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.