Saturday, December 31, 2016
Welcome
The emphasis on this blog, however, is mainly critical of neoclassical and mainstream economics. I have been alternating numerical counterexamples with less mathematical posts. In any case, I have been documenting demonstrations of errors in mainstream economics. My chief inspiration here is the CambridgeItalian 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.
Monday, October 05, 2015
A Bifurcation Diagram for Hahn and Solow
Figure 1: Bifurcation Diagram for Hahn and Solow, Example 1, Generalized 
I have been writing a draft paper, "A Neoclassical Model of Pension Capitalism in which r > g". In my latest iteration, I have developed the bifurcation diagram shown above. This is a generalization for the overlapping generations model, in which the number of households can grow, but specialized to Hahn and Solow's Example 1. Example 1 specifies the form of the utility function.
One can define dynamic equilibrium paths for the model. And given the values of certain parameters, one can locate a steady state in a certain range of parameters. Always being happy to examine a model, whether it can or cannot ever be instantiated in an actually existing economy, I have identified types of steady states and their stability in certain parameter ranges. I was able to establish analytically the boundary between steady Portfolio Indifferent and Liquidity Constrained States. I located the curved dashed and solid lines towards the south east of the diagram through a mixture of analysis and numeric experimentation. This is also true for my identification of types of stability (saddlepoint, locally stable, locally unstable).
I do not fully understand the topological variation in flows for the bifurcations that I have identified. I think I understand the bifurcation, shown by the dashed line, in which a steady Liquidity Constrained State loses stability. This bifurcation most likely results from the steady state ejecting a stable or absorbing an unstable twoperiod business cycle. The former case is analogous to the logistic equation for a parameter a of 3. I can understand the bifurcation in which the steady state disappears in terms of the diagram in this post. But I find it difficult to understand how dynamic equilibrium paths differ across this bifurcation. And I have not previously gone into the details of the analysis of how two dynamic systems  in this case, for Portfolio Indifferent and Liquidity Constrained States are patched together across a bifurcation. But the linked paper illustrates what I have so far.
More complete details are provided in the linked paper. I provide more details than anybody can want in appendices so as to be able to step through the model myself, if I look at this stuff later.
Reference Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press
Wednesday, September 23, 2015
For Technical Discussions Of Cavalry Tactics At The Battle Of Austerlitz?
Figure 1: Steady States As Function Of Effective Return On Savings 
1.0 Introduction
I have previously said I am not thrilled about arguments about whether or not assumptions are realistic. In this post, I describe some analysis I have done with a model of a world that does not exist and analysis I may do in the future with some variation on such a world. The title of this post refers to this quote from Bob Solow, talking about how to respond to Robert Lucas and the new "classical" school:
"Suppose someone sits down where you are sitting right now and announces to me that he is Napoleon Bonaparte. The last thing I want to do with him is to get involved in a technical discussion of cavalry tactics at the battle of Austerlitz."  Robert Solow2.0 Generalization of Hahn and Solow's Model of Overlapping Generations
I have previously outlined a microfounded macroeconomic model of overlapping generations, presented in Hahn and Solow (1995). They use this model to show that claims, from new classical economists and their followers, of the desirability of perfectly flexible prices and wages are unjustified, even on their own theory. They do not think of this model as a good empirical description of any actually existing economy. Hahn and Solow present another model as a prototype of the direction in which they thought macroeconomics should have developed.
Hahn and Solow consider case where one household is born at the start of each year. Under their assumptions, a stationary state is characterized by an equality between a certain function of the effective rate of return on savings and certain model parameters:
g(Q) = [ξ/(ξ  1)] [β/(1  β)]
The parameter ξ relates to the Clower cashinadvance contraint. The parameter β is for the aggregate CobbDouglas production function. Parameters and the form of the utility function are embodied in the function g.
I consider a slight modification to this model. Suppose the number of households born each year is no longer constant. Specifically, let the number of households born at the start of year t, h_{t}, grow at the rate G:
h_{t} = G^{t},
where:
G ≥ 1.
I have worked through this model somewhat. A steady state exists if only if the following equality holds for the effective rate of return on savings:
g(Q) = G [ξ/(ξ  1)] [β/(1  β)]
Along a steady state growth path, the nominal price of corn declines so as to maintain a constant real money supply. Hahn and Solow also have that the supply of money is a fixed quantity. They need this assumption, I guess, for their abstract discussion of policy responses to a shock to make sense.
3.0 Other GeneralizationsHere are some other possible generalizations and explorations one might make to the model:
 Household lives more than two years.
 Endogenous supply of labor, with leisure entering the utility function.
 Introduction of a bequest motive.
 Heterogeneous households.
 Nonhomothetic preferences.
 Various specific forms of utility functions.
 Multiple sectors in production, instead of the production of a single good.
 Introduction of fixed capital (with radioactive depreciation), instead of only circulating capital.
 Various specific forms of production functions.
 Introduction of stochastic noise.
 Analysis of reactions to different kind of shocks.
 Introduction of government, foreign trade.
 More detailed analysis of money, finance, and banks.
The above outlines a research program, not necessarily original. Econometricians can go through models in this family in the literature, trying to find the best fit for some time period and country. From what little I know, one can find models with one generalization and not another, or vice versa. A theoretician might want to try to develop a model that combines some generalizations, thereby advancing the field.
4.0 Empirical Applicability of Generalized Model?This program entails lots of work, some of it empirical. How could an outsider have standing to criticize this approach?
Truthfully, the mathematics is mostly tedious algebra, only not at a high school level because of the length of the derivations. I suppose the concepts I am applying here are deeper than that. Sometimes one gets to the level of high school calculus, what with LaGrangians and all. (If I can develop a fairly comprehensive and interesting bifurcation diagram for some models, I will consider myself to be approaching advanced mathematics.) Some conventional concepts from economics (marginal conditions, excess demand functions, Walras' law, steady states) help organize the approach.
One who has learned the details of such a program might react negatively to criticism. The supposedly unrealistic assumptions you object to are maintained for analytical tractability. Past developments have supposedly shown us how to relax assumptions. One can be confident that future developments will continue to show us how to generalize the models and how to remove more scaffolding, leaving the building untouched. And, if analytical developments, such as tractable models of imperfect competition, lead to widescale changes, we will adopt them if empirical data shows such changes to be warranted.
But are there some assumptions that are untouched by such a program, that are always maintained, and that render all models (admittedly, internally consistent) developed along these lines forever empirically inapplicable?
4.1 How Are Dynamic Equilibrium Paths Found?Under the assumption of perfect competition, prices and wages are assumed to be flexible. This is assumed to imply that markets in each period instantaneously clear. I do not understand why anybody uptodate on economic theory should believe this?
4.2 No Keynesian UncertaintyHouseholds and firms are assumed to know what the usual range of interest rates, for example, will be in 60 years, in only probabilistically. This does not seem to be plausible to me.
5.0 ConclusionsI intend to pursue some generalizations suggested above. (I could be distracted by trying to develop a bifurcation diagram by a Hahn and Solow model in a later chapter.) The point of the mathematics is to tell a story of some fantasy or science fiction world. This sort of project, to me, does not to make empirical claims. Rather I am interested in whether qualitatively similar stories can be told with some complications. Which, if any, generalizations undermine such stories?
Monday, September 14, 2015
Paul Krugman Stumbles
In his editorial in the New York Times this morning (14 September 2015), Paul Krugman writes about Jeremy Corbyn and the British Labour Party. The establishment politicians in Labour are none too happy about Corbyn's victory. Krugman criticizes these establishment politicians for accepting Tory canards on recent economic history in the United Kingdom, with the former Labour government supposedly being at fault. Krugman's concluding paragraph is:
"Beyond that, however, Labour's political establishment seems to lack all conviction, for reasons I don't fully understand. And this means that the Corbyn upset isn't about a sudden left turn on the part of Labour supporters. It's mainly about the strange, sad moral and intellectual collapse of Labour moderates."  Paul Krugman
I have no comment on the substance of Krugman's editorial. However, when I read "lack all conviction", I hear an echo of W. B. Yeat's poem, "The Second Coming". I have in mind the following lines:
"The best lack all conviction, while the worst
Are full of passionate intensity."  W. B. Yeats
This allusion, if intended, is backwards from the article. That is, it would suggest that Labour establishment is composed of the best, contradicting the rest of the article.
I do like Krugman's previous allusions to Talking Heads lyrics.
Thursday, September 03, 2015
Failure To Replicate Hahn And Solow (1995), Figure 2.1
Figure 1: Stationary States As Function Of Effective Return On Savings 
1.0 Introduction
In Chapter 2 of their Critical Essay, Frank Hahn and Robert Solow present an overlapping generations model^{1}. This model exhibits rational expectations and perfectly flexible wages and prices. Thus, all markets, including the labor market clear. Hahn and Solow argue that even in such a model, unacceptable fluctuations in national income can arise. Room arises, even under these severe assumptions, for a national government to pursue macroeconomic policy.
I am interested in how mainstream models can exhibit counterintuitive behavior, including bifurcations of steady states and interesting nonsteady state dynamics. The endogenous generation of cyclical or aperiodic orbits is among the dynamics in which I am interested. Hahn and Solow suggest that this model can have different numbers of stationary states and can have orbits that fail to converge to stationary states.
I have looked at other models of overlapping generations before. So I thought I would look into Hahn and Solow's model. They provide two examples of specific forms of utility functions for their model. This post documents my reasons for thinking their first example cannot replicate certain qualitative properties of their model that they claim can arise in general.
2.0 Overlapping Generations ModelThe model consists of four markets, for a consumer good, for corporate bonds ("real capital"), for money, and for labor. The supply and demands in these markets are generated by two institutions, households and firms. In this section, I basically echo Hahn and Solow's description of their model. I am particularly interested in three parameters, one for the utility function, one for the production function, and the last for characterizing a liquidity constraint.
2.1 HouseholdsEvery year, one household is born. Households live two years. During the first year, they supply one personyear of labor, and they are paid their wages at the end of the year. At the end of the first year, they consume some of their wages and save the rest. They are retired and do not labor^{2} during their second year. At the end of the second year, they consume all of their savings, and then die.
Households can save their income in the form of two assets:
 Money, which earns a real return only if prices decline while a household holds it^{3}.
 Corporate bonds, which at the end of each year are paid off with the full (accounting) profits earned by firms.
Households would prefer to hold their savings only in the form of the asset with the larger real return. However, a transactions demand for money is introduced in the form of a Clower cashinadvance constraint^{4}.
Formally, the household born at the start of year t must choose decision variables to solve the following nonlinear program:
Maximize u(c_{t,t}, c_{t,t + 1})
such that:
c_{t,t} + s_{t} ≤ w_{t}
c_{t,t + 1} ≤ Q_{ξ}(R_{t}) s_{t}
c_{t,t + 1} ≤ ξ m_{t} p_{t}/p_{t + 1}
The first constraint specifies that the sum of the consumption and savings at the end of the household's first year cannot exceed the wages received by the household at that point in time. The second constraint states that the consumption at the end of the second year cannot exceed savings, accumulated during that year at the effective rate of return on savings, Q_{ξ}(R_{t}). The notation for the effective rate of return reflects the dependence of that rate on the real rate of return, R, on corporate bonds and a parameter, ξ, arising in the third constraint. The third constraint is the Clower cashinadvance condition. The household must hold at least some given fraction (namely, 1/ξ) of the consumption planned at the end of the last period in the form of money during this period^{5}, where
ξ > 1
In a state of Portfolio Indifference (PI), the real rate of return for money and for corporate bonds are equal. On the other hand, if households are Liquidity Constrained (LC), they would prefer to hold savings at the higher rate of return provided by corporate bonds, but cannot because of the Clower constraint. The effective rate of return on savings is therefore less than the rate of return on real capital.
2.1.1 Hahn and Solow's First ExampleTo be a bit more concrete, Hahn and Solow gives two examples of possible forms of the utility function. The first is:
u(c_{t,t}, c_{t,t + 1}) = (1/α)(c_{t,t})^{α} + (1/α)(c_{t,t + 1})^{α}
where,
α < 1
Sometimes it is more convenient to express the solution of the household's program in terms of the parameter ε:
ε = α/(α  1)2.2 An Aggregate CobbDouglas Production Function
The firms are characterized by an aggregate production function^{6}. To be concrete, they specify a CobbDouglas form:
y_{t} = (k_{t  1})^{β} (l_{t})^{β + 1}
where:
0 < β < 1
The wage, the real rate of return on corporate bonds, the demand for labor, and the supply of corporate bonds (also known as the demand for capital) come out of the usual profitmaximizing analysis. The demand for labor is constrained to match the households' supply of one personyear per year. That is, with flexible wages and prices, the labor market is assumed to clear.
3.0 Stationary StatesBy solving the above model, one can find excess demands, at the end of each year, for the produced commodity, corporate bonds, and money. Along a dynamic equilibrium path, excess demands in all three markets are zero. As I understand it, solving for one state variable, the rate of return on corporate bonds, in each year is sufficient to trace out such paths. Stationary states, if any exist, are found by dropping time indices.
Stationary states are conveniently expressed in terms of the following function.
g(Q) = Q s(Q)
where s(Q) is the stationary state savings found by solving the household's constrained maximization problem and substituting in a wage of unity in the solution^{7}.
Exactly one real rate of return, R, corresponds to each each stationary state value of Q, and vice versa. The parameters α and ξ enter into this invertible function. The following equation is a necessary and sufficient condition for a stationary state:
g(Q) = [ξ/(ξ  1)] [β/(1  β)]
Figure 1 graphs g(Q) and the Right Hand Side of the above equation for given parameters in Example 1. The horizontal line can be lowered or raised, within a certain range, by varying, β the parameter in the production function, while leaving other curves unchanged. It is a bit more complicated to analyze the effects of varying ξ. α enters into the shapes of the upwardsloping curves. For this example, they all take on a value of 1/2 at Q = 1.
Anyways, Hahn and Solow present a figure showing possible shapes and locations of g(Q). And they comment on the number and types of possible stationary state equilibria. Table 2 summarizes and compares and contrasts their and my results. I have been unable to find an example with two LCS in their example.
Hahn and Solow Possibilities  Example 1 Possibilities 


I was hoping to find a model with multiple equilbria for some subset of the parameter space. Perhaps I have made some simple error in algebra, but I was disappointed to not find such. This post does not say that Hahn and Solow are in error. They do not claim multiple equilibrium can arise for every conventional form of the utility function in their problem. I guess I'll have to focus on their second example^{8}.
Update (10 September 2015): I've convinced myself that neither Hahn and Solow's Example 1 or Example 2 can exhibit one PIS and two LCS. The derivative of g(1) is upwardsloping in both cases, unlike in Hahn and Solow's diagram for the case of three equilibria. (I do not see offhand why Hahn and Solow rule out a case of in which no PIS exists, but two LCS do.)
Footnotes This model is in the style of the macroeconomics that they are criticizing from the inside. Chapter 6 presents a prototype model more in the spirit of how Hahn and Solow think macroeconomics should be pursued. This model is without an exact reduction to microeconomics, with a labor market which is justified by an earlier gametheoretic analysis of social norms, and with imperfect competition in product markets.
 In other models of overlapping generations, how much labor a household supplies each year is a decision variable.
 In a stationary state, prices are stationary and money earns a real return of unity.
 I had not recognized a Clower constraint before. Presumably, it is not original with this book; Robert Clower's work in macroeconomics goes back to at least the 1960s.
 Hahn and Solow suggest this unrealistic approach to the transactions demand for money can be justified by a deeper analysis.
 Sometimes economists justify ignoring the Cambridge Capital Controversy on the grounds that there are so many other problems with mainstream economics that one need not focus on capital theory. This model illustrates this claim.
 This definition only works for homothetic utility functions, another unrealistic assumption justified here by the critical intent of the model.
 I like that their second household has a parameter for timediscounting for households, anyways.
 Hahn, Frank and Robert Solow (1995). A Critical Essay on Modern Economic Theory, MIT Press
Friday, August 21, 2015
Paul Romer Gyring In A CulDeSac
Paul Romer continues to display his confusion. In reverse chronological order, you can look here, here, here, here, and so on. Also see Noah Smith.
Romer continues to put forward ever more false dichotomies and other simpleminded logical fallacies. For example, he seems to say economics has a choice between talky, nonscientific 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, marketbymarket, supplyanddemandish 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 AngloAmerican 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 WalrasCassel 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 AngloAmerican 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 AngloAmerican 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, I thought 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 nonrigorous, roughandready 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 putdowns 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 ArrowDebreu 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 nonergodic 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
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 costminimizing 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 FunctionsConsider 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 alphatype land.
 p_{β} is the rent for betatype land.
 w_{1} is the wage for unskilled labor.
 w_{2} is the wage for skilled labor.
The indirect average cost function for widgets produced on land of type alpha is:
c_{α}(p_{α}, w_{1}, w_{2}) = (1/2)[(w_{1} p_{α})^{1/2} + (w_{1} w_{2})^{1/2}+ (w_{2} p_{α})^{1/2}]
The indirect average cost function for widgets produced on land of type beta is:
c_{β}(p_{β}, w_{1}, w_{2}) = (3/5)(w_{1} p_{α})^{1/2} + (3/10)(w_{1} w_{2})^{1/2}+ (11/20)(w_{2} 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 costminimizing 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}, aw_{1}, aw_{2}) = a c_{α}(p_{α}, w_{1}, w_{2})
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 offdiagonal 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 WageWage FrontierConsider 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, w_{1}, w_{2})
Or:
w_{1, α} = [(2  w_{2}^{1/2})/(1 + w_{2}^{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 personyear, 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 tradeoff in long run equilibrium between the wages of unskilled and skilled labor.
w_{1, β} = [(20  11 w_{2}^{1/2})/(12 + 6 w_{2}^{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: WageWage 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 costminimizing 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 w_{2}  9 w_{2}^{1/2} + 4 = 0
This equation can be factored:
(w_{2}^{1/2}  1)(5 w_{2}^{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, (w_{1}, w_{2}), 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 LaborWe 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}(w_{1}, w_{2}) = (1/4)(w_{1}^{1/2} + w_{2}^{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}(w_{1}, w_{2}) = (1/40)(12 w_{1}^{1/2} + 11 w_{2}^{1/2})
In what I hope is obvious notation, personyears of unskilled labor employed per unit output of widgets is, depending on the type of land used:
l_{1, α}(w_{1}, w_{2}) = (1/4)(1 + w_{2}^{1/2})/(w_{1}^{1/2})
l_{1, β}(w_{1}, w_{2}) = (3/20)(2 + w_{2}^{1/2})/(w_{1}^{1/2})
Finally, personyears of skilled labor employed per unit output of widgets is given by one of the following two functions of wages:
l_{2, α}(w_{1}, w_{2}) = (1/4)(1 + w_{1}^{1/2})/(w_{2}^{1/2})
l_{2, β}(w_{1}, w_{2}) = (1/40)(6 w_{1}^{1/2} + 11)/(w_{2}^{1/2})5.0 Bringing it all Together
The above algebra can be used to generate various graphs. Figure 1 shows personyears 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 personyears and 16/25 widgets per personyear 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 costminimizing 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 costminimizing 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 costminimizing. (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 costminimizing at the other switch point. The coefficients of production vary among, for example, the costminimizing techniques in which alphatype land is used at a switch point. Hence, as noted, no reswitching of techniques exists in this example.
6.0 ConclusionThis example has costminimizing 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 alphatype land, for the same level of output. Thus, land has a higher proportion of total unit cost when betatype 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 Ushaped curves.
So much for explaining wages and employment by wellbehaved 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