'Neoclassical' Economists On The Lack Of Foundation For Some 'Neoclassical' Economics

1.0 Introduction

Last century and into this one, 'neoclassical' economists noted the lack of theoretical foundation for certain widely used models in economics. They noted that the interest rate is generally not equal to the marginal product of capital. This post quotes three prominent 'neoclassical' economics, over decades, noting the lack of theoretical foundation for such an equality.

For the purposes of this post, I have little to say about my disagreements with these authors. I will note that Sraffians have something to say about microeconomics too. I also do not want to go into here why empirical work with these unfounded models is almost always a kind of humbug.

2.0 Frank Hahn

Frank Hahn attacks my favorite school of thought. He says:

"Sraffa ... confined himself to the remark that the [missing] equation cannot be one which demands the equality of the marginal product of 'capital' and the rate of profit. ... the neoclassical economist has the same view but his reasons are not those given by Sraffa." -- Frank Hahn (1982) The neo-Ricardians. Cambridge Journal of Economics. 6(4): 362.

And again:

"The Sraffian picture of neoclassical theory is this. At any moment of time we can observe something physical called the stock of capital (K) as well as the amount of labor (L). There is a concave production function

Y = F(K, L)

where Y is output. In a neoclassical equilibrium all inputs are used and must be paid their marginal products. The latter are known once (K, L) are known. Hence the rate of profit of capital, the real wage and the distribution of income are all known once F(), K and L are known. The concavity of F further implies that the rate of return on capital is non-increasing (generally decreasing) in K. This construction, to be called the parable, Sraffians claim not to be watertight except in the single good economy. In this they are generally correct." -- Frank Hahn (1982: 370)

3.0 Edwin Burmeister

This is from a standard reference work:

"Imposing some set of conditions on the technology T() should be sufficient to ensure that the real Wicksell effect is always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from the one-good model often used in macroeconomics without any theoretical justification for ignoring aggregation problems. Moreover, Burmeister (1977, 1979) has proved that a negative real Wicksell effect is a necessary and sufficient condition for an index of capital, k, and a neoclassical aggregate production function F(k) defined across steady-state equilibria such that (i) c = F(k), (ii) r = F’(k), and (iii) F’’(k) < 0. Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987). Wicksell effects. The New Palgrave.

That index is Champernowne's chain index measure of capital.

4.0 Emmanuel Farhi

Here is Emmanuel Farhi giving a lecture in 2018 agreeing with the above authors. His history of the CCC is in the first half hour. There is an accompanying paper (working paper version here). Farhi's co-author, David Rezza Baqaee, seems to be pursuing this approach.

5.0 Conclusion

Economics presents a problem for the sociology of 'knowledge'.

Socialism Works In Growing Organic Rice In Latin America

The largest producer of organic rice in Latin America, over the last decade, is the Movimento dos Trabalhadores Rurais Sem Terra (MST), or the Landless Rural Workers Movement, in Brazil. They have 1.5 million members, in 23 of 26 Brazilian states.

Article 5, Section XXIII of Brazil's constitution mandates that land serve a social function. I gather that MST was formed by peasants occupying unused land. They have a radical democratic organization, with some practices that remind me of the 1871 Paris commune. Their base consists of many small settlements working together.

They grow more than rice. Their goals include self-sustaining, self-managed agriculture. They avoid pesticides and use bio-fertilizers. (I guess this is a nice way of talking about manure.) This is a model opposed to a few large business owning huge tracts of land dedicated to production for the export market.

Since I have a vestigial interest in software, I'll mention work by Prof. Celso Alexandre Souza de Alvear and others to develop Sementes, a plugin for web sites designed especially with marketing products from the solidarity economy. There is also Ciranda. If you are going to market organic food, you want the customer to be able to easily access information about ingredients and organizations that grow it. The Arvoredo app (I have not read that paper) "facilitates documentation, monitoring, and evaluation of grassroots environmental governance activities, including tree planting, agroforestry practices, tree nursery construction, and seed collection efforts". Recently, MST has launched Iaraa, an AI tool. The software must be in service of the collective's larger goals.

A corresponding urban organization exists, the Movimento dos Trabalhadores Sem-Teto (MTST) or Movement of Homeless Workers. And MST seems to be networked with other organizations providing models for a post-capitalist society.

As usual, I disclaim much knowledge of these organizations. You can check out an article in Jacobin by Joao Paulo Rodrigues, the write up for the 1991 Right Livelihood Award, or a write up from Grassroots International.

Labor Demand In Corn Industry With The Recurrence Of Truncation

Figure 1: Labor Demand In Corn Industry

1.0 Introduction

This post continues my example of the recurrence of truncation without reswitching. Here I present graphs of the economy-wide demand for capital and for labor, as well as a sectorial demand for labor.

2.0 Technology and Techniques

Of the two industries in the model, one produces machines, and the other produces corn. Machines are fixed capital. Their physical life is two production cycles - that is, two years - in each industry. Corn is circulating capital in each industry and also a consumption good. A bushel corn is also the numeraire. Old machines cannot be transferred between industries. Constant returns to scale (CRS) and the free disposal of machines are assumed . Tables 1 and 2 define specific numeric values for a technology that meets these specifications, with a direct labor input in each process.

Table 1: Inputs for The Technology

Input	Industry
	Machine		Corn
	I	II	III	IV
Labor	1/10	8	43/40	1
Corn	1/16	3/20	1/8	53/200
New Machines	1	0	1	0
One-Year Old Machines (1st type)	0	1	0	0
One-Year Old Machines (2nd type)	0	0	0	1

Table 2: Outputs for The Technology

Output	Industry
	Machine		Corn
	I	II	III	IV
Corn	0	0	1	14/25
New Machines	2	5/2	0	0
One-Year Old Machines (1st type)	1	0	0	0
One-Year Old Machines (2nd type)	0	0	1	0

With this specification of the technology, the economic life of the machine must be chosen in each industry. Table 3 lists the available techniques. The machine is truncated in both industries in the Alpha technique. The machine is operated for its full physical life in both industries in the Delta technique. In Beta and Gamma, the machine is truncated in one industry and operated for its full physical life in the other.

Table 3: Specification of Techniques

Technique	Processes	Notes
Alpha	I, III	Machines truncated in both industries.
Beta	I, II, III	Machines truncated in machine-production.
Gamma	I, III, IV	Machines operated at full physical life in both industries.
Delta	I, II, III, IV	Machines truncated in corn-production.

3.0 Price Systems and the Choice of Technique

The choice of technique can be analyzed in a model of pure fixed capital by constructing the wage frontier as the outer envelope of the wage curves for each technique. I take a bushel of corn as the numeraire. I assume that wages are paid out of the surplus at the end of the year, not advanced at the beginning. Figure 2 shows the wage curves for the example. Figure 2 shows an enlargement.

Figure 2: Wage Curves

Figure 3: Wage Curves (Enlarged)

The order of the cost-minimizing techniques, with an increasing rate of profits or a decreasing wage, is Alpha, Gamma, Delta, and Beta. This is also in order of increasing labor intensity. I measure labor-intensity by employment, economy-wide, needed to produce a net product of a bushel corn.

This is not a reswitching example. Techniques do not reswitch even off the frontier. The economic life od the machine is truncated in the corn industry for Alpha and Beta. Alpha and Beta are cost-minimizing at the furtherest extremes of the rate of profits, with Gamma and Delta cost-minimizing for middling rates of profits. So this is an example of the recurrence of truncation in the corn industry.

4.0 Economy-Wide Demand Curves for Capital and Labor

The above supports the drawing of economy-wide demand curves for capital and labor. Suppose net output for the economy as a whole is one bushel corn. Figure 4 shows the demand curve for capital. The value of inputs of corn and machines needed as inputs for the cost-minimizing technique are aggregated to obtain the quantity of capital at each point on the demand curve. Switch points are horizontal line segments in the graph. Around each switch point, a lower rate of profits is associated with a demand for a greater quantity of capital. This property is consistent with outdated marginalist theory. It is not a general property.

Figure 4: Economy-Wide Demand Function for Capital

The demand function for capital is not vertical between switch prices. These curves reflect the variation of prices with the rate of profits, given the technique. This variation is known as the price Wicksell effect. Edwin Burmeister champions David Champernowne's chain index measure of capital. This chain index eliminates price Wicksell effects. Given no capital-reversing, this chain index can be used to show that the rate of profits equals the marginal product of capital. This equality plays no role in solving the price system or in analyzing the choice of technique.

You can also draw an economy-wide demand curve for labor (Figure 5). Switch points are again shown as horizontal line segments. Here, a more labor-intensive technique is adopted at a lower wage. This, too, is a general property. But to emphasize the effects of the recurrence of truncation, I want an example that does not contradict obsolete marginalist theory at an aggregate level.

Figure 5: Economy-Wide Demand Function for Labor

5.0 Sectorial Demand Curves for Labor

I now consider the demand for labor in each industry. Figure 6 plots a sectorial demand curve in the machine industry. Prices of production are assumed to prevail at each level of the wage. One new machine is produced gross by the machine industry. Alpha and Gamma both operate the machine for a single year in producing new machines. Thus, the amount of labor employed with the given gross output in the machine industry is the same for Alpha and Gamma, as shown by the single vertical line to the left. Beta and Delta both operate the machine for its full physical life of two years in the machine industry, also, as shown in the graph. The sectorial labor demand function in the machine industry is a downward-sloping step function approximation to the traditional, non-justified story.

Figure 6: Labor Demand In Machine Industry

Figure 1, at the top of this post is the demand function for labor in the corn industry. Alpha and Beta both operate the machine for one year in the corn industry, while Gamma and Delta operate the machine in this industry for the full two years. The switch point between Beta and Delta exhibits the reverse substitution of labor. A higher wage around this switch point is associated with firms wanting to ultimately employ more labor per bushel corn produced gross in the corn industry.

Conclusion

So much for explaining wages and employment by well-behaved supply and demand functions for labor.

Even without reswitching or capital-reversing, the marginalist textbook stories do not work.

How To Draw Hayekian Triangles In A Model Of The Production Of Commodities

1.0 Introduction

This post revisits an analysis of Hayekian triangles in the context of a circular flow of production. Here I go through the mathematics to show how to construct the "triangle".

2.0 The Technique and Net Output

A technique is specified by a row vector a₀ of direct labor coefficents of a square Leontief input matrix A. Each labor coefficient and corresponding column of the Leontief matrix specify a process to produce one unit of the good produced by that industry. All coefficients are specified in physical units, such as barrel oils per kilowatt. I assume:

• The economy is in a stationary state.
• Constant returns to scale (CRS) prevail.
• All direct labor coefficients are positive.
• Every good enters, either directly or indirectly, into the production of every good.
• The Leontief matrix specifies a productive technique in that a suplus product can be produced.
• Full employment is assumed. More generally, the units in which labor is measured is scaled such that employment is unity.
• Wages are paid at the end of the year, not advanced with the payments for capital goods at the beginning of the year.

These assumptions are stronger than needed.

The proportions of net output are assumed to be specified by a column vector d. This vector is also a numeraire. The level of net output y is specified by the scalar c:

y = c d

This formulation allows for specifying any number of techniques, all with the same numeraire and composition of net output, but at different levels.

3.0 Quantity Flows

The net output vecotr y and the gross output vector q are related as:

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

Total employment is unity:

a₀ q = 1

These equations have a solution. Consumption per worker is:

c = 1/[a₀ (I - A)^-1 d

Gross quantities are:

q = c (I - A)^-1 d = c d + c A d + c A² d + ...

The first term in the infinite expansion on the right-hand side is the net product available at the end of the given year. The second term is the quantities of capital goods being produced in the current year to support the production of the net output in the next year. The third term is the capital goods being produced in the current year to eventually produce the net output two years hence. Note that all of these vectors, of consumption goods, specific capital goods, and so on are heterogeneous.

4.0 Labor and Capital Flows in the Hayekian Triangle

The labor l_i expended in the current year, with previously produced capital goods, to produce the net output, that is, goods of the first order, is defined as:

l₁ = c a₀ d

The labor l_i expended in the current year to produce goods of each of the higher orders is:

l_i = c a₀ A^{i - 1} d, i = 2, 3, ...

The sum of these quantities of labor is unity, that is, the labor force employed in the current year.

The capital goods expended in the current year to produce goods of each order is:

k_i = c Aⁱ d, i = 1, 2, ...

The sum of these quantities of capital goods used in the current year is A q.

5.0 Value Flows in the Hayekian Triangle

Let z_i be the addition in value for each stage in the Hayekian triangle. This is merely the value added by original factors of production, properly time discounted for each stage:

z_i = w(r) l_i (1 + r)^{i - 1}, i = 1, 2, ...

The notation reflects the interdependence of the wage 𝑤(𝑟) and the interest rate 𝑟 in a stationary state.

For goods of first order, the length of this step in the Hayekian triangle is:

z₁ + z₂ + ... = c p(r) d

where p(r) is a row vector of prices. For goods of the second order, the length of the step in the Hayekian triangle is:

z₂ + z₃ + ... = c p(r) d - z₁

For goods of the third order, the length of the step in the Hayekian triangle is:

z₃ + z₄ + ... = c p(r) d - (z₁ + z₂)

These steps can be continued. This completes one derivation of the lengths of the steps in a Hayekian triangle.

The length of the ith step can also be expressed as:

z_i + z_{i + 1} + ... = p(r) k_i (1 + r)ⁱ + w(r) l_i (1 + r)^{i - 1}, i = 1, 2, ...

With a couple of substitutions and factoring, the above becomes:

z_i + z_{i + 1} + ... = c [p(r) A (1 + r) + w(r) a₀] A^{i - 1} d (1 + r)^{i - 1}

Or:

z_i + z_{i + 1} + ... = c p(r) A^{i - 1} d (1 + r)^{i - 1}, i = 1, 2, ...

The Hayekian triangle, with an infinite number of steps, has now been derived, in two ways, from the circulating capital case of a model of the production of commodities by means of commodities.

6.0 Conclusion

The above derivations assume knowledge of the solutions of the price system for the technique. A more complete exposition would present that solution. It would also show that the Hayekian triangle approaches one constructed with a geometric series, as the order of goods increases. The composition of capital goods approaches that of Sraffa's standard system.

On A Transition To Socialism

I have just started David Schweickart's After Capitalism. I have yet to get to his exposition of how transition to economic democracy might be achieved.

To my mind, a transition to Schweickart's preferred system could start with a government fund acquiring enough equity in corporates to prescribe certain organization decisions. As I recall, the United States government bailed out General Motors during the 2008 worldwide depression. These were loans; they could have taken equity. The Norwegian government accumulates a Sovereign Wealth Fund from North Sea oil. The Australian government has superanuation funds.

Once a government has enough ownership of some corporations, they are not required to maintain them as hierarchical, authoritarian institutions. They could, say, get rid of the board of directors. An elected workers' council could appoint management.

The government fund can still own the capital. The firm pays a fee for renting the capital. I once worked for a wholl owned subsidiary, a capitalist firm owned by another capitalist firm. We had to pay such a fee to the owning company.

Schweickart has the revenue obtained by such fees used to create an investment fund. Investment will be directed not solely by profitability. He has the investment fund being paid out on a per capita basis, as a first approximation. Some of the investments are national in scope. The remainder goes to regions on a per capita basis, as a first approximation. This works well with a federal system, like Australia, Canada, and the USA. Public needs would have some influence. For example, some investment might be deliberately directed to less developed regions of the country or to update old technology with more green, ecological technology. New worker-directed firms can be created, and current firms can invest in new processes and new products.

I can see how some of these fees might also pay for, say, head start, pension funds, and so on. Not everybody is in the work force. Schweickart recommends a job guarantee program. He imagines current social programs would continue with funding, I guess, by current tax systems. This is in parallel with new investment fund.

The workers, however, would be the residual claimant under economic democracy. They might not even be paid wages. Profit calculations would differ. Maybe profits are broken down and distributed every week, every two weeks, or every month.

Under Schweickart's economic democracy, the means of production are not privately owned but rented from the public. Workers do not sell their labor power for a wage. Products are produced for buying and selling on markets. As now, large islands exist where the maket is replaced by internal planning. Now those are privately owned corporations. Under this new structure, these firms would be governed by the workers themselves.

We could start transitioning to such a system today.

But I do not see where the political will comes from to do this. On the other hand, I have seen changes in my lifetime that I did not think possible: the fall of the Berlin Wall, the end of apartheid in South Africa, the fall of fascism in Chile, a genuine democracy created for a while in the Philippines.

On The Incoherence Of Austrian Business Cycle Theory

I thought I would try to summarize again some objections to the Austrian school.

Austrian Business Cycle Theory (ABCT) focuses on the consequences of the monetary authority setting the monetary interest rate below the natural rate of interest. Following Knut Wicksell somewhat, the theory argues that capitalist entrepreneurs will lengthen production processes. Since these decisions do not synchronize with household consumption and savings decisions, the artificial boom is unsustainable. A bust is the result.

This theory is built on mistaken capital theory. When Sraffa spanked Hayek, he deliberately put capital theory aside.

What would it mean for a production technique to be more capital-intensive? To examine the (il)logic of this approach, I make various simplifying assumptions.

Accordingly, consider vertically-integrated firms. They produce a given commodity or, rather, a basket of commodities, in fixed proportions. The only input is homogenous labor. All means of production, tools, intermediate goods are produced and used internally.

Under these assumptions, one can talk about (net) output per person-year. Productivity is well-defined. I start by postulating that with a more capital-intensive technique, workers are more productive.

It turns out that a lower interest rate does not induce managers of firms to adopt more capital-intensive techniques. Numerical examples illustrating this point have been available in the literature since the 1960s. And they are accepted by all sides. "The interesting point, however, is the perversity, not the duplicity." -- Robinson and Naqvi (1967).

I have now demonstrated that marginalist capital theory, including the Austrian variant, is invalid. Given typical assumptions, the traditional stories about 'capital' markets do not follow. But what about all that stuff Austrian school economists say about the structure of production?

They are wrong there, too. First, I consider aggregate measures of the period of production. Bohm-Bawerk's measure assumes simple interest, not compound interest. The counterexamples mentioned above demonstrate it is invalid to conclude cost-minimizing entrepreneurs will lengthen the period of production when they anticipate lower interest rates.

Nicolas Cachanosky and Peter Lewin have recently proposed a financial measure of Duration. By this measure, the technique chosen at a lower interest rate, around a switch point, has a larger Duration. But a larger Duration is associated with lower productivity in the counter-examples.

You could also consider how long capitalists will choose to run given machinery. Machines last for more than one production cycle, and capitalists must choose to set their economic life. Surely, a lower interest rate will provide incentives to capitalists to increase their economic life. Well, no. A longer economic life of a machine can be associated with a less capital-intensive technique in the sense that the productivity of labor is decreased. I happen to know that the recurrence of the period of truncation is possible without the reswitching of techniques.

And then there are Hayekian triangles. They do not work either. Hayekian triangles, as Roger Garrison notes, are heuristic pictures, useful for pedagogic purposes. Hayek unsuccessfully tried to put them on rigorous foundations in Prices and Production. But it all fell apart. He even discovered capital-reversing, in some sense. You can find more recent statements with these triangles, but nothing that addresses the difficulties that Hayek found, much less anything that surmounts them.

Jevons Versus Marshal On Ricardo

I have been pointing out that much teaching in most universities and high schools in economics is propaganda. Mistakes that were exposed more than half a century ago continue to be taught. Alternatives have been available, at varying levels, in textbooks for decades.

Does an alternative, building on classical political economy and Marx, exist? Assertions on this topic go back more than a century.

A (bad) way of reading classical political economy is that its proponents were struggling towards developing the one true system, that of marginalist economists. With this incorrect view of continuity, you might say incorrectly that they overemphasized supply. Their theories were corrected by developing theories of utility and demand.

A better reading recognizes that they had their own approach. Jevons held this view, although he was wrong about which approach was better:

"When at length a true system of Economics comes to be established, it will be seen that that able but wrong-headed man, David Ricardo, shunted the car of Economic science on to a wrong line - a line, however, on which it was further urged towards confusion by his equally able and, wrong-headed admirer, John Stuart Mill." -- William Stanley Jevons, The Theory of Political Economy, Preface to the Second Edition, p. li.

Marshall, on the other hand, was an early progenitor of a supposedly generous reading that blurs the distinctiveness of the classical theory of value and distribution:

"1... [Ricardo's] book makes no pretence to be systematic. He was with difficulty induced to publish it; and if in writing it he had in view any readers at all, they were chiefly those statesmen and business men with whom he associated. So he purposely omitted many things which were necessary for the logical completeness of his argument, but which they would regard as obvious. And further, as he told Malthus in the following October, he was 'but a poor master of language.' His exposition is as confused as his thought is profound; he uses words in artificial senses which he does not explain, and to which he does not adhere; and he changes from one hypothesis to another without giving notice.

If then we seek to understand him rightly, we must interpret him generously, more generously than he himself interpreted Adam Smith. When his words are ambiguous, we must give them that interpretation which other passages in his writings indicate that he would have wished us to give them. If we do this with the desire to ascertain what he really meant, his doctrines, though very far from complete, are free from many of the errors that are commonly attributed to them...

...Again, in a profound, though very incomplete, discussion of the difference between 'Value and Riches' he seems to be feeling his way towards the distinction between marginal and total utility. For by Riches he means total utility, and he seems to be always on the point of stating that value corresponds to the increment of riches which results from that part of the commodity which it is only just worth the while of purchasers to buy; and that when the supply runs short, whether temporarily in consequence of a passing accident, or permanently in consequence of an increase in cost of production, there is a rise in that marginal increment of riches which is measured by value, at the same time that there is a diminution in the aggregate riches, the total utility, derived from the commodity. Throughout the whole discussion he is trying to say, though (being ignorant of the terse language of the differential calculus) he did not get hold of the right words in which to say it neatly, that marginal utility is raised and total utility is lessened by any check to supply.

2. But while not thinking that he had much to say that was of great importance on the subject of utility, he believed that the connection between cost of production and value was imperfectly understood; and that erroneous views on this subject were likely to lead the country astray in practical problems of taxation and finance; and so he addressed himself specially to this subject. But here also he made short cuts." -- Alfred Marshall, Principles of Economics, Appendix I

Marshall is wrong here. For example, Ricardo describes riches as a collection of commodities. They were not measured along a single scale, whatever measurement level you might think that scale obtains. Even less could his labor values be said to have been marginal utilities.

Samuel Hollander is the greatest exponent in my lifetime of the view of continuity in the development of theories of value and distribution. Even he, though, recognizes that Marx had reasons for his reading of Ricardo, but I forget where.