Quantity Flows For Structural Dynamics

1.0 Introduction
This post presents an example of a model of structural economic dynamics. I consider what quantity flows would arise for an economy in which agents make decisions in which the economy smoothly reproduces. The solution for this exercise turns out to be dynamically unstable in the special case I use for illustration. I think this means that, if I solve this special case in a future post for one way of setting out the price system, the solution for prices will be stable. The model presented in this post illustrates the difficult discovery problems that are solved in successful economies.

2.0 Technology
This economy consists of two sectors. In the first sector, labor produces means of production with existing means of production. In the second sector, labor produces means of consumption with existing means of production. (I use steel as as a synecdoche for means of production and corn for means of consumption.) The technique in use in both sectors exhibits Constant Returns to Scale (CRS). Only circulating capital is modeled; the means of production are entirely consumed in producing the output. Table 1 shows the coefficients of production for the technique in use during the t-th year.

Table 1: The Technology

	Steel Industry	Corn Industry
Labor	a0,1(t) person-years	a0,2(t) person-years
Steel	a1,1(t) tons	a1,2(t) tons
Outputs	1 ton steel	1 bushel corn

The technique improves each year. That is, each coefficient of production decreases at a constant rate of 100 c_i,j percent per year:

[a_i,j(t) - a_i,j(t + 1)]/a_i,j(t) = c_i,j

The above difference equation can be solved in closed form. The coefficients of production evolve as:

a_i,j(t) = a_i,j(0) (1 - c_i,j)^t

A more complex formulation might have non-constant percentage rates of decrease in the coeffients of production. For example, the percentage rate of decrease might be larger if the level of output of an industry was larger. Then one would be modeling "learning by doing" or endogenous growth, following in the tradition of Nicholas Kaldor and Kenneth Arrow. (Mainstream economists would cite Paul Romer's confused balderdash.)

3.0 Conditions for Smooth Reproduction
Let q₁(t) and q₂(t) be the tons of steel and the bushels of corn, respectively, produced as output and available at the end of the t-th year. I want to consider the case in which the labor force is always fully employed, the proportions in which output is produced always turns out to be appropriate, and no excess capacity is ever created.

The gross output of corn each year is divided up between the workers and the capitalists and then consumed. The gross outputs of steel and corn in a given year determine, along with the coefficients of production, how much steel should have been produced in the previous year:

q₁(t - 1) = a_1,1(t) q₁(t) + a_1,2(t) q₂(t)

The amount of labor employed in the t-th year is:

L(t) = a_0,1(t) q₁(t) + a_0,2(t) q₂(t),

where L(t) is the person-years of labor employed. In a general formulation, one might model the number of workers growing each year, but with increased productivity being taken partly in the form of decreased working hours per worker. For simplicity, I here model the labor force as a given constant:

L(t) = L^*

The above equations specify a dynamic system. An initial condition needs to be specified for any solution path to be completely determined. I take the initial ratio of employment in the two sectors as a given parameter:

a_0,1(0) q₁(0)/a_0,2(0) q₂(0) = h

The model can be simplified by expressing one quantity flow in terms of other by use of the condition that labor is fully employed. Some algebraic manipulation yields a single difference equation for the output of steel:

q₁(t) = [a_1,2(t) L^* - a_0,2(t) q₁(t - 1)]/d(t),

where

d(t) = [a_0,1(t)a_1,2(t) - a_0,2(t)a_1,1(t)]

If the coefficients of production were constant, the above would be a linear difference equation. If I recall my mathematics correctly, linear systems either blow up; decay to an equilibrium; or, for coefficients meeting an exact balance, generate a constant wave.

4.0 The Solution of a Special Case
I tried a numerical experiment to increase my understanding of this dynamical system. Accordingly, I chose some specific values for the model parameters. Table 2 gives the initial coefficients of production. The difference equation for gross steel outputs is simplified in that the coefficients of production in a sector decrease at the same constant rate. I chose the following rates of decrease:

c_0,1 = c_1,1 = 1/20

c_0,2 = c_1,2 = 1/40

Let the labor force be unity:

L^* = 1

Finally, I carefully specified an initial condition:

a_0,1(0) q₁(0)/a_0,2(0) q₂(0) = 0.22335983

Table 2: The Initial Technology

	Steel Industry	Corn Industry
Labor	a0,1(0) = 1	a0,2(0) = 1
Steel	a1,1(0) = 1/10	a1,2(0) = 1/5
Outputs	1 ton steel	1 bushel corn

One can easily step through the first few years of the solution, thereby obtaining the start of a series for q₁(t) and q₂(t).The solution is dynamically unstable. I carefully chose the initial condition to get six years before the solution blows up. For the first five years, the output of steel grows over 3% and the output of corn grows over 14 1/2%, for a constant labor supply. This set of priorities is the reverse of what was typically achieved in no-longer actually existing socialism. When Imre Nagy, for example, tried to put Hungary on a new course, he was deposed. The distribution of labor, shown in Table 2, is not realistic for a developing capitalistic economy either. In practice, the labor force becomes steadily less concentrated in producing means of consumption and more in producing means of production. Still, I think, this model with a better choice of parameters and perhaps some generalizations can be quite interesting.

Figure 1: Dynamic Distribution of the Labor Force

References

• Karl Marx (1885) Capital, Volume 2
• Luigi L. Pasinetti (1977) Lectures on the Theory of Production, Columbia University Press
• Luigi L. Pasinetti (1983) Structural Change and Economic Growth: A Theoretical Essay on the Dynamics of the Wealth of Nations, Cambridge University Press
• Luigi L. Pasinetti (1993) Structural Economic Dynamics: A Theory of the Consequences of Human Learning, Cambridge University Press

To read:

• Dale W. Jorgenson (1960) "A Dual Stability Theorem", Econometrica, V. 28, N. 4 (October): pp. 892-899

Judt On The Influence Of The Austrian School

I continue to find writers characterizing Austrian school economists as influential.

I think some might quarrel with this description of the influence of the Austrian school on Chicago:

"We are the involuntary heirs to a debate with which most people are altogether unfamiliar. When asked what lies behind the new (old) economic thinking, we can reply that it was the work of Anglo-American economists associated overwhelmingly with the University of Chicago. But if we ask where the 'Chigago boys' got their ideas, we shall find that the greatest influence was exercised by a handful of foreigners, all of them immigrants from central Europe: Ludwig von Mises, Friedrich Hayek, Joseph Schumpeter, Karl Popper, and Peter Drucker." -- Tony Judt (2010) Ill Fares the Land, Penguin Press, pp. 97-98

I don't think differences in details (e.g., aggregation in economic models) adequately refutes Judt's point.

Judt does read, for example, Hayek as more nuanced than some of his followers:

"The intellectual refugees - and especially the economists among them - lived in a condition of endemic resentment toward their uncomprehending hosts. All non-individualist social thought - any argument that rested upon collective categories, common objectives or the notion of social goods, justice, etc. - aroused in them troubling recollections of past upheavals... Men like Hayek or von Mises seemed doomed to professional and cultural marginality. Only when the welfare states whose failure they had so sedulously predicted began to run into difficulties did they once again find an audience for their views: high taxation inhibits growth and efficiency, government regulation stifles initiative and entrepreneuship, the smaller the state the healthier the society and so forth.

Thus when we recapitulate conventional clichés about free markets and western liberties, we are in effect echoing - like light from a fading star - a debate inspired and conducted seventy years ago by men born for the most part in the late 19th century...

It is perhaps worth noting here that even Hayek cannot be held responsible for the ideological simplifications of his acolytes. Like Keynes, he regarded economics as an interpretive science, not amenable to prediction or precision. If planning was wrong for Hayek, this was because it was obliged to base itself on calculations and predictions which were essentially meaningless and thus irrational. Planning was not a moral misstep, much less undesirable on some general principle. It was simply unworkable - and, had he been consistent, Hayek would have acknowledged that much the same applied to 'scientific' theories of the market mechanism...

In the United States, among a younger generation of self-confident econometricians (a sub-discipline of whose bostful scientificity both Hayek and Keynes would have had much to say), the belief that democratic socialism is unachievable and has perverse consequences has become something close to a theology. This creed has attached itself to every effort to increase the role of the state - or the public sector - in the daily lives of American citizens." -- Tony Judt (2010): pp. 102-104

Orthodox Economists As Knavish Lickspittles

Bill Mitchell says:

"The mainstream economists hide behind lies. Most of the time they claim their policy recommendations are derived from economic theory. The reality is that they are not and economists lie and obsfucate.

There are many situations where strident policy suggestions – like the austerity packages – cannot be based on the economic theory that they are associated with – on the theories that economists use to give an air of authority and legitimacy to what are otherwise demands based on their blind ideology.

Unfortunately, the public is not in a position to judge and get swamped by the arrogance of economists..."

The arrogance of orthodox economists plays out in many ways. Philip Mirowski, for example, notes:

"But economists could not begin to discuss the major difference between 1929 and 2008: this time around, professional economists had played a much larger role in producing the conditions leading to systemic breakdown, from theorizing the financial innovations and staffing the financial institutions to justifying the deconstruction of regulatory structures held over from the last Great Depression. The profession did not entirely succeed in distracting public attention from that fact, either." -- Philip Mirowski, "The Great Mortification: Economists’ Responses to the Crisis of 2007–(and counting)", The Hedgehog Review, (Summer 2010)

And there is the matter of funding of inconvenient researchers. So we see Mirowski writing, "My home institution declined to provide any support for this research." And Jacob Hacker and Paul Pierson write something similar:

"The authors received no financial support for the research and/or authorship of this article."

Greg Mankiw seems always willing to demonstrate that he is a varlet. In his post labeled, "The Root Cause of the Crisis", he kowtows to another fool:

"According to Raghu Rajan: Skill-biased technological change, followed by ill-advised policies."

If an economist were honest, he could hardly continue to cite skill-biased technological change as a cause income distribution in the United States.

Simple Pepper Steak

Ingredients

3 Tbsps olive oil
1 sliced onion
2 sliced (green & red?) peppers
1 lb sandwich steak
1/2 tsp. salt
1/4 tsp. black pepper
3 Tbsps (ginger-flavored?) soy sauce

1) Fry sliced peppers and onions in olive oil over medium high heat about 7 minutes. (Variant: add chopped garlic clove.) Drain oil and lower heat.

2) Meanwhile, cut meat into 1 inch squares. Season with salt and pepper.

3) Add beef to fry pan. (Variant: consider adding additional spices, e.g. 1 tsp. thyme leaves.) Add soy sauce. Cook on low to medium about 10 minutes, stirring frequently.

4) Serve over boiled white rice.

Makes 2 or 3 servings

Congressional Testimony on Building a Science of Economics

The United States House of Representatives has a Committee on Science and Technology, with a Subcommittee on Investigations and Oversight. On Thursday, 20 July, they heard testimony about macroeconomics from Robert Solow, Sidney Winter, Scott Page, David Colander, and V. V. Chari. You can download the hearing charter, prepared statements, etc. I guess V. V. Chari's role is to defend the orthodox.

'Libertarian' Used Correctly

"Less World! More Bank!"

I have been reading Phil Edwards' 'More Work! Less Pay!' Rebellion and Repression in Italy, 1972-7 (Manchester University Press, 2009). He occasionally uses the word 'libertarian', for example:

"the group re-emerged within the 'area' as the Collettivi Politici Operai (CPO; 'Workers' Political Collectives'), opening itself to left-libertarian as well as Leninist influences." -- Phil Edwards (p. 69)

and in translating:

"The first, numerically in the majority, is the 'creative' wing, libertarians with radical leanings ... These are the 'small-a' autonomists, who at one time or another fight for a 'better quality of life' ... The second large strand is that of the professors, the intellectuals, the theorists of the message ... The third strand, finally, is that of the 'capital-A' autonomists, or Autonomia operaia organizzata ['Organised Workers' Autonomy']" -- M. Monicelli, L'ultrasinistra in Italia 1968-1978 (1978, Phil Edwards' translation)

Apparently, the custom of street performances at protests emerged from the area of autonomia. Antonio Negri was a member of the CPO. Perhaps activists in Detroit might consider the establishment of social centers (centri sociali) by squatting in abandoned buildings.

And Every Stinking Bum Should Wear A Crown

Why so much inequality in the United States? I have been reading Hacker and Pierson (2010). (I haven't yet read the commentaries, available for the moment.) I think I have previously made many of their points, as shown below. Hacker and Pierson present some striking graphs taken from work by Piketty and Saez. I reproduce two.

Figure 1: Richest 1 Percent's Share of National Pretax Income (Excluding Capital Gains) (Based on Piketty and Saez)

Figure 2: Average Actual Tax Incidence for Top Incomes (Based on Piketty and Saez (2007))

Anybody who still thinks the mainstream story of skills-biased technical change is a reasonable hypothesis is a feckless fop.

Given contract law and property law, government cannot leave the economy to itself. Policy has been driving increased inequality.

Another driver of increased inequality is a change in ideas and social norms. These include faulty ideas on corporate governance, incorrect theories of factor markets, and performative models of finance.

How did ideas that never had sound empirical and theoretical backing become dominant? Part of the explanation must be a propaganda campaign by vile reactionaries, including the suppression of progress in explaining actually existing capitalist economies.

There is an aspect of cumulative causation here. A smaller government is associated with more inequality. And more inequality is associated with the rich and powerful promoting an exploded and evil ideology.

Increased inequality also leads to failures in aggregate demand. A steadily growing economy needs a certain balance to be maintained. The consequences of the failure to maintain such a balance since the end of the post war golden age are all around us.

Reference

Jacob S. Hacker and Paul Pierson, "Winner-Take-All Politics: Public Policy, Political Organization, and the Precipitous Rise of Top Incomes in the United States", Politics & Society, V. 38 N. 2 (2010): pp. 152-204

Elsewhere

• Cosma Shalizi states a simple ergodic theorem.
• Cosma Shalizi reviews Yves Smith's Econned: How Unenlightened Self-Interest Undermined Democracy and Corrupted Capitalism.
• Victoria Chick and Ann Pettifor argue that the United Kingdom needs more government spending.
• James Galbraith affirms that "There is no economic justification for deficit reduction."
• Updated: David Harvey gives an animated lecture on the causes of the global financial crisis. I like the use of Volume 2 of Marx's Capital.

But Maybe Everything That Dies Someday Comes Back

I think the following three events, at least, were important in the development of the post (second world)-war model of development:

• The 1942 issuing of the Beveridge Report
• The 1944 Bretton Woods conference
• The April 1955 Bandung conference

Shouldn't the documents issued by such conferences and the whole Beveridge Report now be online somewhere? The first 20 pages of the Beveridge Report are here. Extracts from those pages and from Part V are here (mirrored).

Despite the post title, I don't think a Fordist accumulation regime can be recreated. But we need a new mode of regulation with new institutions and transformative events to set up a system that works. The current one doesn't.

Impact of Capital Controversies On Labor Economics

I think some with some exposure to economics might not be impressed by the Cambridge Capital Controversy because of the abstractness of capital. "Capital" can refer to finance, and it can refer to the means of production. And even if one thought that interest rates were determined by supply and demand, what quantity would be supplied or demanded? Loanable funds? Savings? One might expect capital theory to be complicated, even in the view of the defenders of mainstream theory. So why, some might rationalize, should one be surprised by aggregation problems?

I like to emphasize the labor market. This market might seem more concrete to many. (This is one of the illusions created by competition.) Consider cases in which firms have adapted their capital equipment and production techniques to the prices prevailing on both product and factor markets. A logical implication of capital reversing is that, given the level of output, such firms employ more workers at a higher wage. (I have deliberately worded the above to avoid saying, "Increasing wages lead to greater employment" - which is a claim about dynamics.) The sort of example I have in mind doesn't seem to depend on aggregating labor. I often motivate discussion of these matters with the introductory neoclassical textbook story on minimum wages, which is theoretically unfounded and only taught by bad economists. (Nose counting as a refutation of the above statement is a fallacy.)

I realize lots of literature backs up my position; I am not being original. I like to cite Graham White (2001) and Tony Aspromourgos (2001). I recently stumbled upon another economist who forcefully states a similar position.

John Weeks has been updating his 1989 book, A Critique of Neoclassical Macroeconomics. Two parts of the update-in-progress are currently available online: part I, part II. I extract some random quotes from the relevant chapter in part II:

"Reswitching implies an unexpected conclusion: theory tells us that in general capitalists will not necessarily select more labour-intensive techniques when wages fall.

This result is a potential disaster for the neoclassical macro model and its parable about real wages and employment."

"In the introduction to this book a quotation from The Times was cited, which ventured the assertion that '...few economists would argue with the general proposition that lower real wages will mean higher employment...' If it refers to theoretically competent neo-classical economists, this statement is false."

"One can conclude that when referring to actual economic outcomes, there is no theoretical basis for the generalization that lower real wages will stimulate more employment. The opposite conclusion has equal theoretical merit. The neoclassical parable, upon which so many policy prescriptions are based, is a false guide to real economies."

References

• Tony Aspromourgos (2001) "Is Labour Cheapening a Means to Reducing Involuntary (Labour) Unemployment?", History of Economics Review, V. 34: (Summer) pp. 7-18
• John Weeks (1989) A Critique of Neoclassical Macroeconomics
• Graham White (2001) "The Poverty of Conventional Economic Wisdom and the Search for Alternative Economic and Social Policies", The Drawing Board: An Australian Review of Public Affairs, V. 2, N. 2 (November): pp. 67-68.

Neoclassical Economics Overthrown A Half-Century Ago

Today the 50-year anniversary of the publication of Piero Sraffa's The Production of Commodities by Means of Commodities is being celebrated at Queen's College, Cambridge.

A Political Pamphlet

How many of you have worked through Paul Lafarge’s The Right To Be Lazy? Isn’t that against the spirit of the piece?

Manifestations of Sraffa Effects in General Equilibrium Models?

A Strange Attractor Arises From The Lorenz Equations

I think reswitching, capital reversing, and Sraffa effects may be the source of both dynamic and structural instabilities in General Equilibrium models. I am not so much interested in dynamics of a tâtonnement process in some sort of no-time before the beginning of time in the Arrow-Debreu model of intertemporal equilibrium. Rather, I find more of interest the dynamics of spot prices in models of temporary equilibrium.

My claim that the Cambridge Capital Controversy can be drawn on for examining the dynamics of certain economic models is not original. Barkley Rosser (1983) related reswitching to a cusp catastrophe. A cusp catastrophe, as I understand it, is a kind of structural instability. Overlapping Generation Models (OLGs) provide my favorite neoclassical closure of Sraffian production models. Saverio Fratini (2007) has investigated cases in which reswitching gives rise to multiple stationary state equilibria in OLGs. I've convinced myself that whether multiple equilibria are associated with a "normal" or "perverse" switch point can depend on the form of the utility functions in OLGs.

An issue arises in showing that Sraffa effects are associated with the appearance of complex and chaotic dynamics in models of General Equilibrium. Researchers have already established that complex dynamics can arise in such models anyways, including OLGs, for other reasons. For example, John Geanakoplus states:

"Grandmont ..., following related work of Benhabib and Day ... and Benhabib and Nishimura ..., gave a robust example of a one-commodity, stationary economy ... giving rise to a three-cycle... Of course a cycle ... is also a cyclical equilibrium for the economy, hence there are robust examples of economies with cycles of all orders." -- John Geanakoplos (2008)

Geanakoplos is relying on Theorem 1 in Li and Yorke (1975). In the references, I give sources for identifying literature exploring the dynamics of General Equilibrium models, including OLGs, independently of considerations raised in the CCC.

The consequences of modeling the economy as potentially exhibiting complex non-linear dynamics are far reaching. Rajiv Sethi, in a series of blog posts, has pointed out some implications of a serious concern with non-linear dynamics for mainstream macroeconomics:

• An Outsider's View of Modern Macroeconomics
• On the Consequences of Nominal Wage Flexibility
• On Buiter, Goodwin, and Nonlinear Dynamics
• On Rational Expectations and Equilibrium Paths

I think one can show that Sraffa effects can give rise to complex dynamics in OLGs, even with the knowledge that OLGs can produce chaotic dynamics otherwise. I need to find an OLG model with perhaps a single good being produced in each period and in which complex dynamics do not arise for the specified form of the utility function. Then one should alter the production model to be a two or three-good reswitching example. Finally, one should establish complex dynamics arise in the resulting models. Even if this strategy is not successful, one pursuing it will have to explore and understand already existing models with complex dynamics.

References

• Jess Benhabib (2008) "Chaotic Dynamics in Economics", in The New Palgrave Dictionary of Economics (Ed. by S. N. Durlauf and L. E. Blume), 2nd edition, Palgrave Macmillan
• Jess Benhabib (editor) (1992) Cycles and Chaos in Economic Equilibrium, Princeton University Press
• Saverio M. Fratini (2007) "Reswitching of Techniques in an Intertemporal Equilibrium Model with Overlapping Generations", Contributions to Political Economy, V. 26: pp. 43-59.
• John Geanakoplos (2008) "Overlapping Generations Model of General Equilibrium", in The New Palgrave Dictionary of Economics (Ed. by S. N. Durlauf and L. E. Blume), 2nd edition, Palgrave Macmillan
• John Guckenheimer and Philip Holmes (1983) Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag
• Yijun He and Willam A. Barnett (2006) "Existence of Bifurcation in Macroeconomic Dynamics: Grandmont was Right"
• Tien-Yien Li and James A. Yorke (1975) "Period Three Implies Chaos", American Mathematical Monthly, V. 82, N. 10 (Dec.): pp. 985-992
• J. Barkley Rosser, Jr. (1983) "Reswitching as a Cusp Catastrophe", Journal of Economic Theory, V. 31: pp. 182-193
• Paul A. Samuelson (1958) "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money", Journal of Political Economy, V. 66, N. 6 (December): pp. 467-482
• Robert Shiller (1978) “Rational Expectations and the Dynamic Structure of Macroeconomic Models: A Critical Review”, Journal of Monetary Economics, V. 4: pp. 1-44.

Good For Me, Bad For You

I still hope to publish my critique of Austrian business cycle theory (most recent public version). In my introduction, I need to answer the question why anybody should care about criticism of Austrian business cycle theory or Austrian economics more generally. Mentions of Austrian economics in popular venues helps make my case. And some recent outbreaks are available.

On 8 May 2010, the Maine Republicans adopted a platform with these insane planks:

"V. To Promote the General Welfare

• Return to the principles of Austrian Economics, and redirect the economy back to one of incentives to save and invest.
• Cut spending, balance the budget, and institute a plan for paying down debt. Proclaim that generational debt shifting is immoral and unconscionable and will not be tolerated! ..."

Apparently, nobody has told these maniacs that Von Mises declares praxeology, that is, economic theory, to be Wertfreiheit, that is, value-free (Human Action, Chapter II Section 7).

And an unstable liar on Fox News has made Hayek's The Road to Serfdom the number one most popular book at times on Amazon. So Hayek is being discussed in the popular press and among academics who previously had taken no notice. By the way, other works of Hayek are more important for theories of the business cycle. In The Road to Serfdom, Hayek says his differences with Keynes are a matter of pragmatic judgment, not political principle:

"There is, finally, the supremely important problem of combating general fluctuations of economic activity and the recurrent waves of large-scale unemployment which accompany them. This is, of course, one of the gravest and most pressing problems of our time. But, though its solution will require much planning in the good sense, it does not - or at least need not - require that special kind of planning which according to its advocates is to replace the market. Many economists hope, indeed, that the ultimate remedy may be found in the field of monetary policy, which would involve nothing incompatible even with nineteenth century liberalism. Others, it is true, believe that real success can only be expected from the skillful timing of public works undertaken on a very large scale. This might lead to much more serious restrictions of the competitive sphere, and, in experimenting in this direction, we shall have carefully to watch our step if we are to avoid making all economic activity progressively more dependent on the direction and volume of government expenditure. But this neither the only nor, in my opinion, the most promising way of meeting the gravest threat to economic security. In any case, the very necessary efforts to secure protection from these fluctuations do not lead to the kind of planning which constitutes such a threat to our freedom." -- Friedrich A. Hayek, The Road to Serfdom, Chapter IX: Security and Freedom

The popularity of Austrian economics refracts and contributes to a policy environment in which bad ideas are being adopted with bad consequences.