Another Reswitching Example

1.0 Introduction
I have previously presented examples of reswitching and capital reversing. These phenomena were discovered as part of the Cambridge Capital Controversy, and you will occasionally see commentators bring up the CCC on various blogs. This post presents another example. This example has a different structure than my favorite models. In this example, not all commodities are produced with the aid of commodities. Some capital goods not explicitly shown are produced directly with unassisted labor.

2.0 Some Theory And An Example
Consider a firm choosing among a set of techniques for producing a given commodity. Each technique, as in Table 1, exhibits Constant Returns to Scale (CRS) and can be represented as a series of dated labor inputs. The first element in this series is the amount of labor that must be applied in the technique during the current production cycle to produce one unit of the commodity. The second element is the amount of labor that must be applied one cycle ago to produce the capital goods required for use by the first labor input in the series. And so on. This representation of a technique is known as a flow-input, point-output model.

Table 1: The Technology

Year Before Output	Labor Hired for Each Technique
	Alpha	Beta
0	33 Person-Years	0 Person-Years
1	0 Person-Years	52 Person-Years
2	20 Person-Years	0 Person-Years

Consider the the cost of producing a unit of output with a given technique, where the calculation of the cost is performed at the end of the year in which the output becomes available. The cost is the sum of the cost of the labor inputs over previous years, with the cost of each labor input including an interest charge:

L₀ w + L₁ w (1 + r) + L₂ w (1 + r)² + ... + L_n w (1 + r)ⁿ + ...

where w is the wage for a unit of labor, r is the rate of profits, and (L₀, L₁, L₂, ...) is the series of dated labor inputs representing the technique. The wage is paid at the end of the year for the labor expended during that year.

The best rate of profits (also known as the interest rate) available to the firm should be used in finding the cost of producing a unit of output with each technique. Since the only use of financial capital available to the firm here is to produce with one or another of the techniques of production, the best rate of profits is the rate of profits obtainable with cost-minimizing techniques. The following algorithm traverses the techniques to find the rate of profits associated with cost-minimizing techniques:

1. Choose a technique.
2. For the given wage w, find the rate of profits r that, when used to cost up the labor inputs, equates that cost to the price of a unit of the produced commodity.
3. Cost up the labor inuts for all techniques with this wage and rate of profits.
4. Choose a technique with the minimum of the costs found in Step 3.
5. If the technique for which the rate of profits was obtained in Step 2 can be selected in Step 4, stop. You have found a cost-minimizing technique and the rate of profits.
6. Else, go to Step 2 and repeat with the newly selected technique.

Figures 1, 2, and 3 illustrate the application of the algorithm to the example. Figure 1 shows the wage-rate of profits curves associated with each technique. In flow-input, point-output models, these curves never cross the axis for the rate of profits. Instead, the curves approach it asymptotically. Points at which such curves intersect are called switch points. The "perverse" or "paradoxical" switch point is at a wage of 1/78 units of output per person year and a rate of profits of 50%. The higher-wage switch point is at a wage of 5/286 units of output per person-year and a rate of profits of 10%. A switch point is called "paradoxical" merely because it illustrates behavior contradicting mistaken and outdated neoclassical beliefs. The algorithm can be considerably simplified:

Theorem: Consider a technology specified as a choice among techniques, where each technique is represented by a flow-input, point-output model. Let the outer envelope of the wage-rate of profits curves for each technique be constructed by finding the maximum rate of profits for each wage over all wage-rate of profits curves. For each point on the outer envelope, a technique in which the technique’s corresponding wage-rate of profits curve contains that point is selected by the above algorithm as a cost-minimizing technique at that wage.

Figure 1: Rate of Profits by Technique

The theorem is easily applied to the example. By the theorem, the alpha technique is cost-minimizing for wages between the switch points. The beta technique is cost-minimizing for wages below 1/78 units of output per person-year and between wages of 5/286 and the maximum wage of 1/52 units of output per person-year.

Now to show that the algorithm yields the same answers. Suppose one starts by choosing the alpha technique at Step 1. The rate of profits sought in Step 2 is read off the wage-rate of profits curve for the alpha technique in Figure 1, given the wage. The costs found in Step 3 are shown in Figure 2. Figure 1 shows that the alpha technique is, indeed, cost minimizing for intermediate wages, that is, between the switch points. If the wage is either too low or too high, the beta technique is selected in Step 4, as shown in Figure 2. According to Step 6, the flow of control in the algorithm then goes to Step 2 with the beta technique selected. Figure 3 shows the costs of the techniques using the rate of profits for the beta technique. Here too, the algorithm concludes with the same answer as suggested by the theorem.

Figure 2: Cost of Techniques at Alpha Rate of Profits

Figure 3: Cost of Techniques at Beta Rate of Profit

3.0 Conclusion
Once the cost-minimizing technique has been determined, one can consider how much labor firms will want to hire per unit output at any given wage. Figure 4, which is based on the assumption of a stationary-state output, illustrates this calculation for the example. If a firm is operating the alpha technique, for example, to produce an unchanged output over the next three years, workers producing consumption goods available at the end of the current year, producing capital goods to be used to produce consumption goods available at the end of the next year, and producing capital goods to be used to produce capital goods available at the end of the next year will all be working side-by-side. That is, 53 workers will be employed per unit output under the alpha technique for a stationary state. As the graph shows, around the switch point at the lower wage, a higher wage is associated with a cost-minimizing technique in which firms want to employ more workers for a given stationary-state output.

Figure 4: Labor Intensity of Cheapest Technique

So much for the theory that wages and employment are determined by the intersection of well-behaved supply and demand curves in the labor market.

The Undermining Social Democratic Downhill Slide...

"[Michael] Harrington was a 'democratic socialist', not a social democrat. Whereas social democracy endorsed the welfare state, democratic socialism expanded proletarian decision-making into every corner of society, including the workplace, thereby reversing capitalist priorites. In America, democratic socialism meant taking the New Deal beyond its first two stages [1930s and 1960s?]. By empowering workers, a 'Third New Deal' would complete the process begun in 1929, when control of key economic investments was passed from unfettered boardrooms to a capitalist-controlled welfare state." -- Robert A. Gorman, Michael Harrington: Speaking American, Routledge (1995)

Two Crises Of Economic Theory

The American Economic Association held their annual meeting in New Orleans in 1971. At the invitation of John Kenneth Galbraith, the AEA president that year, Joan Robinson delivered the keynote Ely address. She identified "The Second Crisis of Economic Theory".

The first crisis occurred in the 1930s. Economists had no theory, at the time, to guide policy for addressing the dramatic drop in the volume of output and the increase in unemployment. John Maynard Keynes, as well as Michal Kalecki, developed the theory to address this crisis.

As I recall, the second crisis of economic theory relates to the mix of goods being produced, even when the volume is such that more-or-less full employment is being achieved. Many of her time did not think the balance correct. Conspicuous consumption, positional goods, and the means of destruction are produced in abundance. But as for the production of amenities useful for modern life typically provided by government (e.g., public transportation) - not so much. Mainstream economic theory does not provide a perspective for a thorough-going improvement on what comes out of more-or-less capitalist markets.

Consider the context of Robinson's Ely lecture. Although she had already developed a theory of stagflation, she couldn't have known how poor western economies would perform over the next decades. So at the time of her lecture, the second crisis of economic theory might have been more readily apparent than the first.

It seems to me that both crises are evident today. The world's economic problems are not only how to get people back to work during this worldwide global downturn. We also need to reorient the world's economy to operate with more sustainable and renewable energy resources, encourage the production of more public goods in many economies, decrease the workweek, etc.

Reference

• Joan Robinson (1972) "The Second Crisis of Economic Theory", American Economic Review, Papers and Proceedings, V. 62 (May): 1-10

Empirical Evidence On Minimum Wages

A theme of this blog is that it is illogical to explain wages and employment by the interaction of well-behaved demand and supply curves in the labor market, even under assumptions of perfect competition, no information asymmetries, etc. Since orthodox teaching on the subject is simply incorrect, David Card's empirical results on minimum wages did not astonish me.

For those interested in these results, I notice that researchers at Berkeley have extended them. So this paper goes on my long list of ones to read some day:

"We use policy discontinuities at state borders to identify the effects of minimum wages on earnings and employment in restaurants and other low-wage sectors. Our approach generalizes the case study method by considering all local differences in minimum wage policies between 1990 and 2006. We compare all contiguous county pairs in the U.S. that straddle a state border and find no adverse employment effects. We show that traditional approaches that do not account for local economic conditions tend to produce spurious negative effects due to spatial heterogeneities in employment trends that are unrelated to minimum wage policies. Our findings are robust to allowing for long term effects of minimum wage changes." -- Amdrajit Dube, T. Wiliam Lester, and Michael Reich "Minimum Wage Effects Across State Borders: Estimates Using Contiguous Counties", working paper (2008)

In Honor Of The Day

Today is the bicentennial of Darwin's birth.

I think of evolutionary economics as equivalent to (old) institutional economics.

Clarence Ayres was one prominent institutionalist economist, maybe the most prominent of his generation. He ended up at the University of Texas at Austin and helped develop the Texas school of institutional economics. (James Galbraith is currently at UT.) Ayres most well known book is probably The Theory of Economic Progress, which I have not read. I have read The Divine Right of Capital, in which I found some echoes of Joan Robinson's capital critique, even though it precedes the relevant portions of her work.

But I want to point out that Ayres also wrote Huxley, a 1932 biography of Thomas Huxley, also known as "Darwin's bulldog". In 1860 Huxley debated Bishop Samuel Wilberforce:

"Turning to Huxley he inquired with a charming show of solicitude whether the scientist supposed himself to be descended from an ape on the side of his grandmother or his grandfather... And then [Huxley] delivered his famous counterstroke, the upshot of which is that ancestor apes are preferable to bishops."

Infinite Are The Arguments Of Mages

The following letter comes after five interchanges of letters between Keynes and Hayek and a summary letter by Keynes. Keynes was trying to get at what definitions of saving and investment Hayek was using, what Hayek meant by "forced saving", and why Hayek thought a constant proportion of the money in circulation must be saved to keep the capital stock at the same level. The date of this letter is 29 March 1932, and it is reprinted in The Collected Works of F. A. Hayek: Volume 9: Contra Keynes and Cambridge: Essays, Correspondence (edited by Bruce Caldwell, Chicago University Press, 1995):

Dear Hayek,

I will certainly reserve you space in this June [Economic] Journal for a reply to Sraffa. But let it be no longer than it need be. It is the trouble of controversy - from an editor's point of view - that it is without end. Your MS should reach me not later than May 1.

Having been much occupied in other directions, I have not yet studied your Economica article as closely as I shall. But, unless it be dealt with in isolation from the main issue, I doubt if I shall return to the charge in Economica. I am trying to re-shape and improve my central position, and that is probably a better way to spend one's time than in controversy.

Yours sincerely
J. M. Keynes

Economists As Creationists Who Have Never Heard Of Evolution

I found this quote amusing:

"One of Brad DeLong’s commentators compares what’s going on to the discovery that some eminent biologists are creationists, but it’s actually worse than that: it’s like discovering that some eminent biologists have never heard of the theory of evolution and the concept of natural selection." -- Paul Krugman

I am in agreement with, for instance, Krugman and DeLong that the U.S.A. government currently needs to spend a lot on a stimulus package. But I wondered if such more-or-less liberal economists acknowledge other controversies in economics. Have they heard factor prices (like wages) cannot necessarily be explained by the interaction of well-behaved supply and demand curves in factor markets, even under ideal assumptions of perfect competition, flexible prices, perfect information, etc? I don't know whether I want to put much emphasis on Brad DeLong's failure to note such in this post about immigration.

Simple And Expanded Reproduction

1.0 Introduction
This is a repost. I've switched templates, and the new one interferes with the layout of the mathematics in the previous version. I've made a few minor changes here and there.

This post presents a model in which a capitalist economy smoothly reproduces itself. The purpose of such a model is not to predict that capitalist economies will converge to some such path as illustrated in the model. Rather, the model provides a basis for the analysis of where things can go wrong.

This sort of model has a long history. My exposition is close to Marx (1956), with the difference that Marx sets out the conditions of simple and expanded production in terms of labor values, not in terms of prices of production. Rosa Luxemburg (1951) and Michal Kalecki (1969) used Marx's department break-down to develop a Keynes-like model of the long run and the short run. Shigeto Tsuru (1942) apparently exposed this model to english-speaking academics when few were looking at Marx's analysis. Paul Samuelson (1957) thought this model of interest. Joan Robinson (1962) drew on these ideas, among others, in her models of metallic ages. Goodwin's generalization (1949) of Keynes to a multisectorial model and Pasinetti's (1981, 1993) analyses of vertically integrated sectors also seem to me to bear family resemblances to this model. Doubtless, my references could be extended in many directions.

Table 1: Definition of Variables

Variable	Definition
a01	The person-years of labor hired per unit output (e.g., ton steel) in the first sector.
a02	The person-years of labor hired per unit output (e.g., bushel corn) in the second sector.
a11	The capital goods (measured in tons) used up per unit output in the first (steel-producing) sector.
a01	The capital goods (measured in tons) used up per unit output in the second (corn-producing) sector.
p1	The price of a unit output in the first sector.
p2	The price of a unit output in the second sector.
r	The rate of profits.
s	The savings rate out of profits.
w	The wage, that is, the price of hiring a person-year.
X1	The number of units (ton steel) produced in the first sector.
X2	The number of units produced (bushels corn) in the second sector.
g	The rate of growth.

2.0 Two Departments
This model considers a capitalist economy with no government and no foreign trade. The outputs of this economy are grouped into two great departments. In the first department, capitalists direct workers to produce means of production (also known as capital goods) with the means of production in that department. In the second department, the workers are directed to produce means of consumption (also known as consumption goods) with the means of production in that department.

For ease of exposition, I make certain additional simplifying assumptions. The workers consume all of their wages. Only the capitalists save, and they save only in the case of expanded reproduction. All capital is circulating capital. That is, there is no fixed capital, such as long-lived machinery. In other words, all capital goods are totally used up each year in producing the yearly output. No technological innovations are introduced.

I think introducing technological innovations and fixed capital makes the possibility of smooth reproduction more incredible. A government can be introduced as a third department, or perhaps by dividing government output among the two departments shown. Foreign trade introduces the possibility of correcting imbalances in domestic demand from outside the domestic economy. But then one could recast the model as of the world economy.

3.0 Prices
A necessary condition for smooth reproduction of a competitive capitalist economy is that the same rate of profit be made in all departments. Otherwise, some capitalists are finding that the expectations on which investments were made are being unfulfilled. They would want to have contracted some departments and expanded others. I also impose the condition that spot prices remain stationary. The following equations express these conditions:

(a₁₁ p₁)(1 + r) + a₀₁ w = p₁

(a₁₂ p₁)(1 + r) + a₀₂ w = p₂

I suppose one could put time indices on the prices in the above equations, thereby defining a dynamic system for prices. Suppose distribution and the ratios of physical quantity flows remain unchanged year after year. Then the steady-state prices expressed in the above equations (without time indices) would be a limit point of the dynamic process so defined. It is this caveat, I think, that allows me to ignore that constant prices are, perhaps, not a necessary condition for smooth reproduction.

Table 2: Value of Outputs by Department and Distribution

Department	Capital	Wages	Profits
Capital Goods	a11 X1 p1	a01 X1 w	a11 X1 p1 r
Consumption Commodities	a12 X2 p1	a02 X2 w	a12 X2 p1 r

4.0 In Balance
4.1 Simple Reproduction
The economy is in simple reproduction when it is replicated on the same scale year after year. A necessary condition for an economy in simple reproduction is that the production of capital goods each year be equal to the capital goods used up each year. In the model shown here, the value of the capital goods used up each year must equal the value of the output of the first department:

a₁₁ X₁ p₁ + a₁₂ X₂ p₁ = (a₁₁ X₁ p₁)(1 + r) + a₀₁ X₁ w

The above equation can be simplified:

a₁₂ X₂ p₁ = a₀₁ X₁ w + a₁₁ X₁ p₁ r

The above is easily summarized in words. It states that the value of capital goods demanded from the second department matches the demand for consumption goods from the first department. In a sense, this equation is a generalization of Keynes' idea of effective demand. The condition that all workers looking for a job are able to find one at the going wage is a separate condition, not stated here. This model generalizes Keynes' theory, in some sense, to the long-run.

An alternate method of deriving the last equation is available. Start from the equation of the value of total demand for consumption goods and the value of the output of the department producing consumption goods. This condition, when simplified, yields the same equation.

4.2 Expanded Reproduction
The economy experiences expanded reproduction when it consistently expands each year. In this case, the demand for capital goods from the second department includes the savings of the capitalists receiving profits from that department. Likewise, the demand for consumption goods from the first department excludes the savings of the capitalists in that department. Observing these qualifications, it is easy to mathematically express the condition that the demand for capital goods from the second department match the demand for consumption goods from the first department:

a₁₂ X₂ p₁ + s a₁₂ X₂ p₁ r = a₀₁ X₁ w + (1 - s) a₁₁ X₁ p₁ r

Or:

a₁₂ X₂ p₁(1 + s r) = a₀₁ X₁ w + (1 - s) a₁₁ X₁ p₁ r

Focus on the left-hand side of the above equation. Is it apparent that the rate of growth of the value of the capital goods in the second department is the product of the capitalists' saving propensity and the rate of profit? In expanded reproduction, under these simplifying assumptions, both departments and their components all grow at the same rate. In other words, the rate of profit along a warranted growth path is the quotient of the rate of growth and the saving propensity of the capitalists.

r = g/s

This is the famous Cambridge equation typically arising in a Post Keynesian theory of distribution, especially in, say, Luigi Pasinetti's version.

5.0 Conclusion
In the model, capitalists independently decide on what department to enter, and how much to produce in that department. A collective result of those decisions is the total output of each department. For those decisions to be validated, the value of consumer goods demanded by workers and capitalists in the department producing capital goods must match the value of capital goods demanded by the capitalists in the department producing consumption goods.

The model is silent on how such an equality can come about. Supply and demand seems like an inadequate answer to me.

References

• Richard M. Goodwin (1949). "The Multiplier as Matrix", Economic Journal, V. 59, N. 236 (Dec.): 537-555
• M. Kalecki (1969). Theory of Economic Dynamics: An Essay on Cyclical and Long-Run Changes in Capitalist Economy, Second Edition, Augustus M. Kelly
• Rosa Luxemburg (1951). The Accumulation of Capital (Trans. by Agnes Schwarzschild), Yale University Press
• Karl Marx (1956). Capital, Volume 2, Progress Publishers
• Luigi L. Pasinetti (1981). Structural Change and Economic Growth: A Theoretical Essay on the Dynamics of the Wealth of Nations, Cambridge University Press
• Luigi L. Pasinetti (1993). Structural Economic Dynamics: A Theory of the Economic Consequences of Human Learning, Cambridge University Press
• Joan Robinson (1962). Essays in the Theory of Economic Growth, Macmillan
• Paul A. Samuelson (1957). "Wages and Interest: A Modern Dissection of Marxian Economic Models", American Economic Review, V. 47 (Dec.): 884-912
• Shigeto Tsuru (1942). "On Reproduction Schemes", Appendix A in Paul Sweezy's The Theory of Capitalist Development, Monthly Review Press [This reference I haven't read]

Leijonhufvud's Corridor

The ideas of a number of economists have been cited in popular accounts of the current economic crisis.

Some have been calling this a "Minsky moment" and have been exploring the Post Keynesian economist Hyman Minsky. In building on this trend, Michael Perelman sees connections to Marx's views on fictious capital. I have pointed out Hugh Townshend basically describing (in 1937) the housing crisis as part of his account of

Another popular interpretation points to erroneous Austrian Business Cycle Theory. For some reason - probably political, although I'm willing to accept ignorance also - very few who go on about the ABCT bring up the Austrian concept of the "secondary depression". Yet this epicycle would seem to be precisely relevant as the crisis becomes a general downturn, instead of being more restricted to financial institutions.

Another interesting idea, although one I am not sure I agree with, is Axel Leijonhufvud's corridor. Leijonhufvud became famous for arguing "Keynesians" had misinterpretated Keynes. As I understand it, his interpretation is closely related to that offered by Robert Clower in his 1965 paper, "The Keynesian Counter-Revolution: A Theoretical Appraisal". Clower argues that households, in making consumption and savings decisions, are constrained by realized income, not by the income they would receive at a point of full-employment equilibrium. Clower and Leijonhufvud end up with thinking of Keynes as describing a disequilibrium. I think Keynes claimed to have described an equilibrium that could be consistent with the presence of involuntary unemployment.

I think Leijonhufvud developed his notion of the corridor later. The corridor constitutes time paths near enough to full employment. If the economy falls outside the corridor, agents no longer have well-developed norms and expectations about what happens that are mutually consistent and likely to quickly lead the economy back to full employment.

Influence Of Government Size On Income Distribution

I have been reading Philip A. Klein's Economics Confronts the Economy (Edward Elgar, 2006). He presents data in two tables that I thought, when combined, suggest relationships. So I drew the regression lines below. The data on income distribution is from 1998. Total government expenditures as a percentage of GDP is from 1999. Government expenditures include, for example state and local spending in the United States. The graphed data are for the United States, and, in order of decreasing percentage of government spending as a percentage of Gross Domestic Product, Sweden, Denmark, Belgium, Finland, France, Austria, Italy, Netherlands, Norway, Canada, Germany, New Zealand, the United Kingdom, Spain, Ireland, Australia, and Japan. Apparently, in advanced industrial democracies, as government spending directs a greater percentage of resources, the poorest have relatively more income and the richest have relatively less. One can see why the richest would like lackeys to fight the latter tendency.

Figure 1: Share of Income in Lowest 10% Among Developed Economies

Figure 2: Share of Income in Highest 10% Among Developed Economies

I haven't finished Klein's book, but I think I'll note two points I find of interest. He argues that when economists advocate for positive analysis of existing economies, they implicitly accept the status quo. This is a value judgement that could be contested. Secondly, when economists limit normative theory to Pareto-improving recommendations, they restrict themselves from commenting on such matters as income distribution and the quality of life of the vast population. (Moving along one of the regression lines probably makes some worse off.)

A Brouhaha

Some mainstream economics have been pointing out other well-known economists are simply incompetent. Among those doing the pointing are Paul Krugman and Brad DeLong. Robert Waldmann provides some background. (I'm not sure if this is Waldmann.) Mark Thoma comments. I think Uwe Reinhardt's post somehow fits in this grouping. Matthew Yglesias opts for believing the economists being pointed at are deliberately misleading, instead of merely stupid.

I was amused at the ignorance exhibited by a self identifed "UMN Econ Student" in the comments to Waldmann's post:

"The short of it, and this is coming from a Minnesota Econ PhD student, is that we try hard not to bullshit people by hiding our assumptions, no matter how ridiculous you may find them."

A canonical freshwater macroeconomic model is not merely a General Equilibrium model with perfect competition, perfect information, etc. It will also have a single representative agent and, in each period, a single homogeneous produced good that can be used either as a consumption good or as a capital good. What special-case assumptions on technology and preferences would one like to impose in a multi-good multi-agent model such that such a model behaves like a one agent, one good model? Economists have no answer.

The above posts are a selection. It is only in the comments that anybody points out that both saltwater and freshwater economists totally ignore Post Keynesians. I like this comment by A. Senderowicz.

Update: Somehow I missed Daniel's overview. Neil Sinhababu, at Ezra Klein's blog, has noticed this contretemps.

Samuelson Jests

I might as well document some jibes Kevin Quinn finds amusing. These are from two different articles from two different periods:

"I have dealt with Karl Marx the economist, not Marx the philosopher of history and revolution. A minor Post-Ricardian, Marx was an autodidact cut off in his lifetime from competent criticism and stimulus."-- Paul A. Samuelson (1957) "Wages and Interest: A Modern Dissection of Marxian Economic Models", American Economic Review, V. 47 (Dec.): pp. 884-912

"The 'transformation algorithm' is precisely of the following form: 'contemplate two alternative and discordant systems. Write down one. Now transform by taking an eraser and rubbing it out. Then fill in the other one. Voila! You have completed your transformation algorithm.'" -- Paul A. Samuelson (1971) "Understanding the Marxian Notion of Exploitation: A Summary of the So-Called Transformation Problem Between Marxian Values and Competitive Prices", Journal of Economic Literature, V. 9, N. 2 (June): pp. 399-431

"The truth has now been laid bare. Stripped of logical complication and confusion, anybody's method of solving the famous transformation problem is seen to involve returning from the unnecessary detour taken in Volume I's analysis of values. As I have cited in my mathematical paper, such a 'transformation' is precisely like that in which an eraser is used to rub out an earlier entry, after which we make a new start to end up with the properly calculated entry." -- Paul A. Samuelson (1971) "Understanding the Marxian Notion of Exploitation: A Summary of the So-Called Transformation Problem Between Marxian Values and Competitive Prices", Journal of Economic Literature, V. 9, N. 2 (June): pp. 399-431

Samuelson refers to:

• Paul A. Samuelson (1970) "The 'Transformation' from Marxian 'Values' to Competitive 'Prices': A Process of Rejection and Replacement", Proceedings of the National Academy of Sciences, 67(1) (Sept.): pp. 423-425.

As far as I know, no economist responded to Samuelson’s paper in the 1950s. He had reaction in the 1970s. The JEL published replies by Abba Lerner and Michio Morishima; an editorial comment by Mark Perlman; an analysis by Martin Brofenbrenner of submitted but unpublished reactions by Gordon Bjornson, Jean Cartelier, Bruno Jossa, David Laibman, Paul Massick, and Murray Wolfson; a paper on Marx by William Baumol; and responses by Samuelson, including an interchange with Joan Robinson. I suggest the difference in the number of reactions has something to do with events in non-academia. Perhaps the sixties had some impact on the algebra in which some economists were interested.

Krugman Correct

More than a decade ago, Paul Krugman characterized Austrian Business Cycle Theory (ABCT) as "The hangover theory":

"It is the idea that slumps are the price we pay for booms, that the suffering the economy experiences during a recession is a necessary punishment for the excesses of the previous expansion"

The other day, Steven Horwitz, one of the leading advocates of ABCT offered the following analogy for economics:

"the mistake is drinking too much and being hungover is the correction."

(Steven Horwitz is the secretary and webmaster for the Society of the Development of Austrian Economics and manages to get letters to the editor published in the Watertown Daily Times. Myself, I read the Lowville Journal & Republican if I want to know who is growing the largest pumpkin and who has been visited over the weekend by their children away at college.) I think my critique of ABCT goes more into the nuts and bolts of the theory.

Elsewhere, Krugman adopts a Post Keynesian point:

"I should also point out this, in [Robert] Barro’s article:

'John Maynard Keynes thought that the problem lay with wages and prices that were stuck at excessive levels. But this problem could be readily fixed by expansionary monetary policy, enough of which will mean that wages and prices do not have to fall.'

Is it too much to ask that someone criticizing Keynes actually, you know, read Keynes — at least enough to know that he devoted a whole chapter to explaining why a fall in wages would not expand employment?"

Hart and Zingales, Stupid or Dishonest?

On 3 December 2008, Oliver Hart and Luigi Zingales wrote on the funny pages of the Wall Street Journal:

"This year will be remembered not just for one of the worst financial crises in American history, but also as the moment when economists abandoned their principles. There used to be a consensus that selective intervention in the economy was bad."

What kind of principle of economics is this that tells us what policy should be?

It is obviously untrue that any such consensus existed. Corporations for which their owners have limited liability could not be founded without a selective government intervention in the economy. Any legal protection for Intellectual Property, such as any non-zero period for copyrights and patents, is a selective government intervention. The existence of the Federal Reserve is a selective government intervention. Furthermore, economists have had no consensus for decades on claims that monetary policy is ineffective or that the monetary authorities should conform to the Cassels/Friedman rule of growing the money supply at a constant rate. Bankruptcy law, including its recent tweaking against the interests of debtors, is a selective government intervention. Hart and Zingales demonstrate, with the balderdash in their first paragraph, that they have not given a serious moment's thought to the theory of the firm, industrial organization, or macroeconomics.

What has been demonstrated is, in general, orthodox economists are willing to serve as lackeys to the malefactors of great wealth. (I could pick many more examples.) What effect on elite opinion does this willingness of supposed academic experts to teach nonsense in the news organs of the affluent have, do you think?

A Theory of the Rate of Change in Exchange Rates

1.0 Introduction
I created this model by thinking about what would happen if no basic commodity existed, and yet no commodity could be produced with unassisted labor alone. That is, suppose (seed) corn can be used to produce corn, and rice can be used to produce rice. But corn does not enter either directly or indirectly into the production of rice. Nor does rice enter either directly or indirectly into the production of corn.

The mathematical problem posed by these thoughts can be set out in a model of two countries trading. I end up with an explanation of relative rates of currency appreciation across countries by the interaction of technology and the distribution of income - class struggle, if you will - in each country. I think this model illustrates that non-mainstream ways of thinking about economics can suggest new models and new insights.

I don't claim originality for this model. I was re-reading Samuelson (1957) to confirm my impression that that is where Samuelson describes Marx as "a minor Post-Ricardian". (I'm fairly sure Samuelson sets out his eraser algorithm in his 1971 JEL article.) I did not recall that Samuelson had set out a Marxist scheme of reproduction in his 1957 paper, albeit with a poor supply and demand interpretation. Anyways, I stopped at this passage:

"Without going into the social relations of the past or future, any economist... can evisage a case where Industry III [luxuries] alone, by virtue of having a₃ = 0 and b₃ < 1 will determine its own-rate of profit by itself, and he will realize that if this new r differs from that of (11) what must give is not bourgeois economic theory or the capitalistic institutional economy but rather the assumption of stationary relative prices." -- Paul A. Samuelson (1957)

This quote jostled my memory of a Joan Robinson review of Sraffa's book. I also dimly recall Keynes' 1923 analysis of arbitrage in forward trades in international currency markets and expected rates of inflation. I'd have to review whether one of the papers collected in Steedman (1979) sets out something like this model. I don't recall any conclusion as simple as the one I obtain, but I think there must be something like this in older Marxist models.

2.0 The Model
Consider two countries, each producing one of two commodities, corn and rice. The commodity produced in each country is a basic good in that country's economy. Assume no migration of labor is possible between the countries. Hence, wages can vary across economies. Assume, however, that no barriers to international flows of (financial) capital have been erected. Thus, a tendency exists for the same rate of profits to arise across countries, where financial outlays and revenues are calculated in some common abstract unit of account.

Without loss of generality, assume the price of corn is always one dollar per unit. The price of rice is one yen per unit. And the exchange rate at the start of the year is one yen per dollar.

These assumptions allow one to formulate equations for the prices of production in a common unit of account, for example, dollars. In the corn-producing country, prices of production satisfy the following equation:

a_c,c (1 + r) + a_0,c w_c = 1

where

• a_c,c is the amount of corn needed as input per unit corn produced
• a_0,c is the person-years labor needed as input per unit corn produced
• w_c is the wage in units corn in the corn-producing country
• r is the rate of profits

I here follow Sraffa's approach of regarding the wage as being paid at the end of the cycle of production. Given the assumptions, both sides of the above equation can be considered as being expressed in terms of dollars per unit corn produced. The rate of profits shows how many dollars are returned for each dollar invested.

The remaining equation specifying the model relates the revenues, in terms of dollars per unit rice produced, to costs in the rice-producing country. That equation is:

a_r,r (1 + r) + a_0,r w_r/p = 1/p

where

• a_r,r is the amount of rice needed as input per unit rice produced
• a_0,r is the person-years labor needed as input per unit rice produced
• w_r is the wage in units rice in the rice-producing country
• p is the exchange rate of yens per dollar at the end of the year

The above equations express the idea that the capitalists are indifferent between investing their dollars in producing corn in the corn-producing country or producing rice in the rice-producing country.

One can easily solve the above equations for the exchange rate at the end of the year:

p = [(1 - a_0,r w_r)/a_r,r]/[(1 - a_0,c w_c)/a_c,c]

The left-hand side of the above equation is ratio of the exchange rate at the end of the year to the exchange rate at the start of the year. The right-hand side is the quotient of two ratios, each ratio characterizing one of the two countries. These are the ratios of the net product remaining after compensating the workers for their labor power to the outlay needed to produce that surplus.

3.0 Conclusion
This model suggests that the smaller the rate of surplus value the capitalists are able to extract from the workers in a given country, the stronger their currency tends to become.

Update: I originally had the conclusion incorrect.

References

• John Maynard Keynes (1923) A Tract on Monetary Reform
• Paul A. Samuelson (1957) "Wages and Interest: A Modern Dissection of Marxian Economic Models", American Economic Review, V. 47, N. 6 (December): pp. 884-912
• Paul A. Samuelson (1971) "Understanding the Marxian Notion of Exploitation: A Summary of the So-Called 'Transformation Problem' Between Marxian Values and Competitive Prices", Journal of Economic Literature, V. 9, N. 2: pp. 399-431.
• Ian Steedman (editor) (1979) Fundamental Issues in Trade Theory, Macmillan

Samuelson on Hayek

Barkley Rosser, Jr., has published a piece by Samuelson on Hayek in the current issue of The Journal of Organization and Behavior, as well as an article, by Andrew Farrant and Edward McPhail, about a dispute between Hayek and Samuelson. I here record some thoughts by Samuelson in justifying his tone in an earlier article on Sraffa:

"If a scholar in his ninth decade is to record his considered opinions on an important topic, it had better be a matter not of when but of now... Dr. Samuel Johnson said that being hung in the morning greatly clarifies the mind. Nonsense. It is more likely to paralyze coherent thought. True though that as the days grow shorter, one does dispense with nice diplomancies and ancient jockeyings for victory." -- Paul A. Samuelson, "Sraffa's Hits and Misses", in Critical Essays on Piero Sraffa's Legacy in Economics (edited by Heinz D. Kurz), Cambridge University Press (2000)

Anyway, a number of bloggers have reacted. I noticed Tyler Cowen, Brad DeLong, Peter T. Leeson, Mark Thoma, and Barkley Rosser himself.

The comments sections for these posts is of varying length. I'm in the one on Rosser's co-blog. In discussing Hayek's contribution to the socialist calculation debate on Thoma's blog, Rosser brings up Jean-François Revel's The Totalitarian Temptation. I haven't read this book in decades. I'd have to reread it to see if Revel predicted the fall of the Soviet Union.

I also want to point out Chris Dillow's comments on Keynes' anti-semitism. I don't think much about Sraffa being of Jewish descent; Sraffa angered Mussolini directly anyways, what with his reporting on Italian banking in the December 1922 issue of the Guardian and Sraffa's support for Gramsci. I had known about Keynes' support of Sraffa, including intervention with the British government to obtain his release from internment. Dillow points to documentation of more broad-based support of Keynes for Jewish refugees. (I've previously linked to some other post on that month's discussion on that list.)

Economists Without Ethics

As I understand it, the American Economic Association (AEA) is almost unique among professional organizations. The AEA does not have a code of ethics.

School Rankings

I have been reading Philip A. Klein's Economics Confronts the Economy, which I purchased in the Strand, several weeks ago. This 2006 book is an institutionalist critique of mainstream economics. At one point, Klein wants to establish that mainstream economists at the best schools are, for the most part, no longer taught, for example, the history of economic thought. He provides the following list of schools:

1. University of Chicago
2. Harvard
3. MIT
4. Princeton
5. Stanford
6. Norwestern
7. Yale
8. University of Pennsylvania
9. University of California at Berkeley
10. University of California at Los Angeles
11. University of Wisconsin
12. Columbia
13. University of Rochester
14. Cornell
15. University of Minnesota
16. University of Michigan

I don't know the basis of Klein's ranking. If one were to ask me which schools in the U.S. are considered the most prestigious among mainstream economists, I might have named seven from this list. Maybe Northwestern is higher than I expect, and Columbia is lower. I suppose one could offer further quibbles

Does Depression, V. 2, Require Economists To Reconsider Theory?

Jeff Madrick reports little sign of the consideration of this question at the AEA/ASSA conference last week. Yves Smith comments on this question, and points us to a paper by Daron Acemoglu. I have yet to read Acemoglu's paper. Steve Keen makes a case for such reconsideration. (I wrote this post before reading this particular bit from Keen.) Peter Boettke asserts current events cannot challenge laissez faire economic policy because laissez faire has yet to be tried.

Hicks Invents Arrow-Debreu Model Of Intertemporal Equilibrium

"It is possible, at the other extreme, to conceive of an economy in which, for a considerable period ahead, everything was fixed up in advance. If all goods were bought and sold forward, not only would current demands and supplies be matched, but also planned demand and supplies. In such a 'Futures Economy' ... plans would be co-ordinated; and, for practical purposes, expectations would be co-ordinated too. (The price which would govern a firm's planned output for a particular future week would be the futures price, and not its own individual price-expectation.)" -- J. R. Hicks, Value and Capital: An Inquiry into Some Fundamental Principles of Economic Theory, Second edition, Oxford University Press, p. 136

Hat tip to Pierangelo Garegnani.

A Troll Is A Trail

"[Miss Ingram] entered into a discourse on botany with the gentle Mrs. Dent. It seemed Mrs. Dent had not studied that science: though, as she said, she liked flowers, 'especially wild ones;' Miss Ingram had, and she ran over its vocabulary with an air. I presently perceived she was (what is vernacularly termed) trailing Mrs. Dent; that is, playing on her ignorance--her trail might be clever, but it was decidedly not good-natured." -- Charlotte Bronte, Jane Eyre, Chapter 17

The Meaning Of A Word Is Its Use In The Language

The economy is never in equilibrium, or so many heterodox economists say. Being a Joan Robinson fan, I'm likely to agree. To understand the implications of the idea, and the use and abuse of equilibrium analysis, one must understand what economists mean by "equilibrium".

Tyler Cowen and Richard Fink provide an example of abuse, that is, a confusion about the implications of the economy not being in equilibrium. In the following passage, Cowen and Fink explicitly put aside income effects, false prices, strategic behavior, etc.:

"all that the Rothbard-Mises analysis implies is that there is a tendency towards equilibrium in a world with frozen data. Of course, this implies little or nothing about whether there is a tendency towards equilibrium in a world where the data are not frozen. All that [Evenly Rotating Economy] theorists are saying is that, if we freeze the disequilibrating forces, then the equilibrating forces will prevail. But on this basis we may likewise assert a tendency towards disequilibrium. By allowing the data to change just as it does in the real world, and 'freezing' all individual learning, we can demonstrate that the economy would degenerate into a series of successively less-coordinated states of disequilibrium. However, this would clearly be an illegitimate proof of a real world tendency towards disequilibrium..." -- Tyler Cowen and Richard Fink, "Inconsistent Equilibrium Constructs: The Evenly Rotating Economy of Mises and Rothbard", m V. 75, N. 4 (Sep. 1985): 866-869

(Hat tip to Matthew Mueller.) If there were a tendency, with frozen data, towards a ERE, the time path of the ERE corresponding to the data at each moment of time would show the tendency of the economy as a function of time. This is not the only point at which Cowen and Fink are confused.

What economists mean by equilibrium is not a simple question. Economists use the word "equilibrium" in many ways. The number of such ways has proliferated with the development of game theory. In this post, I compare and contrast only two uses of the word "equilibrium". And I think I don't fully explicate even these two uses.

The economy can be said to be in equilibrium when the following two conditions are met:

• The quantity supplied equals the quantity demanded of all commodities with positive prices
• The quantity supplied does not fall below the quantity demanded of all goods with zero prices.

This definition is specific to a particular neoclassical theory (or model).

Another definition comes out of the mathematical abstraction of a dynamical system. A dynamical system specifies how state variables change at any momement of time as a function of the location in state space. For example, a system of differential equations can define a dynamical system:

dx(t)/dt = f(x(t))

A limit point is a location, x, in the state space such that that location does not change with respect to time as function of the system dynamics. In other words, f(x) is zero at a limit point. Certain loci, other than the set of limit points, are of interest in dynamical systems. I am thinking specifically of limit cycles, strange attractors, and non-wandering sets. Consider a model of the economy as a dynamical system. The economy is in equilibrium, by a dynamical systems definition, when it is at a limit point.

The definition of equilibrium as equating supply and demand can be read as a special case of the definition of equilibrium as a limit point in a dynamical system. The tâtonnement process is a model of a type of dynamical system. Equilibrium, in the sense of a limit point in this system, is equilibrium in the sense of no excess demand for goods.

But Keynes can be read as suggesting the dynamical system definition of equilibrium need not equate supply and demand, particularly in the labor market. That is, Keynes' view of the possibility of the existence of an equilibrium with unemployment is more general and points to a non-neoclassical theory of prices.

Debauched By Economics

The following is a letter Sraffa sent to Robinson in 1936:

King's College, Cambridge
27.10.36

Dear Joan

Many thanks for your letter – it is a valuable addition to my museum and I shall hang it next to an extract from Sidgwick where, after lecturing Ricardo on how meaningless it is to talk of a quantity of labour, goes on cheerfully himself to talk of quantities of utility.

If one measures labour and land by heads or acres the result has a definite meaning, subject to a margin of error: the margin is wide, but it is a question of degree. On the other hand if you measure capital in tons the result is purely and simply nonsense. How many tons is, e.g., a railway tunnel?

If you are not convinced, try it on someone who has not been debauched by economics. Tell your gardener that a farmer employs 10 men – will he not have a pretty accurate idea of the quantities of land and labour? Now tell him that he employs 500 tons of capital, and he will think you are dotty – (not more so, however, than Sidgwick or Marshall).

Yours
P.S.

This Is Not A...

...mural by Diego Rivera. This mural, from which Figure 1 is a detail, replaced one by Diego Rivera. John D. Rockefeller, Jr., was displeased by the commie imagery in Rivera's mural.

Figure 1: Detail Of Jose Maria Sert's American Progress at Rockefeller Center

I also had a chance to see a cubist painting by Rivera, at the Musuem Of Modern Art. In general, I am not a fan of cubism.

Figure 2: Diego Rivera Painting at MOMA

I deliberately did not stay in New York City over new year's eve.

A Prescient Passage From 1937

One can read Keynes as proposing an alternative theory of value. Hyman Minsky advances this reading to some extent. Classic texts for this reading include chapter 17 of the General Theory and Hugh Townshend's 1937 article on Keynes' book. The following passage is from the latter:

"... The following example ... I have made, for the sake of clarity, so extreme as to be absurd if taken literally. Imagine the community, during a given short period, to be all asleep, so that in this period neither exchange nor new production takes place, and prices must be supposed to remain where they were when business closed down the previous evening. Suppose that, on waking up the next morning and resuming business, all wealth-owners find that a fit of optimism about the (prospective) price of residential property has come over them. (I have taken this particular asset as typical of an asset having a high degree of durability, a long period of production and a low degree of substitutability, and am ignoring the complications due to the existence of various types of residential houses, selling at different prices and more or less inter-substitutable; that is to say, we assume only one kind of house available to live in or to deal in or to build.) Immediately the normal exchange of residential house-property resumes in the morning, there will be a sellers' market and the price will rise sharply. If we further assume the increase in liquidity-premium attaching to houses owing to the mental revaluations of owners and potential owners to be equal in all cases - that is, the change in opinion to be unanimous - no more and no less buying and selling will take place than on the day before. (More money will be required, other things being equal, to finance this volume of trade in houses at the higher price-level; we assume this to be forthcoming to all who want to deal, e.g. out of bank-loans.) If opinion is not unanimous, additional exchange of houses between the 'bulls' and the 'bears' will take place and will settle the price, but not in general at its former level; we assume for, the sake of the example, that the bulls preponderate, so that the price rises, the necessary money for the dealing, as before, being forthcoming. House-building will, of course, have become an abnormally profitable occupation; and in time the diversion of resources to this industry will come into play and will tend to readjust the relative prices of houses and of assets and people's expectations about them towards their former levels. But before it can do so completely, in general further (similar or opposite) spontaneous changes in the liquidity-premiums attaching to the existing houses will have taken place; obviously the physical production of new houses can never take place fast enough for its effect on prices to catch up with people's purely mental revaluations of existing ones. For the latter operate without any time-lag at all. Of course, in practice, the possibility or prospect of new production bringing down again the money-price of houses is present to people's minds, and operates to diminish optimism or to cause a wave of optimism to be followed by a wave of pessimism. (It is essential to the argument that people think in term of money-prices ...) But there is in fact no reason why new building should ever bring down the money-price of houses at all; if the price of building materials and/or labour is rising rapidly, the new production of houses may operate to reduce their relative price only, the prices of other valuables rising to the necessary degree - or, of course, intermediately to any extent. Or again, all prices may fall, that of houses more than others; or all prices may rise, that of houses less than others. The course of the actual money-price of houses is thus quite indeterminate, even in the shortest period, unless we know the course of the money-price of some one single or composite valuable (e.g. labour) - i.e. unless we have a 'convention of stability.' And, even so, the relative price, and therefore in spite of the convention of stability, the actual price of houses is still not precisely determined; it remains indeterminate to the extent to which it may be influenced by unknown changes in liquidity-preferences. This holds even in the shortest period." -- Hugh Townshend (1937) "Liquidity-Premium and the Theory of Value", Economic Journal, V. 47, N. 185 (March): pp. 157-169.

Townshend continues by considering this argument as valid for any durable asset, including monetary assets and equitities. He cites, as another example, cotton-mills in Lancashire during the 1920s. According to Townshend, cotton mills were being bought and sold, not with regard to "expectations about the price of cotton goods", but with the intent to "flip" them - to use the jargon of the recent U.S. housing market.

Minsky Versus Sraffa

Kevin "Angus" Grier reminisces about Hyman Minsky's dislike for Piero Sraffa. But he doesn't recall points at issue. Minsky expressed his views in print:

"Given my interpretation of Keynes (Minsky, 1975, 1986) and my views of the problems that economists need to address as the twentieth century draws to a close, the substance of the papers in Eatwell and Milgate (1983) and the neoclassical synthesis are (1) equally irrelevant to the understanding of modern capitalist economies and (2) equally foreign to essential facets of Keynes's thought. It is more important for an economic theory to be relevant for an understanding of economies than for it to be true to the thought of Keynes, Sraffa, Ricardo, or Marx. The only significance Keynes's thought has in this context is that it contains the beginning of an economic theory that is especially relevant to understanding capitalist economies. This relevance is due to the monetary nature of Keynes's theory.

Modern capitalist economies are intensely financial. Money in these economies is endogenously determined as activity and asset holdings are financed and commitments of prior contracts are fulfilled. In truth, every economic unit can create money - this property is not restricted to banks. The main problem a 'money creator' faces is getting his money accepted...

...The title of this session, 'Sraffa and Keynes: Effective Demand in the Long Run', puzzles me. Sraffa says little or nothing about effective demand and Keynes's General Theory can be viewed as holding that the long run is not a fit subject for study. At the arid level of Sraffa, the Keynesian view that effective demand reflects financial and monetary variables has no meaning, for there is no monetary or financial system in Sraffa. At the concrete level of Keynes, the technical conditions of production, which are the essential constructs of Sraffa, are dominated by profit expectations and financing conditions." -- Hyman Minsky "Sraffa and Keynes: Effective Demand in the Long Run", in Essays of Piero Sraffa: Critical Perspectives on the Revival of Classical Theory (edited by Krishna Bharadwaj and Bertram Schefold), Unwin-Hyman (1990)

I gather, from second or third-hand accounts, that debates along these lines became quite acrimonious at the annual summer school in Trieste during the 1980s. I've always imagined Paul Davidson and Pierangelo Garegnani would be the most vocal advocates of the extremes in these debates. And I think of Jan Kregel, Edward Nell, and Luigi Pasinetti as being somewhere in the middle, going off in different directions. I don't know much about monetary circuit theory, but such theory may provide an approach to integrating money into Sraffianism.

Of course, Minsky's theories and Davidson's proposals for national and international reforms are of great contemporary relevance.

Don't Say "There Must Be Something Common, Or They Would Not Be Called 'Games'"

1.0 Introduction
Von Neumann and Morgenstern posed a mathematical problem in 1944: Does every game have a solution, where a solution is defined in their sense? W. F. Lucas solved this problem in 1967. Not all games have such a solution. (It is known that such a solution need not be unique. In fact, the solution to the three person game I use below to illustrate the Von Neumann and Morgenstern solution is not unique.)

I may sometime in the future try to explain the game with ten players Lucas presents as a counterexample, assuming I can grasp it better than I do now. With this post, I try to explain some concepts of cooperative game theory so as to have this post for reference when and if I do. The Nash equilibrium and refinements are notions from the different theory of non-cooperative game theory.

2.0 Definition of a Game
Roughly, a game is specified by:

• The number of players
• The strategies available for each player
• The payoffs to each player for each combination of player strategies

How a strategy is described depends on the specification of the game - whether it is in extensive form, normal form, or characteristic function form. Von Neumann and Morgenstern hoped that all three forms would be equivalent, with less data needing to be specified in the later forms in this series. This hope has arguably not worked out.

2.1 Extensive Form
A game in extensive form is specified as a tree. This is most easily seen for board games, like backgammon or chess. Each node in the tree is a board position, with the root of the tree corresponding to the initial position.

The specification of a node includes which player is to move next, as well as the board position. Each possible move the player whose turn it is can make is shown by a link leading from the node to a node for the board position after that choice of a move. Random moves are specified as moves made by a fictitious player, who might be named "Mother Nature". The roll of a pair of dice or the deal of a randomly selected card are examples of random moves. With a random move, the probability of each move is specified along the line connecting one node to another. Since a move by an actual player is freely chosen, the probabilities of any move by an actual player are not specified in the specification of a game.

The above description of the specification of a game cannot yet handle games like poker. In poker, not every player knows every card that is dealt. Von Neumann and Morgenstern introduce the concept of "information sets" to allow one to specify that, for instance, a player only knows all the cards in his hands and, perhaps, some of the cards in the other players' hands. An information set at a node, specifies for the player whose turn it is, which of the previous choices of moves in the game he has knowledge of. That is, an information set is a subset of the set of links in the tree leading from the initial position to the current node position. Since some of these moves were random, this specification allows for the dealing of hands of cards, for example.

The final element in this specification of a game occurs at the leaves of the tree. These are the final positions in the games. Leaves have assigned the values of the payouts to each player in the game.

It is easy to see how to define a player's strategy with this specification of a game. A strategy states the player's choice of a move at each node in the game denoting a position in which it is the player's move. A play of the game consists of each player specifying their strategy and the random selection of a choice from the specified probability distributions at each node at which a random move is chosen. These strategies and the random moves determine the leaf at which the game terminates. And one can then see the payouts to all players for the play.

One can get rid of the randomness, in some sense, by considering an infinite number of plays of the game for each combination of players' strategies. This will result in a probability distribution for payouts. The assumption is that each player is interested in the expected value, that is, the mean payout, to be calculated from this probability distribution. (All these descriptions of calculations have abstracted from time and space computational constraints.)

2.2 Normal Form
One abstracts from the sequence of moves and from random moves in specifying a game in normal form. The extensive form allows for the definition of strategies for each player, and each strategy can be assigned an arbitrary label. A game in normal form consists of a grid or table. A player's strategies are listed along one dimension of the table, and each dimension corresponds to a player. Each entry in the table consists of a ordered tuple, where the elements of the tuple are the expected payouts to the players for the specified combination of strategies.

Table 1 shows a simple example - the children's game, "Rock, Paper, Scissors." The rules specify the winner. Rock crushes scissors, scissors cut paper, and paper covers rock. This is a two-person zero-sum game. The payouts are shown in the table for the player whose strategies are listed for each row to the left. The payouts to the column player are, in this case, the additive inverse of the table entries.

Table 1: Rock, Paper, Scissors

	Rock	Scissors	Paper
Rock	0	+1	-1
Scissors	-1	0	+1
Paper	+1	-1	0

By symmetry, no pure strategy in Rock, Paper, Scissors is better than any other. A mixed strategy is formed for a player by assigning probabilities to each of that player's pure strategies. Probabilities due to states of nature are removed in the analysis of games by taking mathematical expectations. But probabilities reappear from rational strategization. I also found interesting Von Neumann and Morgenstern's analysis of an idealized form of poker. One wants one's bluffs to be called in bluffing on occasion so that players will be willing to add more to the pot when one raises on a good hand.

Each player's best mixed strategy in a two-person zero-sum game can be found by solving a Linear Program (LP). Let p₁, p₂, and p₃ be the probabilities that the row player in Table 1 chooses strategies Rock, Scissors, and Paper, respectively. The value of the game to the row player is v. The row player's LP is:

Choose p₁, p₂, p₃, v
To Maximize v
Such that
-p₂ + p₃ ≥ v
p₁ - p₃ ≥ v
-p₁ + p₂ ≥ v
p₁ + p₂ + p₃ = 1
p₁ ≥ 0, p₂ ≥ 0, p₃ ≥ 0

The interest of the column player is to minimize the payout to the row player. The left-hand sides of the first three constraints show the expected value to the row player when the column player plays Rock, Scissors, and Paper, respectively. That is, the coefficients by which the probabilities are multiplied in these constraints come from the columns in Table 1. Given knowledge of the solution probabilities, the column player can guarantee the value of the game does not exceed these expected values by choosing the corresponding column strategy. That is, the column player chooses a pure strategy to minimize the expected payout to the row player.

The column player's LP is the dual of the above LP. As a corollary of duality theory in Linear Programming, a minimax solution exists for all two-person zero-sum games. This existence is needed for the definition of the characteristic function form of a game.

2.3 Characteristic Function Form
The characteristic function form of a game is defined in terms of coalitions of players. An n-person game is reduced to a two-person game, where the "players" consist of a coalition of true players and the remaining players outside the coalition. The characteristic function for a game is the value of the corresponding two-person zero-sum game for each coalition of players. The characteristic function form of the game specifies the characteristic function.

As an illustration, Von Neumann and Morgenstern specify the three-person game in Table 2. In this game, coalitions of exactly two people win a unit.

Table 2: Canonical Three Person Game

Coalition	Value
{ }	v( { } ) = 0
{1}	v( {1} ) = -1
{2}	v( {2} ) = -1
{3}	v( {3} ) = -1
{1, 2}	v( {1, 2} ) = 1
{1, 3}	v( {1, 3} ) = 1
{2, 3}	v( {2, 3} ) = 1
{1, 2, 3}	v( {1, 2, 3} ) = 0

3.0 A Solution Concept

Definition: An imputation for an n-person game is an n-tuple (a₁, a₂, ..., a_n) such that:

• For all players i, the payout to that player in the imputation does not fall below the amount that that player can obtain without the cooperation of any other player. That is, a_i ≥ v( {i} ).
• The total in the imputation of the payouts over all players is the payout v( {1, 2, ..., n} ) to the coalition consisting of all players.

Definition: An imputation a = (a₁, a₂, ..., a_n) dominates another imputation b = (b₁, b₂, ..., b_n) if and only if there exists a set of players S such that:

• S is a subset of {1, 2, ..., n}
• S is not empty
• The total in the imputation a of the payouts over all players in S does not exceed the payout v( S ) to the coalition consisting of those players
• For all players i in S, the payouts a_i in a strictly exceed the payouts b_i in b

Definition: A set of imputations is a solution (also known as a Von Neumann and Morgenstern solution or a stable set solution) to a game with characteristic function v( ) if and only if:

• No imputation in the solution is dominated by another imputation in the solution
• All imputations outside the solution are dominated by some imputation in the solution

Notice that an imputation in a stable set solution can be dominated by some imputation outside the solution. The following set of three imputations is a solution to the three-person zero-sum game in Table 2:

{(-1, 1/2, 1/2), (1/2, -1, 1/2), (1/2, 1/2, -1)}

This solution is constructed by considering all two-person coalitions. In each imputation in the solution, the payouts to the winning coalition are evenly divided.

The above is not the only solution to this game. An uncountably infinite number of solutions exist. Another solution is the following uncountable set of imputations:

{(a, 1 - a, -1) | -1 ≤ a ≤ 2}

This solution can be understood in at least two ways:

• Player 3 is being discriminated against.
• The above is a solution to the two-person, non-constant game with the characteristic function in Table 3. A fictitious third player has been appended to allow the game to be analyzed as a three-person zero-sum game.

Von Neumann and Morgenstern present both interpretations.

Table 3: A Two-Person Game With Variable Sum

Coalition	Value
{ }	v( { } ) = 0
{1}	v( {1} ) = -1
{2}	v( {2} ) = -1
{1, 2}	v( {1, 2} ) = 1

The above has defined the Von Neumann and Morgenstern solution to a game. Mathematicians have defined at least one other solution concept to a cooperative game, the core, in which no imputation in the solution set is dominated by any other imputation. I'm not sure I consider the Shapley value as a solution concept, although it does have the structure, I guess, of an imputation.

References

• W. F. Lucas, "A Game With No Solution", Bulletin of the American Mathematical Society, V. 74, N. 2 (March 1968): 237-239
• John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Princeton University Press (1944, 1947, 1953)

Keynes' General Theory As A Long Period Theory

The following paragraph appears in Chapter 5 of The General Theory of Employment Interest and Money:

"If we suppose a state of expectation to continue for a sufficient length of time for the effect on employment to have worked itself out so completely that there is, broadly speaking, no piece of employment going on which would not have taken place if the new expectation had always existed, the steady level of employment thus attained may be called the long-period employment corresponding to that state of expectation. It follows that, although expectations may change so frequently that the actual level of employment has never had time to reach the long-period employment corresponding to the existing state of expectation, nevertheless every state of expectation has its definite corresponding level of long-period employment."

It seems to me that this passage is important in an interpretation of Keynes as claiming that his theory applies in both the long and short periods. Even if the capital equipment in the economy were adjusted to effective demand, Keynes claims, the labor force need not be fully employed.

I think this reading is strengthened by a couple of considerations. One should distinguish between the full utilization of capital equipment and full employment. Distinguishing between these concepts makes most sense if one has dropped the idea of substitution between capital and labor. Likewise, one should drop the idea that the (long run) interest rate equilibrates savings and investment. But the dropping of these ideas is one result of the Cambridge Capital Controversies. On the other hand, it is not clear that Sraffa accepted Keynes' Chapter 17, another important locus for a long period interpretation of Keynes' General Theory.

I conclude with a couple of important Sraffian references on this issue. I could probably find some more recent. But I think Milgate (1982) and Eatwell and Milgate (1983) are key texts in this controversial area (even though I haven't read them in years).

Reference

• John Eatwell and Murray Milgate (editors) (1983) Keynes's Economics and the Theory of Value and Distribution, Duckworth
• Murray Milgate (1982) Capital and Employment: A Study of Keynes's Economics, Academic Press

Stiglitz the Keynesian

Stigliz has an article, "Capitalist Fools", in the January issue of Vanity Fair. He argues that the new depression is the result of:

• Firing Volker after he successfully fought inflation
• Abolishing Glass-Steagall
• Imposing the non-stimulative and regressive Bush tax cuts
• Incentive structure encouraging faulty accounting
• Paulson's faulty October bail out package

In summary, "The truth is most of the individual mistakes boil down to just one: a belief that markets are self-adjusting and that the role of government should be minimal." Stiglitz is consistent. But I thought males read Vanity Fair, like Esquire, for fashion tips.

And he has an 11 December article in Business Day, a South African newspaper. Stiglitz is interested in how to formulate Keynesian policy effectively.

In local news... Last March, Stiglitz wrote the New York State governor recommending that NY address its deficit by raising taxes on the rich.

Here's a characterization of Stiglitz's economic teaching:

"In his lectures, Stiglitz applied the machinery of neoclassical economics to upturn the standard results. Like a magician drawing rabbits from a hat, he could make demand curves slope up, supply curves slope down, markets in competitive equilibrium fail to clear, cross-subsidies make everyone better off, students over-educate themselves, and farmers produce the wrong quantities of goods. And then he would show how the magic reflected some very human and rational response to imperfect information. The theorem that individual rationality leads to social rationality applies to a special case, not the general case." -- Karla Hoff, in Economics for an Imperfect World: Essays in Honor of Joseph E. Stiglitz (ed. by R. Arnott, B. Greenwald, R. Kanbur, & B. Nalebuff) MIT Press, 2003 (quoted by John Lodewijks, "Review", Review of Political Economy, V. 21, N. 1, 2009)

So why is Stiglitz considered a mainstream economist and Ian Steedman a non-mainstream heterodox economist?

Economists Unsuccessful In Teaching

I've previously commented on Philip Ball's opinion of economics. He's recently offered an article in Nature. This is behind a pay wall, but Ball provides an unedited, longer version on his blog, and a comment on it.

Ball notices that mainstream economists often defend their discipline against critics by asserting that critics attack a straw person. Of course introductory courses are simplified, but sophisticated research has long move beyond such models. Ball's point seems to be that, if so, economists have not been successful in getting the public or policy-makers to realize the introductory nature of simplified models or to be aware of more sophisticated lessons. "Knowledgable economists and critics of traditional economics are on the same side."

Designing A Keynesian Stimulus Plan

Some version of this New York Times article contains the following passage:

"A blueprint for such spending can be found in a study financed by the Political Economy Research Institute at the University of Massachusetts and the Center for American Progress, a Washington research organization founded by John D. Podesta, who is a co-chairman of Mr. Obama's transition team.

The study, released in November after months of work, found that a $100 billion investment in clean energy could create 2 million jobs over two years." -- Peter Baker and John M. Broder, New York Times, 7 December 2008 [Links inserted by Robert Vienneau]

I went looking for this study, but was unable to find it. The PERI report, "Green Recovery: A Program to Create Good Jobs and Start Building a Low-Carbon Economy" (by Robert Pollin, Heidi Garrett-Peltier, James Heintz, and Helen Scharber) is dated September 2008. The CAP report, "How to Spend $350 Billion in a First Year of Stimulus and Recovery" (by Will Straw and Michael Ettinger) is dated 5 December 2008.

Based on the reports I found, I doubt the report referred to by the New York Times article goes into details on the analytical justifications for its figures, which I'd like to see. I know that, in principle, one could create a transactions table from use and make tables. From such a transactions table, one can calculate multipliers by sectors and some measure of environmental impact. But I do not understand all the accounting conventions and approximations one would have to make to get such practical analyses from the national income accounts.

How Individuals Can Choose, Even Though They Do Not Maximize Utility

1.0 Introduction
I think of this post as posing a research question. S. Abu Turab Rizvi re-interprets the primitives of social choice theory to refer to mental modules or subroutines in an individual. He then shows that the logical consequence is that individuals are not utility-maximizers. That is, in general, no preference relation exists for an individual that satisfies the conditions equivalent to the existence of an utility function. I have been reading Donald Saari on the mathematics of voting. What are the consequences for individual choice from interpreting this mathematics in Rizvi's terms?

I probably will not pursue this question, although I may draw on these literatures to present some more interesting counter-intuitive numerical examples.

2.0 Arrow's Impossibility Theorem and Work-Arounds
Consider a society of individuals. These individuals are "rational" in that each individual can rank all alternatives, and each individual ranking is transitive. Given the rankings of individuals, we seek a rule, defined for all individual rankings, to construct a complete and transitive ranking of alternatives for society. This rule should satisfy certain minimal properties:

• Non-Dictatorship: No individual exists such that the rule merely assigns his or her ranking to society.
• Independence of Irrelevant Alternatives (IIA): Consider two countries composed of the same number of individuals. Suppose the same number in each country prefer one alternative to another in a certain pair of alternatives, and the same number are likewise indifferent between these alternatives. Then the rule cannot result in societal rankings for the two countries that differ in the order in which these two alternatives are ranked.
• Pareto Principle: If one alternative is ranked higher than another for all individuals, then the ranking for society must rank the former alternative higher than the latter as well.

Arrow's impossibility theorem states that, if there are at least three alternatives, no such rule exists.

Arrow's work has generated lots of critical and interesting research. For example, Sen considers choice functions for society, instead of rankings. A choice function selects the best alternative for every subset of alternatives. That is, for any menu of alternatives, a choice function specifies a best alternative. Consider a rule mapping every set of individual preferences to a choice function. All of Arrow's conditions are consistent for such a map from individual preferences to a choice function.

Saari criticizes the IIA property as requiring a collective choice rule not to use all available information. In particular, the rule makes no use of the number of alternatives, if any, that each individual ranks between each pair. The rule does not make use of enough information to check that each individual has transitive preferences. (Apparently, the IIA condition has generated other criticisms, including by Gibbard.) Saari proposes relaxing the IIA condition to use information sufficient for checking the transitivity of each individual's preference.

Saari also describes a collective choice rule that includes each individual numbering their choices in order, with the first choice being assigned 1, the second 2, and so on. With these numerical assignments, the choices are summed over individuals, and the ranking for society is the ranking resulting from these sums. This aggregation procedure is known as the Borda count. Saari shows that Borda count satisfies the relaxed IIA condition and Arrow's remaining conditions.

3.0 Philosophy of Mathematics
Above, I have summarized aspects of the theory of social choice in fairly concrete terms, such as "individuals" and "society". The mathematics behind these theorems is formulated in set-theoretic terms. The referent for mathematical terms is not fixed by the mathematics:

"One must be able to say at all times - instead of points, straight lines, and planes - tables, chairs, and beer mugs." - David Hilbert (as quoted by Constance Reid, Hilbert, Springer-Verlag, 1970: p. 57)

"Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." -- Bertrand Russell

4.0 An Interpretation
Rizvi re-interprets the social choice formalism as applying to another set of referents. A society’s ranking, in the traditional interpretation, is now an individual’s ranking. An individual’s ranking, in the traditional interpretation, is now an influence on an individual’s ranking. Rizvi’s approach reminds me of Marvin Minsky's society of mind, in which minds are understood to be modular. Rizvi examines the implication’s of Sen’s impossibility of a Paretian liberal for individual preferences under this interpretation of the mathematics of social choice theory.

Constructing natural numbers in terms of set theory allows one to derive the Peano axioms as theorems. Similarly, interpreting social choice theory as applying to decision-making components for an individual allows one to analyze whether the conditions often imposed on individual preferences by mainstream economists can be derived from this deeper structure. And, it follows from Arrow's impossibility theorem, these conditions cannot be so derived in general. Individuals do not and need not maximize utility. On the other hand, Sen's result explains how individuals can choose a best choice from menus with which they may be presented.

References

• Kenneth J. Arrow (1963) Social Choice and Individual Values, Second edition, Cowles Foundation
• Alan G. Isaac (1998) "The Structure of Neoclassical Consumer Theory", working paper (9 July)
• Marvin Minsky (1987) The Society of Mind, Simon and Schuster
• Donald G. Saari (2001) Chaotic Elections! A Mathematician Looks at Voting, American Mathematical Society
• S. Abu Turab Rizvi (2001) "Preference Formation and the Axioms of Choice", Review of Political Economy, V. 13, N. 2 (Nov.): 141-159
• Amartya K. Sen (1969) "Quasi-Transitivity, Rational Choice and Collective Decisions", Review of Economic Studies, V. 36, N. 3 (July): 381-393 (I haven't read this.)
• Amartya K. Sen (1970) "The Impossibility of a Paretian Liberal", Journal of Political Economy, V. 78, N. 1 (Jan.-Feb.): 152-157

On John Maynard Keynes

• Paul Krugman again and again and again
• Brad DeLong basically quotes Krugman
• Tyler Cowen
• Peter Boettke comments on Cowen on Keynes
• Matthew Mueller comments on Boettke on Keynes

I think Chapter 12 of the General Theory is insightful. Chapter 17 draws on Sraffa's concept of own rates of interest, which Sraffa introduced in his critique of Hayek. Apparently Sraffa was not enthusiastic about Keynes' theory of liquidity preference. As I recall, the connections between Keynes and the more junior Sraffa go quite a bit further back. Sraffa translated Keynes' Tract on Monetary Reform into Italian. A major thesis of this book is that monetary authorities should worry more about unemployment and inflation inside their country than exchange rate stability. Apparently this was Sraffa's thesis, presented in Italy at the end of World War I.

Update: I suppose I ought to mention my game, which is closer to hydraulic keynesianism. I did get the idea for the underlying model from Kalecki.