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

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.