Labor Reversing Without Capital: An Example

Figure 1: Skilled Labor Hired by Firms per Unit Output

1.0 Introduction

This example is from Opocher and Steedman (2015). They present many examples in which the reader is expected to work them out, as illustrated in this post.

This is an example in which cost-minimizing firms desire to hire more labor (of a specific type) for an increased wage, around a specific wage. This example is of a firm producing a single commodity from inputs of specific types of land and specific types of labor. No produced capital goods exist in this example, and the interest rate is assumed to be zero. Yet perverse behavior arises on the demand side of markets for factors of production anyway - where results are called perverse merely if they violate neoclassical intuitions shown to be mistaken half a century ago. The most complicated aspect of this example is that some techniques of production are specific to specific types of land.

2.0 Indirect Average Cost Functions

Consider a firm that produces widgets from inputs of skilled labor, unskilled labor, and land of one of two types. Suppose the price of widgets is unity. Define:

• p_α is the rent for alpha-type land.
• p_β is the rent for beta-type land.
• w₁ is the wage for unskilled labor.
• w₂ is the wage for skilled labor.

The indirect average cost function for widgets produced on land of type alpha is:

c_α(p_α, w₁, w₂) = (1/2)[(w₁ p_α)^1/2 + (w₁ w₂)^1/2

+ (w₂ p_α)^1/2]

The indirect average cost function for widgets produced on land of type beta is:

c_β(p_β, w₁, w₂) = (3/5)(w₁ p_α)^1/2 + (3/10)(w₁ w₂)^1/2

+ (11/20)(w₂ p_α)^1/2

The indirect average cost function shows the average cost of producing each widget, when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm.

These indirect average cost functions are homogeneous of the first degree. For the indirect average cost function for land of type alpha, this property is expressed as:

c_α(ap_α, aw₁, aw₂) = a c_α(p_α, w₁, w₂)

This a traditional assumption for cost functions.

Consider the indirect average cost function for a specific type of land. That type of land, unskilled labor, and skilled labor are substitutes. No inputs are complements in this example. In other words, the off-diagonal elements of the Hessian matrices formed from each indirect average cost function are all positive. The elements along the principal diagonal of each Hessian matrix are negative.

3.0 The Wage-Wage Frontier

Consider a long run equilibrium of the firms in which pure economic profits have been competed away and no firm is making a loss. Perhaps, the prospect of firms entering or exiting the industry has caused this situation to arise. Furthermore, suppose rents for both types of land happen to be unity. (Without this assumption, this example would have two more degrees of freedom.) If firms are producing on a given type of land, the indirect average cost function for that type of land will be equal to unity. For alpha type land, one has:

1 = c_α(1, w₁, w₂)

Or:

w_{1, α} = [(2 - w₂^1/2)/(1 + w₂^1/2)]²

As shown in Figure 2, given the type of land employed, the wage for unskilled labor is a declining function of the wage for skilled labor. The maximum wage for unskilled labor, 4 widgets per person-year, corresponds to skilled labor working for free. Symmetrically, the maximum wage for skilled labor likewise corresponds to unskilled labor working for free.

Equating the indirect average cost function for production on land of type beta yields another trade-off in long run equilibrium between the wages of unskilled and skilled labor.

w_{1, β} = [(20 - 11 w₂^1/2)/(12 + 6 w₂^1/2)]²

When land of type beta is used, the maximum wage for unskilled labor is 2 7/9. The maximum wage for skilled labor is 3 37/121.

Figure 2: Wage-Wage Curves and the Frontier

For some combination of wages of skilled and unskilled labor, firms will be indifferent between producing widgets with land of type alpha and type beta. The cost-minimizing technique at these wages, on each type of land, is equally cheap. These combinations can be found by equating the wages of unskilled labor for the expressions above. After some manipulation, one obtains the equation:

5 w₂ - 9 w₂^1/2 + 4 = 0

This equation can be factored:

(w₂^1/2 - 1)(5 w₂^1/2 - 4) = 0

Firms will thus be indifferent to the type of land used in production for ordered pairs of wages of unskilled and skilled labor, (w₁, w₂), of (1/4, 1) and (4/9, 16/25).

Firms produce widgets on land of type alpha for wages for skilled labor between zero and 16/25, and for wages of skilled labor between one and four. For wages for skilled labor between 16/25 and four, firms produce widgets on land of type beta. The outer frontier allows one to determine the wage of unskilled labor for any feasible wage for skilled labor, given the model assumptions. As well soon be apparent, this is not an example of reswitching. The overall indirect average cost function is almost always differentiable. It is not differentiable only at the two points found by the construction of the outer frontier.

4.0 Land and Labor

We have seen that when rents are unity, long run equilibrium of the firm necessitates that the wages of unskilled labor is a declining function of the wages of skilled labor. Shepherd's lemma can be used to find the coefficients of production for each feasible combination of wages of unskilled and skilled labor. The quantity of each input the firm wants to hire per unit output is the derivative of the indirect average cost function with respect to the price of that input. Thus, when land of type alpha is used, the number of acres of land employed per unit output of widgets is:

t_α(w₁, w₂) = (1/4)(w₁^1/2 + w₂^1/2)

The number of acres of land of type beta per unit output of widgets, when land of that type is used, is:

t_β(w₁, w₂) = (1/40)(12 w₁^1/2 + 11 w₂^1/2)

In what I hope is obvious notation, person-years of unskilled labor employed per unit output of widgets is, depending on the type of land used:

l_{1, α}(w₁, w₂) = (1/4)(1 + w₂^1/2)/(w₁^1/2)

l_{1, β}(w₁, w₂) = (3/20)(2 + w₂^1/2)/(w₁^1/2)

Finally, person-years of skilled labor employed per unit output of widgets is given by one of the following two functions of wages:

l_{2, α}(w₁, w₂) = (1/4)(1 + w₁^1/2)/(w₂^1/2)

l_{2, β}(w₁, w₂) = (1/40)(6 w₁^1/2 + 11)/(w₂^1/2)

5.0 Bringing it all Together

The above algebra can be used to generate various graphs. Figure 1 shows person-years of skilled labor firms desire to hire per unit output. As one moves to the right in the figure, the wage of skilled labor rises and the wage of unskilled labor falls. But at every point in the figure, the wages of the two types of labor are such as to maintain wages as on the outer frontier in Figure 2. That is, firms are minimizing costs, and the output price and input prices are such as to enforce the equilibrium condition that no pure economic profits are available in this industry.

Figure 3 shows the analogous graph for unskilled labor. The point for wages of 4/9 widgets per person-years and 16/25 widgets per person-year for unskilled and skilled labor, respectively, is emphasized. At any point to the left, wages for unskilled labor are higher, and wages for skilled labor are lower. And an infinitesimal variation around this point is associated with firms wanting to employ unskilled labor more intensively when their wage is relatively higher.

Figure 3: Unskilled Labor Hired by Firms per Unit Output

Reswitching of techniques arises when one technique of production is cost-minimizing at, say, a high and low wage but not at an intermediate wage. A technique of production is specified by four coefficients of production in this example. The amount of skilled labor and unskilled labor hired per unit output are two of these coefficients. The acres of land of each type rented per unit output are the other two. The latter two coefficients of production obviously vary, depending on which type of land can be used in a cost-minimizing technique. In fact, the coefficients of production for the type of land not employed is zero. As can be seen in Figures 1 and 3, the coefficient of productions for the two types of labor vary monotonically with relative wages, given the type of land employed.

At one of the two switch points highlighted in Figure 2, two techniques of production are cost-minimizing. (This is the definition of a switch point.) In one technique, one type of land is used. And in the other, the other type of land is used. But a different pair of techniques of production is cost-minimizing at the other switch point. The coefficients of production vary among, for example, the cost-minimizing techniques in which alpha-type land is used at a switch point. Hence, as noted, no reswitching of techniques exists in this example.

6.0 Conclusion

This example has cost-minimizing firms in equilibrium in a single industry. Price and quantity relationships among factors of production have been analyzed, where factors of production consist of land of two types and labor of two types. Quantity relationships have been presented in terms of inputs per unit output for a firm. For simplicity, only the case in which the interest rate is zero and rents of land per acre are unity has been considered. When beta type land is adopted, more acres are cultivated for alpha-type land, for the same level of output. Thus, land has a higher proportion of total unit cost when beta-type land is used. Both skilled and unskilled labor are a lower proportion of total unit cost (as seen in Figures 1 and 3) than they would be if alpha type land was employed. A wage has been found for unskilled labor in which a higher relative wage for unskilled labor is associated with firms desiring to hire more unskilled labor per unit output. And a different relative wage for skilled labor has been found with the analogous property.

I wonder whether an example can be found with a continuum of types of land in which the analog of Figures 1 and 2 come out as continuous U-shaped curves.

So much for explaining wages and employment by well-behaved supply and demand curves in competitive labor markets.

Reference

• Opocher, Arrigo and Ian Steedman (2015). Full Industry Equilibrium: A Theory of the Industrial Long Run, Cambridge University Press

Locke's Caveats To His Labor Theory Of Property

1.0 Introduction

A couple of months ago, I read John Locke's Second Treatise of Government. He has a caveat on his theory of property I did not expect, as well as one I did. The caveat I did not expect involves the unjustness of acquiring more than you can use and wasting it. But Locke thought it allowable to have more than you can use, as long as you did not waste it. Ultimately, he justifies such superfluous property by claiming it will lead to economic development and benefit the community as a whole.

I suppose one can read Locke as a defense of British and American conquest of the autochthonous peoples in the Americas. They held the land in common, but were not using it well, as least from a bourgeois perspective.

I prefer to draw an analogy to socialism. Property in possessions more than used in everyday living is justified by thinking of that property as held for the benefit of the commonwealth, so to speak. (For the purpose of this post, I put aside any qualms I have about Robinsonades.)

2.0 The Labor Theory of Property

I suppose this is the most famous statement of justification of ownership on the basis of a right to the fruits of one's labor:

Sect 27. Though the earth, and all inferior creatures, be common to all men, yet every man has a property in his own person: this no body has any right to but himself. The labour of his body, and the work of his hands, we may say, are properly his. Whatsoever then he removes out of the state that nature hath provided, and left it in, he hath mixed his labour with, and joined to it something that is his own, and thereby makes it his property.

Locke begins talking about the ownership of food gained through hunting and gathering. Land is held in common. "Thus in the beginning all the world was America" (Sect. 49). Locke then transitions to ownership of land. He argues that in such a more complex society, one can trace back commodities to dated labor embodied over many activities in the past. The labor embodied in bread includes the labor of the baker, the miller, the farmer, the manufacturer of tools for the use of the farmer, etc. This view of embodied labor (in Sections 42 and 43) was later echoed in the labor theory of value.

3.0 First Caveat: "Enough, and as Good Left"

From secondary literature, I knew that Locke justified the enclosure of common lands into private property only if what was left still was as good as before:

Sect. 33. Nor was this appropriation of any parcel of land, by improving it, any prejudice to any other man, since there was still enough, and as good left; and more than the yet unprovided could use. So that, in effect, there was never the less left for others because of his enclosure for himself.

He writes about, for example, the unjustness of denying somebody a drink from a river when the river would still flow on undiminished.

4.0 Second Caveat: No Waste of Superfluity

I was not previously aware that, in justifying private property, Locke condemned wasting more than your share:

Sect. 31. It will perhaps be objected to this, that if gathering the acorns, or other fruits of the earth, &tc. makes a right to them, then any one may ingross as much as he will. To which I answer, Not so. The same law of nature, that does by this means give us property, does also bound that property too. God has given all things richly, 1 Tim. vi. 12. is the voice of reason confirmed by inspiration. But how far has he given it us? To enjoy as much as any one can make use of to any advantage of life before it spoils, so much he may by his labour fix a property in: whatever is beyond this, is more than his share and belongs to others. Nothing was made by God for man to spoil or destroy.

And again:

Sect. 37. ...Before the appropriation of land, he who gathered as much as the wild fruit, killed, caught, or tamed, as many of the beasts, as he could; he that so imployed his pains about any of the spontaneous products of nature, as any way to alter them from the state which nature put them in, by placing any of his labour on them, did thereby acquire a property in them: but if they perished, in his possession, without their due use; if the fruits rotted, or the venison putrified, before he could spend it, he offend against the common law of nature, and was liable to be punished; he invaded his neighbor's share, for he had no right, farther than his use called for any of them, and they might serve to afford him conveniences of life.

This is obviously not a position to end at if you are justifying property rights in the rising bourgeois society. Locke caveats his caveat by arguing that you are not wasting more than your share if you hold the extra in goods that do not waste away quickly, whether they be useful or pretty baubles:

Sect. 46. ...He was only to look, that he used them before they spoiled, else he took more than his share, and robbed others. And indeed it was a foolish thing, as well as dishonest, to hoard up more than he could make use of. If he gave away a part to any body else, so that it perished not uselessly in his possession, these also he made use of. And if he also bartered away plums, that would have rotted in a week, for nuts that would last good for his eating a whole year, he did not injury; he wasted not the common stock; destroyed no part of the portion of goods that belonged to others, so long as nothing perished uselessly in his hands. Again, if he would give his nuts for a piece of metal, pleased with its colour; or exchange his sheep for shells, or wool for a sparkling pebble or a diamond, and keep those by him all his life he invaded not the right of others, he might heap up as much of these durable things as he pleased; the exceeding of the bounds of his just property not lying in the largeness of his possession, but the perishing of any thing uselessly in at it.

Locke argues that the ability to accumulate money through commerce leads to owners developing their property:

Sect. 48. And as different degrees of industry were apt to give men possessions in different proportions, so this invention of money gave them the opportunity to continue and enlarge them: for supposing an island, separate from all possible commerce with the rest of the world, wherein there were but an hundred families, but there were sheep, horses and cows, with other useful animals, wholesome fruits, and land enough for corn for a hundred thousand times as many, but nothing in the island, either because of its commonness or perishableness, fit to supply the place of money; what reason could any one there to enlarge his possessions beyond the use of his family, and a plentiful supply to its consumption, either in what their own industry produced, or they could barter for the perishable, useful commodities, with others? Where there is not some thing, both lasting and scarce, and so valuable to be hoarded up, there men will not be apt to enlarge their possessions of land, were it never so rich, never so free for them to take: for I ask, what would a man value ten thousand, or an hundred thousand acres of excellent land, ready cultivated, and well stocked too with cattle, in the middle of the inland parts of America, where he had no hopes of commerce with other parts of the world, to draw money to him by the sale of the product?

Locke's defense of private property reminds me of Jesus's parable of the talents (Matthew 25). You should use your property for the benefit of humankind:

Sect. 37. ...To which let me add, that he who appropriates land to himself by his labour, does not lessen, but increase the common stock of mankind: for the provisions serving to the support of human life, produced by one acre of increased and cultivated land, are (to speak much within compass) ten times more than those which are yielded by an acre of land of an equal richness lying waste in common. And therefore he that incloses land, and has a greater plenty of the conveniences of life from ten acres, than he could have from an hundred left to nature, may truly be said to give ninety acres to mankind: for his labour now supplies him with provisions out of ten acres, which were but the product of an hundred lying in common...

5.0 Other Subjects in Locke's Tract

All of the above quotes are from Chapter 5 of Locke's Second Treatise. This book contains nineteen chapters, and treats many other topics. These include a recap of the first treatise, which presumably refutes the claims of Sir Robert Filmer that:

• God gave Adam property in all the earth.
• And current monarchs own their countries through their descent from Adam.

Locke praises William III, the victor of the Glorious Revolution. He also treats of the state of nature, in which humans are free. And they give up that freedom in a social contract so as to end the war of all against all. But, if rulers:

become destructive of these ends, it is the Right of the People to alter or to abolish [the government], and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. (America's Declaration of Independence)

I find the start of Chapter 13, as well as other passages (e.g., Sect. 225) echoed in the declaration. Other topics include natural rights, war, slavery, and parental rights.

References

• John Locke, Second Treatise of Government (1690).

Recurrence Of Capital-Output Ratio Without Reswitching

Figure 1: Recurrence of Capital-Output Ratio

1.0 Introduction

This example is from Arrigo Opocher and Ian Steedman. It illustrates the analysis of an isolated industry in equilibrium. This analysis is therefore more akin to partial equilibrium than to general equilibrium. Sometimes (mainstream?) economists say that the Cambridge Capital Controversies were only about aggregate neoclassical theory, that is, macroeconomics. Or that the CCC has been subsumed by General Equilibrium Theory. The example illustrates that such economists are, as has long been apparent, spouting poppycock.

2.0 Indirect Average Cost Function

Consider a firm that produces widgets from inputs of widgets, unskilled labor, and skilled labor. Let the indirect average cost function be:

c(p, w₁, w₂) = sp + w₁ + w₂

+ 2(pw₁)^1/2 + 2(pw₂)^1/2 + 2γ(w₁w₂)^1/2

where

0 < s < 1

0 < γ

γ ≠ 1

and

• p is the price of a widget. Widgets used as inputs are assumed to be totally consumed in one production period.
• w₁ is the wage for unskilled labor.
• w₂ is the wage for skilled labor.

The indirect average cost function shows the average cost of producing each widget (net), when each firm in the industry is producing the cost-minimizing quantity. That is, each firm is producing at the point where the marginal cost and average cost of production of a widget is the same. Assume all firms face the same indirect average cost function. If a positive rate of (accounting) profit was being earned by any firm, the rate of profit would show up in the arguments of the indirect average cost function for that firm.

This indirect average cost function is homogeneous of the first degree:

c(a p, a w₁, a w₂) = a c(p, w₁, w₂)

This is a conventional assumption for cost functions.

Suppose the firm faces a given price of widgets and given wages for skilled and unskilled labor. By Shephard's lemma, the quantity of each input the firm wants to hire per unit output, given the price of each input, is the derivative of the indirect average cost function with respect to the price of that input. Hence, the capital-output ratio, k(p, w₁, w₂), is:

k(p, w₁, w₂) = ∂c/∂p = s + (w₁/p)^1/2 + (w₂/p)^1/2

Notice that the capital-output ratio is a pure number, unambiguously defined in this example, and independent of prices.

By the same logic, the amount of unskilled labor the managers of the firm desire to hire per widget produced is:

l₁(p, w₁, w₂) = ∂c/∂w₁ = 1 + (p/w₁)^1/2 + γ(w₂/w₁)^1/2

The amount of skilled labor the managers of the firm desire to hire per widget produced is:

l₂(p, w₁, w₂) = ∂c/∂w₂ = 1 + (p/w₂)^1/2 + γ(w₁/w₂)^1/2

The matrix of second derivatives of the indirect average cost function is:

(I am not sure whether it is more common to define the above matrix as the transpose of what I have above.) Anyway, for a positive price of widgets and positive wages, the signs of the second derivatives are as follows:

The signs along the principal diagonal show that the slopes of the per-unit input demand functions slope down. That is, given prices for all but one input, a lower price of that input is associated with a willingness of the firm to employ more of that input per unit produced. The positivity of the off-diagonal elements of the above matrix show that widgets, considered as inputs; unskilled labor; and skilled labor are all substitutes, not complements, in some sense. These signs for the matrix of second derivatives of the indirect average cost function are also conventional properties for cost functions.

3.0 Full Industry Equilibrium

Suppose the industry in which widgets are produced has no barriers to entry or exit. Thus, in the long run, economic profits will have been competed away. For firms to be earning no economic profits, the price of widgets must be equal to the average cost of manufacturing them:

p = c(p, w₁, w₂)

So far, no numeraire has been specified. Let widgets themselves be numeraire. Then:

1 = c(w₁, w₂)

where the argument in the indirect average cost function for widgets has been dropped as otiose.

Consider various levels of w₁, the wage of unskilled labor. For the industry to continue to be in long run equilibrium, the wage of skilled labor, w₂, must vary as well, thereby leaving the average cost of producing a widget as unity. Figure 2 illustrates the resulting wage-wage frontier. (Figures are drawn for s = 1/10 and γ = 2/3.) The highest wage for unskilled labor (when the wage for skilled labor is zero) is ((2 - s)^1/2 - 1)². Since this model is symmetric in skilled and unskilled labor, the highest wage for unskilled labor is likewise ((2 - s)^1/2 - 1)². As long as the rate of accounting profits is zero and technology is given, the wage of unskilled labor can only be higher if the wage of skilled labor is lower.

Figure 2: Wage-Wage Frontier

The wage-wage frontier can be used to find the wage of skilled labor for a given wage of unskilled labor between zero and the maximum. In other words, the frontier is helpful in calculating the ratio of the wage of skilled labor to the wage of unskilled labor, given the wage of unskilled labor. This ratio of wages is independent of the choice of the numeraire.

4.0 Capital and Labor

With the chosen numeraire, the capital-output ratio is:

k(w₁, w₂) = s + (w₁)^1/2 + (w₂)^1/2

Given the wage of unskilled labor, one can find the wage of skilled labor and, consequently, both the ratio of wages of the two types of labor and the capital-output ratio. Figure 1, at the start of this post, graphs the capital-output ratio as the derived function of the ratio of wages.

The capital-output ratio is the same when either skilled or unskilled labor is earning their maximum wage, with the other type of labor being paid a wage of zero. In these two extreme cases, the capital-output ratio is (2 - s)^1/2 - (1 - s). Likewise for any ratio but one of the wage of skilled labor to the wage of unskilled labor between these extremes of zero and infinity, the capital-labor ratio is non-unique. The exception is the ratio of wages at which the function in Figure 1 peaks.

One can see that recurrence of the capital-output ratio is not reswitching. Figures 3 and 4 show, respectively, unskilled labor and skilled labor per unit output as a function of the ratio of wages. As shown in Figure 3, a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ more unskilled labor per unit output. Likewise, a a higher wage of skilled labor accompanied by a lower wage of unskilled labor is associated with firms wanting to employ less skilled labor per unit output. As far as unproduced inputs go, this example of the isolated firm in long run equilibrium does not contradict outdated and exploded neoclassical intuitions about substitution and the mistaken notion of equilibrium prices as scarcity indices. But, since the functions in Figures 3 and 4 are monotonic, no reswitching of techniques arises in this example.

Figure 3: Unskilled Labor Employed per Unit Output

Figure 4: Skilled Labor Employed per Unit Output

5.0 Conclusion

This post has presented an example of an isolated firm in a long period equilibrium. The indirect average cost function, which includes the cost of the use of an input which itself is produced by the firm's industry, otherwise has utterly conventional properties. The analysis of the firm in a long run equilibrium demonstrates that it is an incoherent thought experiment to consider the equilibrium response of the firm to the variation of one price at a time. Only the variation of more than one price at a time can yield an equilibrium analysis that could be at all empirically relevant.

A result of this analysis is to reveal a non-monotonic response of the capital-output ratio to variations in the relative prices of the two unproduced inputs used by this firm in production. In fact, every possible capital-output ratio, except for one, recurs in the example. This is a step in an argument leading to the conclusion that economic theory is consistent with competitive firms wanting to employ more input per unit output for higher prices of that input, a finding that seems consistent with empirical results.

Election Paradoxes And Faustian Agents

I have been trying to reread Donald Saari on election paradoxes. I have previously considered a few parallels between the Condorcet paradox and models of agents as composed of multiple selves. It seems to me that one could draw more analogies here. I do not plan to pursue the research agenda outlined here - I'm not sure how plausible its results would be. Anyways, Saari provides a comprehensive analysis of a range of voting procedures. Could a fuller range of such procedures - as opposed to pairwise majority rule - be applied to models of multiple selves?

For example, consider a model of a person as having multiple selves, where each one of those selves has a set of preferences over commodities. And suppose the individual, in making choices, resolves those selves with a procedure analogous to an election procedure (e.g., plurality vote, antiplurality vote, Borda Count). Suppose which procedure is used is context-dependent. Can an outside agent modify the context somehow such that the individual follows a different procedure, with consequent effects on the individual's choice?

Or consider two people each composed of the same number of multiple selves, with the preferences of those selves the same across these two people. But suppose each person resolves those selves with a different voting procedure. Maybe these two different voting procedures yield the same "best" choice for one specific menu of choices, but order the non-best choices differently. So if a new menu was created with the best choice removed, these two people - who have identical preferences, in some sense - would make different choices.

I suppose if you follow research along these lines, it would be theoretical research. I do not know how an experiment could elicit the required information to determine the preferences of the multiple selves and the election procedure. I guess the challenge would be to come up with an account consistent with some behavioral anomaly arising in economics experiments. Even better might be to suggest a new experiment and to implement it.

References

• Donald G. Saari (2001). Chaotic Elections! A Mathematician Looks at Voting, AMS.

Bertrand Russell, Crank

On the Post Topic

Some great thinkers compare their work to the works of Nicolaus Copernicus or of Galileo:

"The old logic put thought in fetters, while the new logic gives it wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics, making it possible at last to see what kinds of problems may be capable of solution, and what kinds must be abandoned as beyond human powers. And where a solution appears possible, the new logic provides a method which enables us to obtain results that do not merely embody personal idiosyncrasies, but must command the assent of all who are competent to form an opinion." -- Bertrand Russell, Our Knowledge of the External World as a Field For Scientific Method in Philosophy (1914).

"...an imagination better stocked with logical tools would have found a key to unlock the mystery. It is in this way that the study of logic becomes the central study in philosophy: it gives the method of research in philosophy, just as mathematics gives the method in physics. And as physics, which, from Plato to the Renaissance, was as unprogressive, dim, and superstitious as philosophy, became a science through Galileo's fresh observation of facts and subsequent mathematical manipulation, so philosophy, in our own day, is becoming scientific through the simultaneous acquisition of new facts and logical methods.

In spite, however, of the new possibility of progress in philosophy, the first effect, as in the case of physics, is to diminish very greatly the extent of what is thought to be known. Before Galileo, people believed themselves possessed of immense knowledge on all the most interesting questions in physics. He established certain facts as to the way in which bodies fall, not very interesting on their own account, but of quite immeasurable interest as examples of real knowledge and of a new method whose future fruitfulness he himself divined. But his few facts sufficed to destroy the whole vast system of supposed knowledge handed down from Aristotle, as even the palest morning sun suffices to extinguish the stars. So in philosophy: though some have believed one system, and others another, almost all have been of opinion that a great deal was known; but all this supposed knowledge in the traditional systems must be swept away, and a new beginning must be made, which we shall esteem fortunate indeed if it can attain results comparable to Galileo's law of falling bodies." -- Bertrand Russell, ibid.

The "new logic" Russell refers to is set out in, for example, Russell and Whitehead's Principia Mathematica. So Russell is comparing himself to Galileo.

An Approach to a Book Review

I'm glad I read this book, although I think it is basically mistaken. Not surprisingly, given their interactions at Cambridge before World War II, Russell's exposition reminds me of Ludwig Wittgenstein's Tractatus Logico-Philosophicus. Although clearly written, Russell's book has a quite different literary style than Wittgenstein's gnostic utterances and hierarchical structure. Both argue that everyday observations about, say, tables and chairs, should be decomposed into logical conjunctions, negations, and disjunctions of atomic facts, which cannot be further broken down. Russell and Wittgenstein differ on the nature of these atomic facts. For Wittgenstein, the referents for entities in atomic facts are quite mysterious; the specification of what these entities are is not a matter of logic, but of its application. Russell is quite clear that these entities include unintegrated sensations, something like "red patch here now."

Russell outlines how one might combine statements about such entities to construct entities that we see, hear, taste, smell, or feel. He goes on to analyze claims about other minds. The analysis of time leads to comments on Zeno's paradoxes and the mathematical theory of continuity. He also explains the idea of infinity, explaining the then recent theory of Cantor. He tries to present a popular overview of these topics. He acknowledges that some of his exposition is more mathematics than philosophy. But, as you can see above, he thinks previous philosophers and many of his contemporaries stumbled into error because they did not possess these logical and mathematical tools. For later developments along the lines, I gather one can look at such works of logical positivism as Rudolf Carnap's The Logical Structure of the World. I have never read Carnap, but I have read A. J. Ayer's Language, Truth, and Logic.

I recently stumbled somewhere across an argument that Noam Chomsky's approach to linguistics supercedes Russell's application of logic to philosophy. Russell and Chomsky agree that sentences of very different structures can have a close surface appearance, and that the same structure can be exhibited in sentences of different surface appearances. In deciding whether or not propositions are true, or even make sense, one should supposedly concentrate on the meaning captured by this deeper structure. But in trying to analyze the meaning of such propositions as, "The king of France is bald", Russell takes an a priori approach. The adequacy of grammar, however, to characterize sentences in a language is an empirical question. And semantics should be based on the parse trees derived from grammatical analysis of the surface appearances of language, not a logical analysis of the surface appearance. This approach, as I understand it, is analogous to how compilers operate. They apply a semantic analysis to a computer program only after first completing a parsing phase. And Chomsky's approach, I gather, has been influential in Artificial Intelligence.

One can argue that just as Wittgenstein, in Philosophical Investigations, showed his earlier approach in the Tractatus was mistaken, so he also showed Chomsky's approach in linguistics to be mistaken. A fortiori, AI is not possible either. Exposition of the parallelism between Russell and Chomsky's analysis of language makes these claims a bit more clear to me. (I guess Sraffa was not too impressed by Chomsky, either.) I suppose one might look at Norman Malcolm's Wittgenstein: Nothing is Hidden, for a fuller argument against Chomsky along these lines. (I did not get much out of Malcolm when I read him years ago.)

Data By Country On Gross And Net Investment?

My article demonstrating the empirical soundness of a simple labor theory of value needs updating. In particular, I should calculate the rate of profits on total capital. So I need data on both constant and circulating capital, not just circulating capital.

Or, at least, I need data on depreciation expenses by some consistent set of conventions. In other words, I need data on gross and net investment. Perhaps it would be sufficient for empirical approximations to have this data on the country level for every country or region in the world. I do not expect to find such data broken down for each country by industry.

Does anybody have suggestions or comments on where to find such data?

Paul Romer Confused On Capital Theory

I have noted Paul Romer's confusions before. For example, consider the following passage:

"In the conventional specification, total capital K is implicitly defined as being proportional to the sum of all different types of capital. This definition implies that all capital goods are perfect substitutes. One additional dollar of capital in the form of a truck has the same effect on the marginal productivity of mainframe computers as an additional dollar's worth of computers. Equation (1) expresses output as an additively separable function of all the different types of capital goods so that one additional dollar of trucks has no effect on the marginal productivity of computers." -- Paul Romer (1990).

Does Romer think that the so-called factor price curve for all techniques must be an affine function? That price Wicksell effects are always zero? Or maybe he just is trying to buffalo his reader with an ill-thought out use of mathematical symbols.

On his twitter feed, he expresses a disinterest in knowing what he is talking about:

"Sorry, but the capital controversies were a waste of time. No relevance then or now." -- Paul Romer, 16 May 2015, 1:09 PM.

I suppose one might possibly be able to defend this view:

"Economists usually stick to science. Robert Solow (1956) was engaged in science when he developed his mathematical theory of growth. But they can get drawn into academic politics. Joan Robinson (1956) was engaged in academic politics when she waged her campaign against capital and the aggregate production function." -- Paul M. Romer (2015).

One might say Solow was looking to empiricalism when he developed his non-rigorous, loose theory of growth. And, I suppose one could say that some political views were involved in Joan Robinson's insistence that Keynes' theory applies to all runs, both the short run and the long run. And in her attempt to combat the development of pre-Keynesian theories after Keynes, even if such developments were the product of those who called themselves Keynesians in some other context.

But to make such an argument, one would have to have read at least some of Joan Robinson's work from the era. It is clear that Romer has not:

"When I learned mathematical economics, a different equilibrium prevailed. ...when economic theorists used math to explore abstractions, it was a point of pride to do so with clarity, precision, and rigor. Then too, a faction like Robinson’s that risked losing a battle might resort to mathiness as a last-ditch defense, but doing so carried a risk. Reputations suffered.

If we have already reached the lemons market equilibrium where only mathiness is on offer, future generations of economists will suffer... Where would we be now if Robert Solow’s math had been swamped by Joan Robinson’s mathiness?" -- Paul M. Romer (2015).

When, during the Cambridge Capital Controversy, did Robinson try to buffalo readers with pretend rigorous manipulations of imprecisely defined mathematical symbols. How about never? Is never good for you?

Update (21 May 2015): Reactions to Romer from Peter Dorman, Justin Fox, Joshua Gans, Noah Smith, Lars Syll, and Matias Vernengo

Update (24 July 2015): Marc Lavoie and Mario Seccareccia also comment on Romer's confusion.

References

• Romer, Paul M. (1990) Endogenous Technological Change, Journal of Political Economy V. 98, N. 5 (Oct): S71-S102.
• Romer, Paul M. (2015). Mathiness in the Theory of Economic Growth, American Economic Review, V. 105, N. 5: pp. 89-93.

Free Trade In The Popular Consciousness

I think it important to oppose errors, both when they are formulated by supposedly rigorous economic thinkers and when they are popularly repeated. The relationships between ideas on different levels can be complicated.

I like writing about minimum wages because it is a clear case where:

• Textbook teaching is wrong, both on empirical
and on theoretical grounds.
• Until recently at least, surveys of economists showed that they, by and large, accepted the mistaken teaching.

As I understand it, "free trade" is a policy area where economists have even more agreement, based on mistaken theory. (I suppose I should include a link to survey of economists somewhere in this post. Any suggestions?) The book from which the following quote is taken contains a structured literature survey, and is written for the general reader:

"Even if I were wrong about this, and the most sophisticated of mainstream economists did think that there was something flawed about the connection between free trade policies and comparative advantage on its own terms (and not merely where its assumptions have been debased), a further point should be raised... We are here engaged in a project of critiquing the ideological effect generated by comparative advantage - of the rationalization that it provides for certain kinds of policies and socioeconomic arrangements. The views of the most sophisticated of academic economists are one - although not the only - part of this, particularly when a less sophisticated version of their ideas predominate outside the halls of economics departments. The manner in which the ideas of intellectuals permeate education, the media and public debate are often more important to the practices of actual agents, policy-makers, and so on, than the most sophisticated renderings of those ideas that emerge from the academy. It matters little if the most sophisticated of economists doubt the connection between free trade and comparative advantage if politicians, commentators, policy-makers or indeed, the public at large, buy the connection between the two and therefore support free trade policies. Indeed, my point - and this enquiry as a whole - is lent its sharpness by its relevance to the world of action, to policy, to normative concerns, to what people do - and think is right to do - because of the ideas that are peddled by intellectuals." -- Vishaal Kishore (2014). Ricardo's Gauntlet: Economic Fiction and the Flawed Case for Free Trade (Anthem Press).

Growth Versus Levels

As far as I can see, mainstream economists generally believe:

• Policy changes that raise the level of growth of the economy are more important than one over changes to the level of output¹.
• Free trade is a desirable policy.
• This policy preference falls out of the theory of comparative advantage.
• Abolishing protectionist tariffs will result in a one-time increase of the level of output of the economy, but not the rate of growth.

As I understand it, Post Keynesians generally think theories of growth cannot be neatly be separated from theories of business cycles - that is, theories of short run fluctuations in the level of output. This intertwining complicates policy recommendations. Nevertheless, one can look back to Keynes² to see a concern with economic growth.

It is hard to see how the two beliefs in the list above are consistent. Why should mainstream economists these days care much about whether countries put in place so-called free trade agreements? Maybe the question is one of who should govern. Anyways, economists in the public sphere should strive to be clear on what they advocate, and why they do.

Footnotes

1. Robert Lucas famously said (that is, I do not know where), "Once you start thinking about growth, it's hard to think about anything else". I do know that in Lucas (1987), he estimated that consumers "would surrender 42 per cent across the board [in consumption] to obtain an increase in the growth rate from [3 per cent] to [6 per cent." On the other hand, to eliminate aggregate consumption variability of the magnitude seen in the USA from the end of the Second World War to the 1980s would be worth "something less than a tenth of a percentage point" in average consumption.
2. Keynes wrote this paean to economic growth in the midst of the Great Depression:

   "The modern age opened; I think, with the accumulation of capital which began in the sixteenth century... From that time until to-day the power of accumulation by compound interest, which seems to have been sleeping for many generations, was re-born and renewed its strength. And the power of compound interest over two hundred years is such as to stagger the imagination.

   For I trace the beginnings of British foreign investment to the treasure which Drake stole from Spain in 1580. In that year he returned to England bringing with him the prodigious spoils of the Golden Hind. Queen Elizabeth was a considerable shareholder in the syndicate which had financed the expedition. Out of her share she paid off the whole of England’s foreign debt, balanced her Budget, and found herself with about 40,000 [pounds] in hand... Thus, every 1 [pound] which Drake brought home in 1580 has now become 100,000 [pounds]. Such is the power of compound interest!

   If capital increases, say, 2 per cent per annum, the capital equipment of the world will have increased by a half in twenty years, and seven and a half times in a hundred years. Think of this in terms of material things--houses, transport, and the like.

   At the same time technical improvements in manufacture and transport have been proceeding at a greater rate in the last ten years than ever before in history. In the United States factory output per head was 40 per cent greater in 1925 than in 1919. In Europe we are held back by temporary obstacles, but even so it is safe to say that technical efficiency is increasing by more than 1 per cent per annum compound. There is evidence that the revolutionary technical changes, which have so far chiefly affected industry, may soon be attacking agriculture... In quite a few years-in our own lifetimes I mean-we may be able to perform all the operations of agriculture, mining, and manufacture with a quarter of the human effort to which we have been accustomed.

   ...All this means in the long run that mankind is solving its economic problem. I would predict that the standard of life in progressive countries one hundred years hence will be between four and eight times as high as it is to-day. There would be nothing surprising in this even in the light of our present knowledge."

References

• John Maynard Keynes (1931). "Economic Possibilities for our Grandchildren", in Essays in Persuasion, W. W. Norton.
• Robert E. Lucas, Jr. (1987). Models of Business Cycles, Basil Blackwell.
• Dani Rodrik (4 May 2015). The War of Trade Models

Aulë's Children Or Durin's Folk

I have been reading the Younger Edda. I am not sure if I will finish it. Anyways, when I came to an account of the dwarves, I found I recognized many of the names.

"Then the gods set themselves in their high-seats and held counsel. They remembered how the dwarfs had quickened in the mould of the earth like maggots in flesh. The dwarfs had first been created and had quickened in Ymer’s flesh, and were then maggots; but now, by the decision of the gods, they got the understanding and likeness of men, but still had to dwell in the earth and in rocks. Modsogner was one dwarf and Durin another. So it is said in the Vala's Prophecy:

Then went all the gods,
The all-holy gods,
On their judgment seats,
And thereon took counsel
Who should the race
Of dwarfs create
From the bloody sea
And from Blain’s bones.
In the likeness of men
Made they many
Dwarfs in the earth,
As Durin said.

And these, says the Vala, are the names of the dwarfs:

Nye, Nide,
Nordre, Sudre,
Austre, Vestre,
Althjof, Dvalin,
Na, Nain,
Niping, Dain,
Bifur, Bafur,
Bombor, Nore,
Ore, Onar,
Oin, Mjodvitner,
Vig, Gandalf,
Vindalf, Thorin,
File, Kile,
Fundin, Vale,
Thro, Throin,
Thek, Lit, Vit,
Ny, Nyrad,
Rek, Radsvid.

But the following are also dwarfs and dwell in the rocks, while the above-named dwell in the mould:

Draupner, Dolgthvare,
Hor, Hugstare,
Hledjolf, Gloin,
Dore, Ore,
Duf, Andvare,
Hepte, File,
Har, Siar.

But the following come from Svarin’s How to Aurvang on Joruvold, and from them is sprung Lovar. Their names are:

Skirfer, Virfir,
Skafid, Ae,
Alf, Inge,
Eikinslgalde,
Fal, Froste,
Fid, Ginnar."

The above list includes all but one of the thirteen dwarves who travel with Bilbo Baggins in Tolkien's The Hobbit. These are Thorin, Fili (File), Kile (Kile), Oin, Gloin, Dwalin (Dvalin), Ori (Ore), Dori (Dore), Nori (Nore), Bifur, Bofur (Bafur), and Bombur (Bombor). (Presumably, Tolkien had a different opinion on the translation of the alphabet for the Eddas.) Balin, the remaining one of Tolkien's thirteen, is the name of a knight in Thomas Malory's Le Morte d'Arthur. Dain and Durin are two other dwarf names apparently taken from the Eddas. According to the Younger Edda, Gimle (Gimli?) is the name of a palace in the south of the world. Gandalf, Tolkien's greatest wizard, is also a name on the above list.

An Example With Heterogeneous Labor

Figure 1: "Labor Demand" in the Consumer Goods Industry

1.0 Introduction

In this post, I work through an example created by Arrigo Opocher and Ian Steedman. In this example, circulating capital is represented by machines of one of a continuum of types, and I compare stationary states. Unskilled and skilled workers use the machines to produce corn, along with more machines. The output of machines are needed to sustain production in future periods. In the stationary states, the same rate of profits is earned in all industries with a positive output. In fact, only the special case when the rate of profits is zero is considered here.

The (slice of) the so-called factor price frontier in this example resembles Paul Samuelson's surrogate production function. Aggregate relationships in this example are "non-perverse". In other words, they do not violate the outdated and exploded intuition of neoclassical microeconomics. The aggregate production function shows positive, but diminishing, marginal returns, in the relevant range, to inputs of factors of production. Lower wages for unskilled labor are associated with capitalists desiring to employ more unskilled labor in the economy overall.

But a perverse relationship arises in the market for corn. Corn is the only consumer good in the example. If capitalists are to want to employ more unskilled labor directly in the production of corn, the wage for unskilled labor must be higher, not lower (Figure 1). If more unskilled labor is available for production, and markets clear, more corn is produced. But when capitalists choose the cost-minimizing technology, at prices and wages they take as given, the quantity of unskilled labor used as input, in the corn industry, per bushel corn produced, decreases. This decrease overwhelms the increased output of corn, and the employment of unskilled labor in the corn industry declines.

2.0 The Technology

Consider a simple capitalist economy, composed of (unskilled and skilled) workers and capitalists. After replacing (circulating) capital goods, output consists of a single consumption good, corn. Unskilled workers are paid the wage w, and skilled workers are paid the wage W out of the harvest. Both wages are in units of bushels corn per person year. Capitalists obtain the rate of profits r. The technology consists of an infinite number of Constant-Returns-to-Scale (CRS) techniques, indexed by s. Table 1 presents the coefficients of production for a single technique.

Table 1: Inputs Required Per Unit Outputs

Inputs	Machine Industry	Corn Industry
Unskilled Labor	a(s) l(s) Person-Years	l(s) Person-Years
Skilled Labor	a(s) t(s) Person-Year	t(s) Person-Years
Machines	a(s) Machines	1 Machine
Outputs	1 Machine	1 Bushel Corn

Notice that the first column of inputs in Table 1 is proportional to - that is, a constant multiple of the - second column. This is akin to Karl Marx's assumption of a constant Organic Composition of Capital, an unrealistic assumption that simplifies price theory.

The index s for the technology is chosen from a set of real numbers, with √ 6  ≤ s ≤ 3. The parameters of a technique are defined in terms of the index as follows:

a(s) = 2 - (6/s) + (6/s²)

l(s) = 1/s

t(s) = 1/s²

Each different value of the index s is associated with the use of a different type of machine. And different quantites of unskilled and skilled labor must be used with each different type of machine to produce the output.

I compare stationary states under these assumptions:

• L person-years of unskilled labor are available for employment in the economy, with √ 6  ≤ L ≤ 3.
• T = 1 person-years of skilled labor are available for employment in the economy.
• r = 0% is the rate of profits in the stationary states considered here.
• The markets for skilled and unskilled labor both clear.
• The production of machines and corn are adapted to a stationary state. So the endowments of machines (by type) are found by solving the model, not givens.

3.0 Quantity Flows for a Given Technique

Given the type of machine, suppose the quantity of corn, c(s), produced is:

c(s) = [1 - a(s)]/t(s) = s² [1 - a(s)]

Let the number of machines, m(s), produced be:

m(s) = 1/t(s) = s²

Table 2 shows the output of the machine and corn industries, scaled to produce these gross outputs.

Table 2: Quantity FLows

Inputs	Machine Industry	Corn Industry
Unskilled Labor	s a(s) Person-Years	s [1 - a(s)] Person-Years
Skilled Labor	a(s) Person-Year	[1 - a(s)] Person-Years
Machines	s2 a(s) Machines	s2 [1 - a(s)] Machines
Outputs	m(s) Machines	c(s) Bushels Corn

For these quantity flows, the total employment of unskilled labor is s. The total employment of skilled labor is one person-year. The total inputs of machines, which are used up each year, are replaced by the output of the machine industry.

4.0 Stationary State Prices in the Special Case

Section 3 specifies quantity flows in a stationary state, given the type of machine. The capitalists choose the technique, including the machine, based on price. Let corn be numeraire, and suppose workers are paid at the end of the production period. If the same rate of profits is earned in the production of machines and corn, the following pair of equations must be satisfied for the technique in use:

p a(s)(1 + r) + a(s) l(s) w + a(s) t(s) W = p

p(1 + r) + l(s) w + t(s) W = 1

These equations have two degrees of freedom. One is eliminated by only considering the special case in which the rate of profits is zero. The other can be seen by expressing the solution as a function of, say, the wage for unskilled labor. In this sense, the solution of the system of equations for prices in a stationary state, given the special case assumption and the technique, is:

p = a(s)

W = [1 - a(s) - l(s) w]/t(s)

Or:

p = 2 - (6/s) + (6/s²)

W = - s² + s(6 - w) - 6

The wage of skilled labor, given the technique, is an affine function of the wage of unskilled labor. Figure 2 illustrates this function for three different techniques. This figure is akin to Figure 2b on page 197 of Samuelson (1962), which shows how to construct the so-called factor price frontier for Samuelson's surrogate production function.

Figure 2: Wage-Wage Curves

In a stationary state, capitalists will have adopted the cost-minimizing technique. The cost-minimizing technique, given the wage of unskilled labor, corresponds to the technique on the outer envelope (that is, the frontier) formed from all (uncountably infinite) functions that one might plot in Figure 2. One can find the technique on the frontier by setting the derivative, with respect to the index s, of the wage-wage curve equal to zero:

dW/ds = 0

Thus, the machine type used by the cost-minimizing technique, in this special case, is the following function of the wage of unskilled labor:

s = (6 - w)/2

The frontier has the equation:

W = (1/4)w² - 3 w + 3

The wage, w, of skilled labor ranges from 0 to (6 - 2 √ 6 ). The wage of skilled labor, W, ranges from 0 to 3. If the rate of profits were positive, the wage-wage frontier would lie inside the frontier found here.

5.0 Some Aggregate Markets

The results found so far can be combined.

5.1 The Market for Unskilled Labor

I have postulated that L person-years of unskilled labor and one person-year of skilled labor are available for employment in a stationary state. For quantity flows in a stationary state to fully employ both types of labor, the index for the machine type must be:

s = L

For this machine type to correspond to the cost-minimizing technique, given a rate of profits of zero and market clearing for both labor markets, the wage of unskilled labor must be the following function of unskilled labor:

w = 6 - 2 L

Figure 3 plots the wage for unskilled labor, under these assumptions, with the amount of unskilled labor firms want to hire in a stationary state. In this example, for more unskilled labor to be hired in a stationary state, its real wage must be lower. This property is particular to this example; it does not generalize.

Figure 3: Employment of Unskilled Labor

5.2 The Market for Skilled Labor

The analysis so far has shown how to determine the cost minimizing technique and the wage for unskilled labor as a function of the amount of unskilled labor employed in a stationary state. And the wage for skilled labor is a function of the wage for unskilled labor, as shown by the wage-wage frontier. The wage for skilled labor can accordingly be expressed as a function of the amount of unskilled labor employed in a stationary state.

W = L² - 6

Figure 4 shows the wage of skilled labor plotted against the quantity of skilled labor firms desire to hire in this example. In some sense, this function neither slopes up nor down.

Figure 4: Employment of Skilled Labor

5.1 The Market for Capital

Under the above assumptions, one can find the type and number of machines, m(s), produced in a stationary state. For stationary states in which different quantities of unskilled labor are employed, different types of machines will be produced. Quantities of different types of machines are incommensurable; physical measures of different types of capital cannot be plotted together on the same axis. A numeraire measure of the quantity of capital, k, can be found by taking the product of the price of machines and their physical quantity:

K = p m(s) = a(s) m(s)

Under the assumption that markets for unskilled and skilled labor clear, one can express numeraire units of capital as a function of the person-years of unskilled labor employed in a stationary state.

K = 2(L² - 3 L + 3)

Figure 5 shows the rate of profits plotted against the above quantity of capital. In this special case, the rate of profits of capital is a non-increasing function of the quantity of capital.

Figure 5: Value of Capital

6.0 Employment in the Corn Industry

The previous section shows that no phenomena that violates outdated neoclassical price theory arises in aggregate markets for unskilled labor, skilled labor, or capital, in this particular example. But consider how much unskilled labor firms, under these assumptions, want to employ in the production of corn. Figure 1 shows the graph of the wage, w, for unskilled labor against the unskilled labor, l₂, hired in the production of corn. That function can be found as:

l₂ = (-L² + 6 L - 6)/L

And this function slopes up, contrary to what neoclassical economists would have expected about half a century ago.

7.0 Conclusion

If you work through enough examples in production theory, you ought to conclude that it is hard to find any justification for mainstream theories in microeconomics. Why so many economists continue to teach archaic balderdash, and (mis)train their intuition accordingly, is a question.

References

• Arrigo Opocher and Ian Steedman (2013). Unconventional results with surrogate production functions Global and Local Economic Review, V. 17, No. 1: pp. 45-53.
• Paul A. Samuelson (1962). Parable and realism in capital theory: The surrogate production function, Review of Economic Studies, V. 29, No. 3: pp. 193-206.

A Plague On Both Your Houses

In a Bloomberg News piece, Noah Smith makes some false claims. I think his mistakes - what Eatwell and Milgate call an imperfectionist view - are widely shared among many macroeconomists. My belief that these mistakes are widely shared is not overthrown, I think, by the confusions put forth in these later posts by Stephen Williamson and Noah Smith, respectively.

First, we have the mistaken belief that in a perfect world, capitalist economies would move quickly towards equilibrium. Smith starts his column with an anecdote:

"One time, at a dinner, I asked a famous macroeconomist: 'So, what really causes recessions?'

His reply came immediately: 'Unexplained shocks to investment.'"

I take this to be an expression of the freshwater view, as embodied in models of Real Business Cycles. Cycles are to be understood as equilibrium paths responding to exogeneous stochastic shocks. Risk exists, but uncertainty does not. Recessions and depressions occur when workers voluntarily decide to take long vacations.

Second, we have mistaken understandings of price theory and how equilibrium is established:

"The market adjusts by the price mechanism. If the cost of something goes up, the price goes up to match. If demand falls, the price drops until the market clears."

I take this to be a claim that equilibrium prices are indices of relative scarcity, a belief shown to be without logical foundation about half a century ago. Ever since Robert Lucas put forth his critique in the 1970s, mainstream macroeconomists have claimed to be developing models with rigorous microfoundations. And those foundations are supposed to be provided by General Equilibrium Theory, in which agents optimize under constraints.

But many macroeconomists seem to be just ignorant of price theory, as experts in GET, such as Frank Hahn explained long ago. In the most rigorous neoclassical theory, with many commodities and many agents, the assumptions do not lead to the conclusion that prices behave that way. Nor do the theorists have a good story about how equilibrium is established. The mathematics used in mainstream macroeconomists does not allow one to find clear statements of assumptions. At least, I am unable to understand what assumptions mainstream economists think they are making on tastes, technology, and endowments in multicommodity models to justify their macroeconomic modeling. I would rather that economists turn to non-equilibrium modeling, a position that I think Robert Lucas still finds incoherent.

Third, suppose you hold that observed fluctuations in employment and output in capitalist economies can hardly be an equilibrium response. If you held the mistaken ideas about price theory that Noah Smith does, you would think that the empirical behavior of economies could only be explained by introducing some imperfection, some failure of competition, some information asymmetry, or some stickiness or slow adjustment into your theory. And given your empirical beliefs, you would think the development of theory in such a direction is a triumph of science:

"But despite these scattered denunciations and grumbles, sticky prices are enjoying a hard-fought place in the sun. The moral of the story is that if you just keep pounding away with theory and evidence, even the toughest orthodoxy in a mean, confrontational field like macroeconomics will eventually have to give you some respect."

But it is not the case that markets, including the labor market, would rapidly clear if only imperfections did not exist in a market economy. For economists to have reached this as a consensus position is a failure of their profession, not an achievement. Business cycles neither need to be explained as an equilibrium phenomenon, nor need sticky prices be invoked to explain the failure of markets to clear.

Is the topic of the above post orthogonal to a debate Paul Krugman overviews? I am of two minds on Krugman's post. I cannot be too hostile to a blog post illustrated with a homoclinic bifurcation. Maybe a solid appreciation of nonlinearity in macroeconomics is associated these days with heterodox, but not necessarily non-mainstream economics.

References

• John Eatwell and Murray Milgate (2011). The Fall and Rise of Keynesian Economics, Oxford University Press.
• Richard M. Goodwin (1990). Chaotic Economic Dynamics, Oxford University Press.
• Murray Milgate (1982). Capital and Employment: A Study of Keynes's Economics, Academic Press.

How To And How Not To Attack Marx's Economics

1.0 Introduction

I am currently reading John Roemer's Free to Lose. I thought I would outline some areas where Marx can be criticized on economic theory, as well as some areas where I do not think he is not so vulnerable. (I do not think I had previously absorbed Roemer's theory of the emergence of classes from an analysis of reproducible equilibrium. But then the Roemer work I know the best is Analytical Foundations of Marxian Economic Theory, which may predate this explanation.) Another motivation is irritation with a series of post here.

2.0 Labor Theory of Prices

For purposes of this post, I put aside the question of whether prices tend to be proportional to labor values. I think Marx rejected this theory, including in the first volume of Capital. He says so, for example, in this passage:

"From the foregoing investigation, the reader will see that this statement only means that the formation of capital must be possible even though the price and value of a commodity be the same; for its formation cannot be attributed to any deviation of the one from the other. If prices actually differ from values, we must, first of all, reduce the former to the latter, in other words, treat the difference as accidental in order that the phenomena may be observed in their purity, and our observations not interfered with by disturbing circumstances that have nothing to do with the process in question. We know, moreover, that this reduction is no mere scientific process. The continual oscillations in prices, their rising and falling, compensate each other, and reduce themselves to an average price, which is their hidden regulator. It forms the guiding star of the merchant or the manufacturer in every undertaking that requires time. He knows that when a long period of time is taken, commodities are sold neither over nor under, but at their average price. If therefore he thought about the matter at all, he would formulate the problem of the formation of capital as follows: How can we account for the origin of capital on the supposition that prices are regulated by the average price, i. e., ultimately by the value of the commodities? I say 'ultimately,' because average prices do not directly coincide with the values of commodities, as Adam Smith, Ricardo, and others believe." -- Karl Marx, Capital, V. 1 (last footnote in Chapter V.)

I take "average price" in the above passage to be referring to what has also been called "such classical terms as 'necessary price', 'natural price', or 'price of production'" (Piero Sraffa, PCMC: p. 9). And Marx is saying that prices of production do not correspond to labor values, even though he is abstracting from this distinction in the first volume of Capital. Others have also asserted that a contradiction in Marx cannot be found here:

"Writers ... like E. Bohm-Bawerk have asserted that there is a contradiction between the analyses of Volumes I and III which is certainly not to be found there unless one reads into them an interpretation different from that which Marx repeatedly emphasized." -- William J. Baumol, "The Transformation of Values: What Marx 'Really' Meant (An Interpretation)" (, V. 12, N. 1 (Mar. 1974): pp. 51-62,

3.0 Heterogeneous Labor Activities

Employees perform many distinct activities in laboring under the direction of capital. I do not think this observation is sufficient, in itself, to hinder the development of a theory organized around labor values. Consider jobs provided by supposedly unskilled labor, such as stocking shelves in a supermarket or working behind the counter in a fast food restaurant. These sort of jobs are often treated as homogenous, both by workers and employers. Workers in one or other such job can transition among them easily enough in times of high employment.

What are jobs that require vastly different levels or types of skills? I do not think this is a problem for Marx as long as relative wages can be treated as stable:

"We suppose labor to be uniform in quality or, what amounts to the same thing, we assume any differences in quality to have been previously reduced to equivalent differences in quantity so that each unit of labor receives the same wage." -- Piero Sraffa, (1960: p. 10).

As far as I can tell, this is a common position among the classical economists, with Adam Smith providing an early explanation of wage differentials.

A problem can arise here, however. Suppose some skills are acquired through an investment, such as paying for higher education. Perhaps there is a tendency for skilled workers to make decisions based on anticipated rates of return. Then, just as Wicksell effects express the dependence of the price of capital goods on distribution, so relative wages would vary with distribution. And labor values would be dependent on prices. One could then express labor value as a vector of different quantities of different types of non-competing workers. But would the assumption that the economy hangs together - e.g., all commodities are basic - work in this case? Or one could make the claim that even skilled labor is heavily produced in the household and outside of firms run for profits. And, thus, calculations of rates of return for acquisition of many skills for the worker are empirically unimportant. (I think I take this objection, as well as the first response, from Ian Steedman.)

4.0 Labor Values Dependent on Choice of Technique

I take labor values as being found from the processes used in production, as expressed in a Leontief input-output matrix and labor coefficients. The components of such matrices and vectors are given in physical units. The analysis of the choice of technique shows that the cost-minimizing technique varies with distribution. So, here too, labor values depend on prices, instead of vice-versa.

Here one could object that the choice of technique is a highly artificial problem, of interest primarily for an internal critique of neoclassical economics. In actuality, firms do not have a choice at any time of processes from a pre-existing menu. Rather technology evolves as a non-reversible process in historical time.

5.0 Volume III Invariants Cannot All Hold

In the above, I have been concentrating mostly on objections to the premises of Marx's economic theory. Let me consider a conclusion. According to Marx, accounting in labor values allows one to identify certain invariants that hold for the economy as a whole. For example, the sum of labor values for gross outputs of industry is equal to the sum of gross outputs, evaluated at prices of production. And the sum of surplus value across industry is equal to the sum of profits. According to Marx, the competition under which prices of production are formed redistributes total surplus values into aliquot quantities distributed to each industry.

Under the traditional analyses of prices of production, Marx was just wrong. For an arbitrary numéraire, not all invariants can simultaneously hold.

Four answers have been given to this issue. I do not think highly of traditional Marxists who argue that one or the other invariant should be given preference. Typically, such arguments are presented with a lot of Hegelian terminology. I find intriguing the argument that all invariants can hold if one adopts Sraffa's standard commodity as the numéraire. Duncan Foley and Gerard Duménil have proposed the new interpretation, organized around the concept of the Monetary Expression of Labor Value (MELT). As I understand it, the new interpretation makes Marx's claims too much a matter of an accounting tautology for my taste. Finally, there is the Temporal Single System Interpretation (TSSI), which I associate mainly with Alan Freeman and Andrew Kliman, although, I guess, they work with many more scholars. Of course, more invariants can be made to hold if you interpret the theory to have many more degrees of freedom.

6.0 Exploitation of Corn

A theorem in the analysis of prices of production states that the rate of profit is positive if and only if labor is exploited. Exploitation here has a technical definition; it is not an ethical concept. From John Roemer, I learn that one can argue that Marx had both ideas in mind.

Anyways, from the same analysis, one can show that same theorem holds for any commodity (that is basic or in the workers' consumption basket?). So why focus on labor? Answers have been given that deal with matters not in the math at this level of abstraction. Workers, unlike owners of commodities sold as means of production, must be brought under the direction of the capitalists when they hire them. Furthermore, the agreements laborers strike are, at best, incomplete contracts. Not all activities that the workers will be expected to perform in given situations can be prespecified. Furthermore, often some will be unpleasant, and a tug-of-war can arise between the worker and the capitalist's representative in the workplace.

Whatever you think of these rationales for focusing on the exploitation of labor, the issue of working conditions seems like a perennial concern.

7.0 Falling Rate of Profit

I do not have much to say about the theory of the falling rate of profit. I think Marx was mistaken here, but recall this is a volume 3 theory, never published in Marx's lifetime. I am aware of Marx's account of countervailing tendencies. (How is this a theory, if no explanation is given why one tendency should predominate?) And, as usual, theorists in the TSSI tradition disagree.

9.0 Outside the Theory of Value and Distribution

Such a brief overview, compared to the thousands of pages Marx wrote, and the many ways scholars and followers have read (parts of?) this work, obviously cannot cover all issues. I have said nothing about historical materialism, for instance. If this theory is read as mandating economic determinism, with no possibility of the superstructure shaping the evolution of the economic base, I, like many others, think the theory is wrong.

Nor have I said anything much about many of Marx's analyses that can be developed independently of the theory of value and distribution. For example, I like to set out Volume 2 models of simple and expanded reproduction in terms of prices of production. Whether or not Richard Goodwin's theory of the business cycle is Marxist or is descriptive of some capitalist economies at some time seems to be independent of Marx's theory of value. And Marx had many other analyses of concrete situations that might or not be worthwhile. For example, in Volume 1, he presented the introduction in Great Britain of laws regulating maximum hours of work as addressing what we would now call a prisoner's dilemma. Each mill owner would like to work their employees until their health breaks, fire them, and then hire refreshed workers. But if all mill owners are doing this for wokers from a young age, no large population of such refreshed workers will exist in the locality. So the owners need such laws after a certain level of development.

I suppose I should say something about the theory of monopoly. I do not see why prices of production cannot be developed with different markups in different industries. I may not be familiar enough with the literature, but it is my impression that many accounts of markup pricing do not take into account constraints arising from the inter-industry flows emphasized in Sraffian theory and empirical work in Leontief input-output analysis. Furthermore, markups cannot be so high in a viable economy that demands total more than the net output of a viable economy. (A theory of cost-push inflation can arise here.) This is not to say that I do not think those exploring administered, full-cost, or markup pricing are not looking at something empirically important.

And Marx had many detailed empirical observations, including claims about how feudalism evolved into capitalism. I cannot address such matters of history. Finally, I have said nothing above about the sociology of economics. I think the above is quite enough for one post.