Elsewhere

• A blog for a close and critical reading of Mas-Colell, Whinston, and Green. MWG is the most dominant introductory graduate microeconomics textbook.
• Alan Kirman calls for a paradigm shift in economics. (Hat tip to Lars Syll.)
• A NOAA FAQ: Why don't we try to destroy tropical cyclones by nuking them?

What Is Mathematics - And Sraffa

An Unsurveyable Rule For Generating A Real Number In Binary Format

Noah Smith offers a definition: "Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules." ("Unlearning Economics" has also recently written on mathematics in economics). To me, the manipulation of meaningless symbols is a powerful form of reasoning. Taking this definition as is, I think two questions can be raised here:

• What is the interest that mathematicians find in these rules and these symbols in the historical circumstances current at the time?
• What does it mean to follow a rule?

Ludwig Wittgenstein is the philosopher most known, I think, for raising the question of what it means to follow a rule. Any summary of his views will be controversial, but I suppose one can fairly say that he adopted an anthropological point of view, at least for some purposes. Describing how to follow a rule by another rule raises the prospect of an infinite regression. Rather, one might show how people do actually follow a rule, how these uses and practices work pragmatically in some form of life. I find it difficult to see how such description conveys the logical must, so to speak, of many rules. But Wittgenstein was alive to this difficulty. He notes that a judge does not seem to treat a statute book as a manual of anthropology.

Furthermore, Wittgenstein spent quite some time in elaborating how these ideas relate to the philosophy of mathematics. His views on the foundations of mathematics seems to have been constructivist and included questioning whether mathematics needs a foundation. Wittgenstein has frequently been labeled an anti-foundationalist. From this viewpoint, one might question whether existence proofs that do not specify how to construct the relevant object can be reformulated. And one even ends up doubting the meaningfulness of defining the real numbers as, say, any set isomorphic to a set of certain equivalence classes of Cauchy-convergent sequences of rational numbers. The use of the notion of infinity remains, I guess, as a standard topic in the philosophy of mathematics.

It seems one of my favorite economists, Piero Sraffa, was an important stimulus in Wittgenstein's development of these views. Sraffa has been said to have led Wittgenstein to see the importance of an anthropological point of view. Sraffa's masterpiece, The Production of Commodities by Means of Commodities: A Prelude to a Critique of Economic Theory, is written in a unique style, not less in the presentation of the mathematics underlying the economics in the book. Sraffa frequently provides outlines of algorithms for constructive existence proofs, maybe most famously for the Standard Commodity. So Sraffa and Wittgenstein might be said to have shared a certain attitude to the philosophy of mathematics, although I do not expect to ever see oral discussions on this topic to be well documented. Sraffa's book can also be said to address only a limited range of topics in economics. An earlier statement of his seems to suggest that he thought room should exist in economics for non-formal treatment of some topics:

"The causes of the preference shown by any group of buyers for a particular firm are of the most diverse nature, and may range from long custom, personal acquaintance, confidence in the quality of the product, proximity, knowledge of particular requirements and the possibility of obtaining credit, to the reputation of a trademark, or sign, or a name with high traditions, or to such special features of modelling or design in the product as - without constituting it a distinct commodity intended for the satisfaction of particular needs - have for their principal purpose that of distinguishing it from the products of other firms. What these and the many other possible reasons for preference have in common is that they are expressed in a willingness (which may frequently be dictated by necessity) on the part of the group of buyers who constitute a firm's clientele to pay, if necessary, something extra in order to obtain the goods from a particular firm rather than from any other." -- Piero Sraffa (1926). "The Laws of Returns Under Competitive Conditions", Economic Journal (Dec.): pp. 544-545.

Whatever you think of the speculations in this post, I think some conclusions are nearly inarguable. Advocates and opponents of the use of mathematics in economics do not neatly divide between mainstream and non-mainstream economists. In particular, one important non-mainstream economist, Piero Sraffa, demonstrated one approach to mathematical economics, while still being aware of the limits to formalism in economics. Furthermore, any comprehensive scholarly study of the philosophy of mathematics will necessarily look at his work as long as Wittgenstein's later views are considered germane to such scholarship.

A Student's Recollection

More than two decades ago, I took a course in intermediate microeconomics. The textbook was R. Robert Russell and Maurice Wilkinson's Microeconomics: A Synthesis of Modern and Neoclassical Theory. "Modern", in this case, refers to the use of set theory terminology, linear programming, and proofs like those in an introductory real analysis class. In contrast, "Neoclassical" refers to the use of continuously differentiable functions. In any case, the substance of the theory - which is only one possible theory - is unaffected. (I would not have been clear on this at the time.)

One day, our professor was teaching us about oligopoly and the theory of the kinked demand curve. And, in response to a question, the professor said something like, "This is a theory I actually believe". Yet, in the rest of the classes, when he was teaching us to manipulate utility functions or production functions or to take Lagrangians or whatever, he never expressed an opinion of the empirical applicability of what he was teaching us.

I also recall that our professor made a special effort to teach us input-output analysis one week. This topic was not in the textbook, if I recall correctly. But Leontief was coming to give a lecture (not to our class, but in a big lecture hall, that is, CC308). And our professor wanted us prepared. As it was, Leontief's lecture did not concern the details of input-output analysis, but the complaint that most of then contemporary economics was unconcerned with empirical results. Most economists did not even cast their theory in a form where it could be connected up to empirical data that one might collect.

References

• Wassily Leontief (1982). Academic Economics, Science, New Series, V. 217, N0. 4555 (9 July): pp. 104-107.
• Wassily Leonteif (1983). Academic Economics Continued, Science, New Series, V. 219, No. 4587 (25 February): 904.
• R. Robert Russell and Maurice Wilkinson (1979). Microeconomics: A Synthesis of Modern and Neoclassical Theory, John Wiley & Sons.

A Simple, But Perverse, Neoclassical Model

Figure 1: Less Plentiful Supply of Capital Lowers the Interest Rate

1.0 Introduction

I claim that capital reversing can be a source of instability and interesting dynamics in neoclassical models. I am interested in, for example, the convergence or not of equilibrium paths in models of intertemporal and temporary equilibrium to steady states, but not in tâtonnement dynamics. The ill-behaved nature of many neoclassical models is a challenge in demonstrating this claim.

This post is a start on revisiting these issues. I here outline a simple model of overlapping generations with a simple production model that cannot exhibit reswitching, capital reversing, or even price Wicksell effects. Yet, in this model, a greater willingness among the households to save is associated with a higher interest rate. This is inconsistent with the supposedly intuitive stories told in outdated and exploded neoclassical textbooks.

2.0 The Model

The model describes an economy in which a single commodity, corn, is produced. In this model, corn functions as both the consumption good and as the only capital good. In production, all (seed) corn is used up in producing the harvest; that is, all capital is circulating capital. For my purposes in this post, I want to consider an economy in a stationary state.

The point of these assumptions is not to describe any actually existing capitalist economy. Rather, the point is to demonstrate that neoclassical theory does not justify conclusions commonly made. I suppose you can say that these types of models raise the following empirical question: why do mainstream economists continue to teach, both in the classroom and in policy work, conclusions long exposed as nonsense by their own theory?

2.1 Utility-Maximizing Agents

Suppose the population consists of overlapping generations, as in Figure 2. Each generation lives for two years. In a given year, all members of the generation born at the start of that year work a full year. They are paid their wages at the end of the year. Out of their wages, they consume some and they save the remainder at the going interest rate. They are retired during the second year of their life. At the end of their second year, they consume the remainder of their income and die.

Figure 2: Lifespans of Overlapping Generations

Furthermore, assume that each generation consists of a single individual, also known as an agent. Furthermore, suppose all generations are identically characterized by the following Cobb-Douglas utility function:

U(c₀, c₁) = (c₀)^γ(c₁)^{(1 - γ)}

where c₀ is the bushels corn the agent consumes at the end of the first year of their life, c₁ is the bushels corn consumed at the end of the second year, and

0 < γ < 1

A higher value of γ indicates a lesser willingness to defer consumption and a smaller supply of savings. Let w be the wage, and r the interest rate. Under these assumptions, the agent born in each generation solves the following utility-maximization problem:

Given w, r

Choose c₀, c₁

To Maximize U(c₀, c₁)

Such that c₀(1 + r) + c₁ = w(1 + r)

c_i ≥ 0; i = 0, 1.

The constraint states that the total value of consumption, evaluated at a single point in time, equals the income of the agent, also evaluated at the same point in time. The solution to this mathematical programming problem is:

c₀ = γ w

c₁ = (1 - γ) w(1 + r)

S = (1 - γ) w

where S is the bushels corn saved at the end of each year.

2.2 Production

For simplicity, I assume a Leontief, fixed coefficients production function. Let L be the person-years of labor employed during the year, K be the bushels corn used as capital during the year, and q be the bushels corn produced during the year. The production function is:

q = min( L/a₀, K/a₁)

where:

a₀ > 0

0 < a₁ < (1/2)

(Productivity has to exceed a certain threshold for an equilibrium to exist in this model.)

Only consider cases where both constraints bind. In a stationary state, the corn available at the end of the year is divided up into a₁/a₀ bushels to use as capital next year and (1 - a₁)/a₀ corn to consume, per person-year employed.

Given this technology, the wage-rate of profits frontier is easily expressed:

a₁(1 + r) + a₀w = 1

Hence, one can solve for the wage as a function of the interest rate and the coefficients of production:

w = [(1 - a₁)/a₀] - (a₁/a₀)r

When the interest rate is zero, the wage is (1 - a₁)/a₀, that is, the total surplus of corn, after subtracting the seed corn needed to sustain production at the same level. When the wage is zero, the interest rate is (1 - a₁)/a₁.

2.3 Equilibrium

This model is completed by assuming that the households want to hold the capital stock at the end of every year. since only one generation is saving for retirement at the end of this year, this equilibrium condition is:

S = a₁/a₀

I might as well make an aside on marginal productivity. In models in which the firms choose the cost-minimizing technique, marginal productivity conditions are used to specify the coefficients of production. The price of each commodity used as a capital good is equal, in equilibrium, to the present value of the marginal product of that commodity. In models in which the technology is specified as a set of fixed-coefficient techniques, the value of marginal product, as I understand it, is an interval in which left-hand and right-hand derivatives enter. In any case, since prices and the quantities of capital goods are both found by solving the model, one cannot say that the (rental) price of a capital good is determined by its marginal product. Furthermore, wages are not determined by the marginal product of labor. A fortiori, the rate of profits is not determined by the marginal product of finance capital, even if one can concoct some equation involving the return on capital, some measure of the value of capital goods, and its marginal product.

Anyways, one can solve the above model to find the following closed-form expression for the interest rate in a stationary state:

r = [(1 - a₁)/a₁] - [1/(1 - γ)]

Figure 1 above graphs this function. And one can see that, in this model, a stationary state in which households are less willing to save is associated with a lower interest rate. If the interest rate were the price of capital and prices were indices of relative scarcities, this example could not be created. But equilibrium prices are not scarcity indices and neoclassical economics, as taught by most university professors, is nonsensical poppycock.

3.0 Conclusion

This post has presented a simple neoclassical model, a limit point, in some sense, of the kind of model that neoclassical economists advocated as a resolution of the Cambridge Capital Controversies. And this simple model shows that much of mainstream teaching and policy work is theoretically unfounded, by their own logic.

Nick Rowe Teaching Miasma Theory Of Plague...

...and other outdated blatherskite:

"An increase in desired saving will only affect the rate of interest slowly, over time, as the greater flow of investment slowly increases the stock of capital and reduces MPK [Marginal Product of Capital]."

"If people want to save more, the rate of interest will fall, the price of capital goods will rise, and there will be a movement along the PPF as existing resources move away from producing consumption goods towards producing investment goods." -- Nick Rowe

(Some have tried to explain.)

Reproducing Civil Society

"Civil Society - an association of members as self-subsistent individuals in a universality which, because of their self-subsistence, is only abstract. Their association is brought about by their needs, by the legal system - the means to security of person and property - and by an external organization for attaining their particular and common interests." -- G. W. F. Hegel, The Philosophy of Right

"...in the case of the most advanced States, ...civil society has become a very complex structure and one which is resistant to the catastrophic incursions of the immediate economic element (crises, depressions, etc.). The superstructures of civil society are like the trench systems of modern warfare. ... In Russia the State was everything, civil society was primordial and gelatinous; in the West, there was a proper relation between State and civil society, and when the State trembled a sturdy structure of civil society was at once revealed. The State was only an outer ditch, behind which there stood a powerful system of fortresses and earthworks: more or less numerous from one State to the next, it goes without saying - but this precisely necessitated an accurate reconnaissance of each individual country." -- Antonio Gramsci, Prison Notebooks

There exist at least two approaches to economics:

• One focused on the allocation of given scarce resources among alternative ends.
• One focused on the conditions for the reproduction of society.

The first is the approach of the so-called neoclassical theory, and the second is the approach of classical political economy.

I like to write about price theory, a field in which one can formulate certain definite quantitative relationships. But the investigation of conditions that facilitate the reproduction of society can extend well outside of price theory and even of economics, as the above Gramsci quote suggests. I suggest the following are examples of components of civil society, whatever your definition: churches, labor unions, charities, civic groups, professional societies, and athletic clubs.

I am not at all sure that anybody has drawn attention in the literature to this commonality between the works of Gramsci and Sraffa: both analyzed the conditions for the reproduction of society, one concentrating on political theory and the other on price theory.

Your Opinion Does Not Matter

Figure 1: Bottom Decile Irrelevant To Policy

Figure 2: Top Decile Much More Influential on Policy Than Median

These striking figures are from Martin Gilens (2005). Gilens looks at 1,781 questions from opinion surveys given between 1981 and 2002 and soliciting opinions on policy changes that could be implemented by some combination of the president and the Congress. He codes the question based on whether the policy change was implemented in the four years after the survey was given. As I understand it, for each question he performs a regression based on opinions and income. This allows him to analyze the consistency, for each income percentile, of opinions and policy outcome.

Gilens then looks at questions where people at different income percentiles differ in opinion by at least 8% for his scale. He finds 887 such questions for the 10th and 90th percentile, and 498 questions for the 50th and 90th percentile. As you can see, poor people at the 10th percentile have virtually no impact on policy outcomes, and middle income people at the 50th percentile have only slight impact. Empirically, the United States is a plutocracy.

• Martin Gilens (2005). "Inequality and Democratic Responsiveness", , V. 69, N. 5: pp. 778-796.

Production Takes Time

In his book¹, Piero Sraffa presented the elements of an economic theory that takes into consideration the empirical fact stated in my post title.

Suppose capitalists make decisions on what and how much to produce. They base these decisions partly on what they can expect to sell. As a consequence of these decisions, they have created a stock of capital goods, a set of interindustry flows, and a production of the surplus of commodities.

Suppose one takes the share of labor in this surplus as a datum², at least for the purposes of a theory of value. Questions could then arise: for what set of prices would these decisions of the capitalists be justified³? For what prices would capitalists be willing to hold the produced capital goods? Sraffa provides an answer. The solution of his price equations⁴ are the desired prices of production.

In giving this answer, one is not claiming that these prices will prevail in an economy. The question of whether or not they will prevail is known as the realization problem⁵. An analysis of consumer demand enters in in addressing this problem.

I do not consider myself to be original in anything above⁶, other than, perhaps, in exposition. And this theory has widespread empirical application, both in the form of Leontief's Input-Output analysis and through Keynesian economics.

Footnotes

1. Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory, (1960).
2. One could make other decisions on the givens. One might take expectations about the decisions of the monetary authority as a given. Or one could append the Cambridge equation to this model.
3. Keynes examines a closely related issue on pages 48-50 of The General Theory of Employment, Interest, and Money. Anyway, Sraffa and Keynes's theories share, at least, a family resemblance in that decisions on quantities temporally precede the determination of prices.
4. Since Sraffa presents a system of simultaneous equations, one cannot competently describe his theory as one of uni-directional causation.
5. The realization problem is addressed by Keynes' theory of effective demand.
6. See, for example, Alessandro Roncaglia's Sraffa and the Theory of Prices (1978).

Eigenvectors Generalized?

I have occasionally mentioned some of the mathematics that many find useful in reading Sraffa¹. Here I want to raise a question. Consider the following two problem statements:

Problem 1: Given a vector space V and a linear function A mapping that vector space into itself, find a vector v in V such that the image of v under A is merely the original vector lengthened or shortened. In other words:

A(v) = λ v.

Problem 2: Given a vector space V and two linear functions, A and B, mapping that vector space into itself, find a vector v in V such that the images of v under A and B are two vectors, where one such image is the other vector lengthened or shortened.In other words:

A(v) = λ B(v).

The second problem statement is, in some sense, a generalization² of the first. I know of lots of theory for analyzing the first problem and many application areas³ unrelated to economic models of circulating capital. I do not know of any literature on the second problem outside of mathematical economics and the analysis of joint production. Likewise, I have a name, eigenvector, for a solution to the first problem. But I have no such name for a solution to the second. Where, if anywhere, can one find literature on the second problem and other application areas motivating its application?

Maybe this is a question for math overflow.

Footnotes

1. In such discussions, I usually do not worry about whether or not the mathematics on which I draw is constructive. Arguably, Sraffa insisted that his proofs be constructive. This topic should be of interest to Wittgenstein scholars.
2. Singular values are one generalization of eigenvalues. My question relates to another generalization.
3. Since I stated the problem so as not to be limited to finite-dimensional vector spaces, one such application area is Fourier analysis.

Elsewhere

• Jacob Hacker and Nate Loewentheil, under the sponsorship of various labor unions and think tanks, have produced a political program, Prosperity Economics: Building an Economy for All.
• Nick Rowe continues a theme - he finds price theory, including the Cambridge Capital Controversies, incomprehensible. (For what it is worth, I have a number of examples of Cambridge models which are closed by assuming utility maximization, including intertemporally.)

Ayn Rand: Too Stupid To Be A Philosopher

"It cannot be the case that the only universally valid norm refers solely to discourse. It is, after all, possible for someone to recognize truth-telling as a binding norm while otherwise being guided solely by 'enlightened egoism.' (This is, indeed, the way of life that was recommended by the influential if amateurish philosophizer - I cannot call her a philosopher - Ayn Rand.) But such a person can violate the spirit if not the letter of the principle of communicative action at every turn. After all, communicative action is contrasted with manipulation, and as such a person can manipulate people without violating the maxims of 'sincerity, truth-telling, and saying only what one believes to be rationally warranted.' Ayn Rand's capitalist heroes manipulated people all the time (even if she didn't consider it manipulation) via their control of capital, for example. Indeed, the person who says, 'do what I want or I'll shoot you,' need not be violating any maxim concerned solely with discourse. But it would be a mistake to use such examples as objections to Habermasian 'discourse ethics.'" -- Hilary Putnam, The Collapse of the Fact/Value Dichotomy (Harvard University Press, 2002)

See also Lars Syll and, for amusement, Will Wilkinson.

Sraffa Solves Marx's Transformation Problem

"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.)

Peak And Off-Peak Electricity As A Joint Product (Continued)

Figure 1: A Photo Probably From The Same Week I visited A Joint Production Process

1.0 Intrduction

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.

Peak And Off-Peak Electricity As A Joint Product

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

1. Bertram Schefold has written much on this theme, including on applied problems.
2. 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.
3. See the second law of thermodynamics.

Nick Rowe On Reswitching And On Joint Production

1.0 Introduction

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.

6.0 Conclusion

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?

How To Attack Marx's Theory Of Value

Figure 1: Capitalism

1.0 Introduction

Before one can criticize a theory, one must first restate it. I take Marx to have:

1. Developed a theory of value as an aid to demonstrating that returns to propertied classes exist only through the exploitation of labor.
2. 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

1. 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.
2. As measured by labor values or by prices of production at a given rates of profits, for example.
3. 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.
4. 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.
5. Gustavo Lucas and Franklin Serrano have recently commented on Steedman's example.
6. 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.
7. John Roemer has proposed an even more radical definition of exploitation, using game theory concepts and, I guess, dropping labor value accounting.
8. One can consult the work of, for example, Christian Bidard, Michael Mandler, and Bertram Schefold to find quite different perspectives on these issues.
9. Which kind of land is marginal is determined endogenously.
10. 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?

Phyllis Deane, An Editor Of The Modern Cambridge Economics Series

Matias Vernengo has noted the passing of Phyllis Deane.

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

Here is the original introduction to this series of books:

"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

Not Necessarily. I Could Be Arguing In My Spare Time

"... 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.

Footnotes

1. Contemporary practitioners in these disciplines tend to be descriptive, not prescriptive.
2. 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.
3. I can think of exceptions.
4. 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.
5. Notice that Krugman adopted behavior in his debate with Keen that he rightfully castigates in the Republican party.
6. 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.
7. 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.
8. 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.
9. 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?
10. 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.

Selected References

• 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.

Nick Rowe, Fool Or Knave?

I take the following as given:

• Contemporary heterodox economists are continuing communities that have been around for generations in leading universities. And they are and have been publishing in scholarly peer reviewed journals.
• Many in these communities accept theories of endogenous money, that a central monetary authority cannot control the quantity of money in use in the economy.
• Many in these communities reject the loanable funds theory of interest in all runs. Interest rates cannot be explained by the interaction of supply and demand, savings and investment.
• These perspectives are not taught in mainstream textbooks (which are almost all a matter of shedding darkness).

Nick Rowe demonstrates the truth of the last proposition while pretending the opposite. To appreciate Rowe's refusal to honestly state that he and the mainstream textbooks do not teach the existence of the heterodox perspectives on money I list above, one must read the comments to the linked post. For example, Nick Rowe states, "The idea that banks just act as intermediaries between savers/lenders and spenders/borrowers is about long run equilibrium." Nowhere does he state that this idea is controversial.

One sees a lot of other ignorance and stupidity on display in the comments:

• One W. Peden pretends that a "majority" of heterodox critiques of economics are, like climate change denialism, "by outsiders to the discipline".
• Ian Lippert pretends "there is [an] unified methodology to ... microeconomics", even though commentators earlier in the thread demonstrated that heterodox economists severely criticize microeconomics, just as well as macroeconomics.
• DavidN also pretends that "micro and the various subfields" do not get critiques from the heterodox.
• Stephen Gordon, after being told more than once of specific political programs associated with Modern Monetary Theory (MMT), such as a program for an Employer of Last Resort (ELR), continues to pretend MMT has "No policy implications/recommendation". (For his gross stupidity and illiteracy, he owes the commentator DeusDJ an apology.)

I realize that I should not get angry about those who refuse to engage a large number of their fellow economists and who rudely tell lies, maybe first to themselves, about what economists teach. For there will always be many vulgar fools and knaves.

Amartya Sen And The Second Phase Of The Classical Revival

Adam Smith and David Ricardo exemplify classical economics. Economists lost and forgot many good ideas in classical economics with the advent of marginalism.

The first phase of the classical revival clarified the classical theory of value and distribution. Scholars achieved this clarification by building on Piero Sraffa's work. Amartya Sen, a student of Maurice Dobb, Piero Sraffa, and Joan Robinson, has commented (e.g., Sen 1974, Sen 2003) a couple of times on Sraffa. He's even had something appreciative to say of the labor theory of value.

Ricardo and Smith have different writing styles, with Ricardo more formal and Smith treating a wider range of material. Furthermore, the theory of value and distribution is only a component, albeit an essential one, of classical economics. In Smith, individuals are not exclusively self-interested utility-maximizers. They exhibit a variety of types of motivations, including sympathy and a regard to social conventions. Furthermore, Smith's statements cannot be partitioned into statements of facts and statements of values. Smith's judgements on what is desirable are multidimensional, not based solely on increasing, for example, income per head.

I have been reading Sen's Development as Freedom and related literature. Like Smith, Sen treats a wider ranges of ideas than are dealt with in the classical theory of value. Sen argues that economic development must be considered along many dimensions, not just income per head. Some of the goals he considers, such as the empowerment of women, are good in themselves. But his consideration of the interconnected instrumental roles of many of these goods provides a rich empirical research program. As examples, I cite the role of decreased income inequality in raising life expectancy, the role of democracy of preventing famine even when crops fail, and the role of increased female literacy in decreasing infant mortality for girls.

Vivian Walsh, while drawing on Hilary Putnam, points out that facts, values, and conventions are, as in Smith, impossible to separate in Sen's work. He looks to Sen as exemplifying the "second phase" of a revival of classical economics. This second phase is less formal than the first, and it treats a wider range of issues.

Perhaps a historian of ideas would find it of interest to research Wittgenstein's role in all of this. Putnam drew on the later Wittgenstein in developing his pragmatic anti-dualism. And, of course, Wittgenstein acknowledges Sraffa's influence. Perhaps a study along these lines would strengthen claims of the complementary nature of Sraffa's and Sen's economics.

(I want to recall that I have commented before on a limited amount of Sen's early work on choice. I also want to recall Daniel Little's post on the second phase of the classical revival.)

(It seems to me that Sen's work already answers the concerns raised in some recent discussions of propertarian confusions about freedom.)

References

• Nuno Martins (2011). "The Revival of Political Economy and the Cambridge Tradition: From Scarcity Theory to Surplus Theory" Review of Political Economy, V. 23, N. 1 (January): pp. 111-131
• Hilary Putnam (2004). The Collapse of the Fact/Value Dichotomy and Other Essays, Harvard University Press. [To read.]
• Hilary Putnam and Vivian Walsh (2009). "Entanglement through Economic Science: The End of a Separate Welfare Economics", Review of Political Economy, V. 21, N. 2 (April): pp. 291-297.
• Hilary Putnam and Vivian Walsh (editors) (2011). The End of Value-Free Economics, Routledge. [To read; a collection that includes some of my other references.]
• Amartya Sen (1974). "On Some Debates in Capital Theory", Economica, V. 41, Iss. 163 (August): pp. 328-335.
• Amartya Sen (1997a). "Maximization and the Act of Choice", Econometrica, V. 65, N. 4 (July): pp. 745-779.
• Amartya Sen (1997.) "Dobb, Maurice Herbert", in The New Palgrave: A Dictionary of Economics (ed. by J. Eatwell, M. Milgate, and P. Newman), Macmillan Press.
• Amartya Sen (1999). Development as Freedom, Alfred A. Knopf.
• Amartya Sen (2003). "Sraffa, Wittgenstein, and Gramsci", Journal of Economic Literature, V. 41 (December): pp. 1240-1255.
• Amartya Sen (2005). "Walsh on Sen after Putnam", Review of Political Economy, V. 17, N. 1 (January): pp. 107-113
• Amartya Sen (2010). "Adam Smith and the Contemporary World", Erasmus Journal for Philosophy and Economics, V. 3, Iss. 1 (Spring): pp. 50-67.
• Vivian Walsh (2000). "Smith After Sen", Review of Political Economy, V. 12, No. 1.
• Vivian Walsh (2003). "Sen After Putnam", Review of Political Economy, V. 15, No. 3 (July).
• Vivian Walsh (2008). "Freedom, Values and Sen: Towards a Morally Enriched Classical Economic Theory", Review of Political Economy, V. 20, N. 2 (April): pp. 199-232.