Tuesday, June 30, 2015

Recurrence Of Capital-Output Ratio Without Reswitching

Figure 1: Recurrence of Capital-Output Ratio
1.0 Introduction

This example is from Arrigo Opocher and Ian Steedman. It illustrates the analysis of an isolated industry in equilibrium. This analysis is therefore more akin to partial equilibrium than to general equilibrium. Sometimes (mainstream?) economists say that the Cambridge Capital Controversies were only about aggregate neoclassical theory, that is, macroeconomics. Or that the CCC has been subsumed by General Equilibrium Theory. The example illustrates that such economists are, as has long been apparent, spouting poppycock.

2.0 Indirect Average Cost Function

Consider a firm that produces widgets from inputs of widgets, unskilled labor, and skilled labor. Let the indirect average cost function be:

c(p, w1, w2) = sp + w1 + w2
+ 2(pw1)1/2 + 2(pw2)1/2 + 2γ(w1w2)1/2


0 < s < 1
0 < γ
γ ≠ 1


  • p is the price of a widget. Widgets used as inputs are assumed to be totally consumed in one production period.
  • w1 is the wage for unskilled labor.
  • w2 is the wage for skilled labor.

The indirect average cost function shows the average cost of producing each widget (net), when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm.

This indirect average cost function is homogeneous of the first degree:

c(a p, a w1, a w2) = a c(p, w1, w2)

This is a conventional assumption for cost functions.

Suppose the firm faces a given price of widgets and given wages for skilled and unskilled labor. By Shephard's lemma, the quantity of each input the firm wants to hire per unit output, given the price of each input, is the derivative of the indirect average cost function with respect to the price of that input. Hence, the capital-output ratio, k(p, w1, w2), is:

k(p, w1, w2) = ∂c/∂p = s + (w1/p)1/2 + (w2/p)1/2

Notice that the capital-output ratio is a pure number, unambiguously defined in this example, and independent of prices.

By the same logic, the amount of unskilled labor the managers of the firm desire to hire per widget produced is:

l1(p, w1, w2) = ∂c/∂w1 = 1 + (p/w1)1/2 + γ(w2/w1)1/2

The amount of skilled labor the managers of the firm desire to hire per widget produced is:

l2(p, w1, w2) = ∂c/∂w2 = 1 + (p/w2)1/2 + γ(w1/w2)1/2

The matrix of second derivatives of the indirect average cost function is:

(I am not sure whether it is more common to define the above matrix as the transpose of what I have above.) Anyway, for a positive price of widgets and positive wages, the signs of the second derivatives are as follows:

The signs along the principal diagonal show that the slopes of the per-unit input demand functions slope down. That is, given prices for all but one input, a lower price of that input is associated with a willingness of the firm to employ more of that input per unit produced. The positivity of the off-diagonal elements of the above matrix show that widgets, considered as inputs; unskilled labor; and skilled labor are all substitutes, not complements, in some sense. These signs for the matrix of second derivatives of the indirect average cost function are also conventional properties for cost functions.

3.0 Full Industry Equilibrium

Suppose the industry in which widgets are produced has no barriers to entry or exit. Thus, in the long run, economic profits will have been competed away. For firms to be earning no economic profits, the price of widgets must be equal to the average cost of manufacturing them:

p = c(p, w1, w2)

So far, no numeraire has been specified. Let widgets themselves be numeraire. Then:

1 = c(w1, w2)

where the argument in the indirect average cost function for widgets has been dropped as otiose.

Consider various levels of w1, the wage of unskilled labor. For the industry to continue to be in long run equilibrium, the wage of skilled labor, w2, must vary as well, thereby leaving the average cost of producing a widget as unity. Figure 2 illustrates the resulting wage-wage frontier. (Figures are drawn for s = 1/10 and γ = 2/3.) The highest wage for unskilled labor (when the wage for skilled labor is zero) is ((2 - s)1/2 - 1)2. Since this model is symmetric in skilled and unskilled labor, the highest wage for unskilled labor is likewise ((2 - s)1/2 - 1)2. As long as the rate of accounting profits is zero and technology is given, the wage of unskilled labor can only be higher if the wage of skilled labor is lower.

Figure 2: Wage-Wage Frontier

The wage-wage frontier can be used to find the wage of skilled labor for a given wage of unskilled labor between zero and the maximum. In other words, the frontier is helpful in calculating the ratio of the wage of skilled labor to the wage of unskilled labor, given the wage of unskilled labor. This ratio of wages is independent of the choice of the numeraire.

4.0 Capital and Labor

With the chosen numeraire, the capital-output ratio is:

k(w1, w2) = s + (w1)1/2 + (w2)1/2

Given the wage of unskilled labor, one can find the wage of skilled labor and, consequently, both the ratio of wages of the two types of labor and the capital-output ratio. Figure 1, at the start of this post, graphs the capital-output ratio as the derived function of the ratio of wages.

The capital-output ratio is the same when either skilled or unskilled labor is earning their maximum wage, with the other type of labor being paid a wage of zero. In these two extreme cases, the capital-output ratio is (2 - s)1/2 - (1 - s). Likewise for any ratio but one of the wage of skilled labor to the wage of unskilled labor between these extremes of zero and infinity, the capital-labor ratio is non-unique. The exception is the ratio of wages at which the function in Figure 1 peaks.

One can see that recurrence of the capital-output ratio is not reswitching. Figures 3 and 4 show, respectively, unskilled labor and skilled labor per unit output as a function of the ratio of wages. As shown in Figure 3, a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ more unskilled labor per unit output. Likewise, a a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ less skilled labor per unit output. As far as unproduced inputs go, this example of the isolated firm in long run equilibrium does not contradict outdated and exploded neoclassical intuitions about substitution and the mistaken notion of equilibrium prices as scarcity indices. But, since the functions in Figures 3 and 4 are monotonic, no reswitching of techniques arises in this example.

Figure 3: Unskilled Labor Employed per Unit Output

Figure 4: Skilled Labor Employed per Unit Output

5.0 Conclusion

This post has presented an example of an isolated firm in a long period equilibrium. The indirect average cost function, which includes the cost of the use of an input which itself is produced by the firm's industry, otherwise has utterly conventional properties. The analysis of the firm in a long run equilibrium demonstrates that it is an incoherent thought experiment to consider the equilibrium response of the firm to the variation of one price at a time. Only the variation of more than one price at a time can yield an equilibrium analysis that could be at all empirically relevant.

A result of this analysis is to reveal a non-monotonic response of the capital-output ratio to variations in the relative prices of the two unproduced inputs used by this firm in production. In fact, every possible capital-output ratio, except for one, recurs in the example. This is a step in an argument leading to the conclusion that economic theory is consistent with competitive firms wanting to employ more input per unit output for higher prices of that input, a finding that seems consistent with empirical results.

Saturday, June 20, 2015

Election Paradoxes And Faustian Agents

I have been trying to reread Donald Saari on election paradoxes. I have previously considered a few parallels between the Condorcet paradox and models of agents as composed of multiple selves. It seems to me that one could draw more analogies here. I do not plan to pursue the research agenda outlined here - I'm not sure how plausible its results would be. Anyways, Saari provides a comprehensive analysis of a range of voting procedures. Could a fuller range of such procedures - as opposed to pairwise majority rule - be applied to models of multiple selves?

For example, consider a model of a person as having multiple selves, where each one of those selves has a set of preferences over commodities. And suppose the individual, in making choices, resolves those selves with a procedure analogous to an election procedure (e.g., plurality vote, antiplurality vote, Borda Count). Suppose which procedure is used is context-dependent. Can an outside agent modify the context somehow such that the individual follows a different procedure, with consequent effects on the individual's choice?

Or consider two people each composed of the same number of multiple selves, with the preferences of those selves the same across these two people. But suppose each person resolves those selves with a different voting procedure. Maybe these two different voting procedures yield the same "best" choice for one specific menu of choices, but order the non-best choices differently. So if a new menu was created with the best choice removed, these two people - who have identical preferences, in some sense - would make different choices.

I suppose if you follow research along these lines, it would be theoretical research. I do not know how an experiment could elicit the required information to determine the preferences of the multiple selves and the election procedure. I guess the challenge would be to come up with an account consistent with some behavioral anomaly arising in economics experiments. Even better might be to suggest a new experiment and to implement it.

  • Donald G. Saari (2001). Chaotic Elections! A Mathematician Looks at Voting, AMS.

Saturday, June 06, 2015

Bertrand Russell, Crank

On the Post Topic

Some great thinkers compare their work to the works of Nicolaus Copernicus or of Galileo:

"The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds must be abandoned as beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion." -- Bertrand Russell, Our Knowledge of the External World as a Field For Scientific Method in Philosophy (1914).

"...an imagination better stocked with logical tools would have found a key to unlock the mystery. It is in this way that the study of logic becomes the central study in philosophy: it gives the method of research in philosophy, just as mathematics gives the method in physics. And as physics, which, from Plato to the Renaissance, was as unprogressive, dim, and superstitious as philosophy, became a science through Galileo's fresh observation of facts and subsequent mathematical manipulation, so philosophy, in our own day, is becoming scientific through the simultaneous acquisition of new facts and logical methods.

In spite, however, of the new possibility of progress in philosophy, the first effect, as in the case of physics, is to diminish very greatly the extent of what is thought to be known. Before Galileo, people believed themselves possessed of immense knowledge on all the most interesting questions in physics. He established certain facts as to the way in which bodies fall, not very interesting on their own account, but of quite immeasurable interest as examples of real knowledge and of a new method whose future fruitfulness he himself divined. But his few facts sufficed to destroy the whole vast system of supposed knowledge handed down from Aristotle, as even the palest morning sun suffices to extinguish the stars. So in philosophy: though some have believed one system, and others another, almost all have been of opinion that a great deal was known; but all this supposed knowledge in the traditional systems must be swept away, and a new beginning must be made, which we shall esteem fortunate indeed if it can attain results comparable to Galileo's law of falling bodies." -- Bertrand Russell, ibid.

The "new logic" Russell refers to is set out in, for example, Russell and Whitehead's Principia Mathematica. So Russell is comparing himself to Galileo.

An Approach to a Book Review

I'm glad I read this book, although I think it is basically mistaken. Not surprisingly, given their interactions at Cambridge before World War II, Russell's exposition reminds me of Ludwig Wittgenstein's Tractatus Logico-Philosophicus. Although clearly written, Russell's book has a quite different literary style than Wittgenstein's gnostic utterances and hierarchical structure. Both argue that everyday observations about, say, tables and chairs, should be decomposed into logical conjunctions, negations, and disjunctions of atomic facts, which cannot be further broken down. Russell and Wittgenstein differ on the nature of these atomic facts. For Wittgenstein, the referents for entities in atomic facts are quite mysterious; the specification of what these entities are is not a matter of logic, but of its application. Russell is quite clear that these entities include unintegrated sensations, something like "red patch here now."

Russell outlines how one might combine statements about such entities to construct entities that we see, hear, taste, smell, or feel. He goes on to analyze claims about other minds. The analysis of time leads to comments on Zeno's paradoxes and the mathematical theory of continuity. He also explains the idea of infinity, explaining the then recent theory of Cantor. He tries to present a popular overview of these topics. He acknowledges that some of his exposition is more mathematics than philosophy. But, as you can see above, he thinks previous philosophers and many of his contemporaries stumbled into error because they did not possess these logical and mathematical tools. For later developments along the lines, I gather one can look at such works of logical positivism as Rudolf Carnap's The Logical Structure of the World. I have never read Carnap, but I have read A. J. Ayer's Language, Truth, and Logic.

I recently stumbled somewhere across an argument that Noam Chomsky's approach to linguistics supercedes Russell's application of logic to philosophy. Russell and Chomsky agree that sentences of very different structures can have a close surface appearance, and that the same structure can be exhibited in sentences of different surface appearances. In deciding whether or not propositions are true, or even make sense, one should supposedly concentrate on the meaning captured by this deeper structure. But in trying to analyze the meaning of such propositions as, "The king of France is bald", Russell takes an a priori approach. The adequacy of grammar, however, to characterize sentences in a language is an empirical question. And semantics should be based on the parse trees derived from grammatical analysis of the surface appearances of language, not a logical analysis of the surface appearance. This approach, as I understand it, is analogous to how compilers operate. They apply a semantic analysis to a computer program only after first completing a parsing phase. And Chomsky's approach, I gather, has been influential in Artificial Intelligence.

One can argue that just as Wittgenstein, in Philosophical Investigations, showed his earlier approach in the Tractatus was mistaken, so he also showed Chomsky's approach in linguistics to be mistaken. A fortiori, AI is not possible either. Exposition of the parallelism between Russell and Chomsky's analysis of language makes these claims a bit more clear to me. (I guess Sraffa was not too impressed by Chomsky, either.) I suppose one might look at Norman Malcolm's Wittgenstein: Nothing is Hidden, for a fuller argument against Chomsky along these lines. (I did not get much out of Malcolm when I read him years ago.)