Classical Cross Dual Dynamics

"The actual price at which any commodity is commonly sold is called its market price. It may either be above, or below, or exactly the same with its natural price.

The market price of every particular commodity is regulated by the proportion between the quantity which is actually brought to market, and the demand of those who are willing to pay the natural price of the commodity, or the whole value of the rent, labour, and profit, which must be paid in order to bring it thither. Such people may be called the effectual demanders, and their demand the effectual demand; since it may be sufficient to effectuate the bringing of the commodity to market...

...The natural price, therefore, is, as it were, the central price, to which the prices of all commodities are continually gravitating...

The whole quantity of industry annually employed in order to bring any commodity to market, naturally suits itself in this manner to the effectual demand. It naturally aims at bringing always that precise quantity thither which may be sufficient to supply, and no more than supply this demand." - Adam Smith, The Wealth of Nations, Book I, Chapter VII

1.0 Introduction
Contemporary economists have elaborated Smith's metaphor of the gravitational attraction of market prices to natural prices. Elaborations consist of formal models of cross-dual dynamics.

Systems of equations describing prices and quantitites are dual systems in the post Sraffa/von Neumann tradition. Dynamics are cross dual when changes in prices respond to quantities and changes in quantities respond to prices. In particular, industries expand in which rates of profits are high and contract in industries in which profits are low. And prices fall in industries in which the quantity supplied exceeds the effectual demand. Prices rise in industries in which the quantity supplied is below the effectual demand.

Dupertuis and Sinha (2009) is one immediate impetus for my setting out this model of the reallocation of labor for a given wage. I don't think I thoroughly understand models of cross-dual dynamics. I think they are of interest for exploring the possible dynamics of market prices in competitive capitalist economies. I don't see that they are directly empirically applicable. For that, one needs to worry about markup prices and the degree of utilization of capacity in various industries.

2.0 Natural Prices
The data of our problem are:

• The n x n input-output matrix A, where n is the number of industries and a_{i, j} is the amount of the ith commodity used as input per unit output of the jth industry
• The n-element row vector a₀ of labor inputs, where a_{0, j} is the amount of person-hours hired per unit output in the jth industry
• The money wage w_given
• The composition of net output as expressed in the n-element column vector c_given.

This is a circulating capital model in which all production processes use up their inputs in a year. Labor is assumed to be paid their wages at the end of the year. Only economies capable of producing a surplus product are considered. For simplicity, assume Constant Returns to Scale (CRS) and that every commodity is basic, in Sraffa's sense. This section considers the problem of finding:

• The natural prices, as expressed in the n-element row vector p^*
• The corresponding wages w^*
• The corresponding rate of profits r^*
• The effectual demand as expressed in the n-element column vector of gross quantities q^*
• The n-element column vector of net quantities y^*.

In the system of natural prices, the same rate of profits are made in every industry:

p^* A(1 + r^*) + a₀ w^* = p^*

I take the net output as the numeraire.

p^* y^* = 1

In my formulation here, the wage is taken as a given ratio of the net output:

w^* = w_given

The above equations comprise the price system for natural prices.

Net outputs and gross outputs are related by the following equation:

y^* = q^* - A q^* = (I - A) q^*

where I is the identity matrix. I normalize the units of labor such that one unit is employed throughout the economy:

a₀ q^* = 1

Finally, the net output is assumed to be in the specified proportions. That is, there exists a positive constant k such that

y^* = k c_given

The above systems of equations are sufficient to determine gross and net effectual demands, natural prices, and the distribution of income. (An alternative specification would take the composition of gross output as given, instead of the net output. Perhaps outputs should be in units of Sraffa's standard commodity.)

3.0 Initial Condititions
The problem in the remaining sections is to define a dynamic process for the quantities produced q(t) and the market prices p(t) for t = 0, 1, 2, ... The initial quantities q(0) and market prices p(0) are givens. For the sake of the argument, I consider a dynamic process in which the amount of labor employed and the value of net output are invariant. So the initial quantities and prices must satisfy the following equations:

a₀ q(0) = 1

p(0) y(0) = p(0)(I - A)q(0) = 1

4.0 Reallocation of Labor
Define r_average(t), the average rate of profits for the economy as a whole at time t:

r_average(t) = [p(t)(I - A - a₀ w_given)q(t)]/[p(t) A q(t)]

The numerator in the expression on the right hand side above is the value of the surplus product remaining in the capitalists' possession after replacing the means of production and paying laborers their wages. The denominator is the value of the capital goods advanced.

Typically, the rate of profits will vary from the average among the industries. The rate of profits for the jth industry at time t is:

r_j(t) = [p_j(t) - p(t) a_{., j} - a_{0, j} w_given]/[p(t) a_{., j}]

where a_{., j} is the jth column of the input-output matrix A.

Define R_average(t) to be the n-element column vector with each element equal to the average rate of profits. Let R(t) be the n-element column vector with each element being the rate of profits for the corresponding sector.

Now dynamics of the quantities of produced commodities can be specified:

q(t + 1) = [1/f₁(t)]{[R(t) - R_average(t)] + q(t)}

Or, in terms of scalars:

q_i(t + 1) = [1/f₁(t)]{[r_i(t) - r_average(t)] + q_i(t)}

where

f₁(t) = 1 + a₀[R(t) - R_average(t)]

The denominator f₁(t) above is a normalization that ensures the quantity of labor employed is always unity. The numerator ensures that the more the rate of profits in a sector exceeds the average, the faster that sector will expand in comparison with other sectors. (An alternative formulation might compare the rate of profits in each industry with the rate of profits r^* in the system of natural prices.)

5.0 Price Changes
Price dynamics are here set out more directly:

p(t + 1) = [1/f₂(t)]{p(t) - [q^T(t) - q^*T]}

where x^T is the transpose of the vector x. In terms of scalars, prices are given by:

p_i(t + 1) = [1/f₂(t)]{p_i(t)- [q_i(t) - q^*_i]}

The time series f₂(t) is defined as follows:

f₂(t) = [q^*T - q^T(t)](I - A)q(t + 1)} + p(t)(I - A)q(t + 1)

The denominator f₂(t) above is, again, a normalization condition. In this case, the normalization ensures the value of the net output is equal to unity. Since the composition of net output typically changes over the course of the process, the real wage varies in terms of any fixed commodity basket. It does remain, however, a given ratio of the net output.

6.0 Conclusion
I have set out above a model of a dynamic process, but without an analysis of its properties. An obvious theorem is that if initial quantities and prices happen to be equal to the effectual demands and natural prices, they will be left unchanged by the dynamics of market adjustments. In other words, the natural system is a stationary point of this dynamic process.

An interesting question is the trajectory of market prices, given an arbitrary starting point. I don't expect the process to necessarily converge to the natural system. At this point, I don't have any numeric examples of limit cycles or chaotic behavior. A failure of local stability doesn't bother me; I have often thought of Sraffa's work as pointing towards the possibility of complex dynamics arising in models of capitalist economies.

Questions of structural stability are of interest as well. Do dynamic properties of the system depend on the level of wages, especially if one introduces into the model a choice of technique? And how do the answers to these questions vary, if at all, with alternative modeling assumptions, some of which I have indicated? I do not know that the literature has reached definitive answers to these questions.

Update (28 January 2010): I have redefined the dynamics above in a way that seems more reasonable to me.

References

• Michel-Stéphane Dupertuis and Ajit Sinha, "A Sraffian Critique of the Classical Notion of Centre of Gravitation", Cambridge Journal of Economics, V. 33 (2009): 1065-1087.