Foreign Trade And Non-Uniform Rates Of Profits

This post raises a question. Supposedly, the classical concept of prices of production with non-uniform rates of profits can be recast as a theory of foreign trade. I do not see how wages can properly be treated in such recasting.

D'Agata (2018) and Zambelli (2018) are two recent papers that argue prices of production can be formulated with non-uniform rates of profits. They argue that this introduces a certain indeterminateness into prices, as in some of my examples of foreign trade. Both D'Agata and Zambelli cite Adam Smith and David Ricardo to justify their models as of classical inspiration. If somebody is to draw on this research for a theory of foreign trade, I hope they cite this passage from Adam Smith:

… every individual … endeavors as much as he can both to employ his capital in the support of domestic industry, and so to direct that its produce may be of the greatest value; every individual necessarily labours to render the of the society as great as he can. He generally, indeed, intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, he only intends his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in so many other cases, led by an invisible hand to promote an end which was no part of his intention.

I suspect many propertarians are not aware that Smith was arguing that a lack of foreign direct investment is desirable, that enough barriers exist against entrepreneurs investing in other countries that no need exist for certain protectionist laws to be passed by government.

D'Agata models non-uniform rates of profits as arising due to both "objective and idiosyncratic factors affecting producers' investment decisions". Objective factors are modeled by different groups of producers having access to different techniques of production for producing the same commodities. For example, firms in England and Portugal might have access to different techniques for producing corn and wine, as in Ricardo's Principles. I guess countries having different endowments of land and labor, thereby limiting the scale at which some processes can be operated, is also an objective factor important to the theory of foreign trade.

Idiosyncratic factors are formalized by different producers having different valuation functions, where a valuation function is a continuous, strictly increasing function of the rate of profits obtained in a given industry. In terms of the theory of foreign trade, one might model entrepreneurs in England all having identical valuation functions, while entrepreneurs in Portugal have another valuation function, common among the Portuguese. Each valuation function might be assumed not to vary among industries. For example, English entrepreneurs value the rate of profits made in making corn the same as the rate of profits made in making wine.

From these considerations, one can obtain a theory of foreign trade in which:

• Countries differ among themselves in the technology or endowments they have access to.
• In a full employment position with balanced trade, countries specialize in the production of different commodities.
• In such an equilibrium position, the rate of profits varies among countries.

(I do not claim such a theory is complete, since it does not consider Keynesian effective demand, paths with unbalanced trade, fluctuations in exchange rates, and so on.)

When I have tried to develop such a theory of foreign trade, I have created examples in which the wage also varies across countries. This is easy to justify based on an assumption of a lack of a free movement of people across national borders. But how is this idea formalized in D'Agata's approach?

References

• Antonio D'Agata, 2018. Freeing long-period prices from the uniform profit rate hypothesis: A general model of long-period positions. Metroeconomica 69: 847-861.
• Stefano Zambelli, 2018. Production of commodities by means of commodities and uniform rates of profits. Metroeconomica 69: 791-819.

Robert Visits An American Grave: Frederick Douglass

Frederick Douglass was an escaped slave, a great abolitionist orator, and generally a great American. Not too long ago, I read one of his autobiographies. Of his speeches, I am most likely to recognize bits from his 1952 observations on independence day. (Eldridge Cleaver quotes it in Soul on Ice.) This part is fierce:

What, to the American slave, is your 4th of July? I answer: a day that reveals to him, more than all other days in the year, the gross injustice and cruelty to which he is the constant victim. To him, your celebration is a sham; your boasted liberty, an unholy license; your national greatness, swelling vanity; your sounds of rejoicing are empty and heartless; your denunciations of tyrants, brass fronted impudence; your shouts of liberty and equality, hollow mockery; your prayers and hymns, your sermons and thanksgivings, with all your religious parade, and solemnity, are, to him, mere bombast, fraud, deception, impiety, and hypocrisy — a thin veil to cover up crimes which would disgrace a nation of savages. There is not a nation on the earth guilty of practices, more shocking and bloody, than are the people of these United States, at this very hour.

Frederick Douglass is buried in Mount Hope Cemetery in Rochester, NY. This is a family plot, with his widow at his left.

Variation Of Gains From Trade With International Prices

Figure 1: Intercepts of Production Possibilities Frontiers for England

1.0 Introduction

In this example, gains and losses from trade vary with international prices. Given rates of profits are compatible with an interval of relative international prices for linen and corn, when trade exists only in consumer goods. I explore whether, when trade exists in capital and consumer goods, more than one pattern of specialization among countries is possible, depending on relative international prices. I am beginning to think that specialization, in this model, in corn and linen is infeasible, except in knife-edge cases.

The theory of comparative advantage provides no valid justification for the abolition or the lowering of tariffs. Unregulated international trade is not about efficient use of an international allocation of resources. Many existing textbooks, including Krugman and Obstfeld's, should be ripped up, and the authors should start again.

2.0 Technology, Endowments, And The Rate Of Profits

I assume each of two countries (Tables 1 and 2) have a fixed-coefficients technology for producing three commodities. The technology varies between countries, although it has the same structure in both. Steel is the only capital good. Each commodity can be produced, in a year, from inputs of labor and steel. A coefficient of production shows the quantity of an input needed per unit output. For example, in England, one person-year and 1/30 tons of steel must be purchased per square meter of produced linen. Steel is totally used up in production, and constant returns to scale obtains.

Table 1: Coefficients of Production in England

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(E) = 1	a0, 2(E) = 8	a0, 3(E) = 12
Steel	a1, 1(E) = 1/5	a1, 2(E) = 1	a1, 3(E) = 1

Table 2: Coefficients of Production in Portugal

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(P) = 6/5	a0, 2(P) = 12	a0, 3(P) = 20
Steel	a1, 1(P) = 1/4	a1, 2(P) = 2	a1, 3(P) = 3/2

I take endowments of labor as given, as in the Ricardian model of foreign trade. Let England and Portugal both have available a labor force consisting of one person-year. So Production Possibilities Frontiers (PPFs) are found per person-year. By assumption, workers neither immigrate nor emigrate. In this model, full employment is assumed.

I also take the rate of profits as given, at 100 per cent in England and at 20 percent in Portugal. I assume that financial capital cannot flow between countries. So the rate of profits need not be the same across countries.

3.0 Summary

I apply my usual analysis to determine patterns of specialization, given technology, endowments, and rates of profits in each country. When foreign trade is possible in corn and linen, but not steel, the domestic price of steel and the wage in each country must be such that the going rate of profit is earned in producing steel. Likewise, firms in, say, England make neither extra profits nor incur extra costs in producing the consumer good in which England specializes. The firms would incur extra costs if they were to produce the other consumer good. The same principles extend to the case in which foreign trade is possible in all produced commodities.

In this analysis, which is an example of a small country model, prices for goods bought or sold in foreign trade are taken as given by firms in all countries. I find prices and specializations which are consistent with the given parameters. One can draw Production Possibility Frontiers (PPFs) for each country, given prices in foreign markets and specializations. A PPF shows possible baskets of consumer goods when labor is fully employed. In this model, each PPF is a decreasing function in the first sector of the two-dimensional space formed by quantities of corn and linen. Such a PPF is fully specified by the intercepts. The intercept with the corn axis is maximum amount of corn that can be consumer, per employed worker, given that no linen is consumed. Similarly, the intercept with the linen axis is the maximum amount of linen that can be consumed. Figure 1, above, and Figure 2, show the intercepts for the PPFs for England and Portugal, respectively.

Figure 2: Intercepts of PPFs for Portugal

In the example:

• When foreign markets exist only for corn and linen:
• England specializes in the production of linen (and steel), while Portugal specializes in corn (and steel).
• England suffers a loss from trade, except when the international relative price of linen is at its highest feasible level.
• Portugal obtains a gain from trade.
• England’s loss and Portugal’s gain is smaller for larger relative prices of linen on international markets.
• When foreign markets exist for steel, corn, and linen:
• For a relatively small ratio of the international price of linen to the international price of corn, England specializes in corn and linen, and Portugal specializes in steel.
• In this range, prices compatible with England specializing in linen and Portugal specializing in steel and corn provide England with extra profits in producing corn.
• This case is infeasible. England only obtains steel by trading corn for it. England is unwilling to trade linen for steel, and Portugal is unable to acquire linen by selling steel.
• For a relatively large ratio of the international price of linen to the international price of corn, England specializes in linen, and Portugal specializes in steel and corn.
• In this range, prices compatible with England specializing in corn and linen and Portugal specializing in steel provide Portugal with extra profits in producing corn.
• England obtains a gain from trade, as compared to when foreign trade is only possible in consumer goods
• For a low price of linen in this range and a consumer basket heavily weighted to corn, England suffers a loss from trade, as compared to autarky.
• Otherwise, England obtains a gain from trade, as compared to autarky.
• Portugal’s PPF is identical to what it would be if foreign trade were possible only in consumer goods.
• Accordingly, Portugal obtains a gain from trade, as compared to autarky.

Gain or Loss from Trade with Multiple Equilibria

Figure 1: Production Possibility Frontiers

1.0 Introduction

Suppose foreign trade is possible in consumption goods, but not in capital goods. In this example, whether or not England achieves gains from trade depends on relative international prices. If foreign trade were possible in both consumption and capital goods, both England and Portugal would obtain gains from trade. The numeric example in this post is a modification of one in a previous post.

As I understand it, most students of economics are taught this numeric example cannot exist. And it raises questions on, for example, tariffs and the distribution of income that you will be hard-pressed to find discussed.

2.0 Technology, Endowments, And The Rate Of Profits

I assume each of two countries (Tables 1 and 2) have a fixed-coefficients technology for producing three commodities. The technology varies between countries, although it has the same structure in both. Steel is the only capital good. Each commodity can be produced, in a year, from inputs of labor and steel. A coefficient of production shows the quantity of an input needed per unit output. For example, in England, one person-year and 1/30 tons of steel must be purchased per square meter of produced linen. Steel is totally used up in production, and constant returns to scale obtains.

Table 1: Coefficients of Production in England

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(E) = 2	a0, 2(E) = 3	a0, 3(E) = 1
Steel	a1, 1(E) = 1/20	a1, 2(E) = 1	a1, 3(E) = 1/30

Table 2: Coefficients of Production in Portugal

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(P) = 2	a0, 2(P) = 7	a0, 3(P) = 2
Steel	a1, 1(P) = 1/40	a1, 2(P) = 1	a1, 3(P) = 1/100

I take endowments of labor as given, as in the Ricardian model of foreign trade. Let England and Portugal both have available a labor force consisting of one person-year. So Production Possibilities Frontiers (PPFs) are found per person-year. By assumption, workers neither immigrate nor emigrate. In this model, full employment is assumed.

I also take the rate of profits as given, at 25 per cent, in both countries. I originally intended to assume that financial capital cannot flow between countries. So the rate of profits need not be the same across countries.

3.0 One of Two Equilibria

One can analyze each country under autarky, that is, under the assumption that foreign trade is not possible. One can find, given the rate of profits in each country, relative prices of corn and linen in each country. Suppose foreign trade is possible in corn and linen, but not in steel. And suppose the ratio of the international price of linen to the international price of corn is between the corresponding ratio of autarkic prices in England and Portugal. (I have chosen the rates of profits so this ratio is lower in England than in Portugal under autarky.) Then the English specialize in producing linen, and the Portuguese specialize in producing corn. I consider international prices at the two extreme ends of this range. This section presents the first extreme.

3.1 Trade in Corn and Linen

Table 3 present prices and costs when trade is only possible in corn and linen. I follow the notation in a previous post. The rows show the international price of corn, the international price of linen, wages in each country, the domestic price of steel, the cost of producing corn, and the cost of producing linen. If anybody wants to work it out, wages and the price of steel are such that the given rate of profits is made in producing steel in each country.

Table 3: Trade in Consumer Goods

Variable	England	Portugal
P2	$15 per Bushel
P3	$49/17 per Sq. Meter
w(n)	$45/17 Person-Yr.	$155/99 per Person-Yr.
p1(n)	$96/17 per Ton	$320/99 per Ton
p1(n)a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$15 per Bushel	$15 per Bushel
p1(n)a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$49/17 per Sq. Meter	$314/99 per Sq. Meter

Firms in a country will only produce a commodity if its cost of production does not exceed its price. With the prices in the above table, the English are willing to produce both corn and linen, while the Portuguese produce only corn. I want to ignore that the English might want to produce corn. If the price of linen on international markets was just an infinitesimal higher, the English would not be willing to produce corn.

The upper half of the figure at the top of this post illustrates this case. When, at these prices, England specializes in linen, they obtain a loss from trade. Portugal obtains gains from trade throughout.

3.2 Trade in Steel, Corn, and Linen

I now consider this case with foreign trade in steel also. Table 4 shows prices and costs. The first row is for the price of steel on international markets. I also introduce a row for the cost of producing steel. With the same logic as above, I ignore that England can produce steel, as well as corn and linen, with these prices. I take the international prices of corn and linen as unchanged from the previous subsection.

Table 4: Trade in Capital and Consumer Goods

Variable	England	Portugal
P1	$96/17 per Bushel
P2	$15 per Bushel
P3	$49/17 per Sq. Meter
w(n)	$45/17 per Person-Yr.	$93/34 per Person-Yr.
P1 a1,1(n)(1 + r(n)) + a0, 1(n) w(n)	$96/17 per Ton	$96/17 per Ton
P1a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$15 per Bushel	$891/34 per Bushel
P1a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$49/17 per Sq. Meter	$471/85 per Sq. Meter

In this case, both England and Portugal gain from trade. England specializes in corn and linen, and Portugal specializes in steel. The possible consumption baskets for both England and Portugal, under trade in all commodities, is also shown in the upper half of the figure at the top of this page. Even if you click through, it is hard to see that the maximum amount of linen that can be consumed in England is strictly greater than autarky in this case. Samuelson calls the additional gains from trade obtained through foreign trade in capital goods as the "Sraffian bonus". I have previously shown that the Sraffian bonus can be negative.

4.0 A Second Equilibrium

Now suppose the international price of linen is at the opposite extreme, with the same specializations. Again, this is the endpoint of what should be an open interval.

4.1 Trade in Corn and Linen

Table 5 shows prices and costs when foreign trade is possible only in consumer goods. English firms make the going rate of profits in producing steel and linen, but would incur extra costs if they produced corn domestically. Portuguese firms make the going rate of profits in producing any of steel, corn, and linen. But I treat them here as specializing in producing corn for foreign trade and obtaining linen only through foreign trade.

Table 5: Trade in Consumer Goods

Variable	England	Portugal
P2	$15 per Bushel
P3	$314/99 per Sq. Meter
w(n)	$4710/1617 Person-Yr.	$155/99 per Person-Yr.
p1(n)	$10048/1617 per Ton	$320/99 per Ton
p1(n)a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$26690/1617 per Bushel	$15 per Bushel
p1(n)a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$314/99 per Sq. Meter	$314/99 per Sq. Meter

The bottom half of the figure above shows Production Possibility Frontiers for this case. Both England and Portugal obtain gains from trade. (The PPF for England, under trade in consumption goods, is not easy to visually distinguish from the PPF under autarky.) A given technology and given rates of profits is compatible with a country both obtaining gains and suffering losses from foreign trade in consumption goods, depending on international prices.

4.2 Trade in Steel, Corn, and Linen

International prices of corn and linen are the same in Table 6 below and Table 5 above. Table 6 is drawn up for the possibility of foreign trade in steel, corn, and linen. England specializes in corn and linen, and Portugal specializes in steel. As seen in the bottom half of the figure at the top of this post, both England and Portugal have gains in trade, as compared to autarky and to foreign trade in consumer goods, when trade is possible in all produced commodities.

Table 6: Trade in Capital and Consumer Goods

Variable	England	Portugal
P1	$1448/297 per Bushel
P2	$15 per Bushel
P3	$314/99 per Sq. Meter
w(n)	$2645/891 per Person-Yr.	$5611/2376 per Person-Yr.
P1 a1,1(n)(1 + r(n)) + a0, 1(n) w(n)	$11123/1782 per Ton	$1448/297 per Ton
P1a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$15 per Bushel	$181/8 per Bushel
P1a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$314/99 per Sq. Meter	$28417/5940 per Sq. Meter

5.0 Conclusion

In this example, only one process is known in each country for producing each commodity domestically. The possibility of foreign trade creates a choice of technique. I wonder if more processes existed for each country's technology, would the range of international prices for consumer goods consistent with certain national specializations be narrowed? Would the introduction of consumer demand in the model remove the indeterminism? I suppose, for exploring the last question, I should see what has been done with J. S. Mill's approach to analyzing foreign trade.

Elsewhere

• Matthew Klein writes, in Barron's, about "Tarrifs and the Minimum Wage Are More Alike Than You Think". I disagree with some of the stuff in the middle about efficiency and reject the dualistic notion that government intervention is a meaningful concept. But this article otherwise parallels some of my arguments here.
• Josh Mason has made available his piece in Jacobin about the state of economics after the global financial catastrophe.
• The Review of Political Economy has made available Pierangelo Garegnani's posthumous On the Labour Theory of Value in Marx and in the Marxist Tradition. I have yet to read Fabio Petri's introduction. Some points from Garegnani's article:
• Chapter 1 of volume 1 of Capital is not meant to be a proof of the Labor Theory of Value (LTV).
• The LTV fills an instrumental role in providing a calculation of the rate of profits prior to the system of prices of production.
• Much of volume 1 remains valid, even after correcting the mathematical theory. For a given technology, there is a trade-off between wages and the rate of profits. Capitalists try to increase relative and absolute surplus value.
• Marx's account of profits as the result of the exploitation of workers is descriptive, not a moral or ethical judgement.
• Rudolf Hilferding did not have the mathematical machinery (e.g., theorems on the principal Eigenvalue of a matrix) to counter Eugen Böhm von Bawerk's criticism of Marx. Consequently, his attempt is misdirected.

Gains And Losses From Foreign Trade: A Numeric Example

Figure 1: Production Possibility Frontiers

1.0 Introduction

This post presents a numeric example of foreign trade in a model of the production of commodities by means of commodities. This is a modification of the model here, which considers a flow-input, point output technology. As usual, I show neoclassical economics is mistaken. Frictions, increasing returns, information asymmetries, principal agent problems, and so on do not need to be introduced to explain why the outcomes of free markets are not always ideal. Even under ideal conditions, problems can arise.

2.0 Technology, Endowments, And The Rate Of Profits

I assume each of two countries (Tables 1 and 2) have a fixed-coefficients technology for producing three commodities. The technology varies between countries, although it has the same structure in both. Steel is the only capital good. Each commodity can be produced, in a year, from inputs of labor and steel. A coefficient of production shows the quantity of an input needed per unit output. For example, in England, one person-year and 1/30 tons of steel must be purchased per square meter of produced linen. Steel is totally used up in production, and constant returns to scale obtains.

Table 1: Coefficients of Production in England

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(E) = 2	a0, 2(E) = 3	a0, 3(E) = 1
Steel	a1, 1(E) = 1/20	a1, 2(E) = 1	a1, 3(E) = 1/30

Table 2: Coefficients of Production in Portugal

Inputs	Industry
	Steel	Corn	Linen
Labor	a0, 1(P) = 2	a0, 2(P) = 7	a0, 3(P) = 2
Steel	a1, 1(P) = 1/40	a1, 2(P) = 1	a1, 3(P) = 1/100

I take endowments of labor as given, as in the Ricardian model of foreign trade. Let England and Portugal both have available a labor force consisting of one person-year. So Production Possibilities Frontiers (PPFs) are found per person-year. By assumption, workers neither immigrate nor emigrate. In this model, full employment is assumed.

I also take the rate of profits as given, at 300 per cent, in both countries. I originally intended to assume that financial capital cannot flow between countries. So the rate of profits need not be the same across countries. (If you find the rate of profits unacceptably high, read "year" as "decade" throughout this post.)

3.0 Aspects of Autarky

Suppose all three commodities are each produced in each country. Foreign trade is not possible. The technology allows one to calculate the labor embodied in each commodity. For steel, the number of person-years embodied in each ton of steel is:

v₁(n) = a_{0, 1}(n)/(1 - a_{1, 1}(n))

The labor embodied in corn and linen is:

v_j(n) = a_{0, 1}(n) a_{1, j}(n)/(1 - a_{1, 1}(n)) + a_{0, j}(n), j = 2, 3.

Labor values are useful in drawing the PPF for each country, under autarky. Consumers in England can consume 1/v₂(E) bushels of corn per person-year of labor hired, if they consume no linen. Or they can consume 1/v₃(E) square meters per person-year, with no corn. Any linear combination of these two consumption baskets, with positive quantities of both corn and linen, can also be consumed.

Each PPF embodies a rate of transformation between corn and linen, in a comparison of stationary states. For example, in England 61 bushels of corn can be traded off, in some sense, for 291 square meters of line. England has a comparative advantage in linen, as compared to corn, at a rate of profits of zero.

v₃(E)/v₂(E) < v₃(P)/v₂(P)

Portugal has a comparative advantage in steel, as compared to both corn and linen, at a rate of profits of zero.

4.0 Trade in Corn and Linen

In this section, I assume that foreign trade is possible in the consumer commodities, corn and linen. But international markets do not exist in the capital good, steel. The introducition of the possibility of foreign trade creates a choice of technique.

In the small country model, firms take prices, including on international markets, as given. I introduce the following notation:

• P₂: The price of a bushel corn on international markets.
• P₃: The price of a square meter of linen on international markets.
• w(n), n = E, P: The wage.
• r(n), n = E, P: The rate of profits.
• p₁(n), n = E, P: The domestic price of steel.
• p₁(n)a_1,2(n)(1 + r(n)) + a_{0, 2}(n) w(n), n = E, P: The cost of producing a bushel corn domestically.
• p₁(n)a_1,3(n)(1 + r(n)) + a_{0, 3}(n) w(n), n = E, P: The cost of producing a square meter of linen domestically.

Table 3 shows the value of each of these variables. Firms will not produce commodities when their cost of producing it exceeds what it can be purchased for on international markets. Accordingly, England specializes in producing linen and the necessary steel at these prices. Portugal produces corn and linen. The domestic price of steel and wages are such that firms cannot make extra profits in steel, corn, or iron production.

Table 3: Trade in Consumer Goods

Variable	England	Portugal
P2	$6 per Bushel
P3	$2/3 per Sq. Meter
w(n)	$19/32 Person-Yr.	$117/286 per Person-Yr.
p1(n)	$35/64 per Ton	$69/88 per Ton
p1(n)a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$127/32 per Bushel	$6 per Bushel
p1(n)a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$2/3 per Sq. Meter	$1869/2200 per Sq. Meter

The ratio of the prices of linen and corn are a key variable here. Countries specialize in the consumer commodity in which relative international prices exceeds the domestic relative price, as calculated under autarky. Since the rate of profits is positive, relative domestic prices differ from the slope of the autarkic PPF. That is, the relative autarky price is what determines comparative advantage, in some sense. But the slope of the autarkic PPF is important in analyzing whether gains from trade are positive or negative.

The possibility of foreign trade in consumer goods has made consumers in England worse off, in a comparison of stationary states. The English PPF is rotated inwards, when firms specialize as induced by these prices. Consumers in Portugal, on the other hand, are better off. Their PPF is rotated outwards.

4.0 Trade in Steel, Corn, and Linen

I now assume trade is possible in all goods, including the capital good. Let P₁ be the price of steel on international markets. Cost of domestic production are modified in the obvious way in Table 4.

Table 4: Trade in Capital and Consumer Goods

Variable	England	Portugal
P1	$10/9 per Bushel
P2	$6 per Bushel
P3	$2/3 per Sq. Meter
w(n)	$14/27 Person-Yr.	$1/2 per Person-Yr.
P1 a1,1(n)(1 + r(n)) + a0, 1(n) w(n)	$34/27 per Ton	$10/9 per Ton
P1a1,2(n)(1 + r(n)) + a0, 2(n) w(n)	$6 per Bushel	$143/18 per Bushel
P1a1,3(n)(1 + r(n)) + a0, 3(n) w(n)	$2/3 per Sq. Meter	$47/45 per Sq. Meter

England specializes in the production of corn and linen, and Portugal specializes in the production of steel. Firms in England obtain the steel they need to continue production by trading corn and linen in foreign trade. Likewise, consumers in Portugal obtain corn and linen from firms selling the surplus steel product in foreign trade. No firm incurs extra costs or obtains extra profits in any process which they operate. And operated process would incur extra costs.

The opening up of foreign markets in steel has made England better off, both in comparison to autarky and in comparison with foreign trade only being possible in consumer goods. (Although it is difficult to see in Figure 1, the intercept of the PPF, for trade in all commodities, with the ordinate strictly exceeds the intercept for the other PPFs). The opening up of foreign trade in steel has made Portugal worse off, as compared to trade only in consumer goods. Whether the Portuguese are better off as compared to autarky is ambiguous. It depends on the consumption basket.

5.0 Conclusion

Why, oh why, do mainstream economists teach untruths about the theory of trade?

Pattern Analysis Applied to Structural Economic Dynamics with a Choice of Technique: A Numerical Example

I have made a working paper with the above title available on SSRN.

Abstract: This article illustrates the application of pattern analysis to structural economic dynamics with a choice of technique. A numerical example is presented in which technical progress is introduced. Examples of temporal paths through the parameter space illustrate variations of the wage frontier. A single technique is initially uniquely cost-minimizing for all feasible rates of profits. Eventually, the technique for which coefficients of production decrease at the fastest rate is always cost-minimizing. This example illustrates possible variations in the existence of Sraffa effects, which arise during the transition between these positions.

Structural Economic Dynamics, Markups, Real Wicksell Effects, and the Reverse Substitution of Labor

I have made available a working paper with the post title.

Abstract: This article presents an example in which perturbations in relative markups and technical progress result in variations in characteristics of the labor market. Around a switch point with a positive real Wicksell effect, a higher wage is associated with firms wanting to employ more labor per unit output of net product. Around a switch point with a reverse substitution of labor, firms in a particular industry want to hire more labor per unit output of gross product. Technical progress and variations in markups can bring about and take away circumstances favorable for workers wanting to press claims for higher wages.

On Mariana Mazzucato's The Value of Everything

Mariana Mazzucato's The Value of Everything: Making and Taking in the Global Economy is a popular book. She argues that we should change our ideas of what we consider productive and unproductive jobs and activities. She presents a brief overview of the history of economics focused on the classical and neoclassical theories of value, describes how Gross Domestic Product (GDP) is calculated from the System of National Accounts (SNA), examines how finance came to be mistakenly considered productive of value added in the SNA, and argues that government activities have been mistakenly considered as necessarily unproductive.

Value theory is a good way of organizing a popular history of economics, although it leaves out German historical schools and institutionalism. Mazzucato praises Petty for an early attempt at drawing up national accounts. She says he did not have a theory of value. (From secondary literature, I am under the impression he talked about labor being the father and land the mother of value.) She says Quesnay's Tableau is the "first spreadsheet". She has Adam Smith putting forth a labor theory of value. She notes that he had two theories of productive and unproductive labor: work that provides a physical product versus work for out of capital. (This is straight out of Marx's Theories of Surplus Value.) With Smith, Ricardo, Marx, she doesn't get into details of labor embodied, labor commanded, the transformation problem, and so on. She distinguishes between objective versus subjective value theories. Neoclassical theory is subjective. In some sense, it gets rid of value theory for a theory of price. Producing anything that people are willing to pay for becomes productive.

I wished she had included an illustrative table when she turns to national accounts. She notes that they are drawn up with many conventions, not all of which can be derived from neoclassical theory. She popularizes various objections to how the Gross Domestic Product (GDP) is calculated and used. Along with feminist economists, she notes that household production is not counted. Pollution and cleaning up are more productive of GDP than not polluting in the first place. She notes that certain government activities appear in GDP as productive, even though is not clear from neoclassical theory that (some?) government activity can be productive.

Mazzucato has lots to say about finance. In classical thought, finance would probably considered unproductive, with returns to finance akin to rents. But that is not currently the case. She mentions the idea of privatized Keynesianism, in which increased borrowing by those not well off maintains effective demand in an era in which ideology promotes austerity for government. She looks at how Milton Friedman and Michael Jensen promoted the maximizing of shareholder value, disregarding other stakeholders in corporate management.

Mazzucato argues a need exists to transition from looking at government as a provider of public goods to looking at how government can be productive of value. Health care and information technology are two industries that provide examples of this perspective. In both cases, government funds basic research, with private industries taking profits, including through patents and copyrights. In health care, the idea of Quality-adjusted Life Years (QLYs) is used to justify prices with extreme markups, that cover much more than private Research and Development costs. How many QLYs would be sacrificed if a certain drug was not there? In the view of institutionalism, going back to Veblen, technological innovation combines the activities of many, although the opposite view is embodied in current intellectual property law. From Mazzucato's view, government has been and can be productive.

My initial reaction to the overall thesis of this book was that it is too idealistic. The ruling ideas are the ideas of the ruling classes. Changing those ideas requires changing the material base. I am not sure I fully get her idea that how we draw the boundary between activities and jobs that are productive and unproductive is performative. Mazzucato gives examples of finance, of other activities that would be formerly classified as unproductive grabbing of rents, and of government activities. I came to agree that there is a need for the development and a public discussion of a conceptual frame of who produces what.

I think of this book as a popularization of certain practical and policy ideas drawing on heterodox economics. Mazzucato does not make a point of discussing or drawing boundaries between heterodox and orthodox, non-mainstream and mainstream economics. She draws on, say, Duncan Foley or Joseph Stiglitz when each is useful for her points. I like that this shows that heterodox economists are economists. She could have been more explicit, perhaps with loss of rhetorical efficacy, on when she draws on Marx.

Overall, I recommend this book. I wonder what Dean Baker or Robert Reich would make of this book. Anwar Shaikh and Ahmet Tonak would also have a reaction.

More Pattern Analysis For An Example With Fixed Capital

Figure 1: A Two-Dimensional Pattern Diagram

1.0 Introduction

This post continues this example of the application of pattern analysis to the study of fixed capital. I generalize the technology in the previous post. Technical progress can now proceed at different rates in the production and use of machines. I partition the resulting two-dimensional parameter space based on how the distribution of income, in the system of prices of production, alters the economically efficient length to run the machine. And I find an example in this parameter space that conforms to outdated neoclassical intuition on such matters.

2.0 Technology

Table 1 represents the technology for this example. Machines and corn are produced in this economy. Corn is the only consumption good. New machines are produced from inputs of labor and corn. Corn is produced from inputs of labor, corn, and machines. A machine can be worked for two years. After the end of the first year of its working life, it is known as an old machine. I assume each process requires a year to complete and exhibits constant returns to scale.

Table 1: Coefficients of Production

Inputs	Industry
	Machine	Corn
Labor	a0,1 = (1/10) u(t)	a0,2 = (43/40) v(t)	a0,3 = v(t)
Corn	a1,1 = (1/16) u(t)	a1,2 = (1/16) v(t)	a1,3 = (1/4) v(t)
New Machines	0	1	0
Old Machines	0	0	1
Outputs
Corn	0	1	1
New Machines	1	0	0
Old Machines	0	1	0

I model technical progress by constantly decreasing inputs into each process, other than machines:

u(t) = e^{1 - σ t}

v(t) = e^{1 - φ t}

Given values of σ t and φ t, technology is specified.

3.0 Choice of Technique

For specified parameters, including σ t and φ t, a system of equations and inequalities is specified such that:

• The rate of profits is determined, given a non-negative wage not exceeding a certain maximum.
• For such a given wage, it is determined whether or not running the machine for one or two years is cost-minimizing.

Which technique - running the machine for one or two years - is cost-minizimizing can vary with mathematical variations in the wage.

Figure 1 partitions the parameter space such that the characteristics of the variations in the choice of technique technique do not vary within each region. Each region is bounded by thick (non-dotted) lines. I do not show a fifth region to the right of the graphed portion of the parameter space. Somewhere before a value of σ t of three, the locus labeled "Pattern over wage axis" curves down to cross the locus labeled "Pattern over axis for rate of profits". In Region 5, the latter locus lies above the former. Table 1 briefly describes the cost-minimizing technique in each region.

Table 1: Coefficients of Production

Region	Description
1	Machine is run for 1 year, for all feasible wages.
2	Machine is run for 2 years, for low wages; 1 year, for high wages.
3	Reswitching. Machine is run for 2 years, for low and high wages; 1 year for intermediate wages.
4	Machine is run for 2 years, for all feasible wages.
5	Machine is run for 1 year, for low wages; 2 years, for high wages.

4.0 Temporal Paths

Suppose that technical progress in producing and using machines is steady. That is, σ and φ have some fixed values. Each of the two dotted lines in Figure 1 illustrate a path in logical time for such a thought experiment. The 45 degree line corresponds to the case in which σ and φ are equal. The choice of technique along such a path was illustrated in the previous post.

I constructed the dashed line with the smaller slope to pass through Region 5. Technical progress is faster, in this case, in producing machines than in using machines. Figure 2 illustrates this case, which complies with outdated neoclassical intuition. Notice that around the switch point in Region 5, a higher wage is associated with a choice, by managers of firms, to run the machine for a second year. The wage acts like a scarcity index for labor, and the lengths at which firms run machines responds appropriately for a measure of capital intensity. But the example proves that there is no logical necessity for economically efficient decisions to work out like this.

Figure 2: A Pattern Diagram

5.0 Conclusion

This post suggests the tools that I have been developing for post-Sraffian price theory apply, without modification, to models of fixed capital.

Pattern Analysis for a Fixed Capital Example

Figure 1: A Pattern Diagram

1.0 Introduction

In this example, I perturb parameters in an example of Bertram Schefold's. I was disappointed in that, as far as I can see, one can analyze the choice of technique in this example by the construction of the wage-rate of profits frontier. As far as I understand, this is not true for joint production in general. I guess I also need to find an example in which the physical life of a machine is at least three years so as to find a three-technique pattern.

This example does highlight differences in different measures of capital-intensity.

2.0 Technology

Table 1 presents the technology for this example. Machines and corn are produced in this economy. Corn is the only consumption good. New machines are produced from inputs of labor and corn. Corn is produced from inputs of labor, corn, and machines. A machine can be worked for two years. After the end of the first year of its working life, it is known as an old machine. I assume each process requires a year to complete and exhibits constant returns to scale.

Table 1: Coefficients of Production

Inputs	Industry
	Machine	Corn
Labor	a0,1 = (1/10) u(t)	a0,2 = (43/40) u(t)	a0,3 = u(t)
Corn	a1,1 = (1/16) u(t)	a1,2 = (1/16) u(t)	a1,3 = (1/4) u(t)
New Machines	0	1	0
Old Machines	0	0	1
Outputs
Corn	0	1	1
New Machines	1	0	0
Old Machines	0	1	0

I model technical progress by constantly decreasing inputs into each process, other than machines:

u(t) = e^{1 - σ t}

When σ t is unity, this is Bertram Schefold's example of reswitching, at rates of profits of 1/3 and 1/2.

3.0 Prices of Production

The first row in Table 1 can be summarized by a row vector, a₀, of labor coefficients. The next three rows are expressed by a square matrix A. The last three rows form the matrix B. Suppose wages are paid out of the surplus product at the end of the year. If the same rate of profits is to be made in all operating processes, prices must satisfy the following system of equations;

p A (1 + r) + w a₀ = p B

I let corn be the numerator:

p e₁ = 1

where e₁ is the first column of the identity matrix.

Given the wage, w, in a range between zero and some maximum, the above system of price equations can be solved for the rate of profits, r, the price of a new machine, p₂, and the price of an old machine, p₃.

4.0 Choice of Technique

The managers of firms need not run the machine for two years. They could discard the machine after only one year. (I assume free disposal.) The managers will be cost-minimizing if they run the machine for only one year if the price of an old machine is negative.

Alternatively, consider the price system when the machine is operated only two years. The matrices A and B are 2x2 square matrices, and a₀ is a row vector with two elements. With these prices and the price of an old machine of zero, one could calculate the cost of operating the machine for a second year to produce a bushel of corn. When this cost is less than unity (the price of a bushel of corn), it is cost-minimizing to operate the machine for both years.

These two methods of analyzing the choice of technique yield the same answer for this example. Figure 1, above, illustrates the results. Until time reaches the pattern over the axis for the rate of profits, it is cost-minimizing to operate the machine for only one year. In Region 2, the machine is operated for two years when wages are low, and for one year when wages are higher. Region 3 is an example of reswitching. Eventually, it is cost-minimizing to operate the machine for two years, for all feasible wages.

5.0 Capital

In outdated neoclassical intuition, a higher wage indicates that labor is more scarce, in some sense, and capital is relatively more abundant. One might, wrongly, except the price system to encourage capitalists to adopt less labor-intensive or more capital-intensive techniques, in some sense. And, in a simple example like this one, one might expect the more capital-intensive technique to be one in which the machine is run for both years.

The example confounds these expectations in both Region 2 and Region 3. Around the switch point in Region 2, a higher wage is associated with the adoption of a technique in which the machine is only operated for the first year. The same is true of the same switch point - the one at the lower wage - in Region 3. From this viewpoint, the switch point is "perverse" in both regions.

This result contrasts with the usual analysis based on real Wicksell effects. The real Wicksell effect is negative for the switch point in Region 2. It is positive for the same switch point in Region 3. For a switch point with a negative real Wicksell effect, a higher wage is associated with the adoption of a technique with more net output per person-year employed. And that is so in this case too. The switch point is only 'perverse', from this perspective, in Region 3.

6.0 Conclusion

This post has illustrated that what I am calling pattern analysis can be applied to examples of joint production in which joint production is only manifested in production and use of long-lived machines. It has focused attention on the distinction between different intuitions about the capital-intensity of a technique.

A Linear Program for Markup Pricing

Figure 1: A Partition of Price-Wage Space for a Two-Commodity Reswitching Example

1.0 Introduction

This post generalizes my approach in Vienneau (2005). In that article, I present a Linear Programming (LP) problem for the firm. In the case of an economy that produces two commodities, one can present a graphical display that clarifies how Sraffa's equations arise. The dual LP is important in this development. Here, I show how that approach can work for a case in which rates of profits systematically vary among industries.

I was pleased that this approach works out for markup pricing. In a sense, this post derives both a direct and an indirect approach for analyzing the choice of technique, in the context of a model of markup pricing.

2.0 The Model

To begin with, consider a model of the production of N commodities from labor and these commodities. This is a model with circulating capital and no joint production. Assume that managers of firms know of U_j processes for producing the output of that industry.

Each process is defined by:

• a_{0, j}(u), u = 1, 2, ..., U_j, the person-years of labor needed to produce one unit of the jth commodity.
• a_{., j}(u) = [a_{1, j}(u) ..., a_{N, j}(u)]^T, the inputs of each commodity needed to produce one unit of the jth commodity.

Each process exhibits constant returns to scale (CRS), requires a year to complete, and use up all their inputs. I also take a set of weights for industries, 1/s₁, ..., 1/s_N, as givens. Let prices be p = [p₁, ..., p_N]. Also, let e = [e₁, ..., e_N]^T be the numeraire, so that:

p e = 1

I should have some assumptions on coefficients ensuring that the economy can be productive by a suitable choice of technique.

I introduce some variables as abbreviations:

k_j(u) = p a_{., j}(u)

c_j(u) = p a_{., j}(u) + w a_{0, j}(u)

π_j(u) = p_j - c_j(u)

r_j(u) = π_j(u)/k_j(u)

2.1 The Firm's LP

The managers of a firm take the wage, w and prices p as given. Let ω = [ω₁, ..., ω_N]^T be the firm's inventory of each commodity at the start of the year. Let q_j(u) be the quantity of the jth commodity that the firm produces with the uth process known for producing that commodity. Let q_{N + 1} be the value of inventory not used for purchasing inputs into production.

Each year the managers of the firm choose how much to produce of each commodity and with which process so as to maximize the weighted increment of value:

(1/s₁)[π₁(1) q₁(1) + π₁(2) q₁(2) + ... + π₁(U₁) q₁(U₁)]

+ (1/s₂)[π₂(1) q₂(1) + ... + π₂(U₂) q₂(U₂)]

...

+ (1/s_N)[π_N(1) q_N(1) + ... + π_N(U_N) q_N(U_N)]

Such that the firm can purchase all of the inputs into production needed at the beginning of the year:

k₁(1) q₁(1) + k₁(2) q₁(2) + ... + k₁(U₁) q₁(U₁)

+ k₂(1) q₂(1) + k₂(2) q₂(2) + ... + k₂(U₂) q₂(U₂)

...

+ k_N(1) q_N(1) + k_N(2) q_N(2) + ... + k_N(U_N) q_N(U_N) ≤ p ω

For all j, u:

q_j(u) ≥ 0

The weights formalize the concept that managers find some industries more desirable or easier to invest in than others. It works out that an industry that managers are less willing to contest or expand production in has a larger rate of profits, in the system of prices of production.

2.2 The Dual LP

The above LP has a dual problem. It is to choose r to minimize:

p ω r

Such that for all j, u:

p a_{., j}(u) (1 + rs_j) + w a_{0, j}(u) ≥ p_j

r ≥ 0

When a decision variable is positive in a solution to the primal LP, the corresponding constraint is met with equality in the dual LP. Suppose the solution of the primal LP leads to each commodity being produced by a specific process in each industry. The price system defined by the technique composed of those process will be satisfied. The economy will be on the wage curve for that technique.

3.0 Solution of the Primal LP

The solution to the primal LP is illustrated by Table 1. In a solution, only basis variables are positive The table specifies the value of each basis variable, when only it is positive in the solution, and conditions that must hold for it to be in the basis. The decision variable q_{N + 1} is a slack variable, introduced to convert the inequality constraint in the primal LP into an equality. It represents the value of inventory carried over, without supporting production. The conditions for when a decision variable is in the basis are intuitive. Consider the first row. A given commodity is produced with a given process only if the rate of profits made in other processes producing that commodity do not exceed the rate of profits made in the given process. Furthermore, the marked-up rate of profits in producing other commodities must not exceed the marked-up rate of profits in the given process. Finally, the (undiscounted) cost of producing a the given commodity must not exceed the revenue made from selling iron. (I am aware that there is some redundancy in how I have stated conditions in the table.)

Table 2: Solution of Primal LP

Variable in Basis	Value	When Optimal
qJ(V)	p ω/kJ(V)	For u = 1, 2, ...,UJ [pJ - w a0, J(V)]/kJ(V) ≥ [pJ - w a0, J(u)]/kJ(u)
		For all j, u (1/sJ)rJ(V) ≥ (1/sj)rj(u)
		cJ(V) ≤ p
qN + 1	p ω	For all j, u cj(u) ≥ p

The solution to the primal LP, in a two-commodity example, is easily visualized. The second commodity is the numeraire, and the price of the first commodity is graphed on the ordinate. Figure 1 partitions the space formed from the price of iron and the wage. A single decision variable enters the basis inside each region in Figure 1. Each region is labeled by that decision variable, in an obvious notation. On the boundaries, a solution to the LP can be formed from a linear combination of decision variables. In the example, both commodities must be produced for the economy to be self-sustaining. Firms are willing to produce both only if prices lie along the heavy locus. The figure shows that this is a reswitching example. One technique is adopted at low and high wages, while the other technique is used at intermediate wages. The figure also illustrates that the wage cannot exceed a maximum.

4.0 Conclusion

I have thought about how this LP approach generalizes. In a general joint production framework, it is not immediately obviously how to assign processes to industries. So I do not see how to define the weights. I suppose one could have a weight for each process, instead of for each industry.

Land presents another difficulty. One would like to impose additional constraints in the primal LP to specify that overall production cannot require that more than a given quantity of some inputs cannot be used in production. Then multiple processes would be used, in a model of extensive rent, in certain industries. But should not such constraints be imposed above the level of the firm? That is, if a firm's production meets the constraints, they might still be violated in the economy as a whole.

But, I suppose, this LP approach applies to cases of fixed capital, where joint production is such that firms in an industry can choose to operate multiple processes, each jointly with a machine of a specific age.

Reference

• Robert L. Vienneau (2005). On Labour Demand and Equilibria of the Firm. The Manchester School 73 (5): 612-619.

Extending An Example With Markup Pricing

Figure 1: A Two-Dimensional Pattern Diagram

The example in this working paper is of an economy in which two commodities are produced. Technical progress is modeled as decreasing the coefficients of production in one of the processes for producing corn. They decrease at a rate of σ of ten percent.

Figure 2 shows how the pattern of switch points vary with technical progress. Initially, the Beta technique is cost-minimizing. Then it becomes a reswitching example. Around the switch point at the lower rate of profits, a higher wage is associated with more labor being hired, per unit of net output. Also, a higher wage is associated with the adoption of more direct labor being hired in corn production, per bushel corn produced gross. This is called a reverse substitution of labor. The other switch point disappears over the wage axis with more technical progress. The remaining switch point still exhibits a reverse substitution of labor. Eventually, that switch point no longer exhibits such a reverse substitution. Finally, it disappears entirely.

Figure 2: A Pattern Diagram

I have been exploring how this example behaves with full cost pricing. I let the rate of profits in the iron industry be s₁ r, and the rate of profits in the corn industry be s₂ r. Figure 1 illustrates how these modeling choices for technical progress and markup pricing interact when s₂ = 1.

Figure 2 illustrates the characteristics of switch points along a horizontal line, at s₁, in Figure 1. The numbered areas in the two figures correspond. Only one switch point exists in the region numbered 6, and it has a positive real Wicksell effect.

The example illustrates that an increase in the markup in a specific industry can result in the creation of a switch point in which higher wages are associated with firms wanting to employ more workers, both per unit net output in the economy as a whole and per unit gross output in a specific industry. Think of a vertical line going through Regions 6, 2, and 1, and, specifically, the partition between Regions 6 and 2. On the other hand, the transition from Region 5 to Region 3 is associated with creation of a switch point that only exhibits a reverse substitution of labor; it still has a negative real Wicksell effect.

Thanks to the comments of Sturai for encouraging me to write this post and for pointing out a paper by Antonio D'Agata that I'll have to read.

Structural Economic Dynamics, Real Wicksell Effects, and the Reverse Substitution of Labor

I have uploaded another working paper:

This article presents an example in which technical progress results in variations in the labor market. Around a switch point with a positive real Wicksell effect, a higher wage is associated with firms wanting to employer more labor per unit output of net product. Around a switch point with a reverse substitution of labor, firms in a particular industry want to hire more labor per unit output of gross product. Technical progress can bring about and take away circumstances favorable for workers wanting to press claims for higher wages.

My research approach can generate fluke switch points. I have decided that such flukes are more interesting when placed in a story about the perturbation of parameters.

A Visualization of the Choice of Technique

Figure 1: Regions for Basis Variables

1.0 Introduction

I introduced a new way of visualizing the choice of technique for two-commodity models back in 2005. As far as I know, nobody has taken up this idea. I modify my method slightly by having labor advanced; wages are paid out of the surplus at the end of the year. I cite John Roemer in my paper linked previously.

2.0 Technology

Table 1 specifies the technology I use for illustration. Each row lists the inputs needed to produce one unit (ton or bushel) for the indicated industry. As usual, this is a model of circulating capital.

Table 1: Example Technology

Input	Industry
	Iron	Corn
		Alpha	Beta
Labor	a0, 1 = 1	aα0, 2 ≈ 0.9364	aβ0, 2 ≈ 0.6174
Iron	a1, 1 = 9/20	aα1, 2 ≈ 0.02602	aβ1, 2 ≈ 0.001518
Corn	a2, 1 = 2	aα2, 2 ≈ 0.1041	aβ2, 2 ≈ 0.4636

For this economy to be reproducible, both iron and corn must be produced. The iron-producing process can be combined with either of the corn-producing processes. Thus, there are two possible techniques, the Alpha and Beta techniques, each of which include the corn-producing process with the corresponding label. (The approach in this post can be extended to include any number of available processes in either industry.)

3.0 A Linear Program for the Firm

Consider a firm that starts the year with an inventory of ω₁ tons iron and ω₂ bushels corn. I take corn as the numeraire. The firm faces a price for iron of p bushels per ton and a wage of w bushels per person years. The managers of the firm must set the value of the following decision variables:

• q₁: The tons iron produced with the iron-producing process.
• q^α₂: The bushels corn produced with the Alpha corn-producing process.
• q^β₂: The bushels corn produced with the Beta corn-producing process.
• q₃: The value of inventory that the firm carries over unused to the next year.

The firm is constrained by the value of its inventory. Its level of production cannot require it to advance more than the value of its inventory.

The managers of the firm attempt to maximize the increment of value. Their problem can be formulated as a Linear Program (LP). They choose q₁, q^α₂, and q^β₂ to maximize:

z = (p - pa_{1, 1} - a_{2, 1} - a_{0, 1}w)q₁

+ (1 - pa^α_{1, 2} - a^α_{2, 2} - a^α_{0, 2}w)q^α₂

+ (1 - pa^β_{1, 2} - a^β_{2, 2} - a^β_{0, 2}w)q^β₂

Such that:

(pa_{1, 1} + a_{2, 1})q₁

+ (pa^α_{1, 2} + a^α_{2, 2})q^α₂

+ (pa^β_{1, 2} + a^β_{2, 2})q^β₂

≤ p ω₁ + ω₂

q₁ ≥ 0, q^α₂ ≥ 0, q^β₂ ≥ 0

In solving this LP by the simplex method, it is convenient to introduce the slack variable, q₃, to convert the constraint to an equality.

4.0 The Dual LP

The above LP has a dual. It is to choose a non-negative rate of profits so as to minimize the capital charge on the inventory. Constraints are such that the cost of each production process, including a charge for capital, does not fall below the revenue from operating that process. Formally, choose r to minimize:

(p ω₁ + ω₂) r

Such that:

(pa_{1, 1} + a_{2, 1})(1 + r) + a_{0, 1}w ≥ p

(pa^α_{1, 2} + a^α_{2, 2})(1 + r) + a^α_{0, 2}w ≥ 1

(pa^β_{1, 2} + a^β_{2, 2})(1 + r) + a^β_{0, 2}w ≥ 1

r ≥ 0

If the primal LP has a solution, so will the dual LP. And the value of the objective functions will be the same, for a solution, for both the primal and dual LP. When a decision variable is positive in a solution to the primal LP, the corresponding constraint is met with equality in the dual LP. Thus, if the solution of the primal LP leads to corn being produced and iron being produced with the Alpha iron-producing process, the economy will be on the wage curve for the Alpha technique. Similar remarks apply to the Beta technique.

Table 2: Solution of Primal LP

Variable in Basis	Value	When Optimal
q1	(p ω1 + ω2)/(pa1, 1 + a2, 1)	r1 ≥ rα2
		r1 ≥ rβ
		c1 ≤ p
qα2	(p ω1 + ω2)/(paα1, 2 + aα2, 2)	r1 ≤ rα2
		rα2 ≥ rβ2
		cα2 ≤ 1
qβ2	(p ω1 + ω2)/(paβ1, 2 + aβ2, 2)	r1 ≤ rβ2
		rα2 ≤ rβ2
		cβ2 ≤ 1
q3	p ω1 + ω2	c1 ≥ p
		cα2 ≥ 1
		cβ2 ≥ 1

5.0 The Solution of the Primal LP

The solution to the primal LP is illustrated by Table 2. In a solution, only basis variables are positive. The table specifies the value of each basis variable, when only it is positive in the solution, and conditions that must hold for it to be in the basis. These conditions are specified in terms of certain variables introduced as abbreviations. The rates of profits in each process are:

r₁ = (p - a_{0, 1}w)/(pa_{1, 1} + a_{2, 1})

r^α₂ = (1 - a^α_{0, 2}w)/(pa^α_{1, 2} + a^α_{2, 2})

r^β₂ = (1 - a^β_{0, 2}w)/(pa^β_{1, 2} + a^β_{2, 2})

The (undiscounted) costs of each process are:

c₁ = pa_{1, 1} + a_{2, 1} + a_{0, 1}w

c^α₂ = pa^α_{1, 2} + a^α_{2, 2} + a^α_{0, 2}w

c^β₂ = pa^β_{1, 2} + a^β_{2, 2} + a^β_{0, 2}w

The conditions for when a decision variable is in the basis are intuitive. Consider the first row. Corn is produced only if the rate of profits made in either of the iron-producing processes does not exceed the rate of profits made in the corn producing process. Furthermore, the (undiscounted) cost of producing a bushel corn must not exceed the revenue made from selling corn.

6.0 Visualization

The solution to the primal LP, in a two-commodity example, is easily visualized. Figure 1 partitions the space formed from the price of iron and the wage. A single decision variable enters the basis inside each region in the figure, and that region is labeled by that decision variable. On the boundaries, a solution to the LP can be formed from a linear combination of decision variables. Iron and corn must be both produced for the economy to be self-sustaining. Firms are willing to produce both only if prices lie along the heavy locus. The figure shows that this is a reswitching example. The Beta technique is adopted at low and high wages, while the Alpha technique is used at intermediate wages. The figure also illustrates that the wage cannot exceed a maximum.

7.0 Conclusion

If you think about it, the above is a derivation of the usual method of analyzing the choice of technique by constructing the outer frontier of the wage curves for all available techniques. It is not restricted to a two-commodity example, although the diagram is so restricted. The proof follows from duality theory in linear programming. The graph illustrates that equilibrium prices must vary with the wage.

I remain puzzled about why mainstream economists continue to teach that, under the ideal assumptions of free competition, wages and employment are determined by the interaction of supply and demand in labor markets.