Wednesday, January 01, 2020
Welcome
The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.
In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.
I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.
Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.
Thursday, April 20, 2017
Nonstandard Investments as a Challenge for Multiple Interest Rate Analysis?
This post contains some musing on corporate finance and its relation to the theory of production.
2.0 Investments, the NPV, and the IRRIn 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 C_{t}; 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, C_{n - 1} is the cash flow at the end of the first year in the investment, and C_{0} 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:
NPV_{0}(r) = C_{n - 1} + C_{n - 2}/(1 + r) + ... + C_{0}/(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:
NPV_{1}(r) = C_{n - 1}(1 + r)^{n - 1} + C_{n - 2}(1 + r)^{n - 2} + ... + C_{0}
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/C_{n - 1}) NPV_{1}(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:
NPV_{1}(r) = (-C_{n - 1})(r - r_{0}) (r - r_{1})... (r - r_{n - 1})
where r_{0}, r_{1}, ..., r_{n - 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 r_{0} be the IRR.
3.0 Standard Investments in Selected Models of ProductionA 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 IRRIn 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.
Year | Revenue |
0 | 750 |
1 | -500 |
2 | -500 |
3 | -500 |
4 | 1,000 |
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 TerminologySee 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
- 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 |
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 TechnologyThe 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.
Input | Iron Industry | Corn Industry | |
Labor | 1 | 305/494 | 1 |
Iron | 1/10 | 229/494 | 2 |
Corn | 1/40 | 3/1976 | 2/5 |
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, a_{0, β}, is:
a_{0, β} = [305/494, 1]
The components of the Leontief matrix for the Beta technique, A_{β}, are:
a_{1,1, β} = 229/494
a_{1,2, β} = 2
a_{2,1, β} = 3/1976
a_{2,2, β} = 2/5
The labor coefficients for the Alpha technique, a_{0, α}, differ in the first element from those for the Beta technique. The Leontief matrix for the Alpha technique, A_{α}, 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 TechniqueConsider steady-state prices that repeat, year after year, as long as firms adopt the same technique. Let a_{0} 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) + a_{0} 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:
a_{0} w = p [I - (1 + r) A]
Multiply through, on the right, by the inverse of the matrix in square brackets:
a_{0} [I - (1 + r) A]^{-1} w = p
Multiply through, again on the right, by e:
a_{0} [I - (1 + r) A]^{-1} e w = p e = 1
Both sides of the above equation are scalars. The wage is:
w = 1/{a_{0} [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 = a_{0} [I - (1 + r) A]^{-1}/{a_{0} [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:
a_{0} 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 a_{0} [I - (1 + g) A]^{-1} e = a_{0} q = 1
Consumption per person-year is:
c = 1/{a_{0} [I - (1 + g) A]^{-1} e}
Gross quantities are:
q = [I - (1 + g) A]^{-1} e/{a_{0} [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 IdentitiesThe 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) = a_{0} w + p A r
Multiply both sides by the vector of gross outputs:
p (I - A) q = a_{0} 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 EffectsThis 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 MarketA 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 ConclusionThis 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 |
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 TechnologyThe 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.
Input | Iron Industry | Corn Industry | |
Labor | 1 | 305/494 | 1 |
Iron | 1/10 | 229/494 | 11/10 |
Corn | 1/40 | 3/1976 | 2/5 |
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 EquationsThe 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 + rs_{1}) + w = p
[(229/494)p + (3/1976)](1 + rs_{1}) + (305/494)w ≥ p
[(11/10)p + (2/5)](1 + rs_{2}) + w = 1
where p is the price of a unit of iron, and w is the wage.
The parameters s_{1} and s_{2} are given constants, such that rs_{1} is the rate of profits in iron production and rs_{2} is the rate of profits in corn production. The quotient s_{1}/s_{2} 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:
s_{1} = s_{2} = 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 s_{1} and s_{2} 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 TechniqueThe above system can be solved, given s_{1}, s_{2}, 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 CompetitionConsider the special case of free competition. The wage curve for the Alpha technique is:
w_{α} = (41 - 38r + r^{2})/[80(2 + r)]
The price of iron, when the Alpha technique is cost-minimizing, is:
p_{α} = (5 - 3r)/[8(2 + r)]
The wage curve for the Beta technique is:
w_{β} = (6,327 - 9,802r + 3,631r^{2})/[20(1,201 + 213r)]
When the Beta technique is cost-minimizing, the price of iron is:
p_{β} = [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%:
r_{switch} = (1/1,301)[799 - 24 (826^{1/2})]
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:
s_{1} = 4/5
s_{2} = 5/4
In this case, the wage curve for the Alpha technique is:
w_{α} = (4,100 - 4,435r + 100r^{2})/[40(400 + 259r)]
The price of iron, when the Alpha technique is cost-minimizing, is:
p_{α} = (125 - 96r)/(400 + 259r)
The wage curve for the Beta technique is:
w_{β} = 8(126,540 - 195,289r + 72,620r^{2})/[160(24,020 + 9,447r)]
The price of iron, when the Beta technique is cost-minimizing, is:
p_{β} = 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:
s_{2} = 1/s_{1}
One can then consider how the shapes and locations of wage curves and switch points vary with continuous variation in s_{1}/s_{2}. 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 ConclusionSo 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.