Austrian Business Cycle Theory As Uninteresting And Esoteric

I have been unsuccessful in getting my refutation of Austrian Business Cycle Theory accepted for publication in a peer-reviewed journal. My latest publically available version is downloadable from SSRN. I have had a later version rejected a couple of weeks ago. This version's references include Roger Garrison's 2006 "Reflections on Reswitching and Roundaboutness". I hope quoting an extract from a review is acceptable, since some might find this of interest:

"Hayek had washed his hands of the triangles [that ground his theory of the trade cycle] long before the ink was dry on his Pure Theory of Capital... Modern Austrian school economists like Roger Garrison still find the triangle-logic compelling, but I have the impression that, with the possible exception of Leland Yeager’s 1976 Economic Inquiry paper, no systematic attempt has been made to show that Austrian economics is immune to the capital critique; the arguments are intuitive rather than carefully drawn. (Even Yeager’s piece struck me when I read it, many years ago, as largely intuitive.) ...the arguments had never been put forward in a robust way..."

Classical Cross Dual Dynamics

"The actual price at which any commodity is commonly sold is called its market price. It may either be above, or below, or exactly the same with its natural price.

The market price of every particular commodity is regulated by the proportion between the quantity which is actually brought to market, and the demand of those who are willing to pay the natural price of the commodity, or the whole value of the rent, labour, and profit, which must be paid in order to bring it thither. Such people may be called the effectual demanders, and their demand the effectual demand; since it may be sufficient to effectuate the bringing of the commodity to market...

...The natural price, therefore, is, as it were, the central price, to which the prices of all commodities are continually gravitating...

The whole quantity of industry annually employed in order to bring any commodity to market, naturally suits itself in this manner to the effectual demand. It naturally aims at bringing always that precise quantity thither which may be sufficient to supply, and no more than supply this demand." - Adam Smith, The Wealth of Nations, Book I, Chapter VII

1.0 Introduction
Contemporary economists have elaborated Smith's metaphor of the gravitational attraction of market prices to natural prices. Elaborations consist of formal models of cross-dual dynamics.

Systems of equations describing prices and quantitites are dual systems in the post Sraffa/von Neumann tradition. Dynamics are cross dual when changes in prices respond to quantities and changes in quantities respond to prices. In particular, industries expand in which rates of profits are high and contract in industries in which profits are low. And prices fall in industries in which the quantity supplied exceeds the effectual demand. Prices rise in industries in which the quantity supplied is below the effectual demand.

Dupertuis and Sinha (2009) is one immediate impetus for my setting out this model of the reallocation of labor for a given wage. I don't think I thoroughly understand models of cross-dual dynamics. I think they are of interest for exploring the possible dynamics of market prices in competitive capitalist economies. I don't see that they are directly empirically applicable. For that, one needs to worry about markup prices and the degree of utilization of capacity in various industries.

2.0 Natural Prices
The data of our problem are:

• The n x n input-output matrix A, where n is the number of industries and a_{i, j} is the amount of the ith commodity used as input per unit output of the jth industry
• The n-element row vector a₀ of labor inputs, where a_{0, j} is the amount of person-hours hired per unit output in the jth industry
• The money wage w_given
• The composition of net output as expressed in the n-element column vector c_given.

This is a circulating capital model in which all production processes use up their inputs in a year. Labor is assumed to be paid their wages at the end of the year. Only economies capable of producing a surplus product are considered. For simplicity, assume Constant Returns to Scale (CRS) and that every commodity is basic, in Sraffa's sense. This section considers the problem of finding:

• The natural prices, as expressed in the n-element row vector p^*
• The corresponding wages w^*
• The corresponding rate of profits r^*
• The effectual demand as expressed in the n-element column vector of gross quantities q^*
• The n-element column vector of net quantities y^*.

In the system of natural prices, the same rate of profits are made in every industry:

p^* A(1 + r^*) + a₀ w^* = p^*

I take the net output as the numeraire.

p^* y^* = 1

In my formulation here, the wage is taken as a given ratio of the net output:

w^* = w_given

The above equations comprise the price system for natural prices.

Net outputs and gross outputs are related by the following equation:

y^* = q^* - A q^* = (I - A) q^*

where I is the identity matrix. I normalize the units of labor such that one unit is employed throughout the economy:

a₀ q^* = 1

Finally, the net output is assumed to be in the specified proportions. That is, there exists a positive constant k such that

y^* = k c_given

The above systems of equations are sufficient to determine gross and net effectual demands, natural prices, and the distribution of income. (An alternative specification would take the composition of gross output as given, instead of the net output. Perhaps outputs should be in units of Sraffa's standard commodity.)

3.0 Initial Condititions
The problem in the remaining sections is to define a dynamic process for the quantities produced q(t) and the market prices p(t) for t = 0, 1, 2, ... The initial quantities q(0) and market prices p(0) are givens. For the sake of the argument, I consider a dynamic process in which the amount of labor employed and the value of net output are invariant. So the initial quantities and prices must satisfy the following equations:

a₀ q(0) = 1

p(0) y(0) = p(0)(I - A)q(0) = 1

4.0 Reallocation of Labor
Define r_average(t), the average rate of profits for the economy as a whole at time t:

r_average(t) = [p(t)(I - A - a₀ w_given)q(t)]/[p(t) A q(t)]

The numerator in the expression on the right hand side above is the value of the surplus product remaining in the capitalists' possession after replacing the means of production and paying laborers their wages. The denominator is the value of the capital goods advanced.

Typically, the rate of profits will vary from the average among the industries. The rate of profits for the jth industry at time t is:

r_j(t) = [p_j(t) - p(t) a_{., j} - a_{0, j} w_given]/[p(t) a_{., j}]

where a_{., j} is the jth column of the input-output matrix A.

Define R_average(t) to be the n-element column vector with each element equal to the average rate of profits. Let R(t) be the n-element column vector with each element being the rate of profits for the corresponding sector.

Now dynamics of the quantities of produced commodities can be specified:

q(t + 1) = [1/f₁(t)]{[R(t) - R_average(t)] + q(t)}

Or, in terms of scalars:

q_i(t + 1) = [1/f₁(t)]{[r_i(t) - r_average(t)] + q_i(t)}

where

f₁(t) = 1 + a₀[R(t) - R_average(t)]

The denominator f₁(t) above is a normalization that ensures the quantity of labor employed is always unity. The numerator ensures that the more the rate of profits in a sector exceeds the average, the faster that sector will expand in comparison with other sectors. (An alternative formulation might compare the rate of profits in each industry with the rate of profits r^* in the system of natural prices.)

5.0 Price Changes
Price dynamics are here set out more directly:

p(t + 1) = [1/f₂(t)]{p(t) - [q^T(t) - q^*T]}

where x^T is the transpose of the vector x. In terms of scalars, prices are given by:

p_i(t + 1) = [1/f₂(t)]{p_i(t)- [q_i(t) - q^*_i]}

The time series f₂(t) is defined as follows:

f₂(t) = [q^*T - q^T(t)](I - A)q(t + 1)} + p(t)(I - A)q(t + 1)

The denominator f₂(t) above is, again, a normalization condition. In this case, the normalization ensures the value of the net output is equal to unity. Since the composition of net output typically changes over the course of the process, the real wage varies in terms of any fixed commodity basket. It does remain, however, a given ratio of the net output.

6.0 Conclusion
I have set out above a model of a dynamic process, but without an analysis of its properties. An obvious theorem is that if initial quantities and prices happen to be equal to the effectual demands and natural prices, they will be left unchanged by the dynamics of market adjustments. In other words, the natural system is a stationary point of this dynamic process.

An interesting question is the trajectory of market prices, given an arbitrary starting point. I don't expect the process to necessarily converge to the natural system. At this point, I don't have any numeric examples of limit cycles or chaotic behavior. A failure of local stability doesn't bother me; I have often thought of Sraffa's work as pointing towards the possibility of complex dynamics arising in models of capitalist economies.

Questions of structural stability are of interest as well. Do dynamic properties of the system depend on the level of wages, especially if one introduces into the model a choice of technique? And how do the answers to these questions vary, if at all, with alternative modeling assumptions, some of which I have indicated? I do not know that the literature has reached definitive answers to these questions.

Update (28 January 2010): I have redefined the dynamics above in a way that seems more reasonable to me.

References

• Michel-Stéphane Dupertuis and Ajit Sinha, "A Sraffian Critique of the Classical Notion of Centre of Gravitation", Cambridge Journal of Economics, V. 33 (2009): 1065-1087.

Upcoming URPE Conferences

Chris Pepin informs us that the Union for Radical Political Economics (URPE) will participate in the upcoming Eastern Economics Association (EEA) conference. The conference will be held at the Loews Philadelphia Hotel, February 26 - 28. A program is available. This year is the 50th anniversary of the publication of Sraffa's book.

URPE is also participating in the Left Forum, on March 19-21 at Pace University.

Elsewhere

• Eric Rauchway tells us about the Bretton Woods conference, where John Maynard Keynes showed that some economists could be more useful than dentists.
• David Ruccio reprints a cartoon by B. Deutsch making fun of economists prefering a supposedly elegant theory of the minimum wage to empirical results falsifying the theory.
• Bill Mitchell has a negative view of Greg Mankiw's textbook.

Skimming Moshe Adler

A few weeks ago, in a bookstore a couple of hundred miles away from here, I skimmed Moshe Adler's Economics for the Rest of Us: Debunking the Science that Makes Life Dismal. I did not purchase it because I am already too far behind in my reading. It is targeted for those outside the economics profession.

It is a short and approachable book that, as I recall, falls into two main parts.

The first part is about the mainstream economist's concept of (Pareto) efficiency. I hopped over this section fairly quickly, since I see no need to be strongly guided by this criterion in making policy decisions. I gather Adler agrees.

The second part is about income distribution, the theory of marginal productivity, and wages. Adler compares and contrasts neoclassical theory and the more empirically applicable classical theory. If I read this book in more depth, I would probably have some caveats about Adler's interpretation of the classical economists and his assignment to them of one (non-Malthusian) theory of wages. Adler recognizes that in a theory in which wages are determined by well-behaved supply and demand functions for labor, the imposition of higher wages results in less employment. Less security and less employment is a bad thing for many members of that vast majority in capitalist societies who depend on income from labor to live. On the other hand, when wages are the result of class struggle, as in Adam Smith, for example, the theory does not predict that unions, minimum wages, less "flexible" labor markets will result in less employment. And, despite the poppycock mainstream economists teach, that is the world we live in.

I agree with the author. Economics took a mostly wrong turning more than a century ago. I don't think that this book will convince many mainstream economists. If Adler wanted to convince mainstream economists, he would have had to written a more impenetrable book. I think Adler does address some of the questions raised by the current global economic crisis.

(I realize I am behind in responding to comments on previous posts.)

The First Communist

One can find all sorts of things on the internet, such as T-shirts.

And one can find lots of silliness: "Why Jesus Christ is Not a Communist".

Hat tip: My friend Lucas.

Leontief's Work As Applied Sraffianism

Sraffa's critique is formulated in terms of physical quantity flows among industries and labor inputs into each industry. From this data, institutional assumptions, and, for example, the rate of profits, Sraffa deduces the set of constant prices that allow for the smooth reproduction of a capitalist economy.

Leontief independently developed the theory of Input-Output (I-O) analysis, and national income accounts include I-O accounts. The Bureau of Economic Analysis maintains I-O accounts for the United States.

A body of work exists in which the national income accounts are used to examine empirical questions. Sometimes the output of each industry is normalized to one physical unit of each industry, thereby allowing the national accounts to be thought of as closer to Sraffa’s data. Much of this work can be seen as classical in approach, not marginalist. Benjamin H. Mitra-Kahn's "Debunking the Myths of Computable General Equilibrium Models" (2008) shows that the classical nature of one such body of empirical work is often disguised by tendentious and incorrect history.

So Sraffa’s work is empirically applicable, although his own intentions seem to have been more focused on criticism.

Mainstream Economists Unable To Discuss Economics

Over on Economics Job Market Rumors, an anonymous poster asks:

"Despite the neoclassicals admitting that the Post-Keynesians were right, why has the impact of heterogeneous capital on an economy left out of the macro models?"

It will not surprise me if he or she receives no coherent answer. I recently had a chance to skim the transcripts of David Colander's interviews with graduate students at the "best" economics departments in the United States. These are in his book The Making of an Economist Redux. The following phrases seem no connote nothing to such students: "Cambridge Capital Controversy", "Neoclassical Economics", and "Post Keynesian Economics". One student responded to a question about Joan Robinson by asking, "Who's that?" The best students seem to realize that they will have to get an education by themselves in their "spare" time after they receive their doctorate.

Minimum Wages In The U.K.

The Australian Fair Pay Commission's Minimum Wage Research Forum met in Melbourne on 30 and 31 October 2008. Stephen Machin summarized recent experience in the United Kingdom (in the 2008 Minimum Wage Research Forum Proceedings, Volume 1).

Minimum wages were set by industry in the United Kingdom up until 1993. The wage councils were abolished in 1993, except for the Agriculture Wages Board which continues to this day. Outside of agriculture, the UK did not have a minimum wage between 1993 and 1999. From 1999 on, the National Minimum Wage was in effect in the UK, as recommended by the UK Low Pay Commission, established in 1997 by the newly elected Labour government. Notice that the trend in employment visually appears unaffected by the introduction of the national minimum wage and subsequent increases in it. The trend appears the same before as afterwards. This seems like disconfirmatory evidence to me of the simple neoclassical model of wages and employment as determined by supply and demand functions. Some of us know that model is without theoretical foundation anyways.

Hat tip to Bill Mitchell

Parallel Thoughts By Wittgenstein And Sraffa

Apparently Wittgenstein wrote the following in 1937:

"The origin and the primitive form of the language game is a reaction; only from this can more complicated forms develop.

Language - I want to say - is a refinement, 'in the beginning was the deed'." -- Ludwig Wittgenstein, Culture and Value (Translated by Peter Winch) (1980)

And Sraffa, I guess, wrote the following in the early 1930s:

"If the rules of language can be constructed only by observation, there can never be any nonsense said. This identifies the cause and the meaning of a word.

The language of birds, as well as the language of metaphysicians can be interpreted consistently in this way.

It is only a matter of finding the occasion on which they say a thing, just as one finds the occasion on which they sneeze.

And if nonsense is 'a mere noise' it certainly must happen, as sneeze, when there is cause: how can this be distinguished from its meaning?

We should give up the generalities and take particular cases, from which we started. Take conditional propositions: whan are they nonsense, and when are they not?" -- Piero Sraffa as quoted by Heinz D. Kurz, "'If some people looked like elephants and others like cats, or fish...' On the difficulties of understanding each other: the case of Wittgenstein and Sraffa", The European Journal of the History of Economic Thought, V. 16, n. 2 (2009): pp. 361-374

Colander Testimony On Risks Modeling

Last September, the Committee on Science and Technology's Subcommittee on Investigations and Oversight, a subcommittee of the United States House of Representatives heard testimony on the risks of financial modeling. I looked at David Colander's testimony.

Colander advocates modeling economies as complex dynamical systems. He thinks economists should be aware of the limitations of models. Macroeconomists, in settling on the Dynamic Stochastic General Equilibrium (DSGE) model, failed to consider a wide range of models. The assumptions of the DSGE model do not fit the real world. (In objecting to the use of the "assumption" of the existence of a representative agent, I am on the side of such economists as Alan Kirman and Frank Hahn & Robert Solow.)

Colander discusses how mainstream economists are indoctrinated. Colander recommends that peer review for grants from the National Science Foundation for economics research include, "for example, physicists, mathematician[s], statisticans, and even business and govermental representatives".

This bit about the NSF reminds me of a story Paul Davidson tells:

"In 1980 I applied for a grant from the National Science Foundation to write International Money and the Real World... One of the [insider peer reviewers] had the most telling observation of them all. He said something like, 'It is true that Davidson has a very good track record and surprisingly good publications, but he marches to a different drummer. If he's marching to a different drummer, if his music is different, then he ought to get his own money and not use ours.'" -- Paul Davidson in J. E. King, Conversations with Post Keynesians (1995)

Davidson did not get the grant.

Weird Science II

A bit from Avatar reminds me of Ursula K. LeGuin's "Vaster Than Empires and More Slow", a short story republished in her collection The Wind's Twelve Quarters (1975). LeGuin postulates a world in which nodes in tree roots act like synapses. The plant life is one sentience. Maybe even vines and spores partake in it. As before, a cultural work reminds me of some science:

• The longest lived thing is arguably Pando, a grove of aspens in Utah that seems to be one plant, connected at the roots and propagating through runners like strawberries or mrytle.
• Or maybe it is an instance of the fungus Armillaria bulbosa in Oregon.

A Wikipedia article lists other such organisms, for what it's worth. (The references in this post are reminders for me to look up sometime.)

Wage-Rate Of Profits Curves

1.0 Introduction
I have written about so-called factor price curves and frontiers in many posts. They are so-called because the interest rate is not a price of any factor of production. In this post, I use the more neutral expressions "Wage-Rate of Profits Curve" and "Wage-Rate of Profits Frontier". I consider the concepts denoted by these terms to be elements of mathematical economics that arise, in particular, in the analysis of steady states.

2.0 Derivation of a Wage-Rate of Profits Curve
Consider an economy in which n commodities are produced. Each commodity j is produced in a corresponding industry in which it is the sole output of a single process. This process:

• Requires inputs of labor and commodities. These inputs are represented as a_{0, j} person-years per unit output and a_{i, j} units of the ith commodity per unit output.
• Exhibits Constant Returns to Scale (CRS).
• Requires a year to complete.
• Totally uses up its commodity inputs.

A technique consists of a process for each of the n industries. The technique is represented by the row vector a₀ of direct labor coefficients and the square Leontief Input-Output matrix A. Assume:

• Each commodity enters either directly or indirectly into the production of all commodities. That is, all commodities are basic in the sense of Sraffa.
• The economy is viable. That is, there exists a level of operation of all processes such that the outputs can replace the commodities used up in their production and leave a surplus product to be paid out in the form of wages and profits.
• Wages are paid at the end of the year.
• The same rate of profits is earned on advances in all industries.

The assumptions of CRS and of all commodities being basic are made for ease of exposition.

Under these assumptions, the constant prices that allow the economy to smoothly reproduce satisfy the following system of n equations:

p A (1 + r) +w a₀ = p

where p is the row vector of prices, w is the wage, and r is the rate of profits. Given the rate of profits, this is a linear system in n + 1 variables. The last equation imposed in the model sets the value of the numeraire to unity:

p e = 1

where e is a column vector denoting the units of each commodity that comprise the numeraire. Only solutions in which all prices are positive and the wage is non-negative are considered.

The price equation can be transformed into:

w a₀ = p [I - (1 + r)A]

where I is the identity matrix. Or:

w a₀ [I - (1 + r)A]^-1 = p

where the assumption of viability guarantees the existence of the inverse for all rates of profits between zero and a maximum rate of profits. Right multiply both sides of the above equation by the numeraire:

w a₀ [I - (1 + r)A]^-1 e = p e = 1

The wage-rate of profits curve for the technique is then:

w = 1/{a₀ [I - (1 + r)A]^-1 e}

3.0 Properties of Wage-Rate of Profits Curves
The Wage-Rate of Profits Curve for a technique, under the assumptions above, has the following properties:

• There is a finite maximum rate of profits for which the wage is zero. (If no commodity were basic, this maximum would not be finite.)
• There is a maximum wage for which the rate of profits is zero.
• The wage-rate of profits curve is strictly decreasing between the rate of profits of zero and the maximum rate of profits.
• The wage rate of profits curve can be both convex to the origin and concave to the origin. (If the number of commodities n is greater than 2, the convexity can vary throughout the curve.)
• If the vector of direct labor coeffients is a left-hand eigenvector of the Leontief Input-Output matrix, the wage-rate of profits curve is a straight line, that is, affine. (This is Marx's case of equal organic composition of capitals.)
• If the numeraire is a right-hand eigenvector of the Leontief Input-Output matrix, the wage-rate of profits curve is affine. (This is the case of Sraffa's standard commodity.)

Figure 1 illustrates the wage-rate of profits curve for five techniques (α, β, δ, ε, and τ). Pasinetti uses π, not r, to denote the rate of profits. These curves are drawn under the assumption that the organic composition of capitals is not constant for any technique, and the numeraire is not the standard commodity for any of the techniques. Figure 1 also shows the wage-rate of profits frontier, formed from the outer envelope of all the wage-rate of profits curves for the individual techniques. This frontier is used to analyze the choice of technique for long-period, circulating capital models with single production.

Figure 1: The Frontier Formed From Factor-Price Curves (from Pasinetti (1977), p. 157)

Selected References

• Heinz D. Kurz and Neri Salvadori (1995) Theory of Production: A Long-Period Analysis, Cambridge University Press
• Heinz D. Kurz and Neri Salvadori "Production Theory: An Introduction"
• Luigi L. Pasinetti (1977) Lectures on the Theory of Production, Columbia University Press

Paul A. Samuelson, 1915-2009

I've been influenced by Samuelson's work. I've referenced him here on such topics as:

• Aggregate production functions
• Cambridge Capital Controversies, Joan Robinson, and Piero Sraffa
• Growth theory
• International trade, theory of
• Linear programming
• Marginal productivity theory
• Marxist economics
• Revealed preference theory

I don't think I've referenced him on overlapping generations models when I've used them. But I believe he originated such models.

Negative Price Wicksell Effect, Positive Real Wicksell Effect

1.0 Introduction
I have previously suggested a taxonomy of Wicksell effects. This post presents an example with:

• The cost-minimizing technique varying continuously along the so-called factor-price frontier
• Negative price Wicksell effects
• Positive real Wicksell effects
• Price Wicksell effects greater in magnitude than real Wicksell effects.

This example is due to Saverio M. Fratini ("Reswitching and Decreasing Demand for Capital").

2.0 Technology
Suppose technology consists of a continuum of techniques indexed by the parameter θ, where θ is a real number restricted to the interval [0, 1]:

0 ≤ θ ≤ 1

Each technique consists of the three Constant-Returns-to-Scale processes in Table 1. No commodity is basic, in Sraffa's sense, in any technique in this technology. In the first process in a technique, θ-grade iron is produced directly from unassisted labor. In the second process, labor transforms the θ-grade iron into θ-grade steel. Finally, in the third process, labor transforms θ-grade steel into corn, the consumption good in the model. All processes take a year to complete, and all processes totally use up their input.

Table 1: The Technique Indexed by θ

Inputs	Industry Sector
	θ-Grade Iron	θ-Grade Steel	Corn
Labor (Person-Yrs)	1/(1 + θ)	θ	3/(1 + θ)
Iron (Tons)	0	1	0
Steel (Tons)	0	0	1
Corn (Bushels)	0	0	0
Output	1 Ton	1 Ton	1 Bushel

Capital goods are specific in their uses in this example. θ₁-grade steel cannot be made out of θ₂-grade iron when θ₁ ≠ θ₂.

3.0 Stationary-State Quantity Flows
Suppose in Table 1 that:

• The first process is used to produce (1 + θ)/(4 + θ + θ²) tons of θ-grade iron
• The second process is used to produce (1 + θ)/(4 + θ + θ²) tons of θ-grade steel
• The third process is used to produce (1 + θ)/(4 + θ + θ²) bushels corn

Then one person-year would be employed over these three processes. Capital goods would consist of (1 + θ)/(4 + θ + θ²) tons of θ-grade iron and (1 + θ)/(4 + θ + θ²) tons of θ-grade steel. The capital goods would be used up throughout the latter two sectors, but reproduced at the end of the year. Net output would consist of (1 + θ)/(4 + θ + θ²) bushels corn per person-year.

4.0 Prices
Given the technique, stationary state prices must satisfy the following three equations:

[1/(1 + θ)] w = p₁

p₁(1 + r) + θ w = p₂

p₂(1 + r) + [3/(1 + θ)] w = 1

where:

• p₁ is the price of a ton θ-grade iron;
• p₂ is the price of a ton θ-grade steel;
• w is the wage;
• r is the rate of profits.

A bushel corn is the numeraire. The above equations embody the assumption that labor is paid at the end of the year.

The above is a system of three equations in four unknowns, given the technique. It is a linear system, given the rate of profits. The solution in terms of the rate of profits is easily found. The so-called factor-price curve for a technique is:

w(r, θ) = (1 + θ)/[3 + θ(1 + θ)(1 + r) + (1 + r)²]

The price of a ton θ-grade iron is:

p₁(r, θ) = 1/[3 + θ(1 + θ)(1 + r) + (1 + r)²]

The price of a ton θ-grade steel is:

p₂(r, θ) = [(1 + r) + θ(1 + θ)]/[3 + θ(1 + θ)(1 + r) + (1 + r)²]

Given the technique and the rate of profits, these prices can be used to evaluate the value of the capital goods used in a stationary state.

5.0 The Cost-Minimizing Technique
The optimal technique to use at any given rate of profits maximizes the wage. The first-order condition for such maximization is found by equating the derivative of the factor-price curve to zero:

dw/dθ = 0

Or:

3 + θ(1 + θ)(1 + r) + (1 + r)² - (1 + θ)(1 + 2θ)(1 + r) = 0

For 0 ≤ r ≤ 2, the cost-minimizing technique is then:

θ(r) = {[3 + (1 + r)²]/(1 + r)}^1/2 - 1

For r > 2, a corner solution is found:

θ(r) = 1

Figure 1 illustrates the cost-minimizing technique.

Figure 1: The Choice of Technique

The graph in Figure 1 reaches a minimum at a rate of profits of (3^1/2 - 1). For (12^1/4 - 1) < θ < 1, two rate of profits have the corresponding cost-minimizing technique indexed by the given value of θ. In other words, this is an example of reswitching.

The index for the cost-minimizing technique can be plugged into the factor price curve for the technique to which it corresponds at a given rate of profits. Figure 2 displays the resulting so-called factor price frontier. The index θ varies continuously for 0 ≤ r ≤ 200% in Figure 2. As the rate of profits increases without bound, the frontier approaches a wage of zero.

Figure 2: The Factor-Price Frontier

6.0 Capital and Labor "Markets"
Fratini’s notes that this is a reswitching example in which the capital market initially appears to be in accord with out-dated neoclassical intuition. The above analysis has shown how to find physical quantities of capital goods per worker, how to evaluate them at equilibrium prices, and how to find net output per worker. Figure 3 shows the resulting plot of the value of capital per unit output. Fratini looks at the value of capital per worker instead. Either curve is continuous and downward-sloping. The regions above and below the rate of profits of (3^1/2 - 1) appear qualitatively similar and visually indistinguishable. This curve might be said to be a downward-sloping demand function for capital.

Figure 3: The Capital Market

The analogous curve looks very different for the labor market (Figure 4). The region with a positive Wicksell effect is a region with a high rate of profits and thus a low real wage. The demand function for labor might be said to be upward-sloping in the region with a positive real Wicksell effect.

Figure 4: The Labor Market

7.0 Conclusion
The example makes Fratini’s point. The shape of the relationship between the value of capital, either per worker or per unit output, and the rate of profits is not necessarily a good indicator of the presence of reswitching or reverse capital-deepening.

Two Blogs Critical Of Economics

The post-autistic movement now has a blog: The Real-World Economics Review Blog.

I'm much less enthusiastic about the Counter-Economics Blog, which I stumbled over recently. Shaun Snapp claims to be applying critical thinking to economics, but he is too popular and too focused on finance for my taste. His claim that nobody reads either Adam Smith or Karl Marx is belied by the many serious scholars that do. (I've read major works by both.)

Herbert Gintis, Amazon Reviewer

Herb Gintis has now posted 231 reviews to Amazon. He has something to anger everybody.

Here he describes Jerry Cohen as a "supporter of virtually unsupportable Marxian doctrines" and having "studied ignorance of standard social and psychological theory."

He gives only two stars to Keen's Debunking Economics because, according to Gintis, it attacks a straw person. Mainstream economics is not as depicted by Keen, only undergraduate teaching is. "Abjectly brainless", "often just plain wrong", and "like teaching ... phlogiston and ether in physics class" are Gintis' phrases. I like how defenders of the mainstream cannot and will not defend economics as taught.

Gintis also gives only two stars to Ontology and Economics: Tony Lawson and His Critics. Basically, he disagrees that "Lawson's arguments are so powerful that few economists now feel that his case can be ignored." According to Gintis, his case can too be ignored; economists just ignore methodology. Gintis doesn't really engage the give and take in the book. I think he should have noted his agreement with John Davis's take on the openness of mainstream economics to some kinds of heterodox contributions.

I found this review of a recent George Soros book of interest. Some blame the current financial meltdown on failures of either individual or collective rationality. Gintis says that even if everybody were as rational as some (Chicago?) economists posit, market fundamentalism would still be unfounded. He bases this claim on the failure of the Arrow-Debreu model of General Equilibrium to have any attractive dynamical properties. He recommends agent-based modeling to analyze capitalist economies.

Gintis has quite a few positive reviews of rightists. For example, he gives four stars to Hazlitt's Economics in One Lesson. (Despite most of the reviews I'm highlighting, he also has some extremely positive reviews for liberals and leftists.) I think his reviews of right wing books generate more comments, and Gintis replies. (The worst are full of passionate intensity.) One review of a book that I would think is not worth reading currently has 103 comments.

In addition to politics and economics, he has also reviewed books on language, biology, and logic. I want to recall the existence of Torkel Franzen's Godel's Theorem: An Incomplete Guide to Its Use and Abuse.

A Taxonomy Of The Effects Of Wicksell Effects

Consider a typical circulating capital model, in which commodities are produced from commodities and labor. The technique in use is described by a square Leontief input-output matrix and a vector of labor coefficients. In a long run equilibrium, in which prices are stationary, the technique is selected from a set of techniques to minimize production costs at a given interest rate. That set is known as the technology.

Suppose the composition and quantity of output is taken as given, along with the interest rate and the technology. The difference in the value of the capital goods at two different interest rates is the sum of the price Wicksell and real Wicksell effects. The price Wicksell effect is the sum of the differences in prices among the capital goods for a given technique. But the cost-minimizing technique might not be the same at two interest rates. The real Wicksell effect is the difference in the value of the capital goods for two techniques, given the price system at a one interest rate.

I want to compare the relative magnitude of price and real Wicksell effects at a given interest rate. Thus, I want to consider derivatives at a given interest rate. Therefore, suppose the technology consists of a continuum of techniques that might be eligible along the so-called factor price frontier. Table 1 shows all combinations of price and real Wicksell effects. A Wicksell effect is negative when the equilibrium at the higher interest rate has a lower value of capital, from the effects of price and quantity changes, respectively.

Table 1: Possibilities

	Technology Property		Labor Market Response	Capital Market Response
	Price Wicksell Effects	Real Wicksell Effects	Higher Wage	Lower Interest Rate
A	Negative	Negative	Less Employment	Increased Value of Capital
B	Negative	Positive	More Employment	Indeterminate
C	Positive	Negative	Less Employment	Indeterminate
D	Positive	Positive	More Employment	Decreased Value of Capital
E	Zero	Negative	Less Employment	Increased Value of Capital
F?	Zero	Positive	More Employment	Decreased Value of Capital
G	Negative	Zero	Unchanged Employment	Increased Value of Capital
H	Positive	Zero	Unchanged Employment	Decreased Value of Capital
I	Zero	Zero	Unchanged Employment	Unchanged Value of Capital

Row A in Table 1 conforms to the outdated neoclassical intuition of equilibrium prices as indices of relative scarcity. But, as Edwin Burmeister has noted, nobody knows what special case assumptions need to be imposed on technology to ensure that Wicksell effects happen to fall in any given direction.

I have the response in the capital market shown as indeterminate for rows B and C. The claim is that, for the case of a technology representable by a continuum of techniques, the price Wicksell effect can, but need not, swamp the real Wicksell effect. It is essential for this swamping to occur at a single interest rate that the technology be continuous. Pierangelo Garegnani, Heinz D. Kurz & Neri Salvadori, and Saverio M. Fratini have examples that illustrate some possibilities with a continuum of techniques.

Row E is the case of Samuelson's Surrogate Production Function. Price Wicksell effects are zero when the factor price curve for a given technique is a straight line. The question mark after the label for Row F reflects my belief that this row catalogs an impossibility. If factor price curves are straight lines along their entire length, capital-reversing cannot arise.

Rows G, H, and I are cases in which the real Wicksell effect is zero. The real Wicksell effect is zero in the discrete case when the factor price curves are tangent at a switch point. I'm not sure how this extends to the continuous case, in which all points along the factor price frontier are non-switching points. If the Row I case is possible, the technique is not determined by the location of the corresponding factor price curve. I think this may be so for non-straight line factor price curves, but I'm unsure about this case.

These remarks suggest a research program. First, demonstrate that no possibilities exist that are not listed in Table 1. This would seem to be obvious. But I don't understand Andreu Mas-Colell's paper "Capital Theory Paradoxes: Anything Goes" (in Joan Robinson and Modern Economic Theory (ed. by G. Feiwel) (1989)). He shows some multi-valued relations where I would expect functions. Second, for those rows that are impossible in Table 1, demonstrate this impossibility. Third, for each possible row, construct a numeric example. For rows B and C, one should construct at least two examples, one for each direction of the capital market response. I suppose a third example, in which price and real Wicksell effects are exactly matched in magnitude would be amusing. Much of this research would be non-original; many components are in the literature.

Nietzsche On The Individual As A Society

I have previously noted the problems for utility theory created by the application of Arrow's impossibility theorem to a single individual. And I had quoted a number of classic authors who wrote of themselves as being composed of more than one mind. Here's another:

"'Freedom of the will' - that is the expression for the complex state of delight of the person exercising volition, who commands and at the same time identifies himself with the executor of the order - who, as such, enjoys also the triumph over obstacles, but thinks within himself that it was really his will itself that overcame them. In this way the person exercising volition adds the feelings of delight of his successful executive instruments, the useful 'underwills' or undersouls - indeed our body is but a social structure composed of many souls - to his feelings of delight as commander. L'effet c'est moi. What happens here is what happens in every well-constructed and happy commonwealth; namely, the governing class identifies itself with the successes of the commonwealth. In all willing it is absolutely a question of commanding and obeying, on the basis, as already said, of a social structure composed of many 'souls'." -- Friedrich Nietzsche, Beyond Good and Evil: Prelude to a Philosophy of the Future (Kaufmann translation), paragraph 19

By the way, the idea of modeling an individual choice with a structure underlying the textbook treatment of preferences over the elements of a linear space of commodities is not necessarily non-mainstream. I cannot say I know much about the relevant literature. However, I stumbled over an example - a paper, "Multiple Temptations", from John E. Stovall, a graduate student at the University of Rochester.

An Indeterminate Two-Person Zero-Sum Game With Perfect Information

1.0 Introduction
I have stumbled upon some odd mathematics, some mathematics that I have not validated. Consider the claim that all two-person zero-sum games with perfect information have a value. Apparently, this claim is inconsistent with the Axiom of Choice, an axiom in set theory. This inconsistency is shown by the Banach-Mazur game and its variants. I guess it is essential to this demonstration that these games have a potentially countable infinite number of moves.

I don't know that this demonstration is as important for economics as, for instance, W. F. Lucas' example of a cooperative game without an equilibrium.

A game has perfect information if the results of all moves prior to any given move are known to all players. Simple examples of games with imperfect information are card games in which the deal gives a player a hand which only he knows. A two-person zero-sum game is determinate if one can prove either (1) the first player wins some definite amount, (2) the second player wins some definite amount, or (3) the game is a draw. Chess is a determinate game, although it is in practice impossible to expand the tree enough to determine its value.

2.0 A Game
I steal this example from a Usenet post by Herman Rubin.

The game is fully specified by the rules and by defining a set C, where C is a given subset of the real numbers between 0 and 1, inclusive. The two players alternatively select the successive binary digits of the base-two expansion of a number within the interval [0, 1].

In other words, consider the number:

(1/2) x₁ + (1/4) x₂ + (1/8) x₃ + (1/16) x₄ + ...

where, for all i, x_i is in {0, 1}. The first player chooses the binary digits with the odd indices, and the second player chooses the binary digits with the even indices. But they take turns and go in order.

The game ends with the second player paying the first player a unit when it is guaranteed that any further expansion will result in a number within C. The game ends with the second player winning a unit payment from the first player when it is guaranteed that any further expansion will result in a number in the complement of C.

A simple example is C = [0.5, 1]. The first player wins in this case. A more complicated game arises when C is the set of all irrational numbers in the unit interval. I gather this game is determinate, but I don't see offhand who wins. Finally, consider a set C that does not have a Lebesque measure. (The Axiom of Choice is necessary for the definition of such a set.) I gather that in this case, the game is not determinate. Nobody can tell a priori who will win.