A Semi-Idyllic Golden Age

1.0 Introduction

This post presents a model of a steady state with a constant rate of growth in which:

• Total wages and total profits grow at the same rate.
• Neutral technical change increases the productivity of labor in all industries.
• The wage per hour increases with productivity.
• Each worker continues to consume the same quantity of produced commodities.
• But each worker takes advantage of increased productivity to work less hours per year.

In these times, when concerns about global warning are so important, one would also want to see a suggestion of a reduced ecological footprint. So this model of a steady state is only semi-idyllic.

I do not consider anything in the mathematical model below to be original. I outline it to raise the question whether such a growth path is possible under capitalism. The model demonstrates logical consistency, but cannot demonstrate that details abstracted from in the model would not prevent its realization.

2.0 The Model

Consider a closed economy with no foreign trade. Industries are grouped into two great departments. In Department I, firms produce means of production, also known as capital goods. The output of Department I is called ‘steel’ and measured in tons. In Department II, firms produce means of consumption, also known as consumer goods. The output of Department II is called ‘corn’, measured in bushels. Both steel and corn are produced from inputs of steel and labor.

Constant coefficients of production (Table 1) are assumed to characterize production in each year. All capital is circulating capital. Long-lived machines, natural resources, and joint production are abstracted from in this model. Free competition is assumed. Labor is advanced, and wages are paid out of the net output at the end of the year. Workers are assumed to spend all of their wages on means of consumption. Profits are saved at a constant proportion, s.

Table 1: Constant Coefficients of Production

Parameter	Definition	Units
a0, 1(t)	Labor required as input per ton steel produced in year t.	Person-Hrs per Ton
a1, 1	Steel services required as input per ton steel produced.	Tons per Ton
a0, 2(t)	Labor required as input per bushel corn produced in year t.	Person-Hrs per Bushel
a1, 2	Steel services required as input per bushel corn produced.	Tons per Bushel

Suppose coefficients of production for steel inputs are constant through time, but labor coefficients exhibit a growth in labor productivity of 100 ρ percent:

a_{0, j}(t + 1) = (1 - ρ) a_{0, j}(t), j = 1, 2

Let X_i(t), i = 1, 2; represent the physical output produced in each department in year t and available at the end of the year. Furthermore, suppose the price of steel, p, and the rate of profits, r, are constant. Let outputs from each of the two departments grow at a constant rate of 100 g percent:

X_i(t + 1) = (1 + g) X_i(t), i = 1, 2

Certain quantity equations follow from these assumptions. The quantity of capital goods added each year must equal the capital goods remaining after reproducing those used up in producing total output, in both departments:

g [a_1,1 X₁(t) + a_1,2 X₂(t)]

= X₁(t) - [a_1,1 X₁(t) + a_1,2 X₂(t)]

The person-years of labor employed relates to labor coefficients and gross outputs:

L(t) = a_{0, 1}(t) X₁(t) + a_{0, 2}(t) X₂(t)

Price equations are:

p a_{1, 1} (1 + r) + a_{0, 1} w(t) = p

p a_{1, 2} (1 + r) + a_{0, 2} w(t) = 1

These equations embody the use of a bushel corn as numerate. w(t) is the wage per person-hour, paid out at the end of the year out of the surplus.

These assumptions and parameters are enough to depict Table 2. The column labeled "Constant capital" shows the value of advanced capital goods, taking the output of Department II as the numeraire. The column labeled "Variable Capital" depicts the wages paid out of revenues available at the end of the year. The surplus is what remains for the capitalists.

Table 2: A Tableau Economique

	Constant Capital	Variable Capital	Surplus	Output
I	p a1,1 X1(t)	w(t) a0,1 X1(t)	p a1,1 X1(t) r	p X1(t)
II	p a1,2 X2(t)	w(t) a0,2 X2(t)	p a1,2 X2(t) r	X2(t)

Workers spend what they get, and capitalists save a constant ratio, s, of their profits. With these assumptions, one can calculate the bushels corn that the workers and capitalists in Department I want to purchase, at the end of each year, from Department II. Likewise, one can calculate the numeraire value of the steel that capitalists in Department II want to purchase from Department I. Along a steady state, these quantities must be in balance:

[a_{0, 1}(t) w(t) + (1 - s) p a_{1, 1} r] X₁(t)

= p a_{1, 2} [1 + s r] X₂(t)

This completes the specification of this model of expanded reproduction with technical change uniformly increasing the productivity of labor.

3.0 The Solution

Output per labor hour is found by solving the quantity equations:

X₁(t)/L(t) = a_{1, 2} (1 + g)/β(t, g)

X₂(t)/L(t) = [1 - a_{1, 1} (1 + g)]/β(t, g)

where:

β(t, g) = a_{0, 2}(t) + [a_{0, 1}(t) a_{1, 2} - a_{0, 2}(t) a_{1, 1}](1 + g)

That is:

X_i(t)/L(t) = [1/(1 - ρ)^t] [X_i(0)/L(0)], i = 1, 2

The path of employed labor hours falls out as:

L(t) = (1 - ρ)^t (1 + g)^t L(0)

The number of employed person-hours decreases if:

ρ > g

The above expresses the condition that the labor inputs needed to produce a unit of output, in both departments, decrease faster than the rate of growth in both departments.

The price equations are also easily solved. Given a constant rate of profits, the price of steel is constant as well:

p = a_{0, 1}(0)/β(0, r)

The wage per person-hour increases with productivity:

w(t) = [1 - a_{1, 1} (1 + r)/β(t, r) = [1/(1 - ρ)^t] w(0)

The trade-offs between consumption per worker and the steady-state rate of growth and between the wage and the rate of profits have the same form.

These solutions can be substituted into the balance equation. It becomes:

[1 - a_{1, 1} (1 + s r)] (1 + g) = [1 - a_{1, 1} (1 + s r)] (1 + s r)

Suppose the rate of profits falls below its maximum (where the workers ‘live on air’) or not all profits are saved. Then this is a derivation of the "Cambridge equation":

r = g/s

A steady rate of growth, when the workers consume their wage, requires that the rate of profits be the quotient of the rate of growth and the savings rate out of profits.

4.0 Demographics and Institutions

I make some rather arbitrary assumptions about demographics and institutions. Suppose the number of person-years supplied as labor grows at the postulated rate of growth:

L_S(t + 1) = (1 + g) L_S(t)

with L_S(t) measured in person-years. Let the number of hours in a standard labor-year, α(t) decrease at the same constant rate as the growth in productivity:

α(t + 1) = (1 - ρ) α(t)

The rate at which the total supply of labor-hours increases is easily calculated:

α(t + 1) L_S(t + 1) = (1 - ρ) (1 + g) α(t) L_S(t)

Under these assumptions, the supply of labor-hours grows at the same rate as the demand for labor-hours. Total wages and total profits increase at the same rate, 100 g percent. The wage per worker increases at the same rate as the standard length of a labor year declines. Thus, workers consume a constant quantity of commodities, but they take increased productivity in steadily increased free time.

5.0 Discussion and Conclusions

What should one postulate about money in this model? One could assume the money supply grows endogenously, along with commodities. Or, perhaps, the velocity of the circulation of money increases with productivity. A continuous decrease in the money price of corn is another logical possibility. Perhaps Rosa Luxemburg was right, and an external source of demand from less developed regions and countries is needed to support expanded reproduction. Or Kalecki is correct, and military spending by the government will do.

I do not know if this model describes any existing capitalist economy. It does not describe the post-war golden age. In that time, at least in the United States, workers took increased productivity in increased consumer goods. (I think the memory of the Great Depression, the occurrence of World War II, and the existence of the Soviet Union has something to do how this worked out.) Could any capitalist economy function like this? Somehow, an advertising industry is not encouraging workers to consume ever more produced commodities, or they ignore such messages. They continually have more freedom. Yet, they always spend a bit of time under the domination and direction of their employers. Will the capitalists tolerate this?

Samir Amin (1931-2018)

Despite the label at the bottom of this post, this is not really a profile of Amin. I happen to have started reading Modern Imperialism, Monopoly Finance Capital, and Marx's Law of Value (Monthly Review Press, 2018) last month. Here are a couple of quotations:

"Vulgar economics is obsessed with the false concept of 'true prices,' whether for ordinary commodities, for labor, for money, for time, or for natural resources. There are no 'true prices' to be 'revealed' by the genius of the 'market.' Prices are the combined products of rates of exploitation of labor (rates of surplus-value), of competition among fragmented capitals, and the deduction levied in the form of 'oligopoly rents,' and of the political and social conditions that govern the division of surplus-value among profits, interest, ground rents, and extractive rents." -- Amin, p. 99.

"Marx's criticism of the classic bourgeois political economy of Smith and Ricardo concluded by shifting from analysis centered on 'the market' ... to one centered on the depths of production where value and the extraction of on surplus value are determined. Without this shifting of the analysis from the superficial to the essential, from the apparent to the concealed, no radical critique of capitalism is possible...

The law of value formulated by Marx, based on the concept of abstract labor, expresses the rationality of the social utility (the utility for society) of a defined use value. This rationality transcends that which governs the reproduction of a particular mode of production (in this case, the capitalist mode of production). Under capitalism, rationality demands the accumulation of capital, itself based on the extraction of surplus value. The price system frames the operation of this rationality. Economic decisions in this framework ... will be different from those that might be made on the basis of the law of value that would define, in the socialism to come, the mode of social governance over economic decision making.

Bourgeois economic theory attempts to prove that the mode of decision making in the framework of its system of prices and incomes produces a rational allocation of labor and capital resources synonymous with an optimal pattern of output. But it can reach that goal only through cascading tautological arguments. To do so it artificially slices productivity into 'components' attributed to 'factors of production.'

Although this pattern of slices has no scientific value and rests on tautological argument, it is 'useful' because it is the only way to legitimize capital's profits. The operative method of this bourgeois economics to determine 'the wage' by the marginal productivity of 'the last employee hired' stems from the same tautology and breaks up the unity of the collective, the sole creator of value. Moreover, contrary to the unproven affirmations of conventional economics, employers do not make decisions by using such 'marginal calculations.'" -- Amin, pp. 232-234.

I have several other books by Amin on my bookshelf:

• Samir Amin (2006). Samir Amin: A Life Looking Forward: Memoirs of an Independent Marxist. Zed Books.
• Samir Amin (1998). Spectres of Capitalism: A Critique of Current Intellectual Fashions. Monthly Review Press.
• Samir Amin (1997). Capitalism in the Age of Globalization: The Management of Contemporary Society. Zed Books.

As I understand Amin is most well known for inventing the word "Eurocentrism" and for extending the law of value to the law of worldwide value.

Amin builds on the concept of the "surplus", as developed in the work of Paul Baran and Paul Sweezy. One can formalize this notion in a model of a developed country with three departments, for producing capital goods, consumption goods, and luxuries. The last department is not in Marx's models of simple and expanded reproduction. This department is needed to address the problem of realization in an age of monopoly capital.

When it comes to realization problems, there is a long tradition among Marxists of looking at open economies, with advanced industrial capitalist economies trading with less developed peripheral regions or countries. Amin, an Egyptian trained in Paris and working in Dakar, was well positioned to develop these ideas of North-South trade. In the book mentioned above, he often talks about extending Marx's law of value to the law of worldwide value. I gather his ideas are partly the result of a critical engagement with Andre Gunder Frank's work, which I do not know.

To my mind, you can find similar ideas, about monopoly and finance capital and imperialism, going back to the time of the Second International. Amin mentions Rosa Luxembourg, but, as I recall, is critical of her. By the way, he groups Sraffa with bourgeois economists.

I was hoping to find Amin providing an exposition of a mathematical model in Modern Imperialism. He does provide some, but mostly he sticks with numerical examples and historical analysis. He says that this is, partly, to make his work accessible to a larger audience. Also, I am not sure that a mathematical model of the whole is appropriate for monopoly capital. I guess if I want to explore more, I should look at his 1974 book, Accumulation on a World Scale.

Economists In Popular Fiction

Apparently, a character in a current movie, Crazy Rich Asians is an economist. Dan Kopf considers whether she is a good economist. In a couple of recent tweets, Paul Krugman reacts:

"Actually, I can fill this gap.

"There was a movie titled The Internecine Project ... with James Coburn as a chairman of the Council of Economic Advisers who gets a bunch of people to kill each other to hide his evil past. Sounds good to me, but the movie was terrible." -- Paul Krugman, 9 August 2018

I do not know about the movie versions, but I can name a couple of book series with characters who are economists:

• Meyer is the sidekick in John D. MacDonald's Travis McGee mystery series. Meyer's houseboat is the John Maynard Keynes, until it is blown up. He replaces it with the Thorstein Veblen.
• The love interest in the Bourne Identity series is an economist. If I recall correctly, Jason Bourne first meets her by carjacking and kidnapping her, and then forcing her to drive with him to Paris.

I don't think you can count the Marshall Jevons' mystery series, since that is a pen name for two economists.

A Unique Natural Rate Of Interest?

1.0 Introduction

In explaining the policy implications of the Austrian Business Cycle Theory, Hayek argued that the central bank should try to keep the money rate of interest rate equal to the natural rate. Sraffa famously criticized Hayek by describing a model with multiple interest rates, not necessarily all equal. In reply, Hayek asserted that all the interest rates in Sraffa's example would be equilibrium rates. Sraffa had a rejoinder:

"The only meaning (if it be a meaning) I can attach to this is that his maxim of policy now requires that the money rate should be equal to all these divergent natural rates."

This interchange was part of the downfall of the Austrian theory of the business cycle. I thought I would try to shortly explain what is and is not compatible with a unique natural interest rate.

2.0 Multiple Interest Rates Compatible with a Unique Natural Interest Rate

When one talks about the interest rate or the rate of profits, one may be abstracting from all sorts of complications. And these complications may be consistent with multiple interest rates, in some sense. Yet these multiple interest rates were not in dispute between Hayek and Sraffa.

2.1 Interest Rates for Loans of Different Lengths

Suppose at the start of the year, one can obtain a one-year loan of money for an interest rate of 10%. At the same time, one can obtain a two-year loan for 21%. Implicit in these different rates is a prediction that a one-year loan will be available at the start of next year for an unchanged interest rate of 10%. This implication follows from some trivial arithmetic:

1 + 21/100 = (1 + 10/100)(1 + 10/100)

The yield curve generalizes these observations. A certain shape, with the interest rate increasing for longer loans is consistent with the interest rate being expected to be unchanged, for loans of a standard length, over time.

2.2 Interest Rates for Loans of Different Risks

One might also find interest rates being higher for loans deemed riskier, independently of the time period for which the loan is made. This variation is consistent with talk of the interest rate. Often, in finance, one sees something called the risk-free rate of interest defined and used for discounting income streams. In practice, the rate on a United States T-bill is taken as the risk-free rate.

2.3 Rate of Profits

One can also distinguish between finance and business income. One might refer to the interest rate for the former, and the rate of profits for the latter. Kaldor and others, in a dispute over a Cambridge non-marginal theory of the distribution of income, have described a steady state in which the interest rate is lower than the rate of profits. Households lend out finance to businesses and obtain the interest rate. Such a steady state is compatible with the existence of two classes of households. Capitalist households receive income only from their ownership of firms.

2.4 Rates of Profits Varying Among Industries

Steady states are also compatible with the rate of profits varying among industries, as long as relative profit rates are stable. Perhaps some industries require work in more uncomfortable circumstances. Or perhaps firms are able to maintain barriers to entry.

3.0 Interest Rates with Different Numeraires

I have shown above how money interest rates for loans of different lengths embody expectations of the future course of money interest rates. Interest rates need not be calculated in terms of money. They can be calculated for any numeraire. And the ratio of real interest rates embody expectations of how relative prices are expected to change.

As an example, suppose that at a given time t, both spot and forward markets exist for (specified grades of) wheat and steel. One pays out dollars immediately on both spot and forward markets. Consider the following prices:

• p_{W, t}: The spot price of a bushel wheat for immediate delivery.
• p_{S, t}: The spot price of a ton steel for immediate delivery.
• p_{W, t + 1}: The spot price of a bushel wheat for delivery at the end of a year.
• p_{S, t + 1}: The spot price of a ton steel for delivery at the end of a year.

The wheat-rate of interest is defined by:

(1 + r_W) = p_{W, t}/p_{W, t + 1}

I always like to check such equations by looking at dimensions. The units of the numerator on the right-hand side are dollars per spot bushels. The denominator is in terms of dollars per bushel a year hence. Dollars cancel out in taking the quotient. The wheat interest rate is quoted in terms of bushels a year hence per immediate bushels.

Suppose all real interest rates are equal. So one can form an equation like:

p_{W, t}/p_{W, t + 1} = p_{S, t}/p_{S, t + 1}

Or:

p_{W, t}/p_{S, t} = p_{W, t + 1}/p_{S, t + 1}

If spot prices a year hence were expected not to be in the ratio of current forward prices, one would expect to be able to make a pure economic profit by purchasing some goods now for future delivery. Hence, a no-arbitage condition allows one to calculated expected relative prices from quoted prices on complete spot and forward markets.

Anyways, a steady state requires constant ratios of spot prices and, thus, real interest rates to be independent of the numeraire. This is the condition Hayek imposed in his exposition of Austrian business cycle theory in Prices and Production. And this is the condition that he dropped in his argument with Sraffa, leaving his macroeconomics a confused mess.

I might as well note that a steady state is consistent with constant inflation. If all prices go up at, say, ten percent, relative spot prices do not vary. On the other hand, relative spot prices differ with the interest rate in comparisons across steady states.

4.0 Temporary Equilibrium with Consistent Plans and Expectations

Perhaps Hayek was willing to get himself into a muddle about the natural rate because he had already investigated another equilibrium concept in previous work.

Suppose above that real interest rates vary among commodities. Then forward prices show expected movements in spot prices. One might go further and assume a complete set of forward markets do not exist. Markets clear when each agent believes they can carry out their plans, consistent with their expectations, including of future spot prices. Should one call such market-clearing an equilbrium, even if agents plans and expectations are not mutually consistent?

Concepts of temporary, intertemporal, and sequential equilibrium were to become more important in mainstream economics after Hayek quit economics, more or less. John Hicks was a major developer of these ideas, under Hayek's influence at the London School of Economics. He eventually came to accept that the mainstream notions could not be set in historical time and were, at best, of limited help in understanding actual economies.

5.0 Conclusion

The above has outlined multiple ways in which multiple interest rates and multiple rates of profits are compatible with steady states. Nevertheless, such circumstances are often described by models in which one might talk about the rate of interest.

I have also described an equilibrium in which one cannot talk about the interest rate, whether natural or not. Advocates of Austrian business cycle theory have never clarified how it can be set in a temporary equilibrium. One can sometimes find Austrian fanboys asserting that critics do not appreciate distinctions between:

• Sources of exogenous shocks in central banks and supposed determinants (inter temporal preferences, technology) of the natural rate
• Money rates of interest and real rates
• Subjectivism and objectivism
• Interest rates and relative prices.

But assertions do not constitute an argument. One would have to do some work to show that these distinctions can serve to rehabilitate Austrian business cycle theory. No matter how much you send somebody chasing through the literature by Kirzner, Lachmann, Jesus Huerta de Solo, and Garrison, they will find the work has yet to be done. (Robert Murphy probably knows this.)

References

• Hahn, Frank. 1982. The neo-Ricardians. Cambridge Journal of Economics 6: 353-374.
• Hayek, F. A. 1932. Money and Capital: A Reply. Economic Journal 42: 237-249.
• Kaldor, Nicholas. 1966. Marginal Productivity and the Macro-Economic Theories of Distribution: Comment on Samuelson and Modigliani. Review of Economic Studies 33(4): 309-319.
• Sraffa, Piero. 1932. Dr. Hayek on Money and Capital. Economic Journal 42: 42-53.
• Sraffa, Piero. 1932. A Rejoinder. Economic Journal 42: 249-251.