Saturday, December 31, 2016
The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counter-examples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the Cambridge-Italian economist Piero Sraffa.
In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.
I've also started posting recipes for my own purposes. When I just follow a recipe in a cookbook, I'll only post a reminder that I like the recipe.
Comments Policy: I'm quite lax on enforcing any comments policy. I prefer those who post as anonymous (that is, without logging in) to sign their posts at least with a pseudonym. This will make conversations easier to conduct.
Friday, August 21, 2015
Romer continues to put forward ever more false dichotomies and other simple-minded logical fallacies. For example, he seems to say economics has a choice between talky, non-scientific political advocacy or rigorous mathematical economics. And he gets his history wrong:
"Over the five decades from 1890 to 1940 (a time when physicists developed mathematical theories of statistical mechanics, quantum mechanics and both special and general relativity) economists avoided the use even of calculus and spent 50 years mired in the confusion spawned by the talky, market-by-market, supply-and-demand-ish approach to economic analysis codified in 1890 in Alfred Marshall's Principles of Economics." -- Paul Romer
I suppose one can be generous and take Romer to be confining himself to Anglo-American economics. Obviously, economists such as Leon Walras, Gustav Cassel, and Frederick Zeuthen were analyzing mathematical models. (As I understand it, Zeuthen was the first to formulate the Walras-Cassel model with inequalities.) And, I guess in this tradition, Abraham Wald, in 1935, provided the first rigorous proof of the existence of a general equilibrium.
But even when restricted to Anglo-American economics, Romer is not quite correct. J. R. Hicks, with his 1939 edition of Value and Capital and earlier papers with R. G. D. Allen, reintroduced General Equilibrium theory into Anglo-American economics, with as many derivatives, matrices, etc. as you please.
Romer's comments about "talkiness" are silly. I would be embarrassed to dismiss a scholar like Fernand Braudel on the grounds that he did not put forth mathematical models, as in physics.
Romer is just as silly on the other side of his false dichotomy. He's seems to think that as long as a model is put forth in terms of valid mathematics, it is rigorous. Here's what he writes about Solow's growth model:
"Robert Solow (a close colleague of Samuelson's at MIT) ... showed how to describe the behavior of an economy in which things did change. By restricting attention to a single type of output, Solow developed a workable framework for talking about changes in wages, the return to capital, and total output." -- Paul Romer
When I read that in context, Romer was just expressing himself badly. This is in the midst of a short overview about Paul Samuelson's contributions to economics, a task I would find Herculean. Maybe Romer knows that Solow's model is, at best, a non-rigorous, rough-and-ready framework for empirical work. But he really does think otherwise, that Solow's model is rigorous:
"Solow's explicit dynamic model of growth based on an aggregate production function was a solid piece of SAGE [Simple, Applied General Equilibrium] theory. After all, if new Chicago and the rest of the profession agree on one part of good theoretical practice, this has to signal something." -- Paul Romer
The above is just false. The rest of the profession do not agree.
What would have to be the case for Solow's model to apply in a world in which more than one commodity is produced? One set of assumptions is that, in some sense, effectively one commodity is produced. At any given time, the capital stock could be disassembled and costlessly transmuted into either any consumption good or any other collection of capital goods, and vice versa. Then, the historical cost of capital goods, the current prices of capital goods, and their present value would not diverge. On the other hand, these costs do diverge in actual economies set in historical time. The above is a summary of a substantive argument from Joan Robinson, who jokingly claimed that neoclassical economists thought of capital goods as meccano sets or ectoplasm.
Romer resolutely refuses to address the substance of either side of the Cambridge Capital Controversy. (And there are other points than the above. Is Romer even aware of the existence of Piero Sraffa or Pierangelo Garegnani?) Instead, he whines about Robinson's tone:
"...the sarcasm and put-downs that were a part of British intellectual life that Solow had to confront in his exchanges with Joan Robinson." -- Paul Romer
And he attacks Joan Robinson's motives:
"In so doing, he used the same techniques that economists from Cambridge England used to attack his model of output as a function of a stock of capital. Joan Robinson probably had the same concern. What will young Samuelson and Solow do with all their maths? Because an aggregate production function might lend support for a marginal productivity theory of the distribution of income, perhaps we should strangle it in the crib." -- Paul Romer
The above is simply ad hominem. Apparently, some have sent email to Romer with similar points. He then cites Roger Backhouse as an authority, while doubling down on the ad hominem.
I suppose I cannot complain about Romer's treatment of Robinson. Romer's knowledge of General Equilbrium theory seems to be lacking, and he treats Frank Hahn and Robert Solow's objections to macroeconomics after Lucas no more seriously. He complains about their tone, but pretends they had no substance to their complaints. Is Romer even aware of Hahn's attempts to integrate money into the Arrow-Debreu model and his outline of the difficulties? Is Romer even aware of the existence of Hahn and Solow's 1995 monograph? To be generous to Romer, I suppose one could say the latter is only of retrospective importance when considering the controversies in macroeconomics in the 1970s.
I might as well conclude with another example of silliness from Romer. Here Romer tries to explain one of Lucas's contributions:
"Then Robert Lucas showed how to add uncertainty to a version of the Samuelson and Diamond models. This let him pin down loose conjectures from Keynes about the role of expectations." -- Paul Romer
Now, Chapter 12 in the General Theory is often turned to when one wants to read Keynes on expectations. And in that chapter, one finds:
"By 'very uncertain' I do not mean the same thing as 'very improbable'. Cf. my Treatise on Probability..." -- John Maynard Keynes (1936, p. 148).
Romer is equivocating. As far as I know, Lucas did not introduce uncertainty in any mathematical models in economics. (Can anybody find Lucas explicitly discussing the inconsistency between rational expectations and non-ergodic time series?) So Romer should either not reference Keynes at all (with silliness about "loose conjectures") or talk about Lucas modeling probability (also known as risk) or expand on his text to show how Lucas was actually modeling Keynes's uncertainty. That is, Romer should if he has any interest in the truth value of his statements.
I think the above is not one of my better posts. Too uniformly negative even for me and too wandering. But I think Romer should try not to commit simple logical fallacies in his complaints about lack of scholarship and rigor among economists.References
- Braudel, Fernand (). Civilization and Capitalism, 15th - 18th Century, Volume 1: The Structure of Everyday Life.
- Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Macroeconomic Theory, MIT Press.
- Hicks, J. R. (1939). Value and Capital (1st edition).
Monday, July 27, 2015
|Figure 1: Skilled Labor Hired by Firms per Unit Output|
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:
- pα is the rent for alpha-type land.
- pβ 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:
cα(pα, w1, w2) = (1/2)[(w1 pα)1/2 + (w1 w2)1/2+ (w2 pα)1/2]
The indirect average cost function for widgets produced on land of type beta is:
cβ(pβ, w1, w2) = (3/5)(w1 pα)1/2 + (3/10)(w1 w2)1/2+ (11/20)(w2 pα)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:
cα(apα, aw1, aw2) = a cα(pα, 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 = cα(1, w1, w2)
w1, α = [(2 - w21/2)/(1 + w21/2)]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.
w1, β = [(20 - 11 w21/2)/(12 + 6 w21/2)]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 w21/2 + 4 = 0
This equation can be factored:
(w21/2 - 1)(5 w21/2 - 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:
tα(w1, w2) = (1/4)(w11/2 + w21/2)
The number of acres of land of type beta per unit output of widgets, when land of that type is used, is:
tβ(w1, w2) = (1/40)(12 w11/2 + 11 w21/2)
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:
l1, α(w1, w2) = (1/4)(1 + w21/2)/(w11/2)
l1, β(w1, w2) = (3/20)(2 + w21/2)/(w11/2)
Finally, person-years of skilled labor employed per unit output of widgets is given by one of the following two functions of wages:
l2, α(w1, w2) = (1/4)(1 + w11/2)/(w21/2)
l2, β(w1, w2) = (1/40)(6 w11/2 + 11)/(w21/2)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
Wednesday, July 15, 2015
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.
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).
Tuesday, June 30, 2015
|Figure 1: Recurrence of Capital-Output Ratio|
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
0 < s < 1
0 < γ
γ ≠ 1
- 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|
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.
Saturday, June 20, 2015
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 experimental 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.
Saturday, June 06, 2015
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.)