Monday, April 25, 2011

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, x1,1, x2,1, x3,1) = l1
  • qi; i = 1, 2, 3; is the quantity of the ith commodity produced in the period under consideration.
  • fi( ); i = 1, 2, 3; is the production function for the ith commodity.
  • lj; j = 1, 2, 3; is the number of person-years hired in the period under consideration to produce the jth commodity.
  • xi,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, x1,2, x2,2, x3,2) = (l2)D (x1,2)E/(DD EE)
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, x1,3, x2,3, x3,3) = (l3)d (x1,3)e (x2,3)f/(dd eeff)
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 pj; 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, x1,2, x2,2, x3,2
To Minimize wl2 + p1x1,2 + p2x2,2 + p3x3,2
Such that:
f2(l2, x1,2, x2,2, x3,2) = 1
l2 ≥ 0; xi,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 a0,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 ai,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
Labora0,1 = 1a0,2 = D (p1/w)Ea0,3 = dwd - 1 (p1)e(p2)f
Irona1,1 = 0a1,2 = E (w/p1)Da1,3 = ewd (p1)e - 1(p2)f
Steela2,1 = 0a2,2 = 0a2,3 = fwd (p1)e(p2)f - 1
Corna3,1 = 0a3,2 = 0a3,3 = 0
Output1 ton iron1 ton steel1 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:
a0,1 w (1 + r) = p1
[p1 a1,2(w, p1) + w a0,2(w, p1)](1 + r) = p2
[p1 a1,3(w, p1, p2) + p2 a2,3(w, p1, p2) + w a0,3(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.


Anonymous said...

As a neoclassical economist who enjoys your blog, I hope I can offer 2 points of constructive criticism.

1. If I understand you (I didn't doublecheck the math), your result can literally not happen with 'standard' neoclassical production functions and Walrasian GE. There are proofs of this in canonical Econ texts such as Mas-Colell,Winston, and Green. I want to know what in their assumptions you changed/violated. I could derive it myself, but I think it'd be nice for you to present as it would make it more obvious to readers what is driving your result.

Without deriving, I suspect the difference lies in either a) the non-diminishing returns in production of iron or b) the lack of feasibility requirements which would constrict your degrees of freedom down to 1 in section (4.0).This latter especially worries me, because by choosing a numeraire you are also partly choosing a scale of the economy which affects quantities demanded and supplied. It could be both.

2. No offense, but your notation is a disaster. f_i is generally understood as the df/di but not here. Plus you have some repeats - f is a production function and a Cobb-Douglas share parameter. I recommend greek letters.

Similarly confusing is your modeling of time, nowhere is time important in this model but you keep referring to how much time things take and 'long period' and etc etc. Clarity would be greatly enhanced if you just said 'one-period problem' and go from there.

Robert Vienneau said...

Lots to talk about here.

As I understand it, MWG's proof is what I echo in my Manchester School paper, available on the SSRN site on the upper right on the blog. Notice that in Table 1, da_(0,j)/dw; j = 2, 3 is negative. And da_i,j/dp_i is negative when the coefficient of production is non-zero. MGW's proof just ignores the point of Section 4 and after.

I don't think choosing a numeraire fixes the scale of the economy, although I can see the point of analyzing the vertically-integrated industry that produces the numerire.

How these equations look depends on your browser and operating system. Representing the three production functions by a bold f is consistent with my notation. An italics f, without a subscript, should not be a production function. I hope you can see why I would not like to use δ and ε for non-small parameters. I probably won't go to the bother of updating the post if more chip in in agreement with you. But I do not have strong opinions here.

The model is long-period because initial quantities of produced iron and steel are not givens. Assuming production takes (logical) time allows r to be positive while pure economic profits can be zero.

Anonymous said...

I took a quick look at the Manchester School paper and think I figured out where the difference may arise. So let me try a slightly different question:

Why are you focused on linear production technologies? What's wrong with the standard neoclassical CRS and strictly quasiconcave production technologies? Results like the cost function being homogeneous of degree one in input prices (ie the numeraire doesn't matter) are immediate with them.

Relying on linear technologies will just generate a bunch of strange knife-edge/substitution patterns that have more to do with the set-up than the underlying economics.

Anonymous said...

I should add that I know you're production functions here are not linear, but if you're trying to apply lessons from the mentioned paper to this model, the assumptions of linearity make this application strange, to say the least.

Robert Vienneau said...

I was not clear, I guess. I think the proof in the appendix to my Manchester School paper is like the canonical proofs anonymous refers to in the first comment.

A continuously differentiable production function can be approximated arbitrarily closely by a production function formed of linear combinations of Leontief functions. I do not see any difference in principle here. Given the post war turn to topological proofs, I do not recognize my approach as non-standard.

Constant Returns to Scale is also a standard assumption in mainstream economics. I do not see the results I find most interesting in Sraffian economics as an internal critique of neoclassical economics as driven by assumptions on returns to scale.

Anonymous said...

First, constant returns to scale is not the same as linear production technology. Linear is a subset of CRS.

Second, I remain confused, and I suspect it is largely jargon differences. If you want a neoclassically-trained economist to easily understand your critiques, I very much recommend using our language, shortcomings notwithstanding.

My reading of your claim is that you have a counterexample to either (a) homogeneity of degree one in expenditure functions; or (b) Walras' Law. I am not entirely sure, but I believe it is (b).

However, both (a) and (b) are just math with rather trivial proofs. So regardless of which one you are arguing against, you are doing one of two things: (1) arguing that the math translates poorly to a real example; or (2) arguing that the proof is wrong.

I'm sure we can agree that it is almost certainly not (2), and that you are really worried about the applicability of, say, Walras' Law to a particular economy, say, the model you wrote down. A natural question, then, is WHY the application failed. But since the Law is just math, there must be a failing of a premise in your application. And that's all I want to know: which assumption is being violated in your example?

Bruce said...

Also, I just noticed your comment policy preferring pseudonyms. My apologies about that - I have authored all the 'anonymous' posts in this thread. I will be sure to sign my comments going forward.

Robert Vienneau said...

I model neither the supply of labor nor the demand for corn in the post. Clearly Walras' law is not at issue here.

I claim none of my assumptions are non-standard. The proofs that Bruce refers to are not about the locus drawn in the post relating the price of iron and the quantity of iron demanded by corn-producing firms. If prices and quantities were explained by the interaction of well-behaved supply and demand curves, this locus seems one that would be relevant for demand.

Bruce said...

Fair enough if you claim your assumptions are non-standard. Which neoclassical proof is incorrect then? And where is the math wrong?

I am struggling to uncover exactly *which* proposition of neoclassical economics your example sheds light on. The two frequently used 'laws' about the numeraire are that cost/expenditure functions are heterogeneous of degree one and Walras' law. And since you claim your example isn't about those, then what exactly is it about? It would really help me if there were a claim or a sentence in a neoclassical text that you could point to and say that your model is relevant to.