Pattern Analysis Applied to Structural Economic Dynamics with a Choice of Technique: A Numerical Example

I have made a working paper with the above title available on SSRN.

Abstract: This article illustrates the application of pattern analysis to structural economic dynamics with a choice of technique. A numerical example is presented in which technical progress is introduced. Examples of temporal paths through the parameter space illustrate variations of the wage frontier. A single technique is initially uniquely cost-minimizing for all feasible rates of profits. Eventually, the technique for which coefficients of production decrease at the fastest rate is always cost-minimizing. This example illustrates possible variations in the existence of Sraffa effects, which arise during the transition between these positions.

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

I have made available a working paper with the post title.

Abstract: This article presents an example in which perturbations in relative markups and technical progress result in variations in characteristics of the labor market. Around a switch point with a positive real Wicksell effect, a higher wage is associated with firms wanting to employ 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 and variations in markups can bring about and take away circumstances favorable for workers wanting to press claims for higher wages.

On Mariana Mazzucato's The Value of Everything

Mariana Mazzucato's The Value of Everything: Making and Taking in the Global Economy is a popular book. She argues that we should change our ideas of what we consider productive and unproductive jobs and activities. She presents a brief overview of the history of economics focused on the classical and neoclassical theories of value, describes how Gross Domestic Product (GDP) is calculated from the System of National Accounts (SNA), examines how finance came to be mistakenly considered productive of value added in the SNA, and argues that government activities have been mistakenly considered as necessarily unproductive.

Value theory is a good way of organizing a popular history of economics, although it leaves out German historical schools and institutionalism. Mazzucato praises Petty for an early attempt at drawing up national accounts. She says he did not have a theory of value. (From secondary literature, I am under the impression he talked about labor being the father and land the mother of value.) She says Quesnay's Tableau is the "first spreadsheet". She has Adam Smith putting forth a labor theory of value. She notes that he had two theories of productive and unproductive labor: work that provides a physical product versus work for out of capital. (This is straight out of Marx's Theories of Surplus Value.) With Smith, Ricardo, Marx, she doesn't get into details of labor embodied, labor commanded, the transformation problem, and so on. She distinguishes between objective versus subjective value theories. Neoclassical theory is subjective. In some sense, it gets rid of value theory for a theory of price. Producing anything that people are willing to pay for becomes productive.

I wished she had included an illustrative table when she turns to national accounts. She notes that they are drawn up with many conventions, not all of which can be derived from neoclassical theory. She popularizes various objections to how the Gross Domestic Product (GDP) is calculated and used. Along with feminist economists, she notes that household production is not counted. Pollution and cleaning up are more productive of GDP than not polluting in the first place. She notes that certain government activities appear in GDP as productive, even though is not clear from neoclassical theory that (some?) government activity can be productive.

Mazzucato has lots to say about finance. In classical thought, finance would probably considered unproductive, with returns to finance akin to rents. But that is not currently the case. She mentions the idea of privatized Keynesianism, in which increased borrowing by those not well off maintains effective demand in an era in which ideology promotes austerity for government. She looks at how Milton Friedman and Michael Jensen promoted the maximizing of shareholder value, disregarding other stakeholders in corporate management.

Mazzucato argues a need exists to transition from looking at government as a provider of public goods to looking at how government can be productive of value. Health care and information technology are two industries that provide examples of this perspective. In both cases, government funds basic research, with private industries taking profits, including through patents and copyrights. In health care, the idea of Quality-adjusted Life Years (QLYs) is used to justify prices with extreme markups, that cover much more than private Research and Development costs. How many QLYs would be sacrificed if a certain drug was not there? In the view of institutionalism, going back to Veblen, technological innovation combines the activities of many, although the opposite view is embodied in current intellectual property law. From Mazzucato's view, government has been and can be productive.

My initial reaction to the overall thesis of this book was that it is too idealistic. The ruling ideas are the ideas of the ruling classes. Changing those ideas requires changing the material base. I am not sure I fully get her idea that how we draw the boundary between activities and jobs that are productive and unproductive is performative. Mazzucato gives examples of finance, of other activities that would be formerly classified as unproductive grabbing of rents, and of government activities. I came to agree that there is a need for the development and a public discussion of a conceptual frame of who produces what.

I think of this book as a popularization of certain practical and policy ideas drawing on heterodox economics. Mazzucato does not make a point of discussing or drawing boundaries between heterodox and orthodox, non-mainstream and mainstream economics. She draws on, say, Duncan Foley or Joseph Stiglitz when each is useful for her points. I like that this shows that heterodox economists are economists. She could have been more explicit, perhaps with loss of rhetorical efficacy, on when she draws on Marx.

Overall, I recommend this book. I wonder what Dean Baker or Robert Reich would make of this book. Anwar Shaikh and Ahmet Tonak would also have a reaction.

More Pattern Analysis For An Example With Fixed Capital

Figure 1: A Two-Dimensional Pattern Diagram

1.0 Introduction

This post continues this example of the application of pattern analysis to the study of fixed capital. I generalize the technology in the previous post. Technical progress can now proceed at different rates in the production and use of machines. I partition the resulting two-dimensional parameter space based on how the distribution of income, in the system of prices of production, alters the economically efficient length to run the machine. And I find an example in this parameter space that conforms to outdated neoclassical intuition on such matters.

2.0 Technology

Table 1 represents 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

Inputs	Industry
	Machine	Corn
Labor	a0,1 = (1/10) u(t)	a0,2 = (43/40) v(t)	a0,3 = v(t)
Corn	a1,1 = (1/16) u(t)	a1,2 = (1/16) v(t)	a1,3 = (1/4) v(t)
New Machines	0	1	0
Old Machines	0	0	1
Outputs
Corn	0	1	1
New Machines	1	0	0
Old Machines	0	1	0

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

u(t) = e^{1 - σ t}

v(t) = e^{1 - φ t}

Given values of σ t and φ t, technology is specified.

3.0 Choice of Technique

For specified parameters, including σ t and φ t, a system of equations and inequalities is specified such that:

• The rate of profits is determined, given a non-negative wage not exceeding a certain maximum.
• For such a given wage, it is determined whether or not running the machine for one or two years is cost-minimizing.

Which technique - running the machine for one or two years - is cost-minizimizing can vary with mathematical variations in the wage.

Figure 1 partitions the parameter space such that the characteristics of the variations in the choice of technique technique do not vary within each region. Each region is bounded by thick (non-dotted) lines. I do not show a fifth region to the right of the graphed portion of the parameter space. Somewhere before a value of σ t of three, the locus labeled "Pattern over wage axis" curves down to cross the locus labeled "Pattern over axis for rate of profits". In Region 5, the latter locus lies above the former. Table 1 briefly describes the cost-minimizing technique in each region.

Table 1: Coefficients of Production

Region	Description
1	Machine is run for 1 year, for all feasible wages.
2	Machine is run for 2 years, for low wages; 1 year, for high wages.
3	Reswitching. Machine is run for 2 years, for low and high wages; 1 year for intermediate wages.
4	Machine is run for 2 years, for all feasible wages.
5	Machine is run for 1 year, for low wages; 2 years, for high wages.

4.0 Temporal Paths

Suppose that technical progress in producing and using machines is steady. That is, σ and φ have some fixed values. Each of the two dotted lines in Figure 1 illustrate a path in logical time for such a thought experiment. The 45 degree line corresponds to the case in which σ and φ are equal. The choice of technique along such a path was illustrated in the previous post.

I constructed the dashed line with the smaller slope to pass through Region 5. Technical progress is faster, in this case, in producing machines than in using machines. Figure 2 illustrates this case, which complies with outdated neoclassical intuition. Notice that around the switch point in Region 5, a higher wage is associated with a choice, by managers of firms, to run the machine for a second year. The wage acts like a scarcity index for labor, and the lengths at which firms run machines responds appropriately for a measure of capital intensity. But the example proves that there is no logical necessity for economically efficient decisions to work out like this.

Figure 2: A Pattern Diagram

5.0 Conclusion

This post suggests the tools that I have been developing for post-Sraffian price theory apply, without modification, to models of fixed capital.

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

Inputs	Industry
	Machine	Corn
Labor	a0,1 = (1/10) u(t)	a0,2 = (43/40) u(t)	a0,3 = u(t)
Corn	a1,1 = (1/16) u(t)	a1,2 = (1/16) u(t)	a1,3 = (1/4) u(t)
New Machines	0	1	0
Old Machines	0	0	1
Outputs
Corn	0	1	1
New Machines	1	0	0
Old Machines	0	1	0

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

u(t) = e^{1 - σ t}

When σ t is unity, this is Bertram Schefold's example of reswitching, 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.

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 U_j processes for producing the output of that industry.

Each process is defined by:

• a_{0, j}(u), u = 1, 2, ..., U_j, the person-years of labor needed to produce one unit of the jth commodity.
• a_{., j}(u) = [a_{1, j}(u) ..., a_{N, 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/s_N, as givens. Let prices be p = [p₁, ..., p_N]. Also, let e = [e₁, ..., e_N]^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:

k_j(u) = p a_{., j}(u)

c_j(u) = p a_{., j}(u) + w a_{0, j}(u)

π_j(u) = p_j - c_j(u)

r_j(u) = π_j(u)/k_j(u)

2.1 The Firm's LP

The managers of a firm take the wage, w and prices p as given. Let ω = [ω₁, ..., ω_N]^T be the firm's inventory of each commodity at the start of the year. Let q_j(u) be the quantity of the jth commodity that the firm produces with the uth process known for producing that commodity. Let q_{N + 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/s_N)[π_N(1) q_N(1) + ... + π_N(U_N) q_N(U_N)]

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)

...

+ k_N(1) q_N(1) + k_N(2) q_N(2) + ... + k_N(U_N) q_N(U_N) ≤ p ω

For all j, u:

q_j(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 + rs_j) + w a_{0, j}(u) ≥ p_j

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 q_{N + 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

Variable in Basis	Value	When 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 + 1	p ω	For 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.

Reference

• Robert L. Vienneau (2005). On Labour Demand and Equilibria of the Firm. The Manchester School 73 (5): 612-619.

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.