Chaotic Cobwebs (Part 1 1/2)

5.0 Return to a Special Case

This post continues the examination of a cobweb cycle with a non-linear demand curve. In this part, I talk again about the special case examined in Section 3 of that previous post. In that case, the parameters b and e of the model are zero. Furthermore, I continue to assume c is 3/5 and d is 21/20. Figure 6 shows attracting limiting behavior for a whole range of the parameter a. The abscissa in this graph is a, with a set to unity at the right edge. The ordinate is the limiting values of the normalized quantity, Q(t). As I point out towards the right, a two-period cycle shows up on the graph as a plot of two values of the ordinate for the value of a for which that cycle is generated. One can see the period-doubling scenario leading to chaos as one moves to the left on the graph. By the way, this figure is a fractal, repeating on an infinite number of scales. It has both qualitative and quantitative universal features for a certain family of one-dimensional maps (for example, characterized by the Feigenbaum constant).

Figure 6: Structural Dynamics of a Special Case

6.0 An Economically Relevant Special Case

The case above has both demand and supply functions through the origin in the quantity-price space. I want to consider a case in which the demand curves intersects the price axis at a strictly positive price and slopes downward in the first quadrant. Accordingly, consider the case where a is unity, b is 781/960, c is 3/5, d is 1/2, and e is zero. (I put aside questions of whether adaptive expectations make sense here or whether a partial equilibrium framework with monotonic supply and demand curves is justified - see implications of the Sonnenschein-Debreu-Mantel theorem.)

Figure 7 shows how the price and quantity evolve for selected initial values. The red line suggests the point equilibriating of supply and demand is unstable. It will never be observed in this model. Instead, the red line evolves to a two-period limit cycle. The blue line shows that points outside that cycle will evolve inward to that cycle. As a matter of fact, I chose parameter values such that Figures 2 and 3 in Section 3 of the previous post show the behavior of the normalized quantity for this case.

Figure 7: Temporal Dynamics with All Positive Prices and Quantities

I don't see that one can find positive parameters values such that period doubling can lead to chaos in this model with price and quantity remaining positive throughout. Although I am not sure of this, I am willing to accept that the maximum of the quadratic function in Equation 4 (see previous post) must be strictly positive for chaos to arise in the first quadrant. I don't understand what Richard Goodwin graphs in Figures 2.4 and 2.5 of his book. Perhaps he is considering a cubic demand curve or some other higher polynomials for supply and demand.

Update (28 Sep.): A Google search on "cobweb", "chaos", and "economics" shows lots of literature, mostly behind paywalls. I notice particularly work by Barkley Rosser, Jr. and in the Journal of Economic Behavior and Organization, which he edits. So I have reading I could do.

Writing Down Von Neumann's Contributions to Game Theory

Today's New York Times, in the science section, contains an ad for the "Von Neumann Memorial Lectures". This reminded me that, several years ago in the Times, Hal Varian misrepresented Von Neumann's treatment of game theory:

"Modern game theory was developed by the great mathematician John Von Neumann in the mid-1940s. His goal was to understand the general logic of strategic interaction, from military battles to price wars.

Von Neumann, working with the economist Oscar Morgenstern, established a general way to represent games mathematically and offered a systematic treatment of games in which the players' interests were diametrically opposed. Games of this sort - zero-sum games - are common in sporting events and parlor games.

But most games of interest to economists are non-zero sum. When one person engages in voluntary trade with another, both are typically made better off. Although von Neumann and Morgenstern tried to analyze games of this sort, their analysis was not as satisfactory as that of zero-sum games. Furthermore, the tools they used to analyze these two classes of games were completely different.

Mr. Nash came up with a much better way to look at non-zero-sum games. His method also had the advantage that it was equivalent to the von Neumann-Morgenstern analysis if the game happened to be zero sum." -- Hal R. Varian (2002).

I find it hard to read this as saying anything other than:

• Nash generalized the Von Neumann and Morgenstern (VNM) solution to zero-sum games to a solution (the Nash solution) applying to both zero-sum and non-zero-sum games.
• Although VNM had a solution for non-zero-sum games, it was not a generalization of their solution for zero-sum games.

Both claims are false.

Varian's statement only makes sense if one pretends The Theory of Games and Economic Behavior (TGEB) is missing the almost 300 pages on zero-sum n-person games. Under this pretense, the only zero-sum games treated in TGEB would be two-person games. The Nash equilibrium is, in some sense, a generalization of the VNM minimax treatment of two-person zero sum games. And the TGEB treatment of coalitions in non-zero sum games is something else.

VMN do decompose their treatment of games into two phases, but not based on whether or not a game is zero sum. They decompose their treatment into zero-sum two-person games and all other games (All quotations of numbered paragraphs are of the third edition of TGEB):

"66.1.2. Our theory of games divides clearly into two distinct phases: The first one comprising the treatment of the zero-sum two-person game and leading to the definition of its value, the second one dealing with the zero-sum n-person game, based on the characteristic function, as defined with the help of the values of the two-person games."

The TGEB solution of n-person zero-sum games is, in some sense, a generalization of the TGEB minimax solution of zero-sum two person games. One can form two "collective persons" for the n-person game, where each "person" is one of two coalitions:

"25.1.2. Suppose then that we have a game Gamma of n players... Without yet making any predictions or assumptions about the course a play of this game is likely to take, we observe this: if we group the players into two parties, and treat each party as an absolute coalition - i.e. if we assume full cooperation within each party - then a zero-sum two-person game results..."

In the TGEB treatment, a coalition can pool their winnings and then redistribute them to the players in the coalition. VMN define a solution to a game as a set of imputations of payouts to the players. The definition of the set of imputations is concerned with why a player would chose to be in one coalition or the other, and why the remaining members of the winning coalition would chose to woo a player or not.

To help fix intuition, VMN define an interesting zero-sum three person game, the Majority Game:

"21.1...Each player, by a personal move, chooses the number of one of the two other players. Each one makes his choice uninformed about the choices of the two other players.

After this the payments will be made as follows: if two players have chosen each other's numbers we say that they form a couple. Clearly there will be precisely one couple, or none at all. If there is precisely one couple, then the two players who belong to it get one-half unit each, while the third (excluded) player correspondingly loses one unit. If there is no couple, then no one gets anything."

The TGEB analysis of a generalization of the Majority Game is indeterminate in two senses:

• An uncountably infinite number of solution sets of imputations exist (some of which VMN describe as analogous to discrimination).
• In the most obvious solution, { (1/2, 1/2, -1), (1/2, -1, 1/2), (-1, 1/2, 1/2) }, how much a player gets and whether or not he is in the winning two-person coalition is indeterminate (which of the three imputations is realized is unspecified)

VNM generalize their treatment of zero-sum games to non-zero sum games by introducing a powerless dummy:

"56.2.1. ...any given general [not necessarily zero-sum] game can be re-interpreted as a zero-sum game...Our procedure will be to interpret an n-person general game as an n+1-person zero-sum game."

Contrary to Varian, the TGEB treatment of non-zero sum games is a generalization of the TGEB treatment of zero sum games. The VNM solution has come to be known as a solution to cooperative games. (If one sets aside his analysis of bargaining, Nash treats non-cooperative games.) Trivially, only one set of imputations is a solution to a zero-sum two-person game. There is only one imputation in that set, and that imputation is equivalent to the minimax solution.

TGEB has lots of interesting asides and suggestions that relate to later ideas. For example, VNM suggest an alternative treatment in which an external enforcement mechanism for (contracts between players in) cooperative games is not needed. In this alternative treatment of iterative play, cooperation emerges spontaneously:

"21.2.3. If our theory were applied as a statistical analysis of a long series of plays of the same game - and not as the analysis of one isolated play - an alternative interpretation would suggest itself. We should then view agreements and all forms of cooperation as establishing themselves by repetition in such a long series of plays.

It would not be impossible to derive a mechanism of enforcement from the player's desire to maintain his record and to be able to rely on the record of his partner. However, we prefer to view our theory as applying to an individual play. But these considerations, nevertheless, possess a certain significance in a virtual sense. The situation is similar to the one which we encountered in the analysis of the (mixed) strategies of a zero-sum two-person game..."

I don't think this idea works for all cooperative games. But one can see here some ideas of evolutionary game theory.

I read TGEB, particularly the first chapter, as hostile to neoclassical economics. VNM disparage the idea that a model of Robinson Crusoe can tell us much about social phenomena. And they cast doubt on the idea that imitating the mathematical methods used in physics will bring much progress in economics.

References

• Hal R. Varian (2002). "Economic Scene; You've Seen the Movie. Now Just Exactly What was It that John Nash had on his Beautiful Mind", New York Times, 11 April.
• John Von Neumann and Oscar Morgenstern (1953). Theory of Games and Economic Behavior, Third Edition, Princeton University Press.

Chaotic Cobwebs (Part 1)

1.0 Introduction

This post duplicates an example in Richard M. Goodwin's Chaotic Economic Dynamics (Oxford University Press, 1990). At least, I think it does, but without the typographic errors that I think are in Goodwin's book. My Figure 3 is Goodwin's Figure 2.1, and my Figure 2 is Goodwin's Figures 2.2, and 2.3.

I have no plans to prepare a Part 2 to post later. But I describe in the conclusion below why there should be a Part 2.

2.0 Supply and Demand

This model is a partial equilibrium model with well-behaved supply and demand curves. It is an internal exploration of a mainstream textbook model. The demand curve shows the price that must instantaneously prevail if the quantity on the market is to be sold:

(1)

where p(t) is the price of the commodity at time t and q(t) is the quantity supplied or demanded.

Time is discrete in this model, and the supply curve contains a lag. Firms plan the quantity to supply in the next period based on the price in this period:

(2)

The supply curve shows "adaptive expectations". Economists such as Lucas have criticized the assumption of adaptive expectations. I think that critique may be inapplicable in a model with the behavior illustrated in Figure 5 below.

It's easy enough to solve for equilibrium, in which the quantity supplied and the quantity demanded are equal and do not change through time. Equation 3 gives the equilibrium quantity:

(3)

Figure 1 illustrates. The supply and demand curves are shown. The solid dot is the equilibrium. A hint at the dynamics is also shown. At time t, the indicated quantity is thrown on the market. One reads the price at that time off the demand curve. The quantity supplied in the next period is found from drawing a horizontal line from that intersection with the demand curve to the supply curve. This point of intersection with the supply curve is the quantity supplied in the next period. Proceeding in this way, one draws a figure that resembles a cobweb. Thus, this model is known as the cobweb model.

Figure 1: Supply and Demand

I have explained the dynamics of this model above. One can approach the story with algebra and obtain a difference equation:

(4)

Goodwin suggests redefining quantity as the deviation from the equilibrium quantity:

(5)

where Q(t) is the redefined quantity. Equation 6 gives the difference equation in terms of the time path of the redefined quantity variable:

(6)

3.0 Numerical Exploration of a Special Case

Goodwin considers the special case where the parameters b and e are both zero. Under this special case, the difference equation becomes considerably simplified:

(7)

Equation 7 resembles the logistic equation. As a start at exploring the dynamics of Equation 7, consider the case where a is unity, c is 3/5, and d is 21/20. (Update: Parameters were originally specified incorrectly.) Figure 2 shows time paths for two arbitrary initial values of the redefined quantity. Both time paths converge to a two-period limit cycle. The upper extreme of the blue time path continually falls, while the upper extreme of the red path continually rises. They meet in the limit at one point in the limit cycle. The lower extremes, shown in FIgure 2, converge to the other point in the limit cycle.

Figure 2: Time Paths of Redefined Quantity

Figure 3 shows another method for illustrating these paths. The difference equation is graphed along with a 45-degree line through the origin. One starts at an initial point along the abscissa. Drop a line to the graph of the difference equation. The value of the ordinate at this intersection with the difference equation shows the value of the redefined quantity variable at the next instant in time. Draw a horizontal line from this intersection to the 45-degree line. The value of the abscissa at this intersection to the 45-degree line is, of course, the value of the redefined quantity variable at the next instant in time. Continually in this way, one can easily trace out a time path graphically. The limit cycle is drawn in Figure 3.

Figure 3: Phase Space for These Paths

Structural dynamics concerns how the limiting behavior varies with the parameters of a difference or differential equation. Here I only consider the effects of a fall in a. Accordingly, Figure 4 shows the result when a is set to 7/10, with the other parameters as in Figure 3. The limit cycle has split into a cycle of period four. This period doubling sort of bifurcation is a common approach to chaos. For some lower values of a, the cycle will have periods eight, sixteen, thirty-two, etc. Figure 5 shows the result when a is 2/3. I believe Figure 5 is an example of chaos, where the limit is a non-wandering set with a fractal structure. (With more exploration, I would like to see a limit cycle of period three, or some other odd value. (Update: Try a as 0.57.) As I understand the mathematics, period doubling can only lead to such a limit cycle with intervening chaos.)

Figure 4: Phase Space for Period Four Cycle

Figure 5: Phase Space Showing Chaos

4.0 Conclusion

The above exposition begins with a common introductory model in economists. And it ends with mathematical chaos. Chaos is shown in a special case in which both the demand and the supply curves go through the origin. A supply curve going through the origin, although a special case, is quite reasonable in economics. It is not economically sensible for the demand curve to go through the origin.

If there were to be a part 2 for this post, it would demonstrate the possibility of chaos in an economically relevant parameter range. One would want the demand curve to be declining throughout the first quadrant as well as intersecting the price axis at a strictly positive price. And one would want the strange attractor to lie entirely in the first quadrant for the price and untransformed quantity variables. But I haven't done enough numerical exploration yet.

Walras, Poincaré, Jaffé, and Mirowski

A couple of posts from Sean Carroll led me to recall some of the interactions between economists and mathematical physicists. One interaction I find of interest is between Walras and Poincaré. Walras’ translator, William Jaffé describes some of the correspondence between the two. Jaffé even quotes one of Poincaré's letters:

"Your definition of rareté [marginal utility] impresses me as legitimate. And this is how I should justify it. Can satisfaction be measured? I can say that one satisfaction is greater than another, since I prefer one to the other, but I cannot say that the first satisfaction is two or three times greater than the other. That makes no sense by itself and only some arbitrary convention can give it meaning. Satisfaction is therefore a magnitude but not a measurable magnitude. Now is a non-measurable magnitude ipso facto excluded from all mathematical speculation? By no means. Temperature, for example, was a non-measurable magnitude – at least until the advent of thermodynamics which gave meaning to the term absolute temperature. The measurement of temperature by the expansion of mercury rather than the expansion of any other substance was nothing but an arbitrary convention. One could just as well have defined temperature by any function of temperature … provided that the function was monotonically increasing. Similarly you [on your side] can define satisfaction by any arbitrary function provided the function always increases with an increase in the satisfaction it represents.

Among your premises, there are a certain number of arbitrary functions; but once given these premises you have the right to draw consequences from them mathematically. If the arbitrary functions still appear in the conclusions, the conclusions are not false, but they are totally without interest because they depend upon the arbitrary conventions made at the start. You ought, therefore, to do your utmost to eliminate these arbitrary functions and that is what you are doing…

…I can tell whether the satisfaction experienced by the same individual is greater under one set of circumstances than under another set of circumstances; but I have no way of comparing the satisfactions experienced by two different individuals. This increases the number of arbitrary functions to be eliminated.

When I spoke of the 'proper limits', that is not all I wanted to say. What I had in mind was that every mathematical speculation begins with hypotheses, and that if such speculation is to be fruitful, it is necessary (as in applications to physics) that one be aware of these hypotheses. If one forgets this condition, one oversteps the proper limits. For example, in mechanics one often neglects friction and assumes the bodies to be infinitely smooth. You, on your side, regard men as infinitely self-seeking and infinitely clairvoyant. The first hypothesis can be admitted as a first approximation, but the second hypothesis calls, perhaps, for some reservations." -- Henri Poincaré (1901)

Jaffé points out that some of Poincaré’s coments foreshadow later developments in economics: revealed preferences and ordinal utility.

Jaffé published his article in 1977. He mentions that Walras initiated this correspondence after being criticized by the mathematician Hermann Laurent, but Jaffé does not explain Laurent's criticism. Since then, Mirowski (1989) has cast new light on this criticism. According to Mirowski, Laurent, in correspondence with Walras and Pareto, queried these neoclassicals about integrability and why economists felt they were justified in assuming utility was the potential of a conservative vector field.

Mirowski does not mention Poincaré's correspondence with Walras. I’d like to see an analysis of this correspondence fromm Mirowski’s viewpoint. I would not expect to see something from Mirowski. He's gone on to other aspects of the history of mainstream economics.

As I understand it, Poincaré, in addition to all of his other accomplishments, was on the verge of discovering the special theory of relativity, but Einstein arrived there first. Poincaré is also cited in the mathematics of dynamic systems. So I expect that he understood the mathematics of vector fields quite well. Did he raise any questions about integrability and conservation laws in his correspondence with Walras?

References

• William Jaffé (1977). "The Walras-Poincaré Correspondence on the Cardinal Measurability of Utility", Canadian Journal of Economics, V. 10 (May): 300-307.
• Neil De Marchi (editor) Non-Natural Social Science: Reflecting on the Enterprise of 'More Heat than Light', Duke University Press.
• Philip Mirowski (2002). Machine Dreams: Economics Becomes a Cyborg Science, Cambridge University Press.
• Philip Mirowski (1989). More Heat Than Light: Economics as Social Physics, Physics as Nature's Economics, Cambridge University Press.
• Léon Walras (1954). Elements of Pure Economics (Trans. by William Jaffé), Ricard D. Irwin.

Neither Econophysics Nor Neoclassical Economics

Some time ago I mentioned Philip Ball's commentary on Gallegati, Keen, and Lux's critique of econophsyics. Cosma Shalizi now has related comments on econophysics and neoclassical economics.

Price Theory Unstudied By Mainstream Economists

This seems hard to credit:

"if current graduate students know anything of price theory, it would have had to have been self-taught, because it is no longer on the curriculum on the 'best' American departments. (Except Chicago where it hangs in by a whisker...)" -- Angus Deaton (2007). "Letter from America - Random Walks by Young Economists", Royal Economic Society Newsletter (April)

Instead, economists study how to statistically analyze intrument variables. (I hope D-Squared will find the reference of interest.)

Judging A Book By Its Back Materials

I simple way of deciding whether one wants to persue reading a paper or book is to look at its bibliography. If it is on a topic that one is interested in and lacks certain references, one might put a low priority on reading it.

I get an ambiguous conclusion in the case of Steven Landsburg's "The Methodology of Normative Economics". From the introduction, I see that Landsburg argues that normative goals cannot be imposed exogenously. People care what a central goal-setting authority does, and the authority must account for that in setting his goals. The reference I look for here is Amartya Sen's "The Impossibility of a Paretian Liberal" (Journal of Political Economy, V. 78, N. 1 (Jan.-Feb. 1970): 152-157). And Landsburg is lacking. But he does reference a later Sen paper that I do not know. Perhaps Sen summarizes his earlier work there.

I get a negative conclusion when looking at Roger Farmer's draft book on old Keynesian economics. Farmer says he presents a model in which the level of economic activity is determined by "animal spirits." This is an allusion to chapter 12 of the General Theory, but Farmer's bibliography lacks any references putting forth a Post Keynesian reading, as far as I can see. Authors I look for include A. Asimakopulos, Victoria Chick, Coddington, Paul Davidson, and Joan Robinson.

And I get a negative conclusion for Claudia Goldin and Lawrence Katz's "Long-Run Changes in the U.S. Wage Structure: Narrowing, Widening, Polarizing". This paper looks at skill-biased technological change. Here I find lacking the failure to reference James Galbraith, such as his book Created Unequal. (Goldin and Katz do reference a number of authors I respect.)

I realize that authors put out drafts just to get these sort of comments. One wants to know if there are elements of a literature on topic that one has missed.

For Updating My Blog Roll

I should alphabetize my list of blogs. The distinction between "Blogs I Like" and others has collapsed. The roll is for me, and I'm aware of some of the blogs on the rolls of blogs I link to. So if I know how to get to a blog in two links, I haven't add it.

I might want to add Deirdre McCloskey's blog (Hat tip to Gabriel Mihalache). McCloskey's blog is not a commonplace book, like mine is. She seems to concentrate more on her life and less on economics.

EconoSpeak is the successor to MaxSpeak. It's Max Sawicky's co-bloggers, organized mostly by Barkley Rosser, Jr. (That was his father I mentioned in my program for studying philosophy of math.) EconoSpeak seems to be more about economic theory than its predecessor. On no, competition for Eric Nilsson and me. I should also mention Relentlessly Progressive Economics in this category.

Insight from Blaug's History

Mark Blaug, in "The Fundamental Theorems of Modern Welfare Economics, Historically Contemplated" (History of Political Economy, V. 39, N. 2 (2007): 185-203), offers some ideas relevant to comments on other economic blogs.

Blaug considers the widely repeated claim that the first fundamental theorem of welfare economics formalizes Adam Smith's notion of the "invisible hand". And he finds this claim mistaken. Blaug cites Gavin Kennedy favorably.

Dani Rodrik recently kicked off a discussion of two policy stances - "first best" and "second best" economists. And Rodrik organized this discussion around the two fundamental theorems. As I understand it, first best economists, according to Rodrik, think the general equilibrium world in which these theorems follow is close enough to, say, the United States economy to justify a bias against government intervention. Second best economists, according to Rodrik, think that the deviations of actual economies from the ideal general equilibrium model justify a bias towards intelligent government intervention. In either case, a static efficiency ideal is what we should be aiming at. (I admit to a bias - towards suspicion of dualistic thinking.)

Blaug considers whether the first and second fundamental theorems can provide policy guidance. And he concludes that static ineffiency is not very relevant to the "real-world dynamic performance of a competitive economy", which is what we should be interested in. This is just one more paper Blaug has produced over the years attacking the Arrow-Debreu formalism.