Friday, November 16, 2018

Pattern Analysis for a Fixed Capital Example

Figure 1: A Pattern Diagram
1.0 Introduction

In this example, I perturb parameters in an example of Bertram Schefold's. I was disappointed in that, as far as I can see, one can analyze the choice of technique in this example by the construction of the wage-rate of profits frontier. As far as I understand, this is not true for joint production in general. I guess I also need to find an example in which the physical life of a machine is at least three years so as to find a three-technique pattern.

This example does highlight differences in different measures of capital-intensity.

2.0 Technology

Table 1 presents the technology for this example. Machines and corn are produced in this economy. Corn is the only consumption good. New machines are produced from inputs of labor and corn. Corn is produced from inputs of labor, corn, and machines. A machine can be worked for two years. After the end of the first year of its working life, it is known as an old machine. I assume each process requires a year to complete and exhibits constant returns to scale.

Table 1: Coefficients of Production
Labora0,1 = (1/10) u(t)a0,2 = (43/40) u(t)a0,3 = u(t)
Corna1,1 = (1/16) u(t)a1,2 = (1/16) u(t)a1,3 = (1/4) u(t)
New Machines010
Old Machines001
New Machines100
Old Machines010

I model technical progress by constantly decreasing inputs into each process, other than machines:

u(t) = e1 - σ t

When σ t is unity, this is Bertram Schefold's example of restitching, at rates of profits of 1/3 and 1/2.

3.0 Prices of Production

The first row in Table 1 can be summarized by a row vector, a0, of labor coefficients. The next three rows are expressed by a square matrix A. The last three rows form the matrix B. Suppose wages are paid out of the surplus product at the end of the year. If the same rate of profits is to be made in all operating processes, prices must satisfy the following system of equations;

p A (1 + r) + w a0 = p B

I let corn be the numerator:

p e1 = 1

where e1 is the first column of the identity matrix.

Given the wage, w, in a range between zero and some maximum, the above system of price equations can be solved for the rate of profits, r, the price of a new machine, p2, and the price of an old machine, p3.

4.0 Choice of Technique

The managers of firms need not run the machine for two years. They could discard the machine after only one year. (I assume free disposal.) The managers will be cost-minimizing if they run the machine for only one year if the price of an old machine is negative.

Alternatively, consider the price system when the machine is operated only two years. The matrices A and B are 2x2 square matrices, and a0 is a row vector with two elements. With these prices and the price of an old machine of zero, one could calculate the cost of operating the machine for a second year to produce a bushel of corn. When this cost is less than unity (the price of a bushel of corn), it is cost-minimizing to operate the machine for both years.

These two methods of analyzing the choice of technique yield the same answer for this example. Figure 1, above, illustrates the results. Until time reaches the pattern over the axis for the rate of profits, it is cost-minimizing to operate the machine for only one year. In Region 2, the machine is operated for two years when wages are low, and for one year when wages are higher. Region 3 is an example of reswitching. Eventually, it is cost-minimizing to operate the machine for two years, for all feasible wages.

5.0 Capital

In outdated neoclassical intuition, a higher wage indicates that labor is more scarce, in some sense, and capital is relatively more abundant. One might, wrongly, except the price system to encourage capitalists to adopt less labor-intensive or more capital-intensive techniques, in some sense. And, in a simple example like this one, one might expect the more capital-intensive technique to be one in which the machine is run for both years.

The example confounds these expectations in both Region 2 and Region 3. Around the switch point in Region 2, a higher wage is associated with the adoption of a technique in which the machine is only operated for the first year. The same is true of the same switch point - the one at the lower wage - in Region 3. From this viewpoint, the switch point is "perverse" in both regions.

This result contrasts with the usual analysis based on real Wicksell effects. The real Wicksell effect is negative for the switch point in Region 2. It is positive for the same switch point in Region 3. For a switch point with a negative real Wicksell effect, a higher wage is associated with the adoption of a technique with more net output per person-year employed. And that is so in this case too. The switch point is only 'perverse', from this perspective, in Region 3.

6.0 Conclusion

This post has illustrated that what I am calling pattern analysis can be applied to examples of joint production in which joint production is only manifested in production and use of long-lived machines. It has focused attention on the distinction between different intuitions about the capital-intensity of a technique.

Saturday, November 10, 2018

A Linear Program for Markup Pricing

Figure 1: A Partition of Price-Wage Space for a Two-Commodity Reswitching Example
1.0 Introduction

This post generalizes my approach in Vienneau (2005). In that article, I present a Linear Programming (LP) problem for the firm. In the case of an economy that produces two commodities, one can present a graphical display that clarifies how Sraffa's equations arise. The dual LP is important in this development. Here, I show how that approach can work for a case in which rates of profits systematically vary among industries.

I was pleased that this approach works out for markup pricing. In a sense, this post derives both a direct and an indirect approach for analyzing the choice of technique, in the context of a model of markup pricing.

2.0 The Model

To begin with, consider a model of the production of N commodities from labor and these commodities. This is a model with circulating capital and no joint production. Assume that managers of firms know of Uj processes for producing the output of that industry.

Each process is defined by:

  • a0, j(u), u = 1, 2, ..., Uj, the person-years of labor needed to produce one unit of the jth commodity.
  • a., j(u) = [a1, j(u) ..., aN, j(u)]T, the inputs of each commodity needed to produce one unit of the jth commodity.

Each process exhibits constant returns to scale (CRS), requires a year to complete, and use up all their inputs. I also take a set of weights for industries, 1/s1, ..., 1/sN, as givens. Let prices be p = [p1, ..., pN]. Also, let e = [e1, ..., eN]T be the numeraire, so that:

p e = 1

I should have some assumptions on coefficients ensuring that the economy can be productive by a suitable choice of technique.

I introduce some variables as abbreviations:

kj(u) = p a., j(u)
cj(u) = p a., j(u) + w a0, j(u)
πj(u) = pj - cj(u)
rj(u) = πj(u)/kj(u)

2.1 The Firm's LP

The managers of a firm take the wage, w and prices p as given. Let ω = [ω1, ..., ωN]T be the firm's inventory of each commodity at the start of the year. Let qj(u) be the quantity of the jth commodity that the firm produces with the uth process known for producing that commodity. Let qN + 1 be the value of inventory not used for purchasing inputs into production.

Each year the managers of the firm choose how much to produce of each commodity and with which process so as to maximize the weighted increment of value:

(1/s1)[π1(1) q1(1) + π1(2) q1(2) + ... + π1(U1) q1(U1)]
+ (1/s2)[π2(1) q2(1) + ... + π2(U2) q2(U2)]
+ (1/sN)[πN(1) qN(1) + ... + πN(UN) qN(UN)]

Such that the firm can purchase all of the inputs into production needed at the beginning of the year:

k1(1) q1(1) + k1(2) q1(2) + ... + k1(U1) q1(U1)
+ k2(1) q2(1) + k2(2) q2(2) + ... + k2(U2) q2(U2)
+ kN(1) qN(1) + kN(2) qN(2) + ... + kN(UN) qN(UN) ≤ p ω

For all j, u:

qj(u) ≥ 0

The weights formalize the concept that managers find some industries more desirable or easier to invest in than others. It works out that an industry that managers are less willing to contest or expand production in has a larger rate of profits, in the system of prices of production.

2.2 The Dual LP

The above LP has a dual problem. It is to choose r to minimize:

p ω r

Such that for all j, u:

p a., j(u) (1 + rsj) + w a0, j(u) ≥ pj
r ≥ 0

When a decision variable is positive in a solution to the primal LP, the corresponding constraint is met with equality in the dual LP. Suppose the solution of the primal LP leads to each commodity being produced by a specific process in each industry. The price system defined by the technique composed of those process will be satisfied. The economy will be on the wage curve for that technique.

3.0 Solution of the Primal LP

The solution to the primal LP is illustrated by Table 1. 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. The decision variable qN + 1 is a slack variable, introduced to convert the inequality constraint in the primal LP into an equality. It represents the value of inventory carried over, without supporting production. The conditions for when a decision variable is in the basis are intuitive. Consider the first row. A given commodity is produced with a given process only if the rate of profits made in other processes producing that commodity do not exceed the rate of profits made in the given process. Furthermore, the marked-up rate of profits in producing other commodities must not exceed the marked-up rate of profits in the given process. Finally, the (undiscounted) cost of producing a the given commodity must not exceed the revenue made from selling iron. (I am aware that there is some redundancy in how I have stated conditions in the table.)

Table 2: Solution of Primal LP
in Basis
ValueWhen Optimal
qJ(V)p ω/kJ(V)For u = 1, 2, ...,UJ
[pJ - w a0, J(V)]/kJ(V) ≥ [pJ - w a0, J(u)]/kJ(u)
For all j, u
(1/sJ)rJ(V) ≥ (1/sj)rj(u)
cJ(V) ≤ p
qN + 1p ωFor all j, u
cj(u) ≥ p

The solution to the primal LP, in a two-commodity example, is easily visualized. The second commodity is the numeraire, and the price of the first commodity is graphed on the ordinate. 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 Figure 1. Each region is labeled by that decision variable, in an obvious notation. On the boundaries, a solution to the LP can be formed from a linear combination of decision variables. In the example, both commodities must be 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. One technique is adopted at low and high wages, while the other technique is used at intermediate wages. The figure also illustrates that the wage cannot exceed a maximum.

4.0 Conclusion

I have thought about how this LP approach generalizes. In a general joint production framework, it is not immediately obviously how to assign processes to industries. So I do not see how to define the weights. I suppose one could have a weight for each process, instead of for each industry.

Land presents another difficulty. One would like to impose additional constraints in the primal LP to specify that overall production cannot require that more than a given quantity of some inputs cannot be used in production. Then multiple processes would be used, in a model of extensive rent, in certain industries. But should not such constraints be imposed above the level of the firm? That is, if a firm's production meets the constraints, they might still be violated in the economy as a whole.

But, I suppose, this LP approach applies to cases of fixed capital, where joint production is such that firms in an industry can choose to operate multiple processes, each jointly with a machine of a specific age.


Friday, November 02, 2018

Extending An Example With Markup Pricing

Figure 1: A Two-Dimensional Pattern Diagram

The example in this working paper is of an economy in which two commodities are produced. Technical progress is modeled as decreasing the coefficients of production in one of the processes for producing corn. They decrease at a rate of σ of ten percent.

Figure 2 shows how the pattern of switch points vary with technical progress. Initially, the Beta technique is cost-minimizing. Then it becomes a reswitching example. Around the switch point at the lower rate of profits, a higher wage is associated with more labor being hired, per unit of net output. Also, a higher wage is associated with the adoption of more direct labor being hired in corn production, per bushel corn produced gross. This is called a reverse substitution of labor. The other switch point disappears over the wage axis with more technical progress. The remaining switch point still exhibits a reverse substitution of labor. Eventually, that switch point no longer exhibits such a reverse substitution. Finally, it disappears entirely.

Figure 2: A Pattern Diagram

I have been exploring how this example behaves with full cost pricing. I let the rate of profits in the iron industry be s1 r, and the rate of profits in the corn industry be s2 r. Figure 1 illustrates how these modeling choices for technical progress and markup pricing interact when s2 = 1.

Figure 2 illustrates the characteristics of switch points along a horizontal line, at s1, in Figure 1. The numbered areas in the two figures correspond. Only one switch point exists in the region numbered 6, and it has a positive real Wicksell effect.

The example illustrates that an increase in the markup in a specific industry can result in the creation of a switch point in which higher wages are associated with firms wanting to employ more workers, both per unit net output in the economy as a whole and per unit gross output in a specific industry. Think of a vertical line going through Regions 6, 2, and 1, and, specifically, the partition between Regions 6 and 2. On the other hand, the transition from Region 5 to Region 3 is associated with creation of a switch point that only exhibits a reverse substitution of labor; it still has a negative real Wicksell effect.

Thanks to the comments of Sturai for encouraging me to write this post and for pointing out a paper by Antonio D'Agata that I'll have to read.

Wednesday, October 24, 2018

Structural Economic Dynamics, Real Wicksell Effects, and the Reverse Substitution of Labor

I have uploaded another working paper:

This article presents an example in which technical progress results in variations in the labor market. Around a switch point with a positive real Wicksell effect, a higher wage is associated with firms wanting to employer more labor per unit output of net product. Around a switch point with a reverse substitution of labor, firms in a particular industry want to hire more labor per unit output of gross product. Technical progress can bring about and take away circumstances favorable for workers wanting to press claims for higher wages.

My research approach can generate fluke switch points. I have decided that such flukes are more interesting when placed in a story about the perturbation of parameters.

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
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
in Basis
ValueWhen Optimal
q1(p ω1 + ω2)/(pa1, 1 + a2, 1)r1rα2
qα2(p ω1 + ω2)/(paα1, 2 + aα2, 2)r1rα2
cα2 ≤ 1
qβ2(p ω1 + ω2)/(paβ1, 2 + aβ2, 2)r1rβ2
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.)

  • 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.

Saturday, September 29, 2018

Cambridge Capital Controversy Applied At The Level Of The Firm

For a number of decades, Arrigo Opocher and Ian Steedman have been developing arguments that apply the CCC to industries and even individual firms. They also draw on mainstream literature in microeconomics, from the 1960s and 1970s. Their 2015 book is a major statement of their position. Since their book's publication, they have continued research in this vein.

The CCC applies whenever you see a production function with capital measured in numeraire-units. This can be an aggregate production function for the economy as a whole, at the level of an individual industry, or even for an individual firm. Arguably, any model with such a component is incoherent. Opocher and Steedman, in their book, however emphasize a representation of technology in terms of cost functions. They consider cases with a rate of profits of zero and issues that arise even with disaggregated capital inputs or, even, no capital inputs.

I recently skimmed Steedman (2018). This looks at the incoherence of representing technology with capital-labor isoquants. Since constant returns to scale are assumed, the distinction between analysis at the firm and industry level is not definite. Many of Steedman's papers, with and without Opocher, present exercises for the reader. But I reacted to the following as if he was trolling me:

As was noted above for the general case, each kj and each lj is a known function of r; in the specific case of our example, (4), (5) and (7) permit the explicit calculation of k1(r) and l1(r). Since l1(r) is always increasing, (d2k1/dl12) has the same sign as [(dl1/dr)(d2k1/dr2) - (dk1/dr)(d2l1/dr2)] [Notation changed] ..., and some calculation leads to the conclusion stated in the text; unfortunately, the equations involved are rather long and tedious, so they are left for the amusement of the interested reader. In our example, it is possible to find k1(l1) explicitly but, again, the equation is not a pleasant one.

I suppose I may someday take up this challenge.

  • Arrigo Opocher and Ian Steedman (2015). Full industry equilibrium: A theory of the industrial long run. Cambridge University Press
  • Arrigo Opocher and Ian Steedman (2016). Recurrence: A neglected aspect of the Sraffian critique of marginalism. Metroeconomica 67(3): pp. 1-6.
  • Ian Steedman (2018). Industry-level capital-labour isoquants. Metroeconomica: pp. 1-6.

Saturday, September 22, 2018

Two Kinds Of Economists

Or, rather, I classify economists into two kinds on each of three dimensions (Table 1).

Table 1: Classifications For Economists
Emphasis on social reproductionEmphasis on allocating scarce resources
Money non-neutralMoney as a veil
Economic issues arise under competitive model, with all agents in possession of all information actually existingEconomic issues to be explained as a result of deviation from an ideal, competitive model

I have written about the first dimension before. Classical political economy, and economists in related traditions, focus on what needs to hold such that society is reproduced. Neoclassical economics is defined, by many, as being about the allocation of scarce resources.

Post Keynesians and others describe money as having real effects. Many mainstream economists, on the other hand, model capitalist economies as, basically, barter economies. They hold money to be neutral, at least in the long run. It is not clear that such models can be extended to contain money.

My third dimension, above, relates to attitudes to two types of models. In one, an economy is described, at a high level of abstraction, as characterized by free competition, with no agent being able to influence market prices, and all agents having complete information about what can be known. In the other model, one introduces rigidities and stickiness in prices; oligopolies, monopolies, and monopsonies; information asymmetries; and so on. One group of economists thinks the former model can describe an economy that need not tend to an equilibrium with desirable properties. Many mainstream economists, however, think actually existing economies are to be described by deviations from perfect competition and that it is the goal of policy to try to make actual economies function like the ideal. (I was inspired to try to define this dimension by Palermo (2016).)

Theoretically, the above taxonomy yields eight kinds of economists. I do not know that one can find important economists at every node of the cube so defined. But, to see how this works out, consider Joseph Schumpeter. He emphasized scarcity, thought money and finance impact real variables, and saw issues with a perfectly competitive economies. For the latter, consider his argument - later taken up by John Kenneth Galbraith - that large corporations were needed for the research and development needed for growth in a mature economy.

John Maynard Keynes is another economist that emphasized the real effects of money and argued issues can arise in the ideal economy. He argued, in the General Theory, that a perfectly competitive economy would be violently unstable. Rigidities in wages are desirable, for they provide stability. I am not sure where I would put him on the first dimension, but followers at Cambridge, such as Kaldor and Robinson, developed models of warranted growth in the 1950s that lie in the upper left box in the figure.

Obviously, this post should go on to explore more nodes in the cube I have outlined.

  • John H. Finch and Robert McMaster (2018). History Matters: On the mystifying appeal of Bowles and Gintis. Cambridge Journal of Economics.
  • Giulio Palermo (2016). Post-Walrasian Economics: A Marxist Critique. Science & Society 80(3): 346-378.

Sunday, September 16, 2018

Normal Forms for Switch Point Patterns

My article with the post title is now available at the Review of Behavioral Economics. The abstract follows:

Abstract: The choice of technique can be analyzed, in a circulating-capital model of prices of production, by constructing the wage frontier. Switch points arise when more than one technique is cost-minimizing for a specified rate of profits. This article defines four normal forms for variations in the number and sequence of switch points with a perturbation of, for example, a coefficient of production. The 'perversity' of switch points that appear on and disappear from the wage frontier is analyzed. The conjecture is made that no other normal forms for local patterns of co-dimension one exist.

Saturday, September 15, 2018


  • Maeve Cohen, the director of Rethinking Economics, notes the absurdism of undergraduate economics teaching, even after the Global Financial Crisis. In this one page article in Nature, she calls for greater pluralism in teaching. (This has led to the usual whining and silliness in the usual places.)
  • The American Economic Association has a moderated discussion board. I suspect much of the discussion will be too focused on narrow questions for interest by non-economists.
  • Thomas Piketty, Emmanuel Saez, and Gabriel Zucman have estimates for income in the United States, over time, for various percentiles. These can be called distributional national accounts.
  • Yanis Varoufakis calls for an international movement to fight both a re-insurgent fascism and establishment globalists.
  • Georgist single-taxers are not too my taste. I found this website for the New Physiocratic League colorful. I find intriguing the concept of certifying a political party's platform.

Saturday, September 01, 2018

Theses For Debate In Reading Marx

I present four claims about Marx's Capital. I strive for topics more general than, for example, squabbles about the transformation problem. I suggest that some of these claims present a useful focus for reading Marx's book, even if part of your focus is arguing why the claim is wrong. If this were more than a blog post, I would need to cite various Marxists and scholars that inspired me.

Thesis I: Capital is organized around a model of a pure, two-class capitalist economy.

I think the above claim is helpful in making sense of the opening chapters of Volume 1 and of Volume 2. In Volume 2, I am thinking of the analysis of the analysis of various circuits, as well as the models of simple and expanded reproduction.

This claim separates out the historical material and the analysis more sharply than some commentators on Marx accept. I guess it is consistent with some of Marx's use of Blue Books filed by factory inspectors in Britain. Historical material that goes beyond a model of pure capitalism includes the analyses of primitive accumulation in pre-capitalist formations and of the development of machinery and manufacture. I think of the replacement of the putting-out system, handicraft, and domestic industry by factories.

Thesis II: Capital continues the tradition of classical political economy; it does not represent a sharp break with this tradition.

One can argue Marx saw William Petty, Francois Quesnay, Adam Smith, and David Ricardo, for example, as having applied a scientific method of abstraction to identify essences that lie behind the surface phenomena of market prices. Of course, Marx had many criticisms of his predecessors. He thought Smith had not sufficiently distinguished labor that was and was not productive of surplus value. Even Ricardo did not distinguish (abstract, social) labor from labor power. Marx argued his distinction between constant and variable capital was more fundamental, in some sense, that the classical political economy distinction between fixed and circulating capital. And the classical did not talk about surplus value in general, instead of manifestations in the form of profits, interest, and rent.

This claim of continuity can also be argued to be consistent with Marx's contrast of vulgar and scientific political economy. Not everybody in the time of the classics, including Adam Smith, were thoroughgoing in the application of their scientific method.

But some of what Marx has to say about illusions generated by competition is in tension with this claim of continuity. He was interested in what social conditions made possible the development of political economy. The classical political economists championed the rising bourgeois before the social question became sufficiently biting. And what about the sarcasm and irony in Capital.

Thesis III: The system of labor values is a reality behind the appearance of freedom in market transactions.

In some sense, labor values provide a sub-basement underlying a building more obvious to our sight.

A counter thesis would be based on a Wittgenstein-like reading of Capital. Nothing is hidden, but markets, like languages, are befuddling. Marx is presenting arrangements in a therapeutic treatment to dissolve confusions. This also gets into some readings of Sraffa's work.

Thesis IV: One can accept the analysis in Capital as a way of understanding the world, independently of a any position on the desirability of changing it, either through a revolution or otherwise.

Friday, August 24, 2018

A Semi-Idyllic Golden Age

1.0 Introduction

This post presents a model of a steady state with a constant rate of growth in which:

  • Total wages and total profits grow at the same rate.
  • Neutral technical change increases the productivity of labor in all industries.
  • The wage per hour increases with productivity.
  • Each worker continues to consume the same quantity of produced commodities.
  • But each worker takes advantage of increased productivity to work less hours per year.

In these times, when concerns about global warning are so important, one would also want to see a suggestion of a reduced ecological footprint. So this model of a steady state is only semi-idyllic.

I do not consider anything in the mathematical model below to be original. I outline it to raise the question whether such a growth path is possible under capitalism. The model demonstrates logical consistency, but cannot demonstrate that details abstracted from in the model would not prevent its realization.

2.0 The Model

Consider a closed economy with no foreign trade. Industries are grouped into two great departments. In Department I, firms produce means of production, also known as capital goods. The output of Department I is called ‘steel’ and measured in tons. In Department II, firms produce means of consumption, also known as consumer goods. The output of Department II is called ‘corn’, measured in bushels. Both steel and corn are produced from inputs of steel and labor.

Constant coefficients of production (Table 1) are assumed to characterize production in each year. All capital is circulating capital. Long-lived machines, natural resources, and joint production are abstracted from in this model. Free competition is assumed. Labor is advanced, and wages are paid out of the net output at the end of the year. Workers are assumed to spend all of their wages on means of consumption. Profits are saved at a constant proportion, s.

Table 1: Constant Coefficients of Production
a0, 1(t)Labor required as input per ton steel produced in year t.Person-Hrs per Ton
a1, 1Steel services required as input per ton steel produced.Tons per Ton
a0, 2(t)Labor required as input per bushel corn produced in year t.Person-Hrs per Bushel
a1, 2Steel services required as input per bushel corn produced.Tons per Bushel

Suppose coefficients of production for steel inputs are constant through time, but labor coefficients exhibit a growth in labor productivity of 100 ρ percent:

a0, j(t + 1) = (1 - ρ) a0, j(t), j = 1, 2

Let Xi(t), i = 1, 2; represent the physical output produced in each department in year t and available at the end of the year. Furthermore, suppose the price of steel, p, and the rate of profits, r, are constant. Let outputs from each of the two departments grow at a constant rate of 100 g percent:

Xi(t + 1) = (1 + g) Xi(t), i = 1, 2

Certain quantity equations follow from these assumptions. The quantity of capital goods added each year must equal the capital goods remaining after reproducing those used up in producing total output, in both departments:

g [a1,1 X1(t) + a1,2 X2(t)]
= X1(t) - [a1,1 X1(t) + a1,2 X2(t)]

The person-years of labor employed relates to labor coefficients and gross outputs:

L(t) = a0, 1(t) X1(t) + a0, 2(t) X2(t)

Price equations are:

p a1, 1 (1 + r) + a0, 1 w(t) = p
p a1, 2 (1 + r) + a0, 2 w(t) = 1

These equations embody the use of a bushel corn as numerate. w(t) is the wage per person-hour, paid out at the end of the year out of the surplus.

These assumptions and parameters are enough to depict Table 2. The column labeled "Constant capital" shows the value of advanced capital goods, taking the output of Department II as the numeraire. The column labeled "Variable Capital" depicts the wages paid out of revenues available at the end of the year. The surplus is what remains for the capitalists.

Table 2: A Tableau Economique
Ip a1,1 X1(t)w(t) a0,1 X1(t)p a1,1 X1(t) rp X1(t)
IIp a1,2 X2(t)w(t) a0,2 X2(t)p a1,2 X2(t) rX2(t)

Workers spend what they get, and capitalists save a constant ratio, s, of their profits. With these assumptions, one can calculate the bushels corn that the workers and capitalists in Department I want to purchase, at the end of each year, from Department II. Likewise, one can calculate the numeraire value of the steel that capitalists in Department II want to purchase from Department I. Along a steady state, these quantities must be in balance:

[a0, 1(t) w(t) + (1 - s) p a1, 1 r] X1(t)
= p a1, 2 [1 + s r] X2(t)

This completes the specification of this model of expanded reproduction with technical change uniformly increasing the productivity of labor.

3.0 The Solution

Output per labor hour is found by solving the quantity equations:

X1(t)/L(t) = a1, 2 (1 + g)/β(t, g)
X2(t)/L(t) = [1 - a1, 1 (1 + g)]/β(t, g)


β(t, g) = a0, 2(t) + [a0, 1(t) a1, 2 - a0, 2(t) a1, 1](1 + g)

That is:

Xi(t)/L(t) = [1/(1 - ρ)t] [Xi(0)/L(0)], i = 1, 2

The path of employed labor hours falls out as:
L(t) = (1 - ρ)t (1 + g)t L(0)

The number of employed person-hours decreases if:

ρ > g

The above expresses the condition that the labor inputs needed to produce a unit of output, in both departments, decrease faster than the rate of growth in both departments.

The price equations are also easily solved. Given a constant rate of profits, the price of steel is constant as well:

p = a0, 1(0)/β(0, r)

The wage per person-hour increases with productivity:

w(t) = [1 - a1, 1 (1 + r)/β(t, r) = [1/(1 - ρ)t] w(0)

The trade-offs between consumption per worker and the steady-state rate of growth and between the wage and the rate of profits have the same form.

These solutions can be substituted into the balance equation. It becomes:

[1 - a1, 1 (1 + s r)] (1 + g) = [1 - a1, 1 (1 + s r)] (1 + s r)

Suppose the rate of profits falls below its maximum (where the workers ‘live on air’) or not all profits are saved. Then this is a derivation of the "Cambridge equation":

r = g/s

A steady rate of growth, when the workers consume their wage, requires that the rate of profits be the quotient of the rate of growth and the savings rate out of profits.

4.0 Demographics and Institutions

I make some rather arbitrary assumptions about demographics and institutions. Suppose the number of person-years supplied as labor grows at the postulated rate of growth:

LS(t + 1) = (1 + g) LS(t)

with LS(t) measured in person-years. Let the number of hours in a standard labor-year, α(t) decrease at the same constant rate as the growth in productivity:

α(t + 1) = (1 - ρ) α(t)

The rate at which the total supply of labor-hours increases is easily calculated:

α(t + 1) LS(t + 1) = (1 - ρ) (1 + g) α(t) LS(t)

Under these assumptions, the supply of labor-hours grows at the same rate as the demand for labor-hours. Total wages and total profits increase at the same rate, 100 g percent. The wage per worker increases at the same rate as the standard length of a labor year declines. Thus, workers consume a constant quantity of commodities, but they take increased productivity in steadily increased free time.

5.0 Discussion and Conclusions

What should one postulate about money in this model? One could assume the money supply grows endogenously, along with commodities. Or, perhaps, the velocity of the circulation of money increases with productivity. A continuous decrease in the money price of corn is another logical possibility. Perhaps Rosa Luxemburg was right, and an external source of demand from less developed regions and countries is needed to support expanded reproduction. Or Kalecki is correct, and military spending by the government will do.

I do not know if this model describes any existing capitalist economy. It does not describe the post-war golden age. In that time, at least in the United States, workers took increased productivity in increased consumer goods. (I think the memory of the Great Depression, the occurrence of World War II, and the existence of the Soviet Union has something to do how this worked out.) Could any capitalist economy function like this? Somehow, an advertising industry is not encouraging workers to consume ever more produced commodities, or they ignore such messages. They continually have more freedom. Yet, they always spend a bit of time under the domination and direction of their employers. Will the capitalists tolerate this?

Monday, August 20, 2018

Samir Amin (1931-2018)

Despite the label at the bottom of this post, this is not really a profile of Amin. I happen to have started reading Modern Imperialism, Monopoly Finance Capital, and Marx's Law of Value (Monthly Review Press, 2018) last month. Here are a couple of quotations:

"Vulgar economics is obsessed with the false concept of 'true prices,' whether for ordinary commodities, for labor, for money, for time, or for natural resources. There are no 'true prices' to be 'revealed' by the genius of the 'market.' Prices are the combined products of rates of exploitation of labor (rates of surplus-value), of competition among fragmented capitals, and the deduction levied in the form of 'oligopoly rents,' and of the political and social conditions that govern the division of surplus-value among profits, interest, ground rents, and extractive rents." -- Amin, p. 99.

"Marx's criticism of the classic bourgeois political economy of Smith and Ricardo concluded by shifting from analysis centered on 'the market' ... to one centered on the depths of production where value and the extraction of on surplus value are determined. Without this shifting of the analysis from the superficial to the essential, from the apparent to the concealed, no radical critique of capitalism is possible...

The law of value formulated by Marx, based on the concept of abstract labor, expresses the rationality of the social utility (the utility for society) of a defined use value. This rationality transcends that which governs the reproduction of a particular mode of production (in this case, the capitalist mode of production). Under capitalism, rationality demands the accumulation of capital, itself based on the extraction of surplus value. The price system frames the operation of this rationality. Economic decisions in this framework ... will be different from those that might be made on the basis of the law of value that would define, in the socialism to come, the mode of social governance over economic decision making.

Bourgeois economic theory attempts to prove that the mode of decision making in the framework of its system of prices and incomes produces a rational allocation of labor and capital resources synonymous with an optimal pattern of output. But it can reach that goal only through cascading tautological arguments. To do so it artificially slices productivity into 'components' attributed to 'factors of production.'

Although this pattern of slices has no scientific value and rests on tautological argument, it is 'useful' because it is the only way to legitimize capital's profits. The operative method of this bourgeois economics to determine 'the wage' by the marginal productivity of 'the last employee hired' stems from the same tautology and breaks up the unity of the collective, the sole creator of value. Moreover, contrary to the unproven affirmations of conventional economics, employers do not make decisions by using such 'marginal calculations.'" -- Amin, pp. 232-234.

I have several other books by Amin on my bookshelf:

  • Samir Amin (2006). Samir Amin: A Life Looking Forward: Memoirs of an Independent Marxist. Zed Books.
  • Samir Amin (1998). Spectres of Capitalism: A Critique of Current Intellectual Fashions. Monthly Review Press.
  • Samir Amin (1997). Capitalism in the Age of Globalization: The Management of Contemporary Society. Zed Books.

As I understand Amin is most well known for inventing the word "Eurocentrism" and for extending the law of value to the law of worldwide value.

Amin builds on the concept of the "surplus", as developed in the work of Paul Baran and Paul Sweezy. One can formalize this notion in a model of a developed country with three departments, for producing capital goods, consumption goods, and luxuries. The last department is not in Marx's models of simple and expanded reproduction. This department is needed to address the problem of realization in an age of monopoly capital.

When it comes to realization problems, there is a long tradition among Marxists of looking at open economies, with advanced industrial capitalist economies trading with less developed peripheral regions or countries. Amin, an Egyptian trained in Paris and working in Dakar, was well positioned to develop these ideas of North-South trade. In the book mentioned above, he often talks about extending Marx's law of value to the law of worldwide value. I gather his ideas are partly the result of a critical engagement with Andre Gunder Frank's work, which I do not know.

To my mind, you can find similar ideas, about monopoly and finance capital and imperialism, going back to the time of the Second International. Amin mentions Rosa Luxembourg, but, as I recall, is critical of her. By the way, he groups Sraffa with bourgeois economists.

I was hoping to find Amin providing an exposition of a mathematical model in Modern Imperialism. He does provide some, but mostly he sticks with numerical examples and historical analysis. He says that this is, partly, to make his work accessible to a larger audience. Also, I am not sure that a mathematical model of the whole is appropriate for monopoly capital. I guess if I want to explore more, I should look at his 1974 book, Accumulation on a World Scale.

Saturday, August 11, 2018

Economists In Popular Fiction

Apparently, a character in a current movie, Crazy Rich Asians is an economist. Dan Kopf considers whether she is a good economist. In a couple of recent tweets, Paul Krugman reacts:

"Actually, I can fill this gap.

"There was a movie titled The Internecine Project ... with James Coburn as a chairman of the Council of Economic Advisers who gets a bunch of people to kill each other to hide his evil past. Sounds good to me, but the movie was terrible." -- Paul Krugman, 9 August 2018

I do not know about the movie versions, but I can name a couple of book series with characters who are economists:

  • Meyer is the sidekick in John D. MacDonald's Travis McGee mystery series. Meyer's houseboat is the John Maynard Keynes, until it is blown up. He replaces it with the Thorstein Veblen.
  • The love interest in the Bourne Identity series is an economist. If I recall correctly, Jason Bourne first meets her by carjacking and kidnapping her, and then forcing her to drive with him to Paris.

I don't think you can count the Marshall Jevons' mystery series, since that is a pen name for two economists.

Friday, August 03, 2018

A Unique Natural Rate Of Interest?

1.0 Introduction

In explaining the policy implications of the Austrian Business Cycle Theory, Hayek argued that the central bank should try to keep the money rate of interest rate equal to the natural rate. Sraffa famously criticized Hayek by describing a model with multiple interest rates, not necessarily all equal. In reply, Hayek asserted that all the interest rates in Sraffa's example would be equilibrium rates. Sraffa had a rejoinder:

"The only meaning (if it be a meaning) I can attach to this is that his maxim of policy now requires that the money rate should be equal to all these divergent natural rates."

This interchange was part of the downfall of the Austrian theory of the business cycle. I thought I would try to shortly explain what is and is not compatible with a unique natural interest rate.

2.0 Multiple Interest Rates Compatible with a Unique Natural Interest Rate

When one talks about the interest rate or the rate of profits, one may be abstracting from all sorts of complications. And these complications may be consistent with multiple interest rates, in some sense. Yet these multiple interest rates were not in dispute between Hayek and Sraffa.

2.1 Interest Rates for Loans of Different Lengths

Suppose at the start of the year, one can obtain a one-year loan of money for an interest rate of 10%. At the same time, one can obtain a two-year loan for 21%. Implicit in these different rates is a prediction that a one-year loan will be available at the start of next year for an unchanged interest rate of 10%. This implication follows from some trivial arithmetic:

1 + 21/100 = (1 + 10/100)(1 + 10/100)

The yield curve generalizes these observations. A certain shape, with the interest rate increasing for longer loans is consistent with the interest rate being expected to be unchanged, for loans of a standard length, over time.

2.2 Interest Rates for Loans of Different Risks

One might also find interest rates being higher for loans deemed riskier, independently of the time period for which the loan is made. This variation is consistent with talk of the interest rate. Often, in finance, one sees something called the risk-free rate of interest defined and used for discounting income streams. In practice, the rate on a United States T-bill is taken as the risk-free rate.

2.3 Rate of Profits

One can also distinguish between finance and business income. One might refer to the interest rate for the former, and the rate of profits for the latter. Kaldor and others, in a dispute over a Cambridge non-marginal theory of the distribution of income, have described a steady state in which the interest rate is lower than the rate of profits. Households lend out finance to businesses and obtain the interest rate. Such a steady state is compatible with the existence of two classes of households. Capitalist households receive income only from their ownership of firms.

2.4 Rates of Profits Varying Among Industries

Steady states are also compatible with the rate of profits varying among industries, as long as relative profit rates are stable. Perhaps some industries require work in more uncomfortable circumstances. Or perhaps firms are able to maintain barriers to entry.

3.0 Interest Rates with Different Numeraires

I have shown above how money interest rates for loans of different lengths embody expectations of the future course of money interest rates. Interest rates need not be calculated in terms of money. They can be calculated for any numeraire. And the ratio of real interest rates embody expectations of how relative prices are expected to change.

As an example, suppose that at a given time t, both spot and forward markets exist for (specified grades of) wheat and steel. One pays out dollars immediately on both spot and forward markets. Consider the following prices:

  • pW, t: The spot price of a bushel wheat for immediate delivery.
  • pS, t: The spot price of a ton steel for immediate delivery.
  • pW, t + 1: The spot price of a bushel wheat for delivery at the end of a year.
  • pS, t + 1: The spot price of a ton steel for delivery at the end of a year.

The wheat-rate of interest is defined by:

(1 + rW) = pW, t/pW, t + 1

I always like to check such equations by looking at dimensions. The units of the numerator on the right-hand side are dollars per spot bushels. The denominator is in terms of dollars per bushel a year hence. Dollars cancel out in taking the quotient. The wheat interest rate is quoted in terms of bushels a year hence per immediate bushels.

Suppose all real interest rates are equal. So one can form an equation like:

pW, t/pW, t + 1 = pS, t/pS, t + 1


pW, t/pS, t = pW, t + 1/pS, t + 1

If spot prices a year hence were expected not to be in the ratio of current forward prices, one would expect to be able to make a pure economic profit by purchasing some goods now for future delivery. Hence, a no-arbitage condition allows one to calculated expected relative prices from quoted prices on complete spot and forward markets.

Anyways, a steady state requires constant ratios of spot prices and, thus, real interest rates to be independent of the numeraire. This is the condition Hayek imposed in his exposition of Austrian business cycle theory in Prices and Production. And this is the condition that he dropped in his argument with Sraffa, leaving his macroeconomics a confused mess.

I might as well note that a steady state is consistent with constant inflation. If all prices go up at, say, ten percent, relative spot prices do not vary. On the other hand, relative spot prices differ with the interest rate in comparisons across steady states.

4.0 Temporary Equilibrium with Consistent Plans and Expectations

Perhaps Hayek was willing to get himself into a muddle about the natural rate because he had already investigated another equilibrium concept in previous work.

Suppose above that real interest rates vary among commodities. Then forward prices show expected movements in spot prices. One might go further and assume a complete set of forward markets do not exist. Markets clear when each agent believes they can carry out their plans, consistent with their expectations, including of future spot prices. Should one call such market-clearing an equilbrium, even if agents plans and expectations are not mutually consistent?

Concepts of temporary, intertemporal, and sequential equilibrium were to become more important in mainstream economics after Hayek quit economics, more or less. John Hicks was a major developer of these ideas, under Hayek's influence at the London School of Economics. He eventually came to accept that the mainstream notions could not be set in historical time and were, at best, of limited help in understanding actual economies.

5.0 Conclusion

The above has outlined multiple ways in which multiple interest rates and multiple rates of profits are compatible with steady states. Nevertheless, such circumstances are often described by models in which one might talk about the rate of interest.

I have also described an equilibrium in which one cannot talk about the interest rate, whether natural or not. Advocates of Austrian business cycle theory have never clarified how it can be set in a temporary equilibrium. One can sometimes find Austrian fanboys asserting that critics do not appreciate distinctions between:

  • Sources of exogenous shocks in central banks and supposed determinants (inter temporal preferences, technology) of the natural rate
  • Money rates of interest and real rates
  • Subjectivism and objectivism
  • Interest rates and relative prices.

But assertions do not constitute an argument. One would have to do some work to show that these distinctions can serve to rehabilitate Austrian business cycle theory. No matter how much you send somebody chasing through the literature by Kirzner, Lachmann, Jesus Huerta de Solo, and Garrison, they will find the work has yet to be done. (Robert Murphy probably knows this.)

  • Hahn, Frank. 1982. The neo-Ricardians. Cambridge Journal of Economics 6: 353-374.
  • Hayek, F. A. 1932. Money and Capital: A Reply. Economic Journal 42: 237-249.
  • Kaldor, Nicholas. 1966. Marginal Productivity and the Macro-Economic Theories of Distribution: Comment on Samuelson and Modigliani. Review of Economic Studies 33(4): 309-319.
  • Sraffa, Piero. 1932. Dr. Hayek on Money and Capital. Economic Journal 42: 42-53.
  • Sraffa, Piero. 1932. A Rejoinder. Economic Journal 42: 249-251.

Saturday, July 28, 2018

Complex Numbers As A Field Extension

1.0 Introduction

This is some well-established mathematics. I do not know of any use in economics.

2.0 The Field of Real Numbers

I start by taking real numbers R, with binary operations for addition and multiplication, as known.

3.0 A Ring of Polynomials

Consider polynomials with coefficients taken from the reals as a formal object:

R[X] = { an Xn + ... + a1 X + a0 | n ≥ 0,
coefficients from the reals. }

The symbol X is known as an indeterminate. One does not consider it here as a member of some set. Polynomial addition is defined to yield another polynomial, with addition of coefficients in the reals. Polynomial multiplication is also defined as usual.

3.1 Evaluation Homomorphisms

Still, one would like to talk about evaluating polynomials. For every real number r, there exists an evaluation homomorphism φr( ) that maps R[X] into the reals. This homomorphism is defined by:

φr( an Xn + ... + a1 X + a0 ) = an rn + ... + a1 r + a0

Addition and multiplication on the right-hand side above is performed in the reals. The map is a homomorphism because it preserves addition and multiplication:

φr( f(X) + g(X) ) = φr( g(X) ) + φr( g(X) )
φr( f(X) g(X) ) = φr( g(X) ) φr( g(X) )

In words, it does not matter, in evaluating the sum or product of polynomials if:

  • You perform the operation in the polynomial ring first and then evaluate the sum, or
  • You evaluate the polynomials and then sum or multiply in the reals.

A homomorphism that is one-to-one is an isomorphism. These evaluation homomorphisms are not isomorphisms since more than one polynomial may be evaluated to have the same value. An example follows:

φ2(X2) = φ2(X + 2) = 4

3.2 A Polynomial Without a Multiplicative Inverse

The constant polynomials 0 and 1 are the additive and multiplicative identities for polynomial addition and multiplication, respectively. These identities are distinct.

Not all polynomials have a multiplicative inverse. A simple example is the polynomial X. Suppose f(X) were a polynomial in R[X] that was the multiplicative inverse of X. Then:

X f(X) = 1

Consider the evaluation homomorphism for the additive identity in the reals.

1 = φ0(X f(X)) = φ0(X) φ0(f(X)) = 0 φ0(f(X)) = 0

So the non-existence of a multiplicative inverse for X is proven by a proof by contradiction.

It has been demonstrated that R[X] cannot be a field, since not every non-zero element has a multiplicative inverse. I believe it is actually an integral domain. Just as the field of rational numbers can be constructed as equivalence classes of ordered pairs of integers, a field of rational polynomials with real coefficients can be constructed. I do not pursue this construction here.

4.0 Polynomial Addition and Multiplication Modulo p(X)

One can define the quotient q(X) and remainder r(X) for any polynomials f(X) and g(X) in R[X]:

f(X) = q(X) g(X) + r(X)

where r(X) is of degree less than g(X). Since the reals are a field, the quotient and remainder are unique.

The above theorem allows one to define polynomial addition and multiplication modulo p(X). In particular, consider:

p(X) = X2 + 1

p(X) is irreducible. There do not exist non-constant polynomials f(X) and g(X) in R[X such that:

p(X) = f(X) g(X)

I now define the set C of polynomials

C = { r(X) | there exists a f(X) in R[X]
such that r(X) = f(X) mod p(X)}

All polynomials in C are at most of degree one.

4.1 C as a Two-Dimensional Vector Space

Each element of C can be expressed as a linear combination of the elements of the basis {X, 1}:

C = { (a1, a0) | a1 X + a0 is in R[X]}

4.2 C as a Field Extension of the Reals

Consider C with addition and multiplication defined modulo p(X). I claim this is a field. Consider:

f(X) = a1 X + a0
g(X) = (-a1/(a02 + a12)) X + a0/(a02 + a12)

Their product in R[X] is:

f(X) g(X) = ((-a12/(a02 + a12)) X2 + a02/(a02 + a12))

The quotient and remainder are found from:

f(X) g(X) = p(X) (-a12/(a02 + a12)) + 1


(f(X) g(X)) mod p(X) = 1

So every non-zero element of C has a multiplicative inverse.

The set of constant polynomials in C is isomorphic to the reals. Thus, C extends the reals in a precise sense.

4.3 Evaluation Homomorphisms in C

For every a1 X + a0 in C, one can define an evaluation homomorphism φ(a1 X + a0)( ) that maps R[X] into C. For every constant polynomial in C, this evaluation homomorphism yields the same answer as the corresponding evaluation homomorphism in Section 3.1.

As an example of this evaluation homomorphism, consider:

φX(p(X)) = φX(X2 + 1)


φX(p(X)) = φX(X2) + φX(X1)


φX(p(X)) = (XX) mod p(X) + 1

With addition and multiplication in R[X]:

X2 = p(X) - 1


φX(p(X)) = -1 + 1 = 0

In other words, the polynomial X in the field extension C is the square root of -1.

5.0 Summary

The above has extended the field of reals to the field of complex numbers. This field extension contains a zero for the equation:

p(X) = X2 + 1 = 0


  • The imaginary numbers are polynomials of degree one and no constant term, with addition and multiplication defined modulo p(X).
  • The real numbers are isomorphic to constant polynomials, with addition and multiplication defined modulo p(X).

That is, the extension field C is the field of complex numbers. The complex numbers are only defined up to isomorphism. But their existence is constructed here, not postulated.

6.0 Other Field Extensions

One need not begin this exposition with polynomials with coefficients from the real numbers. Coefficients can be drawn from other fields.

For example, consider the set {0, 1, 2, ..., p - 1}, with addition and multiplication defined modulo p, and p prime. This set with these operations is a field. Let p(X) be, as above, an irreducible polynomial in the ring of polynomials with coefficients in the set. Suppose p(X) is of degree n. Then the field extension is the Galois Field, GF(pn). The set of elements of GF(2n) - {0}, with multiplication, is a cyclic group. GF(2n) has application in the Advanced Encryption System (AES) and in Reed-Solomon error correction codes. (The latter has something to do with how checkout scanners work in your neighborhood supermarket.)

On the other hand, consider the field of rational numbers with addition and subtraction defined as usual. There are at most a countably infinite number of polynomials with rational coefficients. An irreducible polynomial leads to an extension field for use in constructing real numbers. But this construction leaves out an uncountably infinite number of real numbers, namely the transcendental real numbers. A real number is algebraic if it is the root of some polynomial with rational coefficients. The real numbers, including transcendentals, can be constructed, instead, as Dedekind cuts or as equivalence classes of Cauchy-convergent sequences of rational numbers. (Cauchy often comes across as a villain in accounts of Galois and Abel's short lives.)

Finally, consider polynomials with coefficients drawn from the field of complex numbers. (Since, under the above construction, a complex number is, in some sense, a first-degree polynomial with real coefficients, this may be a somewhat confusing construction to think about.) Suppose one defines polynomial addition and multiplication modulo p(X), where p(X) is a first degree polynomial in the ring C[X]. Then one obtains a field "extension" isomorphic to the field of complex numbers.

To find a bigger field extension, one needs to find an irreducible polynomial of at least degree two in C[X]. But no such polynomial exists. Proof: Abel was something else, wasn't he?