An Example With Heterogeneous Labor

Figure 1: "Labor Demand" in the Consumer Goods Industry

1.0 Introduction

In this post, I work through an example created by Arrigo Opocher and Ian Steedman. In this example, circulating capital is represented by machines of one of a continuum of types, and I compare stationary states. Unskilled and skilled workers use the machines to produce corn, along with more machines. The output of machines are needed to sustain production in future periods. In the stationary states, the same rate of profits is earned in all industries with a positive output. In fact, only the special case when the rate of profits is zero is considered here.

The (slice of) the so-called factor price frontier in this example resembles Paul Samuelson's surrogate production function. Aggregate relationships in this example are "non-perverse". In other words, they do not violate the outdated and exploded intuition of neoclassical microeconomics. The aggregate production function shows positive, but diminishing, marginal returns, in the relevant range, to inputs of factors of production. Lower wages for unskilled labor are associated with capitalists desiring to employ more unskilled labor in the economy overall.

But a perverse relationship arises in the market for corn. Corn is the only consumer good in the example. If capitalists are to want to employ more unskilled labor directly in the production of corn, the wage for unskilled labor must be higher, not lower (Figure 1). If more unskilled labor is available for production, and markets clear, more corn is produced. But when capitalists choose the cost-minimizing technology, at prices and wages they take as given, the quantity of unskilled labor used as input, in the corn industry, per bushel corn produced, decreases. This decrease overwhelms the increased output of corn, and the employment of unskilled labor in the corn industry declines.

2.0 The Technology

Consider a simple capitalist economy, composed of (unskilled and skilled) workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. Unskilled workers are paid the wage w, and skilled workers are paid the wage W out of the harvest. Both wages are in units of bushels corn per person year. Capitalists obtain the rate of profits r. The technology consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques, indexed by s. Table 1 presents the coefficients of production for a single technique.

Table 1: Inputs Required Per Unit Outputs

Inputs	Machine Industry	Corn Industry
Unskilled Labor	a(s) l(s) Person-Years	l(s) Person-Years
Skilled Labor	a(s) t(s) Person-Year	t(s) Person-Years
Machines	a(s) Machines	1 Machine
Outputs	1 Machine	1 Bushel Corn

Notice that the first column of inputs in Table 1 is proportional to - that is, a constant multiple of the - second column. This is akin to Karl Marx's assumption of a constant Organic Composition of Capital, an unrealistic assumption that simplifies price theory.

The index s for the technology is chosen from a set of real numbers, with √ 6  ≤ s ≤ 3. The parameters of a technique are defined in terms of the index as follows:

a(s) = 2 - (6/s) + (6/s²)

l(s) = 1/s

t(s) = 1/s²

Each different value of the index s is associated with the use of a different type of machine. And different quantites of unskilled and skilled labor must be used with each different type of machine to produce the output.

I compare stationary states under these assumptions:

• L person-years of unskilled labor are available for employment in the economy, with √ 6  ≤ L ≤ 3.
• T = 1 person-years of skilled labor are available for employment in the economy.
• r = 0% is the rate of profits in the stationary states considered here.
• The markets for skilled and unskilled labor both clear.
• The production of machines and corn are adapted to a stationary state. So the endowments of machines (by type) are found by solving the model, not givens.

3.0 Quantity Flows for a Given Technique

Given the type of machine, suppose the quantity of corn, c(s), produced is:

c(s) = [1 - a(s)]/t(s) = s² [1 - a(s)]

Let the number of machines, m(s), produced be:

m(s) = 1/t(s) = s²

Table 2 shows the output of the machine and corn industries, scaled to produce these gross outputs.

Table 2: Quantity FLows

Inputs	Machine Industry	Corn Industry
Unskilled Labor	s a(s) Person-Years	s [1 - a(s)] Person-Years
Skilled Labor	a(s) Person-Year	[1 - a(s)] Person-Years
Machines	s2 a(s) Machines	s2 [1 - a(s)] Machines
Outputs	m(s) Machines	c(s) Bushels Corn

For these quantity flows, the total employment of unskilled labor is s. The total employment of skilled labor is one person-year. The total inputs of machines, which are used up each year, are replaced by the output of the machine industry.

4.0 Stationary State Prices in the Special Case

Section 3 specifies quantity flows in a stationary state, given the type of machine. The capitalists choose the technique, including the machine, based on price. Let corn be numeraire, and suppose workers are paid at the end of the production period. If the same rate of profits is earned in the production of machines and corn, the following pair of equations must be satisfied for the technique in use:

p a(s)(1 + r) + a(s) l(s) w + a(s) t(s) W = p

p(1 + r) + l(s) w + t(s) W = 1

These equations have two degrees of freedom. One is eliminated by only considering the special case in which the rate of profits is zero. The other can be seen by expressing the solution as a function of, say, the wage for unskilled labor. In this sense, the solution of the system of equations for prices in a stationary state, given the special case assumption and the technique, is:

p = a(s)

W = [1 - a(s) - l(s) w]/t(s)

Or:

p = 2 - (6/s) + (6/s²)

W = - s² + s(6 - w) - 6

The wage of skilled labor, given the technique, is an affine function of the wage of unskilled labor. Figure 2 illustrates this function for three different techniques. This figure is akin to Figure 2b on page 197 of Samuelson (1962), which shows how to construct the so-called factor price frontier for Samuelson's surrogate production function.

Figure 2: Wage-Wage Curves

In a stationary state, capitalists will have adopted the cost-minimizing technique. The cost-minimizing technique, given the wage of unskilled labor, corresponds to the technique on the outer envelope (that is, the frontier) formed from all (uncountably infinite) functions that one might plot in Figure 2. One can find the technique on the frontier by setting the derivative, with respect to the index s, of the wage-wage curve equal to zero:

dW/ds = 0

Thus, the machine type used by the cost-minimizing technique, in this special case, is the following function of the wage of unskilled labor:

s = (6 - w)/2

The frontier has the equation:

W = (1/4)w² - 3 w + 3

The wage, w, of skilled labor ranges from 0 to (6 - 2 √ 6 ). The wage of skilled labor, W, ranges from 0 to 3. If the rate of profits were positive, the wage-wage frontier would lie inside the frontier found here.

5.0 Some Aggregate Markets

The results found so far can be combined.

5.1 The Market for Unskilled Labor

I have postulated that L person-years of unskilled labor and one person-year of skilled labor are available for employment in a stationary state. For quantity flows in a stationary state to fully employ both types of labor, the index for the machine type must be:

s = L

For this machine type to correspond to the cost-minimizing technique, given a rate of profits of zero and market clearing for both labor markets, the wage of unskilled labor must be the following function of unskilled labor:

w = 6 - 2 L

Figure 3 plots the wage for unskilled labor, under these assumptions, with the amount of unskilled labor firms want to hire in a stationary state. In this example, for more unskilled labor to be hired in a stationary state, its real wage must be lower. This property is particular to this example; it does not generalize.

Figure 3: Employment of Unskilled Labor

5.2 The Market for Skilled Labor

The analysis so far has shown how to determine the cost minimizing technique and the wage for unskilled labor as a function of the amount of unskilled labor employed in a stationary state. And the wage for skilled labor is a function of the wage for unskilled labor, as shown by the wage-wage frontier. The wage for skilled labor can accordingly be expressed as a function of the amount of unskilled labor employed in a stationary state.

W = L² - 6

Figure 4 shows the wage of skilled labor plotted against the quantity of skilled labor firms desire to hire in this example. In some sense, this function neither slopes up nor down.

Figure 4: Employment of Skilled Labor

5.1 The Market for Capital

Under the above assumptions, one can find the type and number of machines, m(s), produced in a stationary state. For stationary states in which different quantities of unskilled labor are employed, different types of machines will be produced. Quantities of different types of machines are incommensurable; physical measures of different types of capital cannot be plotted together on the same axis. A numeraire measure of the quantity of capital, k, can be found by taking the product of the price of machines and their physical quantity:

K = p m(s) = a(s) m(s)

Under the assumption that markets for unskilled and skilled labor clear, one can express numeraire units of capital as a function of the person-years of unskilled labor employed in a stationary state.

K = 2(L² - 3 L + 3)

Figure 5 shows the rate of profits plotted against the above quantity of capital. In this special case, the rate of profits of capital is a non-increasing function of the quantity of capital.

Figure 5: Value of Capital

6.0 Employment in the Corn Industry

The previous section shows that no phenomena that violates outdated neoclassical price theory arises in aggregate markets for unskilled labor, skilled labor, or capital, in this particular example. But consider how much unskilled labor firms, under these assumptions, want to employ in the production of corn. Figure 1 shows the graph of the wage, w, for unskilled labor against the unskilled labor, l₂, hired in the production of corn. That function can be found as:

l₂ = (-L² + 6 L - 6)/L

And this function slopes up, contrary to what neoclassical economists would have expected about half a century ago.

7.0 Conclusion

If you work through enough examples in production theory, you ought to conclude that it is hard to find any justification for mainstream theories in microeconomics. Why so many economists continue to teach archaic balderdash, and (mis)train their intuition accordingly, is a question.

References

• Arrigo Opocher and Ian Steedman (2013). Unconventional results with surrogate production functions Global and Local Economic Review, V. 17, No. 1: pp. 45-53.
• Paul A. Samuelson (1962). Parable and realism in capital theory: The surrogate production function, Review of Economic Studies, V. 29, No. 3: pp. 193-206.

A Plague On Both Your Houses

In a Bloomberg News piece, Noah Smith makes some false claims. I think his mistakes - what Eatwell and Milgate call an imperfectionist view - are widely shared among many macroeconomists. My belief that these mistakes are widely shared is not overthrown, I think, by the confusions put forth in these later posts by Stephen Williamson and Noah Smith, respectively.

First, we have the mistaken belief that in a perfect world, capitalist economies would move quickly towards equilibrium. Smith starts his column with an anecdote:

"One time, at a dinner, I asked a famous macroeconomist: 'So, what really causes recessions?'

His reply came immediately: 'Unexplained shocks to investment.'"

I take this to be an expression of the freshwater view, as embodied in models of Real Business Cycles. Cycles are to be understood as equilibrium paths responding to exogeneous stochastic shocks. Risk exists, but uncertainty does not. Recessions and depressions occur when workers voluntarily decide to take long vacations.

Second, we have mistaken understandings of price theory and how equilibrium is established:

"The market adjusts by the price mechanism. If the cost of something goes up, the price goes up to match. If demand falls, the price drops until the market clears."

I take this to be a claim that equilibrium prices are indices of relative scarcity, a belief shown to be without logical foundation about half a century ago. Ever since Robert Lucas put forth his critique in the 1970s, mainstream macroeconomists have claimed to be developing models with rigorous microfoundations. And those foundations are supposed to be provided by General Equilibrium Theory, in which agents optimize under constraints.

But many macroeconomists seem to be just ignorant of price theory, as experts in GET, such as Frank Hahn explained long ago. In the most rigorous neoclassical theory, with many commodities and many agents, the assumptions do not lead to the conclusion that prices behave that way. Nor do the theorists have a good story about how equilibrium is established. The mathematics used in mainstream macroeconomists does not allow one to find clear statements of assumptions. At least, I am unable to understand what assumptions mainstream economists think they are making on tastes, technology, and endowments in multicommodity models to justify their macroeconomic modeling. I would rather that economists turn to non-equilibrium modeling, a position that I think Robert Lucas still finds incoherent.

Third, suppose you hold that observed fluctuations in employment and output in capitalist economies can hardly be an equilibrium response. If you held the mistaken ideas about price theory that Noah Smith does, you would think that the empirical behavior of economies could only be explained by introducing some imperfection, some failure of competition, some information asymmetry, or some stickiness or slow adjustment into your theory. And given your empirical beliefs, you would think the development of theory in such a direction is a triumph of science:

"But despite these scattered denunciations and grumbles, sticky prices are enjoying a hard-fought place in the sun. The moral of the story is that if you just keep pounding away with theory and evidence, even the toughest orthodoxy in a mean, confrontational field like macroeconomics will eventually have to give you some respect."

But it is not the case that markets, including the labor market, would rapidly clear if only imperfections did not exist in a market economy. For economists to have reached this as a consensus position is a failure of their profession, not an achievement. Business cycles neither need to be explained as an equilibrium phenomenon, nor need sticky prices be invoked to explain the failure of markets to clear.

Is the topic of the above post orthogonal to a debate Paul Krugman overviews? I am of two minds on Krugman's post. I cannot be too hostile to a blog post illustrated with a homoclinic bifurcation. Maybe a solid appreciation of nonlinearity in macroeconomics is associated these days with heterodox, but not necessarily non-mainstream economics.

References

• John Eatwell and Murray Milgate (2011). The Fall and Rise of Keynesian Economics, Oxford University Press.
• Richard M. Goodwin (1990). Chaotic Economic Dynamics, Oxford University Press.
• Murray Milgate (1982). Capital and Employment: A Study of Keynes's Economics, Academic Press.

How To And How Not To Attack Marx's Economics

1.0 Introduction

I am currently reading John Roemer's Free to Lose. I thought I would outline some areas where Marx can be criticized on economic theory, as well as some areas where I do not think he is not so vulnerable. (I do not think I had previously absorbed Roemer's theory of the emergence of classes from an analysis of reproducible equilibrium. But then the Roemer work I know the best is Analytical Foundations of Marxian Economic Theory, which may predate this explanation.) Another motivation is irritation with a series of post here.

2.0 Labor Theory of Prices

For purposes of this post, I put aside the question of whether prices tend to be proportional to labor values. I think Marx rejected this theory, including in the first volume of Capital. He says so, for example, in this passage:

"From the foregoing investigation, the reader will see that this statement only means that the formation of capital must be possible even though the price and value of a commodity be the same; for its formation cannot be attributed to any deviation of the one from the other. If prices actually differ from values, we must, first of all, reduce the former to the latter, in other words, treat the difference as accidental in order that the phenomena may be observed in their purity, and our observations not interfered with by disturbing circumstances that have nothing to do with the process in question. We know, moreover, that this reduction is no mere scientific process. The continual oscillations in prices, their rising and falling, compensate each other, and reduce themselves to an average price, which is their hidden regulator. It forms the guiding star of the merchant or the manufacturer in every undertaking that requires time. He knows that when a long period of time is taken, commodities are sold neither over nor under, but at their average price. If therefore he thought about the matter at all, he would formulate the problem of the formation of capital as follows: How can we account for the origin of capital on the supposition that prices are regulated by the average price, i. e., ultimately by the value of the commodities? I say 'ultimately,' because average prices do not directly coincide with the values of commodities, as Adam Smith, Ricardo, and others believe." -- Karl Marx, Capital, V. 1 (last footnote in Chapter V.)

I take "average price" in the above passage to be referring to what has also been called "such classical terms as 'necessary price', 'natural price', or 'price of production'" (Piero Sraffa, PCMC: p. 9). And Marx is saying that prices of production do not correspond to labor values, even though he is abstracting from this distinction in the first volume of Capital. Others have also asserted that a contradiction in Marx cannot be found here:

"Writers ... like E. Bohm-Bawerk have asserted that there is a contradiction between the analyses of Volumes I and III which is certainly not to be found there unless one reads into them an interpretation different from that which Marx repeatedly emphasized." -- William J. Baumol, "The Transformation of Values: What Marx 'Really' Meant (An Interpretation)" (, V. 12, N. 1 (Mar. 1974): pp. 51-62,

3.0 Heterogeneous Labor Activities

Employees perform many distinct activities in laboring under the direction of capital. I do not think this observation is sufficient, in itself, to hinder the development of a theory organized around labor values. Consider jobs provided by supposedly unskilled labor, such as stocking shelves in a supermarket or working behind the counter in a fast food restaurant. These sort of jobs are often treated as homogenous, both by workers and employers. Workers in one or other such job can transition among them easily enough in times of high employment.

What are jobs that require vastly different levels or types of skills? I do not think this is a problem for Marx as long as relative wages can be treated as stable:

"We suppose labor to be uniform in quality or, what amounts to the same thing, we assume any differences in quality to have been previously reduced to equivalent differences in quantity so that each unit of labor receives the same wage." -- Piero Sraffa, (1960: p. 10).

As far as I can tell, this is a common position among the classical economists, with Adam Smith providing an early explanation of wage differentials.

A problem can arise here, however. Suppose some skills are acquired through an investment, such as paying for higher education. Perhaps there is a tendency for skilled workers to make decisions based on anticipated rates of return. Then, just as Wicksell effects express the dependence of the price of capital goods on distribution, so relative wages would vary with distribution. And labor values would be dependent on prices. One could then express labor value as a vector of different quantities of different types of non-competing workers. But would the assumption that the economy hangs together - e.g., all commodities are basic - work in this case? Or one could make the claim that even skilled labor is heavily produced in the household and outside of firms run for profits. And, thus, calculations of rates of return for acquisition of many skills for the worker are empirically unimportant. (I think I take this objection, as well as the first response, from Ian Steedman.)

4.0 Labor Values Dependent on Choice of Technique

I take labor values as being found from the processes used in production, as expressed in a Leontief input-output matrix and labor coefficients. The components of such matrices and vectors are given in physical units. The analysis of the choice of technique shows that the cost-minimizing technique varies with distribution. So, here too, labor values depend on prices, instead of vice-versa.

Here one could object that the choice of technique is a highly artificial problem, of interest primarily for an internal critique of neoclassical economics. In actuality, firms do not have a choice at any time of processes from a pre-existing menu. Rather technology evolves as a non-reversible process in historical time.

5.0 Volume III Invariants Cannot All Hold

In the above, I have been concentrating mostly on objections to the premises of Marx's economic theory. Let me consider a conclusion. According to Marx, accounting in labor values allows one to identify certain invariants that hold for the economy as a whole. For example, the sum of labor values for gross outputs of industry is equal to the sum of gross outputs, evaluated at prices of production. And the sum of surplus value across industry is equal to the sum of profits. According to Marx, the competition under which prices of production are formed redistributes total surplus values into aliquot quantities distributed to each industry.

Under the traditional analyses of prices of production, Marx was just wrong. For an arbitrary numéraire, not all invariants can simultaneously hold.

Four answers have been given to this issue. I do not think highly of traditional Marxists who argue that one or the other invariant should be given preference. Typically, such arguments are presented with a lot of Hegelian terminology. I find intriguing the argument that all invariants can hold if one adopts Sraffa's standard commodity as the numéraire. Duncan Foley and Gerard Duménil have proposed the new interpretation, organized around the concept of the Monetary Expression of Labor Value (MELT). As I understand it, the new interpretation makes Marx's claims too much a matter of an accounting tautology for my taste. Finally, there is the Temporal Single System Interpretation (TSSI), which I associate mainly with Alan Freeman and Andrew Kliman, although, I guess, they work with many more scholars. Of course, more invariants can be made to hold if you interpret the theory to have many more degrees of freedom.

6.0 Exploitation of Corn

A theorem in the analysis of prices of production states that the rate of profit is positive if and only if labor is exploited. Exploitation here has a technical definition; it is not an ethical concept. From John Roemer, I learn that one can argue that Marx had both ideas in mind.

Anyways, from the same analysis, one can show that same theorem holds for any commodity (that is basic or in the workers' consumption basket?). So why focus on labor? Answers have been given that deal with matters not in the math at this level of abstraction. Workers, unlike owners of commodities sold as means of production, must be brought under the direction of the capitalists when they hire them. Furthermore, the agreements laborers strike are, at best, incomplete contracts. Not all activities that the workers will be expected to perform in given situations can be prespecified. Furthermore, often some will be unpleasant, and a tug-of-war can arise between the worker and the capitalist's representative in the workplace.

Whatever you think of these rationales for focusing on the exploitation of labor, the issue of working conditions seems like a perennial concern.

7.0 Falling Rate of Profit

I do not have much to say about the theory of the falling rate of profit. I think Marx was mistaken here, but recall this is a volume 3 theory, never published in Marx's lifetime. I am aware of Marx's account of countervailing tendencies. (How is this a theory, if no explanation is given why one tendency should predominate?) And, as usual, theorists in the TSSI tradition disagree.

9.0 Outside the Theory of Value and Distribution

Such a brief overview, compared to the thousands of pages Marx wrote, and the many ways scholars and followers have read (parts of?) this work, obviously cannot cover all issues. I have said nothing about historical materialism, for instance. If this theory is read as mandating economic determinism, with no possibility of the superstructure shaping the evolution of the economic base, I, like many others, think the theory is wrong.

Nor have I said anything much about many of Marx's analyses that can be developed independently of the theory of value and distribution. For example, I like to set out Volume 2 models of simple and expanded reproduction in terms of prices of production. Whether or not Richard Goodwin's theory of the business cycle is Marxist or is descriptive of some capitalist economies at some time seems to be independent of Marx's theory of value. And Marx had many other analyses of concrete situations that might or not be worthwhile. For example, in Volume 1, he presented the introduction in Great Britain of laws regulating maximum hours of work as addressing what we would now call a prisoner's dilemma. Each mill owner would like to work their employees until their health breaks, fire them, and then hire refreshed workers. But if all mill owners are doing this for wokers from a young age, no large population of such refreshed workers will exist in the locality. So the owners need such laws after a certain level of development.

I suppose I should say something about the theory of monopoly. I do not see why prices of production cannot be developed with different markups in different industries. I may not be familiar enough with the literature, but it is my impression that many accounts of markup pricing do not take into account constraints arising from the inter-industry flows emphasized in Sraffian theory and empirical work in Leontief input-output analysis. Furthermore, markups cannot be so high in a viable economy that demands total more than the net output of a viable economy. (A theory of cost-push inflation can arise here.) This is not to say that I do not think those exploring administered, full-cost, or markup pricing are not looking at something empirically important.

And Marx had many detailed empirical observations, including claims about how feudalism evolved into capitalism. I cannot address such matters of history. Finally, I have said nothing above about the sociology of economics. I think the above is quite enough for one post.

On Mainstream Economists' Ignorance Of Real Analysis

"Logic sometimes makes monsters. Since half a century we have seen a crowd of bizarre functions which seem to try to resemble as little as possible the honest functions which serve some purpose. No longer continuity, or perhaps continuity, but no derivatives, etc. Nay more, from the logical point of view, it is these strange functions which are the most general, those one meets without seeking no longer appear except as particular cases. There remains for them only a small corner.

Heretofore when a new function was invented, it was for some practical end; to-day they are invented expressly to put at fault the reasonings of our fathers, and one never will get from them anything more than that." -- Henri Poincaré (1908, as quoted in Lakatos 1976, pp. 22-23).

Mainstream economists these days seem unwilling to accept claims about economics that are not backed up by mathematical models. (I think that views on mathematical formalism are pluralistic among non-mainstream economists. Mathematical models are just one of several approaches to acceptable claims about economics, and some non-mainstream economists are quite good at producing mathematical models.) Generally speaking, mainstream economists seem to me to reject norms common among mathematicians.

Anybody taking a standard undergraduate sequence in mathematics at a reasonably good university has an opportunity to be introduced to real analysis. Often, such a class is where the mathematician is introduced to a certain style of definitions and proofs, particularly epsilon-delta proofs. Besides this style, these classes teach a certain content, that is, the theory of limits, the differential calculus, and the integral calculus, from a rigorous standpoint. (I also draw on measure theory below, which, for me, was not taught at the undergraduate level.) In such a class, one should see various examples and purported counter-examples. The examples help the student to understand the range of behavior consistent with certain axioms. The supposed counter-examples help the student understand why theorems contain certain assumptions and why certain concepts useful for stating these assumptions were introduced into mathematics. Given an example inconsistent with the conclusion of a theorem, the student should identify a clause in the assumptions of the theorem that rules out the example.

To make my point, I'll list some examples. For my amusement, I'm not (initially) looking up anything for this post. Just as when someone criticizes somebody else's grammar, the probability approaches unity that they will make a typographic error, so I'll almost certainly be mistaken somewhere below. Does anybody have suggestions for additions to the following list of examples from real analysis?

1. Define a function that is discontinuous at some point.
2. Define a function that is continuous everywhere, but differentiable nowhere.
3. Define a sequence of functions that converges pointwise, but is not uniformly convergent. (Or is it the other way 'round?)
4. Define a function that is Lebesque integrable, but not Riemann integrable.
5. Provide an example of a non-(Lebesque) measurable set.

The style of reasoning introduced in courses on real analysis has been important in economics since, at least, Debreu (1959). And economics provides many examples analogous to the answers to the above problems. Lexicographic preferences can provide an example of a complete order on a commodity space - that is, rational preferences - that cannot be represented by an utility function. Such preferences highlight the need for an assumption on the continuity of preferences, given that the commodity space is a continuum; "rationality" is not sufficient. Menu-dependent preferences suggest the possibility of specifying deeper structures that do and do not allow the construction of binary preference relation providing an order for a commodity space. I suppose the concept of hemi-continuity is proof generated in economics.

Sraffians have also provided many examples not consistent with outdated mainstream teaching. Ian Steedman's work, over the last quarter century, is particularly good on examples illustrating that the Cambridge critique is not exhausted by the possibilities highlighted by reswitching and capital-reversing. As of yet, economists have not specified any general assumption on production processes that rules out these sort of Sraffian examples and yields neoclassical conclusions. Yet many economists - who, I guess, treat their training in mathematics as a hazing ceremony for induction into the brotherhood of economists - proceed as if they have some such theorem.

Obviously, despite my generalization, some economists, both mainstream and non-mainstream understand and accept mathematical analysis. Maybe more mainstream economists understand than my generalization would suggest. The refusal I have seen of economists to accept their own logic may be the manifestation of anti-intellectualism and boundary-patrolling that I think is so common among properly socialized economists. The general public must not come to understand how vulnerable the conclusions of mainstream economists are to slight perturbations in model assumptions. Demonstrations of the failure of the logic in the teaching and public pronouncements of economists must be distracted in blather about credentials or (false?) irrelevancies about empirical results. What economists say in public and what they say in professional seminars need not be consistent. (This is not quite the right link from Dani Rodrik making his point.) I can easily be led to believe that explanation for some behavior I have seen is more a matter of the sociology of economics and less a lack of understanding of mathematics. So, in general, are economists still exhibiting a century-outdated attitude to mathematics?

Answers

1. This is an easy question. For amusement, I'll name a function that exhibits a discontinuity of the second kind, if I correctly remember the terminology. Consider the limit of the following function of the reals as x approaches zero: f(x) = sin(1/x), if x ≠ 0; 0, if x = 0.
2. Various space filling curves provide examples. I think both Hilbert and Sierpinski provide examples.
3. I'm vague on this one, but consider the Fourier series for a square wave, where the value of the square wave at points of discontinuity is the midpoint of the left-hand and right-hand limits. I think mathematicians greeted Fourier's work on functions that were only piecewise continuous with some degree of incredulity.
4. f(x) = 0, for x rational; 1 for x irrational.
5. Consider a decomposition of the real numbers between zero and unity, inclusive, into equivalence classes. For this example, two real numbers in the range are considered equivalent if the difference between them, modulo one, more or less, is a rational number. The axiom of choice allows one to select a real number in each equivalence class. Take the union, with the index set for the union formed by the choice from each equivalence class. The index set contains an infinite number of elements, and the union is the desired closed interval. Furthermore, each equivalence class can be put into a one-to-one correspondence with any other equivalence class. Thus, the measure of each equivalence class must be the same. And these measures must add up to one, since that is the Lebesque measure of the closed interval. But assigning a measure of zero to each equivalence class will not do, and the sum over equivalence claess for any finite measure would be positive infinity. So any equivalence class formed in this way in non-measurable.

References

• Gerard Debreu (1959). Theory of Value: An Axiomatic Analysis of Economic Equilibrium. John Wiley & Sons.
• Imre Lakatos (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
• Walter Rudin (1976). Principles of Mathematical Analysis, Third edition. McGraw-Hill.

Purge of Heterodox Economists Underway at Manitoba?

I stumbled across an article published yesterday in "The students' newspaper of the University of Manitoba". Apparently, the Canadian Association of University Teachers (CAUT) published a report, Report of the Ad Hoc Investigatory Committee into the Department of Economics at the University of Manitoba. They are concerned with the violation, in the economics department, of the academic freedom of professors of economics.

Newton Method, Re-Iterated

Figure 1: Cube Roots Of Unity, Rotated, Newton's Method

I have been re-visiting my program for drawing fractals with Newton's method. Newton's method is an iterative method for finding the roots of non-linear systems of equations. That is, it is used to find zeros of functions. For my purposes, Newton's method can be used to draw fractals, although I was pleased to learn a bit more about methods in numerical analysis. I made various improvements to my program, including the the implementation of:

• More polynomial functions whose zeros are desired.
• Rotations and reflections.
• Two additional iterative methods for root finding.

I was pleased that I had thought to define a Java interface for functions whose zeros were sought. (When one looks at one's own code from a couple years ago, one might as well as be looking at code by somebody else.) Each new function could be added by defining a class implementing this interface. Besides specific functions, I defined a general polynomial, with complex coefficients, that maps complex numbers into complex numbers. I defined rotations and reflections by the transformations to the zeros of this general polynomial. A different strategy would need to be specified if one wanted to create a program for drawing fractals for functions that are not limited to being polynomials.

Halley's method is derived from a second-order Taylor approximation. (Newton's method is derived from a first order approximation.) As nearly, as I can see, Halley's method does not produce as interesting fractals. In implementing the method, I had to review a bit about tensors, since the second derivative of a function mapping the real plane into the real plane is a tensor.

Figure 2: Cube Roots Of Unity, Rotated, Halley's Method

I do not have much of an understanding of the rationale for the Chun-Neta method. I can see that it takes less iterations than Newton or Halley's method, although more calculations per iteration than either of those two methods. (The visual result of less iterations is a lighter color around the roots in the image below, as compared with above.) As I understand it, the black lines in the figure are an artifact of my implementation, probably resulting from dividing by zero.

Figure 3: Cube Roots Of Unity, Rotated, Chun-Neta Method

I conclude with an example from a general polynomial, where I defined roots so that the resulting figures would have no obvious symmetries.

Figure 4: A Fourth Degree Polynomial, Halley's Method

Figure 5: A Fourth Degree Polynomial, Chun-Neta Method

References

• Chun, C. and B. Neta (2011). A new sixth-order scheme for nonlinear equations. Applied Mathematics Letters.
• Scott, Melvin, B. Neta, and C. Chun (2011). Basin attractors for various methods. Applied Mathematics and Computation, V. 218: pp. 2584-2599.
• Yau, Lily and A. Ben-Israel (1998). The Newton and Halley methods for complex roots. American Mathematical Monthly, V. 105: pp. 806-818

Bad Math In Good Math

1.0 Introduction and Overview of the Book

Mark C. Chu-Carroll's blog is Good Math, Bad Math. His book is Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation.

A teenager recently asked me about what math he should learn if he wanted to become a computer programmer or game developer. One cannot recommend a textbook (on discrete mathematics?) to answer this, I think. If you do not mind the errors, this popular presentation will do. I like how it presents the building up of all kinds of numbers from set theory. And the order of this presentation seems right, starting with the natural numbers, but then later providing a set theoretic construction in which the Peano axioms were derived. (I suppose Chu-Carroll could also present a complementary explanation of the need for more kinds of numbers by starting out with the problem of finding roots for polynomial equations in which all coefficients are natural numbers. Eventually, you would get to the claim that an nth degree polynomial with coefficients in the complex numbers has n zeros (some possibly repeating) in the complex numbers.)

The book also has an introduction to the theory of computation, with descriptions of Finite State Machines, lambda calculus, and Turing machines. There is an outline of how the universal Turing machine cannot be improved, in terms of what functions can be computed. It doesn't help to add a second or more tapes. Nor does it help to add a two-dimensional tape. The book concludes with a presentation of a function that cannot be computed by a universal Turing machine. The halting problem, as is canonical, is used for an illustration.

2.0 Bad Math Not In Good Math

Besides being interested in popular presentations of mathematics, I was interested in seeing a book developed from blog posts. Chu-Carroll wisely leaves out a large component of his blog, namely the mocking of silly presentations of bad math. I could not do that with this blog. But there is a contrast here. The bad economics I attempt to counter is presented by supposed leaders of the field and heads of supposed good departments. The bad math Chu-Carroll usually writes about is not being to used to make the world a worse place, to obfuscate and confuse the public, to disguise critical aspects of our society. Rather, it is generally presented by people with less influence than Chu-Carroll or academic mathematicians.

2.1 Not a Proof

Anyways, I want to express some sympathy for why some might find some propositions in mathematics hard to accept. I do not want to argue such nonsense as the idea that Cantor's diagonalization argument fails, by conventional mathematical standards; that different size infinities do not exist; or that 0.999... does not equal 1. Anyways, consider the following purported proof of a theorem.

Theorem:

Proof: Define S by the following:

Then a S is:

Subtract a S from S:

Or:

Thus:

The above was what was to be shown.

Corollary: 0.999... = 1

Proof: First note the following:

Some simple manipulations allow one to apply the theorem:

Or:

That is:

2.2 Comments on the Non-Proof and a Valid Proof

I happen to think of the above supposed proof as a heuristic than I know yields the right answer, sort of. A student, when first presented with the above by an authority, say, in high school, might be inclined to accept it. It seems like symbols are being manipulated in conventional ways.

I do not know that I expect a student to notice how various questions are begged above. What does it mean to take an infinite sum? To multiply an infinite sum by a constant? To take the difference between two infinite sums? To define an infinitely repeating decimal number? But suppose one does ask these questions, questions whose answers are presupposed by the proof. And suppose one is vaguely aware of non-standard analysis. Besides how does inequality in the statement of the theorem arise? One might think the wool is being pulled over one's eyes.

How could one prove that 0.999... = 1? First, one might prove the following by mathematical induction:

Then, after defining what it means to take a limit, one could derive the previously given formula for the infinite geometric series as a limit of the finite sum. (Notice that the restriction in the theorem follows from the proof.) Finally, the claim follows, as a corollary, as shown above.

3.0 Errata and Suggestions

I think that this is the most useful part of this post for Chu-Carroll, especially if this book goes through additional printings or editions.

• p. 7, last line: "(n + 1)(n + 2)/n" should be "(n + 1)(n + 2)/2"
• p. 11, 7 lines from bottom: "our model" should be "our axioms".
• p. 19: Associativity not listed in field axioms.
• p. 20: Since the rational numbers are a field, continuity is not part of the axioms defining a field.
• Sections 2.2 and 3.3: Does the exposition of these constructions already presume the existence of integers and real numbers, respectively?
• p. 21: Shouldn't the definition of a cut be (ignoring that this definition already assumes the existence of the real number r) something like (A, B) where:

A = {x | x rational and x ≤ r}

B = {x | x rational and x > r}

• p. 84, footnote: If one is going to note that exclusive or can be defined in terms of other operations, why not note that one of and or or can be defined in terms of the other and not? Same comment applies to if ... then.
• p. 85, last 2 lines: the line break is confusing.
• p. 95, proof by contradiction of the law of the excluded middle: Is this circular reasoning? Maybe thinking of the proof as being in a meta-language saves this, but maybe this is not the best example.
• p. 97, step 1: Unmatched left parenthesis.
• p. 106: Definition of parent is not provided, but is referenced in the text.
• p. 114, base case: Maybe this should be "partition([], [], [], []).
• p. 130: In definitions of union, intersection, and Cartesian product, logical equivalence is misprinted as some weird character. This misprinting seems to be the case throughout the book (e.g., see pp. 140, 141, and 157).
• p. 133 equation: Right arrow misprinted as ">>".
• Chapter 17: Has anybody proved ZFC consistent? I thought it was the merely the case that nobody has found an inconsistency or can see how one would come about.
• p. 148: Might mention that the order being considered in the well-ordering principle is NOT necessarily the usual, intuitive order.
• p. 148: Drop "larger" in the sentence ending as "...there's a single, unique value that is the smallest positive real number larger!"
• p. 163" "powerset" should be "power set".
• p. 164, line 6: "our choice on the continuum as an axiom" is awkward. How about, "our choice about the continuum hypothesis as an axiom"?
• p. 168, Table 3: g + d = e should be g + d = g.
• p. 171-172: Maybe list mirror symmetry or write, "in addition to mirror symmetry".
• Part VI: Can we have something on the Chomsky hierarchy?
• p. 185; p. 186, Figure 15; p. 193): Labeling state A as a final state is inconsistent with the wording on p. 185, but not the wording on p. 193. On p. 185, write "...that consist of any string containing at least one a, followed by any number of bs."
• p. 190: Would not D_a(ab*) be b*, not ab*?
• p. 223: "second currying example" should be "currying example". No previous example has been presented.
• p. 225, towards bottom of page: I do not understand why α does not appear in formal definition of β.
• p. 229: Suggestion: Refer back to recursion in Section 14.2 or to chapter 18.
• p. 244, 5 lines from bottom: Probably γ should not be used here, since γ was just defined to represent Strings, not a generic type. Same comment goes for α.
• p. 245, last bullet: It seems here δ is being used for the boolean type. On the previous page, β was promised to be used for booleans, as in the first step of the example on the bottom of p. 247.
• p. 249 (Not an error): The reader is supposed to understand what "Intuitionistic logic" means, with no more background than that?
• p. 257: Are the last line of the second paragraph and the last line of the page consistent in syntax?
• Can we have an index?

What Is A "Special Interest"?

I do not want to compare and contrast analytically precise definitions that answer the question in the post title. (Socrates, as reported by Plato, always asked for a definition after being given examples.) Instead, I give two lists, where I trust the reader to see family resemblances among the items on each list:

• Ethnic groups like African-Americans; women; the poor; organized labor; and lesbians, gays, bisexuals, and transgenders.
• Corporations, especially those operating in specific industries (e.g., big oil); Corporate Executive Officers; and owners of small businesses.

I suggest that the policies and culture of a country would be quite different, when the dominant understanding of the phrase, "special interests" was consistent with one or another list.

I think somewhere or other Noam Chomsky has asserted that the second understanding reflects the true meaning or the term, or at least a meaning consistent with what the Founding Fathers of the United States wrote. This quote does not have the look back to classical liberals:

"...these questions have been asked for a long time in polls, a little differently worded so you get some different numbers, but for a long time about half the population was saying, when asked a bunch of open questions - like, Who do you think the government is run for? would say something like that: the few, the special interests, not the people. Now it's 82%, which is unprecedented. It means that 82% of the population don't even think we have a political system, not a small number.

What do they mean by special interests? Here you've got to start looking a little more closely. Chances are, judging by other polls and other sources of information, that if people are asked, Who are the special interests? they will probably say, welfare mothers, government bureaucrats, elitists professionals, liberals who run the media, unions. These things would be listed. How many would say, Fortune 500, I don't know. Probably not too many. We have a fantastic propaganda system in this country. There's been nothing like it in history. It's the whole public relations industry and the entertainment industry. The media, which everybody talks about, including me, are a small part of it. I talk about mostly that sector of the media that goes to a small part of the population, the educated sector. But if you look at the whole system, it's just vast. And it is dedicated to certain principles. It wants to destroy democracy. That's its main goal. That means destroy every form of organization and association that might lead to democracy. So you have to demonize unions. And you have to isolate people and atomize them and separate them and make them hate and fear one another and create illusions about where power is. A major goal of this whole doctrinal system for fifty years has been to create the mood of what is now called anti-politics." -- Noam Chomsky, Class Warfare: Interviews with David Barsamian Common Courage Press (1966): p. 138.

But there is another literature, a post modern literature, that also looks at how people come to associate examples with words. People generally do not think logically, following the rules of predicate calculus. One trying to understand culture should realize this. One might talk about the The politics of the signifier. How does one or another definition, or set of examples, become hegemonic? (For what it is worth, I think Slavoj Zizek is a very intelligent, very well-read, self-aware clown.)

Income Inequality In OECD Countries

I recently took another look at data, available from the Organization for Economic Co-operation and Development (OECD), on income inequality. The Gini coefficient is available on countries in the database, under measures of Social Protection and Well-being. Under that menu, expand the sub menu for Income distribution and poverty, and select inequality. You can see the Gini coefficient (at disposable income, post taxes and transfers) displayed, by country, for various years. Table 1 shows the most recent numbers, sorted from countries with the most equal distribution to the least equal. For one way of thinking about it, the United States is not number 1, since the US is exceeded by Turkey, Mexico, and Chile.

Table 1: Gini Coefficient

Country	Gini Coefficient (Non Provisional)	Year
Slovenia	0.245	2011
Norway	0.250	2011
Iceland	0.251	2011
Denmark	0.253	2011
Czech Republic	0.256	2011
Finland	0.261	2012
Slovak Republic	0.261	2011
Belgium	0.264	2010
Sweden	0.273	2011
Luxembourg	0.276	2011
Netherlands	0.278	2012
Austria	0.282	2011
Switzerland	0.289	2011
Hungary	0.290	2012
Germany	0.293	2011
Poland	0.304	2011
Korea	0.307	2012
France	0.309	2011
Ireland	0.312	2009
Canada	0.316	2011
Italy	0.321	2011
Estonia	0.323	2011
New Zealand	0.323	2011
Australia	0.324	2012
Greece	0.335	2011
Japan	0.336	2009
United Kingdom	0.341	2010
Portugal	0.341	2011
Spain	0.344	2011
Israel	0.377	2011
United States	0.389	2012
Turkey	0.412	2011
Mexico	0.482	2012
Chile	0.503	2011

The Gini coefficient is a measure of inequality, with a higher Gini coefficient denoting a more unequal distribution of income. It is defined as follows: sort the population in order of increasing income. Plot the percentage of income received by those poorer than each value of income against the percentage of the population with less than that value of income. This is the Lorenz curve, and it will fall below a line with a slope of 45 degrees going through the origin. The Gini coefficient is the ratio of the area between the 45 degree line and the Lorenz curve to the area under the 45 degree line. A Gini coefficient of zero indicates perfect equality, while a Gini coefficient of unity arises when one person receives all income and everybody else gets nothing. Consequently, the Gini coefficient lies between zero and one.

A Cynical Take By Greece's Finance Minister On Mainstream Economists

I have found Yanis Varoufakis' 2014 book, Economic Indeterminacy: A personnel encounter with economists' peculiar nemesis a bit too abstract for my tastes. I am not sure that game theory counts as a subset of neoclassical economics, although I can see how some game theory meets Varoufakis' definition. One might see how a lot of game theory illustrates the idea that economists, collectively, exhibit weakness of will. That is, a lot of game theory can be used to develop models with multiple equilibria and of nondeterministic outcomes. One might expect economists to shy away from these conclusions.

I find it hard to accept Varoufakis's argument that in games, one might want to deliberately be irrational. I wondered if that was so, wouldn't an opponent see this? And, thus, would not this irrational behavior therefore be rational at a meta-level? Varoufakis' argument is structured to address this objection.

But my point in this post is to quote from the preface:

"...my project's failure was predetermined, at least in the sense that it was never going to cause a shift in the attitudes and demeanour of a profession which operates like a priesthood, dedicated solely to preservation of its dogmas... as well as to the recapitulation of its authority within the universities, the financial sector and the government. Indeed, at no point did I harbour any significant hope that this priesthood would take kindly to the demons of doubt and indeterminacy which my work was bound to give rise to. But it did not matter, at least not at a personal level. My intimate familiarity with the neoclassical models was sufficient to keep me on the roster of neoclassical economics departments, where a capacity to teach these models, and produce academic papers based on them is all that matters.

Looking back at these long years of tampering with, and delving into, the complex models of the neoclassical tradition, I cannot but question my decision to keep pushing, Sisyphus-like, the theoretical rock up the neoclassical hill. Why did I stick to this task, when I knew it would end up in failure? In retrospect, there were two reasons, neither of which was predicated upon any hope of influencing a profession utterly uninterested in the truth status of its models. First, I deeply enjoyed toying with these models as an end-in-itself, just as a clockmaker enjoys taking apart and then re-assembling some old clock for the hell of it. Secondly, and more importantly, I felt it necessary to make it hard for my colleagues to pretend to themselves that the models they were being forced to with, by a particularly authoritarian profession, were logically coherent. Bringing them, even fleetingly, to the point when they had to confess to their models' internal contradictions was, I felt, a victory of sorts; the equivalent of a lone sniper behind enemy lines making life difficult for an army of cocupation." -- Yannis Varoufakis (2004: p. xxiv.)

Varoufakis has some other books that sound interesting and more popular. I think his book; The Global Minotaur: America, Europe and the Future of the Global Economy; might be especially topical at the moment.

Update: Steve Keen provides a link to one exposition of Varoufakis' argument that, in game theory, agents can and will deliberately choose irrational behavior.

Approximating a Continuous Time Markov Process

Figure 1: Rate of Transitions Between States in a Three-State Markov Chain

1.0 Introduction

This post, about Markov processes, does not have much to do with economics. I here define how to approximate a continuous time Markov chain with a discrete time Markov chain. This mathematics is useful for one way of implementing computer simulations involving Markov chains. That is, I want to consider how to start with a continuous time model and synthesize a realization with a small, constant time step.

2.0 Continuous Time Markov Chains

Consider a stochastic process that, at any non-negative time t is in one of N states. Assume this process satisfies the Markov process: the future history of the process after time t depends only on the state of the process at time t, independently of how the process arrived at that state. I consider here only processes with stationary probability distributions for state transitions and for times between transitions. A continuous time Markov chain is specified by a state transition matrix. In this section, I define such a matrix as well as specifying two additional assumptions.

Formally, let P_{i, J} denote the conditional probability that the next transition will be into state j, given that the process is in state i at time zero. (As seen below, in the notation adopted here it matters that these conditional probabilities are not a function of time.) Assume that for each state, the next transition when the process is in that state is into a different state:

P_{i, i} = 0; i = 0, 1, ..., N - 1

Further, assume that for each state, the time to the next transition is from an exponential distribution with the rate of occurrence of state transitions dependent only on the initial state:

F_{i, j}(t) = 1 - e^{- λ_i t}; i, j = 0, 1, ..., N - 1;

where F_{i, j}(t) is the conditional probability that the next transition will be before time t, given that the chain is in state i at time zero and that the next transition will be into state j. In other words, F_{i, j}(t) is the Cumulative Distribution Function (CDF) for the specified random variable. Under the above definitions, the stochastic process is a continuous time, finite state Markov chain.

Let P_{i, j}(t) be the conditional probability that the chain is in state j at time t, given that the chain is in state i at time zero. These conditional probabilities satisfy Kolmogorov's forward equation:

,

where the transition rate matrix Q is defined to be:

The elements in each row of the transition rate matrix sum to zero. Kolmogorov's forward equation can be expressed in scalar form:

The above equation applies to continuous time Markov chains with a countably infinite number of states only under certain special conditions.

Steady state probabilities of this Markov chain satisfy:

p Q = 0,

where p is a row vector in which each element is the steady-state probability that the chain is in the corresponding state.

3.0 Discrete Time Approximation

A discrete time Markov chain is specified by a state transition matrix A, where a_{i, j} is the probability that the chain will transition in a time step from state i to state j, given that the chain is in state i at the start of the time step. Steady state probabilities for a discrete time Markov chain satisfy:

p A = p

The above equation compares and contrasts with how steady state probabilities relate to the transition rate matrix in a continuous time Markov chain.

Let the time step h be small enough that the probability of the continuous time Markov chain undergoing two or more transitions in a single time step is negligible. In other words, the following probability, calculated from a Poisson distribution, is close to unity for all states i:

P(0 or 1 transitions in time h | Chain in state i at time 0) =

(1 + λ_i h) e^{- λ_i h}

The probability that the chain remains in a given state for a time step is the probability that no transitions occur during that time step, given the state of the chain at the start of the time step. This probability is also found from a Poisson distribution:

a_{i, i} = e^{- λ_i h} = e^{- q_{i, i} h}; i = 0, 1, ..., N - 1

The probability that the chain transitions to state j, given the chain is in state i at the start of the time step, is the product of:

• The probability that a transition occurs during that time step, and
• The conditional probability that the next transition will be into state j, given the chain is in state i at the start of the time step.

The following equation specifies this probability:

a_{i, j} = (1 - a_{i, i})P_{i, j} = (1 - a_{i, i}) q_{i, j}/(- q_{i, i}); i ≠ j

These equations allow one to write a computer program to synthesize a realization from a finite state Markov chain, given the parameters of a continuous time, finite state Markov chain. Such a program will be based on a discrete time approximation.

4.0 An Example

Consider a three-state, continuous time Markov chain. Figure 1 shows the rate of transitions between the various states. The transition rate matrix is:

To discretize time, choose a small time step h such that, for all states i, the following probabilities are approximately unity:

P(0 or 1 transitions in time h | Chain in state 0 at time 0) =

[1 + (λ_{0, 1} + λ_{0, 2})h] e^{-(λ_{0, 1} + λ_{0, 2})h}

P(0 or 1 transitions in time h | Chain in state 1 at time 0) =

[1 + (λ_{1, 0} + λ_{1, 2})h] e^{-(λ_{1, 0} + λ_{1, 2})h}

P(0 or 1 transitions in time h | Chain in state 2 at time 0) =

[1 + (λ_{2, 0} + λ_{2, 1})h] e^{-(λ_{2, 0} + λ_{2, 1})h}

The state transition matrix A for the discrete-time Markov chain is:

I have not tested the above with concrete values for a continuous time Markov chain.

Reference

• S. M. Ross (1970). Applied Probability Models with Optimization Applications. San Francisco: Holden-Day

Laughing At Neoclassical Economists, Elsewhere

• Matthew Yglesias lists "Nine Things Only Neoclassical Economists Will Understand". Strangely, his twitter announcement of this article is about a tenth.
• Noah Smith purports to explain each thing in only a couple of sentences. Stranegly, only for the Modiliani-Miller theorem does he note, "Obviously this doesn't work in the real world".
• Tyler Cowen attempts to clarify the Heckscher-Ohlin theorem, but fails to note that "capital" cannot be a factor of production in the Heckscher-Ohlin-Samuelson model. (He does note Leontief's empirical demonstration that the theory fails.)

Because Something Is Happening Here/But You Don’t Know What It Is/Do You, Mister Jones?

Strangely, some prominent, somewhat liberal, economics bloggers have decided simultaneously to complain about (unnamed) left-leaning heterodox economists:

• Noah Smith, at Bloomberg View and at his blog.
• Simon Wren-Lewis, at his blog.
• Paul Krugman, at his New York Times blog.

All three, incorrectly in my view, think the heterodox economists who they object to are only arguing politics. As far as I know, many, including me, do not take issue with Krugman's short-term policy views. Smith, in his trollish approach, raises a side comment about Austrian economists and the Mont Pelerin society. (I will state the proper label for Friedrich Hayek and Ludwig Von Mises is "economist", not "quasi-economist", as Smith would have it. But I've seen for some time that I am more well-informed on Austrian economics than Smith is.)

I think more pertinent issues center around modeling approaches, the image the profession projects in the public sphere, and the sociology of the profession. How is it than so many rightists have been able to push the view that their politics is good economics, while simultaneously insisting that economics is a positive science? The involvement of economists with neoliberal politics is not confined to some fringe. Consider, for example, the Chicago school, the lack of a strong ethics policy in the American Economic Association, the Washington consensus, and even Paul Samuelson's 1960s research that led to to Efficient Market Hypothesis.

There is probably also a personnel element here. Non-mainstream, heterodox economists would like more acknowledgement by mainstream economists. If your knowledge of heterodox economics is confined to what you can get off the Internet, aside from what professional literature is now available there, you might not know what you are talking about when you talk about heterodox economics. (And this includes the Austrian school.) Furthermore, when you develop parallel ideas, or draw on heterodox economics, you should acknowledge it. In the linked post above, Krugman makes the point that "a country that borrows in its own currency" cannot easily become like Greece, under attack from "bond vigilantes", without saying anything about endogenous money or the economists at the University of Missouri Kansas City. (I could also say something about the research for which Krugman won the "Nobel Prize".) If you know where to look, you can find Joseph Stiglitz acknowledging that he learned a lot from such Cambridge economists as Nicky Kaldor and Joan Robinson.

Maybe economics would be a better place if the center of gravity in economics in the United States were arguments between mainstream economists and, say, economists at the New School and the University of Massachusetts at Amherst. If the profession were to move in this direction, young doctorates would need to be socialized to not dismiss viewpoints because of the rankings of the universities and the journals in which they were advanced. Methodology would continue to need to be broadened to include more than mathematical models of optimizing agents.

Update: Reactions from Chris Dillow, Peter Dorman, and Alex Marsh.

Greg Mankiw, Fool Or Knave?

Greg Mankiw seems determined to continually attempt to bring his supposed profession into disrepute. Last week, at the annual meeting of American economists (the Allied Social Science Associations), Greg Mankiw chaired a session on Thomas Piketty's Capital in the 21st Century. In his draft of his prepared remarks, Mankiw writes:

"Equation (3) says that capital earns its marginal product." -- Greg Mankiw, "Yes, r > g. So what?" (24 November 2014).

Because of price Wicksell effects, the marginal product of finance capital is generally unequal, in equilibrium, to the rate of profits. Even Champernowne's chain index, which abstracts from price Wicksell effects, cannot generally be used to defend the equality in aggregate models of the rate of profits and the marginal product of capital. Economic theory imposes no restriction on the direction of price and real Wicksell effects, and the chain index is not well-defined in the presence of positive Wicksell effects. Neoclassical theory claims, at best, that the price of each capital good is equal, in equilibrium, to its marginal product. But marginal productivity is not a theory of the functional income distribution, since every point on the wage-rate of profits frontier is compatible with all valid marginal productivity conditions. Even if the returns to capital could be explained by marginal productivity, this would not justify any particular size of the tolls that capitalists are able to impose. A conceptual distinction can and should be made between the cost of capital goods and the returns to capitalists.

As far as I am concerned, the above is just good economics, agreed to by all non-ignorant economists, neoclassical or otherwise. But the confusion and general muddleheadness promoted by such as Mankiw, seems to serve a functional purpose in the sublunary world.