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 In other words, they do not violate the outdated and exploded intuition of neoclassical microeconomics.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.