Robert Nozick, The Refutation Of Rational Choice, Etc.

"Robert Nozick has a unique place in the annals of rational choice theory: he refuted it." -- Ian Hacking (1994)

My reaction, when reading this, was, "What?" Hacking is referring to a paper by Robert Nozick¹ on Newcomb's Paradox. I'm fairly sure I've read something about this paradox, but I had to look it up.

Suppose there exists a psychic that has shown themselves to be extremely reliable in their predictions. And the psychic has presented you with a choice, based on one of their predictions. You are presented two boxes, one transparent and one wrapped such that you cannot see the contents. The rules are that you can take either:

• Just the opaque box, or
• Both boxes.

The transparent box contains $1,000, as you can plainly see. If the psychic has predicted you will pick just the opaque box, they have placed $1,000,000 in it. If they have predicted you will pick both boxes, they have ensured that the opaque box contains nothing. The prediction has been made, and the boxes have been sealed. You know all these conditions but not what the prediction was. What should you do?

Apparently many initially are very decided on what they would do. But people split half-and-half on what that is. Anyways, Hacking states that this example shows that two principles of rational decision-making are not necessarily consistent². I guess he is correct, and I'm in no position to challenge that this is of philosophical interest³. But, since no such psychic can exist, I find other examinations of rational choice theory of more practical import.

By the way, I want to give a qualified defense of Stephen Metcalfe's comments in Slate on Nozick's Wilt Chamberlin example⁴. Strictly speaking, Metcalf's confusion about which Keynes comment was on which Hayek book is irrelevant to these comments later in the article⁵. And I accept that he doesn't describe the logic of Nozick's argument⁶. Neither did I. It is perfectly legitimate to argue that the rhetorical force of the argument comes from elements of the argument extraneous to its strict logic. And that is what Metcalf does⁷.

Footnotes

1. Nozick's "Reflections On Newcomb's Paradox" (in Knotted Doughnuts and Other Mathematical Entertainments (ed. by M. Gardner), W. H. Freeman, 1986).
2. Choose dominant strategies. Maximize mathematical expected utility.
3. I find Wittgenstein perennially fascinating.
4. Metcalf's Slate followup is here.
5. So is the fact that Nozick was smoking dope during the period in which he wrote Anarchy, State, and Utopia; I was startled to find he mentions in his book his experiences while under the influence. More by Brad DeLong on Nozick is here. Even more can be found in the Delong's blog archives.
6. By the way, Yglesias is mistaken in concluding, "Since as best I can tell nobody does hold such a [patterned] theory [of distribution]". Nozick explicitly states that marginal productivity gives such a patterned theory. Nozick is confused, since marginal productivity, correctly understood, is a theory of the choice of technique, not a theory of distribution.
7. Although I am not convinced appealing to guilty regret over the history of race relations in the United States has anything to do with Nozick's rhetoric.

Really, Really Free Market

I stumbled upon this idea recently. Not that I've ever physically seen such a market.

Elsewhere

• Alejandro Nadal asks, "Whatever happened to stability analyisis?" (h/t: Brad DeLong).
• A blogger with the pseudonym of "Lord Keynes" critiques Austrian Business Cycle Theory, based on Sraffa's demonstration of the non-uniqueness of the "natural" rate of interest in an intertemporal equilibrium.
• Jayati Ghosh provides access, from here, to an essay surveying Michal Kalecki's contributions to development economics.

A Popular Exposition Of Post Keynesianism

I have stumbled across a commentator, Tim Bending, at Open Democracy has been explaining some practical implications of Post Keynesianism. I particularly like this exposition of the theoretical incoherence of marginal productivity as a theory of the distribution of income.

Numeraire-Free Tests Of The Labor Theory Of Value

A reminder to my self - the following article belongs on this list.

• Mariolis, Theodore and George Soklis (2011) "On Constructing Numeraire-Free Measures of Price-Value Deviation: A Note on the Steedman-Tomkins Distance" Cambridge Journal of Economic, V. 35, N. 3: 613-618.

Three Routes To Choice

A theme of this blog is the incorrectness of the neoclassical textbook description of how agents choose. The assumptions of this view can be stated as:

1. An agent knows the complete list of choices from which they must select.
2. Given any two elements from this space of choices, the agent knows whether one of these elements is not preferred to the other.
3. Any element from this space is not preferred to itself.
4. The ranking obtained from the preference relation is transitive.
5. If the space of choices is a continuum, a certain continuity assumption must hold for the preference relation so as to rule out lexicographic preferences.

These assumptions supposedly imply the claim that utility attains at most an ordinal measurement scale level¹. And they allow one to derive the demand for consumer goods and the supply of factors of production.

Economists have transcended this framework. I have previously pointed out models of agents as consisting of multiple selves. I think this approach exhibits a consilience with theories in, for example, cognitive psychology. I have recently stumbled upon two other ways of modeling choice, generalizing the textbook view to an approach more consistent with empirical evidence from behavioral economics and that cannot be justifiably characterized as "irrational".

Nadeem Naqvi has developed an approach of incorporating tertiary information into choice. In the outdated neoclassical theory, one might represent the relationship y is not preferred to x for agent i by:

x R_i y

Naqvi and his colleaques introduce the relation R_i(V_i^j), where V_i^j is the background set for agent i. Parametric variation in the agent’s background set can alter the agent’s preferences. That is, one can have, for l ≠ m:

x R_i(V_i^l) y

and

y R_i(V_i^m) x

One interesting consequence of this modeling strategy is that racial discrimination is formally consistent with Pareto optimality. This "is a surprising, though serious, indictment of relying exclusively on the Pareto principle in social evaluation."

Gul and Pesendorfer consider choice among menus. They consider an agent who is a vegetarian for health reasons, but who is tempted to choose hamburgers, if available. In choosing a restaurant at noon, they would prefer a restaurant with hamburgers on the menu. But in choosing in the morning a restaurant to visit at noon, they will select one with an all-vegtable menu. I hope you can see how this approach allows one to analyze time-consistency of choices.

How long do you think before such approaches are presented in mainstream textbooks in widespread use?

Footnotes:
¹ Nominal, ordinal, interval, and ratio are well-known measurement scale level, where a level is defined up to a set of transformations. I find curious the claim that the expression of the marginal rate of transformation as a ratio of marginal utilities is consistent with an ordinal scale. Mirowski, in More Heat Than Light has also raised questions about the claim that utility only attains an ordinal scale level. I recently stumbled upon Mandler (2006), where he suggests, not necessarily for related reasons, utility be considered to attain a measurement scale level between ordinal and interval.

References

• George A. Akerlof and Rachel E. Kranton (2010) Identity Economics: How Our Identities Shape Our Work, Wages, and Well-Being, Princeton University Press (TO READ).
• Arian Berdellima and Nadeem Naqvi (2011) "Existence of a Pareto optimal social interaction with non-binary preferences".
• Frank Gul and Wolgang Pesendorfer (November 2001) "Temptation and Self-Control", Econometrica, V. 69, N. 6: 1403-1435.
• Michael Mandler (September 2006) "Cardinality versus Ordinality: A Suggested Compromise", American Economic Review, V. 96, N. 4: 1114-1136.
• Nadeem Naqvi (2010) "On Non-binary Personal Preferences, Economic Theory and Racial Discrimination".

Play It Cool, Daddy-O

1.0 Introduction "Is Von Neumann Square?" is one of my favorite titles for an article in economics¹. This post is about a case in which Von Neumann is more hep². Sraffa’s book presents a succession of models in which, after the second chapter, the system of price equations have one degree of freedom. This is usually taken to be a trade-off between wages and the rate of profits. Once the distribution of the surplus product is exogenously specified, prices are determined. Some, such as Michael Mandler and Paul Samuelson, have criticized Sraffian economics on the basis that this number of degrees of freedom is arbitrary. Cases can arise in which the system of price equations has either more or less than one degree of freedom. This post illustrates a case in which more than one degree of freedom exists. 2.0 The Example 2.1 Technology and Quantity Flows Consider a very simple economy in which laborers produce corn from seed corn on lands of definite types. Two types of land are available. Assume that this economy has 100 acres of land of each type available. Two Constant-Returns-to-Scale processes are known for producing corn. As shown in Table 1, each process requires inputs of a single type of land, as well as labor and seed corn. The technology is such that the order in which types of land will be rented can be read off directly from the technology. As I have previously pointed out, this is not a general property in long period models analyzing rent. I think this special case property, however, is not what drives the existence of possibly more than one degree of freedom.

Table 1: The Technology

	α Process	β Process
Labor	1 person-year	1 person-year
Type I Land	1 acre	0 acre
Type II Land	0 acre	1 acre
Corn	1/5 bushels	1/4 bushels
Outputs	1 bushel corn	1 bushel corn

Under the assumptions, anywhere from zero to 200 bushels of corn can be produced as gross output in this economy. Assume that the gross output of this economy is 100 bushels of corn. Then cost-minimizing firms will cultivate all of type I land, and all of type II land will lie fallow. 2.2 The Price System For stationary-state prices, no process can earn pure economic prices. This condition imposes the following inequalities:

(1/5)(1 + r) + ρ₁ + w ≥ 1

(1/4)(1 + r) + ρ₂ + w ≥ 1

• w is the wage (bushels per person-year), paid at the end of the year
• r is the rate of profits
• ρ1 is the rent (bushels per acre) on type I land, paid at the end of the year
• ρ2 is the rent (bushels per acre) on type II land, paid at the end of the year

An equality applies for any process in use. Land of a given type can be modeled, in an alternative specification of the technology, as jointly produced at the end of the period from the inputs of labor, seed corn, and that type of land³. As long as less than 200 bushels of corn are produced, at least one type of land will pay no rent:

ρ₁ ρ₂ = 0

2.2.1 First Special Case Consider what would happen if the gross output was infinitesimally less. Both types of land would be in excess supply. The rent on both would be zero:

ρ₁ = ρ₂ = 0

The solution in this case is:

0 ≤ r ≤ 4

w = (1/5)(4 - r)

Only type I land is cultivated. The number of processes in use is equal to the number of produced commodities, that is produced goods with a positive price. The system of price equations has one degree of freedom.

(1/5)(1 + r) + ρ₁ + w = 1

(1/4)(1 + r) + ρ₂ + w > 1

2.2.2 Second Special Case Consider, however, what would happen if the gross output was infinitesimally more. Both types of land would be cultivated. Type I land would not be able to produce all the output quantity needed for the requirements for use, and it would have a positive rent:

ρ₁ > 0

Type II land would be in excess supply, and it would have a rent of zero.

ρ₂ = 0

The price system becomes:

(1/5)(1 + r) + ρ₁ + w = 1

(1/4)(1 + r) + ρ₂ + w = 1

The solution is:

0 ≤ r ≤ 3

w = (1/4)(3 - r)

ρ₁ = (1/20)(1 + r)

Two produced commodities with positive prices exist: corn and the first type of land. And two processes are activated. 2.2.3 The Case With Two Degrees of Freedom But type II land does not need to be cultivated in the case under consideration. Thus, the costs of cultivating type II land can exceed the revenues, and the rent on Type I land is not determined by the price equations. Only the first equation in the system of equations for prices need obtain.

(1/5)(1 + r) + ρ₁ + w = 1

The second process is still characterized by an inequality:

(1/4)(1 + r) + ρ₂ + w ≥ 1

This system has the solution:

0 ≤ r ≤ 4

0 ≤ ρ₁ ≤ (1/20)(1 + r)

ρ₂ = 0

w = (1/5)(4 - 5ρ₁ - r)

Figure 1 illustrates one projection of this solution into two dimensions. The lines closer to the origin are drawn for a higher rent on the first type of land.

Figure 1: Variation in the Wage-Rate of Profits Frontier with Rent

3.0 Conclusions I am loath to argue that the extra degree of freedom in this example is negligible since it arises only for a knife-edge. If the quantity produced is a hair larger or a hair smaller, the input-output matrices for commodities with positive prices are square. But in a larger model, the quantity produced is a choice variable. I also don't see why Sraffian models must not have more than one degree of freedom. Footnotes ¹ If I’ve actually read this article, it must have been in a reprint in some collection. ² Some posts take me a while to write. I began this one the day after Arthur Laurents died. ³ I don't here show the derivation of rent from such a model. References

• Christian Bidard (1986) "Is Von Neumann Square?" Journal of Economics, V. 46: pp. 407-419.
• Michael Mandler (20xx) "Sraffian Economics (new developments)" New Palgrave, 2nd edition.

Austrian Welfare Economics Confused

In my critique of Austrian Business Cycle Theory, I cite some critiques of the Austrian school. Hill (2004) and Gloria-Palermo & Palermo (2005) critiques I do not cite.

Palermo and Palermo focus on how Austrian school economists reach normative conclusions. They put aside the influence of values in, for example, choosing the questions one addresses in one's positive analysis. For Austrian school economists, the idea of coordinated plans acts as a bridge from their positive theory to their normative claims. A state in which all agent's plans are coordinated is thought to be a desirable state by Austrian school economists. They claim that a market system has a tendency towards such a state without ever reaching it. Since then market systems are always in an undesirable state in which some agents' plans are mutually inconsistent and uncoordinated, why do Austrian school economists, in their nascent normative analysis, not conclude that market systems are undesirable?

• Greg Hill (2004). "From Hayek to Keynes: G.L.S. Shackle and Our Ignorance of the Future". Critical Review.
• Sandye Gloria-Palermo & Giulio Palermo (2005). "Austrian Economics and Value Judgements: A Critical Comparison with Neoclassical Economics". Review of Political Economy, V. 17, N. 1 (Jan.): pp. 63-78.

Quiggin's Optimism Of The Intellect

Quiggin says much in Zombie Economics: How Dead Ideas Still Walk Among Us I agree with. I see no reason, however, to believe that economists will follow this recommendation or that this is the best way for economists to model actually-existing more-or-less capitalist economies:

"A new project in the D[namic] S[tochastic] G[eneral] E[quilibrium] framework will typically, as Blanchard indicates, begin with the standard general equilibrium model, disregarding the modifications made to that model in previous work examining the other ways in which the real economy deviated from the modeled ideal.

By contrast, a scientifically progressive program would require a cumulative approach, in which empirically valid adjustments to the optimal general equilibrium framework were incorporated into the standard model taken as the starting point for research. Such an approach would imply the development of a model that moved steadily further and further away from the standard general equilibrium framework, and therefore became less and less amenable to the standard techniques of analysis associated with that model." -- John Quiggin, Zombie Economics: How Dead Ideas Still Walk Among Us (Princeton University Press, 2010)

I was inspired by this Crooked Timber thread to post this.

Picoeconomics: A New Vocabulary Word For Me

I have previously described models of agents divided in mind. And I have noted that akrasia is defined as the phenomenon of acting against one's own best judgement.

I find that George Ainslie uses the term picoeconomics to describe the study of the interaction of components of a mind in individual behavior and decision-making. Microeconomics is, in some sense, the study of the interactions of individuals in determining economic behavior. Picoeconomics is an analysis on an even smaller scale. I also found a website for this subject¹.

By the way, picoeconomics is not necessarily a non-mainstream field of economics. For example, Glen Weyl (2009), a very young mainstream economist trained at some of the most prestigious economics departments in the United States, adopts a model of an agent as a community. He uses this model to examine political individualism. If a community cannot have group rights and cannot have an unique ordering of choices², how can an individual have such rights when he may be just as divided in mind as a community?

One criticism of mainstream economists relates to their treatment of the literature. A mainstream economist can ignore long-established analytical tools to treat their subject, introduce some related analysis into orthodox models in an ad-hoc way, and never reference the previously-existing heterodox literature. I do not feel I have enough understanding of picoeconomics to say whether this criticism applies to mainstream and non-mainstream contributions to the field³.

Footnotes

• ¹ Is this Ainslie's website? I could not quickly find a name associated with the site?
• ² See the Arrow impossibility theorem.
• ³ I'm not even sure I know the field boundaries. My blogs posts on divided minds build on some literature by Amartya Sen. Some recent papers from Nadeem Naqvi and others build on later literature from Sen. They analyze agent decision-making, but, as I understand it, do not model the mind as composed of subagents. Does this literature fall within picoeconomics?

References

• George Ainslie (1992) Picoeconomics: The Strategic Interaction of Successive Motivation States within the Person, Cambridge University Press. (I haven't read this.)
• George Ainslie (2001) "Breakdown of Will" , Cambridge University Press. (I haven't read this.)
• G. Ainslie (October 2005) "Précis of Breakdown of Will" Behav. Brain Sci., V. 28, N. 5: 650-673.
• Arian Berdellima and Nadeem Naqvi (2011) "Existence of a Pareto optimal social interaction with with non-binary preferences".
• E. Glen Weyl (May 2009) "Whose rights? A critique of individual agency as the basis of rights" Politics, Philosophy & Economics, V. 8, N. 2: 139-171.

Monetarism And Marxism

I want to draw a parallelism between monetarism and marxist economics. I do not refer to the attempt by the United States Federal Reserve to implement Marx's theory of the reserve army of the unemployed. Rather, I think an analogy may exist between money and productive labor and their relationships to available empirical data.

If monetarists have a rigorous definition of money that picks out one unique time series, I do not know of it. Suppose they find some correlation between, say, M2 and a price level. And suppose that correlation subsequently breaks down. They need not take this as evidence against their theory. For they can always search for another time series for the quantity of money and a different price deflator.

Likewise, economists building on Marxism, as I understand it, have not settled upon a rigorous theoretical definition of the distinction between productive and non-productive labor. For example, I don't think Marx - especially in the first volume of Theories of Surplus Value - argues that all services are unproductive of surplus value. Rather, his distinction is between labor that produces surplus value and other labor. Duncan Foley finds a tighter correlation between real national income, excluding services, and non-farm employment than he does between all national income and non-farm employment. By my argument, I do not take this as dispositive evidence for Marx's distinction. I worry that which correlation works best can vary by time period and time series.

Nevertheless, I find Foley's analysis of interest. Finance, Insurance, and Real Estate (FIRE) certainly seems to need more analysis by economists in these days.

References

• Duncan K. Foley (2011) "The Political Economy of Output and Employment 2001-2010"
• James K. Galbraith, Olivier G. Giovannoni, and Ann J. Russo (2007) "The Fed's Real Reaction Function: Monetary Policy, Inflation, Unemployment, Inequality - and Presidential Politics"

Galbraith On Sociology Of Economics

I thought this short comment was interesting:

Figure 1: James Galbraith on Sociology of Economics

Provenance is, of course, no guarantee of the quality of art. But one thing that strikes me about dissenters and non-mainstream economists is that many have doctorates in economics from elite schools (e.g., Harvard, Yale, Cambridge), have taught at such elite schools, or might even be professors at such places. Is there any other discipline in which members treated as on the fringe have so many with such credentials?

Elsewhere

With the exception of the first two, these links seem more closely related than most I list. I ought to add something about the flash crash.

• Richard Thaler compares utility to aether, an imaginary substance that 19th century scientists thought existed.
• Matthew Yglesias misunderstands; he thinks Thaler’s comments apply only to macroeconomics.
• Donald MacKenzie describes the effects of automated trading algorithms on microsecond variations in stock market volumes:. Some of this sounds like network security applications. You have sniffers detecting what bots are doing, spoofers attempting to fool the sniffers, etc.
• Kieran Healy reviews MacKenzie's book describing finance theory as performative.
• A news story reports "Some Users Find the Speed of Light Too Slow for Their Networks", (IEEE Computer, V. 44, N. 4 (Apr. 2011): 18-19). These users are ones trying to decide where to locate their automated trading algorithms.
• K. J. Ray Liu describes how radio devices will use game theoretical algorithms to allocate spectrum. ("Cognitive Radio Game", IEEE Spectrum, V. 48, Iss. 4, Apr 2011: 40-56)

Models Building On Minsky

Some recent resources on Hyman Minsky:

• Gauti B. Eggertsson and Paul Krugman (2010) "Debt, Deleveraging, and the Liquidity Trap: A Fisher-Minsky-Koo Approach"
• Steve Keen (15 March 2011) "The Debt Krugman Would Rather Forget", Business Spectator.
• Steve Keen (2011) "A Monetary Minsky Model of the Great Moderation and Great Recession", Journal of Economic Behavior & Organization
• Thomas I. Palley (April 2010)"The Limits of Minsky's Financial Instability Hypothesis as an Explanation of the Crisis", Monthly Review, V. 61, Iss. 11.
• Thomas I. Palley (2011) "A Theory of Minsky Super-Cycles and Financial Crises", Contributions to Political Economy.
• Lance Taylor and Stephen A. O'Connell (1985) "A Minsky Crisis", Quarterly Journal of Economics, V. 100, Supplement: pp. 871-885.

With some work, one can find downloadable copies of the above papers containing the formal models. In the more popular paper above, Keen criticizes Krugman for retaining flawed assumptions of mainstream macroeconomics. Why should anyone care at this point how a Dynamic Stochastic General Equilibrium (DSGE) model can be tweaked? Palley's Monthly Review article is also deliberately designed to generate controversy. He argues that Minsky's approach needs to be supplemented by more structuralist explanations.

Internet Sites For Discussing Economics

I find dispiriting the inability of most mainstream economists to discuss economics. Here are some emerging sites:

• Research Gate has a part set aside for economics.
• Stack Overflow has some proposed topics. (Apparently, mathematical economics occasionally comes up on Math Overflow.)
• Economics Questions apparently uses the Stack Overflow software.
• Quantitative Finance has its own Stack Overflow site.
• The Test Magic forum for PhD applicants in economics has been around for awhile.

Off topic: Matt Rognlie, who strikes me on first perusal as unjustifiably arrogant, casually dismisses Modern Monetary Theory.

Slope Of "Demand Curve" Varying With Numeraire

1.0 Introduction
As I have previously blogged, Ian Steedman has a number of articles explaining price theory. These articles typically explain the implications, for example, for the slopes of certain functions, but often do not contain graphical illustrations. Here then is an opportunity for me to develop blog posts. For instance, this post works through the example in Section 4 of Arrigo Opocher and Ian Steedman's article, "Input price-input quantity relations and the numéraire" (Cambridge Journal of Economics, V. 33, N. 5 (2009): 937-948).

2.0 Technology
Consider a simple economy in which three commodities - iron, steel, and corn - are produced. Corn is the only commodity people consume, and corn is not used as an input in the production of any commodity. The problem considered here is how much of each input into the production of corn will be demanded by the firms in the corn-production industry.

Iron is produced by unassisted labor. The production function for iron is:

q₁ = f₁(l₁, x_1,1, x_2,1, x_3,1) = l₁

where:

• q_i; i = 1, 2, 3; is the quantity of the ith commodity produced in the period under consideration.
• f_i( ); i = 1, 2, 3; is the production function for the ith commodity.
• l_j; j = 1, 2, 3; is the number of person-years hired in the period under consideration to produce the jth commodity.
• x_i,j; i = 1, 2, 3; j = 1, 2, 3; is the quantity of the ith commodity used in the production of the jth commodity in the period under consideration.

Steel is produced by labor from iron. Steel production is modeled by a Cobb-Douglas production function:

q₂ = f₂(l₂, x_1,2, x_2,2, x_3,2) = (l₂)^D (x_1,2)^E/(D^D E^E)

where D and E are positive parameters and D + E = 1.

Corn is produced by labor from inputs of iron and steel. The production function for corn is:

q₃ = f₃(l₃, x_1,3, x_2,3, x_3,3) = (l₃)^d (x_1,3)^e (x_2,3)^f/(d^d e^ef^f)

where d, e, and f are positive parameters and d + e + f = 1.

All production functions exhibit Constant Returns to Scale (CRS). All capital goods are totally used up in production, and all production processes require the same amount of time.

3.0 Cost Functions and Coefficients of Production
Consider competitive firms producing a commodity who want to adopt a cost-minimizing technique. For definitiveness, consider a firm producing steel. Let w be the wage (paid at the beginning of the period) and p_j; j = 1, 2, 3; be the spot price of the jth commodity. The unit cost function, c₂( ), for producing steel is the value of the objective function in the solution to the following mathematical programming problem:

Given w, p₁, p₂, p₃

Choose l₂, x_1,2, x_2,2, x_3,2

To Minimize wl₂ + p₁x_1,2 + p₂x_2,2 + p₃x_3,2

Such that:

f₂(l₂, x_1,2, x_2,2, x_3,2) = 1

l₂ ≥ 0; x_i,2 ≥ 0, i = 1, 2, 3.

Solving this programming problem, one finds the unit cost function for producing steel is:

c₂(w, p₁, p₂, p₃) = (w)^D (p₁)^E

In working out the cost function, I also figured out how much labor and iron a steel-producing firm would hire to produce one ton of steel. Opocher and Steedman pose the problem with given cost functions, not production functions. They derive the coefficients of production by invoking Shephard’s Lemma.

A similar cost-minimizing problem arises for corn-making firms.
The unit cost function, c₃( ), for producing corn is:

c₃(w, p₁, p₂, p₃) = (w)^d(p₁)^e(p₂)^f

The derivatives of the cost functions are summarized by the coefficients of production, a₀ and A, where:

• a₀ is a three-element row vector such that a_0,j; j = 1, 2, 3; is the person-years of labor hired per unit output of the jth industry.
• A is a 3x3 matrix such that a_i,j; i = 1, 2, 3; j = 1, 2, 3; is the quantity of the ith commodity purchased as an input per unit output in the jth industry.

Table 1 displays coefficients of production.

Table 1: The Cost-Minimizing Technique

	Iron Industry	Steel Industry	Corn Industry
Labor	a0,1 = 1	a0,2 = D (p1/w)E	a0,3 = dwd - 1 (p1)e(p2)f
Iron	a1,1 = 0	a1,2 = E (w/p1)D	a1,3 = ewd (p1)e - 1(p2)f
Steel	a2,1 = 0	a2,2 = 0	a2,3 = fwd (p1)e(p2)f - 1
Corn	a3,1 = 0	a3,2 = 0	a3,3 = 0
Output	1 ton iron	1 ton steel	1 bushel corn

4.0 Long Period Price Equations
The above shows the coefficients of production that perfectly competitive firms choose, given prices. Each column in Table 1 has been derived independently of the others. Firms will continue to produce corn, the consumption good, period after period only if some firms also choose to produce iron and steel. Cost-minimizing firms will not make these choices for any configuration of prices. The long-period condition that firm choices be self-sustaining yields the following system of three equations:

a_0,1 w (1 + r) = p₁

[p₁ a_1,2(w, p₁) + w a_0,2(w, p₁)](1 + r) = p₂

[p₁ a_1,3(w, p₁, p₂) + p₂ a_2,3(w, p₁, p₂) + w a_0,3(w, p₁, p₂)](1 + r) = p₃

where r is the rate of profits. Since production takes time, the rate of profits is generally positive. These equations, when solved, define long-period equilibrium prices for produced commodities as functions of the wage and the rate of profits:

p₁ = w(1 + r)

p₂ = w(1 + r)^{1 + E}

p₃ = w(1 + r)^{1 + e + f + (1 + E)f}

5.0 The Numeraire
The above system of price equations has two degrees of freedom. One degree of freedom is removed when the numeraire is specified. Let the numeraire consist of σ₁ units of iron, σ₂ units of steel, and λ person-years:

σ₁p₁ + σ₂p₂ + λw = 1

Prices of the commodities comprising the numeraire are then:

w = 1/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

p₁ = (1 + r)/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

p₂ = (1 + r)^{1 + E}/[σ₁(1 + r) + σ₂(1 + r)^{1 + E} + λ]

Figure 1 shows the relationship between the wage and the rate of profits. This relationship is known as the wage-rate of profits frontier. In this example, coefficients of production vary continuously along the frontier.

Figure 1: Wage-Rate Of Profits Frontier

6.0 Quantity Demanded for Inputs
The above derivations allow one to draw various graphs for specific parameter values. I set the parameters for the production functions as follows: D = 3/4; E = 1/4; d = 2/3; e = 1/6; and f = 1/6. And I considered two numeraires. For the first numeriare, σ₁ = 1/3; σ₂ = 1/3; and λ = 1/3. For the second numeriare, σ₁ = 1/3; σ₂ = 1/2; and λ = 1/6.

For each numeraire, the wage and other prices in a long-period position are defined as functions of the rate of profits. And the cost-minimizing coefficients of production are functions of these prices, that is, ultimately of the rate of profits. Figure 2 shows a locus constructed out of these functions. Curves at the top of the figure plot the price of iron against the quantity of iron demanded by the (non-vertically integrated) corn-producing industry. In drawing this figure, the quantity of corn produced is constrained to be unity. The curves are analogous to conditional demand curves in neoclassical economic theory. And one can see that, for certain regions, whether the slope of such a curve is positive or negative depends on the numeraire. But is not a main point of neoclassical economics to argue that demand curves are downward-sloping (for arbitrary numeraires)?

Figure 2: A Locus for Firms in Long Period Equilibrium

7.0 Conclusion
So much for explaining the price of a capital good by well-behaved supply and demand curves in the market for that commodity.

Book Series

Any number of publishers have published series of books over the last couple of decades of interest to heterodox economists. Some are, I think, targeted for reference libraries - who else can afford them¹? Some collect papers for specific themes or schools of thought. And various publishers specialize in such.

But here I want to focus on the Modern Cambridge Economics Series. These books seem to be targeted for the more introductory student. They consist of, as far as I can tell:

• A. Asimakopulos (1991) Keynes's General Theory and Accumulation²
• Amiya Kumar Bagchi (1982) The Political Economy of Underdevelopment
• John Cornwall and Wendy Cornwall (2007) Capitalist Development in the Twentieth Century
• Phyllis Deane(1978) The Evolution of Economic Ideas²
• Michael Ellman (1989) Socialist Planning
• Eprime Eshag (1984) Fiscal and Monetary Policies and Problems in Developing Countries
• Frederick S. Lee (2006) Post Keynesian Price Theory
• Joan Robinson (1979) Aspects of Development and Underdevelopment
• Colin Rogers (1989) Money, Interest and Capital: A Study in the Foundations of Monetary Theory²

Phyllis Deane and Joan Robinson were the original editors. Phyllis Deane, Geoffrey Harcourt, and Jan Kregel were the editors after a later relaunch.

¹Edward Elgar has a series on Schools of Thought in Economics, another series called The Elgar Companion to ..., and The International Library of Critical Writings in Economics. Palgrave Macmillan had themed extracts from the New Palgrave (with some original entries) and is currently publishing their Great Thinkers in Economics series. (This last series is more introductory and affordable by some students.) Routledge has their multi-volume Critical Assessments of Leading Economists collections and, for the Austrian school, the Foundations of the Market Economy Series. Back in the 1970s, Penguin had the Penguin Modern Economic Readings collections. (Links are to information about random books in a series.)

² I have read and enjoyed these.

Cyril Hédoin On The Failure Of Macroeconomics

Some time ago, I noted James Galbraith's article, "Who Are These Economists, Anyway?" Galbraith suggests that it might be worthwhile to on the work of those economists who were researching "the nature and causes of financial collapse" before the present crisis and had presciently warned of potential problem areas. These empirically successful economists happen to be non-orthodox economists.

The other day, I stumbled upon a somewhat old response to Galbraith,
"Vers un changement de paradigme en économie" (English version), from Cyril Hédoin. As I understand it, he thinks neoclassical economics is collapsing and - what with behavioral economics, evolutionary game theory, complexity economics, etc. - mainstream economics is pluralist. Given these changes, he thinks economists have no need to look among traditionally heterodox economists for good ideas. Hédoin cites articles I like by David Colander, Richard Holt, & Barkley Rosser and by John B. Davis.

Galbraith replies in the comments. He unequivocally rejects the idea that his recommended alternative is on the periphery of economics. He says he cites the economists he does because he thinks their work is good.

Mainstream economists certainly have a problem in explaining - especially to outsiders - why one should not cite certain past presidents of the American Economic Society, various Harvard-trained economists or professors in the Harvard economics department, articles published in certain economics journals associated with the University of Cambridge, some "Nobel Prize" winners, and so on. (For any Austrian-school economists who may be reading this - certain economists trained at New York University don't count either.)

Nozick's Anarchy, State, and Utopia

1.0 Overview
Propertarianism, misleadingly called "libertarianism" by its fans, is a political philosophy. Do any supposedly rigorous arguments exist for this philosophy? Some cite Robert Nozick's 1974 tome, Anarchy, State, and Utopia as a demonstration that this question can be answered in the affirmative.

The book has three main parts. In the first part, Nozick argues that a state can emerge from an anarchistic state of nature without violating anybody's rights. The state, according to Nozick, evolves from a private protection agency; its clients become citizens. In the second part, Nozick argues that a state that tries be to more than a minimal state will violate somebody's rights, especially their right to property. For the distribution of property to be just, according to Nozick, three conditions must be met:

• The original acquisition of property must not have violated anybody's rights.
• The transfer of holdings must be likewise just.
• Whatever injustices may nevertheless have arisen in original acquisition or transfer must be rectified justly.

In the last part, Nozick provides a recipe for cookbooks of the future. He describe an association of voluntary communities, where people are free to join whichever commune they like. Only some of these communities may initially be propertarian.

This obvious decomposition of the book explains the three-word title. Despite Nozick's pretense to be presenting an analytic argument, I did not find nearly as much structure at a lower level. The first part seems to beg how people in an original anarchy would behave. Nozick postulates bourgeois contract-making; I think feudalism would more likely result. He continually brings up objections and needed refinements, pursues the argument to an arbitrary level, and then declares the resolution of these details beyond his scope since he is not writing a work of psychology or epistemology or whatever. So I find it difficult to summarize much more of the book.

2.0 Popular Bits
I think Nozick originated many sayings now popular among propertarians. Maybe some he reformulated or re-emphasized.

Nozick's calls his version of Descartes' demon "The Experience Machine". His story is a science fiction story with technology, tanks to float in, and computers. It raises the question, "What else can matter to us, other than how our lives feel from the inside?"

Nozick comes fairly close to saying that there is no such thing as society:

"But there is no social entity with a good that undergoes some sacrifice for its own good. There are only individual people, different individual people, with their own individual lives."

Nozick formulates a non-aggression principle, now called the No-Initiation-of-Force principle in propertarian polemics:

"An underlying notion sufficiently powerful to support moral side constraints against the powerful intuitive force of the end-state maximizing view will suffice to derive a libertarian constraint on aggression against another."

Nozick creates a well-known example with Wilt Chamberlain. Wilt Chamberlain's contract with a basketball team gives him a share of the gate. People "cheerfully attend his team's game". Nozick's imagines that when they buy their tickets, part of the price is that each person "drops" 25 cents "of their admission price into a special box with Chamberlain's name on it." What can be wrong with all of these voluntary transactions making Chamberlin rich? In the course of this exposition, he comes up with a clever turn of phrase:

"The socialist society would have to forbid capitalist acts between consenting adults." (my emphasis)

3.0 Some Absurdities
Nozick's book is a work of political philosophy containing much economic reasoning. But Nozick makes many mistakes in economics, and ignores what seems to me some major problems in political philosophy.

For example, I think a major question in political philosophy is how people with different ideas of good and evil can live together. Nozick acknowledges he doesn't address this question:

"I have proceeded in this essay (as much as possible) without questioning or focusing upon the assumption common to much utopian and anarchist theorizing, that there is some set of principles obvious enough to be accepted by all men of good will, precise enough to give unambiguous guidance on particular situations, clear enough so that all will realize its dictates, and complete enough to cover all problems that actually will arise. To have rested the case for the state on the denial of such an assumption would have left the hope that the future progress of humanity (and moral philosophy) might yield such agreement, and so might undercut the rationale for the state.

... the day seem[s] distant when all men of good will shall agree to libertarian principles... People who prefer peace to the enforcement of their view of right will unite together in one state."

In contrast to my opinion, Nozick thinks Marxian exploitation is a normative idea¹:

"One traditional socialist view is that workers are entitled to the product and full fruits of their labor; they have earned it; a distribution is unjust if it does not give the workers what they are entitled to."

Nozick doesn't understand marginal productivity. He incorrectly thinks that it is a theory of distribution:

"Almost every suggested principle of distributive justice is patterned: to each according to his moral merit, or needs, or marginal product, or how hard he tries, or..."

I could cite many more instances in which Nozick makes this mistake.

Nozick erroneously ignores the possibilities of multiple equilibria and of path dependence:

"Let us suppose that we know from economic theory that under the standard assumptions defining a competitive market economy, income and wealth will be distributed in an efficient way, and that the particular efficient distribution which results in any period of time is determined by the initial distribution of assets, that is, by the initial distribution of income and wealth, and of natural talents and abilities. With each initial distribution, a definite efficient outcome is arrived at."

Also, Nozick begs the question of the definition of property rights, of how a society determines what can be a commodity and what cannot. I could cite even more examples of Nozick getting economics wrong.

Nozick has a definite view on the always burning question of whether or not slavery is compatible with propertarianism. He thinks it is:

"Perhaps no persons completely sell themselves into slavery... Since this very extensive domination of some persons by others arises by a series of legimate steps, via voluntary exchanges, from an initial situation that is not unjust, it itself is not unjust."

And again:

"The comparable question about an individual is whether a free system will allow him to sell himself into slavery. I believe that it would."

4.0 Conclusion
I do not find Nozick's book convincing². You might want to read it to understand some context for certain debates in political philosophy. Nozick treats Rawls in the second part of his book. Before having read Nozick, I already knew about some of the popular bits from having read Alan Haworth. Nozick tends to be cited by many soi-disant libertarians - I suspect by more than have actually read him.

Footnotes
¹Nozick does foreshadow John Roemer's game-theoretic definition of exploitation:

"An individual benefits from the wider system of extensive cooperation between the better and the worse endowed to the extent of his incremental gain from this wider cooperation; namely, the amount by which his share under a scheme of general cooperation is greater than it would be under one of limited intra-group (but not cross-group) cooperation."

Nozick formulates "a condition of stable associations" much like this in the third part of his book.

²This is probably not surprising, given my preconceptions.

Robert Lucas On The Impossibility Of Involuntary Employment

As I understand it, so-called Dynamic Stochastic General Equilibrium (DSGE) models are central to contemporary mainstream macroeconomics. Some may want to criticize DSGE on the grounds that they failed to predict the recent global financial crisis. I think a stronger criticism would be that they led their users to believe in the impossibility of current circumstances arising.

I think of DSGE models as having evolved out of trends in macroeconomics started by Robert Lucas. Here is Lucas being sarcastic:

"The first sentence in Malinvaud, The Theory of Unemployment Reconsidered, is 'The term involuntary unemployment makes it obvious from the start that the labor market is one in which supply exceeds demand.' Thus we acquire factual information about labor markets from the terminology earlier theorists have used to describe them!" -- Robert E. Lucas, Jr. Models of Business Cycles. Basil Blackwell (1987).

And here is more:

"A theory that does deal successfully with unemployment needs to address two quite distinct problems. One is the fact that job separations tend to take the form of unilateral decisions - a worker quits, or is laid off or fired - in which negotiations over wage rates play no explicit role. The second is that workers who lose jobs, for whatever reason, typically pass through a period of unemployment instead of taking temporary work on the 'spot' labor market jobs that are readily available in any economy. Of these, the second seems to me much the more important: it does not 'explain' why someone is unemployed to explain why he does not have a job with company X. After all, most employed people do not have jobs with company X either. To explain why people allocate time to a particular activity - like unemployment - we need to know why they prefer it to all other available activities: to say that I am allergic to strawberries does not 'explain' why I drink coffee. Neither of these puzzles is easy to understand within a Walrasian framework, and it would be good to understand both of them better, but I suggest we begin by focusing on the second of the two." -- Robert E. Lucas, Jr. Models of Business Cycles. Basil Blackwell (1987).