Wednesday, December 31, 2014

Welcome

I study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.

The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.

In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.

I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.

Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.

Monday, August 18, 2014

Even If The Workers Could Live On Air

The Maximum Rate Of Growth Around The World

Consider a model of an economy in which all commodities are produced from inputs of labor and previously produced commodities. And suppose the commodities needed as inputs in the production of commodities are described through a Leontief input-output matrix in which no commodity can be produced with (unassisted) direct labor alone. Consider the special case in which wages are zero. In a sense, this special case can be seen as a description of a futuristic economy in which all production is automated, and robots are used to produce robots.

In the theory, the input-output relations determine a finite maximum rate of profits, corresponding to the maximum eigenvalue of the Leontief matrix. This maximum rate of profits is also the maximum rate of growth that arises in the Von Neumann growth model. A composite commodity, proportional to the associated eigenvector, arises from the Leontief matrix. Along the Von Neumann ray, the output of the economy each year consists of an evenly expanding output of this standard commodity, as Piero Sraffa called it. The standard commodity, in some sense, is a generalization of "corn" in David Ricardo's corn model (which was expounded in his 1815 Essay on the Influence of a Low Price of Corn on the Profits of Stock). The commodities with positive quantities in the standard commodity are known as basic commodities, once again in Sraffa's terminology.

As this post demonstrates, this is an operational model. The graph above is based on an eigenvector decomposition of Leontief matrices. Each Leontief matrix was derived from a transaction table for a country or region. The transactions tables, in turn, are derived from the GTAP 6 Data Base, compiled by the Global Trade Analysis Project at Purdue. (I had help extracting the database and putting it in a format that I can use.) GTAP 6 data is meant to cover the year 2001. Quantities of each commodities are measured such that a unit of each commodity can be purchased with one billion dollars at prices observed when the data was taken.

The graph above and the table below show the maximum rate of profits or growth for each country or region for the snapshot yielding the data. The actual rate of profits for prices that allow for the smooth reproduction of the economy falls below the maximum, sometimes considerably, because the workers do not live on air. The larger the proportion of the net output of the economy paid out in wages, the lower the corresponding rate of profits. At any rate, prices of production fall out, given some information on the distribution of income and production conditions.

Along with calculating the maximum rate of profits, I found the standard commodity and identified which commodities are basic for each country or region. For example, the commodities produced by the following industries are basic commodities in the United States: Cereal Grains; Vegetables, Fruits, Nuts; Crops; Bovine Cattle, Sheep and Goats, Horses; Animal Products; Raw Milk; Coal; Oil; Minerals; Bovine Meat Products; Meat Products; Dairy Products; Sugar; Food Products; Beverages and Tobacco Products; Textiles; Wearing Apparel; Wood Products; Paper Products; Publishing; Petroleum, Coal Products; Chemical, Rubber, Plastic Products; Mineral Products; Ferrous Metals; Metals; Metal Products; Motor Vehicles and Parts; Transport Equipment; Electronic Equipment; Machinery and Equipment; Manufactures; Electricity; Gas Manufacture, Distribution; Water; Construction; Trade; Transport; Water Transport; Air Transport; Communication; Financial Services; Insurance; Business Services; Recreational and Other Services; and Public Administration, Defense, Education, Health. Which commodities are basic varies among countries, and I typically found a few non-basic commodities in each country.

I think this data is fairly comprehensive, and I hope that I can do further believable analyses with it.

Maximum Rate Of Growth By Country
CountryRate of Growth
(Percent)
Peru144.8
Turkey132.5
Rest of Southeast Asia127.4
Albania125.3
Uganda122.5
Zambia121.6
Rest of Southern Africa Development Community120.7
Mozambique119.3
Greece117.8
Mexico117.3
Argentina116.9
Columbia115.0
France109.8
Sri Lanka109.6
Chile108.8
United States107.1
Bangladesh106.4
Zimbabwe106.3
Rest of Sub-Saharan Africa105.6
Spain104.4
Madagascar104.1
Rest of South Asia104.1
Venezula104.1
Japan103.4
Switzerland102.5
Italy101.0
Botswana98.2
United Kingdom97.7
India93.8
Indonesia93.0
Rest of Free Trade Area of the Americas92.8
Rest of EFTA92.0
Portugal91.4
Canada90.7
Malta89.3
Rest of the Caribean88.9
Denmark88.7
Rest of Central America88.4
Tanzania87.7
Rest of South America87.1
Australia87.1
Rest of Europe86.2
Brazil85.6
Sweden85.2
Rest of North Africa84.3
South Africa83.5
New Zealand82.7
Rest of Middle East82.5
Tunisia82.3
Taiwan82.1
Netherlands81.7
Finland80.0
Poland76.8
Germany76.6
Latvia75.8
Rest of South African Customs Union75.0
Malawi72.9
South Korea72.9
Hungary72.4
Austria70.3
Luxembourg67.6
Romania66.6
Russia66.2
Lithuania66.0
Rest of East Asia64.4
Philippines64.2
Estonia62.3
Thailand61.6
Malaysia60.5
Vietnam60.0
Rest of Oceania57.6
Ireland55.2
Central America54.3
Slovakia54.0
China53.8
Slovenia53.7
Croatia51.1
Czech Republic47.1
Belguim46.9
Morocco45.9
Hong Kong40.6
Singapore35.0
Cypress24.6
Rest of Former Soviet Union12.7
Uruguay12.4
Bulgaria8.6
Rest of North America4.7

Friday, August 15, 2014

Political Intervention in Faculty Selection at the UIUC

This is a post about the University of Illinois at Urbana Champaign (UIUC)1. It is not about current events.

In the late 1940s, UIUC attempted to revamp their economics department. They hired many new economists, including, for example, Jacob Marschak and Franco Modigliani2. A bunch of economists previously at UIUC resisted these modernizing changes. They ended up calling for political support in the press, complaining about New Deal politics. And the department was purged, in a violation of academic freedom, of these new-fangled economists3.

I thought I knew about this incident originally from reading an Esther Merjam Sent article about why both rational expectations and bounded rationality could have emerged from research at Carnegie Mellon during the 1950s - maybe, "Sargent versus Simon: Bounded Rationality Unbound" (Cambridge Journal of Economics, V. 21, No. 3 (1996): pp. 323-338). Or maybe I am recalling Fred Lee's 2009 book, A History of Heterodox Economics: Challenging the mainstream in the twentieth century. Googling, I find a draft of a paper from Antonella Rancan, who I have not otherwise read.

Footnotes
  1. I have many positive impressions of UIUC. As I recall, the first graphical web browser was made there.
  2. Modigliani, in a 1944 paper, extended the Hicksian IS/LM interpretation of Keynesianism to include a labor market with sticky wages. This was a critical contribution towards a politically powerful approach that Post Keynesians quarrel with on theoretical grounds (while agreeing, mostly, on short term political implications).
  3. Modigliani ended up at Carnegie Mellon, which I guess was once not called that.

Friday, August 08, 2014

Labor Demand In A Fog

Figure 1: Labor Demanded Per Unit Output in a Stationary State
1.0 Introduction

As a Sraffian, I have no problem with open models in which room exists for exogenous political forces to determine distribution. The example here, though, has more indeterminancy than I expect.

2.0 Technology

Consider a simple capitalist economy, composed of workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. The workers are paid a wage, w (in units of bushels corn per person year) out of the harvest. Capitalists obtain the rate of profits, r. The technology1 consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques. In each technique, a bushel of corn is produced from inputs of:

  • l0 person-years of labor performed in the year of the harvest.
  • l1 person-years of labor performed one year before the harvest-year.
  • l2 person-years of (unassisted) labor performed two years before the harvest-year.

Each technique is determined, given the values of the two index variables s and t. s is a non-negative real number less than or equal to the parameter c. t is a non-negative real number.

l0(t, s) = A - B + (t + 1)(B - s)/2
l1(t, s) = s
l2(t, s) = (B - s)/[2 (t + 1)]

where A, B, and c are positive constants and

cBA

In effect, the above has traced out isoquants for a production function, where the quantity of output is a function of dated labor inputs2. For a given value of the index variable s, labor inputs in the harvest year and two years before can be traded off. That is, if the amount of labor two years before is lower, then more labor must be expended in the harvest year. Likewise, for a given value of the index variable t, more labor being expended one year before the harvest mandates less labor being expended in the harvest year and two years before. So this specification of technology allows for substitution among inputs, at least in comparing steady states3, 4.

3.0 Choice of Technique

As usual, I consider a competitive, steady state economy in which capitalists have chosen the cost-minimizing technique, at an exogenously specified wage or rate of profits. Consider the function v(r, t, s):

v(r, t, s) = (1 + r)2 l2(t, s) + (1 + r) l1(t, s) + l0(t, s)

Take a bushel of corn as numeraire. The condition that all income be paid out to workers and capitalists leads to a wage-rate of profits curve, as a function of the rate of profits and the technique (specified by the values of the two index variables):

w(r, t, s) = 1/v(r, t, s)

A wage-rate of profits curve can be drawn for each technique. The wage-rate of profits frontier, consistent with a competitive steady-state, is the outer envelope (Figure 2) of all these curves. That is, for a given wage, one finds the values of the index variables that maximizes the wage among all techniques. This maximization does not fix s. But, for each value of s, the maximum is found by setting the index variable t equal to the rate of profits r. The equation for the frontier is:

w(r) = 1/(A + Br)

Notice the frontier is independent of the labor input, s, in the first year before the harvest. In this case, each point on the frontier is consistent with a continuum of profit-maximizing techniques. And these techniques vary continuously along the frontier. None of this indeterminancy is apparent by looking at the frontier5.

Figure 2: The Wage-Rate of Profits Frontier
4.0 Labor Inputs

The analysis of the choice of technique allows one to plot labor inputs versus selected variables from the price system. In any year in a stationary state, some workers will be gathering the harvest, some will be working on preparing for the harvest one year out, and some will be working on preparing for the harvest two years out. So employment, per the unvarying net output, is the sum of l0, l1, and l2. And these labor inputs can be found from a given rate of profits and a choice of s. From the wage-rate of profits frontier, one can calculate the wage for any given rate of profits. Thus, one has the two dimensions needed to draw the curves in Figure 1. One sees that, for any given wage in an interval from zero to a maximum, the quantity of labor demanded by the firms per unit output is a relation, not a function of the wage. If the relation shown were considered to be a labor demand curve, the curve would have a certain (varying) thickness.

5.0 Capital Inputs

The analysis of the choice of technique also allows one to plot the value of capital goods6 versus selected variables from the price system. I define the value of capital per unit output, given the rates of profit and the technique like so:

k(r, t, s) = (1 + r) l2(t, s) w + l1(t, s) w

This definition is such that the value of capital advanced, discounted to harvest time, and the wages paid out of the harvest add up to unity:

k(r, t, s) (1 + r) + l0(t, s) w = 1

Impose the condition here, too, that only cost-minimizing techniques are considered for a given rate of profits. Then one obtains the curves shown in Figure 3. Here, too, the analysis yields an obvious indeterminancy.

Figure 3: Capital Demanded Per Unit Output in a Stationary State
6.0 Conclusion

Does this example undermine Sraffian analysis, as well as introductory textbook labor economics?

Footnotes
  1. Notation and numerical values are chosen to be consistent with a past post.
  2. I am unsure how to explicitly represent such a production function.
  3. With three or more inputs, some complementarity among inputs is possible. I am not sure how to express this formally.
  4. I suppose the production function consistent with the data exhibits non-negative marginal returns. I am not sure it would exhibit non-increasing marginal returns. If not, I would like to see either a proof, in the general case with n dated labor inputs, that the shaded violet regions cannot arise, given such conventional properties for a production function. Or, I would like to see a concrete numerical illustration like mine, but with such conventional properties shown to hold.
  5. Also, notice the analysis of the choice of technique leads to simpler equations than those in the specification of the technology. This is not an accident.
  6. I gather that, for any given value of s, unassisted labor two years before the harvest can be used to produce one of a continuum of capital goods, depending on the value of t. And once one of these capital goods is selected, the minimum dated labor inputs in each of the three years are fixed. Maybe this way of thinking about capital goods makes issues of convexity raised in Footnote 4 of little interest.
References
  • Enrico Bellino (1993). Continuous Switching in Linear Production Models, Manchester School, V. 61, Iss. 2 (June): pp. 185-201.
  • Christian Bidard (2014). The Wage Curve in Austrian Models, Centro Sraffa Working Papers n. 3 (June).

Wednesday, July 30, 2014

The Generality Of The Sraffian Analysis Of The Choice Of Technique

Figure 1: Labor Demanded Per Unit Output in a Stationary State
1.0 Introduction

This post illustrates the analysis of the choice of technique in a case in which marginal products cannot be defined, even in the sense of an interval. It is one more example of the falsity of Austrian and vulgar neoclassical teaching. Perhaps this example suggests the possibility of, say, labor demand "functions" that have a certain thickness or cloudiness.

As far as I know, nobody has used an Austrian flow-input point-output technology to make the point about indeterminacy illustrated by this post. Bellino (1993) presents three examples in which a continuously differentiable, smooth wage-rate of profits curve is consistent with multiple technologies, with the resulting non-differentiability of micro-economic production functions. The technology in none of the three is of the structure used here. Bidard (2014) develops tools for an analysis of Austrian production functions that I found key to developing this post.

2.0 A Simple Economy

Consider a simple capitalist economy, composed of workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. The workers are paid a wage, w (in units of bushels corn per person year) out of the harvest. Capitalists obtain the rate of profits, r.

I specify two technologies, in some sense. Each technology consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques. In each technique, a bushel of corn is produced from inputs of:

  • l0 person-years of labor performed in the year of the harvest.
  • l1 person-years of labor performed one year before the harvest-year.
  • l2 person-years of (unassisted) labor performed two years before the harvest-year.

A technology is fully determined here by specifying all possible values of these dated labor inputs.

2.1 First Technology

Let s be, roughly, an element of a set Q, with Q a subset of the real numbers to be fully specified below. A capitalist knows the minimum labor requirements for each technique in this technology, where a technique is indexed by s. These labor inputs are, in obvious notation:

l1,0(s) = a + b s
l1,1(s) = c
l1,2(s) = 1/(s + 1)

where a, b, and c are positive constants, b is less than one, b is the square of a rational number, and:

b - b1/2a,

(In drawing graphs throughout this post, I use values of a, b, and c of 2, 9/16, and 3, respectively.)

2.2 Second (Sekt) Technology

I learned the word "Sekt" from Bidard; maybe he enjoys champagne. Anyways, in this technology, positive labor inputs only occur in the year of the harvest and two years before. The labor inputs one year before the harvest are zero. For convenience, define the following non-negative constants:

A = a - b + B
B = 2b1/2 + c
C = a - b + D
D = B/2

In this technology, the techniques are indexed by the variable t, where t is from the subset of the real numbers obtained by removing all elements of Q from the real numbers. The labor inputs are:

l2,0(t) = C + D t
l2,1(t) = 0
l2,2(t) = D/(t + 1)
3.0 The Choice of Technique

As usual, I consider a competitive, steady state economy in which capitalists have chosen the cost-minimizing technique, at an exogenously specified wage or rate of profits. For a given wage, the cost of a technique in the first technology is proportional to v1(r, s):

v1(r, s) = (1 + r)2 l1,2(s) + (1 + r) l1,1(s) + l1,0(s)

The corresponding function, v2(r, t), for the second technology is:

v2(r, t) = (1 + r)2 l2,2(t) + l2,0(t)

The capitalists choose the technique to minimize v1(r, s) or v2(r, t), depending on whether such minimization results in an index being selected from Q or not. The wage-rate of profits curve for a given technique in the first technology is:

w1(r, s) = 1/v1(r, s)

I hope the corresponding notation for the wage-rate of profits curve for a technique in the second technology is obvious.

The choice of the cost-minimizing technique results in specifying the indices for the technique in the two technologies as functions of the rate of profits:

s(r) = (1/b1/2)(1 + r) - 1
t(r) = r

After working out this analysis of the choice of technique for the first technology, I could have re-specified the first technology such that the index for the cost-minimizing technique was always equal to the rate of profits, as in the second technology. I worked backwards, in some sense, following Bidard, such that this nice property obtained for the second technology.

The wage-rate of profits frontier is the outer envelope of the wage-rate of profits curves for the techniques. This frontier (Figure 2) can be shown to be:

w(r) = 1/(A + Br)

I do not specify the technology for the frontier. This example has been constructed such that the both technologies have the identical frontier, when they are extended such that the index for the technique can be any real number. Since the technique varies continuously with the rate of profits, no point on the frontier is a switch point. Further, no technique on the frontier appears more than once. So this is not an example of the reswitching of techniques.

Figure 2: The Wage-Rate of Profits Frontier

I conjecture that a continuum of technologies exist with this frontier. Think of each one of these technologies as corresponding to a constant labor input in the first year before the harvest in the interval [0, c]. I guess, given Bidard's paper, this is an obvious idea.

4.0 Labor Inputs

The analysis of the choice of technique allows one to plot labor inputs, given a complete specification of the technology, versus selected variables from the price system. Accordingly, suppose the first technology is specified only when the index for the technique is a rational number. That is, the set Q is the set of rational numbers. And the set from which the index for the technique in the second technology is the set of irrational numbers.

Figure 3 is an attempt to visualize the labor demanded by firms, per unit output, in the harvest year as a function of the rate of profits. Despite appearances, neither curve in this graph is continuous; they both have an infinite number of holes. The upper curve has a countable infinity of gaps, while the lower curve has an uncountably infinite number of holes. Since these curves are discontinuous everywhere, the marginal product of labor in the harvest year is undefined. One might think of the employment for harvesters as leaping up and down between the curves as the rate of profit varies among economies.

Figure 3: Labor Inputs Per Unit Output in Harvest Year

Perhaps this example is an argument for adopting constructive mathematics in economics. But one can still get leaps between technologies with a continuous variation in the rate of profits, given a more straightforward set Q. Anyways, the labor demanded by firms, per unit output, one year before the harvest leaps between zero and the constant c. Figure 4 shows the labor demanded, per unit output, two years before the harvest. These curves are also discontinuous everywhere.

Figure 4: Labor Inputs Per Unit Output Two Years Before Harvest Year

Suppose, as in Austrian and neoclassical capital theory, that the interest rate (that is, supposedly the rate of profits) were a scarcity index for capital. A higher interest rate would indicate the availability of less capital per worker. Consequently, capitalists would supposedly be encouraged, for this sort of Austrian model, to adopt a technique in which more labor is hired during the harvest year and less during succeeding years (for example, two years beforehand). This idea is consistent with, for example, the first technology. Figures 3 and 4 show that, if one focuses solely on the blue lines, at a higher interest rate, firms want to hire less labor in the given year before the harvest and more during the harvest year. But this idea is inconsistent with the possibility of leaping from one technology to another, as illustrated in Figures 3 and 4. So this example provides another logical proof of the incorrectness of Austrian theory.

As a final step, I want to describe how to generate Figure 1, at the top of this post. In any year in a stationary state, some workers will be gathering the harvest, some will be working on preparing for the harvest one year out (except in the case of the sekt technology), and some will be working on preparing for the harvest two years out. So employment, per the unvarying net output, is the sum of l0, l1, and l2. And these labor inputs can be found from a given rate of profits. From the wage-rate of profits frontier, one can calculate the wage for any given rate of profits. Thus, one has the two dimensions needed to draw the curves in Figure 1. And these curves, as usual here, are discontinuous everywhere. One can think of the labor demanded leaping left and right in the figure as the wage varies. So much for textbook teaching about competitive labor markers.

5.0 Conclusion

The above example has demonstrated, once again, the incoherence of vulgar neoclassical theory. If I thought economists cared about the truth or falsity of their claims, I would be puzzled about mainstream teaching about labor markets and about price theory, more generally.

References
  • Enrico Bellino (1993). Continuous Switching in Linear Production Models, Manchester School, V. 61, Iss. 2 (June): pp. 185-201.
  • Christian Bidard (2014). The Wage Curve in Austrian Models, Centro Sraffa Working Papers n. 3 (June).

Tuesday, July 22, 2014

Political Philosophers On How To Read The Bible

Some time ago, I read Hobbes's Leviathan, a classic argument for the existence of a social contract. I recently became aware of the existence of Spinoza's Treatise, which argues for freedom of thought, speech, and religion. I was surprised to discover a common theme in these early modern works of political philosophy, which I did not expect. I knew from second-hand literature that both contain rational arguments about what powers secular authorities should and should not be recognized to possess.

But I was surprised to find both philosophers engaged in interpreting the bible. Both Hobbes and Spinoza quote passages warning about false prophets. Spinoza gives naturalistic principles of reading. For example, he thinks philosophical doctrines in the bible should be read with the understanding that the authors were writing to engage the understanding of the common people of the day. I did not expect that Spinoza would cite the Israelites under Moses as the canonical example of a social contract. Spinoza, kicked out of their community by the Amsterdam Jews, writes quite a bit about the New Testament. I guess this excommunication may have had something to do with more than his identification of God with the cosmos, the Creator with the creation. (I tried to read the Ethics, with its geometric proofs, long ago.)

Hobbes and Spinoza have another commonality. They spend quite a bit of time cataloging, explaining, and analyzing human sensations, emotions, and qualities. I resist the idea that certain perspectives on human nature map directly to political positions. Mayhaps, this sort of analysis of the human psyche was part of a naturalizing Enlightenment project.

I suppose addressing the topic of religion makes sense in these books. The political authorities of the day were often claiming to rule in the name of God, I gather. I do not know much about, for example, the Spanish Inquisition, but, from what I understand, the Jewish community in Amsterdam contained many families that had fled Spain. The political situation in England often saw an entanglement of religion and politics, what with the beheading of Charles I, the rule of the puritan Oliver Cromwell, the English Revolution, and so on. So if you are going to write on politics, you might want to explain how the reader need not accept the unargued proclamations of supposed authorities. You might want to explain, also, how your ideas are consistent with religion, rightly understood.

As far as I know, many later writers arguing for liberalism in thought and speech did not feel the need to argue about what secular authority can and cannot be deduced from the Bible. As examples, I do not recall any such themes in either Rousseau or Mill.

Caveat: writers on general philosophy will include matters of epistemology and ethics. How can we come to know what ethical principles to follow, if we can? I gather that Hume is an example of such an argument. I do not know if his characters take authority from Bible verses.

References
  • Thomas Hobbes (1651). Leviathan Or The Matter, Forme, & Power Of A Common-Wealth Ecclesiastical and Civill.
  • David Hume (1779). Dialogues Concerning Natural Religion [TO READ].
  • John Locke (1689). A Letter Concerning Toleration. [TO READ].
  • John Stuart Mill (1859). On Liberty.
  • Jean Jacques Rousseau (1755). A Discourse on the Origin of Inequality.
  • Jean Jacques Rousseau (1762). The Social Contract.
  • Benedict de Spinoza (16669-1670). Theological-Political Treatise.

Wednesday, July 16, 2014

Vagueness With Mathematical Economics

"Mathematics is a study which, when we start from its most familiar portions, may be pursued in either of two opposite directions. The more familiar direction is constructive, towards gradually increasing complexity: from integers to fractions, real numbers, complex numbers; from addition and multiplication to differentiation and integration, and on to higher mathematics. The other direction, which is less familiar, proceeds, by analysing, to greater abstractness and logical simplicity; instead of asking what can be defined and deduced from what is assumed to begin with, we ask instead what more general ideas and principles can be found, in terms of which what was our starting-point can be defined or deduced." -- Bertrand Russell, Introduction to Mathematical Philosophy

Many economists may be under the mistaken impression that casting economics into mathematical models ensures assumptions are explicitly stated. This is manifestly false. Here are some examples of questions about mathematical assumptions that might puzzle some economists:

  • What quantity is being conserved in typical models with agents maximizing under constraints?
  • Do equilibrium models with rational expectations apply when economic time series are non-ergodic?
  • How would the agents in such models learn to estimate model parameters if dynamics are chaotic?
  • Do models linearized around an equilibrium apply to models with multiple equilibria? Would not interesting bifurcations arise if the number of equilibria varies with model parameters?
  • Have economic models been successfully tested by dimensional analysis? (One of my favorite critiques directs one to question using a measure of capital goods in numeraire units in production functions.)
  • Can economic models with utility-maximizing agents handle preferences changing not randomly, but by agents imitating what they see other agents consuming?
  • What more general ideas and principles (consider, for example, menu independence and the absence of lexicographic preferences) underlie utility maximization?