Saturday, May 27, 2017

Some Main Points of the Cambridge Capital Controversy

For the purposes of this very simplified and schematic post, I present the CCC as having two sides.

  • Views and achievements of Cambridge (UK) critics:
    • Joan Robinson's argument for models set in historical time, not logical time.
    • Mathematical results in comparing long-run positions:
    • Rediscovery of the logic of the Classical theory of value and distribution.
    • Arguments about the role a given quantity of capital in disaggregated neoclassical economic theory between 1870 and 1930.
    • Arguments that neoclassical models of intertemporal and temporary equilibrium do not escape the capital critique.
    • A critique of Keynes' marginal efficiency of capital and other aspects of The General Theory.
    • The recognition of precursors in Thorstein Veblen and earlier capital controversies in neoclassical economics.
  • Views of neoclassical defenders:
    • Paul Samuelson and Frank Hahn's, for example, acceptance and recognition of logical difficulties in aggregate production functions.
    • Recognition the equilibrium prices in disaggregate models are not scarcity indices; rejection of the principle of substitution.
    • Edwin Burmeister's championing of David Champerowne's chain index measure of aggregate capital, useful for aggregate theory when, by happenstance, no positive real Wicksell effects exist.
    • Adoption of models of inter temporal and temporary general equilibrium.
    • Assertion that such General Equilibrium models are not meant to be descriptive and, besides, have their own problems of stability, uniqueness, and determinateness, with no need for Cambridge critiques.
    • Samuel Hollander's argument for more continuity between classical and neoclassical economics than Sraffians see.

I think I am still ignoring large aspects of the vast literature on the CCC. This post was inspired by Noah Smith's anti-intellectualism. Barkley Rosser brings up the CCC in his response to Smith. I could list references for each point above. I am not sure I could even find a survey article that covered all those points, maybe not even a single book.

So the CCC presents, to me, a convincing demonstration, through a counter-example to Smith's argument. In the comments to his post, Robert Waldmann brings up old, paleo-Keynesian as an interesting rebuttal to a specific point.

Thursday, May 25, 2017

Some Resources on Neoliberalism

Here are three:

  • Anthony Giddens, in The Third Way: The Renewal of Social Democracy (1999), advocates a renewed social democracy. He contrasts what he is advocating with neoliberalism, which he summarizes as, basically, Margaret Thatcher's approach. Giddens recognizes that more flexible labor markets will not bring full employment and argues that unregulated globalism, including unregulated international financial markets, is a danger that must be addressed. He stresses the importance of environmental issues, on all levels from the personal to international. I wish he had something to say about labor unions, which I thought had an institutionalized role in the Labour Party, before Blair and Brown's "new labour" movement.
  • Charles Peters had a A Neo-Liberal's Manifesto in 1982. (See also 1983 piece in Washington Monthly.) This was directed to the Democratic Party in the USA. It argues that they should reject the approach of the New Deal and the Great Society. Rather, they should put greater reliance on market solutions for progressive ends. I do not think Peters was aware that the term "neoliberalism" was already taken. Contrasting and comparing other uses with Peters' could occupy much time.
  • I have not got very far in reading Michel Foucault. The Birth of Biopolitics: Lectures at the Collège de France, 1978-1979. Foucault focuses on German ordoliberalism and the Chicago school of economics.

Anyways, neoliberalism is something more specific than any centrist political philosophy, between socialist central planning and reactionary ethnic nationalism. George Monbiot has some short, popular accounts. Read Noah Smith if you want confusion, incoherence, and ignorance, including ignorance of the literature.

Friday, May 19, 2017

Reversing Figure And Ground In Life-Like Celluar Automata

Figure 1: Random Patterns in Life and Flip Life
1.0 Introduction

I have occasionally posted about automata. A discussion with a colleague about Stephen Wolfram's A New Kind of Science reminded me that I had started this post some time last year.

This post has nothing to do with economics, albeit it does illustrate emergent behavior. And I have figures that are an eye test. I am subjectively original. But I assume somebody else has done this - that I am not objectively original.

This post is an exercise in combinatorics. There are 131,328 life-like Celluar Automata (CA), up to symmetry.

2.0 Conway's Game of Life

John Conway will probably ever be most famous for the Game of Life (GoL). I wish I understood monstrous moonshine.

The GoL is "played", if you can call it that, on an infinite plane divided into equally sized squares. The plane looks something like a chess board, extended forever. See the left side of Figure 1, above. Every square, at any moment in time, is in one of two states: alive or dead. Time is discrete. The rules of the game specify the state of each square at any moment in time, given the configuration at the previous instant.

The state of a square does not depend solely on its previous state. It also depends on the states of its neighbors. Two types of neighborhoods have been defined for a CA with a grid of square cells. The Von Neumann neighbors of a cell are the four cells above it, below it, and to the left and right. The Moore neighborhood (Figure 2) consists of the Von Neumann neighbors and the four cells diagonally adjacent to a given cell.

Figure 2: Moore Neighborhood of a Dead Cell

The GoL is defined for Moore neighborhoods. State transition rules can be defined in terms of two cases:

  • Dead cells: By default, a dead cell stays dead. If a cell was dead at the previous moment, it becomes (re)born at the next instant if the number of live cells in its Moore neighborhood at the previous moment was x1 or x2 or ... or xn.
  • Alive Cells: By default, a live cell becomes dead. If a cell was alive at the previous moment, it remains alive if the number of live cells in its Moore neighborhood at the previous moment was y1 or y2 or ... or ym.

The state transition rules for the GoL can be specified by the notation Bx/Sy. Let x be the concatenation of the numbers x1, x2, ..., xn. Let y be the concatenation of y1, y2, ..., ym. The GoL is B3/S23. In other words, if exactly three of the neighbors of a dead cell are alive, it becomes alive for the next time step. If exactly two or or three of the neighbors of a live cell are alive, it remains alive at the next time step. Otherwise a dead cell remains dead, and a live cell becomes dead.

The GoL is an example of recreational mathematics. Starting with random patterns, one can predict, roughly, the distributions of certain patterns when the CA settles down, in some sense. On the other hand, the specific patterns that emerge can only be found by iterating through the GoL, step by step. And one can engineer certain patterns.

3.0 Life-Like Celluar Automata

For the purposes of this post, a life-like CA is a CA defined with:

  • A two dimensional grid with square cells and discrete time
  • Two states for each cell
  • State transition rules specified for Moore neighborhoods
  • State transition rules that can be specified by the Bx/Sy notation.

How many life-like CA are there? This is the question that this post attempts to answer.

The Moore neighborhood of cell contains eight cells. Thus, for each of the digits 0, 1, 2, 3, 4, 5, 6, 7, and 8, they can appear in Bx. For each digit, one has two choices. Either it appears in the birth rule or it does not. Thus, there are 29 birth rules.

The same logic applies to survival rules. There are 29 survival rules.

Each birth rule can be combined with any survival rule. So there are:

29 29 = 218

life-like CA. But this number is too large. I am double counting, in some sense.

4.0 Reversing Figure and Ground

Figure 1 shows, side by side, grids from the GoL and from a CA called Flip Life. Flip Life is specified as B0123478/S01234678. Figure 3 shows a window from a computer program. In the window on the left, the rules for the GoL are specified. The window on the right is used to specify Flip Life.

Figure 3: Rules for Life and Flip Life

Flip Life basically renames the states in the GoL. Cells that are called dead in the GoL are said to be alive in Flip Life. And cells that are alive in the GoL are dead in Flip Life. In counting the number of life-like CA, one should not count Flip Life separately from the GoL. In some sense, they are the same CA.

More generally, suppose Bx/Sy specifies a life-like CA, and let Bu/Sv be the life-like CA in which figure and ground are reversed.

  • For each digit xi in x, 8 - xi is not in v, and vice versa.
  • For each digit yj in y, 8 - yj is not in u, and vice versa.

So for any life-like CA, one can find another symmetrical CA in which dead cells become alive and vice versa.

5.0 Self Symmetrical CAs

One cannot just divide 218 by two to find the number of life-like CA, up to symmetry. Some rules define CA that are the same CA, when one reverses figure and ground. As an example, Figure 4 presents a screen snapshot for the CA called Day and Night, specified by the rule B1/S7.

Figure 4: Day and Night: An Example of a Self-Symmetrical Cellular Automaton

Given rules for births, one can figure out what the rules must be for survival for the CA to be self-symmetrical. Thus, there are as many self-symmetrical life-like CAs as there are rules for births.

6.0 Combinatorics

I bring all of the above together in this section. Table 1 shows a tabulation of the number of life-like CAs, up to symmetry.

Table 1: Counting Life-Like Celluar Automata
Number
Birth Rules29
Survival Rules29
Life-Like Rules29 29 = 262,144
Self-Symmetric Rules29
Non-Self-Symmetric Rules29(29 - 1)
Without Symmetric Rules28(29 - 1)
With Self-Symmetric Rules Added Back28(29 + 1) = 131,328

7.0 Conclusion

How many of these 131,328 life-like CA are interesting? Answering this question requires some definition of what makes a CA interesting. It also requires some means of determining if some CA is in the set so defined. Some CAs are clearly not interesting. For example, consider a CA in which all cells eventually die off, leaving an empty grid. Or consider a CA in which, starting with a random grid, the grid remains random for all time, with no defined patterns ever forming. Somewhat more interesting would be a CA in which patterns grow like a crystal, repeating and duplicating. But perhaps an interesting definition of an interesting CA would be one that can simulate a Turing machine and thus may compute any computable function. The GoT happens to be Turing complete.

Acknowledgements: I started with version 1.5 of Edwin Martin's implementation, in Java, of John Conway's Game of Life. I have modified this implementation in several ways.

References

Saturday, May 13, 2017

Innovation and Input-Output Matrices

Figure 1: National Income and Product Accounts
1.0 Introduction

This post contains some speculation about technical progress.

2.0 Non-Random Innovations and Almost Straight Wage Curves

The theory of the production of commodities by means of commodities imposes one restriction on wage-rate of profits curves: They should be downward-sloping. They can be of any convexity. They are high-order polynomials, where the order depends on the number of produced commodities. So no reason exists why they should not change convexity many times in the first quadrant, where the the rate of profits is positive and below the maximum range of profits. The theory of the choice of technique suggests that, if multiple processes are available for producing many commodities, many techniques will contribute to part of the wage-rate of profits frontier.

The empirical research does not show this. When I looked at all countries or regions in the world, I found very little visual deviation from straight lines for most wage curves, for the ruling technique1. The exceptions tended to be undeveloped countries. Han and Schefold, in their empirical search for capital-theoretic paradoxes in OECD countries, also found mostly straight curves. And only a few techniques appeared on the frontier.

I have a qualitative explanation of this discrepancy between expectations from theory and empirical results. The theory I draw on above takes technology as given. It is as if economies are analyzed based on an instantaneous snapshot. But technology evolves as a dynamic process. The flows among industries and final demands have been built up over decades, if not centuries.

In advanced economies, technology does not change randomly. Large corporations have Research and Development departments, universities form extensive networks, and the government sponsors efforts to advance Technology Readiness Levels2. Sponsored research is not directed randomly. Technical feasibility is an issue, albeit that changes over time. Another concern is what is costly at the moment, with cost being defined widely. I suggest a constant effort to lower a reliance on high cost inputs in production process, over time, results in coefficients of production being lowered such that wage curves become more straight3.

The above story suggests that one should develop some mathematical theorems. I am aware of two areas of research in Sraffian economics that seem promising for further inquiry along these lines. First, consider Luigi Pasinetti's structural economic dynamics. I have an analysis of hardware and software costs in computer systems, which might be suggestive. Second, Bertram Schefold has been analyzing the relationship between the shape of wage curves; random matrices; and eigenvalues, including eigenvalues other than the Perron-Frobenius root.

3.0 Innovations Dividing Columns in Input-Output Table, Not Adjoining Completely New Ones

I have been moping during my day job how I cannot keep up with some of my fellow software developers. I return to, say, Java programming after a few years, and there is a whole new set of tools. And yet, much of what I have learned did not even exist when I received either of my college degrees. For example, creating an Android app in Android Studio or IntelliJ involves, minimally, XML, Java, and Virtual Machines for testing. Back in the 1980s, I saw some presentations from Marvin Zelkowitz for what might be described as an Integrated Development Environment (IDE). He had an editor that understood Pascal syntax, suggested statement completions, and, if I recall correctly, could be used to set breakpoints and examine states for executing code. I do not know how this work fed, for example, Eclipse.

Nowadays, you can specialize in developing web apps4. Some of my co-workers are Certified Information Systems Security Professionals (CISSPs). They know a lot of concepts that are sort of orthogonal to programming5. I also know people that work at Security Operations Centers (SOCs)6. And there are many other software specialities.

In short, software should no longer be considered a single industry. Glancing quickly at the web site for the Bureau of Economic Analysis, I note the following industries in the 2007 benchmark input-output tables:

  • Software publishers (511200)
  • Data processing, hosting, and related services (518200)
  • Internet publishing and broadcasting and Web search portals (518200)
  • Custom computer programming services (541511)
  • Computer systems design services (541512)
  • Other computer related services, including facilities management (54151A)

Coders, programmers, and software engineers definitely provide labor inputs in many other industries. Cybersecurity does not even appear above.

What would input-tables looked like, for software, in the 1970s? I speculate you might find industries for the manufacture of computers, telecommunication equipment, and satellites & space vehicles. And data processing would probably be an industry.

I am thinking that new industries come about, in modern economies, more by division and greater articulation of existing industries, not by suddenly creating completely new products. And this can be seen in divisions and movements in industries in National Income and Product Accounts (NIPA). One might explore innovation over the last half-century or so by looking at the evolution of industry taxonomies in the NIPA.7.

4.0 Conclusion

This post suggests some research directions8. At this point, I do not intend to pursue either.

Footnotes
  1. Reviewers, several years ago, had three major objections to this paper. One was that I had to offer some suggestion why wage curves should be so straight. The other two were that I needed to offer a more comprehensive explanation of how to map from the raw data to the input-output tables I used and that I had to account for fixed capital and depreciation.
  2. John Kenneth Galbraith's The New Industrial State is a somewhat dated analysis of these themes.
  3. They also move outward.
  4. The web is not old. Tools like Glassfish, Tomcat, and JBoss, and their commercial competitors are neat.
  5. Such as Confidentiality, Integrity, and Availability; two-factor identification; Role-Based Access Control; taxonomies for vulnerabilities and intrusions; Public Key Infrastructure; symmetric and non-symmetric encryption; the Risk Management Framework (RMF) for Information Assurance (IA) Certification and Accreditation; and on and on.
  6. A SOC differs from a Network Operations Center. Operators of a SOC have to know about host-based and network-based Intrusion Detection, Security Incident and Event Management (SIEM) systems, Situation Awareness, forensics, and so on.
  7. One should be aware that part of the growth on the tracking of industries might be because computer technology has evolved. Von Neumann worried about numerical methods for calculating matrix inverses. Much bigger matrices are practical now.
  8. I do not think my ideas in Section 3 are expressed well.

Saturday, May 06, 2017

Distribution of Maximum Rate of Profits in Simulation

Figure 1: Blowup of Distribution of Maximum Rate of Profits

This post extends the results from my last post. I think of the results presented here as providing information about the implementation of my simulation. I do not claim any implications about actually existing economies. I did not have any definite anticipations about what I would see. I suppose it could be of interest to regenerate these results where coefficients of production are randomly generated from some non-uniform distribution.

I continue to use a capability to generate a random economy, where such an economy is characterized by a single technique. A technique is specified by a row vector of labor coefficients and a corresponding square Leontief input-output matrix. The labor coefficients are randomly generated from a uniform distribution on (0.0, 1.0]. Each coefficient in the Leontief input-output matrix is randomly generated from a uniform distribution on [0.0, 1.0). The random number generator is as provided by the class java.util.Random, in the Java programming language. I am running Java version 1.8.

Each random economy is tested for viability. Non-viable economies are discarded. Table 1 shows how many economies needed to be generated, given the number of produced commodities, to end up with a sample size of 300 viable economies. The maximum rate of profits is calculated for each viable economy. The maximum rate of profits occurs when the wage is zero, and the workers live on air. Thus, labor coefficients do not matter for the calculation of the maximum rate of profits.

Table 1: Number of Simulated Economies
Seed for
Random
Generator
Number of
Commodities
Number of
Economies
368,424,2342610
345,65736,124
4,566,8434826,471
547,5275> 231 - 1

I looked at the distribution of the maximum rate of profits, calculated as a percentage, in several ways. Figure 2 presents four histograms, superimposed on one another. Figure 1 expands the left tails of these histograms. I suppose Figure 2 is somewhat easier to make sense of than Figure 1, when you click on the image. Maybe the statistics in Tables 2 and 3 are clearer. One can see, for example, in random economies in which two commodities are produced, the mean of the maximum rate of profits is 43.9%. The minimum, in these 300 random economies, of the maximum rate of profits is about 0.03% and the maximum is 318%. If I wanted to be more thorough, I would have to review how skewness and kurtosis are calculated by default in the Java class org.apache.commons.math3.stat.descriptive.DescriptiveStatistics. The coefficient of variation is the ratio of the standard deviation to the mean. The nonparametric analogy, reported in the last row in Table 3, is the ratio of the Inter-Quartile Range to the median. Anyways, the distribution of the maximum rate of profits, in random viable economies generated by the simulation, is non-Gaussian and highly skewed, with a tail extending to the right.

Figure 2: Distribution of Maximum Rate of Profits

Table 2: Parametric Statistics
Number of Produced Commodities
TwoThree FourFive
Sample Size300300 300300
Mean43.915.7 8.284.95
Std. Dev.50.219.3 7.535.90
Skewness2.103.89 1.222.63
Kurtosis5.1422.2 0.8829.64
Coef. of Var.0.8750.811 1.100.839

Table 3: Nonparametric Statistics
Number of Produced Commodities
TwoThree FourFive
Minimum0.03270.113 0.01070.00405
1st Quartile9.354.51 2.521.17
Median25.39.72 5.702.99
3rd Quartile57.319.9 11.36.27
Maximum318168 36.244.2
IQR/Median1.901.58 1.541.70

With the simulation, the maximum rate of profits tends to be smaller, the more commodities are produced. I wish I could extend these results to a lot more produced commodities. National Income and Product Accounts (NIPAs), at the grossest level of aggregation have on the order of 100 produced commodities. Even if results with the assumption of an arbitrary probability distribution for coefficients of production could be directly applied empirically, one would like confirmation that trends seen with a very small number of produced commodities continue.

Wednesday, May 03, 2017

I Just Simulated 6 Billion Random Economies

Figure 1: Probability a Random Economy Will Be Viable

I have begun working towards replicating certain simulation results reported by Stefano Zambelli's.

At this point, I have implemented a capability to generate a random economy, where such an economy is characterized by a single technique. A technique is specified by a row vector of labor coefficients and a corresponding square Leontief input-output matrix. The labor coefficients are randomly generated from a uniform distribution on (0.0, 1.0]. Each coefficient in the Leontief input-output matrix is randomly generated from a uniform distribution on [0.0, 1.0). The random number generator is as provided by the class java.util.Random, in the Java programming language. I am running Java version 1.8.

A Monte Carlo simulation, in the results reported here, tests each random economy for viability, where the technique, for each economy, is used to produce a specified number of commodities. A viable economy can reproduce the inputs used up in producing the outputs. If the economy is just viable, nothing is left over to pay the workers and the capitalists. The Hawkins-Simon condition can be used to check for viability.

Table 1 reports the results. The number of Monte Carlo runs, for each row, is 1,000,000,000. The seed is reported so I can replicate my results, if I want. I think I can provide a symmetry argument for why the probability for the first row should be 1/2. I reran the simulation for the last row with 2,000,000,000 runs and the same seed. I still found zero viable economies.

Table 1: Simulation Results
Seed for
Random
Generator
Number of
Commodities
Number of
Viable
Economies
Probability
46,576,8892499,967,47649.9967476%
89,058,538350,198,6905.019869%
7,586,3384372,3390.0372339
784,0545990.0000099%
568,233,269600%

Zambelli suggests randomly specifying a rescaled output, in some sense, for the technology so as to ensure viability. I have a rough conceptual understanding of this step, but I need a better understanding to reduce it to source code. I think I'll go on to further analyses before revisiting the issue of viability. The above results certainly suggest that my analyses will be limited, in the mean time, to economies that produce only two, three, or maybe four commodities.

I think that Zambelli's approach is worthwhile for pursuing the results in which he is interested. One limitation arises with applying a probability distribution to one particular description of technology. In practice, coefficients of production evolve in a non-random manner. Pasinetti's structural dynamics is a good way of exploring technical progress in the tradition of Sraffa.

References
  • Stefano Zambelli (2004). The 40% neoclassical aggregate theory of production. Cambridge Journal of Economics 28(1): pp. 99-120.

Thursday, April 20, 2017

Nonstandard Investments as a Challenge for Multiple Interest Rate Analysis?

1.0 Introduction

This post contains some musing on corporate finance and its relation to the theory of production.

2.0 Investments, the NPV, and the IRR

In finance, an investment project or, more shortly, an investment, is a sequence of dated cash flows. Consider an investment in which these cash flows take place at the end of n successive years. Let Ct; t = 0, 1, ..., n - 1; be the cash flow at the end of the tth year here, counting back from the last year in the investment. That is, Cn - 1 is the cash flow at the end of the first year in the investment, and C0 is the last cash flow.

The Net Present Value (NPV) of an investment is the sum of discounted cash flows in the investment. Let r be the interest rate used in time time discounting, and suppose all cash flows are discounted to the end of the first year in the investment. Then the NPV of the illustrative investment is:

NPV0(r) = Cn - 1 + Cn - 2/(1 + r) + ... + C0/(1 + r)n - 1

If the above expression is multiplied by (1 + r)n - 1, one obtains the NPV of the investment with every cash flow discounted to the last year in the investment:

NPV1(r) = Cn - 1(1 + r)n - 1 + Cn - 2(1 + r)n - 2 + ... + C0

For the next step, I need some sign conventions. Let a positive cash flow designate revenues, and a negative cash flow be a cost. Suppose, for now, that the (temporally) first cash flow is a cost, that is negative. Then (-1/Cn - 1) NPV1(r) is a polynomial in (1 + r), with unity as the coefficient for the highest-order term. All other terms are real.

Such a polynomial has n - 1 roots. These roots can be real numbers, either negative, zero, or positive. They can be complex. Since all coefficients of the polynomial are real, complex roots enter as conjugate pairs. Roots can be repeating. At any rate, the polynomial can be factored, as follows:

NPV1(r) = (-Cn - 1)(r - r0) (r - r1)... (r - rn - 1)

where r0, r1, ..., rn - 1 are the roots of the polynomial. Note that the interest rate appears only in terms in which the difference between the interest rate and one root is taken. And all roots appear on the Right Hand Side. I am going to call an specification of NPV with these properties an Osborne expression for NPV.

Suppose, for now, that at least one root is real and non-negative. The Internal Rate of Return (IRR) is the smallest real, non-negative root. For notational convenience, let r0 be the IRR.

3.0 Standard Investments in Selected Models of Production

A standard investment is one in which all negative cash flows precede all positive cash flows. Is there a theorem that an IRR exists for each standard investment? Perhaps this can be proven by discounting all cash flows to the end of the year in which the last outgoing cash flow occurs. Maybe one needs a clause that the undiscounted sum of the positive cash flows does not fall below the undiscounted sum of the negative cash flows.

At any rate, an Osborne expression for NPV has been calculated for standard investments characterizing two models of production. As I recall it, Osborne (2010) illustrates a more abstract discussion with a point-input, flow-output example. Consider a model in which a machine is first constructed, in a single year, from unassisted labor and land. That machine is then used to produce output over multiple years. Given certain assumptions on the pattern of the efficiency of the machine, this example is of a standard investment, with one initial negative cash flow followed by a finite sequence of positive cash flows.

On the other hand, I have presented an example for a flow-input, point-output model. Techniques of production are represented as finite series of dated labor inputs, with output for sale on the market at a single point in the time. Each technique is characterized by a finite sequence of negative cash flows, followed by a single positive cash flow.

In each of these two examples, the NPV can be represented by an Osborne expression that combines information about all roots of a polynomial. Thus, basing an investment decision on the NPV uses more information than basing it on the IRR, which is a single root of the relevant polynomial.

4.0 Non-standard Investments and Pitfalls of the IRR

In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments can highlight three pitfalls in basing an investment decision on the IRR:

  • Multiple IRRs: The polynomial defining the IRR may have more than one real, non-negative root. What is the rationale for picking the smallest?
  • Inconsistency in recommendations based on IRR and NPV: The smallest real non-negative root may be positive (suggesting a good investment), with a negative NPV (suggesting a bad investment).
  • No IRR: All roots may be complex.

Berk and DeMarzo (2014) present the example in Table 1 as an illustration of the third pitfall. They imagine an author who receives an advance of $750 hundred thousands, sacrifices an income of $500 hundred thousand in each year of writing a book, and, finally, receives a royalty of one million dollars upon publication. The roots of the polynomial defining the NPV are -1.71196 + 0.78662 j, -1.71196 - 0.78662 j, 0.04529 + 0.30308 j, 0.04529 - 0.30308 j. All of these roots are complex; none satisfy the definition of the IRR.

Table 1: A Non-Standard Investment
YearRevenue
0750
1-500
2-500
3-500
41,000

5.0 Issues for Multiple Interest Rate Analysis

Osborne, in his 2014 book, extends his 2010 analysis of the NPV to consider the first and second pitfall above. Nowhere do I know of is an Osborne expression for the NPV derived for an example in which the third pitfall arises.

The idea that the pitfalls above for the use of the IRR might be a problem for multiple interest rate analysis was suggested to me anonymously. On even hours, I do not see this. Why should I care about how many roots there are in an Osborne expression for the NPV, their sign, or even if they are complex?

On the other hand, I wonder about how non-standard investments relate to the theory of production. I know that an example can be constructed, in which the price of a used machine becomes negative before it becomes positive. Can the varying efficiency of the machine result in a non-standard investment? After all, the cash flow, in such an example of joint production, is the sum of the price of the conventional output of the machine and the price of the one-year older machine. Even when the latter is negative, the sum need not be negative. But, perhaps, it can be in some examples.

Not all techniques in models with joint production, of the production of commodities by means of commodities, can be represented as dated labor flows. I guess one can still talk about NPVs. Can one formulate an algorithm, based on NPVs, for the choice of technique? How would certain annoying possibilities, such as cycling be accounted for? Can one always formulate an Osborne expression for the NPV? Do properties of multiple interest rates have implications for, for example, a truncation rule in a model of fixed capital? Perhaps a non-standard investment, for a fixed capital example and one pitfall noted above, always has a cost-minimizing truncation in which the pitfall does not arise. Or perhaps the opposite is true.

Anyway, I think some issues could support further research relating models of production in economics and finance theory. Maybe one obtains, at least, a translation of terms.

Appendix: Technical Terminology

See body of post for definitions.

  • Flow Input, Point Output
  • Investment
  • Investment Project
  • Internal Rate of Return (IRR)
  • Net Present Value (NPV)
  • Non Standard Investment
  • Osborne Expression (for NPV)
  • Point-Input, Flow Output model
  • Standard Investment
References
  • Jonathan Berk and Peter DeMarzo (2014). Corporate Finance, 3rd edition. Boston: Pearson Education
  • Michael Osborne (2010). A resolution to the NPV-IRR debate? Quarterly Review of Economics and Finance, V. 50, Iss. 2: 234-239.
  • Michael Osborne (2014). Multiple Interest Rate Analysis: Theory and Applications. New York: Palgrave Macmillan
  • Robert Vienneau (2016). The choice of technique with multiple and complex interest rates, DRAFT.

Thursday, April 13, 2017

Elsewhere

  • Beatrice Cherrier suggests that Paul Samuelson originated the term "mainstream economics", in his textbook. (h/t I think I found this by Unlearning Economics's twitter feed.)
  • Jo Michell reviews The Econocracy, by Joe Earle, Cahal Moran, and Zach Ward-Perkins.
  • On twitter, Cameron Murray finds, of the 46 who responded, 78% did not "learn about the Cambridge Capital Controversy at point in [their] degree".
  • In the Guardian, Kate Raworth argues that new economics is needed to replace the old economics and its foundation on false laws of physics.

Saturday, April 01, 2017

Bifurcations With Variations In The Rate Of Growth

Figure 1: Perversity and Non-Perversity in the Labor Market Varying with the Rate of Growth
1.0 Introduction

I have been considering how the existence and properties of switch points vary with parameters specifying numerical examples of models of the production of commodities by means of commodities. Here are some examples of such analyses of structural stability. This post adds to this series.

I consider a change in sign of real Wicksell effects to be a bifurcation. In the model in this post, the steady state rate of growth is an exogenous parameter. So a change of sign of real Wicksell effects, associated with a variation in the steady state rate of growth, is a bifurcation.

2.0 Technology

The technology for this example is as usual. Suppose two commodities, iron and corn, are produced in the example economy. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes.

Table 1: The Technology for a Two-Industry Model
InputIron
Industry
Corn
Industry
Labor1305/4941
Iron1/10229/4942
Corn1/403/19762/5

For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the sole corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process.

Each technique is represented by a two-element row vector of labor coefficients and a 2x2 Leontief input-output matrix. For example, the vector of labor coefficients for the Beta technique, a0, β, is:

a0, β = [305/494, 1]

The components of the Leontief matrix for the Beta technique, Aβ, are:

a1,1, β = 229/494
a1,2, β = 2
a2,1, β = 3/1976
a2,2, β = 2/5

The labor coefficients for the Alpha technique, a0, α, differ in the first element from those for the Beta technique. The Leontief matrix for the Alpha technique, Aα, differs from the Leontief matrix for the Beta technique in the first column.

(The mathematics in this post is set out in terms of linear algebra. I needed to remind myself of how to work out quantity flows with a positive rate of growth. I solved the example with Octave, the open-source equivalent of Matlab for the example. I haven't checked the graphs by also working them out by hand. You can click on the figures to see them somewhat larger.)

3.0 Prices and the Choice of Technique

Consider steady-state prices that repeat, year after year, as long as firms adopt the same technique. Let a0 and A be the labor coefficients and the Leontief matrix for that technique. Suppose labor is advanced and wages are paid out of the surplus at the end of the year. Then prices satisfy the following system of equations:

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

where p is a two-element row vector of prices, w is the wage, and r is the rate of profits. Let e be a column vector specifying the commodities constituting the numeraire. Then:

p e = 1

For the numerical example, a bushel corn is the numeraire, and e is the second column of the identity matrix. I think of the numeraire as in the proportions in which households consume commodities.

The system of equations for prices of production, including the equation for the numeraire, has one degree of freedom. Formally, one can solve for prices and the wage as functions of an externally given rate of profits. The first equation above can be rewritten as:

a0 w = p [I - (1 + r) A]

Multiply through, on the right, by the inverse of the matrix in square brackets:

a0 [I - (1 + r) A]-1 w = p

Multiply through, again on the right, by e:

a0 [I - (1 + r) A]-1 e w = p e = 1

Both sides of the above equation are scalars. The wage is:

w = 1/{a0 [I - (1 + r) A]-1 e}

The above equation is called the wage-rate of profits curve or, more shortly, the wage curve. Prices of production are:

p = a0 [I - (1 + r) A]-1/{a0 [I - (1 + r) A]-1 e}

The above two equations solve the price system, in some sense.

Figure 2 plots the wage curves for the example. The downward-sloping blue and red curves show that, for each technique, a lower steady-state real wage is associated with a higher rate of profits. The two curves intersect at the two switch points, at rates of profits of 20% and 80%. For rates of profits between the switch points, the Alpha technique is cost-minimizing and its wage curve constitutes the outer envelope of the wage curves in this region. For feasible rates of profits outside that region, the Beta technique is cost-minimizing. (I talk more about this figure at least twice below.)

Figure 2: Wage Curves also Characterize Tradeoff Between Consumption per Worker and Steady State Rate of Growth

4.0 Quantities

Suppose the steady-state rate of growth for this economy is 100 g percent. A system of equations, dual to the price equations, arises for quantity flows. Let q denote the column vector of gross quantities, per labor-year employed, produced in a given year. Let y be the column vector of net quantities, per labor-year. Net quantities constitute the surplus once the (circulating) capital goods advanced at the start of the year, for a given technique, are replaced:

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

Since quantities are defined per person-year, employment with these quantities is unity:

a0 q = 1

By hypothesis, net quantities are the sum of consumption and capital goods to accumulate at the steady state rate of profits:

y = c e + g A q

Substituting into the first equation in this section and re-arranging terms yields:

c e = [I - (1 + g) A] q

Or:

c [I - (1 + g) A]-1 e = q

Multiply through on the left by the row vector of labor coefficients:

c a0 [I - (1 + g) A]-1 e = a0 q = 1

Consumption per person-year is:

c = 1/{a0 [I - (1 + g) A]-1 e}

Gross quantities are:

q = [I - (1 + g) A]-1 e/{a0 [I - (1 + g) A]-1 e}

Interestingly enough, the relationship between consumption per worker and the rate of growth is identical to the relationship between the wage and the rate of profits. Thus, Figure 1 is also a graph of the trade-off, for the two technique, between steady-state consumption per worker and the rate of growth. One can think of the abscissa as relabeled the rate of growth and the ordinate as relabeled consumption per person-year. In the graph, the grey point illustrates consumption per worker at a rate of growth of 10% for the Beta technique.

The ordinate on this graph is consumption throughout the economy. If the rate of profits exceeds the rate of growth, both those obtaining income from wages and those obtaining income from profits will be consuming. When the rates of growth and profits are equal, all profits are accumulated.

5.0 Some Accounting Identities

The value of capital per worker is:

k = p A q

The value of net income per worker is:

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

(I hope the distinction between the scalar y and the vector y is clear in this notation.)

The value of net income per worker can be expressed in terms of the sum of income categories. Rewrite the first equation in Section 3:

p (I - A) = a0 w + p A r

Multiply both sides by the vector of gross outputs:

p (I - A) q = a0 q w + p A q r

Or:

y = w + k r

In this model, net income per worker is the sum of wages and profits per worker.

Net income per worker can also be decomposed by how it is spent. For the third equation in Section 4, multiply both sides by the price vector:

p y = c p e + g p A q

Or:

y = c + g k

Net income per worker is the sum of consumption per worker and investment per worker.

Equating the two expressions for net income per worker allows one to derive an interesting graphical feature of Figure 1. This equation is:

w + r k = c + g k

Or:

(r - g) k = c - w

Or solving for the value of capital per worker:

k = (c - w)/(r - g)

Capital per worker, for a given technique, is the additive inverse of the slope of two points on the wage curve for that technique. Figure 1 illustrates for the Beta technique, with a rate of growth of 10% and a rate of profits of 80%, as at the upper switch point.

6.0 Real Wicksell Effects

This section and the next presents an analysis confined to prices at the switch point for a rate of profits of 80%.

For a rate of profits infinitesimally lower than 80%, the Alpha technique is cost-minimizing. And for a rate of profits infinitesimally higher, the Beta technique is cost minimizing. I have explained above how to calculate the value of capital per worker, for the two techniques, at any given rate of growth.

Abstract from any change in prices of production associated with a change in the rate of profits. The difference between capital per head for the Beta technique and capital per head for the Alpha technique, both calculated at the prices for the switch point, is the change in "real" capital around the switch point associated with an increase in the rate of profits. Figure 3 graphs this real Wicksell effect as a function of the rate of steady state growth.

Figure 3: Variation in Real Wicksell Effect with Steady State Rate of Growth

Two regions are apparent in Figure 3. The intersection, at the left, of the downward-sloping graph with the axis for the change in the value of capital per worker shows that the real Wicksell effect is positive, for this switch point, in a stationary state. Around the given switch point, a higher rate of profits is associated, in a stationary state, with firms wanting to adopt a more capital-intensive technique. If a greater scarcity of capital caused the rate of profits to rise, so as to ration the supply of capital, such a logical possibility could not be demonstrated.

The real Wicksell effect, for the switch point at the higher rate of profits, is zero when the rate of growth is equal to the rate of profits at the other switch points. The value of capital per person-year is the same for the two techniques, in this case. Consider a line, in Figure 1, connecting the two switch points. It also connects the points on the wage curve for the Alpha technique for a rate of profits of 80% and a rate of growth of 20%. And the same goes for the wage curve for the Beta technique.

7.0 Real Wicksell Effects in the Labor Market

A variation in real Wicksell effects with the steady state rate of growth is also manifested in the labor market. I have echoed above some mathematics which shows that the value of national income is the dot product of a vector of prices with the vector of net quantity flows. The price vector depends, given the technique, on the rate of profits at which prices of production are found. The quantity vector depends on the steady state rate of growth. The reciprocal, (1/y), is the amount of labor firms want to hire, per numeraire unit of national income, for a given technique. The difference at a switch point between these reciprocals, for the two techniques, is another way of looking at real Wicksell effects.

Around the switch point at a rate of profits of 80%, a lower wage is associated with firms adopting the Beta technique. And a higher wage is associated with firms adopting the Alpha technique. The difference of the above reciprocals, between the Alpha and Beta techniques, is the increase in labor, per numeraire-unit net output, associated with an infinitesimal increase in wages, at the prices for the switch point. Figure 1 shows this difference, as a function of the steady state rate of growth, at the switch point with the higher rate of profits in the example.

Figure 1 qualitatively resembles Figure 3. For a stationary state, a higher wage is associated with firms wanting to employ more labor, per numeraire unit of net output. This effect is reversed for a high enough steady state rate of growth. The bifurcation, here too, occurs at the rate of growth for the switch point at 20%.

8.0 Conclusion

This post has illustrated a comparison among steady state growth paths at rates of profits associated with a switch point. And this switch point is "perverse" from the perspective of outdated neoclassical theory, at least at a low rate of growth. But the perversity of this switch point varies with the rate of growth. In the example, when the rate of growth is between the rate of profits at the two switch points, the second switch point becomes non-perverse.

And it can go the other way. Real Wicksell effects do not even need to be monotonic. I need to find an example with at least three commodities, two techniques, and three switch points. In such an example, the switch point with the largest rate of profits will have a negative real Wicksell effect for a stationary state, a positive real Wicksell effect for steady state rates of growth between the first two switch points, and a negative real Wicksell effect for higher rates of growth, between the second and third switch points.

(I want to look up Gandolfo (2008) in the light of past posts. Can I tell this tale in terms of increasing returns, instead of exogenous technical change?)

References
  • Giancarlo Gandolfo (2008). Comment on "C.E.S. production functions in the light of the Cambridge critique". Journal of Macroeconomics, V. 30, No. 2 (June): pp. 798-800.
  • Nell (1970). A note on Cambridge controversies in capital theory. Journal of Economic Literature V. 8, No. 1 (March): 41-44.

Tuesday, March 28, 2017

Niall Kishtainy's Pluralist, Popular History Of Economics

This post calls attention to A Little History of Economics, by Niall Kishtainy. Since I only skimmed this in a local bookstore, this post is not a proper review.

Kishtainy's book, in successive chapters, focuses on the lives of specific economists. The concluding chapter asks why one would want to be an economist, a question that is answered, I gather, by the preceding chapters. The book is in the same genre as Robert Heilbroner's The Worldly Philosophers. Since many more economists are covered in a short span, individual chapters are shorter. As I recall, economists (not in this order) covered include:

  • Augustine and Thomas Aquinas
  • Mercantilists
  • Francois Quesnay, Mirabeau, and other Physiocrats
  • Adam Smith
  • David Ricardo
  • Charles Fourier, Robert Owen, and other utopian socialists
  • Thomas Malthus
  • Friedrich List
  • Karl Marx
  • William Stanley and Alfred Marshall
  • German historical school, Austrian school, and the methodenstreit
  • Thorstein Veblen
  • Vladimir Lenin and John Hobson
  • Ludwig von Mises
  • Joan Robinson and Edward Chamberlin
  • John Maynard Keynes
  • Paul Samuelson, J. R. Hicks
  • Friedrich Hayek
  • Arthur Lewis, Paul Rosenstein-Rodan, and Raul Prebisch
  • Robert Solow, Trevor Swan, and Paul Romer
  • Joseph Schumpeter
  • Gary Becker
  • John Von Neumann and John Nash
  • Ken Arrow and Gerard Debreu
  • Fidel Castro, Che Guevara, and Andre Frank
  • James Buchanan
  • Milton Friedman
  • George Akerlof and Joseph Stiglitz
  • Hyman Minsky
  • John Muth, Eugene Fama, and Robert Lucas
  • Ed Prescott and Finn Kyland
  • Behavioral economics
  • Thomas Piketty

Doubtless, if I read the book in detail, I would have objections to specific details. The attempted coverage, however, seems quite impressive. This book extends from before to after most histories. And it covers a wider range of economists than most.

Wednesday, March 22, 2017

Krugman Confused On Trade, Capital Theory

Over on EconSpeak, Bruce Wilder provides some comments on a post. He notes that economists wanting to criticize glib free-market ideology in the public discourse often seem unwilling to discard neoclassical economic theory.

Paul Krugman illustrates how theoretically conservative and neoclassical a liberal economist can be. (I use "liberal" in the sense of contemporary politics in the USA.) I refer to Krugman's post from earlier this week, in which he adapts an analysis from the theory of international trade to consider technological innovation (e.g., robots). Krugman presents a diagram, in which endowments of capital and labor are measured along the two axes. Krugman does not seem to be aware that one cannot, in general, coherently talk about a quantity of capital, prior to and independently of prices. He goes on to talk about "capital-intensive" and "labor-intensive" techniques of production.

I point to my draft paper, "On the loss from trade", to illustrate my point that one cannot meaningfully talk about the endowment of capital.

(I did submit this paper to a journal. A reviewer said it was not original enough. I emphasized that I was illustrating my points in a flow-input, point output model, with a one-way flow from factors of production to consumption goods, not a model of production of commodities by means of commodities. Steedman & Metcalfe (1979) also has a one-way model, albeit with a point-input, point-output model. So the reviewer's comments were fair. Embarrassingly, I cite other papers from that book. Apparently, I had forgotten that paper, if I ever read it. I suppose that, given the chance, I could have distinguished some of my points from those made in Steedman & Metcalfe (1979). Also, I close my model with utility-maximization; if I recall correctly, Steedman leaves such an exercise to the reader in papers in that book.)

Reference
  • Ian Steedman and J. S. Metcalfe (1979). 'On foreign trade'. In Fundamental Issues in Trade Theory (ed. by Ian Steedman).

Saturday, March 18, 2017

Reswitching Only Under Oligopoly

Figure 1: Rates of Profits for Switch Points for Differential Rates of Profits
1.0 Introduction

Suppose one knows the technology available to firms at a given point in time. That is, one knows the techniques among which managers of firms choose. And suppose one finds that reswitching cannot occur under this technology, given prices of production in which the same rate of profits prevails among all industries. But, perhaps, barriers to entry persist. If one analyzes the choice of technique for the given technology, under the assumption that prices of production reflect stable (non-unit) ratios of profits, differing among industries, reswitching may arise for the technology. The numerical example in this post demonstrates this logical possibility.

The numerical example follows a model of oligopoly I have previously outlined. In some sense, the example is symmetrical to the example in this draft paper. That example is of a reswitching example under pure competition, which becomes an example without reswitching and capital reversing, if the ratio of the rates of profits among industries differs enough. The example in this post, on the other hand, has no reswitching or capital reversing under pure competition. But if the ratios of the rates of profits becomes extreme enough, it becomes a reswitching example.

2.0 Technology

The technology for this example resembles many I have explained in past posts. Suppose two commodities, iron and corn, are produced in the example economy. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes.

Table 1: The Technology for a Two-Industry Model
InputIron
Industry
Corn
Industry
Labor1305/4941
Iron1/10229/49411/10
Corn1/403/19762/5

For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the lone corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process.

3.0 Price Equations

The choice of technique is analyzed on the basis of cost-minimization, with prices of production. Suppose the Alpha technique is cost minimizing. Then the following system of equalities and inequalities hold:

[(1/10)p + (1/40)](1 + rs1) + w = p
[(229/494)p + (3/1976)](1 + rs1) + (305/494)wp
[(11/10)p + (2/5)](1 + rs2) + w = 1

where p is the price of a unit of iron, and w is the wage.

The parameters s1 and s2 are given constants, such that rs1 is the rate of profits in iron production and rs2 is the rate of profits in corn production. The quotient s1/s2 is the ratio, in this model, of the rate of profits in iron production to the rate of profits in corn production. Consider the special case:

s1 = s2 = 1

This is the case of free competition, with investors having no preference among industries. In this case, r is the rate of profits. I call r the scale factor for the rate of profits in the general case where s1 and s2 are unequal.

The above system of equations and inequalities embody the assumption that a unit corn is the numeraire. They also show labor as being advanced and wages as paid out of the surplus at the end of the period of production. If the second inequality is an equality, both the Alpha and the Beta techniques are cost-minimizing; this is a switch point. The Alpha technique is the unique cost-minimizing technique if it is a strict inequality. To create a system expressing that the Beta technique is cost-minimizing, the equality and inequality for iron production are interchanged.

4.0 Choice of Technique

The above system can be solved, given s1, s2, and the scale factor for the rate of profits. I record the solution for a couple of special cases, for completeness. Graphs of wage curves and a bifurcation diagram illustrate that stable (non-unitary) ratios of rates of profits can change the dynamics of markets.

4.1 Free Competition

Consider the special case of free competition. The wage curve for the Alpha technique is:

wα = (41 - 38r + r2)/[80(2 + r)]

The price of iron, when the Alpha technique is cost-minimizing, is:

pα = (5 - 3r)/[8(2 + r)]

The wage curve for the Beta technique is:

wβ = (6,327 - 9,802r + 3,631r2)/[20(1,201 + 213r)]

When the Beta technique is cost-minimizing, the price of iron is:

pβ = [5(147 - 97r)]/[2(1,201 + 213r)]

Figure 2 graphs the wage curves for the two techniques, under free competition and a uniform rate of profits among industries. The wage curves intersect at a single switch point, at a rate of profits of, approximately, 8.4%:

rswitch = (1/1,301)[799 - 24 (8261/2)]

The wage curve for the Beta technique is on the outer envelope, of the wage curves, for rates of profits below the switch point. Thus, the Beta technique is cost-minimizing for low rates of profits. The Alpha technique is cost minimizing for feasible rates of profits above the switch point. Around the switch point, a higher rate of profits is associated with the adoption of a less capital-intensive technique. Under free competition, this is not a case of capital-reversing.

Figure 2: Wage Curves for Free Competition

4.2 A Case of Oligopoly

Now, I want to consider a case of oligopoly, in which firms in different industries are able to ensure long-lasting barriers to entry. These barriers manifest themselves with the following parameter values:

s1 = 4/5
s2 = 5/4

In this case, the wage curve for the Alpha technique is:

wα = (4,100 - 4,435r + 100r2)/[40(400 + 259r)]

The price of iron, when the Alpha technique is cost-minimizing, is:

pα = (125 - 96r)/(400 + 259r)

The wage curve for the Beta technique is:

wβ = 8(126,540 - 195,289r + 72,620r2)/[160(24,020 + 9,447r)]

The price of iron, when the Beta technique is cost-minimizing, is:

pβ = 2(3,675 - 3,038r)/(24,020 + 9,447r)

Figure 3 graphs the wage curves for the Alpha and Beta techniques, for the parameter values for this model of oligopoly. This is now an example of reswitching. The Beta technique is cost minimizing at low and high rates of profits. The Alpha technique is cost minimizing at intermediate rates. The switch points are at, approximately, a value of the scale factor for rates of profits of 12.07% and 77.66%, respectively.

Figure 3: Wage Curves for a Case of Oligopoly

4.3 A Range of Ratios of Profit Rates

The above example of oligopoly can be generalized. I restrict myself to the case where the parameters expressing the ratio of rates of profits between industries satisfy:

s2 = 1/s1

One can then consider how the shapes and locations of wage curves and switch points vary with continuous variation in s1/s2. Figure 1, at the top of this post, graphs the wage at switch points for a range of ratios of rates of profits. Since the Beta technique is cost-minimizing, in the graph, at all high feasible wages and low scale factor for the rates of profits, I only graph the maximum wage for the Beta technique. I do not graph the maximum wage for the Alpha technique.

As the ratio of the rate of profits in the iron industry to rate in the corn industry increases towards unity, the model changes from a region in which the Beta technique is dominant to a reswitching example to an example with only a single switch point. As expected, only one switch point exists when the rate of profits is uniform between industries.

5.0 Conclusion

So I have created and worked through an example where:

  • No reswitching or capital-reversing exists under pure competition, with all industries earning the same rate of profits.
  • Reswitching and capital-reversing can arise for oligopoly, with persistent differential rates of profits across industries.

No qualitative difference necessarily exists, in the long period theory of prices, between free competition and imperfections of competition. Doubtless, all sorts of complications of strategic behavior, asymmetric information, and so on are empirically important. But it seems confused to blame the failure of markets to clear or economic instability on such imperfections.

Wednesday, March 15, 2017

Bifurcations in a Reswitching Example

Figure 1: Rates of Profits for Switch Points in One Dimension in Parameter Space
1.0 Introduction

This post presents an example of structural variation in the qualitative behavior of a reswitching example, at different values for selected parameters. I know of few applications of bifurcation analysis to the Cambridge Capital Controversy. Most prominently, I think of Rosser (1983). I suppose I could also point to some of my draft papers. Although not presented this way, one could read Laibman and Nell (1977) as a bifurcation analysis, where the steady state rate of growth is the parameter being varied.

I guess one could read this post as a response to the empirical results in Han and Schefold (2006). Schefold has been developing a theoretical explanation, based on random matrices, of why capital-theoretic paradoxes might be empirically rare. I seem to have stumbled on an explanation of why such paradoxes might arise in practice, and yet might not be observable without more data. To fully address recent results from Schefold, on reswitching and random matrices, one should analyze the spectra of Leontief input-output matrices, which I do not do here.

2.0 Technology

Suppose two commodities, iron and corn, are produced in the economy in the numerical example. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes.

Table 1: The Technology for a Two-Industry Model
InputIron
Industry
Corn
Industry
Labor1305/494a0,2
Iron1/10229/494a1,2
Corn1/403/19762/5

Assume a0,2 is non-negative, and that a1,2 is strictly positive. For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the lone corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process.

3.0 Choice of Technique

Managers of firms choose the technique to adopt based on cost-minimization. I take a bushel of corn as the numeraire. Assume that labor is advanced, and that wages are paid out of the surplus at the end of the year. For this post, I do not bother setting out equations for prices of production; I have done that many times in the past.

3.1 Reswitching for one Set of Parameter Values

Figure 2 illustrates that this is a reswitching example. This figure is drawn for the following values of the labor coefficient in the process for producing corn:

a0,2 = 1

The coefficient of production for iron in corn-production, in drawing Figure 2, is set to the following value:

a1,2 = 2

The economy exhibits capital-reversing around the switch point at an 80% rate of profits.

Figure 2: Wage-Rate of Profits Curves

3.2 Bifurcations with Variations in a Labor Coefficient

Wage-rate of profits curves are drawn for given coefficients of production. And they will be moved elsewhere for different levels of coefficients of production. Consequently, the existence and location of switch points differ, depending on the values for coefficients of production.

Accordingly, suppose all coefficients of production, except a0,2, are as in the above reswitching example. Consider values of the labor coefficient for corn-production ranging from zero to three. The labor coefficient is plotted along the abscissa in Figure 3. The points on the blue locus in the figure show the rate of profits for the switch points, as a correspondence for the labor coefficient. The maximum rates of profits for the Alpha and Beta techniques are also graphed.

Figure 3: Rates of Profits for Switch Points as One Labor Coefficient Decreases

Figure 3 shows a structural change in the example. Up to a value of a0,2 of approximately 2.74, this is a reswitching example. For parameter values strictly greater than that, no switch points exist. The maximum rates of profits for the two techniques are constant in Figure 3. The maximum rates of profits are found for a wage of zero, and they do not vary with the labor coefficient. In some sense, only the maximum rate of profits for the Beta technique is relevant in the figure.

3.3 Bifurcations with Variations in a Coefficient of Production for Iron

Figure 1, at the top of this post, also shows structural changes. The coefficient of production for iron in corn-production varies in the figure. a1,2 ranges from one to three. The other coefficients of production are as in the reswitching example in Section 3.1 above. And the blue locus shows the rate of profits at switch points.

The example can seen to have structural variations here, also, with three distinct regions for a1,2, with the same qualitative behavior in each region. For a low enough value of the coefficient of production under consideration, only one switch point exists. The model remains a reswitching example for an intermediate range of this parameter. And for values of this coefficient of production strictly greater than approximately 2.53, the Beta technique is cost-minimizing for all feasible wages and rates of profits.

The maximum rates of profits, for the Alpha and Beta techniques, are also graphed in Figure 1.

4.0 A Story of Technological Process

Using the above example, one can tell a story of technological progress. Suppose at the start of the story, corn production requires a relatively large input of direct labor and iron, per (gross) unit corn produced. Prices of production associated with this technology are such that only one technique is cost-minimizing. For all feasible wages and rates of profits, firms will want to adopt the Beta technique.

Suppose iron production is relatively stagnant, as compared to corn-production. Innovation in the corn industry reduces the labor and iron coefficients defining the single dominant corn-producing process. After some time, either or both coefficients will be reduced enough that the technology for this economy will have become a reswitching example. And around the switch point at the lower wage (and higher rate of profits), a higher wage is associated with the cost-minimizing technique requiring more labor to be hired, in the overall economy, per given bushel of corn produced (net).

But technological innovation continues to proceed apace. At a even lower coefficient of production for the iron input in the corn industry, the structural behavior of the economy changes again. Now a single switch point exists. And the results of the choice of technique around that switch point conforms to outdated neoclassical intuition.

5.0 Conclusion

This example has two properties that I think worth emphasizing.

The choice of technique in the example corresponds to a choice of a production process in the iron industry. As I have told the story, the technology is fixed in iron production. Innovation occurs in corn production. Thus, innovation in one industry can change the dynamics in another industry.

Second, suppose the technology is observed at a single point of time. Suppose the economy is more or less stationary, and that observation is taken at either the start or the end of the above story. Then neither reswitching nor capital reversing will be observed. Yet such phenomena might arise in the future or have arisen in the past.

References

Saturday, March 11, 2017

Here and Elsewhere

  • A commentator informs me that the True Levelers revived some ideas put forth in the Peasants Revolt.
  • Another commentator points me to Naoki Yoshihara's review of Opocher and Steedman's recent book. Yoshihara has a point, but I think the practice of treating inputs and physically identical outputs as different dated commodities is less applicable in partial models, as opposed to full General Equilibrium. Accountants need guidelines that resist easy manipulation in calculating profits and losses.
  • Antonella Palumbo has a post, "Can 'It' Happen Again? Defining the Battlefield for a Theoretical Revolution in Economics", at the Institute for New Economic Thinking. Palumbo argues that a revival of classical economics, without Say's law, can provide an alternative to neoclassical economics. And Keynes' macroeconomics can be usefully be combined with this revival.

Wednesday, March 08, 2017

A Fluke Switch Point

Figure 1: The Choice of Technique in a Model with Four Techniques
1.0 Introduction

I think I may have an original criticism of (a good part of) neoclassical economics. For purposes of this post, I here define the use of continuously differential production functions as an essential element in the neoclassical theory of production. (This is a more restrictive characterization than I usually employ.) Consider this two-sector example, in which coefficients of production in both sectors varies continuously along the wage-rate of profits frontier. It would follow from this post, I guess, that neoclassical theory is a limit, in some sense, of an analysis in which all switch points are flukes.

I have presented many other, often unoriginal, examples with a continuum of techniques:

I have an example with an uncountably infinite number of techniques along the wage-rate of frontier, but discontinuities for (all?) marginal relationships.

2.0 Technology

I want to compare and contrast two models. The technology in the second model is an example in Salvadori and Steedman (1988).

Households consume a single commodity, called "corn", in both models. In both models, two processes are known for producing corn. And these processes require inputs of labor and a capital good to produce corn. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. Both models are models of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes.

2.1 First Model

The technology for the first model is shown in Table 1. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. The two processes for producing corn require inputs of distinct capital goods. One corn-producing process requires inputs of labor and iron, and the other requires inputs of labor and tin.

Table 1: The Technology for a Three-Industry Model
InputIron
Industry
Tin
Industry
Corn
Industry
(a)(b)(c)(d)
Labor1212
Iron002/30
Tin0001/2
Corn2/31/200

Two techniques, as shown in Table 2, are available for producing a net output of corn. A choice of a process for producing corn also entails a choice of which capital good is produced. When the processes are each operated on a appropriate scale, the gross output of the process producing the specific capital good exactly replaces the quantity of the capital good used up as an input, summed over both industries operated in the technique.

Table 2: Techniques in a Three-Commodity Model
TechniqueProcesses
Alphaa, c
Betab, d

2.2 Second Model

The technology for the second model is shown in Table 3. Two processes are known for producing corn. Both corn-producing processes require inputs of labor and iron, but in different proportions.

Table 3: The Technology for a Two-Industry Model
InputIron
Industry
Corn
Industry
(a)(b)(c)(d)
Labor1212
Iron002/31/2
Corn2/31/200

Table 4 lists the techniques available in the second model. The first two techniques superficially resemble the two techniques available in the first model. But, in this model, the first process for producing a capital good can be combined, in a technique, with the second corn-producing producing process. This combination of processes is called the Gamma technique. Likewise, the Delta technique combines the second process for producing a capital good with the first corn-producing processes. Nothing like the Gamma and Delta techniques are available in the first model.

Table 4: Techniques in a Two-Commodity Model
TechniqueProcesses
Alphaa, c
Betab, d
Gammaa, d
Deltab, c

3.0 Prices of Production

Suppose the Alpha technique is cost-minimizing. Prices of production, which permit smooth reproduction of the economy, must satisfy the following system of two equations in three unknowns:

(2/3)(1 + r) + wα = pα
(2/3) pα(1 + r) + wα = 1

These equations are based on the assumption that labor is advanced, and wages are paid out of the surplus at the end of the year. The same rate of profits are generated in both industries. A unit quantity of corn is taken as the numeraire.

One of the variables in these equations can be taken as exogenous. The first row in Table 5 specifies the wage and the price of the appropriate capital good, as a function of the rate of profits. The equation in the second column is called the wage-rate of profits curve, also known as the wage curve, for the Alpha technique. Table 5 also shows solutions of the systems of equations for the prices of production for the other three techniques in the second model, above. I have deliberately chosen a notation such that the first two rows can be read as applying to either one of the two models.

Table 5: Wages and Prices by Technique
TechniqueWage CurvePrices
Alphawα = (1 - 2 r)/3pα = 1
Betawβ = (1 - r)/4pβ = 1
Gammawγ = 2(2 - 2r - r2)
/[3(5 + r)]
pγ = 2(7 + 4r)
/[3(5 + r)]
Deltawδ = (2 - 2r - r2)/(7 + 4r)pδ = 3(5 + r)/[2(7 + 4r)]

Figure 1, at the top of this post, graphs all four wage-curves. The wage curves for the Alpha and Beta techniques are straight lines. In the jargon, the processes comprising these techniques exhibit the same organic composition of capital. The wage curves for the Gamma and Delta techniques are not straight lines. All four wage-curves intersect at a single point, (r, w) = (20%, 1/5). (The wage curves for the Gamma and Delta techniques have the same intersection with the axis for the rate of profits.)

3.0 Choice of Technique

The cost-minimizing techniques form the outer envelope of the wage curves. For a given wage, the cost minimizing technique is the technique with the highest wage curve in Figure 1. A switch point is a point on the outer envelope at which more than one technique is cost-minimizing. All four wage curves intersect, in the figure, at the single switch point.

The Beta technique is cost-minimizing for wages to the left of the single switch point. The Alpha technique is cost-minimizing for all feasible wages greater than the wage at the switch point. Managers of firms replace both processes in the Alpha technique at the switch point with both processes in the Beta technique.

This is no problem for the first model above. The adoption of a new process for producing corn requires, if the economy is capable of self-replacement before and after the switch, that the process for producing iron or tin be replaced by the process for producing the other.

But consider the other model. For all processes in the Alpha technique to be replaced at a switch point, the wage curves for all techniques composed of all combinations of processes in the Alpha and Beta techniques. In other words, in the second model, wage curves for all four techniques must intersect at the switch point. The example in the second model is a fluke.

I have previously explained what makes a result a fluke, in the context of the analysis of the choice of technique. Qualitative properties, for generic results, continue to persist for some small variation in model parameters.

Consider a model with a discrete number of switch points. Consider the cost-minimizing techniques on both sides of a switch point. And suppose that same commodities are produced in both techniques, albeit in different proportions. Generically, only one process is replaced at such a switch point. All processes, except for that one, are common in both techniques.

5.0 A Generalization to An Uncountably Infinite Number of Processes in Each Industry

Consider a model with more than one industry, but a finite number. Suppose each industry has available an uncountably infinite number of processes. And, in each industry, the processes available for that industry can be described by a continuously differentiable production function. Here I present a two-commodity example with Cobb-Douglas production functions.

There are no switch points in such a model. The cost-minimizing technique varies continuously along the outer-envelope of wage curves. In fact, the processes in each industry, in the cost-minimizing technique varies continuously. Since there are no switch points at all, there is not a single switch point in which more than one process varies, as a fluke, with the cost-minimizing technique.

Nevertheless, cannot one see such "smooth" production functions as a limiting case? If so, it would be a generalization or extension of a discrete model, in which all switch points are flukes, to a continuum. From the perspective of the analysis of the choice of technique in discrete models, typical neoclassical models are nothing but flukes.

6.0 Conclusions

I actually found my negative conclusion surprising. I have tried to be conscious of the distinction between the structure of the two models in Section 2 above. I think at least some examples I have presented cannot be attacked by the above critique. They are examples of the first, not the second model. I tend to read Samuelson (1962) in the same way, as not sensitive to the critique in this post.

References
  • Neri Salvadori and Ian Steedman (1988). No reswitching? No switching! Cambridge Journal of Economics, V. 12: pp. 481-486.
  • Samuelson, P. A. (1962). Parable and Realism in Capital Theory: The Surrogate Production Function, V. 29, No. 3: pp. 193-206.
  • J. E. Woods 1990. The Production of Commodities: An Introduction to Sraffa, Humanities Press International.