Sunday, May 31, 2009

Friedman Blinded Me With Science

Scientists aspire to develop theories that observations can potentially demonstrate to be wrong. Here I examine whether this aspiration can possibly be achieved when economics is practiced in keeping with one of two views on methodology, the deductive-nomological or the instrumental view. I get the argument below from Donald P. Green and Ian Shapiro, Pathologies of Rational Choice Theory: A Critique of Applications in Political Science (Yale University Press, 1994).

Consider the covering law model, also known as the deductive-nomological view of scientific methodology. In this view, scientists formulate universal laws, in some sense. In an application of a scientific law, the hypotheses or antecedents are asserted to be true. That is, the statement of scientific law is conjoined with initial conditions. One then checks that the consequent holds. If observation is inconsistent with the consequent and one is sure that the initial conditions are true, the law is refuted.

Milton Friedman advocates instrumentalism, in which the assumptions of a scientific theory are false. (Actually, his famous essay, "The Methodology of Positive Economics", is so incoherent, Friedman can be interpreted as advocating almost any methodology you care to name. But let's stick with a widely argued view.) In Friedman's view the antecedents are always false in a significant theory:
"Truly important and significant hypotheses will be found to have 'assumptions' that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic the assumptions" -- Milton Friedman
Thus, if one holds that economic theories state covering laws and that economists are and should be instrumentalists, economic theories cannot be refuted by observation. The logical implications of false antecedents need not be true.

Can economics be a science if it is practiced in keeping with Friedman's strictures?

Thursday, May 28, 2009

Capital Is Dead Labor, That, Vampire-Like, Only Lives By Sucking Living Labor

1.0 Introduction
Stupidity about Marx is never-ending. So I thought I would put up a post about Marx as a mathematical economist. This is exposition of unoriginal ideas. To amuse myself, I didn't review Sraffa or any other author when writing this.

2.0 The Technology
Consider an economy in which n commodities are produced, each in a separate industry. The technique in use is represented by the nxn Leontief matrix A and the n-element row vector a0 of labor inputs. A column, a.,j, in A and an element of a0,j represent an industry. The ith, jth element of A is the quantity of the ith commodity input per unit output of the jth industry. Quantities are here measured in physical units (e.g., bushels corn per ton steel). The jth element of the row vector of labor inputs, a0,j, is the person-years of labor services hired in the jth industry per unit output.

By assumption, all industries require at least some positive amount of labor to produce their outputs. All industries produce their outputs in a year, and they consume all their inputs in producing their output. This is a model of circulating capital alone; no fixed capital (e.g., long-lasting machines) appears in the model.

Assume that the economy is viable, that is, some levels of operation exist for the industries such that there is a surplus product available, after replacing used-up means of production, when industries are operated at that level. For simplicity, assume all commodities are basic. In other words, every commodity enters either directly or indirectly into the production of every other industry. Presumably, steel enters directly into the production of automobiles. Iron would enter indirectly into the production of automobiles through its use in the production of steel.

No choice of technique occurs in the model.

3.0 Quantity Flows

3.1 Labor Values
Let q be an n-element column vector, where each entry is the gross output of that industry. Each entry is measured in the corresponding physical units (tons, bushels, kilograms, etc.). Let y be an n-element column vector of net outputs. Gross and net outputs are related like so:
y = q - A q
q = (I - A)-1 y
where I is the identity matrix. The existence of the inverse follows from viability. If industries were operated at levels to produce the gross outputs, the net output would be available for consumption or accumulation after replacing exactly the inputs consumed in production.

The amount of labor hired to produce the net output y is:
L = a0 q = a0 (I - A)-1 y
Suppose net output consisted of only one unit of the commodity produced in the jth industry:
y = ej
where ej is the jth column in the identity matrix. The labor value of the jth commodity, that is, the amount of labor hired to produce one more unit of the jth commodity net, is:
vj = a0 (I - A)-1 ej
Labor values are expressed as an n-element row vector:
v = a0 (I - A)-1
Labor values then have a sensible meaning; nothing radical is involved in defining them.

3.2 The Standard Commodity
One might expect any arbitrary basket with a large number of commodities to have both some labor-intensive and some capital-intensive commodities, in some sense. On average, these will approximately cancel in an arbitrary basket. Accordingly, let's assume that gross outputs, net outputs, and the commodity in which wages are paid are all of average capital-intensity, in some sense.

Since both gross and net outputs are of average capital intensity, it seems sensible to assume they are composed of the same proportions, just different in amount:
q* = (1 + 1/R) y*
where the asterisks indicate a basket in standard proportions. R is a strictly positive constant. As a normalization condition, the standard system is assumed to employ a unit amount of labor:
a0 q* = 1
It follows that the gross output of the standard system is a right-hand eigenvector of the Leontief input-output matrix:
A q*= [1/(1 + R)] q*
The outputs of the standard system are guaranteed to be positive by setting [1/(1 + R)] to the maximum eigenvalue of the Leontief matrix, also known as the Perron-Frobenius root of the Leontief matrix.

The net output of the standard system, y*, is the standard commodity.

3.3 The Rate of Exploitation
The labor embodied in the gross output of the standard system is easily found. One has:
1 = a0 q* = v(I - A)q* = v q* - v A q*
Or, taking advantage of the fact that the gross output of the standard system is an eigenvector of the Leontief matrix:
1 = [R/(1 + R)] v q*
Hence, the labor value of gross output of the standard system is found as:
v q* = 1 + 1/R
Marx expressed the labor value of gross output as C + V + S. Constant capital C is vAq*, the labor value of the means of production used up in producing the net output. Variable capital V is the labor value of the value added by labor paid out in wages. Surplus value S is the labor value of the remaining net output, which is obtained by the capitalists. Since one person-year is employed, the sum of variable capital and surplus value in the standard system is unity:
V + S = 1

Let w denote the proportion of the net output of the standard system (that is, the standard commodity) that is paid out in wages. Hence:
0 ≤ w ≤ 1
And variable capital is given as:
V = w
It follows that surplus value is now defined:
S = 1 - w
Marx denoted the ratio of surplus value to variable capital as the rate of exploitation:
e = S/V = (1 - w)/w = (1/w) - 1
where e is the rate of exploitation.When the whole value of the net product is paid out to workers as wages, workers are not exploited and the rate of exploitation is zero. The rate of exploitation is otherwise positive, and increases without bound as the wage becomes a lesser proportion of the value of the net product.

4.0 Price Equations
Let p denote a row vector of prices of production. Prices of production permit smooth reproduction in a competitive capitalist economy. They are defined by the condition that the same rate of profits is obtained in each industry:
p A(1 + r) + a0 w = p
where r is the rate of profits. Since profits are not earned on wages, the workers are paid at the end of the year. Wages are not advanced in this model. Since the same symbol for wages is used in calculating the labor value of quantities in the standard system, the standard commodity is the numeraire. Thus, the price of the standard commodity is unity:
p y* = 1

Recall that the net and gross outputs of the standard system are in proportion. One can thus calculate the price of the gross output of the standard system:
p q* = 1 + 1/R
Postmultiply the price equations by the gross output of the standard system:
p A q* (1 + r) + a0 q* w = p q*
p q* [1/(1 + R)] (1 + r) + w = (1 + 1/R)
(1 + 1/R)[1/(1 + R)] (1 + r) + w = (1 + 1/R)
The rate of profits is an affine function of the wage:
r = R(1 - w)
The above equation can also be expressed as:
w = 1 - r/R
The rate of profits ranges from zero to the maximum R. The maximum rate of profits is obtained when workers live on air, with a wage of zero. A higher wage is associated with a lower rate of profits, with a very simple relationship with this numeraire.

Total wages are a0 q* w. But, since one person-year is employed in the standard system, total wages are simply w.

Total profits are p A q* r, that is:
p A q* r = [1/(1 + R)] p q* R(1 - w)
p A q* r = [1/(1 + R)](1 + 1/R)R(1 - w)
p A q* r = 1 - w
The above is hardly surprising. The ratio of the rate of profits to the maximum rate is an increasing function of the rate of exploitation:
r/R = e/(1 + e)
When the rate of exploitation is zero, the rate of profits in the system of prices of production is also zero. As the rate of exploitation increases without bound, the ratio of the rate of profits to the maximum rate monotonically increases to unity.

5.0 Invariants
The following statements hold, whether the quantities mentioned are evaluated in labor values or in prices of production:
  • The gross output of the standard system is valued at 1 + 1/R
  • The net output of the standard system is unity
  • Variable capital is valued at w
  • Surplus value, that is, profits are (1 - w)
Furthermore, the rate of profits is positive if and only if workers are exploited.

This model certainly suggests that market phenomena are a veil over the exploitation inherent in capitalism. And calculations with labor values exhibit that exploitation.

Monday, May 25, 2009

"Capital As Power"

Maybe I'll purchase the book containing this truism:
"It should be noted upfront that economics – or, more precisely, the neoclassical branch of political economy – is not an objective reality. In fact, for the most part it is not even a scientific inquiry into objective reality. Instead, neoclassical political economy is largely an ideology in the service of the powerful. It is the language in which the capitalist ruling class conceives and shapes society. Simultaneously, it is also the tool with which this class conceals its own power and the means with which it persuades others to accept that power." -- Jonathan Nitzan and Shimshon Bichler, Capital As Power: A Study of Order and Creorder (Routledge 2009)

The above is not a novel idea.
"...the absurd aphorisms of a political economy controlled by property have puzzled the most generous minds." -- P.-J. Proudhon

Wednesday, May 20, 2009

A Neoclassical Response To The Cambridge Capital Controversy

1.0 Introduction
Around 1980, Edwin Burmeister could have justly thought that he was expressing the most prominent neoclassical response to the Cambridge Capital Controversy. He had championed David Champernowne's chain index as a defense of the aggregate neoclassical model, and continued to do so. Nowadays, though, mainstream economists make claims based on the aggregate model apparently in complete ignorance that they had ever been competently challenged:
"However, the damage had been done, and Cambridge, UK, 'declared victory': Levhari was wrong, Samuelson was wrong, Solow was wrong, MIT was wrong and therefore neoclassic economics was wrong. As a result there are some groups of economists who have abandoned neoclassical economics for their own refinements of classical economics. In the United States, on the other hand, mainstream economics goes on as if the controversy had never occurred." -- Edwin Burmeister (2000)
This post illustrates, by means of an example, elements of Burmeister's approach to the neoclassical aggregate model. It is exposition, with next to no criticism.

2.0 Technology
Consider a very simple economy in which a single consumption good, corn, is produced from inputs of labor, iron, and (seed) corn. All production processes in this example require a year to complete. Two production processes are known for producing corn, and two processes are known for producing iron. These processes require inputs to be available at the beginning of the year for each unit output produced and available at the end of the year. Each corn-producing process produces one bushel corn at the scale of operations shown in Table 1. Similarly, each iron-producing process produces one ton iron at the scale shown in Table 1.
Table 1: The CRS Technology
InputsCorn IndustryIron Industry
Labor (Person-Years):2312
Iron (Tons):3/501/103/51/2
Corn (Bushels):1/21/41/23/5
Output (Various):1111
Apparently, inputs of iron and corn can be traded off in producing corn outputs. The process that requires more iron also requires more labor. Inputs of iron and corn are also traded off in producing iron. But in iron production, the process requiring a greater quantity of iron input requires less labor.

A technique consists of a process for producing iron and a process for producing corn. Thus, there are four techniques in this example. They are defined in Table 2.
Table 2: Techniques and Processes
AlphaA, C
BetaA, D
GammaB, C
DeltaB, D

3.0 Quantity Flows
Suppose firms have adopted the alpha technique and they produce 20/43 bushels corn with process A and 3/43 tons iron with process C. One can see, from Table 1, that these firms will employ 40/43 person-years in the corn industry and 3/43 person-years in the iron industry - that is, a total of one person-year throughout the economy. Likewise, firms in the corn industry will purchase inputs of 6/215 tons iron, while firms in the iron industry will purchase inputs of 9/215 tons iron in the iron industry, for a total of 3/43 tons iron inputs throughout the economy. The produced iron at the end of the year exactly replaces the iron used as input, leaving a net output of 17/86 bushels corn. (Calculating corn inputs in the two industries is left as an exercise for the reader.)

Since these processes can be equally scaled up to any desired level, I have described a stationary economy on a per person-year basis. Table 3 shows the results of these calculations, as well as similar calculations for the gamma and delta technique. The beta technique is never cost-minimizing and is not shown in Table 3.
Table 3: Quantities Per Person-Year
AlphaGross Outputs(3/43 Tons, 20/43 Bushels)
Capital Goods(3/43 Tons, 23/86 Bushels)
Net Output17/86 Bushels Corn
GammaGross Outputs(1/13 Tons, 4/13 Bushels)
Capital Goods(1/13 Tons, 3/26 Bushels)
Net Output5/26 Bushels Corn
DeltaGross Outputs(1/17 Tons, 5/17 Bushels)
Capital Goods(1/17 Tons, 37/340 Bushels)
Net Output63/340 Bushels Corn

4.0 Prices
In a steady state, the same rate of profits is earned on all processes in use. Furthermore, charging that rate of profits on a process not eligible for use results in costs in that process exceeding the revenues. That is, we seek steady state prices corresponding with the cost-minimizing technique.

Suppose the alpha technique is cost minimizing. Prices for the iron-producing process (C) must satisfy the following equation:
(3/5 pα + 1/2)(1 + r) + wα = pα,
where pα is the price of iron, wα is the wage, and r is the rate of profits. The wage is paid at the end of the year, and corn is taken as the numeraire (so the price of a bushel corn is unity). Likewise, prices for the corn producing process (A) satisfy the following equation:
(3/50 pα + 1/2)(1 + r) + 2 wα = 1

I have specified a system of two equations in three variables. The wage and price of iron can be found as a function of the third variable, that is, the rate of profits. Table 4 displays this solution, as well as the solutions for the corresponding systems of equations for the other three techniques.

Table : Solutions to Price Equations
AlphaWagewα(r) = (27 r2 - 56 r + 17)/[2 (43 - 57 r)]
Price of Ironpα(r) = 25 (3 + r)/(43 - 57 r)
BetaWagewβ(r) = (107 r2 - 286 r + 107)/[40 (14 - 11 r)]
Price of Ironpβ(r) = 5 (11 +r)/[2 (14 - 11 r)]
GammaWagewγ(r) = (2 r2 - 13 r + 5)/[2 (13 - 17 r)]
Price of Ironpγ(r) = 5(9 + 5 r)/[2 (13 - 17 r)]
DeltaWagewδ(r) = (13 r - 7)(r - 9)/[20 (17 - 13 r)]
Price of Ironpδ(r) = (33 + 13 r)/(17 - 13 r)

Figure 1 graphs the wage-rate of profits curves for each technique. The cost-minimizing technique corresponds to the curve on the outer envelope. The wage-rate of profits curves for the alpha, gamma, and delta technique comprise the wage-rate of profits frontier. Alpha is cost-minimizing at low rate of profits, delta is cost-minimizing at high rates, and gamma is cost-minimizing at intermediate rates. Notice that for each pair of techniques, the wage-rate of profits curves cross at most once in the first quadrant. There is no reswitching, either on or off the frontier, in this example.
Figure 1: Wage-Rate of Profits Frontier

5.0 Champernowne's Chain Index

The above analysis specifies for each rate of profits (or for each wage) which technique will be adopted by cost minimizing firms. At switch points, linear combinations of techniques are cost-minimizing. The above analysis also determines the price of each capital good (e.g. corn and iron) for each rate of profits, as well as the composition of capital goods used in each technique per person-year. Figure 2 can thus be drawn based on this analysis.
Figure 2: Value of Capital and the Rate of Profits

Figure 2 shows the effects of both real and price Wicksell effects. The two horizontal lines arise from switch points. At switch points the composition of capital goods varies with the technique, while the rate of profits and the prices of capital goods are fixed. In other words, "real" capital varies in some sense. So the horizontal lines show real Wicksell effects. The curved, non-horizontal, segments display price Wicksell effects. That is, at non-switching points, the composition of capital goods remains invariant, but the prices of capital goods vary. Consequently, the numeraire value of the basket of capital goods varies here also.

Champernowne's chain index (Figure 3) sums up real Wicksell effects alone. Price Wicksell effects are abstracted from. The value of capital goods at the rate of profits of zero is taken in Figure 3 from Figure 2. Horizontal lines are drawn in Figure 3 at the same rates of profits at which they appear in Figure 2. The horizontal lines are also the same length. Vertical lines are drawn between horizontal lines.
Figure 3: Chain Index Value of Capital and the Rate of Profits

Champernowne's chain index only makes sense of the neoclassical parable in this case because all steps in Figure 3 slope down to the right. In other words, for an infinitesimal variation of the rate of profits around a switch point, the capital intensity of the cost-minimizing technique at the lower rate of profits exceeds the capital intensity of the cost-minimizing technique at the higher rate of profits. That is, Burmeister's defense of the neoclassical parable only applies in cases in which real Wicksell effects happen to be always negative:
"It follows, then, that a negative real Wicksell effect is the appropriate concept of 'capital deepening' in a model with many heterogeneous capital goods... Imposing some set of conditions on the technology ... should be sufficient to assure that the real Wicksell effect is always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from one-good models often used in macroeconomics without any theoretical justification... Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987)

6.0 A Pseudo-Production Function
I finally turn to the aggregate neoclassical production function used in the neoclassical parable:
Y = F(K, L),
where Y is net income, K is capital, and L is labor. Since Constant Returns to Scale are assumed, one can divide through by the labor input:
Y/L = F(K/L, 1)
y = f(k),
where y is net output per worker and k is capital per worker, in some sense. Figure 4 graphs this function for the example, where Champernowne's chain index is used to measure capital per worker. (If net output consisted of more than the numeraire good, a chain index would be used to measure output also.)
Figure 4: Pseudo-Production Function for the Example

Using this construction, the equilibrium condition that the rate of profits equal the marginal product of capital holds at switch points:
r = f ' (k)
This analysis has accepted that the value of capital goods (that is, the "quantity of capital") depends on the rate of profits. Recall, however, that this analysis only applies to examples in which real Wicksell effects happen to be always negative.

  • Salvatore Baldone (1984). "From Surrogate to Pseudo Production Functions", Cambridge Journal of Economics, V. 8: 271-288
  • Edwin Burmeister (1980) Capital Theory and Dynamics, Cambridge University Press
  • Edwin Burmeister (1987) "Wicksell Effects", in The New Palgrave, (ed. by J. Eatwell, M. Milgate, and P. Newman), Macmillan
  • Edwin Burmeister (2000). "The Capital Theory Controversy" in Critical Essays on Piero Sraffa's Legacy in Economics (ed. by H. D. Kurz), Cambridge University Press
  • D. G. Champernowne (1953-1954). "The Production Function and the Theory of Capital: A Comment", Review of Economic Studies, V. 21: 112-135

Tuesday, May 19, 2009

Toxic Textbooks

Edward Fullbook, a post autistic economist, is organizing a community to encourage students to protest orthodox economics textbooks.

Sunday, May 17, 2009

Reflections On "Sraffian Economics (New Developments)"

Michael Mandler has an article, "Sraffian Economics (New Developments)" in the latest edition of The New Palgrave Dictionary of Economics. I have been trying to read this. (Paul Samuelson's article, "Sraffian Economics", in the original New Palgrave is also heavy going.)

I have previously read Mandler as an anti-Sraffian willing to take the views he opposes seriously. I wonder if he is more positive now. Perhaps he feels that, although Sraffians are mistaken in theory, their mistakes are worthwhile to explore.

That is all subjective on my part, of course. Mandler is explicit on the issues of the indeterminateness of equilibrium and of tâtonnement stability. An indeterminate equilibrium is not merely a case of multiple equilibria. Rather, a continuum of equilibria arise. Perturbations of an equilibrium along this continuum would not set up stable or unstable forces driving the economy back towards or away from the original equilibrium. Rather the economy would just be in another equilibrium. The tâtonnement is a particular kind of exchange process that arises before the beginning of time in the Arrow-Debreu model of intertemporal equilibrium. Mandler argues that Sraffa has failed to demonstrate indeterminateness, and that issues of tâtonnement instability are not essentially connected to Sraffa's model of production; they arise from elements of utility-maximization.

Mandler has certainly been engaged by Sraffians (or vice versa) on exactly these issues. But I'm not sure that I agree that Mandler has picked out the essential points of Sraffa's book. The distribution of income is indeterminate in Sraffa's open model. I do not read Sraffa as claiming this property would still obtain if he closed his model by appending a specification of utility-maximizing consumers, including intertemporally. Rather, I take Sraffa as offering an open model demonstrating non-neoclassical theories of value and distribution can be constructed. If one insists on a closed mathematical model (for example), an empirical issue arises. I think Sraffa did not insist that his model be closed, at least, with elements of a model at the same level of abstractness and generality.

While tâtonnement (in)stability is interesting, I take Sraffian analysis to point towards stability isses elsewhere in, say, the Arrow-Debreu model. One can construct models of spot prices corresponding to the forward prices in the Arrow-Debreu model. These spot prices have their own dynamics that would arise even if spot markets always cleared instantaneously over time. Sraffa's model of production supports an exploration of limit points of this dynamics.

I have constructed examples with bifurcations, pointing to possibilities of complex dynamics in models of temporary or momentary equilibrium. (I don't claim to have a good grasp of the distinction, if any.) One can also show, through an analysis of structural stability, that many of the stories applied economists like to tell are without logical foundation.

Variations in the supply of labor can be modeled by perturbing a parameter in utility functions. An increased supply of labor is modeled by an increased desire for consumption, as opposed to leisure. Nevertheless, the corresponding equilibrium associated with an increased supply of labor, all other parameters held constant, might have a higher wage. The increased supply of labor need not drive the equilibrium wage down.

Likewise, variations in the supply of savings can be modeled by perturbations in a parameter describing intertemporal utility-maximizing. And greater savings can be associated, all other parameters held constant, with a higher equilibrium interest rate.

Relating the structural (in)stability of equilibrium limit points to the dynamics of temporary or momentary equilibria is a challenge to me. I am not sure whether interesting bifurcations are tied to capital-theoretic "paradoxes" such as reswitching and capital-reversing. I think it may depend on details of the model. In one reswitching example, I have found that whether the normal or "perverse" switch is associated with bifurcations depends on whether intertemporal maximizing representative agents are also modeled as choosing between leisure and consumption. Whether the latter choice is included or not seems to flip the result. But perhaps in some model where one has fixed the modeling choice, the existence of interesting dynamic behavior, in some sense, may be tied to the existence of perverse switches.

I may never resolve these theoretical issues to my own satisfaction.

Friday, May 15, 2009

No Thanks, Alessandro. I'm Full.

Alessandro Roncaglia has written a book, Piero Sraffa, for Palgrave's "Great Thinkers in Economics" series. (This is the same series containing Paul Davidson's John Maynard Keynes, available in hardcover and paperback.) I have always enjoyed Roncaglia's take on Sraffa, but I think I'll pass for now. I've seen quite a bit of what he has had to write in the past.

  • Alessandro Roncaglia (1978) Sraffa and the Theory of Prices, John Wiley & Sons.
  • Alessandro Roncaglia (1979) "The Sraffian Contribution", in A Guide to Post-Keynesian Economics (edited by Alfred S. Eichner), M. E. Sharpe
  • Alessandro Roncaglia (2000) Piero Sraffa: His Life, Thought and Cultural Heritage, Routledge
  • Alessandro Roncaglia (2001) "Production of Commodities by Means of Commodities Between Criticism and Reconstruction: The Given Quantities Assumption", in Piero Sraffa's Political Economy: A Centenary Estimate (edited by Terenzio Cozzi and Roberto Marchionatti), Routledge
  • Alessandro Roncaglia (2005) The Wealth of Ideas: A History of Economic Thought Cambridge University Press

Sunday, May 10, 2009

On Austrian Business Cycle Theory, Recently

Brad DeLong offers an empirical criticism based on order of magnitude estimates. For some reason, John Quiggin's blog crashes this browser on this platform. So I look to Mark Thoma to echo Quiggin, who doesn't seem to understand the (failed) concepts. Quiggin doesn't mention Wicksell, the idea of a natural rate of interest, or capital structure, for example. DeLong's post was in response to Roger Garrison. Peter Boettke adds a post. In comments to some of these posts, I link to a recent iteration of my critique, which may have some influence on Roger Koppl.

Friday, May 08, 2009

Is Utility Theory Tautological? An Old Argument

"What does [Jevon's theory] really amount to? In my apprehension to this, and no more - that value depends upon utility, and that utility is whatever effects value. In other words, the name 'utility' is given to the aggregate of unknown conditions which determine the phenomenon, and then the phenomenon is stated to depend upon what this name stands for." -- J. E. Cairnes, Some Leading Principles of Political Economy (1874) (quoted by G. Myrdal in The Politcal Element in the Development of Economic Theory)

"...that value was determined by the conditions which determine it - an announcement, the importance of which, even though presented under the form of abtruse mathematical symbols, I must own myself unable to discern." -- J. E. Cairnes, Some Leading Principles of Political Economy (1874) (quoted by G. Myrdal in The Politcal Element in the Development of Economic Theory)

Sunday, May 03, 2009

An Intervention By Kurt Gödel Into Economics

Kurt Gödel attended Karl Menger’s colloquium in Vienna in the 1930s. Sraffians should be interested in this colloquium since Von Neumann, in 1937, presented his classically-inspired economics model to the attendees. (Von Neumann had previously presented an earlier version to a Princeton mathematics seminar.) The later English translation of Von Neumann’s terse article is accompanied by a note from David Champernowne with the following acknowledgement.
"This note is the outcome of conversations with Mr. N. Kaldor, to whom many of the ideas in it are due. I am also indebted to Mr. P. Sraffa of Cambridge and to Mr. Crum of New College, Oxford, for instruction in subjects discussed in this article" -- D. G. Champernowne

Gödel’s published comment, however, was part of a discussion of a marginalist model. Walras’ models of the exchange of several commodities for one another and of production contain equations in which the quantity demanded of each good is a function of prices:
q1 = d1(p1, ..., pn)
qn = dn(p1, ..., pn)
Abraham Wald, in 1934, presented to the colloquium a sort of inverse or dual model, building on a paper from Karl Schlesinger. This model contains equations expressing the prices at which the quantities of commodities are demanded:
p1 = d1(q1, ..., qn)
pn = dn(q1, ..., qn)
I gather this model also contained inequalities, an important development in the theory of general equilibrium. At any rate, Kurt Gödel commented:
"Actually, for each individual entrepreneur the demand also depends on the price of factors of production. One can formulate an appropriate system of equations and investigate whether it is solvable." -- Kurt Gödel
According to John Dawson, Jr., Gödel’s remark is not well-taken; there is no obvious way to introduce prices of factors of production in this "inverse" model with many consumers. By contrast, when Gödel decided to say something about relevatistic physics, his remarks about rotating universes and world-lines traveling into the past, I guess, challenge physicists even decades later.

I stumbled on Gödel’s remark last week by noticing E. Roy Weintraub (1983) referenced in the first volume of Gödel’s collected works and wondering why that should be. I now see that Weintraub also quotes Gödel's remark, and Dawson is concurring with Weintraub.

  • D. G. Champernowne (1945-1946). “A Note on J. v. Neumann’s Article on ‘A Model of Economic Equilibrium’”, Review of Economic Studies, V. 13, N. 1: 10-18
  • S. Feferman et al (editors) (1986). Kurt Gödel: Collected Works: Volume 1: Publications 1929-1936, Oxford University Press
  • J. v. Neumann (1945-1946). “A Model of Economic Equilibrium”, Review of Economic Studies, V. 13, N. 1: 1-9
  • E. Roy Weintraub (1983). “On the Existence of a Competitive Equilibrium: 1930-1954”, Journal of Economic Literature, V. 21, N. 1 (March): 1-39

Friday, May 01, 2009

He Might As Well Have Said He Was Hungry

"Maureen: (getting less confused and quite interested): You mean to say that Socrates talks philosophy, knowing that he is going to die?

Leslie: Weird! A professor who talks and talks although he knows that the executioners are waiting for him, right outside his classroom. How does it all hang together?

Seidenberg (excited): Not only that. The two main characters of the dialogue Professor Cole wants to read with us, Theaetetus and Theodorus, were historical figures, both outstanding mathematicians. And Theaetetus, it says in the introduction, has been severly wounded in a battle and shortly after died from dysentery... there is an 'existential dimension' as one might call it - the way in which the entire conversation is inserted into extreme situations of real life. I feel this is very different from large parts of modern philosophy where you analyse only the logical properties of concepts and think that is all that can be said about them.

David (hesitatingly): I have read the dialogue because I wanted to be prepared for this class. I, too, wondered about the ending. But I don't see that it has any effect on the debate. The debate sounds very much like a philosophy class I just had; there is somebody who says that knowledge is experience...

Dr Cole: Perception...

David: ...well, that knowledge is perception, somebody else has counter-examples and so on. True, the dialogue is a little long-winded - but one doesn't notice anything about death in it. At the end Socrates suddenly says he has to go to court. He might as well have said he is hungry and wants to have dinner. At any rate, this seems just to be added for effect, it doesn't give any existential dimension to the concepts." -- Paul Feyerabend, Three Dialogues On Knowledge (1991)
I have some books written in prison. Antonio Negri's Marx Beyond Marx: Lessons on the Grundrisse, including the 1991 author's preface to the English translation, are written in circumstances beyond my understanding of Italian politics. The book, apparently, is based on lectures Negri gave in the mid 1970s to the École Normale in Paris. He was there at the invitation of Louis Althusser, in exile from Italy, under charges of having incited a riot in Padua.

This is, I gather, before Althusser murdered his wife by running her over. I never got much out of Althusser's For Marx. It seemed to me all methodological preliminaries, never illustrating or demonstrating that these preliminaries were worthwhile.

Negri's later introduction to the English edition is written from prison. I gather he was found guilty of having conspired with the Red Brigades to have kidnapped and murdered Aldo Moro, an Italian ex-prime minister. Negri's group was Potere Operaio (Worker's Power), not the Red Brigades.

I might as well summarize my understanding of the point of Autonomia. For Negri, previous Marxisms depicted workers as objects reacting to the machinations of capital, never as subjects initiating history themselves. Negri emphasizes the subjectivity of workers imprisoned throughout their lives, not just during their work time. Subjectivities will be organized around, for instance, ethnicity and gender, not just class. You can see how this is relevant to debates over Marx's outlines for Capital, whether he ever abandoned his plan for a volume on wages, and just did not get around to that volume. But I don't fully understand either Marx Beyond Marx or the more recent Empire.