Frugal Science

Carolyn Kormann has an article, Through the Looking Glass, in this week's New Yorker. This article profiles Manu Prakash, a biophysicist at Stanford and his invention of the Foldscope. The Foldscope is a small, foldable microscope, with the case made of paper. It is an example of frugal science. Prakash hopes to make these microscopes widely available to people in third world countries. One impact might be that residents in, say, African countries will be more conscious of disease-causing micro-organisms, since they can now see such. But, it is not clear to me, what the overall impact of this project might be.

Frugal science reminds me somewhat of E.F. Schumacher's "appropriate technology". It seems to me that in the last few years I've read articles about people developing new stoves and toilets without water targeted to have very low cost and for distribution among the global poor. (THose links are the result of googling now - not where I first read about them.) It seems to me solar power now gives isolated communities a capability to have power without being hooked up to an extensive infrastructure. I like to look for hopeful stories.

Obscure Postmodern Language

I try here to outline certain postmodern¹ doctrines that, in a full development, might result in one using obscure terminology. None of this is to say that every postmodern writer using polysyllabic terminology is expressing complicated ideas in the most effective way. Nor do I want to argue that it is impossible to ever write clearly² about (some subset) of these ideas.

People have a tendency towards reification³, towards talking as if certain abstract ideas are concrete realities. For example, they might tend to confuse relationships between people with relationships between things⁴. And people tend to think dualistically, or at least to categorize things into pre-existing categories. And with dividing things into two categories, one may tend to elevate one over the other, or to define the inferior in terms of the negation of the properties of the superior⁵. One might think that these confusions become embedded in our language⁶. It is not as if we have access to a language appropriate for a "view from nowhere", where nature is carved at its joints⁷.

Furthermore, current classifications and fundamental ideas embodied in current language have a history; our current language does not reflect how people always thought. In looking at past patterns of language and governance, one should try not to read our current way of thinking into the past⁸.

One might also think current classifications have a functional relationship to class structure, hegemonic⁹ ethnicities, patriarchal relationships, or whatever¹⁰.

I have deliberately been abstract here. But, I suppose, I might mention some examples. In economics, I think one is confused if one looks at capitalism as catallaxy, that is, purely in terms of market relationships, in which all parties are free. Furthermore, many things have been said to be socially constructed. I think here of money¹¹, race¹², gender¹³, and sex¹⁴.

In fully trying to explicate these ideas, one can be expected to struggle with bewitchments brought about by language. One might look for multivocalities in past texts. How have current suppositions been read into them? How might they be read from a subaltern position? How might language be expanded so as not to deny normalcy to currently marginalized groups? So reasons exist why academics thinking along postmodern trends might express themselves obscurely.

The above is not to say that these ideas cannot be criticized¹⁵.

Update (21 December 2015):

• Am I agreeing or disaggreeing with what Robert Paul Wolff says here?
• Noah Smith has a knee-jerk reaction to postmodernism.
• The blogger with the pseudonym "Lord Keynes" has often complained about left-leaning postmoderns.

Footnotes

1. For purposes of this post, I do not distinguish between deconstruction, post structuralism, various trends in the social studies of science, etc.
2. Richard Rorty is an example of a postmodern philosopher known for clear - but not necessarily easy - writing.
3. The popularity of the term "reification", in postmodern discourse, comes from Georg Lukás.
4. This is how Marx defined commodity fetishism.
5. I am thinking of how Simone de Beauvoir, early in The Second Sex, describes women being defined as the Other.
6. Here I point to Ludwig Wittgenstein's later work, unpublished in his lifetime.
7. I guess this relates to Jacques Derrida's claim, "There is no outside the text."
8. Michel Foucault, in particular, offers provocative studies of changing European thought in the classical age, between the Renaissance and the nineteenth century.
9. The popularity of the term "hegemony", in postmodern discourse, comes from Antonio Gramsci.
10. As Marx said, "The ruling ideas are the ideas of the ruling classes."
11. This is an example of how something can both be socially constructed and real. Obviously, money has quite real effects in modern societies.
12. Think of the use of the words "Black" and "Colored" in South Africa and in the USA. In the former, they are not synonyms, while among older Americans of a certain sort, they are.
13. I gather Judith Butler originated the concept of gender as performative.
14. Judith Butler also questions whether sex is necessarily a biological division. People might be classified based on chromosomes, hormones, genitalia, and secondary sex characteristics. More than two categories exist in many of these classifications, and they do not always line up. Philip Mirowski observes somewhere that, for the International Olympic Committee (and the International Association of Athletics Federations), these classifications are a quite practical issue. After all, they are structured to find exceptional humans.
15. For explicit references below, I only give critiques. I am sympathetic to the idea that the popularity of postmodernism among academics was connected to an inability to successfully improve material conditions for many.

References

• Samir Amin (1998). Spectres of Capitalism: A Critique of Current Intellectual Fashions, Monthly Review Press.
• Terry Eagleton (1996). The Illusions of Postmodernism, Blackwell.

Keynes On Rational Expectations And Policy Ineffectiveness

John Maynard Keynes' famous saying, "In the long run we are all dead", is from Chapter III of A Tract on Monetary Reform. He describes, in Chapter II of this 1924 book, how governments can obtain resources from their citizens through a deliberate policy of inflation. In this sense, inflation is like taxation. He also discusses how people might react to such a policy, making it difficult for the government to "tax" at the same rate without constantly raising the rate of inflation.

In Chapter III, Keynes states a general principle:

"...a large change in [the quantity of cash], which rubs away the initial friction, and especially a change in [the quantity of cash] due to causes which set up a general expectation of a further change in the same direction, may produce a more than proportionate effect on the [price level]. After the general analysis of Chapter I. and the narratives of catastrophic inflations given in Chapter II., it is scarcely necessary to illustrate this further, - it is a matter more readily understood than it was ten years ago. A large change in [the price level] greatly affects individual fortunes. Hence a change after it has occurred, or sooner in so far as it is anticipated, may greatly affect the monetary habits of the public in their attempt to protect themselves from a similar loss in future, or to make gains and avoid loss during the passage from the equilibrium corresponding to the old value of [the quantity of cash] to the equilibrium corresponding to its new value. Thus after, during, and (so far as the change is anticipated) before a change in the value of [the quantity of money], there will be some reactions on the values of [the parameters of the quantity equation in Keynes' Cambridge formulation], with the result that changes in the value of [the price level], at least temporarily and perhaps permanently (since habits and practices, once changed with not revert to exactly their old shape), will not be precisely in proportion to the change in [the quantity of cash]." -- J. M. Keynes, pp. 81-82.

It seems to me that the above is the Lucas critique, but with a more realistic understanding of human behaviour. What exactly did Lucas contribute again?

Herbert Scarf (1930-2015)

Herbert Scarf died this 15th of November. I think of Scarf as the economist who first demonstrated that general equilibria need not be stable. Something more, some special case assumption or another approach entirely, is needed.

From his Wikipedia page, I learned that have been exposed to more of Scarf's work than I knew. Long ago I took a course in Operations Research, in which we were taught queuing theory and how to find policies for optimal inventory management. Apparently, that approach to the study of inventory policies comes from Scarf.

I did not find the New York Times obituary enlightening. I wish they had mentioned that his algorithm was for finding so-called Computable General Equilibrium (CGE). I have never quite got CGE models. The ones I have seen do not have the dated commodities of the Arrow-Debreu model of intertemporal equilibrium. I have never been sure that they really belong with that tradition, or, like Leontief's model, really fit with a revival of classical economics. Perhaps they are an example of temporary equilibria, as put forth by J. R. Hicks in Value and Capital.

Quite some time ago, Rajiv Sethi discussed Duncan Foley's appreciation of Scarf as a teacher.

Update: Barkley Rosser provides some comments on Scarf (hat tip to Blissex). Here is an obituary from the blog, Leisure of the Theory Class.

(Unrelated to the above, Cameron Murray recently comments on economists confusion about what is meant by "capital".)

"Those to whom evil is done/Do evil in return"

"...I spent the evening walking round the streets, especially in the neighbourhood of Trafalgar Square, noticing cheering crowds, and making myself sensitive to the emotions of passers-by. During this and the following days I discovered to my amazement that average men and women were delighted at the prospect of war. I had fondly imagined what most pacifists contended, that wars were forced upon a reluctant population by despotic and Machiavellian governments. I had noticed during previous years how carefully Sir Edward Grey lied in order to prevent the public from knowing the methods by which he was committing us to support France in the event of war. I naïvely imagined that when the public discovered how he had lied to them, they would be annoyed; instead of which, they were grateful to him for having spared them the moral responsibility..."

Meanwhile, I was living at the highest emotional tension. Although I did not foresee anything like the full disaster of the war, I foresaw a great deal more than most people did. The prospect filled me with horror, but what filled me with even more horror was the fact that the anticipation of carnage was delightful to something like ninety percent of the population. I had to review my views on human nature. At that time I was wholly ignorant of psychoanalysis, but I arrived for myself at a view of human passions not unlike that of the psychoanalysts. I arrived at this view in an endeavour to understand popular feeling about the War. I had supposed until that time that it was quite common for parents to love their children, but the War persuaded me that it is a rare exception. I had supposed that most people liked money better than almost anything else, but I discovered that they liked destruction even better. I had supposed that intellectuals loved truth, but I found here again that not ten per cent of them prefer truth to popularity. Gilbert Murray, who had been a close friend of mine since 1902, was a pro-Boer when I was not. I therefore naturally expected that he would again be on the side of peace; yet he went out of his way to write about the wickedness of the Germans, and the superhuman virtue of Sir Edward Grey. I became filled with despairing tenderness towards the young men who were to be slaughtered, and with rage against all the statesmen of Europe. For several weeks I felt that if I happen to meet Asquith or Grey I should be unable to refrain from murder. Gradually, however, these personal feelings disappeared. They were swallowed up by the magnitude of the tragedy, and by the realization of the popular forces which the statesmen merely let loose.

-- Bertrand Russell (1951). The Autobiography of Bertrand Russell: The Middle Years: 1914-1944

Update to my Paper on Pension Capitalism

I have updated my paper, "A Neoclassical Model of Pension Capitalism in Which r > g". Changes include:

• Deletion of the claim that, in general, inequality increases in a steady state when the real rate of return on finance exceeds the rate of growth.
• Deletion of states of portfolio indifference, in which the real rates of return on money and on bonds are equal, from the model.
• Addition of illustrations of the solution to the (nonlinear) model with some graphs of some state variables along dynamic equilibrium paths.
• Inclusion of a description of one method for finding such solutions numerically.
• Many minor corrections and rewording.

In general, I try to write papers so anybody, including me several months hence, can follow all the details all they want. I realize in submissions to publication, my appendices would have to be drastically shortened or deleted altogether. My typesetting of the mathematics in this paper needs modification, but it is kind to those with old eyes.

Feels Like Science

Figure 1: Evolution of Two State Variables along Two Dynamic Equilibrium Paths

I continue to explore a micro-founded macroeconomic model from Frank Hahn and Robert Solow, generalized to allow a positive rate of growth of households. Hahn and Solow put forth this model as a strawman, to show that even with perfectly flexible prices and wages, markets clearing always, and rational expectations, room for government macroeconomic management can arise. In their book, they then move on to consider imperfectly competitive markets, norms for wages, and so on.

A dynamic equilibrium path, in the model, defines the values of three state variables at the end of each time period in the model. One of these state variables, the real quantity of money in circulation is easily calculated from the other two. The other two, taken here as the real rate of return on corporate bonds and on money, must be found, in general, by solving a recursive system of two equations at each point in time. I found the code I wrote for this post helpful here.

Figure 1 illustrates the evolution of two state variables for two dynamic equilibrium paths. (The model parameters are β = 2/5, ξ = 2.11, and G = 2. The household utility function is of the form specified by Example 1 in Hahn and Solow, with ε = -1/2.) The stationary, dashed-line, path is for a steady state, which is asymptotically approached by the other dynamic equilibrium path. The oscillations seen in this approach are not in the linear approximation about the steady state. One might view these oscillations as a decaying business cycle. One should be clear, however, that even though economic output varies along such a path, neither unemployment nor disappointed plans arise in this model. Households foresee all future variations in prices and quantities along a dynamic equilibrium path.

One could add various complications to make the model more realistic. Households could live for multiple periods more than two, thereby perhaps modifying the time period for the business cycle. One could add leisure into the utility function and model the supply of labor as the result of trading off the earning of wages for consumption and leisure. Employment would then vary along a business cycle; in this theory, recessions are long vacations. One could add noise terms, from known probability distributions, for various terms. So agents would be continually adjusting their plans to accommodate realizations of stochastic processes. One could add imperfect competition, as modeled by Avinash Dixit and Joseph Stiglitz. I suppose one could describe the parameters of utility functions as lying along a continuum, therefore adding a sort of diversity in the model of households. And so on.

I suppose one would find it difficult to add all of these refinements at once. So one could empirically compare a basic model with each refinement. And a model with one refinement might fit better here and with another there. Room for technical innovation for modelling then arises. Can you add two or more refinements, perhaps simplified, where others could could only add one before? Can you take a model that previously was only described for a linear approximation and analyze at least some non-linearities (as I do above)?

I gather I have just briefly outlined the direction of research in mainstream macroeconomics over the last third of a century, albeit the freshwater school did not start, I take it, with overlapping generations models and a Clower constraint.

None of these refinements would even hint at an approach to addressing the question of how economies get into equilibrium. At the end of each year, the economy is automatically in equilibrium in the model, and this instantaneous magic has been foreseen for all time. Head of households and managers of firms have no need to learn a model of the economy. Agents never have disagreements among themselves about what is the true model. And they never change their minds about the structure of the model. J. R. Hicks, the inventor of the model of temporary equilibrium, came to see that it was set in logical time, not historical time. In other words, John Hicks chose to ally himself with Joan Robinson on this theoretical point.

Without an acceptable understanding of disequilibria, mainstream economists should be tolerant of polyvocality in methodology. Why should some economists not be exploring models that are not microfounded? Why not consider the impact and evolution of social norms, without first insisting that they they be justified by methodological individualism? I consider some work in complexity and agent based modeling to be of interest along these lines and not even all that non-mainstream.

A Bifurcation Diagram for Hahn and Solow

Figure 1: Bifurcation Diagram for Hahn and Solow, Example 1, Generalized

I have been writing a draft paper, "A Neoclassical Model of Pension Capitalism in which r > g". In my latest iteration, I have developed the bifurcation diagram shown above. This is a generalization for the overlapping generations model, in which the number of households can grow, but specialized to Hahn and Solow's Example 1. Example 1 specifies the form of the utility function.

One can define dynamic equilibrium paths for the model. And given the values of certain parameters, one can locate a steady state in a certain range of parameters. Always being happy to examine a model, whether it can or cannot ever be instantiated in an actually existing economy, I have identified types of steady states and their stability in certain parameter ranges. I was able to establish analytically the boundary between steady Portfolio Indifferent and Liquidity Constrained States. I located the curved dashed and solid lines towards the south east of the diagram through a mixture of analysis and numeric experimentation. This is also true for my identification of types of stability (saddle-point, locally stable, locally unstable).

I do not fully understand the topological variation in flows for the bifurcations that I have identified. I think I understand the bifurcation, shown by the dashed line, in which a steady Liquidity Constrained State loses stability. This bifurcation most likely results from the steady state ejecting a stable or absorbing an unstable two-period business cycle. The former case is analogous to the logistic equation for a parameter a of 3. I can understand the bifurcation in which the steady state disappears in terms of the diagram in this post. But I find it difficult to understand how dynamic equilibrium paths differ across this bifurcation. And I have not previously gone into the details of the analysis of how two dynamic systems - in this case, for Portfolio Indifferent and Liquidity Constrained States are patched together across a bifurcation. But the linked paper illustrates what I have so far.

More complete details are provided in the linked paper. I provide more details than anybody can want in appendices so as to be able to step through the model myself, if I look at this stuff later.

Reference

• Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press

For Technical Discussions Of Cavalry Tactics At The Battle Of Austerlitz?

Figure 1: Steady States As Function Of Effective Return On Savings

1.0 Introduction

I have previously said I am not thrilled about arguments about whether or not assumptions are realistic. In this post, I describe some analysis I have done with a model of a world that does not exist and analysis I may do in the future with some variation on such a world. The title of this post refers to this quote from Bob Solow, talking about how to respond to Robert Lucas and the new "classical" school:

"Suppose someone sits down where you are sitting right now and announces to me that he is Napoleon Bonaparte. The last thing I want to do with him is to get involved in a technical discussion of cavalry tactics at the battle of Austerlitz." -- Robert Solow

2.0 Generalization of Hahn and Solow's Model of Overlapping Generations

I have previously outlined a micro-founded macroeconomic model of overlapping generations, presented in Hahn and Solow (1995). They use this model to show that claims, from new classical economists and their followers, of the desirability of perfectly flexible prices and wages are unjustified, even on their own theory. They do not think of this model as a good empirical description of any actually existing economy. Hahn and Solow present another model as a prototype of the direction in which they thought macroeconomics should have developed.

Hahn and Solow consider case where one household is born at the start of each year. Under their assumptions, a stationary state is characterized by an equality between a certain function of the effective rate of return on savings and certain model parameters:

g(Q) = [ξ/(ξ - 1)] [β/(1 - β)]

The parameter ξ relates to the Clower cash-in-advance contraint. The parameter β is for the aggregate Cobb-Douglas production function. Parameters and the form of the utility function are embodied in the function g.

I consider a slight modification to this model. Suppose the number of households born each year is no longer constant. Specifically, let the number of households born at the start of year t, h_t, grow at the rate G:

h_t = G^t,

where:

G ≥ 1.

I have worked through this model somewhat. A steady state exists if only if the following equality holds for the effective rate of return on savings:

g(Q) = G [ξ/(ξ - 1)] [β/(1 - β)]

Along a steady state growth path, the nominal price of corn declines so as to maintain a constant real money supply. Hahn and Solow also have that the supply of money is a fixed quantity. They need this assumption, I guess, for their abstract discussion of policy responses to a shock to make sense.

3.0 Other Generalizations

Here are some other possible generalizations and explorations one might make to the model:

• Household lives more than two years.
• Endogenous supply of labor, with leisure entering the utility function.
• Introduction of a bequest motive.
• Heterogeneous households.
• Non-homothetic preferences.
• Various specific forms of utility functions.
• Multiple sectors in production, instead of the production of a single good.
• Introduction of fixed capital (with radioactive depreciation), instead of only circulating capital.
• Various specific forms of production functions.
• Introduction of stochastic noise.
• Analysis of reactions to different kind of shocks.
• Introduction of government, foreign trade.
• More detailed analysis of money, finance, and banks.

The above outlines a research program, not necessarily original. Econometricians can go through models in this family in the literature, trying to find the best fit for some time period and country. From what little I know, one can find models with one generalization and not another, or vice versa. A theoretician might want to try to develop a model that combines some generalizations, thereby advancing the field.

4.0 Empirical Applicability of Generalized Model?

This program entails lots of work, some of it empirical. How could an outsider have standing to criticize this approach?

Truthfully, the mathematics is mostly tedious algebra, only not at a high school level because of the length of the derivations. I suppose the concepts I am applying here are deeper than that. Sometimes one gets to the level of high school calculus, what with LaGrangians and all. (If I can develop a fairly comprehensive and interesting bifurcation diagram for some models, I will consider myself to be approaching advanced mathematics.) Some conventional concepts from economics (marginal conditions, excess demand functions, Walras' law, steady states) help organize the approach.

One who has learned the details of such a program might react negatively to criticism. The supposedly unrealistic assumptions you object to are maintained for analytical tractability. Past developments have supposedly shown us how to relax assumptions. One can be confident that future developments will continue to show us how to generalize the models and how to remove more scaffolding, leaving the building untouched. And, if analytical developments, such as tractable models of imperfect competition, lead to widescale changes, we will adopt them if empirical data shows such changes to be warranted.

But are there some assumptions that are untouched by such a program, that are always maintained, and that render all models (admittedly, internally consistent) developed along these lines forever empirically inapplicable?

4.1 How Are Dynamic Equilibrium Paths Found?

Under the assumption of perfect competition, prices and wages are assumed to be flexible. This is assumed to imply that markets in each period instantaneously clear. I do not understand why anybody up-to-date on economic theory should believe this?

4.2 No Keynesian Uncertainty

Households and firms are assumed to know what the usual range of interest rates, for example, will be in 60 years, in only probabilistically. This does not seem to be plausible to me.

5.0 Conclusions

I intend to pursue some generalizations suggested above. (I could be distracted by trying to develop a bifurcation diagram by a Hahn and Solow model in a later chapter.) The point of the mathematics is to tell a story of some fantasy or science fiction world. This sort of project, to me, does not to make empirical claims. Rather I am interested in whether qualitatively similar stories can be told with some complications. Which, if any, generalizations undermine such stories?

Paul Krugman Stumbles

In his editorial in the New York Times this morning (14 September 2015), Paul Krugman writes about Jeremy Corbyn and the British Labour Party. The establishment politicians in Labour are none too happy about Corbyn's victory. Krugman criticizes these establishment politicians for accepting Tory canards on recent economic history in the United Kingdom, with the former Labour government supposedly being at fault. Krugman's concluding paragraph is:

"Beyond that, however, Labour's political establishment seems to lack all conviction, for reasons I don't fully understand. And this means that the Corbyn upset isn't about a sudden left turn on the part of Labour supporters. It's mainly about the strange, sad moral and intellectual collapse of Labour moderates." -- Paul Krugman

I have no comment on the substance of Krugman's editorial. However, when I read "lack all conviction", I hear an echo of W. B. Yeat's poem, "The Second Coming". I have in mind the following lines:

"The best lack all conviction, while the worst
Are full of passionate intensity." -- W. B. Yeats

This allusion, if intended, is backwards from the article. That is, it would suggest that Labour establishment is composed of the best, contradicting the rest of the article.

I do like Krugman's previous allusions to Talking Heads lyrics.

Failure To Replicate Hahn And Solow (1995), Figure 2.1

Figure 1: Stationary States As Function Of Effective Return On Savings

1.0 Introduction

In Chapter 2 of their Critical Essay, Frank Hahn and Robert Solow present an overlapping generations model¹. This model exhibits rational expectations and perfectly flexible wages and prices. Thus, all markets, including the labor market clear. Hahn and Solow argue that even in such a model, unacceptable fluctuations in national income can arise. Room arises, even under these severe assumptions, for a national government to pursue macroeconomic policy.

I am interested in how mainstream models can exhibit counter-intuitive behavior, including bifurcations of steady states and interesting non-steady state dynamics. The endogenous generation of cyclical or aperiodic orbits is among the dynamics in which I am interested. Hahn and Solow suggest that this model can have different numbers of stationary states and can have orbits that fail to converge to stationary states.

I have looked at other models of overlapping generations before. So I thought I would look into Hahn and Solow's model. They provide two examples of specific forms of utility functions for their model. This post documents my reasons for thinking their first example cannot replicate certain qualitative properties of their model that they claim can arise in general.

2.0 Overlapping Generations Model

The model consists of four markets, for a consumer good, for corporate bonds ("real capital"), for money, and for labor. The supply and demands in these markets are generated by two institutions, households and firms. In this section, I basically echo Hahn and Solow's description of their model. I am particularly interested in three parameters, one for the utility function, one for the production function, and the last for characterizing a liquidity constraint.

2.1 Households

Every year, one household is born. Households live two years. During the first year, they supply one person-year of labor, and they are paid their wages at the end of the year. At the end of the first year, they consume some of their wages and save the rest. They are retired and do not labor² during their second year. At the end of the second year, they consume all of their savings, and then die.

Households can save their income in the form of two assets:

• Money, which earns a real return only if prices decline while a household holds it³.
• Corporate bonds, which at the end of each year are paid off with the full (accounting) profits earned by firms.

Households would prefer to hold their savings only in the form of the asset with the larger real return. However, a transactions demand for money is introduced in the form of a Clower cash-in-advance constraint⁴.

Formally, the household born at the start of year t must choose decision variables to solve the following non-linear program:

Maximize u(c_t,t, c_{t,t + 1})

such that:

c_t,t + s_t ≤ w_t

c_{t,t + 1} ≤ Q_ξ(R_t) s_t

c_{t,t + 1} ≤ ξ m_t p_t/p_{t + 1}

The first constraint specifies that the sum of the consumption and savings at the end of the household's first year cannot exceed the wages received by the household at that point in time. The second constraint states that the consumption at the end of the second year cannot exceed savings, accumulated during that year at the effective rate of return on savings, Q_ξ(R_t). The notation for the effective rate of return reflects the dependence of that rate on the real rate of return, R, on corporate bonds and a parameter, ξ, arising in the third constraint. The third constraint is the Clower cash-in-advance condition. The household must hold at least some given fraction (namely, 1/ξ) of the consumption planned at the end of the last period in the form of money during this period⁵, where

ξ > 1

In a state of Portfolio Indifference (PI), the real rate of return for money and for corporate bonds are equal. On the other hand, if households are Liquidity Constrained (LC), they would prefer to hold savings at the higher rate of return provided by corporate bonds, but cannot because of the Clower constraint. The effective rate of return on savings is therefore less than the rate of return on real capital.

2.1.1 Hahn and Solow's First Example

To be a bit more concrete, Hahn and Solow gives two examples of possible forms of the utility function. The first is:

u(c_t,t, c_{t,t + 1}) = (1/α)(c_t,t)^α + (1/α)(c_{t,t + 1})^α

where,

α < 1

Sometimes it is more convenient to express the solution of the household's program in terms of the parameter ε:

ε = α/(α - 1)

2.2 An Aggregate Cobb-Douglas Production Function

The firms are characterized by an aggregate production function⁶. To be concrete, they specify a Cobb-Douglas form:

y_t = (k_{t - 1})^β (l_t)^{β + 1}

where:

0 < β < 1

The wage, the real rate of return on corporate bonds, the demand for labor, and the supply of corporate bonds (also known as the demand for capital) come out of the usual profit-maximizing analysis. The demand for labor is constrained to match the households' supply of one person-year per year. That is, with flexible wages and prices, the labor market is assumed to clear.

3.0 Stationary States

By solving the above model, one can find excess demands, at the end of each year, for the produced commodity, corporate bonds, and money. Along a dynamic equilibrium path, excess demands in all three markets are zero. As I understand it, solving for one state variable, the rate of return on corporate bonds, in each year is sufficient to trace out such paths. Stationary states, if any exist, are found by dropping time indices.

Stationary states are conveniently expressed in terms of the following function.

g(Q) = Q s(Q)

where s(Q) is the stationary state savings found by solving the household's constrained maximization problem and substituting in a wage of unity in the solution⁷.

Exactly one real rate of return, R, corresponds to each each stationary state value of Q, and vice versa. The parameters α and ξ enter into this invertible function. The following equation is a necessary and sufficient condition for a stationary state:

g(Q) = [ξ/(ξ - 1)] [β/(1 - β)]

Figure 1 graphs g(Q) and the Right Hand Side of the above equation for given parameters in Example 1. The horizontal line can be lowered or raised, within a certain range, by varying, β the parameter in the production function, while leaving other curves unchanged. It is a bit more complicated to analyze the effects of varying ξ. α enters into the shapes of the upward-sloping curves. For this example, they all take on a value of 1/2 at Q = 1.

Anyways, Hahn and Solow present a figure showing possible shapes and locations of g(Q). And they comment on the number and types of possible stationary state equilibria. Table 2 summarizes and compares and contrasts their and my results. I have been unable to find an example with two LCS in their example.

Table 1: Number of Stationary States

Hahn and Solow Possibilities	Example 1 Possibilities
None. No PIS, Exactly one LCS. Exactly one PIS, No LCS. Exactly one PIS, two LCS.	None. No PIS, Exactly one LCS. Exactly one PIS, No LCS.

4.0 Conclusion

I was hoping to find a model with multiple equilbria for some subset of the parameter space. Perhaps I have made some simple error in algebra, but I was disappointed to not find such. This post does not say that Hahn and Solow are in error. They do not claim multiple equilibrium can arise for every conventional form of the utility function in their problem. I guess I'll have to focus on their second example⁸.

Update (10 September 2015): I've convinced myself that neither Hahn and Solow's Example 1 or Example 2 can exhibit one PIS and two LCS. The derivative of g(1) is upward-sloping in both cases, unlike in Hahn and Solow's diagram for the case of three equilibria. (I do not see off-hand why Hahn and Solow rule out a case of in which no PIS exists, but two LCS do.)

Footnotes

1. This model is in the style of the macroeconomics that they are criticizing from the inside. Chapter 6 presents a prototype model more in the spirit of how Hahn and Solow think macroeconomics should be pursued. This model is without an exact reduction to microeconomics, with a labor market which is justified by an earlier game-theoretic analysis of social norms, and with imperfect competition in product markets.
2. In other models of overlapping generations, how much labor a household supplies each year is a decision variable.
3. In a stationary state, prices are stationary and money earns a real return of unity.
4. I had not recognized a Clower constraint before. Presumably, it is not original with this book; Robert Clower's work in macroeconomics goes back to at least the 1960s.
5. Hahn and Solow suggest this unrealistic approach to the transactions demand for money can be justified by a deeper analysis.
6. Sometimes economists justify ignoring the Cambridge Capital Controversy on the grounds that there are so many other problems with mainstream economics that one need not focus on capital theory. This model illustrates this claim.
7. This definition only works for homothetic utility functions, another unrealistic assumption justified here by the critical intent of the model.
8. I like that their second household has a parameter for time-discounting for households, anyways.

Reference

• Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press

Paul Romer Gyring In A Cul-De-Sac

Paul Romer continues to display his confusion. In reverse chronological order, you can look here, here, here, here, and so on. Also see Noah Smith.

Romer continues to put forward ever more false dichotomies and other simple-minded logical fallacies. For example, he seems to say economics has a choice between talky, non-scientific political advocacy or rigorous mathematical economics. And he gets his history wrong:

"Over the five decades from 1890 to 1940 (a time when physicists developed mathematical theories of statistical mechanics, quantum mechanics and both special and general relativity) economists avoided the use even of calculus and spent 50 years mired in the confusion spawned by the talky, market-by-market, supply-and-demand-ish approach to economic analysis codified in 1890 in Alfred Marshall's Principles of Economics." -- Paul Romer

I suppose one can be generous and take Romer to be confining himself to Anglo-American economics. Obviously, economists such as Leon Walras, Gustav Cassel, and Frederick Zeuthen were analyzing mathematical models. (As I understand it, Zeuthen was the first to formulate the Walras-Cassel model with inequalities.) And, I guess in this tradition, Abraham Wald, in 1935, provided the first rigorous proof of the existence of a general equilibrium.

But even when restricted to Anglo-American economics, Romer is not quite correct. J. R. Hicks, with his 1939 edition of Value and Capital and earlier papers with R. G. D. Allen, reintroduced General Equilibrium theory into Anglo-American economics, with as many derivatives, matrices, etc. as you please.

Romer's comments about "talkiness" are silly. I would be embarrassed to dismiss a scholar like Fernand Braudel on the grounds that he did not put forth mathematical models, as in physics.

Romer is just as silly on the other side of his false dichotomy. He's seems to think that as long as a model is put forth in terms of valid mathematics, it is rigorous. Here's what he writes about Solow's growth model:

"Robert Solow (a close colleague of Samuelson's at MIT) ... showed how to describe the behavior of an economy in which things did change. By restricting attention to a single type of output, Solow developed a workable framework for talking about changes in wages, the return to capital, and total output." -- Paul Romer

When I read that in context, I thought Romer was just expressing himself badly. This is in the midst of a short overview about Paul Samuelson's contributions to economics, a task I would find Herculean. Maybe Romer knows that Solow's model is, at best, a non-rigorous, rough-and-ready framework for empirical work. But he really does think otherwise, that Solow's model is rigorous:

"Solow's explicit dynamic model of growth based on an aggregate production function was a solid piece of SAGE [Simple, Applied General Equilibrium] theory. After all, if new Chicago and the rest of the profession agree on one part of good theoretical practice, this has to signal something." -- Paul Romer

The above is just false. The rest of the profession do not agree.

What would have to be the case for Solow's model to apply in a world in which more than one commodity is produced? One set of assumptions is that, in some sense, effectively one commodity is produced. At any given time, the capital stock could be disassembled and costlessly transmuted into either any consumption good or any other collection of capital goods, and vice versa. Then, the historical cost of capital goods, the current prices of capital goods, and their present value would not diverge. On the other hand, these costs do diverge in actual economies set in historical time. The above is a summary of a substantive argument from Joan Robinson, who jokingly claimed that neoclassical economists thought of capital goods as meccano sets or ectoplasm.

Romer resolutely refuses to address the substance of either side of the Cambridge Capital Controversy. (And there are other points than the above. Is Romer even aware of the existence of Piero Sraffa or Pierangelo Garegnani?) Instead, he whines about Robinson's tone:

"...the sarcasm and put-downs that were a part of British intellectual life that Solow had to confront in his exchanges with Joan Robinson." -- Paul Romer

And he attacks Joan Robinson's motives:

"In so doing, he used the same techniques that economists from Cambridge England used to attack his model of output as a function of a stock of capital. Joan Robinson probably had the same concern. What will young Samuelson and Solow do with all their maths? Because an aggregate production function might lend support for a marginal productivity theory of the distribution of income, perhaps we should strangle it in the crib." -- Paul Romer

The above is simply ad hominem. Apparently, some have sent email to Romer with similar points. He then cites Roger Backhouse as an authority, while doubling down on the ad hominem.

I suppose I cannot complain about Romer's treatment of Robinson. Romer's knowledge of General Equilbrium theory seems to be lacking, and he treats Frank Hahn and Robert Solow's objections to macroeconomics after Lucas no more seriously. He complains about their tone, but pretends they had no substance to their complaints. Is Romer even aware of Hahn's attempts to integrate money into the Arrow-Debreu model and his outline of the difficulties? Is Romer even aware of the existence of Hahn and Solow's 1995 monograph? To be generous to Romer, I suppose one could say the latter is only of retrospective importance when considering the controversies in macroeconomics in the 1970s.

I might as well conclude with another example of silliness from Romer. Here Romer tries to explain one of Lucas's contributions:

"Then Robert Lucas showed how to add uncertainty to a version of the Samuelson and Diamond models. This let him pin down loose conjectures from Keynes about the role of expectations." -- Paul Romer

Now, Chapter 12 in the General Theory is often turned to when one wants to read Keynes on expectations. And in that chapter, one finds:

"By 'very uncertain' I do not mean the same thing as 'very improbable'. Cf. my Treatise on Probability..." -- John Maynard Keynes (1936, p. 148).

Romer is equivocating. As far as I know, Lucas did not introduce uncertainty in any mathematical models in economics. (Can anybody find Lucas explicitly discussing the inconsistency between rational expectations and non-ergodic time series?) So Romer should either not reference Keynes at all (with silliness about "loose conjectures") or talk about Lucas modeling probability (also known as risk) or expand on his text to show how Lucas was actually modeling Keynes's uncertainty. That is, Romer should if he has any interest in the truth value of his statements.

I think the above is not one of my better posts. Too uniformly negative even for me and too wandering. But I think Romer should try not to commit simple logical fallacies in his complaints about lack of scholarship and rigor among economists.

References

• Braudel, Fernand (). Civilization and Capitalism, 15th - 18th Century, Volume 1: The Structure of Everyday Life.
• Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Macroeconomic Theory, MIT Press.
• Hicks, J. R. (1939). Value and Capital (1st edition).

Labor Reversing Without Capital: An Example

Figure 1: Skilled Labor Hired by Firms per Unit Output

1.0 Introduction

This example is from Opocher and Steedman (2015). They present many examples in which the reader is expected to work them out, as illustrated in this post.

This is an example in which cost-minimizing firms desire to hire more labor (of a specific type) for an increased wage, around a specific wage. This example is of a firm producing a single commodity from inputs of specific types of land and specific types of labor. No produced capital goods exist in this example, and the interest rate is assumed to be zero. Yet perverse behavior arises on the demand side of markets for factors of production anyway - where results are called perverse merely if they violate neoclassical intuitions shown to be mistaken half a century ago. The most complicated aspect of this example is that some techniques of production are specific to specific types of land.

2.0 Indirect Average Cost Functions

Consider a firm that produces widgets from inputs of skilled labor, unskilled labor, and land of one of two types. Suppose the price of widgets is unity. Define:

• p_α is the rent for alpha-type land.
• p_β is the rent for beta-type land.
• w₁ is the wage for unskilled labor.
• w₂ is the wage for skilled labor.

The indirect average cost function for widgets produced on land of type alpha is:

c_α(p_α, w₁, w₂) = (1/2)[(w₁ p_α)^1/2 + (w₁ w₂)^1/2

+ (w₂ p_α)^1/2]

The indirect average cost function for widgets produced on land of type beta is:

c_β(p_β, w₁, w₂) = (3/5)(w₁ p_α)^1/2 + (3/10)(w₁ w₂)^1/2

+ (11/20)(w₂ p_α)^1/2

The indirect average cost function shows the average cost of producing each widget, when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm.

These indirect average cost functions are homogeneous of the first degree. For the indirect average cost function for land of type alpha, this property is expressed as:

c_α(ap_α, aw₁, aw₂) = a c_α(p_α, w₁, w₂)

This a traditional assumption for cost functions.

Consider the indirect average cost function for a specific type of land. That type of land, unskilled labor, and skilled labor are substitutes. No inputs are complements in this example. In other words, the off-diagonal elements of the Hessian matrices formed from each indirect average cost function are all positive. The elements along the principal diagonal of each Hessian matrix are negative.

3.0 The Wage-Wage Frontier

Consider a long run equilibrium of the firms in which pure economic profits have been competed away and no firm is making a loss. Perhaps, the prospect of firms entering or exiting the industry has caused this situation to arise. Furthermore, suppose rents for both types of land happen to be unity. (Without this assumption, this example would have two more degrees of freedom.) If firms are producing on a given type of land, the indirect average cost function for that type of land will be equal to unity. For alpha type land, one has:

1 = c_α(1, w₁, w₂)

Or:

w_{1, α} = [(2 - w₂^1/2)/(1 + w₂^1/2)]²

As shown in Figure 2, given the type of land employed, the wage for unskilled labor is a declining function of the wage for skilled labor. The maximum wage for unskilled labor, 4 widgets per person-year, corresponds to skilled labor working for free. Symmetrically, the maximum wage for skilled labor likewise corresponds to unskilled labor working for free.

Equating the indirect average cost function for production on land of type beta yields another trade-off in long run equilibrium between the wages of unskilled and skilled labor.

w_{1, β} = [(20 - 11 w₂^1/2)/(12 + 6 w₂^1/2)]²

When land of type beta is used, the maximum wage for unskilled labor is 2 7/9. The maximum wage for skilled labor is 3 37/121.

Figure 2: Wage-Wage Curves and the Frontier

For some combination of wages of skilled and unskilled labor, firms will be indifferent between producing widgets with land of type alpha and type beta. The cost-minimizing technique at these wages, on each type of land, is equally cheap. These combinations can be found by equating the wages of unskilled labor for the expressions above. After some manipulation, one obtains the equation:

5 w₂ - 9 w₂^1/2 + 4 = 0

This equation can be factored:

(w₂^1/2 - 1)(5 w₂^1/2 - 4) = 0

Firms will thus be indifferent to the type of land used in production for ordered pairs of wages of unskilled and skilled labor, (w₁, w₂), of (1/4, 1) and (4/9, 16/25).

Firms produce widgets on land of type alpha for wages for skilled labor between zero and 16/25, and for wages of skilled labor between one and four. For wages for skilled labor between 16/25 and four, firms produce widgets on land of type beta. The outer frontier allows one to determine the wage of unskilled labor for any feasible wage for skilled labor, given the model assumptions. As well soon be apparent, this is not an example of reswitching. The overall indirect average cost function is almost always differentiable. It is not differentiable only at the two points found by the construction of the outer frontier.

4.0 Land and Labor

We have seen that when rents are unity, long run equilibrium of the firm necessitates that the wages of unskilled labor is a declining function of the wages of skilled labor. Shepherd's lemma can be used to find the coefficients of production for each feasible combination of wages of unskilled and skilled labor. The quantity of each input the firm wants to hire per unit output is the derivative of the indirect average cost function with respect to the price of that input. Thus, when land of type alpha is used, the number of acres of land employed per unit output of widgets is:

t_α(w₁, w₂) = (1/4)(w₁^1/2 + w₂^1/2)

The number of acres of land of type beta per unit output of widgets, when land of that type is used, is:

t_β(w₁, w₂) = (1/40)(12 w₁^1/2 + 11 w₂^1/2)

In what I hope is obvious notation, person-years of unskilled labor employed per unit output of widgets is, depending on the type of land used:

l_{1, α}(w₁, w₂) = (1/4)(1 + w₂^1/2)/(w₁^1/2)

l_{1, β}(w₁, w₂) = (3/20)(2 + w₂^1/2)/(w₁^1/2)

Finally, person-years of skilled labor employed per unit output of widgets is given by one of the following two functions of wages:

l_{2, α}(w₁, w₂) = (1/4)(1 + w₁^1/2)/(w₂^1/2)

l_{2, β}(w₁, w₂) = (1/40)(6 w₁^1/2 + 11)/(w₂^1/2)

5.0 Bringing it all Together

The above algebra can be used to generate various graphs. Figure 1 shows person-years of skilled labor firms desire to hire per unit output. As one moves to the right in the figure, the wage of skilled labor rises and the wage of unskilled labor falls. But at every point in the figure, the wages of the two types of labor are such as to maintain wages as on the outer frontier in Figure 2. That is, firms are minimizing costs, and the output price and input prices are such as to enforce the equilibrium condition that no pure economic profits are available in this industry.

Figure 3 shows the analogous graph for unskilled labor. The point for wages of 4/9 widgets per person-years and 16/25 widgets per person-year for unskilled and skilled labor, respectively, is emphasized. At any point to the left, wages for unskilled labor are higher, and wages for skilled labor are lower. And an infinitesimal variation around this point is associated with firms wanting to employ unskilled labor more intensively when their wage is relatively higher.

Figure 3: Unskilled Labor Hired by Firms per Unit Output

Reswitching of techniques arises when one technique of production is cost-minimizing at, say, a high and low wage but not at an intermediate wage. A technique of production is specified by four coefficients of production in this example. The amount of skilled labor and unskilled labor hired per unit output are two of these coefficients. The acres of land of each type rented per unit output are the other two. The latter two coefficients of production obviously vary, depending on which type of land can be used in a cost-minimizing technique. In fact, the coefficients of production for the type of land not employed is zero. As can be seen in Figures 1 and 3, the coefficient of productions for the two types of labor vary monotonically with relative wages, given the type of land employed.

At one of the two switch points highlighted in Figure 2, two techniques of production are cost-minimizing. (This is the definition of a switch point.) In one technique, one type of land is used. And in the other, the other type of land is used. But a different pair of techniques of production is cost-minimizing at the other switch point. The coefficients of production vary among, for example, the cost-minimizing techniques in which alpha-type land is used at a switch point. Hence, as noted, no reswitching of techniques exists in this example.

6.0 Conclusion

This example has cost-minimizing firms in equilibrium in a single industry. Price and quantity relationships among factors of production have been analyzed, where factors of production consist of land of two types and labor of two types. Quantity relationships have been presented in terms of inputs per unit output for a firm. For simplicity, only the case in which the interest rate is zero and rents of land per acre are unity has been considered. When beta type land is adopted, more acres are cultivated for alpha-type land, for the same level of output. Thus, land has a higher proportion of total unit cost when beta-type land is used. Both skilled and unskilled labor are a lower proportion of total unit cost (as seen in Figures 1 and 3) than they would be if alpha type land was employed. A wage has been found for unskilled labor in which a higher relative wage for unskilled labor is associated with firms desiring to hire more unskilled labor per unit output. And a different relative wage for skilled labor has been found with the analogous property.

I wonder whether an example can be found with a continuum of types of land in which the analog of Figures 1 and 2 come out as continuous U-shaped curves.

So much for explaining wages and employment by well-behaved supply and demand curves in competitive labor markets.

Reference

• Opocher, Arrigo and Ian Steedman (2015). Full Industry Equilibrium: A Theory of the Industrial Long Run, Cambridge University Press

Locke's Caveats To His Labor Theory Of Property

1.0 Introduction

A couple of months ago, I read John Locke's Second Treatise of Government. He has a caveat on his theory of property I did not expect, as well as one I did. The caveat I did not expect involves the unjustness of acquiring more than you can use and wasting it. But Locke thought it allowable to have more than you can use, as long as you did not waste it. Ultimately, he justifies such superfluous property by claiming it will lead to economic development and benefit the community as a whole.

I suppose one can read Locke as a defense of British and American conquest of the autochthonous peoples in the Americas. They held the land in common, but were not using it well, as least from a bourgeois perspective.

I prefer to draw an analogy to socialism. Property in possessions more than used in everyday living is justified by thinking of that property as held for the benefit of the commonwealth, so to speak. (For the purpose of this post, I put aside any qualms I have about Robinsonades.)

2.0 The Labor Theory of Property

I suppose this is the most famous statement of justification of ownership on the basis of a right to the fruits of one's labor:

Sect 27. Though the earth, and all inferior creatures, be common to all men, yet every man has a property in his own person: this no body has any right to but himself. The labour of his body, and the work of his hands, we may say, are properly his. Whatsoever then he removes out of the state that nature hath provided, and left it in, he hath mixed his labour with, and joined to it something that is his own, and thereby makes it his property.

Locke begins talking about the ownership of food gained through hunting and gathering. Land is held in common. "Thus in the beginning all the world was America" (Sect. 49). Locke then transitions to ownership of land. He argues that in such a more complex society, one can trace back commodities to dated labor embodied over many activities in the past. The labor embodied in bread includes the labor of the baker, the miller, the farmer, the manufacturer of tools for the use of the farmer, etc. This view of embodied labor (in Sections 42 and 43) was later echoed in the labor theory of value.

3.0 First Caveat: "Enough, and as Good Left"

From secondary literature, I knew that Locke justified the enclosure of common lands into private property only if what was left still was as good as before:

Sect. 33. Nor was this appropriation of any parcel of land, by improving it, any prejudice to any other man, since there was still enough, and as good left; and more than the yet unprovided could use. So that, in effect, there was never the less left for others because of his enclosure for himself.

He writes about, for example, the unjustness of denying somebody a drink from a river when the river would still flow on undiminished.

4.0 Second Caveat: No Waste of Superfluity

I was not previously aware that, in justifying private property, Locke condemned wasting more than your share:

Sect. 31. It will perhaps be objected to this, that if gathering the acorns, or other fruits of the earth, &tc. makes a right to them, then any one may ingross as much as he will. To which I answer, Not so. The same law of nature, that does by this means give us property, does also bound that property too. God has given all things richly, 1 Tim. vi. 12. is the voice of reason confirmed by inspiration. But how far has he given it us? To enjoy as much as any one can make use of to any advantage of life before it spoils, so much he may by his labour fix a property in: whatever is beyond this, is more than his share and belongs to others. Nothing was made by God for man to spoil or destroy.

And again:

Sect. 37. ...Before the appropriation of land, he who gathered as much as the wild fruit, killed, caught, or tamed, as many of the beasts, as he could; he that so imployed his pains about any of the spontaneous products of nature, as any way to alter them from the state which nature put them in, by placing any of his labour on them, did thereby acquire a property in them: but if they perished, in his possession, without their due use; if the fruits rotted, or the venison putrified, before he could spend it, he offend against the common law of nature, and was liable to be punished; he invaded his neighbor's share, for he had no right, farther than his use called for any of them, and they might serve to afford him conveniences of life.

This is obviously not a position to end at if you are justifying property rights in the rising bourgeois society. Locke caveats his caveat by arguing that you are not wasting more than your share if you hold the extra in goods that do not waste away quickly, whether they be useful or pretty baubles:

Sect. 46. ...He was only to look, that he used them before they spoiled, else he took more than his share, and robbed others. And indeed it was a foolish thing, as well as dishonest, to hoard up more than he could make use of. If he gave away a part to any body else, so that it perished not uselessly in his possession, these also he made use of. And if he also bartered away plums, that would have rotted in a week, for nuts that would last good for his eating a whole year, he did not injury; he wasted not the common stock; destroyed no part of the portion of goods that belonged to others, so long as nothing perished uselessly in his hands. Again, if he would give his nuts for a piece of metal, pleased with its colour; or exchange his sheep for shells, or wool for a sparkling pebble or a diamond, and keep those by him all his life he invaded not the right of others, he might heap up as much of these durable things as he pleased; the exceeding of the bounds of his just property not lying in the largeness of his possession, but the perishing of any thing uselessly in at it.

Locke argues that the ability to accumulate money through commerce leads to owners developing their property:

Sect. 48. And as different degrees of industry were apt to give men possessions in different proportions, so this invention of money gave them the opportunity to continue and enlarge them: for supposing an island, separate from all possible commerce with the rest of the world, wherein there were but an hundred families, but there were sheep, horses and cows, with other useful animals, wholesome fruits, and land enough for corn for a hundred thousand times as many, but nothing in the island, either because of its commonness or perishableness, fit to supply the place of money; what reason could any one there to enlarge his possessions beyond the use of his family, and a plentiful supply to its consumption, either in what their own industry produced, or they could barter for the perishable, useful commodities, with others? Where there is not some thing, both lasting and scarce, and so valuable to be hoarded up, there men will not be apt to enlarge their possessions of land, were it never so rich, never so free for them to take: for I ask, what would a man value ten thousand, or an hundred thousand acres of excellent land, ready cultivated, and well stocked too with cattle, in the middle of the inland parts of America, where he had no hopes of commerce with other parts of the world, to draw money to him by the sale of the product?

Locke's defense of private property reminds me of Jesus's parable of the talents (Matthew 25). You should use your property for the benefit of humankind:

Sect. 37. ...To which let me add, that he who appropriates land to himself by his labour, does not lessen, but increase the common stock of mankind: for the provisions serving to the support of human life, produced by one acre of increased and cultivated land, are (to speak much within compass) ten times more than those which are yielded by an acre of land of an equal richness lying waste in common. And therefore he that incloses land, and has a greater plenty of the conveniences of life from ten acres, than he could have from an hundred left to nature, may truly be said to give ninety acres to mankind: for his labour now supplies him with provisions out of ten acres, which were but the product of an hundred lying in common...

5.0 Other Subjects in Locke's Tract

All of the above quotes are from Chapter 5 of Locke's Second Treatise. This book contains nineteen chapters, and treats many other topics. These include a recap of the first treatise, which presumably refutes the claims of Sir Robert Filmer that:

• God gave Adam property in all the earth.
• And current monarchs own their countries through their descent from Adam.

Locke praises William III, the victor of the Glorious Revolution. He also treats of the state of nature, in which humans are free. And they give up that freedom in a social contract so as to end the war of all against all. But, if rulers:

become destructive of these ends, it is the Right of the People to alter or to abolish [the government], and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. (America's Declaration of Independence)

I find the start of Chapter 13, as well as other passages (e.g., Sect. 225) echoed in the declaration. Other topics include natural rights, war, slavery, and parental rights.

References

• John Locke, Second Treatise of Government (1690).

Recurrence Of Capital-Output Ratio Without Reswitching

Figure 1: Recurrence of Capital-Output Ratio

1.0 Introduction

This example is from Arrigo Opocher and Ian Steedman. It illustrates the analysis of an isolated industry in equilibrium. This analysis is therefore more akin to partial equilibrium than to general equilibrium. Sometimes (mainstream?) economists say that the Cambridge Capital Controversies were only about aggregate neoclassical theory, that is, macroeconomics. Or that the CCC has been subsumed by General Equilibrium Theory. The example illustrates that such economists are, as has long been apparent, spouting poppycock.

2.0 Indirect Average Cost Function

Consider a firm that produces widgets from inputs of widgets, unskilled labor, and skilled labor. Let the indirect average cost function be:

c(p, w₁, w₂) = sp + w₁ + w₂

+ 2(pw₁)^1/2 + 2(pw₂)^1/2 + 2γ(w₁w₂)^1/2

where

0 < s < 1

0 < γ

γ ≠ 1

and

• p is the price of a widget. Widgets used as inputs are assumed to be totally consumed in one production period.
• w₁ is the wage for unskilled labor.
• w₂ is the wage for skilled labor.

The indirect average cost function shows the average cost of producing each widget (net), when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm.

This indirect average cost function is homogeneous of the first degree:

c(a p, a w₁, a w₂) = a c(p, w₁, w₂)

This is a conventional assumption for cost functions.

Suppose the firm faces a given price of widgets and given wages for skilled and unskilled labor. By Shephard's lemma, the quantity of each input the firm wants to hire per unit output, given the price of each input, is the derivative of the indirect average cost function with respect to the price of that input. Hence, the capital-output ratio, k(p, w₁, w₂), is:

k(p, w₁, w₂) = ∂c/∂p = s + (w₁/p)^1/2 + (w₂/p)^1/2

Notice that the capital-output ratio is a pure number, unambiguously defined in this example, and independent of prices.

By the same logic, the amount of unskilled labor the managers of the firm desire to hire per widget produced is:

l₁(p, w₁, w₂) = ∂c/∂w₁ = 1 + (p/w₁)^1/2 + γ(w₂/w₁)^1/2

The amount of skilled labor the managers of the firm desire to hire per widget produced is:

l₂(p, w₁, w₂) = ∂c/∂w₂ = 1 + (p/w₂)^1/2 + γ(w₁/w₂)^1/2

The matrix of second derivatives of the indirect average cost function is:

(I am not sure whether it is more common to define the above matrix as the transpose of what I have above.) Anyway, for a positive price of widgets and positive wages, the signs of the second derivatives are as follows:

The signs along the principal diagonal show that the slopes of the per-unit input demand functions slope down. That is, given prices for all but one input, a lower price of that input is associated with a willingness of the firm to employ more of that input per unit produced. The positivity of the off-diagonal elements of the above matrix show that widgets, considered as inputs; unskilled labor; and skilled labor are all substitutes, not complements, in some sense. These signs for the matrix of second derivatives of the indirect average cost function are also conventional properties for cost functions.

3.0 Full Industry Equilibrium

Suppose the industry in which widgets are produced has no barriers to entry or exit. Thus, in the long run, economic profits will have been competed away. For firms to be earning no economic profits, the price of widgets must be equal to the average cost of manufacturing them:

p = c(p, w₁, w₂)

So far, no numeraire has been specified. Let widgets themselves be numeraire. Then:

1 = c(w₁, w₂)

where the argument in the indirect average cost function for widgets has been dropped as otiose.

Consider various levels of w₁, the wage of unskilled labor. For the industry to continue to be in long run equilibrium, the wage of skilled labor, w₂, must vary as well, thereby leaving the average cost of producing a widget as unity. Figure 2 illustrates the resulting wage-wage frontier. (Figures are drawn for s = 1/10 and γ = 2/3.) The highest wage for unskilled labor (when the wage for skilled labor is zero) is ((2 - s)^1/2 - 1)². Since this model is symmetric in skilled and unskilled labor, the highest wage for unskilled labor is likewise ((2 - s)^1/2 - 1)². As long as the rate of accounting profits is zero and technology is given, the wage of unskilled labor can only be higher if the wage of skilled labor is lower.

Figure 2: Wage-Wage Frontier

The wage-wage frontier can be used to find the wage of skilled labor for a given wage of unskilled labor between zero and the maximum. In other words, the frontier is helpful in calculating the ratio of the wage of skilled labor to the wage of unskilled labor, given the wage of unskilled labor. This ratio of wages is independent of the choice of the numeraire.

4.0 Capital and Labor

With the chosen numeraire, the capital-output ratio is:

k(w₁, w₂) = s + (w₁)^1/2 + (w₂)^1/2

Given the wage of unskilled labor, one can find the wage of skilled labor and, consequently, both the ratio of wages of the two types of labor and the capital-output ratio. Figure 1, at the start of this post, graphs the capital-output ratio as the derived function of the ratio of wages.

The capital-output ratio is the same when either skilled or unskilled labor is earning their maximum wage, with the other type of labor being paid a wage of zero. In these two extreme cases, the capital-output ratio is (2 - s)^1/2 - (1 - s). Likewise for any ratio but one of the wage of skilled labor to the wage of unskilled labor between these extremes of zero and infinity, the capital-labor ratio is non-unique. The exception is the ratio of wages at which the function in Figure 1 peaks.

One can see that recurrence of the capital-output ratio is not reswitching. Figures 3 and 4 show, respectively, unskilled labor and skilled labor per unit output as a function of the ratio of wages. As shown in Figure 3, a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ more unskilled labor per unit output. Likewise, a a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ less skilled labor per unit output. As far as unproduced inputs go, this example of the isolated firm in long run equilibrium does not contradict outdated and exploded neoclassical intuitions about substitution and the mistaken notion of equilibrium prices as scarcity indices. But, since the functions in Figures 3 and 4 are monotonic, no reswitching of techniques arises in this example.

Figure 3: Unskilled Labor Employed per Unit Output

Figure 4: Skilled Labor Employed per Unit Output

5.0 Conclusion

This post has presented an example of an isolated firm in a long period equilibrium. The indirect average cost function, which includes the cost of the use of an input which itself is produced by the firm's industry, otherwise has utterly conventional properties. The analysis of the firm in a long run equilibrium demonstrates that it is an incoherent thought experiment to consider the equilibrium response of the firm to the variation of one price at a time. Only the variation of more than one price at a time can yield an equilibrium analysis that could be at all empirically relevant.

A result of this analysis is to reveal a non-monotonic response of the capital-output ratio to variations in the relative prices of the two unproduced inputs used by this firm in production. In fact, every possible capital-output ratio, except for one, recurs in the example. This is a step in an argument leading to the conclusion that economic theory is consistent with competitive firms wanting to employ more input per unit output for higher prices of that input, a finding that seems consistent with empirical results.

Election Paradoxes And Faustian Agents

I have been trying to reread Donald Saari on election paradoxes. I have previously considered a few parallels between the Condorcet paradox and models of agents as composed of multiple selves. It seems to me that one could draw more analogies here. I do not plan to pursue the research agenda outlined here - I'm not sure how plausible its results would be. Anyways, Saari provides a comprehensive analysis of a range of voting procedures. Could a fuller range of such procedures - as opposed to pairwise majority rule - be applied to models of multiple selves?

For example, consider a model of a person as having multiple selves, where each one of those selves has a set of preferences over commodities. And suppose the individual, in making choices, resolves those selves with a procedure analogous to an election procedure (e.g., plurality vote, antiplurality vote, Borda Count). Suppose which procedure is used is context-dependent. Can an outside agent modify the context somehow such that the individual follows a different procedure, with consequent effects on the individual's choice?

Or consider two people each composed of the same number of multiple selves, with the preferences of those selves the same across these two people. But suppose each person resolves those selves with a different voting procedure. Maybe these two different voting procedures yield the same "best" choice for one specific menu of choices, but order the non-best choices differently. So if a new menu was created with the best choice removed, these two people - who have identical preferences, in some sense - would make different choices.

I suppose if you follow research along these lines, it would be theoretical research. I do not know how an experiment could elicit the required information to determine the preferences of the multiple selves and the election procedure. I guess the challenge would be to come up with an account consistent with some behavioral anomaly arising in economics experiments. Even better might be to suggest a new experiment and to implement it.

References

• Donald G. Saari (2001). Chaotic Elections! A Mathematician Looks at Voting, AMS.

Bertrand Russell, Crank

On the Post Topic

Some great thinkers compare their work to the works of Nicolaus Copernicus or of Galileo:

"The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds must be abandoned as beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion." -- Bertrand Russell, Our Knowledge of the External World as a Field For Scientific Method in Philosophy (1914).

"...an imagination better stocked with logical tools would have found a key to unlock the mystery. It is in this way that the study of logic becomes the central study in philosophy: it gives the method of research in philosophy, just as mathematics gives the method in physics. And as physics, which, from Plato to the Renaissance, was as unprogressive, dim, and superstitious as philosophy, became a science through Galileo's fresh observation of facts and subsequent mathematical manipulation, so philosophy, in our own day, is becoming scientific through the simultaneous acquisition of new facts and logical methods.

In spite, however, of the new possibility of progress in philosophy, the first effect, as in the case of physics, is to diminish very greatly the extent of what is thought to be known. Before Galileo, people believed themselves possessed of immense knowledge on all the most interesting questions in physics. He established certain facts as to the way in which bodies fall, not very interesting on their own account, but of quite immeasurable interest as examples of real knowledge and of a new method whose future fruitfulness he himself divined. But his few facts sufficed to destroy the whole vast system of supposed knowledge handed down from Aristotle, as even the palest morning sun suffices to extinguish the stars. So in philosophy: though some have believed one system, and others another, almost all have been of opinion that a great deal was known; but all this supposed knowledge in the traditional systems must be swept away, and a new beginning must be made, which we shall esteem fortunate indeed if it can attain results comparable to Galileo's law of falling bodies." -- Bertrand Russell, ibid.

The "new logic" Russell refers to is set out in, for example, Russell and Whitehead's Principia Mathematica. So Russell is comparing himself to Galileo.

An Approach to a Book Review

I'm glad I read this book, although I think it is basically mistaken. Not surprisingly, given their interactions at Cambridge before World War II, Russell's exposition reminds me of Ludwig Wittgenstein's Tractatus Logico-Philosophicus. Although clearly written, Russell's book has a quite different literary style than Wittgenstein's gnostic utterances and hierarchical structure. Both argue that everyday observations about, say, tables and chairs, should be decomposed into logical conjunctions, negations, and disjunctions of atomic facts, which cannot be further broken down. Russell and Wittgenstein differ on the nature of these atomic facts. For Wittgenstein, the referents for entities in atomic facts are quite mysterious; the specification of what these entities are is not a matter of logic, but of its application. Russell is quite clear that these entities include unintegrated sensations, something like "red patch here now."

Russell outlines how one might combine statements about such entities to construct entities that we see, hear, taste, smell, or feel. He goes on to analyze claims about other minds. The analysis of time leads to comments on Zeno's paradoxes and the mathematical theory of continuity. He also explains the idea of infinity, explaining the then recent theory of Cantor. He tries to present a popular overview of these topics. He acknowledges that some of his exposition is more mathematics than philosophy. But, as you can see above, he thinks previous philosophers and many of his contemporaries stumbled into error because they did not possess these logical and mathematical tools. For later developments along the lines, I gather one can look at such works of logical positivism as Rudolf Carnap's The Logical Structure of the World. I have never read Carnap, but I have read A. J. Ayer's Language, Truth, and Logic.

I recently stumbled somewhere across an argument that Noam Chomsky's approach to linguistics supercedes Russell's application of logic to philosophy. Russell and Chomsky agree that sentences of very different structures can have a close surface appearance, and that the same structure can be exhibited in sentences of different surface appearances. In deciding whether or not propositions are true, or even make sense, one should supposedly concentrate on the meaning captured by this deeper structure. But in trying to analyze the meaning of such propositions as, "The king of France is bald", Russell takes an a priori approach. The adequacy of grammar, however, to characterize sentences in a language is an empirical question. And semantics should be based on the parse trees derived from grammatical analysis of the surface appearances of language, not a logical analysis of the surface appearance. This approach, as I understand it, is analogous to how compilers operate. They apply a semantic analysis to a computer program only after first completing a parsing phase. And Chomsky's approach, I gather, has been influential in Artificial Intelligence.

One can argue that just as Wittgenstein, in Philosophical Investigations, showed his earlier approach in the Tractatus was mistaken, so he also showed Chomsky's approach in linguistics to be mistaken. A fortiori, AI is not possible either. Exposition of the parallelism between Russell and Chomsky's analysis of language makes these claims a bit more clear to me. (I guess Sraffa was not too impressed by Chomsky, either.) I suppose one might look at Norman Malcolm's Wittgenstein: Nothing is Hidden, for a fuller argument against Chomsky along these lines. (I did not get much out of Malcolm when I read him years ago.)