Fred Lee

Barkley Rosser, David Ruccio, and Matias Vernengo have obituaries. I find I had not known much about Lee's life.

I have been influenced by Lee's work on markup pricing (also known as full-cost pricing), the history of heterodox economics, and the suppression of heterodox economics by the mainstream through bullying and bureaucratic measures. I think highly of Lee's 2004 paper (written in collaboration with Steve Keen), "The Incoherent Emperor: A Heterodox Critique of Neoclassical Microeconomic Theory". I can only find one blog post of mine referencing this paper. Lee promoted pluralism in economics.

Marginal Productivity Theory of Distribution: Acknowledged Blatherskite

I was surprised at how many reviews of Thomas Piketty's Capital in the 21st Century draw on the Cambridge Capital Controversy to argue that Piketty's theoretical framework is grossly inadequate.

• James K. Galbraith's Spring 2014 review in Dissent.
• Dean Baker last May.
• October 2014 special issue of the Real-World Economics Review.
• Tony Aspromourgos's Thomas Piketty, the future of capitalism and the theory of distribution: a review essay (October 2014) (H/T: David Fields).
• Javier Lopez Bernardo, Felix Lopez Martinez, and Engelbert Stockhammer's A Post-Keynesian Response to Piketty's 'Fundamental Contradiction of Capitalism' (October 2014) (H/T Lars P. Syll). This response brings up, in addition to the CCC, the Post Keynesian theory of distribution.
• John Bellamy Foster and Michael D. Yates' review
in Monthly Review

I like this Aspromourgos quote:

However classical the questions Piketty addresses, when he turns to explain the determination of r he has recourse to the conventional, post-classical marginal productivity theory of distribution: diminishing marginal capital productivity is 'natural' and 'obvious' (212–16). (He is much less willing to have recourse to time preference: 358–61; cf. 399–400.) The logical critique of capital aggregates – applied either at the macro or micro level – as supposed independent explanatory variables in the theory of profit rates, first coherently stated by Piero Sraffa (1960, pp. 81–7; see also Kurz and Salvadori 1995, pp. 427–67), is nowhere acknowledged or addressed. That such a relatively well-read economist as Piketty can so unhesitatingly apply this bankrupt approach, is testament to how completely a valid body of critical theoretical analysis can be submerged and forgotten in social science (a phenomenon for the sociologists of knowledge to contemplate). This is so, notwithstanding that Piketty offers a brief interpretation of the 'Cambridge' capital debates, making them turn upon the issues of whether there is substitutability in production (and associated flexibility of capital-output ratios), and whether or not 'growth is always perfectly balanced [i.e., full-employment growth]' (230–32). In fact, the participants on both sides of those debates were concerned with production systems in which substitution and capital-output variability occurred; and continuous full-employment growth was not entailed by recourse to orthodox, marginalist production functions, a point perfectly understood by the participants on both sides. -- Tony Aspromourgos

Update (27 October 2014): Added the Bernardo, Martinez, and Stockhammer reference.

Update (1 December 2014): Added the Foster and Yates reference.

r > g In A Steady State

1.0 Introduction

This post presents a model of distribution that Luigi Pasinetti developed. It is one of a family of models. Other important models in this family were developed by Richard Kahn, Nicholas Kaldor, and Joan Robinson. These models have been extended in various ways and presented in textbooks. One can see this family as having extended work by Roy Harrod, and as being related to the work of Michal Kalecki and even of Karl Marx.

2.0 The Model

2.1 Definitions

Consider a simple closed economy with no government. All income is paid out in the form of either wages or profits:

Y = W + P,

where W is total wages, P is total profits, and Y is national income. Total savings is composed of savings by workers and by capitalists, where capitalists are a class whose members receive income only from profits:

S = S_w + S_c

S is total savings. S_w is workers' savings, and S_c is capitalist savings. Profits are also split into two parts:

P = P_w + P_c,

where P_w is returns on the capital owned by the workers, and P_c is the return on the capital owned by the capitalists. The behavior assumption is made that both workers and capitalists save a (different) constant proportion of their income:

S_c = s_c P_c

S_w = s_w (W + P_w)

s_c is the capitalists' (marginal and average) propensity to save. s_w is the workers' (marginal and average) propensity to save. The propensities to save are assumed to lie between zero and one and to be in the following order:

0 ≤ s_w < s_c ≤ 1

Workers' savings are assumed to be insufficient to fund all the investment occurring along a steady-state growth path.

The value of the capital stock is divided up into that owned by the workers and by the capitalists:

K = K_w + K_c,

where K is the value of the capital stock, K_w is the value of the capital stock owned by the workers, and K_c is the value of the capital stock owned by the capitalists

2.2 Steady State Equilibrium Conditions

Along a steady-state growth path, in this model, all capital earns the same rate of profits, r:

r = P/K = P_c/K_c = P_w/K_w

It follows from the above set of equations that the ratio of the profits received from the workers to the profits received by the capitalists is equal to the ratio of the value of capital that each class owns:

P_w/P_c = K_w/K_c

Likewise, one can find the ratio of total profits to the profits obtained by the capitalists:

P/P_c = K/K_c

The analysis is restricted to steady-state growth paths where the value of the capitalists' capital and the value of the workers' capital is growing at the same rate:

S/K = S_c/K_c = S_w/K_w

The ratio of profits to savings is the same for the economy as a whole and for workers:

P/S = (P/K)/(S/K) = (P_c/K_c)/(S_c/K_c) = P_c/S_c

Or, after a similar logical deduction for workers:

P/S = P_c/S_c = P_w/S_w

Along a steady-state growth path, planned investment, I equals savings:

I = S

2.3 Deduction of the Cambridge Equation

The following is a series of algebraic substitutions based on the above:

P/I = P/S = P_c/S_c = P_c/(s_c P_c) = 1/s_c

Or:

P = (1/s_c) I

The share of profits in national income is determined by the savings propensity of the capitalists and the ratio of investment to national income:

(P/Y) = (1/s_c) (I/Y)

Recall that the rate of profits is the ratio of profits to the value of capital:

r = P/K = (1/s_c) (I/K)

Recognizing that I/K is the rate of growth, g, one obtains the famous Cambridge equation:

r = g/s_c

As long as the capitalists consume at least some of their income, the rate of profits is greater than the rate of growth along a steady-state growth path. And along such a path the share of income going to profits will be constant.

3.0 Discussion

If one assumes given investment decisions, the Cambridge Equation tells us what rate of profit is compatible with a steady state growth path in which the expectations underlying those investment decisions are satisfied.

Consider two steady states in which the same rate of growth is being obtained. Suppose that along one path workers have a higher propensity to save. Within broad limits, this greater willingness to save among workers has no effect on determining either the share of profits in income or the rate of profits. Only the capitalists' saving propensity matters for the steady state rate of profits, given the rate of growth. Would a capitalist economy have a tendency to approach such a growth path, given a sufficient length of time? I think such stability would entail the evolution of institutions, conventions, the labor force, and what is seen as common sense, including among dominant political parties.

The above model might have some relevance to current political economy discussions elsewhere.

Jean Tirole, A Practitioner Of New Industrial Organization

I have occasionally summarized certain aspects of microeconomics, concentrating on markets that are not perfectly competitive. Further developments along these lines can be found in the theory of Industrial Organization.

One can distinguish in the literature two approaches to IO know as old IO and new IO. Old IO extends back to the late 1950s. Joe Bain and Paolo Sylos Labini laid the foundations to this approach, and they were heralded by Franco Modigliani. I have not read any of Bain and only a bit of Sylos Labini. Sylos was a Sraffian and quite critical of neoclassical economics. He also had interesting things to say about economic development.

As I understand it, new IO consists of applying game theory to imperfectly competitive and oligopolistic markets. I gather new IO took off in the 1980s. Jean Tirole, the winner of this year's "Nobel" prize in economics, is a prominent exponent of new IO.

One can tell interesting stories about corporations with both old IO and new IO. For example, Tirole has had something to say about vertical integration which, based on what I've read in the popular press, might be of interest to me. (Typically, when I explore the theory of vertical integration, following Luigi Pasinetti, the integration is only notional, not at the more concrete level of concern in IO.)

I wonder, though, whether economists can point to empirical demonstrations of the superiority of new IO over old IO. Or have economists studying IO come to embrace new IO more because of the supposed theoretical rigor of game theory? Are specialists in IO willing to embrace the indeterminism that arises in game theory, what with the variety of solution concepts and the existence of multiple equilibria in many games? Or do they insist on closed models with unique equilibria?

References

• Franco Modigliani (1958). New developments on the Oligopoly Front, Journal of Political Economy, V. 66, No. 3: pp. 215-232.

Update (same day): Corrected a glitch in the title. Does this Paul Krugman post read as a direct response to my post?

Noncommunicating Literatures?

During the 20th century, a number of economists more or less independently developed ideas associated with input-output analysis, activity analysis, modeling the economy as a self-sustaining circular flow, and the revival of classical political economy. I think of:

• Leonid Kantorovich: The Soviet economist who shared the 1975 Nobel Memorial Prize in Economic Sciences with Tjalling Koopmans.
• Wassily Leontief: Always emphasized developing an empirically operational version of these ideas.
• Father Maurice Potron: I stumbled across two references to him. I know nothing otherwise about his work.
• Walter Isard: Extended input-output analysis to regional economics.
• Richard Stone: Developed the idea of a Social Accounting Matrix and conventions for national income accounting.
• Jacob Schwartz: Criticized the mainstream economics of his time on the basis of linear economic models.
• Piero Sraffa: Criticized the mainstream economics of his time on the basis of linear economic models.
• John Von Neumann: A mathematician, not an economist.

I wonder how many make connections between the scholarly literature building on the work of each of these researchers. I am not at all sure anybody explicitly and consciously built on Potron or Schwartz.

References

• Wassily W. Leontief (1936). Quantitative Input and Output Relations in the Economic Systems of the United States, Review of Economic Statistics, V. 18, N. 3 (Aug). pp. 105-125.
• Walter Isard (1951) Interregional and Regional Input-Output Analysis: A Model of a Space-Economy, Review of Economics and Statistics, V. 33, No. 4 (Nov.): pp. 318-328.
• Jacob T. Schwartz (1961). Lectures on the Mathematical Method in Analytical Economics, Gordon and Breach.
• Piero Sraffa (1960). Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory, Cambridge University Press.
• J. Ricard N. Stone (1966). The Social Accounts from a Consumer Point of View, Review of Income and Wealth, V. 12, Iss. 1 (Mar.): pp. 1-33. [I HAVEV'T READ THIS OR ANYTHING ELSE BY STONE]
• John von Neumann (1945-1946) A Model of General Economic Equilibrium, Review of Economic Studies, V. 13, No. 1: pp. 1-9.

Hayek Not Opposed To Keynes On Political Principle

With characteristic cheerful carelessness, Noah Smith misinforms hapless Bloomberg readers:

"Friedrich Hayek tried to argue against Keynes' theories, but for whatever reason, he lost the debate among economists in the 1930s. But Hayek would have the last laugh, because in his book, 'The Road to Serfdom,' he attacked Keynes from a very different angle. Instead of saying Keynes' theories were wrong, Hayek prophesied that Keynesian stabilization policies would lead down the slippery slope to totalitarianism."

As a matter of fact, Hayek said nearly the opposite:

"There is, finally, the supremely important problem of combating general fluctuations of economic activity and the recurrent waves of large-scale unemployment which accompany them. This is, of course, one of the gravest and most pressing problems of our time. But, though its solution will require much planning in the good sense, it does not - or at least need not - require that special kind of planning which according to its advocates is to replace the market. Many economists hope, indeed, that the ultimate remedy may be found in the field of monetary policy, which would involve nothing incompatible even with nineteenth-century liberalism. Others, it is true, believe that real success can be expected only from the skilful timing of public works undertaken on a very large scale. This might lead to much more serious restrictions of the competitive sphere, and, in experimenting in this direction, we shall have to carefully watch our step if we are to avoid making all economic activity progressively more dependent on the direction and volume of government expenditure. But this is neither the only nor, in my opinion, the most promising way of meeting the gravest threat to economic security. In any case, the very necessary efforts to secure protection against these fluctuations do not lead to the kind of planning which constitutes such a threat to our freedom." -- Frierich A. Hayek, The Road to Serfdom (1944), Chapter IX.

Both Hayek and Keynes drew on nineteenth-century Liberalism. They agreed that the inherited lines limiting government action needed to be redrawn. Keynes said as much in the 1920s, in his essays republished in Essays in Persuasion. Hayek's reference above, to the "timing of public works" is to Keynes' ideas. Keynes doubtless would have redrawn the lines more broadly then Hayek. But Hayek explicitly says above that Keynes' approach is neither necessarily a threat to freedom, nor a station on the way to totalitarianism. Hayek says his differences with Keynes are pragmatic, a dispute over what is likely to be effective.

On And Off The Wage-Rate Of Profits Frontier

Figure 1: Wage-Rate of Profits Frontier for Seven Countries

This post reports on the analysis of wage-rate of profits frontiers drawn for each of 87 countries or regions. The input-output tables used for this analysis are derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. Figure 1, above, presents seven examples of such frontiers. Figure 1 also shows two points:

• The observed wage share and rate of profits as a point, typically off the frontier.
• The nearest point on the frontier, in some sense, to the observed point.

The wage-rate of profits frontiers is a decreasing function relating the wage to the rate of profits. The wage, in this case, is expressed as a proportion of the output of the unit output of the industry producing the numeraire commodity basket. I take the numeraire to be in the same proportions as observed net outputs (also known as final demands) in the data. The numeraire-producing industry is conceptually scaled to a level such that the system that produces it employs one unit labor. Since different countries produce commodities in different proportions, the wage is measured for a different numeraire for each wage-rate of profits frontier on my graphs.

The wage-rate of profits frontier is drawn based on several assumptions. First, one assumes the existence of steady state prices. That is, relative prices are the same for inputs and outputs. Under this assumption, the same rate of profits is earned in all industries in a country or region. I also assume wages are paid out of the output at the end of the year, not advanced at the beginning of the year. Prices, with the distribution of income under these assumptions, are known as prices of production.

One might expect the curvature of empirically-developed wage-rate of profits frontiers to deviate from a straight line, with the convexity even being different for different parts of a frontier. Such curvature arises from variations in capital-intensities, so to speak, between net output and the intermediate goods used in producing net output.

The observed wage and rate of profits might be off the frontier for a number of reasons. Wages are paid throughout the year, so even if prices of production prevailed, the assumptions with which I am drawing the frontiers are not exact. But points will also lie off the frontier because prices of production cannot be expected to prevail. Entrepreneurs will have different expectations. Some of these expectations will be disappointed, and some will not be optimistic enough. I also wonder about the importance of foreign trade. If a country is thoroughly integrated in the global economy, might its rate of profits be somewhat independent of the system formed by domestic production?

Anyways, this data allows one to explore the empirical adequacy of the theory of prices of production. How far away do the countries or regions, as described by this dataset, lie from the wage-rate of profits frontier? In the data, nine countries or regions had an actual rate of profits exceeding the theoretical maximum: the Philippines, Sri Lanka, the Rest of North America, Uruguay, Austria, Belgium, Croatia, Cyprus, and the Rest of Middle East. These countries are excluded from the histogram and the statistics given below.

Using the observed rate of profits, one can predict the wage from the wage-rate of profits frontier. Figure 1 shows the distribution of the absolute error in such predictions, while Table 1 provides descriptive statistics for this distribution. Uganda, Singapore, Vietnam, Hong Kong, Luxembourg, and Central America are the countries or regions with the wage on the frontier, at the observed rate of profits, furthest from the observed wage. I find encouraging how the countries or regions that stick out as most anomalous are, mostly, either regions that, for purposes of data collection, consist of disparate countries aggregated together; small countries that presumably have economies that cannot be regarded as systems separate from the economies of their neighbors; or countries and ports that are notable for heavy involvement in international trade.

It seems that most countries lie close to the wage-rate of profits frontier constructed from their observed input-output relations and produced commodities.

Figure 2: Distribution of Distance to Wage-Rate of Profits Frontier

Table 1: Descriptive Statistics for Wages (Four Countries Removed)

Statistic	Distance to Frontier
Sample Size	78
Mean	0.06912
Std. Dev.	0.08998
Coeff. of Var.	1.30187
Skewness	2.59744
Kurtosis	6.75223
Minimum	0.00025
1st Quartile	0.01915
Median	0.03919
3rd Quartile	0.08330
Maximum	0.42903
Interquartile Range/Median	1.63703

Survey Of Empirical Evidence Showing Nonexistence Of Supply And Demand Curves

A theme of this blog is that wages and employment are not determined by, and cannot be determined by, the interaction of well-behaved supply and demand curves in the so-called labor market. I here bring to your attention two new papers supporting this claim:

• Steve Fleetwood, Do labour supply and demand curves exist?, Cambridge Journal of Economics, V. 38, Iss. 5 (Sep. 2104): pp. 1087-1113.

The objective of this paper is to show that circumstantial and empirical evidence for the existence of labour supply and demand curves is at best inconclusive and at worst casts doubt on their existence. Because virtually all orthodox models of labour markets, simple and complex, are built upon the foundation stones of labour supply and demand curves, these models lack empirically supported foundations. Orthodox labour economists must, therefore, either provide stronger evidence or stop using labour supply and demand curves as the foundation stones of their models. The conclusion discusses implications for future orthodox and heterodox labour economics.

• Daniel Kuehn, The importance of study design in the minimum wage debate, Economic Policy Institute (4 Sep. 2014).

This paper reviews the empirical literature on the employment effects of increases in the minimum wage. It organizes the most prominent studies in this literature by their use of two different empirical approaches: studies that match labor markets experiencing a minimum-wage increase with an appropriate comparison labor market, and studies that do not. A review of this literature suggests that:

• The studies that compare labor markets experiencing a minimum-wage increase with a carefully chosen comparison labor market tend to find that minimum-wage increases have little or no effect on employment.
• The studies that do not match labor markets experiencing a minimum-wage increase with a comparison labor market tend to find that minimum-wage increases reduce employment.

A better understanding of which approach is more rigorous is required to make reliable inferences about the effects of the minimum wage. This paper argues that:

• Labor market policy analysts strongly prefer studies that match "treatment" with "comparison" cases in a defensible way over studies that simply include controls and fixed effects in a regression model.
• The studies using the most rigorous research designs generally find that minimum-wage increases have little or no effect on employment.
• Application of these findings to any particular minimum-wage proposal requires careful consideration of whether the proposal is similar to other minimum-wage policies that have been studied. If a proposal occurs under dramatically different circumstances, the empirical literature on the minimum wage should be invoked with caution.

Failing to Empirically Render Visible What Was Hidden

Figure 1: Wage Share versus Ratio of Rate of Profits

1.0 Introduction

Consider the theory that Sraffa's standard system can be used to empirically predict distribution and prices in existing economies. Although individual commodities might be produced with extremely labor-intensive or capital-intensive (at a given rate of profits?) processes, large bundles of commodities chosen for technical characteristics, such as net output or wage goods, would be expected to be of average labor intensity. And the standard commodity formalizes the idea of a commodity of average capital intensity.

The data I looked at rejected this theory as a universal description of economies around the world.

2.0 Theory

The standard system is here defined for a model of an economy in which all commodities are produced from labor and previously produced commodities. The technique in use is characterized by the Leontief input-output matrix A and the vector a₀ of direct labor coefficients. The gross output, q, of the standard system is a (right hand) eigenvector of the Leontief input-output matrix, corresponding to the maximum eigenvalue of the matrix:

(1 + R) A q = q,

where R is the maximum rate of growth (also known as the maximum rate of profits). The maximum rate of profits is related to the maximum eigenvalue, λ_m, by the following equation:

R = (1λ_m) - 1

From previous empirical work, I know that the maximum rate of profits is positive for all countries or regions in my data. The standard system is defined to operate on a scale such that the labor employed in the standard system is a unit quantity of labor:

a₀ q = 1

The standard commodity, y, is the net output of the standard system:

y = q - A q

In the standard system, such aggregates as gross output, the flow of capital goods consumed in producing the gross output, the net output, the commodities paid in wages, and the commodities consumed out of profits all consist of different amounts of a single commodity basket, fixed in relative proportions. Those proportions spring out of the technical conditions of production in the actual economy.

Prices of production represent a self-reproducing system in which tendencies for capitalists to disinvest in some industries and disproportionally invest in other industries do not exist. In some sense, they arise in an economy in which all industries are expanding so as to maintain the same proportions. Such prices can be represented by a row vector, p, satisfying the following equation:

p A(1 + r) + a₀ w = p,

where r is the rate of profits and w is the wage paid out of the net product. The adoption of the standard commodity as numeraire yields the following equation:

p y = 1

One can derive an affine function for the wage-rate of profits. (Hint: multiply both sides of the first equation above for prices of production above on the right by the standard commodity.) This relationship is:

w = 1 - (r/R)

Prices of production in the standard system can easily be found for a known rate of profits.

p = a₀ [I - (1 + r) A]^-1 [1 - (r/R)]

If wages were zero, the rate of profits would be equal to its maximum in the standard system. If the rate of profits were zero, the wage would be equal to unity. The wage represents a proportion of the net output of the standard system. It declines linearly with an increased rate of profits.

The gross and net outputs of any actually existing capitalist economy cannot be expected to be in standard proportions, particularly since some (non-basic) commodities are produced that do not enter into the standard commodity. But do conclusions that follow from the standard system hold empirically? in particular, the average rate of profits, the proportion of the net output paid out in wages, and market prices are observable. Given the average rate of profits for the economy as a whole, the proportion of the standard commodity paid out in wages can be calculated. Is this proportion approximately equal to the observed proportion of wages? Do the corresponding relative prices of production calculated with the standard commodity closely resemble actual relative market prices? This post answers the question about wages. The empirical adequacy of prices of production is left to a later post.

3.0 Results and Discussion

I looked at data on 87 countries or regions, derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. The data covers up to 57 industries. (Not all industries exist in each country.)

For each country or region, I calculated:

• The observed proportion of the net output paid out on wages.
• The observed rate of profits, as the proportion of the difference between net output and wages to the total prices of intermediate inputs.
• The maximum rate of profits for the standard system.
• The ratio of the observed rate of profits to the maximum rate.

Figure 2 shows the distributions of the observed and maximum rate of profits.

Figure 2: Distribution of Actual Rate of Profits and Maximum in Standard System

Four countries or regions in the data had an actual rate of profits exceeding the theoretical maximum rate of profits: The rest of North America, Uruguay, Belgium, and Cyprus. The rest of North America is a region consisting of Bermuda, Greenland, and Saint Pierre and Miquelon. The four countries and regions are excluded from the linear regression and statistics given below.

Figure 1 shows the results of a linear regression of the wage on the ratio of the rate of profits. If, for each country or region, the standard system were empirically applicable to that country or region the intercept of the regression line would be near one, and the slope would be approximately negative one. But the 99% confidence intervals of the intercept and slope do not include these values. In this sense, the theory is rejected by the data.

Figure 1 points out the twelve countries with the wage furthest away from the prediction from the standard system. Why might the theory be off for these countries and the four excluded from the regression? Perhaps the net output is not near standard proportions. This possible variation of between the proportions of the standard commodity and the actual net output is abstracted from when plugs the observed rate of profits into the wage-rate of profits function for the standard system. I have looked at wage-rate of profits curves, drawn with the observed technique in use and the observed net output as numeraire. And countries far from the theory generally stick out as having wage-rate of profits curves with extreme curvatures.

Another possibility is that the industries in an economy are not earning nearly the same rate of profits, not merely because of barriers to entry but because of the economy not being in equilibrium. Prices of production, for any numeraire do not prevail.

Another possibility is that the Leontief matrix and the vector of direct labor coefficients do not capture the economic potential of the country or region. For example, the calculation of the rate of profits abstracts from the existence of land and fixed capital. Most interestingly, suppose the country or region does not characterize an isolated economic system. A region in the data combines several countries for which data is difficult to get. And the above analysis highlights several of these regions: the rest of North America, Central America, and the rest of Middle East (which consist of all of the Middle East besides Turkey). Or the country under consideration might be small and heavily dependent on imports and exports. You might notice Hong Kong and Singapore, which are important international ports. Think also of small countries that provide off-shore banking facilities. Recent events have alerted me to Cyprus serving this purpose for the countries that were formerly in the Soviet Union. I do not know much about Ireland, but recent discussion of how Apple shields its profits makes me wonder about the reported profits for its economy.

I do not know what to fully make of this analysis. The empirical use of the standard commodity seems to be more of a heuristic than the application of a claimed universal law. And the failure of its application seems to point out aspects of the deviating countries that seem of economic interest.

Appendix: Data Tables

Table 1: Descriptive Statistics for Rate of Profits (Four Countries Removed)

Statistic	Maximum Rate of Profits	Observed Rate of Profits	Ratio of Observed Rate To Maximum
Sample Size	83	83	83
Mean	84.852	48.623	0.591
Std. Dev.	26.088	14.898	0.138
Coeff. of Var.	0.307	0.306	0.234
Skewness	-0.374	-0.044	0.623
Kurtosis	0.326	0.591	0.134
Minimum	8.623	5.495	0.356
1st Quartile	66.195	39.947	0.476
Median	86.242	47.385	0.575
3rd Quartile	104.139	58.124	0.662
Maximum	144.818	84.822	0.967
Interquartile Range/Median	0.440	0.384	0.323

Table 2: Descriptive Statistics for Wages (Four Countries Removed)

Statistic	Wage in Standard System	Observed Wage
Sample Size	83	83
Mean	0.409	0.431
Std. Dev.	0.138	0.085
Coeff. of Var.	0.338	0.198
Skewness	-0.623	-0.397
Kurtosis	0.134	-0.597
Minimum	0.033	0.246
1st Quartile	0.338	0.360
Median	0.425	0.453
3rd Quartile	0.524	0.491
Maximum	0.644	0.597
Interquartile Range/Median	0.438	0.289

Update (16 September 2014): The analysis reported above is based on Leontief input-output matrices which include investment as a sector. Apparently, it is common in Computational General Equilibrium (CGE) models to treat investment as endogenous, in some sense. I plan on redoing the analysis with this sector removed and with disaggregated investment included in final demands.

The Temporal Single System Interpretation and Marx's History of Political Economy

I associate the Temporal Single System Interpretation (TSSI) of Marx's Capital most notably with Alan Freeman and Andrew Kliman. The TSSI must be addressed today by those grappling with the mathematics of the Transformation Problem, with how prices and labor values are related. But I think the TSSI makes much of Marx's work incomprehensible.

Whatever else Marx was, he was very well read. And he had many comments on the political economy of his predecessors and contemporaries. You can see this most obviously in Theories of Surplus Value, the so-called fourth volume of Capital. But, really, you can find such comments throughout Marx's work, extending back even to the Economic and Philosophical Manuscripts of 1844.

Arguably, Marx was not trying to create a scientific theory of capitalist economies¹, although he did extend classical political economy along these lines. Rather Marx thought that even the best work of British political economy - that is, David Ricardo - took too much for granted. How does capitalism create the illusion that labor is a commodity, freely bought and sold on the market like any other commodity? Why do so many come to believe that profits are a return to capitalists for the contribution of capital to production? How did the institutions of capitalist economies emerge from a feudal past? These are central questions for Marx. He addressed them through a process of immanent criticism.

I am not sure that Marx was always fair to Smith and Ricardo. He often castigates them for not recognizing distinctions that Marx himself created. (On the other hand, I can see the point of arguing that Ricardo was not clear on the difference between relative natural prices and a notion of absolute value that he was struggling to develop.) Marx's unfairness, if that is what it is, strengthens my point. Does he argue that Ricardo should have been developing the sort of supposedly dynamic concepts essential to the TSSI? Or does he accept that Ricardo has adopted an approach consistent with TSSI, with his difficulties being located elsewhere? On the other hand, a dual system interpretation, in some formulation or other, has no problem with understanding the differences between market and natural prices and Smith's idea, for example, that natural prices act as centers of gravitational attraction for market prices.

One can find many proponents of the TSSI writing in a style drawing on Hegel, whether on his head or right-side up. But I am not aware of any detailed work by such proponents exploring Marx's comments on, say, William Petty, Francois Quesnay, Adam Smith, Ricardo, with an emphasis on if or how they disagreed with the TSSI.

Footnotes

1. I recognize a tension here with the empirical work I have been presenting in the last couple of weeks.

Estimates Based On Labor Values More Precise Than Those Based On Direct Labor Coefficients

Table 1: Variations Across Countries

1.0 Introduction

This post is an empirical exploration of a simple labor theory of value as a theory of price. The precision of estimates of labor values is compared with the precision of estimates based on direct labor coefficients. The question of the accuracy of the labor theory of value is left to later posts.

I think of precision and accuracy in terms of darts. Suppose all your dart throws cluster together. Then they are precise, even if that cluster is not near the bulls eye. But if they are also in the bulls eye, then your throws are accurate, as well.

2.0 Direct Labor Coefficients and Labor Values

Labor values are calculated in the manner I find most straightforward, from a pure circulating capital model. Each industry in a modeled country, in the year in which the country is observed, produces a flow of a single commodity. Inputs for each industry consist of labor power and a flow of commodity inputs. The quantity of labor directly used, per unit output of the industry, constitutes the direct labor coefficient for that industry.

The labor value embodied in a commodity consists of all labor directly or indirectly used as an input for producing it. In the model, all inputs into production can be reduced to an infinitely long, dated stream of labor inputs. For example, the input into the industry for wearing apparel includes labor directly employed in the given year, as well as some labor directly employed in the textile industry in the previous year. (In calculating such dated labor inputs, one abstracts from changes from technology, at least in the approach that I am using. The same technique is assumed to have been used forever in the past.) Inputs directly used in the textile industry include outputs of the industry for wool and silk worm cocoons. Thus, the labor inputs into the industry for wearing apparel include some labor directly employed in that industry two years ago, as well as some labor employed three years ago in the industry for bovine cattle, sheep and goats, and horses. Given that the technique for the economy is viable, the sum of the infinite sequence of labor inputs constructed in the way outlined converges to a finite sum. I know that the techniques for all countries that I am considering are viable, based on previous empirical work.

3.0 Source of the Data

Labor values are found, for each of one of 87 countries or regions, as calculated from a Leontief matrix and vector of direct labor coefficients for a country. Each Leontief matrix was derived from a transaction table. The transactions tables, in turn, are derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. The data covers up to 57 industries. (Not all industries exist in each country.)

Quantities of each commodities, including labor power, are measured such that a unit of each commodity can be purchased with one billion dollars at prices observed when the data was taken. With this choice of units, and the adoption of one billion dollars as the numeraire, observed market prices are unity for each produced commodity.

4.0 Results and Discussion

Figures 2 and 3 show direct labor coefficients and labor values, as calculated from the data. Each point in, say, Figure 2, represents the direct labor coefficient in a specific country for the industry with the label on the X axis. Many points are plotted for each industry, since that industry exists in many countries.

Table 2: Direct Labor Coefficients By Industry

Table 3: Labor Values By Industry

The labor value for each industry, in a given country, exceeds the corresponding direct labor coefficient. I was surprised to see that any direct labor coefficients or labor values exceed unity. The largest labor coefficient and labor value is for the industry producing oil seeds in Greece. Looking at the transactions tables, I see value added includes rows for a value-added tax, as well as income for labor, returns to capital, and rents on land. In Greece, the value-added tax for oil seeds is negative. Perhaps the government of Greece has decided that, for example, the olive oil industry is important to them for cultural reasons. And they subsidize it. So this most extreme point on my graph points to something of economic interest.

The labor values, for example, for a specific industry constitute a sample, with each country contributing a sample point. For the labor values for that industry, one can calculate various statistics, including the sample size, the mean, the standard deviation, skewness, and kurtosis. The sample size will never exceed 87, since Leontief matrices were calculated, in the analysis reported here, for 87 countries.

The coefficient of variation is a dimensionless number. It is defined as the quotient of the standard deviation to the mean. Since the coefficient of variation is dimensionless, it does not depend on the choice of physical units in which to measure the quantities of the various commodities.

Figure 1, at the top of this post, shows the distributions of the coefficient of variation, for labor values and direct labor coefficients, across countries. The variation in labor values tends to be smaller and more clustered than the variation in direct labor coefficients. Consider two theories, where one states that prices in a country tend to be proportional to labor values. The other theory is that prices tend to be proportional to direct labor coefficients. This post is an empirical demonstration that the first theory is more precise.

Even If The Workers Could Live On Air

The Maximum Rate Of Growth Around The World

Consider a model of an economy in which all commodities are produced from inputs of labor and previously produced commodities. And suppose the commodities needed as inputs in the production of commodities are described through a Leontief input-output matrix in which no commodity can be produced with (unassisted) direct labor alone. Consider the special case in which wages are zero. In a sense, this special case can be seen as a description of a futuristic economy in which all production is automated, and robots are used to produce robots.

In the theory, the input-output relations determine a finite maximum rate of profits, corresponding to the maximum eigenvalue of the Leontief matrix. This maximum rate of profits is also the maximum rate of growth that arises in the Von Neumann growth model. A composite commodity, proportional to the associated eigenvector, arises from the Leontief matrix. Along the Von Neumann ray, the output of the economy each year consists of an evenly expanding output of this standard commodity, as Piero Sraffa called it. The standard commodity, in some sense, is a generalization of "corn" in David Ricardo's corn model (which was expounded in his 1815 Essay on the Influence of a Low Price of Corn on the Profits of Stock). The commodities with positive quantities in the standard commodity are known as basic commodities, once again in Sraffa's terminology.

As this post demonstrates, this is an operational model. The graph above is based on an eigenvector decomposition of Leontief matrices. Each Leontief matrix was derived from a transaction table for a country or region. The transactions tables, in turn, are derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. Quantities of each commodities are measured such that a unit of each commodity can be purchased with one billion dollars at prices observed when the data was taken.

The graph above and the table below show the maximum rate of profits or growth for each country or region for the snapshot yielding the data. The actual rate of profits for prices that allow for the smooth reproduction of the economy falls below the maximum, sometimes considerably, because the workers do not live on air. The larger the proportion of the net output of the economy paid out in wages, the lower the corresponding rate of profits. At any rate, prices of production fall out, given some information on the distribution of income and production conditions.

Along with calculating the maximum rate of profits, I found the standard commodity and identified which commodities are basic for each country or region. For example, the commodities produced by the following industries are basic commodities in the United States: Cereal Grains; Vegetables, Fruits, Nuts; Crops; Bovine Cattle, Sheep and Goats, Horses; Animal Products; Raw Milk; Coal; Oil; Minerals; Bovine Meat Products; Meat Products; Dairy Products; Sugar; Food Products; Beverages and Tobacco Products; Textiles; Wearing Apparel; Wood Products; Paper Products; Publishing; Petroleum, Coal Products; Chemical, Rubber, Plastic Products; Mineral Products; Ferrous Metals; Metals; Metal Products; Motor Vehicles and Parts; Transport Equipment; Electronic Equipment; Machinery and Equipment; Manufactures; Electricity; Gas Manufacture, Distribution; Water; Construction; Trade; Transport; Water Transport; Air Transport; Communication; Financial Services; Insurance; Business Services; Recreational and Other Services; and Public Administration, Defense, Education, Health. Which commodities are basic varies among countries, and I typically found a few non-basic commodities in each country.

I think this data is fairly comprehensive, and I hope that I can do further believable analyses with it.

Maximum Rate Of Growth By Country

Country	Rate of Growth (Percent)
Peru	144.8
Turkey	132.5
Rest of Southeast Asia	127.4
Albania	125.3
Uganda	122.5
Zambia	121.6
Rest of Southern Africa Development Community	120.7
Mozambique	119.3
Greece	117.8
Mexico	117.3
Argentina	116.9
Columbia	115.0
France	109.8
Sri Lanka	109.6
Chile	108.8
United States	107.1
Bangladesh	106.4
Zimbabwe	106.3
Rest of Sub-Saharan Africa	105.6
Spain	104.4
Madagascar	104.1
Rest of South Asia	104.1
Venezula	104.1
Japan	103.4
Switzerland	102.5
Italy	101.0
Botswana	98.2
United Kingdom	97.7
India	93.8
Indonesia	93.0
Rest of Free Trade Area of the Americas	92.8
Rest of EFTA	92.0
Portugal	91.4
Canada	90.7
Malta	89.3
Rest of the Caribean	88.9
Denmark	88.7
Rest of Central America	88.4
Tanzania	87.7
Rest of South America	87.1
Australia	87.1
Rest of Europe	86.2
Brazil	85.6
Sweden	85.2
Rest of North Africa	84.3
South Africa	83.5
New Zealand	82.7
Rest of Middle East	82.5
Tunisia	82.3
Taiwan	82.1
Netherlands	81.7
Finland	80.0
Poland	76.8
Germany	76.6
Latvia	75.8
Rest of South African Customs Union	75.0
Malawi	72.9
South Korea	72.9
Hungary	72.4
Austria	70.3
Luxembourg	67.6
Romania	66.6
Russia	66.2
Lithuania	66.0
Rest of East Asia	64.4
Philippines	64.2
Estonia	62.3
Thailand	61.6
Malaysia	60.5
Vietnam	60.0
Rest of Oceania	57.6
Ireland	55.2
Central America	54.3
Slovakia	54.0
China	53.8
Slovenia	53.7
Croatia	51.1
Czech Republic	47.1
Belguim	46.9
Morocco	45.9
Hong Kong	40.6
Singapore	35.0
Cypress	24.6
Rest of Former Soviet Union	12.7
Uruguay	12.4
Bulgaria	8.6
Rest of North America	4.7

Political Intervention in Faculty Selection at the UIUC

This is a post about the University of Illinois at Urbana Champaign (UIUC)¹. It is not about current events.

In the late 1940s, UIUC attempted to revamp their economics department. They hired many new economists, including, for example, Jacob Marschak and Franco Modigliani². A bunch of economists previously at UIUC resisted these modernizing changes. They ended up calling for political support in the press, complaining about New Deal politics. And the department was purged, in a violation of academic freedom, of these new-fangled economists³.

I thought I knew about this incident originally from reading an Esther Merjam Sent article about why both rational expectations and bounded rationality could have emerged from research at Carnegie Mellon during the 1950s - maybe, "Sargent versus Simon: Bounded Rationality Unbound" (Cambridge Journal of Economics, V. 21, No. 3 (1996): pp. 323-338). Or maybe I am recalling Fred Lee's 2009 book, A History of Heterodox Economics: Challenging the mainstream in the twentieth century. Googling, I find a draft of a paper from Antonella Rancan, who I have not otherwise read.

Footnotes

1. I have many positive impressions of UIUC. As I recall, the first graphical web browser was made there.
2. Modigliani, in a 1944 paper, extended the Hicksian IS/LM interpretation of Keynesianism to include a labor market with sticky wages. This was a critical contribution towards a politically powerful approach that Post Keynesians quarrel with on theoretical grounds (while agreeing, mostly, on short term political implications).
3. Modigliani ended up at Carnegie Mellon, which I guess was once not called that.

Labor Demand In A Fog

Figure 1: Labor Demanded Per Unit Output in a Stationary State

1.0 Introduction

As a Sraffian, I have no problem with open models in which room exists for exogenous political forces to determine distribution. The example here, though, has more indeterminancy than I expect.

2.0 Technology

Consider a simple capitalist economy, composed of workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. The workers are paid a wage, w (in units of bushels corn per person year) out of the harvest. Capitalists obtain the rate of profits, r. The technology¹ consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques. In each technique, a bushel of corn is produced from inputs of:

• l₀ person-years of labor performed in the year of the harvest.
• l₁ person-years of labor performed one year before the harvest-year.
• l₂ person-years of (unassisted) labor performed two years before the harvest-year.

Each technique is determined, given the values of the two index variables s and t. s is a non-negative real number less than or equal to the parameter c. t is a non-negative real number.

l₀(t, s) = A - B + (t + 1)(B - s)/2

l₁(t, s) = s

l₂(t, s) = (B - s)/[2 (t + 1)]

where A, B, and c are positive constants and

c ≤ B ≤ A

In effect, the above has traced out isoquants for a production function, where the quantity of output is a function of dated labor inputs². For a given value of the index variable s, labor inputs in the harvest year and two years before can be traded off. That is, if the amount of labor two years before is lower, then more labor must be expended in the harvest year. Likewise, for a given value of the index variable t, more labor being expended one year before the harvest mandates less labor being expended in the harvest year and two years before. So this specification of technology allows for substitution among inputs, at least in comparing steady states^{3, 4.}

3.0 Choice of Technique

As usual, I consider a competitive, steady state economy in which capitalists have chosen the cost-minimizing technique, at an exogenously specified wage or rate of profits. Consider the function v(r, t, s):

v(r, t, s) = (1 + r)² l₂(t, s) + (1 + r) l₁(t, s) + l₀(t, s)

Take a bushel of corn as numeraire. The condition that all income be paid out to workers and capitalists leads to a wage-rate of profits curve, as a function of the rate of profits and the technique (specified by the values of the two index variables):

w(r, t, s) = 1/v(r, t, s)

A wage-rate of profits curve can be drawn for each technique. The wage-rate of profits frontier, consistent with a competitive steady-state, is the outer envelope (Figure 2) of all these curves. That is, for a given wage, one finds the values of the index variables that maximizes the wage among all techniques. This maximization does not fix s. But, for each value of s, the maximum is found by setting the index variable t equal to the rate of profits r. The equation for the frontier is:

w(r) = 1/(A + Br)

Notice the frontier is independent of the labor input, s, in the first year before the harvest. In this case, each point on the frontier is consistent with a continuum of profit-maximizing techniques. And these techniques vary continuously along the frontier. None of this indeterminancy is apparent by looking at the frontier⁵.

Figure 2: The Wage-Rate of Profits Frontier

4.0 Labor Inputs

The analysis of the choice of technique allows one to plot labor inputs versus selected variables from the price system. In any year in a stationary state, some workers will be gathering the harvest, some will be working on preparing for the harvest one year out, and some will be working on preparing for the harvest two years out. So employment, per the unvarying net output, is the sum of l₀, l₁, and l₂. And these labor inputs can be found from a given rate of profits and a choice of s. From the wage-rate of profits frontier, one can calculate the wage for any given rate of profits. Thus, one has the two dimensions needed to draw the curves in Figure 1. One sees that, for any given wage in an interval from zero to a maximum, the quantity of labor demanded by the firms per unit output is a relation, not a function of the wage. If the relation shown were considered to be a labor demand curve, the curve would have a certain (varying) thickness.

5.0 Capital Inputs

The analysis of the choice of technique also allows one to plot the value of capital goods⁶ versus selected variables from the price system. I define the value of capital per unit output, given the rates of profit and the technique like so:

k(r, t, s) = (1 + r) l₂(t, s) w + l₁(t, s) w

This definition is such that the value of capital advanced, discounted to harvest time, and the wages paid out of the harvest add up to unity:

k(r, t, s) (1 + r) + l₀(t, s) w = 1

Impose the condition here, too, that only cost-minimizing techniques are considered for a given rate of profits. Then one obtains the curves shown in Figure 3. Here, too, the analysis yields an obvious indeterminancy.

Figure 3: Capital Demanded Per Unit Output in a Stationary State

6.0 Conclusion

Does this example undermine Sraffian analysis, as well as introductory textbook labor economics?

Footnotes

1. Notation and numerical values are chosen to be consistent with a past post.
2. I am unsure how to explicitly represent such a production function.
3. With three or more inputs, some complementarity among inputs is possible. I am not sure how to express this formally.
4. I suppose the production function consistent with the data exhibits non-negative marginal returns. I am not sure it would exhibit non-increasing marginal returns. If not, I would like to see either a proof, in the general case with n dated labor inputs, that the shaded violet regions cannot arise, given such conventional properties for a production function. Or, I would like to see a concrete numerical illustration like mine, but with such conventional properties shown to hold.
5. Also, notice the analysis of the choice of technique leads to simpler equations than those in the specification of the technology. This is not an accident.
6. I gather that, for any given value of s, unassisted labor two years before the harvest can be used to produce one of a continuum of capital goods, depending on the value of t. And once one of these capital goods is selected, the minimum dated labor inputs in each of the three years are fixed. Maybe this way of thinking about capital goods makes issues of convexity raised in Footnote 4 of little interest.

References

• Enrico Bellino (1993). Continuous Switching in Linear Production Models, Manchester School, V. 61, Iss. 2 (June): pp. 185-201.
• Christian Bidard (2014). The Wage Curve in Austrian Models, Centro Sraffa Working Papers n. 3 (June).

The Generality Of The Sraffian Analysis Of The Choice Of Technique

Figure 1: Labor Demanded Per Unit Output in a Stationary State

1.0 Introduction

This post illustrates the analysis of the choice of technique in a case in which marginal products cannot be defined, even in the sense of an interval. It is one more example of the falsity of Austrian and vulgar neoclassical teaching. Perhaps this example suggests the possibility of, say, labor demand "functions" that have a certain thickness or cloudiness.

As far as I know, nobody has used an Austrian flow-input point-output technology to make the point about indeterminacy illustrated by this post. Bellino (1993) presents three examples in which a continuously differentiable, smooth wage-rate of profits curve is consistent with multiple technologies, with the resulting non-differentiability of micro-economic production functions. The technology in none of the three is of the structure used here. Bidard (2014) develops tools for an analysis of Austrian production functions that I found key to developing this post.

2.0 A Simple Economy

Consider a simple capitalist economy, composed of workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. The workers are paid a wage, w (in units of bushels corn per person year) out of the harvest. Capitalists obtain the rate of profits, r.

I specify two technologies, in some sense. Each technology consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques. In each technique, a bushel of corn is produced from inputs of:

• l₀ person-years of labor performed in the year of the harvest.
• l₁ person-years of labor performed one year before the harvest-year.
• l₂ person-years of (unassisted) labor performed two years before the harvest-year.

A technology is fully determined here by specifying all possible values of these dated labor inputs.

2.1 First Technology

Let s be, roughly, an element of a set Q, with Q a subset of the real numbers to be fully specified below. A capitalist knows the minimum labor requirements for each technique in this technology, where a technique is indexed by s. These labor inputs are, in obvious notation:

l_1,0(s) = a + b s

l_1,1(s) = c

l_1,2(s) = 1/(s + 1)

where a, b, and c are positive constants, b is less than one, b is the square of a rational number, and:

b - b^1/2 ≤ a,

(In drawing graphs throughout this post, I use values of a, b, and c of 2, 9/16, and 3, respectively.)

2.2 Second (Sekt) Technology

I learned the word "Sekt" from Bidard; maybe he enjoys champagne. Anyways, in this technology, positive labor inputs only occur in the year of the harvest and two years before. The labor inputs one year before the harvest are zero. For convenience, define the following non-negative constants:

A = a - b + B

B = 2b^1/2 + c

C = a - b + D

D = B/2

In this technology, the techniques are indexed by the variable t, where t is from the subset of the real numbers obtained by removing all elements of Q from the real numbers. The labor inputs are:

l_2,0(t) = C + D t

l_2,1(t) = 0

l_2,2(t) = D/(t + 1)

3.0 The Choice of Technique

As usual, I consider a competitive, steady state economy in which capitalists have chosen the cost-minimizing technique, at an exogenously specified wage or rate of profits. For a given wage, the cost of a technique in the first technology is proportional to v₁(r, s):

v₁(r, s) = (1 + r)² l_1,2(s) + (1 + r) l_1,1(s) + l_1,0(s)

The corresponding function, v₂(r, t), for the second technology is:

v₂(r, t) = (1 + r)² l_2,2(t) + l_2,0(t)

The capitalists choose the technique to minimize v₁(r, s) or v₂(r, t), depending on whether such minimization results in an index being selected from Q or not. The wage-rate of profits curve for a given technique in the first technology is:

w₁(r, s) = 1/v₁(r, s)

I hope the corresponding notation for the wage-rate of profits curve for a technique in the second technology is obvious.

The choice of the cost-minimizing technique results in specifying the indices for the technique in the two technologies as functions of the rate of profits:

s(r) = (1/b^1/2)(1 + r) - 1

t(r) = r

After working out this analysis of the choice of technique for the first technology, I could have re-specified the first technology such that the index for the cost-minimizing technique was always equal to the rate of profits, as in the second technology. I worked backwards, in some sense, following Bidard, such that this nice property obtained for the second technology.

The wage-rate of profits frontier is the outer envelope of the wage-rate of profits curves for the techniques. This frontier (Figure 2) can be shown to be:

w(r) = 1/(A + Br)

I do not specify the technology for the frontier. This example has been constructed such that the both technologies have the identical frontier, when they are extended such that the index for the technique can be any real number. Since the technique varies continuously with the rate of profits, no point on the frontier is a switch point. Further, no technique on the frontier appears more than once. So this is not an example of the reswitching of techniques.

Figure 2: The Wage-Rate of Profits Frontier

I conjecture that a continuum of technologies exist with this frontier. Think of each one of these technologies as corresponding to a constant labor input in the first year before the harvest in the interval [0, c]. I guess, given Bidard's paper, this is an obvious idea.

4.0 Labor Inputs

The analysis of the choice of technique allows one to plot labor inputs, given a complete specification of the technology, versus selected variables from the price system. Accordingly, suppose the first technology is specified only when the index for the technique is a rational number. That is, the set Q is the set of rational numbers. And the set from which the index for the technique in the second technology is the set of irrational numbers.

Figure 3 is an attempt to visualize the labor demanded by firms, per unit output, in the harvest year as a function of the rate of profits. Despite appearances, neither curve in this graph is continuous; they both have an infinite number of holes. The upper curve has a countable infinity of gaps, while the lower curve has an uncountably infinite number of holes. Since these curves are discontinuous everywhere, the marginal product of labor in the harvest year is undefined. One might think of the employment for harvesters as leaping up and down between the curves as the rate of profit varies among economies.

Figure 3: Labor Inputs Per Unit Output in Harvest Year

Perhaps this example is an argument for adopting constructive mathematics in economics. But one can still get leaps between technologies with a continuous variation in the rate of profits, given a more straightforward set Q. Anyways, the labor demanded by firms, per unit output, one year before the harvest leaps between zero and the constant c. Figure 4 shows the labor demanded, per unit output, two years before the harvest. These curves are also discontinuous everywhere.

Figure 4: Labor Inputs Per Unit Output Two Years Before Harvest Year

Suppose, as in Austrian and neoclassical capital theory, that the interest rate (that is, supposedly the rate of profits) were a scarcity index for capital. A higher interest rate would indicate the availability of less capital per worker. Consequently, capitalists would supposedly be encouraged, for this sort of Austrian model, to adopt a technique in which more labor is hired during the harvest year and less during succeeding years (for example, two years beforehand). This idea is consistent with, for example, the first technology. Figures 3 and 4 show that, if one focuses solely on the blue lines, at a higher interest rate, firms want to hire less labor in the given year before the harvest and more during the harvest year. But this idea is inconsistent with the possibility of leaping from one technology to another, as illustrated in Figures 3 and 4. So this example provides another logical proof of the incorrectness of Austrian theory.

As a final step, I want to describe how to generate Figure 1, at the top of this post. In any year in a stationary state, some workers will be gathering the harvest, some will be working on preparing for the harvest one year out (except in the case of the sekt technology), and some will be working on preparing for the harvest two years out. So employment, per the unvarying net output, is the sum of l₀, l₁, and l₂. And these labor inputs can be found from a given rate of profits. From the wage-rate of profits frontier, one can calculate the wage for any given rate of profits. Thus, one has the two dimensions needed to draw the curves in Figure 1. And these curves, as usual here, are discontinuous everywhere. One can think of the labor demanded leaping left and right in the figure as the wage varies. So much for textbook teaching about competitive labor markers.

5.0 Conclusion

The above example has demonstrated, once again, the incoherence of vulgar neoclassical theory. If I thought economists cared about the truth or falsity of their claims, I would be puzzled about mainstream teaching about labor markets and about price theory, more generally.

References

• Enrico Bellino (1993). Continuous Switching in Linear Production Models, Manchester School, V. 61, Iss. 2 (June): pp. 185-201.
• Christian Bidard (2014). The Wage Curve in Austrian Models, Centro Sraffa Working Papers n. 3 (June).

Political Philosophers On How To Read The Bible

Some time ago, I read Hobbes's Leviathan, a classic argument for the existence of a social contract. I recently became aware of the existence of Spinoza's Treatise, which argues for freedom of thought, speech, and religion. I was surprised to discover a common theme in these early modern works of political philosophy, which I did not expect. I knew from second-hand literature that both contain rational arguments about what powers secular authorities should and should not be recognized to possess.

But I was surprised to find both philosophers engaged in interpreting the bible. Both Hobbes and Spinoza quote passages warning about false prophets. Spinoza gives naturalistic principles of reading. For example, he thinks philosophical doctrines in the bible should be read with the understanding that the authors were writing to engage the understanding of the common people of the day. I did not expect that Spinoza would cite the Israelites under Moses as the canonical example of a social contract. Spinoza, kicked out of their community by the Amsterdam Jews, writes quite a bit about the New Testament. I guess this excommunication may have had something to do with more than his identification of God with the cosmos, the Creator with the creation. (I tried to read the Ethics, with its geometric proofs, long ago.)

Hobbes and Spinoza have another commonality. They spend quite a bit of time cataloging, explaining, and analyzing human sensations, emotions, and qualities. I resist the idea that certain perspectives on human nature map directly to political positions. Mayhaps, this sort of analysis of the human psyche was part of a naturalizing Enlightenment project.

I suppose addressing the topic of religion makes sense in these books. The political authorities of the day were often claiming to rule in the name of God, I gather. I do not know much about, for example, the Spanish Inquisition, but, from what I understand, the Jewish community in Amsterdam contained many families that had fled Spain. The political situation in England often saw an entanglement of religion and politics, what with the beheading of Charles I, the rule of the puritan Oliver Cromwell, the English Revolution, and so on. So if you are going to write on politics, you might want to explain how the reader need not accept the unargued proclamations of supposed authorities. You might want to explain, also, how your ideas are consistent with religion, rightly understood.

As far as I know, many later writers arguing for liberalism in thought and speech did not feel the need to argue about what secular authority can and cannot be deduced from the Bible. As examples, I do not recall any such themes in either Rousseau or Mill.

Caveat: writers on general philosophy will include matters of epistemology and ethics. How can we come to know what ethical principles to follow, if we can? I gather that Hume is an example of such an argument. I do not know if his characters take authority from Bible verses.

References

• Thomas Hobbes (1651). Leviathan Or The Matter, Forme, & Power Of A Common-Wealth Ecclesiastical and Civill.
• David Hume (1779). Dialogues Concerning Natural Religion [TO READ].
• John Locke (1689). A Letter Concerning Toleration. [TO READ].
• John Stuart Mill (1859). On Liberty.
• Jean Jacques Rousseau (1755). A Discourse on the Origin of Inequality.
• Jean Jacques Rousseau (1762). The Social Contract.
• Benedict de Spinoza (16669-1670). Theological-Political Treatise.

Vagueness With Mathematical Economics

"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced." -- Bertrand Russell, Introduction to Mathematical Philosophy

Many economists may be under the mistaken impression that casting economics into mathematical models ensures assumptions are explicitly stated. This is manifestly false. Here are some examples of questions about mathematical assumptions that might puzzle some economists:

• What quantity is being conserved in typical models with agents maximizing under constraints?
• Do equilibrium models with rational expectations apply when economic time series are non-ergodic?
• How would the agents in such models learn to estimate model parameters if dynamics are chaotic?
• Do models linearized around an equilibrium apply to models with multiple equilibria? Would not interesting bifurcations arise if the number of equilibria varies with model parameters?
• Have economic models been successfully tested by dimensional analysis? (One of my favorite critiques directs one to question using a measure of capital goods in numeraire units in production functions.)
• Can economic models with utility-maximizing agents handle preferences changing not randomly, but by agents imitating what they see other agents consuming?
• What more general ideas and principles (consider, for example, menu independence and the absence of lexicographic preferences) underlie utility maximization?