Classification of Finite Simple Groups: A Proved Theorem?

Figure 1: Lattice Diagram for Group of Symmetries of the Square

"I shall now mention something I obviously do not understand." - Ian Hacking (2014, p. 18)

1.0 Introduction

This has nothing to do with economics. It is my attempt to get my mind around a place where I can get a glimmer of some exciting mathematics being done in my lifetime.

Mathematicians have stated a theorem for classifying finite simple groups. Whether they have proven this theorem is an intriguing question in the philosophy of mathematics.

A finite simple group is a group with a finite number of elements and no proper normal subgroup. This definition contains several technical terms. In this post, I try to explain these terms and the setting of the theorem for classifying simple groups. This preamble raises several questions:

• What is a group? A proper subgroup? A normal subgroup?
• How can a finite, non-simple group be factored into a composition of simple groups?

I try to clarify the answers to these questions by means of a lengthy example. You can probably find this better expressed elsewhere. In working this out, I relied heavily on Fraleigh's textbook, which is the only book in the references that I have read, albeit mostly in the second edition.

2.0 The Group of Symmetries of the Square

A group is a generalization, in some sense, of a multiplication table. Formally, it is a set with a binary operation, in which the binary operation satisfies three axioms. A finite group is a group in which the set contains a finite number of elements.

To illustrate, I consider the set of symmetries of the square (Figure 2). These eight elements of the set are like the numbers along the top and left side of a multiplication table. Each element is an operation that can be performed on a square, leaving the square superimposed on itself. Each operation is described in the right column of Figure 2. The third column provides a picture of the operation. The four vertices of the square are numbered so that one can see the result of the operation. The second column specifies each operation as a permutation of the numbered vertices. The first row in each permutation lists the vertices, while the second row shows which of the original vertices ends up in the place of each vertex. The first column introduces a notation for naming each operation. The remainder of this post is expressed in this notation.

Figure 2: Elements of a Group

The group operation, *, is function composition. Let a and b be elements of the set {ρ₀, ρ₁, ρ₂, ρ₀, μ₀, μ₁, σ₀, σ₁}. The product a*b is defined to be the single operation that is equivalent to first performing the operation a on the square and then performing the operation b on the result. (Many textbooks define functional composition from right-to-left, instead.) Table 1 is the multiplication table for this group, under these definitions. For example, rotating a square 90 degrees clockwise twice is equivalent to rotating the square clockwise through 180 degrees. Thus:

ρ₁ * ρ₁ = ρ₂

Table 1: The Group D₄

*	ρ0	ρ1	ρ2	ρ3	μ0	μ1	σ0	σ1
ρ0	ρ0	ρ1	ρ2	ρ3	μ0	μ1	σ0	σ1
ρ1	ρ1	ρ2	ρ3	ρ0	σ0	σ1	μ1	μ0
ρ2	ρ2	ρ3	ρ0	ρ1	μ1	μ0	σ1	σ0
ρ3	ρ3	ρ0	ρ1	ρ2	σ1	σ0	μ0	μ1
μ0	μ0	σ1	μ1	σ0	ρ0	ρ2	ρ3	ρ1
μ1	μ1	σ0	μ0	σ1	ρ2	ρ0	ρ1	ρ3
σ0	σ0	μ0	σ1	μ1	ρ1	ρ3	ρ0	ρ2
σ1	σ1	μ1	σ0	μ0	ρ3	ρ1	ρ2	ρ0

A group is defined by the following three axioms:

• The binary operation in the group is associative. That is, for all a, b, and c in the group:

(a * b) * c = a * (b * c)

• The group contains an identity element. There exists an element e in the group such that for all a in the group:

e * a = a * e = a

• Every element of the group has an inverse. For all a in the group, there exists an element a^-1 in the group such that:

a * a^-1 = a^-1 * a = e

Associativity is tedious to check for D₄. Associativity implies that one can drop parenthesis below. ρ₀ is the identity element. Every row and column in the multiplication table for D₄ contains ρ₀; thus, every element has an inverse.

An Abelian group is one in which the binary operation is commutative. The group of symmetries of the square is not Abelian. For an Abelian group, the multiplication table is symmetric across the principal diagonal; it does not matter to the result in which order one performs the operation for two arguments. The following two equations illustrates that D₄ is not Abelian:

μ₀*ρ₁ = σ₁

ρ₁*μ₀ = σ₀

In words, flipping a square around its horizontal axis of symmetry and then rotating it ninety degrees clockwise is not equivalent to rotating it ninety degrees clockwise and then then reflecting it across that axis. The result of the first composition of operations is equivalent to reflecting the square across the diagonal axis of symmetry running from the south west to the north east. The second composition of operations is equivalent to flipping the square across the other diagonal.

One can also set up equations in a group, for example:

ρ₁*ρ₂*x = μ₀

Then x must be σ₀. Solving a Rubik's cube is analogous to solving such an equation.

3.0 Proper and Improper Subgroups

Some rows and columns in Table 1 can stand alone as a group. The entries in these restricted row and columns all appear as headings in the rows and columns. These entries form a subgroup of the original group. One-fourth of the table in the upper left of Table 1 provides an example. {ρ₀, ρ₁, ρ₂, ρ₃} is a subgroup of D₄ (Table 2).

Table 2: A Subgroup of D₄ with Four Elements

*	ρ0	ρ1	ρ2	ρ3
ρ0	ρ0	ρ1	ρ2	ρ3
ρ1	ρ1	ρ2	ρ3	ρ0
ρ2	ρ2	ρ3	ρ0	ρ1
ρ3	ρ3	ρ0	ρ1	ρ2

The group D₄ has ten subgroups, as shown in the Lattice Diagram in Figure 1 above. Subgroups have been defined such that, for any group G, the group G is a subgroup of itself. Another trivial case, the one-element group consisting of the identity element, also provides a subgroup of G. These two subgroups are known as improper subgroups. All other subgroups are proper subgroups.

One can make a couple of observations about subgroups. The binary operation in the group is the same as the binary operation in the subgroup. The property of associativity carries over from the group to the subgroup. Since a subgroup is a group, it must contain an identity element. And that identity element must also be the identity element for the group containing the subgroup. Thus, every subgroup of D₄ contains ρ₀. Likewise, for every element of a subgroup, the subgroup must also contain its inverse. Finally, the number of elements in a subgroup must evenly divide the number of elements in the group.

I have shown above how the eight elements of D₄ can be defined in terms of permutations. As a matter of fact, the set of permutations of (1, 2, ..., n) form a group under the operation of function composition. This permutation group is designated as S_n, and it contains n! elements. Thus, S₄ contains 24 (= 4x3x2x1) elements. Not only can one find all the subgroups of D₄, one can extend the group such that D₄ is a subgroup of that extended group.

4.0 Isomorphic Groups

In a group, the order of rows and columns in the multiplication table are of no matter. Likewise, the names of the elements are irrelevant to the structure of the group. Two groups are isomorphic if the multiplication table for one group can be mapped into the multiplication table for another group by reordering and renaming the elements of, say, the first group. As an example, consider the groups {ρ₀, ρ₂, μ₀, μ₁} and {ρ₀, ρ₂, σ₀, σ₁}. They each have the same number of elements, which is necessary for an isomorphism. Table 3 defines the group operation for the first group. Suppose that, in Table 3, μ₀ is renamed σ₀, and μ₁ is renamed σ₁ throughout. The resulting table will match the operation for the second group. Thus, the two groups are isomorphic.

Table 3: The Group {ρ₀, ρ₂, μ₀, μ₁}

*	ρ0	ρ2	μ0	μ1
ρ0	ρ0	ρ2	μ0	μ1
ρ2	ρ2	ρ0	μ1	μ0
μ0	μ0	μ1	ρ0	ρ2
μ1	μ1	μ0	ρ2	ρ0

The groups in Tables 2 and 3 are NOT isomorphic. They each contain four elements. Each element, however, in the group in Table 3 is its own inverse. This is an algebraic property, preserved no matter how the elements of the group are renamed. And the group in Table 2 does not have this property. As a matter of fact, only two groups containing four elements exist, up to an isomorphism. In other words, any group with four elements is isomorphic to either the group in Table 2 or to the group in Table 3.

Furthermore, only one group, up to isomorphism, contains two elements. Its operation is defined by Table 4. All the subgroups of D₄ containing two elements are isomorphic to this group and, ipso facto, to each other. The text colors of the subgroups in the lattice diagram (Figure 1) express these isomorphisms.

Table 4: The Unique Group (Up To Isomorphism) With Two Elements

*	0	1
0	0	1
1	1	0

5.0 Normal Subgroups, Factor Groups, and Homomorphisms

Certain additional patterns are apparent in Table 1. I have already pointed out that the first four rows and columns constitute the subgroup with the operation shown in Table 2. Notice that none of the entries in the last four columns for the first four rows are in this subgroup. Likewise, none of the entries in the first four columns for the last four rows are in this subgroup. On the other hand, the entries in the remaining rows and columns in the lower right are all in this subgroup. Can you see that these observations reveal the pattern expressed in Table 4? Mathematicians express this by saying that the factor group D₄/{ρ₀, ρ₁, ρ₂, ρ₃} is isomorphic to the group with two elements.

A subgroup is normal if it can be used to divide up the rows and columns in the multiplication table for the group like this. For another example, consider the subgroup {ρ₀, ρ₂}. Table 5 shows a reordering of the rows and columns in Table 1 to facilitate the calculation of the factor group for this subgroup. Consider dividing this grid up into 16 blocks of two rows and two columns each. Each block will contain two elements of the group D₄, and which element is paired with each element does not vary among these blocks.

Table 5: The Group D₄ Reordered

*	ρ0	ρ2	ρ1	ρ3	μ0	μ1	σ0	σ1
ρ0	ρ0	ρ2	ρ1	ρ3	μ0	μ1	σ0	σ1
ρ2	ρ2	ρ0	ρ3	ρ1	μ1	μ0	σ1	σ0
ρ1	ρ1	ρ3	ρ2	ρ0	σ0	σ1	μ1	μ0
ρ3	ρ3	ρ1	ρ0	ρ2	σ1	σ0	μ0	μ1
μ0	μ0	μ1	σ1	σ0	ρ0	ρ2	ρ3	ρ1
μ1	μ1	μ0	σ0	σ1	ρ2	ρ0	ρ1	ρ3
σ0	σ0	σ1	μ0	μ1	ρ1	ρ3	ρ0	ρ2
σ1	σ1	σ0	μ1	μ0	ρ3	ρ1	ρ2	ρ0

These observations can be formalized by the function defined in Table 6. For an element a of D₄, let f(a) denote the map defined in Table 6. To find the value of this function, locate a in the first column. Whether this value is 0, 1, 2, or 3 is determined by the corresponding entry in the second column. For all a and b in D₄:

f(a * b) = f(a) o f(b)

A map from one group to another with this property is a homomorphism. An isomorphism is a homomorphism, but a homomorphism is a more general concept. Homomorphisms do not need to leave the number of elements in the group invariant.

Table 6: A Homomorphism from D₄ to {0, 1, 2, 3}

Elements of D4	Image
ρ0, ρ2	0
ρ1, ρ3	1
μ0, μ1	2
σ0, σ1	3

The factor group D₄/{ρ₀, ρ₂} is easily calculated. Replace each element of D₄ in Table 5 by its image under the homomorphism in Table 6. Collapse each pair of rows and columns. One ends up with Table 7, where I have renamed the group operation, as above. The factor group D₄/{ρ₀, ρ₂} is isomorphic to the group with four elements with the operation shown in Table 3 above. The number of elements in a factor group is the quotient of the number of elements in the original group and the number of elements in the subgroup used to form the factor group.

Table 7: The Factor Group D₄/{ρ₀, ρ₂}

o	0	1	2	3
0	0	1	2	3
1	1	0	3	2
2	2	3	0	1
3	3	2	1	0

The two improper subgroups for any group are normal and yield trivial factor groups. The factor group D₄/D₄ is isomorphic to the one-element group whose only member is the identity element. The factor group D₄/{ρ₀} is isomorphic to D₄. The factor groups for improper subgroups provide no information about the structure of a group.

6.0 A Subgroup that is Not Normal

Not all subgroups are normal. The subgroup {ρ₀, μ₀}, for example, is not a normal subgroup of D₄. Table 8 proposes a map from the elements of the group to the first four natural numbers. And Table 9 illustrates another reordering of the rows and columns in Table 1, with the entries replaced by the natural numbers to which they map. If one confines oneself to the first two columns, each pair of rows could be collapsed into one, with the label from the row taken from the map. But this process breaks down for the next two and the last two columns.

Table 8: A Map from D₄ to {0, 1, 2, 3} that is Not a Homomorphism

Elements of D4	Image
ρ0, μ0	0
ρ1, σ0	1
ρ2, μ1	2
ρ3, σ1	3

Table 9: Another Reodering of The Group D₄

*	ρ0	μ0	ρ1	σ0	ρ2	μ1	ρ3	σ1
ρ0	0	0	1	1	2	2	3	3
μ0	0	0	3	3	2	2	1	1
ρ1	1	1	2	2	3	3	0	0
σ0	1	1	0	0	3	3	2	2
ρ2	2	2	3	3	0	0	1	1
μ1	2	2	1	1	0	0	3	3
ρ3	3	3	0	0	1	1	2	2
σ1	3	3	2	2	1	1	0	0

Suppose a subgroup contains n elements. To determine if the subgroup is normal, it is sufficient to examine the first n rows and the first n columns in the reordered table. This capability follows from a theorem about what are known as left and right cosets for a subgroup.

The permuation group S₄ provides another example of a subgroup that is not normal. By my calculations, D₄ is NOT a normal subgroup of S₄.

7.0 The Composition Series of a Group

At this point, I have completed my explanation of the lattice diagram at the top of this post, including circles, text colors, and boxes. I draw from these results to illustrate how a non-simple group, namely D₄, can be expressed as a composition of factor groups.

Table 10 lists twelve series of subgroups of the group of symmetries of the square. Each series has the following properties:

• The leftmost group in the series is the one-element group containing the identity element.
• The rightmost group is D₄.
• Each group in the series (except D₄) is a proper normal subgroup of the group immediately to the right of it in the series.

A series with these properties is known as a subnormal series of the group D₄. If every group in the series is also a normal subgroup of D₄, the series is a normal series of the group D₄. By the last property in the bulleted list, one can calculate a factor group for each pair of immediately successive groups in the series.

Table 10: Twelve Normal and Subnormal Series for D₄

Number Factor Groups	Series	Normal Series
1	{ρ0} < D4	Yes
2	{ρ0} < {ρ0, ρ1, ρ2, ρ3} < D4	Yes
2	{ρ0} < {ρ0, ρ2, μ0, μ1} < D4	Yes
	{ρ0} < {ρ0, ρ2, σ0, σ1} < D4	Yes
	{ρ0} < {ρ0, ρ2} < D4	Yes
3	{ρ0} < {ρ0, ρ2} < {ρ0, ρ1, ρ2, ρ3} < D4	Yes
	{ρ0} < {ρ0, ρ2} < {ρ0, ρ2, μ0, μ1} < D4	Yes
	{ρ0} < {ρ0, ρ2} < {ρ0, ρ2, σ0, σ1} < D4	Yes
	{ρ0} < {ρ0, μ0} < {ρ0, ρ2, μ0, μ1} < D4	No
	{ρ0} < {ρ0, μ1} < {ρ0, ρ2, μ0, μ1} < D4	No
	{ρ0} < {ρ0, σ0} < {ρ0, ρ2, σ0, σ1} < D4	No
	{ρ0} < {ρ0, σ1} < {ρ0, ρ2, σ0, σ1} < D4	No

The definition of an isomorphism for a subnormal series builds on the definition of isomorphism for groups. Consider the factor groups arising in each series from successive pairs of subgroups in each series. Two series are isomorphic if they contain the same of number of factor groups, in this sense, and these factor groups are isomorphic. The order in which the factor groups arise can vary among isomorphic subnormal series.

I have collected isomorphic series together, in Table 10, by means of horizontal lines in the first column. The series with one factor group is not isomorphic to any other series. The first series shown with two factor groups is not isomorphic to the other three series with two factor groups. And those three series are isomorphic to one another. All of the series with three factor groups are isomorphic to one another.

The series with three factor groups have another property. All factor groups in these series with three factor groups are simple groups. That is, they contain no proper normal subgroups. A subnormal series of a group in which all factor groups formed by the series are simple is known as a composition series. By the Jordan-Hölder Theorem, all compositions series for a group are isomorphic series. This theorem justifies one in speaking of THE composition series for a group. Finding the factor groups in a the composition series for a group is somewhat analogous to factoring a natural number. Note that D₄ contains eight elements and each of the three factor groups in the composition series contain two elements. Furthermore,

8 = 2³

For a natural number, the prime factors can be combined to yield the original number. Here the analogy apparently breaks down. The factor groups in a composition series for a group constrain the structure of the group, but two non-isomorphic groups can have the same composition series. But still, mathematicians have solved various problems in group theory for finite non-simple groups by use of the classification of finite simple groups.

Composition series apparently have an application in solving polynomial equations. The composition series for the permutation group S₅ contains a factor group that is non-Abelian. This is connected with the insolvability of the quintic. There are formulas for zeros for cubic and fourth order polynomial, analogous to the quadratic formula. But there is no such formulas for poynomials of the fifth degree and higher.

8.0 Classification of Finite Simple Groups

At this point, I have explained how finite simple groups arise as factor groups for the composition series of any finite group. I hope that this gives some hint of why the following theorem is of interest.

Theorem: Each finite simple group is one of the following, up to an isomorphism:

• A group of prime order.
• An alternating group.
• A Lie group.
• One of 26 sporadic groups not otherwise classified.

I am aware that this this theorem uses technical terms I still have not explained, including one that I simply do not understand myself.

The sporadic groups are finite simple groups that do not fall into the other categories, although, I gather, some sporadic groups are related to one another.The sporadic group with the largest number of elements is called the Monster group. It has 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements.

9.0 History of the Theorem

In 1972, Daniel Gorenstein proposed that mathematicians could complete a classification of all simple groups. By the early 1980s, mathematicians had stated the theorem and those specialists who had pursued Gorenstein's program believed they had proven it. The proof, however, was scattered among (tens of?) thousands of pages in hundreds(?) of papers in many mathematics journals. No one person had probably ever understood the proof or read it in its entirety.

The proof, however, was discovered even then to be incomplete. Steve Smith and Michael Aschbacher worked on closing this gap, relating to quasithin groups. They succeeded by 2004.

Meanwhile, a number of mathematicians have been trying to simplify the proof and to restate it in one location. The ambition of these mathematicians is to produce a "second generation" proof of only a couple thousand pages.

Has a theorem been proven if only one or two mathematicians have read the proof in its entirety? How about if nobody has, which would have been the case in the 1980s if the proof had indeed been valid? Certainly, the proof of the classification theorem is not surveyable, in Wittgenstein's sense. Do mathematical results need to be established by a social process? If so, how can such social processes be characterized?

Appendix: Terms Defined or Illustrated Above

Abelian group, Associativity, Composition Series, Factor Group, Finite Group, Group, Homomorphism, Identity Element, Improper Subgroup, Inverse, Isomorphic Groups, Isomorphic Subnormal Series, Lattice Diagram, Normal Series, Normal Subgroup, Permutation Group, Proper Subgroup, Subgroup, Subnormal Series.

References

• Michael Aschbacher (2004). The Status of the Classification of the Finite Simple Groups, Notices of the AMS, V. 51, No. 7 (Aug.): pp. 736-740.
• Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon (2011). The Classification of Finite Simple Groups: Groups of Characteristic 2 Type, American Mathematical Society.
• Nicolas Bourbaki (1943). Elements of Mathematics: Algebra I: Chapters 1-3.
• J. H. Conway and S. P. Norton (1979). Monstrous Moonshine, Bulletin of the London Mathematical Society, V. 11, no. 3: pp. 308-339.
• John B. Fraleigh (2002). A First Course in Abstract Algebra, 7th Edition, Pearson.
• Daniel Gorenstein, Richard Lyons, and Ronald Solomon (1994). The Classification of the Finite Simple Groups, American Mathematical Society.
• Ian Hacking (2014). Why is there Philosophy of Mathematics at all?, Cambridge University Press.
• Daniel Kunkle and Gene Cooperman (2007). Twenty-Six Moves Suffice for Rubik's Cube, ISSAC'07, 29 Jul. - 1 Aug., Waterloo, Canada.
• Tomas Rokicki (2008). Twenty Five Moves Suffice for Rubik's Cube.

European Monetary Union Without Political Union

I recently read Richard Davenport-Hines' Universal Man: The Lives of John Maynard Keynes. One thing I learned was of the existence of the Latin Monetary Union.

Apparently, in the latter half of the nineteenth century, gold and silver coins circulated in a number of European countries in which they speak Romance languages. And the amount of gold or silver in these coins was specified. I guess this is part of being on the gold standard. I gather the countries in the Latin Monetary Union agreed on a fixed ratio of silver to gold. As part of this agreement, coins from all these countries circulated freely throughout these countries. You could spend a franc coin in Italy just as conveniently as a lira coin.

I am surprised that this union lasted past World War I. From Keynes' Tract on Monetary Reform (1924), I recall something about the European inflations and deflations that hit Europe after World War I. Yet from my limited reading, I do not recall much about the stresses that must have arisen in this monetary union. Larger issues seem to me to revolve around how the allies in the United States in the war could pay off their loans and how Germany could pay their reparations, agreed to at Versailles, while abiding by the limitations on their economy - such as the occupation of the Ruhr - imposed by the allies. My interest here might be biased by my interest in Keynes, since these issues were a major point of Economic Consequences of the Peace.

Two Views On Introductory Economics

Recently, two bloggers have commented on what is taught in college classes for introduction to economics¹. Noah Smith accepts simple partial equilibrium models of perfect competition as internally valid². He argues, however, that "Economics 101" models should be complemented, especially in policy applications, with complications introduced in more advanced models. Robert Paul Wolff, on the other hand, uses introductory economics as an example of ideological bullshit, to use Frankfort's technical term.

As far as I am concerned, simplistic supply-and-demand reasoning has been shown to be an incoherent mishmash decades ago. Like Prof. Wolff, I like to justify this view by referring to accepted findings of research literature. I particularly like to emphasize the supposed market for labor. Why do economists not revise their teaching³ so it is not susceptible to being criticized as ideology? I offer three suggestions to complement Wolff's treatment.

First, perhaps economists who teach outdated nonsense are just doing their job. Introductory courses are followed by later courses. And teachers of later courses expect students who have satisfied the prerequisites to have been exposed to graphs of supply and demand functions, the theory of utility maximization, marginal cost, marginal revenue, the First Order Conditions for maximization, consumer and producer surplus, etc. You might hope for teachers who introduce a bit of pluralism. But even economists who agree with me might find it challenging for the students to be both exposed to critiques and alternatives, and yet gain a command over the conventional material.

Second, perhaps the situation might be thought of as a type of coordination game, as in modeling a totalitarian society. Maybe the majority of economists privately think that they are being asked to teach balderdash. But, with the profession being the way it is, they see little benefit in saying so. Each sees others as publicly accepting what is being taught. So they put their doubts aside. If all were to be forthright at once, the situation would be different. But how could teaching transverse from the current equilibrium to that new one?

Third, maybe many economists come to accept what they are teaching as a way of managing cognitive dissonance. It must be an uncomfortable feeling to know one is spouting nonsense and, if one wants to advance in the profession, to be impotent to change it. Better come to accept the nonsense⁴.

Footnotes

1. Both bloggers seem to be concentrating on microeconomics.
2. Is Noah's conflation of elasticity with the slope of a function an acceptable simplification for a mass audience? Or just muddle?
3. I do not teach.
4. I guess this is related to the just world fallacy.

Krugman On Robert Reich's New Book

1.0 Introduction

Robert B. Reich has a new book,Saving Capitalism: For the Many, Not the Few out last year. Paul Krugman reviewed it, on 17 December 2015, in The New York Review of Books. In this post, I record a negative reaction I have to this review. I do not think I am formulating a strong argument, rather merely making a claim that needs more justification.

2.0 Review of Reich's Book

Reich notes that many people portray the major political economic choice in the United States of America as between free markets and government intervention. Reich rightfully rejects this false dichotomy and argues that government creates the markets. Consider such matters as the definition of property rights; what practices are permitted in the market by, say, antitrust law; what contracts will be enforced in courts of law; what legal options, such as bankruptcy, agents can resort to when unforeseen circumstances arise; and the distribution of the allotment of resources to enforcement of various laws. Decisions along these lines create markets, and government can choose various sides. These choices are not necessarily interventions, but constitutive of the definition of markets.

Many examples can be cited. Think of intellectual property, such as copyrights and patents. Consider how markets arise, from cap and trade polices, for pollution permits. Or think of the labor market. Some states will not permit corporations and unions to agree to contracts in which every worker at some specified rank must be a union member; rather, corporations are permitted to hire workers that get the benefit of union wages without making contributions. One could simplify voting for unions by instituting card check. And, if workers choose to join a union, why shouldn't that union be able to freely choose the portion of their budget to spend on political lobbying?

Various myths follow from an acceptance of the false dichotomy. For example, the theory of marginal productivity has been read by many since its creation to say workers are paid in the market what they are worth. Reich also looks at the reality of how corporate executives have increased their pay.

Market processes and their outcomes refract social and legal norms, not natural laws. These norms and their outcomes differ a lot between the post-(World) war (II) golden age and the neoliberal world established after the end of Bretton Woods. Capitalism is a dynamic system, and the current rules are always changing. I do not see why, with lots of struggle, vicious circles currently enriching the few cannot be overthrown and shared prosperity be re-established to some extent.

I have some suggestions for how Reich could strengthen his arguments. I think Reich slips into polemics sometimes when I would prefer more analysis¹. I wish Reich would reference more scholars and traditions developing similar points². I think John Kenneth Galbraith shows an awareness of traditions I like, and Reich does have Galbraith's notion of countervailing power as a major theme in his book. Maybe explorations of these traditions would lead Reich to more radical conclusions³. I think Reich still has a hankering for the theory of perfect competition. Even if markets were perfect and corporate boards did not consist of overlapping sets of cronies, neither wages nor executive pay would be determined by marginal productivity.

4.0 Krugman's Review

Paul Krugman's review is generally positive⁴. This contrasts with how Krugman used to write about Reich back in the 1980s and 1990s. For Krugman then, Reich was a policy entrepreneur who did not measure up to the supposedly rigorous standards of mainstream economists.

A major theme of Reich's book is power. Krugman, by casting this theme in terms of market power, asserts (mainstream) economists have long addressed this issue. I agree that mainstream economists have models addressing this idea:

"Market power has a precise definition: it’s what happens whenever individual economic actors are able to affect the prices they receive or pay, as opposed to facing prices determined anonymously by the invisible hand." -- Paul Krugman

Given this orientation, Krugman can argue against Stigler's claim that Chicago school models of perfectly competitive markets are empirically adequate. Krugman also takes the opportunity of Reich's book to argue that the theory of Skill-Biased Technical Change (SBTC) is mistaken. I think Krugman is reading Reich's book in a more mainstream economist's world of discourse⁵ than, in fact, is and should be the case.

Footnotes

1. Maybe this is a matter of contrasting tastes. I'm less likely to draw policy conclusions. Reich certainly knows more about Washington than I do.
2. For example, institutional economics; Karl Polanyi's The Great Transformation; Hacker and Pierson's Winner Take All Politics; theories of adminstrative, full-cost, or markup pricing.
3. What substantive disagreement is involved in saying your goal is saving capitalism, as opposed to instituting social democracy?
4. The back cover of Reich's book features blurbs from Laura D'Andrea Tyson, Joseph Stiglitz, and Lawrence Summers, economists all.
5. Reich does, in fact, address the (incoherent and incorrect) theory of SBTC.

Marxist-Feminist-Anti-racist-Ecological Economics

I have recently read Julie Nelson's 1995 essay in the Journal of Economic Perspectives. She thinks - and this is a well-established idea among academics - that gender and sex are not the same. One is socially constructed, and the other relates more to a physical substratum. This concept goes back as far as Simone de Beauvoir's The Second Sex. She argues that woman is defined as the negative of man:

"Humanity is male, and man defines woman, not in herself, but in relation to himself; she is not considered an autonomous being... she is nothing other than what man decides; she is thus called 'the sex,' meaning that the male sees her essentially as a sexed being; for him she is sex, so she is it in the absolute. She is determined and differentiated in relation to man, while he is not in relation to her; she is the inessential in front of the essential. He is the Subject; he is the Absolute. She is the Other." -- Simone de Beauvoir

Table 1: Gender-Coded Dualisms

Male	Female
Objectivity	Subjectivity
Strength	Weakness
Self-Interested	Caring
Thinking	Feeling

In this way of analyzing social customs, one might see homo economicus as gendered male. One might wonder if the traditional neoclassical analysis of the optimizing, but constrained, agent is only a partial viewpoint. Do the objective functions in typical neoclassical models miss goals that are often coded as feminine, for example, altruism? (Might your answer have varied since the publication of Nelson's essay?)

Thinking about how certain dualisms are gender-coded might lead one to thinking about other groups that are taken by hegemonic groups as Other. Socially constructed race is an obvious category in the United States in my lifetime. Looking about, I might think that intellect versus physicality is an analogous dualism for race, with intellect being assigned to whites and physicality assigned to blacks. But reading Eldridge Cleaver's Soul on Ice long ago taught me that such assignments vary with time and space. Cleaver thought that both superior intellect and superior physical fitness were assigned to whites. I suppose you can see such tropes in old books, say, Edgar Rice Burroughs' Tarzan series.

Feminist economics also points to the need for economists to analyze the household. This idea of looking outside a narrow definition of economic activity for a full understanding of markets reminds me of another argument, namely Schumacher's in Small is Beautiful. Economists need to also look outside markets to natural ecologies to fully understand markets.

Suppose one is interested in how an advanced capitalist economy, such as in the United States, can sustain itself. How is reproduction, either on the same or an expanded scale, possible? This question was explored by Marx. Furthermore, to fully address this question, one must look beyond the economy of the advanced country, narrowly defined. For an economy to be reproduced, preconditions must be met in:

• The households, in which workers are reproduced and cared for. Households are outside markets, but provide a necessary foundation on which markets rest.
• Other economies, particularly in the third world, where many resources are extracted and production for the market is off-shored these days. That is, the activities in other countries, outside the United States, provide a foundation on which American capitalism rests.
• Nature, which also lies outside markets and provides a necessary foundation on which markets rest.

Thus, there is a need for a Marxist-Feminist-Anti-racist-Ecological economics.

References

• Simone de Beauvoir (1949, 2009). The Second Sex, Trans. by Constance Borde and Sheila Malovany-Chevallier.
• Eldridge Cleaver (). Soul on Ice.
• Robin Hahnel (2016). Environmental Sustainability in a Sraffa Framework, Proceedings of the American Economic Association.
• Julie A. Nelson (1995). Feminism and Economics, Journal of Economic Perspectives, V. 9, No. 2 (Spring): pp. 131-148
• E. F. Schumacher (1973). Small is Beautiful: Economics as if People Mattered.

Frugal Science

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

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

Obscure Postmodern Language

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

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

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

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

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

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

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

Update (21 December 2015):

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

Footnotes

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

References

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

Keynes On Rational Expectations And Policy Ineffectiveness

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

In Chapter III, Keynes states a general principle:

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

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

Herbert Scarf (1930-2015)

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

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

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

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

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

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

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

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

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

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

Update to my Paper on Pension Capitalism

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

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

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

Feels Like Science

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

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

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

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

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

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

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

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

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

A Bifurcation Diagram for Hahn and Solow

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

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

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

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

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

Reference

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

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

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

1.0 Introduction

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

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

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

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

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

g(Q) = [ξ/(ξ - 1)] [β/(1 - β)]

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

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

h_t = G^t,

where:

G ≥ 1.

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

g(Q) = G [ξ/(ξ - 1)] [β/(1 - β)]

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

3.0 Other Generalizations

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

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

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

4.0 Empirical Applicability of Generalized Model?

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

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

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

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

4.1 How Are Dynamic Equilibrium Paths Found?

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

4.2 No Keynesian Uncertainty

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

5.0 Conclusions

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

Paul Krugman Stumbles

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

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

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

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

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

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

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

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

1.0 Introduction

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

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

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

2.0 Overlapping Generations Model

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

2.1 Households

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

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

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

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

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

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

such that:

c_t,t + s_t ≤ w_t

c_{t,t + 1} ≤ Q_ξ(R_t) s_t

c_{t,t + 1} ≤ ξ m_t p_t/p_{t + 1}

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

ξ > 1

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

2.1.1 Hahn and Solow's First Example

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

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

where,

α < 1

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

ε = α/(α - 1)

2.2 An Aggregate Cobb-Douglas Production Function

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

y_t = (k_{t - 1})^β (l_t)^{β + 1}

where:

0 < β < 1

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

3.0 Stationary States

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

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

g(Q) = Q s(Q)

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

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

g(Q) = [ξ/(ξ - 1)] [β/(1 - β)]

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

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

Table 1: Number of Stationary States

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

4.0 Conclusion

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

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

Footnotes

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

Reference

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

Paul Romer Gyring In A Cul-De-Sac

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

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

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

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

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

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

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

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

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

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

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

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

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

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

And he attacks Joan Robinson's motives:

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

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

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

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

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

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

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

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

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

References

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