"if we want to follow in Marx's footsteps and pass from values to prices of production and from rate of surplus value to rate of profits, the Standard System is a necessary adjunct: for that passage implies going through certain averages and if these are calculated without weights (or with the weights of the real system), a result which is only approximately numerically correct is obtained. If an exact result is wanted the proportions of the St[andard] Syst[em] of eq[uation]s q's [quantities] must be applied as weights. - This is not stated explicitly in the book, but is implied. " -- Piero Sraffa (as quoted by Heinz D. Kurz. "Obituary: Aiming for a 'Higher Prize': Paul Anthony Samuelson (1915-2009)", European Journal for History of Economic Thought, V. 17, n. 3 (August 2010): pp. 513-520.)
Thursday, August 30, 2012
Saturday, August 25, 2012
|Figure 1: A Photo Probably From The Same Week I visited A Joint Production Process|
I thought I would continue thinking about the joint production example in the previous post. I want to consider the price equations for three processes that might be operated with this apparatus in the course of a full day. In the first process, labor operates the main generator and pump for 12 off-peak hours. The second and third processes execute in parallel during 12 peak hours. Labor operates the main generator alone in the second process. And, in the third process, labor operates the secondary generator alone.
Assume this electric company takes the wage, the costs of operating the main and secondary generators, and the cost of operating the pump as given. What rate of profits and relative prices of peak and off-peak hours of electricity justifies the utility in operating these processes (when this apparatus is new and no quasi-rent is being charged)? This post gives an incomplete outline answering this accounting question.2.0 Assumptions And Price Equations
Some definitions follow:
- p1 = cost of operating main generator for 12 hours.
- p2 = cost of operating pump for 12 hours.
- p3 = cost of operating secondary generator for 12 hours.
- p4 = 1 = price of a unit of non-peak hours of electricity.
- p5 = price of a unit of peak hours of electricity.
- p6 = price of a unit of pumped and stored water.
- w = the wage.
- r = the rate of profits (for a 12 hour period).
- b41 = Units of off-peak hours of electricity produced in 12 hours when the pump is operating
- b53 = Units of peak-hours of electricity produced in 12 hours by the secondary generator.
- a01 = Person-hours of labor needed to operate the main generator and pump for 12 hours
- a02 = Person-hours of labor needed to operate the main generator alone.
- a03 = Person-hours of labor needed to operate the secondary generator alone.
I have taken a unit of non-peak hours of electricity as the numeraire. Assume that electricity is measured in units such that the output of the main generator operating alone is one unit of electricity. Since the pump is operating during the production of off-peak hours of electricity, the electricity generated during this period is less than one-unit:
0 < b41 < 1.
Measure pumped and stored water in units such that the amount pumped in 12 hours is a unit. The second law of thermodynamics implies the following additional constraint:
0 < b41 + b53 < 1.
Finally, I assume that less labor is required to operate the main generator alone than is required to operate it with the pump:
0 < a02 < a01.
These assumptions allow one to specify the following price equations:
(p1 + p2)(1 + r) + a01w = b41 + p6
(p1)(1 + r) + a02w = p5
(p3 + p6)(1 + r) + a03w = b53p5
The price equations show that wages are paid out of the surplus, not advanced. The price equation for the first process shows that it produces a joint product.3.0 The Solution Prices
The solution prices are:
w = [(b53p1 - p3 + b41)(1 + r) - (p1 + p2)(1 + r)2]/[a01(1 + r) - a02b53 + a03]
p5 = (p1)(1 + r) + a02w
p6 = [(b53p5 - a03w)/(1 + r)] - p3
In a more thorough analysis, one would consider when the wage-rate of profits curve is downward sloping, when the price of peak-hours electricity is positive, and when the price of pumped and stored water is positive. As is typical in price theory, prices depend on the distribution of income. The analysis uncovers the accounting price for pumped and stored water. Since this is a long-period model, consumer demand enters only in determining the scale at which this facility is constructed. Prices can be found without ever considering consumer demand schedules.4.0 Discussion and Conclusions
A fuller development would look at the depreciation of the pump and generators. If one were to look at the economy as a whole, instead of just this electric company, one would want to include processes for producing pumps and generators, perhaps with inputs that include electricity. And one could add further complications. Anyways, I think I have justified, in this post, the (unoriginal) claim that Sraffa's book has empirical implications.
Friday, August 24, 2012
|Figure 1: A Hydro-Electric Facility|
The appearance and effects of joint production are sometimes hard to see, and they often require a degree of abstraction to understand1. For example, suppose only one process exists in the Leontief input-output matrix to produce a certain pair of joint products. And no other process produces either one of them alone. It does not necessarily follow that the Leontief matrix is non-square. It could be that two processes exist for producing another product, but with different ratios of inputs. The inputs in this pair of processes consist, among others of the pair of joint products. And so prices of production still can be explained without specifying demand schedules for consumers. The net output of the economy can vary in some range of proportions with the same prices of production. Demand and supply remain asymmetrical.
But I want to concentrate in this post in describing a specific combination of processes for making a joint product. The joint products, at some level of abstraction in this case, are peak-time and off peak-time electricity2. The apparatus illustrated in the figure above produces these joint products.
The dam has an associated generator. During off-peak hours, some of the resulting electricity is used to pump water up the hill and into the storage area. Only some of the off peak-time electricity is delivered to the grid.
On the other had, during peak hours, two generators are operated, and all of the generated electricity is delivered to the grid. The underground pipe to the storage area flows backwards from how it flows during off-peak hours. This water flowing downwards is used to operate one of the generators, the one not operating during off-peak hours.
It seems to me these are not fixed coefficient processes. I imagine more off-peak hours electricity can be delivered to the grid if not as much water is pumped up to the storage area. So peak and off-peak electricity can be traded off to some extent, but not one for one. Some of the off-peak electricity would be lost to operating the pump and necessary3 inefficiencies in operating the generators. So one unit of off-peak hours electricity would be sacrificed for less than one-unit of peak hours electricity. But the configuration of the apparatus, I gather, sets a limit to maximum amount of electricity that can be generated.
So we see here an application of Sraffian economics in energy economics.Footnotes
- Bertram Schefold has written much on this theme, including on applied problems.
- Milk and gasoline are both measured in gallons. But nobody would say the ratio of the price of milk for delivery at one point of time to the price of gasoline at another point of time is an interest rate, despite what a superficial and mistaken dimensional analysis might say. Likewise, the ratio of the price of peak-time electricity to off-peak time electricity is not an interest rate.
- See the second law of thermodynamics.
Wednesday, August 22, 2012
Nick Rowe has recently posted about two of my themes, reswitching and about joint production. He goes through some of the baseless defensiveness of economists who do not know their ideas on price theory were shown decades ago to be mistaken.
2.0 Multiple Rates Of Return?
Rowe points out the importance of worrying about uniqueness in certain contexts:
"How does the rate of interest affect his decision? (But watch out for that "the", because it hides a massive implicit assumption.)"
As I understand it, reswitching is compatible with a unique price solution to the problem of the choice of technique, properly formulated. I demonstrate that with this example, in which I give an algorithm for finding steady-state prices, given the real wage.
That algorithm does raise questions for the mathematician. Under what conditions will the equation for Net Present Value yield a unique, economically relevant rate of interest? And will the algorithm converge to a cost-minimizing technique? Perhaps the cost-minimizing technique will cycle through α, δ, γ, and back to α. These questions are particularly salient in the case of joint production. I have addressed these questions for one such example.Let me turn to another remark from Rowe:
"Suppose the price of fertiliser goes down, holding the prices of all the different types of food constant. Will the farmer use more fertiliser?"
A question, for me, is whether this is a coherent thought experiment. The analysis of prices of production shows that it is not. If firms adopt cost-minimizing techniques, one price cannot be varied independently of all others. Otherwise, firms will refuse to produce some of the inputs needed for the next production period. Plans will be mutually incompatible. As Ian Steedman has shown, the answer to Rowe's question is indeterminate in an open model of firm equilibrium in which account is taken of which prices can be exogenous and which must be endogenous.
3.0 Analysis Of Fixed Points As A Start On Dynamic Analysis
Rowe writes as if it is a point in favor of neoclassical theory that a comparison of steady states differs from an analysis of a traverse path:
"But first notice something important. When I said 'as the rate of interest starts out high and slowly falls' I am not talking about a process that is happening over time. I am not saying 'suppose r is 100% in the first year, 99% in the second year, 98% in the third year...'. I can't be saying that, because In doing the NPV calculation I have assumed that r stays exactly the same in all years. I have assumed a perfectly flat term structure of interest rates. It's that assumption which lets us talk about 'the' rate of interest. Rather, I am imagining different possible worlds, and asking what happens as we slowly traverse from the first possible world, where r is and always will be 100%, to a second possible world where r is and always will be 99%, etc. And I am looking at what technique a farmer would choose in each of those many possible worlds."
Cambridge economists, such as Geoff Harcourt or Ian Steedman, were always clear that the analysis of the choice of techniques was about a logical point, not a process in historical time. Joan Robinson, of course, would not accepts Rowe's fudge about "slowly" traversing. This is the mistake she accused Samuelson of, although he denied that he ever meant his words to be taken in that way.
In response to capital-theoretic difficulties, neoclassical economics increasingly turned to analysis of temporary and intertemporal equilibrium. Two kinds of dynamics arise in such models:
- The dynamics of equilibrium paths.
- Instantaneous out-of-equilbrium processes that might approach such paths, for example, a tatonnement process.
Mathematicians begin the analysis of dynamics with an examination of bifurcations and the stability of limit points. Steady states, as examined in the analysis of the choice of technique, are limit points for temporary and intertemporal equilibrium paths.
I think it an open question whether capital-reversing and other Sraffa effects can be used to reveal the instability of either dynamics. Both defenders and Cambridge-Italian critics of mainstream economics have asserted that capital-reversing examples are not necessary to expose such instability. Basically, neither J. R. Hicks' model of temporary equilibria nor the Arrow-Debreu model of intertemporal equilibria are descriptive of actually-existing capitalist economies.
4.0 Reswitching With Continuous Substitution
In discussing joint production, Rowe suggests the usual confused claim that the issue is between substitutability and fixed-coefficients of production, as in Leontief production functions. He does not say that continuous substitution rules out reswitching. But, given the context, it would not be surprising if some of his readers took away that muddled view.Of course, reswitching examples have been available for a long time in which the cost-minimizing technique varies continuously along the so-called factor-price frontier. In these examples, each capital good can be used and produced only with fixed coefficients. A continuum of capital goods exist however.
Furthermore, a continuously differentiable production function can be approximated as close as you like by a linear combination of fixed coefficient processes. So I do not know why some economists cannot let go of this canard.
5.0 Land And Fixed Capital As Examples Of Joint Production
Rowe does not seem to know about some standard analyses of joint production. The wool-mutton cases provides room for firms to simultaneously adopt two processes for producing both, but in different proportions. The quantity demanded, also known as requirements for use, if you will, enters into the story. But one still does not need to talk about schedules for supply and demand.
Some of Rowe's commentators bring up netput vectors. Nobody over there notes that fixed capital and land are special cases of joint products. I find joint production useful for analyzing depreciation and for analyzing rent. These special cases show why one cannot ignore joint production; it is ubiquitous in actual economies, even apart from oil refineries and other industrial processes that might be of interest to some chemical engineers. One might also turn to American institutionists for an analysis of overhead costs. Issues of joint production and the resulting accounting conventions have something to do with why industrial firms often adopt administrative pricing.
An analysis of joint production also presents an opportunity to construct more examples of Sraffa effects, which, of course, encompass more than reswitching. I do happen to have handy an example with fixed capital. This case illustrates that, given technology, a lower interest rate will not necessarily induce firms to operate machinery for a longer number of production periods. Sometimes the cost-minimizing technique at the lower interest rate mandates that the firm junk old machinery sooner.
I do not see why mainstream economists cannot learn price theory. Will what is entailed by intertemporal equilibria or how to analyze depreciation in the Von Neumann model always be a mystery?
Friday, August 17, 2012
|Figure 1: Capitalism|
Before one can criticize a theory, one must first restate it. I take Marx to have:
- Developed a theory of value as an aid to demonstrating that returns to propertied classes exist only through the exploitation of labor.
- Argued that:
- The net national product, when evaluated at labor values, is equal to the net national product, when evaluated at prices of production.
- The total labor value of the commodities expropriated by the propertied classes is equal to the total exchange value of these same commodities, when evaluated at prices of production.
- The rate of profits in the system of labor values is equal to the rate of profits in the system of prices of production.
I take Matias Vernengo, Fabio Petri, and others to be arguing1 over whether or not one must accept (2) to defend the conclusion in (1) that labor is exploited. In particular, many of those working with formalizations of a revived classical theory of value and distribution seem to defend (1) while noting that, in general, all three invariants in (2) cannot hold.2.0 An Empirical Sraffian Defense of the Invariants
I start with another defense of the invariants, closer to Sraffa and different from the defenses that Petri argues cannot stand. Consider large aggregates of commodities mentioned in the invariants: the capital stock used throughout the economy, net national income, the total of all wage goods, luxury consumption bought by the capitalists out of their income, etc. One might expect an individual commodity to be highly capital-intensive2 or labor-intensive. But would not such extreme cases average out in these aggregates? So cannot one assume, as a first approximation at least, that such aggregates have an average capital intensity, in some sense?
Sraffa's standard commodity formalizes this argument. The standard commodity is a commodity of average capital intensity for the production technology expressed in the ruling Leontief input-output matrix. Consider the circulating capital case, in which:
- All production processes produce one commodity as an output, and
- Abstract, homogeneous labor is the only non-produced input for all production processes.
Furthermore, assume that net national output consists of the standard commodity and that wages are measured in units of the standard commodity. Then all of Marx's invariants hold. Labor value accounting seems to be prior to and revealing of fundamental features of value and distribution under capitalism.3
I have a question about this approach. It seems to introduce an empirical element into Marxism where neither Marx nor his followers might accept such an element4. Are claims about exploitation of the worker being the source of profits dependent on how close the composition of national output is to that of the standard commodity? Would the truth or falsity of these claims be altered by technological innovations or change in consumption patterns that result in some aggregate becoming more or less capital-intensive?3.0 The Fundamental Theorem of Marxism
I here consider another rationale for paying attention to labor value accounting, while accepting that all three invariants cannot be expected to hold in general. I refer to the so-called fundamental theorem of Marxism, that profits are positive in the system of prices of production if and only if labor is exploited.
The theorem is perfectly valid in the circulating capital case. But Ian Steedman, quite some time ago, produced an example with fixed capital in which profits are positive even though surplus value is negative5. Mishio Morishima's reaction was to redefine labor values in the case of joint production6. My reaction to this redefinition is much like Petri's to the New Interpretation and the Temporal Single System Interpretation (TSSI). It seems to retain Marx's invariants as uninteresting tautologies while muddying up how labor value accounting can be explanatory of price phenomena7.4.0 Rectangular Input-Output Matrices
I next want to consider a more fundamental mathematical objection to the surplus approach, at least as reconstructed by Sraffa. Under what cases might the Leontief matrix corresponding to prices of production turn out to be non-square8? In other words, when might the number of cost-minimizing processes be more or less than the number of produced commodities? Under these cases, a unique standard commodity does not exist. If the number of processes is less than the number of commodities, the system does not yield a solution for prices of production, given the wage. Furthermore, if the number of processes is more than the number of commodities, the system does not provide a degree of freedom for distribution.
First, consider cases when requirements for use become more important because of the lack of enough processes to specify prices of production, given the wage. Suppose inputs into production include more than one non-produced input (for example, labor and different kinds of land). And suppose the marginal land9 happens to be fully employed (that is, not in excess supply). Then the marginal land may have a positive rent10. Prices of production now have, at least, a second degree of freedom.
At a switch point, the number of cost-minimizing processes is one more than the number of produced commodities. Michael Mandler imposes an arbitrary assumption that labor markets clear in one example. This assumption then results in distribution being fixed at a switch point in the example.
I believe there are other cases of rectangular Leontief input output matrices associated with joint production. The golden rule of growth considers smoothly expanding growth paths in which:
- Prices of production prevail, and
- The rate of profits equals the rate of growth.
As I understand it, a theorem about the Von Neumann model states that the cost-minimizing technique yields a square matrix along such a path. So, I guess, rectangular matrices can arise along such a growth path when the rate of profits differs from the rate of growth.5.0 Conclusion
I have considered above different ways of complicating the story even more. My conclusion is that Marxist political economy should remain a live and exciting field of scholarly research.Footnotes
- The argument extends to what other aspects of Marx's thought depends on labor value accounting. For example, does his doctrine of commodity fetishism still retain an interest without such accounting? How about the distinction between classical and vulgar political economy? I have trouble seeing how historical materialism is implicated in these discussions.
- As measured by labor values or by prices of production at a given rates of profits, for example.
- Notice how under this reading, Sraffa's book, unlike, arguably, the Cambridge Capital Controversies, is not confined to an internal critique of neoclassical theory. By reconstructing classical and Marxist economic theory, Sraffa puts forward an (unmet) external critique of neoclassical theory.
- I am not saying that Marxist economics cannot be empirically tested or does not have empirical implications. A lot of work has been performed looking at how close labor values and prices of production are to actual prices. And Marx directs one to look at struggles over wages, variations in the quality of wage goods, struggles over the length of the working day and working conditions, the formation of industrial reserve army, etc.
- Gustavo Lucas and Franklin Serrano have recently commented on Steedman's example.
- Under joint production, the output of some production processes consists of more than one commodity. Fixed capital and non-produced land-like natural resources can be analyzed as special cases of joint production.
- John Roemer has proposed an even more radical definition of exploitation, using game theory concepts and, I guess, dropping labor value accounting.
- One can consult the work of, for example, Christian Bidard, Michael Mandler, and Bertram Schefold to find quite different perspectives on these issues.
- Which kind of land is marginal is determined endogenously.
- I am not at all sure that this corresponds to the case of Marx's absolute rent. Anyways, if one accepts the existence of another degree of freedom here, has one located another potential contradiction between Volumes 1 and 3 of Capital?
Monday, August 13, 2012
I think of Deane as a historian of economics, based on her 1978 book The Evolution of Economic Ideas. In it, she portrays economics as a succession of struggles between competing paradigms, in Thomas Kuhn's sense of the word. Economics was in a pre-paradigm state before Adam Smith. She contrasts this approach to history with an approach emphasizing refinements of analysis, as in Joseph Schumpeter's posthumous history.But I want to point out another role she took on. Her book was the first in the Modern Cambridge Economics series. And she was co-editor, with Joan Robinson, of that series. She was later co-editor with Geoffrey Harcourt and Jan Kregel. As of Asimakopulos 1991 book, the series consisted of:
- Phyllis Deane, The Evolution of Economic Ideas
- Joan Robinson, Aspects of Development and Underdevelopment
- A. K. Bagchi, The Political Economy of Underdevelopment
- Éprime Eshag, Fiscal and Monetary Policies and Problems in Developing Countries
- Michael Ellman, Socialist Planning
- Colin Rogers, Money, Interest and Capital
- A. Asimakopulos, Keynes's General Theory and Accumulation
"The modern Cambridge Economic series...is designed in the same spirit and with the similar objectives to the series of Cambridge Economic Handbooks launched by Maynard Keynes soon after the First World War. Keynes' series, as he explained in his introduction, was intended 'to convey to the ordinary reader and to the uninitiated student some conception of the general principles of thought which economists now apply to economic problems'. He went on to describe its authors as, generally speaking, 'orthodox members of the Cambridge School of Economics' drawing most of their ideas and prejudices from 'the two economists who have chiefly influenced Cambridge thought for the past fifty years, Dr. Marshall and Professor Pigou' and as being 'more anxious to avoid obscure forms of expression than difficult ideas'.
This series of short monographs is also aimed at the intelligent undergraduate and interested general reader, but it differs from Keynes' series in three main ways: first in that it focuses on aspects of economics which have attracted the particular interest of economists in the post Second War World era; second in that its authors, though still sharing a Cambridge tradition of ideas, would regard themselves as deriving their main inspiration from Keynes himself and his immediate successors, rather than from the neoclassical generation of the Cambridge school; and third in that it envisages a wider audience than readers in mature capitalist economies, for it is equally aimed at students in developing countries whose problems and whose interactions with the rest of the world have helped to shape the economic issues which have dominated economic thinking in recent decades.
Finally, it should be said that the editors and authors of this Modern Cambridge Economics series represent a wider spectrum of economic doctrine than the Cambridge School of Economics to which Keynes referred in the 1920s. However, the object of the series is not to propagate particular doctrines. It is to stimulate students to escape from conventional theoretical ruts and to think for themselves on live and controversial issues."-- Joan Robinson and Phyllis Deane
Sunday, August 05, 2012
"... that cooperation must be of a certain kind in order to be effective. It must, for example, obey the principles of discourse ethics. Where there is no opportunity to challenge accepted hypotheses by criticizing the evidence upon which their acceptance was based, or the application of the norms of scientific inquiry to that evidence, or by offering rival hypotheses, and where questions and suggestions are systematically ignored, then the scientific enterprise always suffers. When relations among scientists become relations of hierarchy and dependence, or when scientists instrumentalize other scientists, again the scientific enterprise suffers." -- Hilary Putnam (1995).
Most mainstream teaching in economics has been known for generations to be some combination of incoherent, empirically false, and theoretically unfounded. A central problem posed by economics these days is to explain how and why so many economists persist for so long in promoting ignorance and error. Historians of ideas, philosophers of science, and sociologists might explore this issue; and some have.1.
In some sense, we are not having a debate2 on the correctness or not of particular theories or propositions in economics. Generally speaking, mainstream economists3 do not read the critical literature4, do not respond to it, and do not attempt to accurately represent it. I consider it rude to continually spout lies and nonsense, whatever tone one may adopt. In other words, mainstream economists are socialized to be unable and unwilling to conform to civilized discourse ethics5. Should those who talk about economists adopt a tone where they pretend that this is not so?6
Perhaps it distracts from my point to give examples of propositions mainstream economists refuse to teach. Nevertheless, I will briefly list some in micoeconomics7:
- Manufacturing firms do not face U-shaped cost curves8.
- Pricing for the output of such firms is explained by theories of administrative (also known as full-cost) prices.
- Wages and employment cannot be explained by the supply and demand for labor.
- People are not utility-maximizers9.
- Pareto optimization is not a value-neutral maxim for evaluating policy proposals.
- General equilibrium theory is not a description of a decentralized capitalist economy in which commodities are purchased and sold with and for money.
- Equilibrium prices are not indices of relative scarcity10.
- Contemporary practitioners in these disciplines tend to be descriptive, not prescriptive.
- Frederick Lee and Matias Vernengo, on one side, and David Colander, Richard Holt, and J. Barkley Rosser, on the other are having a debate. Their thesis, polemically stated is: Resolved: Heterodox economists should knowingly teach as fact and develop ideas that they believe to be false so as to have professional standing and an influential audience for the true propositions that they may be able to advance from time to time.
- I can think of exceptions.
- I am talking about more than the heterodox literature. Studies show that heterodox economists cite both heterodox and mainstream literature, while mainstream economists do not cite heterodox literature. But what is heterodox about Alan Kirman on the Sonnenschein-Mantel-Debreu theorem or Franklin Fisher on the stability of General Equilibrium? Those are merely two examples of mainstream critical literature that challenge broad categories of mainstream teaching.
- Notice that Krugman adopted behavior in his debate with Keen that he rightfully castigates in the Republican party.
- I can see how others may answer this question differently. A candid look at the history of economics might lead you to wonder how that will work out for you.
- Benicourt and Guerrien (2008), Lee and Keen (2004), and Varoufakis (2011) are, to some extent, surveys of such mistakes. I want to remember to look up these lecture and slides from James Mirrlees.
- Milton Friedman wrote his incoherent essay on method specifically to rule out the early empirical evidence on this point. Lawrence Boland published an essay in 1979 considering why mainstream textbook authors recommended the methodological approach of Friedman and ignored the many refutations in the literature.
- I am sure that you can find some discussion in undergraduate textbooks of behavioral laboratory experiments. How has the take-up been on theories of divided minds and multiple selves?
- Daniel Hausman wrote a book on the Cambridge Capital Controversy in 1981 that addresses my question. I remained amused by the mainstream ignorance of what the mainstream defense concluded in that supposed controversy.
- Benicourt, Emmanuelle and Bernard Guerrien (2008). Is Anything Worth Keeping in Microeconomics? Review of Radical Political Economics, V. 40, N. 3 (Summer).
- Lee, Frederic S. and Steve Keen (2004). The Incoherent Emperor: A Heterodox Critique of Neoclassical Microeconomic Theory. Review of Social Economy, V. 62, N. 2 (June).
- Varoufakis, Yanis (2011). The Dance of the Meta-Axioms: On the Dynamic Mechanism by Which the Inescapable Theoretical Failures of Neoclassical Economics Reinforce Its Dominance.