Saturday, October 20, 2018

A Visualization of the Choice of Technique

Figure 1: Regions for Basis Variables
1.0 Introduction

I introduced a new way of visualizing the choice of technique for two-commodity models back in 2005. As far as I know, nobody has taken up this idea. I modify my method slightly by having labor advanced; wages are paid out of the surplus at the end of the year. I cite John Roemer in my paper linked previously.

2.0 Technology

Table 1 specifies the technology I use for illustration. Each row lists the inputs needed to produce one unit (ton or bushel) for the indicated industry. As usual, this is a model of circulating capital.

Table 1: Example Technology
InputIndustry
IronCorn
AlphaBeta
Labora0, 1 = 1aα0, 2 ≈ 0.9364aβ0, 2 ≈ 0.6174
Irona1, 1 = 9/20aα1, 2 ≈ 0.02602aβ1, 2 ≈ 0.001518
Corna2, 1 = 2aα2, 2 ≈ 0.1041aβ2, 2 ≈ 0.4636

For this economy to be reproducible, both iron and corn must be produced. The iron-producing process can be combined with either of the corn-producing processes. Thus, there are two possible techniques, the Alpha and Beta techniques, each of which include the corn-producing process with the corresponding label. (The approach in this post can be extended to include any number of available processes in either industry.)

3.0 A Linear Program for the Firm

Consider a firm that starts the year with an inventory of ω1 tons iron and ω2 bushels corn. I take corn as the numeraire. The firm faces a price for iron of p bushels per ton and a wage of w bushels per person years. The managers of the firm must set the value of the following decision variables:

  • q1: The tons iron produced with the iron-producing process.
  • qα2: The bushels corn produced with the Alpha corn-producing process.
  • qβ2: The bushels corn produced with the Beta corn-producing process.
  • q3: The value of inventory that the firm carries over unused to the next year.

The firm is constrained by the value of its inventory. Its level of production cannot require it to advance more than the value of its inventory.

The managers of the firm attempt to maximize the increment of value. Their problem can be formulated as a Linear Program (LP). They choose q1, qα2, and qβ2 to maximize:

z = (p - pa1, 1 - a2, 1 - a0, 1w)q1
+ (1 - paα1, 2 - aα2, 2 - aα0, 2w)qα2
+ (1 - paβ1, 2 - aβ2, 2 - aβ0, 2w)qβ2

Such that:

(pa1, 1 + a2, 1)q1
+ (paα1, 2 + aα2, 2)qα2
+ (paβ1, 2 + aβ2, 2)qβ2
p ω1 + ω2
q1 ≥ 0, qα2 ≥ 0, qβ2 ≥ 0

In solving this LP by the simplex method, it is convenient to introduce the slack variable, q3, to convert the constraint to an equality.

4.0 The Dual LP

The above LP has a dual. It is to choose a non-negative rate of profits so as to minimize the capital charge on the inventory. Constraints are such that the cost of each production process, including a charge for capital, does not fall below the revenue from operating that process. Formally, choose r to minimize:

(p ω1 + ω2) r

Such that:

(pa1, 1 + a2, 1)(1 + r) + a0, 1wp
(paα1, 2 + aα2, 2)(1 + r) + aα0, 2w ≥ 1
(paβ1, 2 + aβ2, 2)(1 + r) + aβ0, 2w ≥ 1
r ≥ 0

If the primal LP has a solution, so will the dual LP. And the value of the objective functions will be the same, for a solution, for both the primal and dual LP. When a decision variable is positive in a solution to the primal LP, the corresponding constraint is met with equality in the dual LP. Thus, if the solution of the primal LP leads to corn being produced and iron being produced with the Alpha iron-producing process, the economy will be on the wage curve for the Alpha technique. Similar remarks apply to the Beta technique.

Table 2: Solution of Primal LP
Variable
in Basis
ValueWhen Optimal
q1(p ω1 + ω2)/(pa1, 1 + a2, 1)r1rα2
r1rβ
c1p
qα2(p ω1 + ω2)/(paα1, 2 + aα2, 2)r1rα2
rα2rβ2
cα2 ≤ 1
qβ2(p ω1 + ω2)/(paβ1, 2 + aβ2, 2)r1rβ2
rα2rβ2
cβ2 ≤ 1
q3p ω1 + ω2c1p
cα2 ≥ 1
cβ2 ≥ 1

5.0 The Solution of the Primal LP

The solution to the primal LP is illustrated by Table 2. In a solution, only basis variables are positive. The table specifies the value of each basis variable, when only it is positive in the solution, and conditions that must hold for it to be in the basis. These conditions are specified in terms of certain variables introduced as abbreviations. The rates of profits in each process are:

r1 = (p - a0, 1w)/(pa1, 1 + a2, 1)
rα2 = (1 - aα0, 2w)/(paα1, 2 + aα2, 2)
rβ2 = (1 - aβ0, 2w)/(paβ1, 2 + aβ2, 2)

The (undiscounted) costs of each process are:

c1 = pa1, 1 + a2, 1 + a0, 1w
cα2 = paα1, 2 + aα2, 2 + aα0, 2w
cβ2 = paβ1, 2 + aβ2, 2 + aβ0, 2w

The conditions for when a decision variable is in the basis are intuitive. Consider the first row. Corn is produced only if the rate of profits made in either of the iron-producing processes does not exceed the rate of profits made in the corn producing process. Furthermore, the (undiscounted) cost of producing a bushel corn must not exceed the revenue made from selling corn.

6.0 Visualization

The solution to the primal LP, in a two-commodity example, is easily visualized. Figure 1 partitions the space formed from the price of iron and the wage. A single decision variable enters the basis inside each region in the figure, and that region is labeled by that decision variable. On the boundaries, a solution to the LP can be formed from a linear combination of decision variables. Iron and corn must be both produced for the economy to be self-sustaining. Firms are willing to produce both only if prices lie along the heavy locus. The figure shows that this is a reswitching example. The Beta technique is adopted at low and high wages, while the Alpha technique is used at intermediate wages. The figure also illustrates that the wage cannot exceed a maximum.

7.0 Conclusion

If you think about it, the above is a derivation of the usual method of analyzing the choice of technique by constructing the outer frontier of the wage curves for all available techniques. It is not restricted to a two-commodity example, although the diagram is so restricted. The proof follows from duality theory in linear programming. The graph illustrates that equilibrium prices must vary with the wage.

I remain puzzled about why mainstream economists continue to teach that, under the ideal assumptions of free competition, wages and employment are determined by the interaction of supply and demand in labor markets.

Wednesday, October 17, 2018

William Nordhaus, 2018 "Nobel" Laureate, On Labor Values

Suppose one wants to quantitatively measure the growth in productivity over centuries. And one wants to look at specific commodities that can be said to have existed over such a long time. Think of a lumen of light or a food calorie. How can one do this? The definition of a price index over such a long time period is questionable.

Adam Smith addressed this problem. Some would find his approach common sense. One could ask how long must a common laborer work to be able to afford the commodity in question. More recently, William Nordhaus considered the question. He, too, advocated the use of a Smithian labor-commanded standard to measure technological change. (I haven't read the reference below in decades.)

Monday, October 08, 2018

Paul Romer, 2018 "Nobel" Laureate

Despite his ignorance of the Cambridge Capital Controversy, Paul Romer's recent criticisms of mainstream macroeconomics have some good points. Typical Dynamic Stochastic General Equilibrium (DSGE) model time series, with exogenous shocks to certain parameters with specified probability distributions. And those parameters are named to suggest they have common language meanings. But there is no reason to think any such correspondence between the mathematics and the labels exist.

I assume, however, that his Nobel prize is for explaining economic growth as the result of some combination of endogenous innovation, the accumulation of human capital, and an increasing variety of capital goods embodying technical progress. Admirers of Adam Smith, Karl Marx, Piero Sraffa, Nicholas Kaldor and those with some grasp of economic history should applaud this emphasis on increasing returns to scale. Mainstream economists, however, claim not to be producing mere descriptive prose, but rigorous formal models that embody their ideas. And mainstream endogenous growth models, including those developed by Paul Romer are, deficient. They:

  1. Depend on knife-edge relationships between model parameters.
  2. Pretend to model nonhomogenous capital goods, but measure such goods in numeraire units in production functions, thereby ignoring price Wicksell effects.
  3. Are unclear on the meaning of human capital and of designs, including on measurement scales.

If you want formal models that emphasize entanglements between increasing returns to scale and the growth of capitalist economies, I recommend the work of Luigi Pasinetti on structural economic dynamics. By the way, these deficiencies in the work of Paul Romer should be well known among scholars. These points have been made in the literature, a selection of which I point out below. (I could probably find something from Solow on the first point. The Rogers' paper is on the failure of DSGE models to coherently include money and banks, that addresses the Hahn problem.)

References
  • Sergio Cesaratto (1999). New and Old Neoclassical Growth Theory: A Critical Assessment, in Value, Distribution and Capital: Essays in Honour of Pierangelo Garegnani (ed. by G. Mongiovi and F. Petri), Routledge.
  • Sergio Cesaratto (2009). Endogenous Growth Theory Twenty Years On: A Critical Assessment, working paper.
  • Man-Seop Park (2007). Homogeneity Masquerading as Variety: The Case of Horizontal Innovation Models, Cambridge Journal of Economics, V. 37 (Nov.): pp. 379-392.
  • Man-Seop Park (2010). How to give up wrestling with time: The case of horizontal Innovation Models, in Economic Theory and Economic Thought: Essays in Honour of Ian Steedman (ed. by Vint et al.), Routledge.
  • Colin Rogers (2018). The conceptual flaw in the macroeconomic foundations of Dynamic Stochastic General Equilibrium models. Review of Political Economy V. 30 (1): 72-83.
  • Ian Steedman (2003). On Measuring Knowledge in Old and New Growth Theories: An Assessment (ed. by Neri Salvadori, Edward Elgar.

Saturday, October 06, 2018

Normal Forms For Switch Point Patterns: A Research Agenda

I have been looking at the effects of perturbing parameters in models of the choice of technique. Now that I have one paper out of this research published, I thought I would recap where I am. I think I should be able to get at least another paper out of this. A challenge for me is to draw interesting economics out of these findings. In a sense, what I am doing is applied mathematics, albeit with more an emphasis on numerical exploration than proof of theorems.

I claim that the development of a taxonomy of fluke (or non-generic) switch points is of some importance in understand how reswitching, capital-reversing, and other Sraffa effects can arise. In pursuit of such a taxonomy, I have developed the concept of a pattern of switch points. The switch points and the wage curves along the wage frontier can alter with parameters, in a model of the production of commodities. Such parameters can be coefficients of production; time, where a number of parameters are functions of time; or the markup in an industry or a number of industries. A normal form exists for each pattern. The normal form describes how the techniques and switch points along the frontier vary with a selected parameter value. Each pattern is defined by the equality of wage curves at a switch point and one or more additional conditions. The co-dimension of a pattern is the number of additional conditions.

I claim that local patterns of co-dimension one, with a switch point at a non-negative, feasible rate of profits can be described by four normal forms. I have defined these patterns as a pattern over the axis for the rate of profits, a pattern across the wage axis, a three-technique pattern, and a reswitching pattern. This post is an update to an update. I continue to examine global patterns, local patterns with a co-dimension higher than unity, and sequences of local patterns. Some examples are:

  • A switch point that is simultaneously a pattern across the wage axis and a reswitching pattern (a case of a real Wicksell effect of zero). This illustrates a pattern of co-dimension two.
  • A reswitching example with one switch point being a pattern across the wage axis (another case of a real Wicksell effect of zero). This is a global pattern.
  • The last two examples, written up as a working paper. (I've already had one rejection of this paper.)
  • An example with a pattern across the wage axis and a pattern over the axis for the rate of profits. This is a global pattern.
  • A pattern like the above, but with both switch points being defined by intersections of wage curves for the same two techniques. This is a global pattern.
  • Two switch points, with both being reswitching patterns, can be found from a partition of a parameter space where two loci for reswitching patterns intersect. This gestures towards a global pattern.
  • A pattern across the point where the rate of profits is negative one hundred percent, combined with a switch point, for the same techniques, with a positive rate of profits (of interest for the reverse substitution of labor). This is a global pattern.
  • An example where every point on the frontier is a switch point. This is a global pattern of an uncountably infinite co-dimension.
  • A working paper, writing up the above, to some extent. (I've already had one rejection of this paper.)
  • Speculation on three sequences of patterns of co-dimension one that result in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
  • A switch point for a four-technique pattern (due to Salvadori and Steedman). This is a local pattern of co-dimension two.
  • Further analysis of the above example.
  • Another four-technique pattern, in which the wage curves for four techniques are tangent at a single switch point.
  • A generalization, in which the wage curves for a continuum of techniques are tangent at a single switch point, written up as a working paper.
  • An example of a four-technique pattern in a model with three produced commodities. This local pattern of co-dimension two results in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
  • Further analysis of the above example. Two normal forms are identified for four-technique patterns.
  • A working paper for the above example. (I think my personal revised copy is ready to submit.)
  • Speculation about common features of many of these examples.

The above list is not complete. More types of fluke switch points exist. Some, like the examples of a real Wicksell effect of zero, I thought, should be of interest for themselves to economists. Others show examples of parameters where the appearance of the wage frontier, at least, changes with perturbations of the parameters. I have used these patterns to tell stories about how technical change or a change in markups (that is, structural economic dynamics) can result in reswitching, capital reversing, or the reverse substitution of labor appearing on or disappearing from the wage frontier.

I would like to see that in at least some cases, short run dynamics changes qualitatively with such perturbations. But this seems to be beyond my capabilities.