x_3,2) = (l₂)^D (x_1,2)^E/(D^D E^E)

where D and E are positive parameters and D + E = 1.

Corn is produced by labor from inputs of iron and steel. The production function for corn is:

q₃ = f₃(l₃, x_1,3, x_2,3, x_3,3) = (l₃)^d (x_1,3)^e (x_2,3)^f/(d^d e^ef^f)

where d, e, and f are positive parameters and d + e + f = 1.

All production functions exhibit Constant Returns to Scale (CRS). All capital goods are totally used up in production, and all production processes require the same amount of time.

3.0 Cost Functions and Coefficients of Production
Consider competitive firms producing a commodity who want to adopt a cost-minimizing technique. For definitiveness, consider a firm producing steel. Let w be the wage (paid at the beginning of the period) and p_j; j = 1, 2, 3; be the spot price of the jth commodity. The unit cost function, c₂( ), for producing steel is the value of the objective function in the solution to the following mathematical programming problem:

Given w, p₁, p₂, p₃

Choose l₂, x_1,2, x_2,2, x_3,2

To Minimize wl₂ + p₁x_1,2 + p₂x_2,2 + p₃x_3,2

Such that:

f₂(l₂, x_1,2, x_2,2, x_3,2) = 1

l₂ ≥ 0; x_i,2 ≥ 0, i = 1, 2, 3.

Solving this programming problem, one finds the unit cost function for producing steel is:

c₂(w, p₁, p₂, p₃) = (w)^D (p₁)^E

In working out the cost function, I also figured out how much labor and iron a steel-producing firm would hire to produce one ton of steel. Opocher and Steedman pose the problem with given cost functions, not production functions. They derive the coefficients of production by invoking Shephard’s Lemma.

A similar cost-minimizing problem arises for corn-making firms.
The unit cost function, c₃( ), for producing corn is:

c₃(w, p₁, p₂, p₃) = (w)^d(p₁)^e(p₂)^f

The derivatives of the cost functions are summarized by the coefficients of production, a₀ and A, where:

• a₀ is a three-element row vector such that a_0,j; j = 1, 2, 3; is the person-years of labor hired per unit output of the jth industry.
• A is a 3x3 matrix such that a_i,j; i = 1, 2, 3; j = 1, 2, 3; is the quantity of the ith commodity purchased as an input per unit output in the jth industry.

Table 1 displays coefficients of production.

Table 1: The Cost-Minimizing Technique

	Iron Industry	Steel Industry	Corn Industry
Labor	a0,1 = 1	a0,2 = D (p1/w)E	a0,3 = dwd - 1 (p1)e(p2)f
Iron	a1,1 = 0	a1,2 = E (w/p1)D	a1,3 = ewd (p1)e - 1(p2)f
Steel	a2,1 = 0	a2,2 = 0	a2,3 = fwd (p1)e(p2)f - 1
Corn	a3,1 = 0	a3,2 = 0	a3,3 = 0
Output	1 ton iron	1 ton steel	1 bushel corn

4.0 Long Period Price Equations
The above shows the coefficients of production that perfectly competitive firms choose, given prices. Each column in Table 1 has been derived independently of the others. Firms will continue to produce corn, the consumption good, period after period only if some firms also choose to produce iron and steel. Cost-minimizing firms will not make these choices for any configuration of prices. The long-period condition that firm choices be self-sustaining yields the following system of three equations:

a_0,1 w (1 + r) = p₁

[p₁ a_1,2(w, p₁) + w a_0,2(w, p₁)](1 + r) = p₂

[p₁ a_1,3(w, p₁, p₂) + p₂ a_2,3(w, p₁, p₂) + w a_0,3(w, p₁, p₂)](1 + r) = p₃

where r is the rate of profits. Since production takes time, the rate of profits is generally positive. These equations, when solved, define long-period equilibrium prices for produced commodities as functions of the wage and the rate of profits:

p₁ = w(1 + r)

p₂ = w(1 + r)^{1 + E}

p₃ = w(1 + r)^{1 + e + f + (1 + E)f}

5.0 The Numeraire
The above system of price equations has two degrees of freedom. One degree of freedom is removed when the numeraire is specified. Let the numeraire consist of σ₁ units of iron, σ₂ units of steel, and λ person-years:

σ₁p₁ + σ₂p₂ + λw = 1

Prices of the commodities comprising the numeraire are then:

w = 1/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

p₁ = (1 + r)/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

p₂ = (1 + r)^{1 + E}/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

Figure 1 shows the relationship between the wage and the rate of profits. This relationship is known as the wage-rate of profits frontier. In this example, coefficients of production vary continuously along the frontier.

Figure 1: Wage-Rate Of Profits Frontier

6.0 Quantity Demanded for Inputs
The above derivations allow one to draw various graphs for specific parameter values. I set the parameters for the production functions as follows: D = 3/4; E = 1/4; d = 2/3; e = 1/6; and f = 1/6. And I considered two numeraires. For the first numeriare, σ₁ = 1/3; σ₂ = 1/3; and λ = 1/3. For the second numeriare, σ₁ = 1/3; σ₂ = 1/2; and λ = 1/6.

For each numeraire, the wage and other prices in a long-period position are defined as functions of the rate of profits. And the cost-minimizing coefficients of production are functions of these prices, that is, ultimately of the rate of profits. Figure 2 shows a locus constructed out of these functions. Curves at the top of the figure plot the price of iron against the quantity of iron demanded by the (non-vertically integrated) corn-producing industry. In drawing this figure, the quantity of corn produced is constrained to be unity. The curves are analogous to conditional demand curves in neoclassical economic theory. And one can see that, for certain regions, whether the slope of such a curve is positive or negative depends on the numeraire. But is not a main point of neoclassical economics to argue that demand curves are downward-sloping (for arbitrary numeraires)?

Figure 2: A Locus for Firms in Long Period Equilibrium

7.0 Conclusion
So much for explaining the price of a capital good by well-behaved supply and demand curves in the market for that commodity.