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.

Saturday, March 04, 2017

Bifurcations Of Roots Of A Characteristic Equation

Figure 1: Rates of Profits for Beta Technique

I have previously considered all roots of a polynomial equation for the rate of profits in a model, of the choice of technique, in which each technique is specified by a finite series of dated labor inputs. One root is the traditional rate of profits, but there are uses for the other roots:

  • All roots appear in an equation defining the Net Present Value (NPV) for the technique, given the wage and the rate of profits.
  • All roots can be combined in an accounting identity for the difference between labor commanded and labor embodied, given the wage.
I thought it of interest to know whether these non-traditional roots are real or complex, as they vary with the wage. I am considering multiple roots in an attempt to build on and critique Michael Osborne's approach to multiple interest rate analysis.

I also have considered examples of models of the production of commodities by means of commodities, in which at least one commodity is basic, in the sense of Sraffa. And I have attempted to apply or extend my critique of multiple interest rate analysis to these models. The point of this post is to illustrate possibilities on the complex plane for multiple interest rates in these models.

A technique in models of the production of commodities by means of commodities, as least in the case when all capital is circulating capital, is specified as a vector of labor coefficients and a Leontief input-output matrix. In parallel with my approach to techniques specified by a finite sequence of dated labor inputs, consider wages as being advanced - that is, not paid at the end of the year out of the surplus - in such models. Given the wage and the numeraire, one can construct a square matrix in which each coefficient is the sum of the corresponding coefficient in the Leontief input-output matrix and the quantity of the commodity produced by that industry that is advanced to the workers, per unit output produced. I call this matrix the augmented input-output matrix.

A polynomial equation, called the characteristic equation, is solved to find eigenvalues of the augmented input-output matrix. The power of this polynomial is equal to the number of commodities produced by the technique. The number of roots for the polynomial is therefore equal to the number of commodities. A rate of profits corresponds to each root. Assume the Leontief input-output matrix is a Sraffa matrix and that the wage does not exceed a certain maximum. Under these conditions, the Perron-Frobenius theorem picks out the maximum eigenvalue of the augmented input-output matrix. The corresponding rate of profits is non-negative, and the prices of production of these commodities are positive at the given wage. I was not able to find an application for the other, non-traditional rates of profits.

I present a numerical example in this working paper. This is a three-commodity example with two techniques. Figure 1 graphs the three roots, at different level of wages, for the Beta technique in that example.

In a previous blog post, I extend that example such that managers of firms have a choice of process for producing each of the three commodities. As a consequence, a choice among eight techniques arises. And one can draw a graph like Figure 1 for each technique in that example. Figure 2 shows the corresponding graph for the Delta technique.

Figure 2: Rates of Profits for Delta Technique

In Figures 1 and 2, the rate of profits picked out by the Perron-Frobenius theorem and used to draw the wage-rate of profits curve for the technique lies along the line segment on the real axis on the left in the figure. A lower wage corresponds to a higher traditional rate of profits. Thus, points further to the right on this line segment correspond to a lower wage. A wage of zero leads to the right-most point on this line segment. The highest feasible wage corresponds to left-most point, at a rate of profits of zero, on this segment.

Two non-traditional rates of profits arise for the other two solutions of the characteristic equation. They are plotted to the right on the graphs in Figures 1 and 2. When complex, they are complex conjugates. I thought it of interest that, in Figure 2, they are purely real for two non-overlapping, distinct ranges of feasible levels of the wage.

I draw no practical, applied implications from the non-traditional rates of profits. I just think the graphs are curious.

Thursday, March 02, 2017

Some Obituaries for Kenneth Arrow

Bill Black has one here, emphasizing Arrow's impossibility theorem. The blog, A Fine Theorem, has two of a planned four-part series. The first is on the impossibility theorem. The second is about General Equilibrium. The two planned, I gather, are to be about learning-by-doing and health economics, respectively.

I have written several posts drawing on Arrow's work. This one, on a sophisticated neoclassical response to the Cambridge Capital Controversy, is among my most popular posts.

Thursday, February 09, 2017

A Reswitching Example in a Model of Oligopoly

The Roots of a Cubic Polynomial Defining Switch Points

I have a draft paper up at SSRN. The abstract:

This paper illustrates, through a numerical example of reswitching under oligopoly, the existence of implications from the Cambridge Capital Controversy for the theory of industrial organization. Oligopoly is modeled by given and persistent ratios in rates of profits among industries, as expressed in a system of equations for prices of production. The numerical example illustrates that this model of oligopoly is a pertubation of free competition. Some comparisons and contrasts are drawn to a model of free competition.

In some sense, this paper shows a somewhat more comprehensive description of value through exogenous distribution than in Sraffa's book. The model can depict capitalists as squabbling over the division of the surplus that their class gets, as well as their struggle against the workers. I'd like to see an example of reswitching or capital reversing in this model, with all (price and real) Wicksell effects as negative in the example in the special case of free competition. I do not see why one cannot arise. Such an example would suggest that "perverse" examples can obtain empirically, even if they are not found in an analysis that presumes one common rate of profits among all industries.

The graph at the top of this post does not appear in the paper. In the model, the ratios of rates of profits among industries are given parameters. A cubic polynomial is defined for a given set of such ratios. Non-negative, real zeroes of that polynomial below a certain maximum define a scale factor for switch points. The location of the zeros varies with the ratios. I happen to be able to solve for the zeros. They are shown in the graph above.

Monday, January 23, 2017

Festschrifts for Sraffians

We are on, maybe, the third generations of Sraffians. An annoyance and a delight of publicly taking up a topic is that one must continually read advances, whether large or small, in your topic. I've read some festschrifts over the last several decades:

  • Competing Economic Theories: Essays in memory of Giovanni Carvale, edited by Serio Nisticó and Domenico Tosato.
  • Value, Distribution and Capital: Essays in honour of Pierangelo Garegnani, edited by Gary Mongiovi and Fabio Petri.
  • Economic Theory and Economic Thought: Essays in Honour of Ian Steedman, edited by John Vint, J. Stanley Metcalfe, Heinz D. Kurz, Neri Salvadori, and Paul A. Samuelson.

I'm aware of some I have not read:

  • Social Fairness and Economics: Essays in the spirit of Duncan Foley, edited by Lance Taylor, Armon Rezai, and Thomas Michl.
  • Keynes, Sraffa and the Criticism of Neoclassical Theory: Essays in honour of Heinz Kurz, edited by Neri Salvadori and Christian Gehrke.
  • Classical Political Economy and Modern Theory: Essays in honour of Heinz Kurz, edited by Christian Gehrke, Neri Salvadori, Ian Steedman and Richard Sturn.
  • Economic Theory and its History, edited by Guiseppe Freni, Heinz D. Kurz, Andrea Mario Lavezzi, and Rudolfo Signorino. (This apparently is a celebration of Neri Salvadori's work.)
  • Production, Distribution and Trade: Essays in honour of Sergio Parrinello, edited by Adriano Birolo, Duncan K. Foley, Heinz D. Kurz, Bertram Schefold and Ian Steedman
  • The Evolution of Economic Theory: Essays in honour of Bertram Schefold, edited by Volker Caspari.

This post provides another demonstration that at least one school of heterodox economists looks, from the outside, like any other group of academics with common research interests. I have posted about Post Keynesianism in this respect. Likewise, I once listed textbooks.

Monday, January 16, 2017

A Story Of Technical Innovation

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

I often present examples of the choice of technique as an internal critique of neoclassical economics. The example in this post, however, is closer to how I think techniques evolve in actually existing capitalist economies. Managers of firms know a limited number of processes in each industry, sometimes only the one in use. Accounting techniques specify a set of prices. An innovation provides a new process in a given industry. The first firm to adopt that process may obtain supernormal profits, whatever the wage or normal rate of profits. Other firms will strive to move into the industry with supernormal profits and to use the new process. Prices associated with the new technique result in the wage-rate of profits frontier being moved outward, perhaps along its full extent.

This example was introduced by Fujimoto (1983). I know it most recently from problem 22 in Woods (1990: p. 126). It is also a problem in Kurz and Salvadori (1995). Fujimoto probably labels it a curiosum because of details more specific than the above overview of how Sraffians might treat technical change.

2.0 Technology

This example is a two-commodity model, in which both commodities, called iron and corn, are basic. Suppose iron is used exclusively as a capital good, and corn is used for both consumption and as a capital good. Consider the processes shown in Table 1. Each process exhibits Constant Returns to Scale. The coefficients in each column show required inputs, per unit output, in each industry for each process. Each process requires a year to complete, and outputs become available at the end of the year. This is a circulating capital model. All commodity inputs are totally used up in the year by providing their services during the course of the year.

Table 1: The Technology
InputIron
Industry
Corn
Industry
(a)(b)(c)(d)
Labor1/21/101/23/5
Iron002/51/5
Corn2/51/1000

For this economy to be reproduced, both iron and corn must be (re)produced. A technique consists of an iron-producing and a corn-producing process. Table 2 lists the four techniques that can be formed from the processes listed in Table 1. In this example, not all processes or techniques are known at the start of the dynamic process under consideration.

Table 2: Techniques
TechniqueProcesses
Alphaa, c
Betaa, d
Gammab, c
Deltab, d

3.0 Price Systems

A system of prices of production characterize smooth reproduction with a given technique. Suppose a unit of corn is the numeraire. Let w be the wage, r be the (normal) rate of profits, and p be the price of a unit of iron. Suppose labor is advanced, and wages are paid out of the surplus. If the Alpha technique is in use, prices of production satisfy the following system of two equations in three unknowns:

(2/5)(1 + r) + (1/2) w = p
(2/5) p (1 + r) + (1/2) w = 1

A non-negative price of iron and wage can be found for all rates of profit between zero and a maximum associated with the technique. Figure 1 illustrates one way of depicting this single degree of freedom, for each technique.

4.0 Innovations

I use the above model to tell a story of technological progress. Suppose at the start, managers of firms only know one process for producing iron and one process for producing corn. Let these be the processes comprising the Alpha technique. In this story, the rate of profits is exogenous, at a level below the rate of profits associated with the switch point between the Gamma and Delta technique, not that that switch point is relevant at the start of this story.

Somehow or other, prices of production provide a reference for market prices. For such prices, the economy is on the wage-rate of profits curve for the Alpha technique in Figure 1. This curve is closest to the origin in the figure.

Suppose researchers in the corn industry discover a new process for producing corn, namely process (d). A choice of technique arises. Corn producers see that they can earn extra profits by adopting this technique at Alpha prices. The Beta technique becomes dominant. Eventually, the extra profits are competed away, and the economy lies on the wage-rate of profits curve for the Beta technique. Under the assumption of an externally specified rate of profits, the wage has increased.

Next, an innovation occurs in the iron industry. Firms discover process (b). At Beta prices, it pays for iron-producing firms to adopt this new process. The wage-rate of profits curve for the Delta technique lies outside the wage-rate of profits curve for the Beta technique. Thus, the Delta technique dominates the Beta technique. But prices of production associated with the Delta technique cannot rule. If the Delta technique were prevailing, corn-producing firms would find they can earn extra profits by discarding process (d) and reverting to process (c). The Gamma technique is dominant at the given rate of profits, and workers will end up earning a still higher wage.

I guess this story does not apply to the United States these days. In the struggle over the increased surplus provided by technological innovation, workers do not seem to be gaining much. At any rate, Table 3 summarizes the temporal sequence of the dominant technique in this story.

Table 3: A Temporal Series of Innovations
EventsDominant
Technique
Processes
in Use
Processes (a) and (c) knownAlphaa, c
Processes (d) introducedBetaa, d
Process (b) introducedGammab, c

I do not see why one could not create an example with a single switch point between the Gamma and Delta techniques, where that switch point is at a wage below the maximum wage for the Alpha technique. For such a postulated example, one could tell story, like the above, with a given wage. The capitalists would end up with all the benefits from technological progress.

5.0 Conclusion

This example illustrates that innovation in one industry (that is, the production of iron) can result in the managers of firms in another industry (corn-production) discarding a previously introduced innovation and reverting to an old process of production.

References
  • T. Fujimoto 1983. Inventions and Technical Change: A Curiosum, Manchester School, V. 51: pp. 16-20.
  • Heinz D. Kurz and Neri Salvadori 1995. Theory of Production: A Long Period Analysis, Cambridge University Press.
  • J. E. Woods 1990. The Production of Commodities: An Introduction to Sraffa, Humanities Press International.

Saturday, January 14, 2017

A Model Of Oligopoly

1.0 Introduction

Suppose barriers to entry exist in an economy. Entrepreneurs and capitalists find that they cannot freely enter or exit some industries. And these barriers are manifested by stable ratios of rates of profits among industries. This post presents equations for prices of production under these assumptions.

I suggest that the model presented here fits into the tradition of Old Industrial Organization, as formulated by Joe Bain and Paolo Sylos Labini. As I understand it, Sylos Labini may have once written down equations like these, but never presented them or published them. I suppose this model is also related to work Piero Sraffa published in the 1920s.

2.0 The Model

Consider an economy consisting of n industries. Suppose the rate of profits in the jth industry is (sj r), where r is the base rate of profits, sj is positive, and:

s1 + s2 + ... + sn = 1

For simplicity, I limit my attention to a circulating capital model of the production of commodities by means of commodities. For the technique in use, let ai, j be the quantity of the ith commodity used to produce a unit of output in the jth industry. Homogeneous labor is the only unproduced input in each industry. Let a0, j be the person years of labor used to produce a unit output in the jth industry. I assume labor is advanced, and wages are paid out of the surplus at the end of production period, say, a year. Then prices of production, which ensure a smooth reproduction of the economy, satisfy the following system of equations:

(a1, 1 p1 + a2, 1 p2 + ... + an, 1 pn)(1 + s1 r) + w a0, 1 = p1
(a1, 2 p1 + a2, 2 p2 + ... + an, 2 pn)(1 + s2 r) + w a0, 2 = p2
. . .
(a1, n p1 + a2, n p2 + ... + an, n pn)(1 + sn r) + w a0, n = pn

The coefficients of production, including labor coefficients, and the ratios of the rate of profits are given parameters in the above system of equations. The unknowns are the prices, the wage, and the base rate of profits. Since only relative prices matter in this model, one degree of freedom is eliminated by choosing a numeraire:

p1 q*1 + ... + pn q*n = 1

Since there are n price equations, appending the above equation for the specified numeraire yields a model with (n + 1) equations and (n + 2) unknowns. One degree of freedom remains.

3.0 In Matrix Form

The above model can be expressed more concisely in matrix form. Define:

  • I is the identity matrix.
  • e is a column vector in which each element is 1.
  • S is a diagonal matrix, with s1, s2, ..., sn along the principal diagonal.
  • p is a row vector of prices.
  • q* is the column vector representing the numeraire.
  • A is the Leontief input-output matrix, representing the technique in use.
  • a0 is the row vector of labor coefficients for the technique.

The model consists of the following equations:

eT S e = 1
p A (I + r S) + w a0 = p
p q* = 1

4.0 Conclusion

One could develop the above model in various directions. For example, one could plot the wage-base rate of profits curve for the technique in use. Of interest to me would be presenting examples of the choice of technique, including reswitching and capital-reversing. The Sraffian critique of neoclassical economics is not confined to the theory of perfect competition.

Update (16 January 2017): I find I have outlined this model before.