Wednesday, January 01, 2020
Welcome
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.
Saturday, March 17, 2018
Economists Critical Of Mainstream Economics Not Going Into Academia
Here are three books I have read:
- Paul Ormerod (1994). The Death of Economics. St. Marin's Press.
- Stanley Wong (1978, 2006). Foundations of Paul Samuelson's Revealed Preference Theory: A Study by the Method of Rational Reconstruction. Routledge
- J. E. Woods (1990). The Production of Commodities: An Introduction to Sraffa. Humanities Press.
These authors have two things in common. All three were critical of some aspects of mainstream economics. And they all ended up in business. Looking at the blurbs on their books, I see some spent more time in academia than I recalled.
I wonder if one can find something like a trend. Are there many economists that have come out of well-regarded economics department and had too critical a mind? And they ended up either in business or in departments less well-regarded? Maybe Thomas Palley (Yale?) fits in here.
Of course, there is another phenomenon of engineers, mathematicians, and scientists looking at economics from the outside. Mirowski is good on this theme. John Blatt is somebody Post Keynesians might cite here. Notice, this goes back well before the recent enthusiasm for econophysics.
Saturday, March 10, 2018
A Generalized Reswitching Pattern
Figure 1: Switch Points Varying with Time |
This post presents a perturbation of a fluke switch point. At this switch point, the wage curves for four techniques are tangent. In the jargon I have been inventing, this is another four-technique, local pattern. In other words, a perturbation of appropriately selected parameters - for example, coefficients of production - changes the sequence of wage curves and switch points on the wage frontier. The perturbation can be viewed as the result of technical progress. When I worked the example, I was surprised to find some other fluke cases.
The numeric example is an instance of the Samuelson-Garegnani model. Of interest to me, is a generalization to a continuum of techniques with wage curves tangent at the switch point. A perturbation leads to an example with no switch points, but the cost-minimizing technique varies continuously along the wage frontier, and techniques recur. So this generalization will have the structure of an example in Garegnani (1970). Kurz and Salvador (2003) later simplified this famous example. In some sense, I am offering a further simplification. But, perhaps, my example is more complicated along other dimensions, insofar as my pattern analysis is original.
2.0 TechnologyIn this economy, a single consumption good - called corn - is produced from inputs of labor and a specified grade of iron. The grade of iron is indexed by v, u_{1}, u_{2}, and u_{3}. Each grade of iron is itself produced from inputs of labor and that grade of iron. Table 1 shows the processes available, at each point in time, for producing iron. Similarly, Table 2 defines the processes available for producing corn. This is a circulating capital model. The iron inputs are totally used up in a single production period.
Input | Process | |||
v | u_{1} | u_{2} | u_{3} | |
Labor | (2/5) | (5/18)e^{-(t - 1)σ1} | (49/360)e^{-(t - 1)σ2} | (2/45)e^{-(t - 1)σ3} |
v Iron | (1/5) | 0 | 0 | 0 |
u_{1} Iron | 0 | (1/4)e^{-(t - 1)σ1} | 0 | 0 |
u_{2} Iron | 0 | 0 | (13/40)e^{-(t - 1)σ2} | 0 |
u_{3} Iron | 0 | 0 | 0 | (2/5)e^{-(t - 1)σ3} |
Input | Process | |||
v | u_{1} | u_{2} | u_{3} | |
Labor | 2 | (20/9) | (23/9) | (26/9) |
v Iron | 1 | 0 | 0 | 0 |
u_{1} Iron | 0 | 1 | 0 | 0 |
u_{2} Iron | 0 | 0 | 1 | 0 |
u_{3} Iron | 0 | 0 | 0 | 1 |
Four techniques for producing a net output of corn exist. Each technique consists of a process for producing iron of a specific grade and a process for producing corn with that grade of iron. For my notes when extending this example to a continuum of techniques, I want to note the following restriction and relationships among coefficients of production:
(1/5) ≤ a_{1,1}(u, 1) < (1/2)
a_{0,1}(u, 1) = (10/9)[1 - 2 a_{1,1}(u, 1)]^{2}
a_{0,2}(u, 1) = (10/9)[1 + 4 a_{1,1}(u, 1)]3.0 Prices
Suppose the technique defined by the u grade of iron is in use. Consider the associated prices of production, for the period of production ending at time t. Let r be the rate of profits, w_{u}(r, t) be the wage, and p_{u}(r, t) the price of u-grade iron. Prices are production satisfy the following system of two equations:
p_{u}(r, t) a_{1,1}(u, t) R + a_{0,1}(u, t) w_{u}(r, t) = p_{u}(r, t)
p_{u}(r, t) R + a_{0,2}(u, t) w_{u}(r, t) = 1
where:
R = 1 + r
A bushel corn is the numeraire.
One can solve this system for the wage and the price of corn, each as a function of the rate of profits and time. The wage, as a function of the rate of profits, is called the wage curve for the technique. A different wage curve is defined for the technique defined by each grade of iron. The wage curve for the v-grade of iron does not shift with time.
4.0 Choice of TechniqueThe wage curves, for each of the techniques defined by a grade of iron, can be plotted on the same graph. This graph depicts wage curves and the wave frontier at a given point in time. The wage frontier is the outer envelope of the wage curves. The technique(s) that contribute their wage curve(s) to the frontier are cost minimizing for the corresponding rate of profits or wage.
4.1 The Wage Frontier at t = 1Figure 2 shows the wage frontier at t = 1. The technique defined by v-grade iron is cost-minimizing for all feasible rates of profits. All four techniques are cost-minimizing at the single switch point. All four wage curves are tangent at the switch point. This is a fluke.
Figure 2: Wage Curves Tangent at Switch Points |
In Figure 2, I have indicated the rate of technical progress for the three techniques defined by u_{1}, u_{2}, and u_{3}. But, with the way I have defined technical progress, these rates do not matter for prices of production at time t = 1. Furthermore, for any time less than unity, the wage frontier is the same. The wage curve for v-grade iron does not shift, and the technique defined by v-grade iron is uniquely cost-minimizing for all feasible rates of profits. The story is different for as time goes on after t = 1.
4.2 The Shift in Wage Frontier when Technical Change is Faster for Smaller a_{1,1}(u, 1)First, consider the case when σ is smaller for a larger index u for the grade of iron. Figure 3 graphs such a case, for a time larger than t = 1. This is an example of reswitching, between the techniques defined by v-grade and u_{1}-grade iron. The wage curves for the techniques defined for u_{2}-grade and u_{3}-grade iron appear on the frontier only at t = 1 and only at the switch point. Otherwise, this is a perturbation analysis, for these rates of technical progress, that yields a traditional reswitching example.
Figure 3: A Reswitching Example |
4.2 The Shift in Wage Frontier when Technical Change is Faster for Larger a_{1,1}(u, 1)
I created this example more with this case in mind. In obvious notation, define the rates of technical progress by:
σ_{u} = (1/10) a_{1,1}(u, 1)
Figure 4 graphs the wage frontier shortly after t = 1. The wage curves for the techniques defined by v-grade, u_{1}, and u_{2} each appear in two separate regions on the wage frontier. The single switch point has become six switch points. Perturbation of the coefficients of production for the fluke switch point has yielded an example of the recurrence of techniques.
Figure 4: Recurrence of Techniques |
Around the three switch points at the larger rates of profits, a higher wage is associated with the adoption of a cost-minimizing technique where more labor is employed per unit of the consumption good produced. When will mainstream economists stop telling lies to students about price theory and the logic of minimum wages?
Figure 5 graphs the wage frontier at the following time:
t = 1 - 40 ln(4/5)
This is an example that is simultaneously a pattern across the wage axis and over the axis for the rate of profits. The technique defined by v-grade iron, and the associated switch points, is disappearing from the wage frontier. I did some work to previously create such a global pattern. I do not know what specific, presumably special case conditions, I have imposed to make this pattern come about. These numeric examples keep
Figure 5: Switch Points on Both Axes |
Figure 6 graphs the wage frontier at an even later point in time. Three switch points appear on the frontier. Three wage curves intersect at the switch point at the larger rate of profits. This is what I call a three-technique pattern. The wage curve for the u_{2}-grade iron is disappearing from the wage frontier.
Figure 6: A Three-Technique Pattern |
The diagram at the top of this post summarizes my analysis for this case. The rates of profits for switch points are plotted against time. The maximum rate of profits is also shown.
5.0 ConclusionI have exhibited a numerical example in which four-wage curves are tangent at a single switch point. Technical progress alters certain parameters - that is, coefficients of production - for three of the four techniques. For any time less than the time at which the fluke switch point occurs, no switch point exists. Given a certain simple specification of the rates of technical progress, the switch point breaks up into six switch points for a small increase in time. Three of the four technique recur on the wage frontier. In my jargon, the fluke case is a four-technique pattern. It generalizes the reswitching pattern I have previously defined. I claim that I can create a n-technique generalized reswitching pattern, for any finite n greater than unity. I can also create generalized reswitching pattern with a continuum of wage curves tangent at the single switch point.
My next steps, if I go on, might be to explicitly write up the generalization to a continuum of techniques. I should also find a closed-form for the time at which the above three-technique pattern occurs. (I found it through a bisection algorithm.)
References- P. Garegnani (1970). Heterogeneous Capital, the Production Function and the Theory of Distribution. Review of Economic Studies, V. 37, no. 3: pp. 407-436.
- Heinz D. Kurz and Neri Salvadori (2003). Reswitching - Simplifying a Famous Example. In Kurz and Salvadori (eds.) Classical Economics and Modern Theory: Studies in Long-Period Analysis Routledge.
Saturday, March 03, 2018
Update To A Start On A Catalog Of Switch Point Patterns Of High Co-Dimension
I have been looking at patterns of switch points. A pattern is a configuration of switch points helpful for perturbation analysis for the choice of technique. I am curious how the switch points and the wage curves along the wage frontier can alter with parameters, in a model of the production of commodities. Such a parameter can be a coefficient of production; time, where a number of parameters are functions of time; or the markup in an industry or a number of industries. A normal form exists for each pattern. The normal form describes how the techniques and switch points along the frontier vary with a selected parameter value. Each pattern is defined by the equality of wage curves at a switch point and one or more additional conditions. The co-dimension of a pattern is the number of additional conditions.
I claim that local patterns of co-dimension one, with a switch point at a non-negative, feasible rate of profits can be described by four normal forms. I have defined these patterns as a pattern over the axis for the rate of profits, a pattern across the wage axis, a three-technique pattern, and a reswitching pattern. This post is an update, and continues to examine global patterns, local patterns with a co-dimension higher than unity, and sequences of local patterns. Some examples are:
- A switch point that is simultaneously a pattern across the wage axis and a reswitching pattern (a case of a real Wicksell effect of zero). This illustrates a pattern of co-dimension two.
- A reswitching example with one switch point being a pattern across the wage axis (another case of a real Wicksell effect of zero). This is a global pattern.
- An example with a pattern across the wage axis and a pattern over the axis for the rate of profits. This is a global pattern.
- A pattern like the above, but with both switch points being defined by intersections of wage curves for the same two techniques. This is a global pattern.
- Two switch points, with both being reswitching patterns, can be found from a partition of a parameter space where two loci for reswitching patterns intersect. This gestures towards a global pattern.
- A pattern across the point where the rate of profits is negative one hundred percent, combined with a switch point, for the same techniques, with a positive rate of profits (of interest for the reverse substitution of labor). This is a global pattern.
- An example where every point on the frontier is a switch point. This is a global pattern of an uncountably infinite co-dimension.
- Speculation on three sequences of patterns of co-dimension one that result in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
- A switch point for a four-technique pattern (due to Salvadori and Steedman). This is a local pattern of co-dimension two.
- Further analysis of the above example.
- An example of a four-technique pattern in a model with three produced commodities. This local pattern of co-dimension two results in one technique replacing another, in an intermediate range of the rate of profits, along the wage frontier.
- Further analysis of the above example. Two normal forms are identified for four-technique patterns.
The above list is not complete. More types of fluke switch points exist. Some, like the examples of a real Wicksell effect of zero, I thought, should be of interest for themselves to economists. Others show examples of parameters where the appearance of the wage frontier, at least, changes with perturbations of the parameters. I have used these patterns to tell stories about how technical change or a change in markups (that is, structural economic dynamics) can result in reswitching, capital reversing, or the reverse substitution of labor appearing on or disappearing from the wage frontier.
I would like to see that in at least some cases, short run dynamics changes qualitatively with such perturbations. But this seems to be beyond my capabilities.
Thursday, March 01, 2018
Workers Benefiting From Increased Markups In Selected Industries
Figure 1: Variation in Switch Points with the Markup in the Iron Industry |
I finally use the tools of pattern analysis that I have been inventing to tell a practical story. I build on the example which I began in my previous post.
Workers would be better off if an increase in wages led to greater employment, not less. A long-period change in relative markups among industries can result in firms in some industries obtaining a greater rate of profits at the expense of firms in other industries. But such a change can also create a switch point that exhibits capital-reversing. Around such a switch point, a higher wage is associated with firms adopting a technique of production in which more labor is hired, in the economy as a whole, to produce a given net output. Thus, the change in relative markups leaves workers in a better position to press for a greater share of the surplus product.
2.0 PostulatesIn telling this story, I am assuming that possibilities in a simple model - but not too simple - can enlighten us on possibilities in the actual economy. My example is one of an economy in which three commodities (iron, steel, and corn) are produced by means of commodities. I take corn as numeraire and assume wages are paid out of the surplus at the end of the year. Firms have a choice of two processes in each industry for producing the output of that industry. Details are in the last post
Any actually-existing economy cannot ever be expected to be in equilibrium. Nevertheless, I assume that prices of production cast light on tendencies over time for market prices. I realize that this is a contentious claim, especially for Post Keynesians that take issues of fundamental uncertainty seriously.
I assume, for this story, that some sort of long period wage is taken as given when calculating prices of production. It reflects norms about consumption, institutions like how widespread labor unions are, minimum wages, conventions on how bargaining for wages, worker militancy, the policy of the monetary authority, and so on. Some of these influences can be changed, with consequent effects on long period wages and prices of production.
Prices of production also reflect conventions on relative rates of profits. In the example, the rate of profits in the iron industry is 100 s_{1} r, 100 s_{2} r in the steel industry, and 100 s_{3} r in the corn industry. I take it that these conventions can be changed, also with resulting effects on the rate of profits.
3.0 Results and DiscussionFigure 1, at the top of this post, graphs the scale factor for the rate of profits, r, against the markup, s_{1}, in the iron industry. In drawing this graph, the markups in the steel and corn industries, s_{2} and s_{3}, are taken as unity. The maximum rate scale factor for the rate of profits is also graphed, with the region above it in the graphed labeled as an infeasible region. The thin vertical lines show markups in the iron industry at which four-technique patterns occur.
Switch points between the Delta and Gamma technique appear on the wage frontier in two parameter ranges for the markup in the iron industry. For s_{1} between approximately 0.66653 and 1.6195, the switch point between these techniques exhibits capital-reversing. If the markup in the iron industry is slightly below this range, a persisting increasing alters prices of production so that workers pressing wage claims can be more advantageous to them. If the markup in the iron industry is slightly above this range, an increase in the markups in the steel and corn industries can benefit the workers in the same way.
3.1 Selected Examples of Wage FrontiersI presented three examples of wage frontiers in the previous post. I might as well present two more examples of the wage frontiers here. Figure 2 shows the wage frontier for the value of markups at the point where the switch point between the Gamma and Delta techniques exhibits capital-reversing. The Delta and Theta techniques are only cost-minimizing at the switch point. (One cannot visually distinguish between the wage curves for these two techniques around the switch point.)
Figure 2: The Wage Frontier at a Four-Technique Pattern, with Capital Reversing Appearing |
Figure 3 shows the wage frontiers for the value of markups at the point in parameter space where the "perverse" switch point between the Gamma and Delta techniques disappears from the wage frontier. In this illustration, the Alpha and Gamma techniques are cost-minimizing only at the the switch point. For a slightly larger markup in the iron industry, the wage curves for neither the Alpha nor the Gamma technique appear on the wage frontier.
Figure 3: The Wage Frontier at a Four-Technique Pattern, with Capital Reversing Disappearing |
3.2 Four Technique Patterns
The graph in Figure 1 demonstrates that at least two patterns over the axis for r and four four-technique patterns arise in this example.
The outer two four-technique patterns resemble one another in some ways. In both cases, processes in two industries are changed, for the (middle) technique that is replaced between the left and the right of the pattern. The Gamma and Epsilon techniques differ in the processes used to produce iron and steel. The same process is used, in both techniques, to produced corn. Similarly, the Gamma and Theta techniques differ in the processes used to produce iron and corn. They share the same process in the production of steel.
The inner four-technique patterns differ from the outer two, but resemble one another. They both show a single switch-point on one side of the pattern being transformed to three switch points on the other side. The wage curves for two new techniques, at least in the region around the switch point, appear on the wage frontier for the parameter values of the markup with the three switch points. I have not previously presented such a pattern. (The structure of the example, in which two processes, but no more than two, are defined for each industry ensure that a three-technique pattern cannot arise in the example. If the wage curves for three techniques intersect in a single switch point, the wage curve for a fourth technique must also go through that switch point as well.)
This example points out the need for normal forms for patterns. I want to formalize the idea that some patterns are topological equivalent, yet differ for others. The presentation of two normal forms for four-technique patterns, which I have only gestured at here, does that.
4.0 By Way of ConclusionI like how this story combines ideas I take from Kalecki and Sraffa. I do not worry about the labor theory of value, but just take as given that capitalist firms are able to acquire some part, over above what is paid out in wages, of the surplus product.
AppendixI document that two mainstream economists stated that reswitching has implications for labor markets and income distribution:
"One final and somewhat fanciful remark may be made with reference to this [reswitching] example. Two mixed types of stationary state ... are possible... Both use the same equipment, but the question of ... what income-distribution between labour and capital is fixed, is left in this model for political forces to decide. It is interesting to speculate whether more complex situations retaining this feature are ever found in the real world." -- D. G. Champernowne (1953- 1954)
"By contrast, one who believes technology to be more like my 1966 reswitching example than like its orthodox contrast, will have a more sanguine view about how successful militant power by organized labor can be in causing egalitarian shifts in the distribution of income away from property even in the long run." -- Paul A. Samuelson (1975)References
- D. G. Champernowne (1953-1954). The Production Function and the Theory of Capital: A Comment. The Review of Economic Studies, V. 21, No. 2: pp. 112-135.
- Paul A. Samuelson (1975). Steady-State and Transient Relations: A Reply on Reswitching. Quarterly Journal of Economics, V. 89, No. 1: pp. 40-47.
- Graham White (2001). The Poverty of Conventional Economic Wisdom and the Search for Alternative Economic and Social Policies. The Drawing Board: An Australian Review of Public Affairs, V. 2, No. 2: pp. 67-87.
Friday, February 23, 2018
One Technique Replacing Another: An Example
Figure 1: The Wage Frontier at a Four-Technique Patterns |
This post presents another numerical example of one technique replacing another, along the wage frontier, with a perturbation of a model parameter.
In a previous post, I identified three sequences of patterns of switch points in which the wage curves for one technique replaces the wage curve of another. In one of these sequences, a three-technique pattern removes the middle technique from three techniques with wage curves on the wage frontier. A further perturbation of the model parameter of concern results in another three technique pattern, in which the wage curve for a new technique appears in the middle of the wage frontier. In a limiting case of this sequence of patterns, the distance in the parameter space between the two three-technique patterns reduces to zero. A four-technique pattern results.
The parameter that is increased, in this case, is not a coefficient of production. Rather, I consider a model in which barriers to entry, or some such idiosyncratic property of investment in specific industries, results in maintaining specified ratios of the rates of profits among industries. One of these ratios is the parameter that is varied in the numerical example. I have applied my pattern analysis in a limited way to such a model before. By the way, although others have recently analyzed such a model, I find that Ian Steedman outlines this model in his 1977 book, Marx after Sraffa.
This example is more complicated than previous examples of four-technique patterns. At the switch point in which the wage curves for four techniques intersect, processes in two industries are both changed. This is a fluke case - a pattern of co-dimension two, in my terminology. But, in the example, the process in a third industry does not vary with the cost-minimizing techniques around the switch point. The example also illustrates variation in the sequence of switch points, at another region in the wage frontier, with the perturbation of a model parameter. I am not totally happy with this example. The wage curves are often curved more sharply than is seen in real-world data. If I am going to look at a three-commodity example, I would like to find one where at least two wage curves intersect three times for positive, feasible rates of profits.
When I applied the terminology of co-dimension to patterns, I expected that it would be difficult to find, through numerical experimentation, examples of patterns of higher co-dimension. I expected I would have to consciously try to create them, as I did for this example of a global pattern. I argued at one point that patterns of co-dimension one are important for seeing how the sequence of switch points along the wage frontier can change with technical progress or changes in the strength of barriers to entry and so on. I am now leaning towards thinking this argument applies to at least some patterns of higher co-dimension.
Anyways, this example illustrates complications that can arise in price theory that I do not think have been previously noted.
2.0 TechnologyThe technology in this example is almost the same as in one of my previous examples. I modified one labor coefficient. The economy produces a single consumption good, called corn. Corn is also a capital good, that is, a produced commodity used in the production of other commodities. In fact, iron, steel, and corn are capital goods in this example. So three industries exist. One produces iron, another produces steel, and the last produces corn. Two processes exist in each industry for producing the output of that industry. Each process exhibits Constant Returns to Scale (CRS) and is characterized by coefficients of production. Coefficients of production (Table 1) specify the physical quantities of inputs required to produce a unit output in the specified industry. All processes require a year to complete, and the inputs of iron, steel, and corn are all consumed over the year in providing their services so as to yield output at the end of the year.
Input | Iron Industry | Steel Industry | Corn Industry | |||
a | b | c | d | e | f | |
Labor | 1/3 | 1/10 | 5/2 | 7/20 | 1 | 3/2 |
Iron | 1/6 | 2/5 | 1/200 | 1/100 | 1 | 0 |
Steel | 1/200 | 1/400 | 1/4 | 3/10 | 0 | 1/4 |
Corn | 1/300 | 1/300 | 1/300 | 0 | 0 | 0 |
A technique consists of a process in each industry. Table 2 specifies the eight techniques that can be formed from the processes specified by the technology. If you work through this example, you will find that to produce a net output of one bushel corn, inputs of iron, steel, and corn all need to be produced to reproduce the capital goods used up in producing that bushel.
Technique | Processes |
Alpha | a, c, e |
Beta | a, c, f |
Gamma | a, d, e |
Delta | a, d, f |
Epsilon | b, c, e |
Zeta | b, c, f |
Eta | b, d, e |
Theta | b, d, f |
Each technique is represented by coefficients of production. For the Alpha technique, let a_{0, α} be a three-element row vector representing the labor coefficients, and let A_{α} be the 3 x 3 Leontief matrix for this technique. The first element of a_{0, α}, (1/3) person-years per ton, represents the labor input needed to produce a ton of iron. The first column of A_{α} represents the inputs of iron, steel, and corn needed to produce a ton of iron. A parallel notation is used for the other seven techniques.
3.0 The Price SystemPrices of production are defined to be constant spot prices that allow the smooth reproduction of the economy. Suppose Alpha is the cost-minimizing technique. Let p be the three-element row matrix designating the prices of iron, steel, and corn. I make the assumption that markets are such that the rate of profits in the iron, steel, and corn industries are (r s_{1}), (r s_{2}), and (r s_{3}), respectively. Suppose S is a diagonal matrix with the obvious elements along the diagonal, and I designates the identity matrix. Then prices of production satisfy the following system of equations:
p_{α} A_{α} (I + r S) + w_{α} a_{0, α} = p_{α}
I choose a bushel of corn to be the numeraire. If e_{3} is the last column of the identity matrix, the following equation specifies the numeraire:
p_{α} e_{3} = 1
As is not surprising, the above system of equations has one degree of freedom. One can solve for the wage, w_{α}(r), as a function of the scale factor for the rate of profits, r. The is a downward-sloping curve that intercepts both the axis for the wage and the scale factor at positive values. A similar function can be derived the other techniques, and they can be graphed in the same diagram.
4.0 The Choice of TechniqueFigure 2 graphs the wage curves for all eight techniques, given specific values for the mark-ups, s_{i}, i = 1, 2, and 3. The outer envelope, called the wage frontier, represents the cost-minimizing technique for any given wage or scale factor for the rate of profits. (Although it is difficult to see in the graph, the Theta technique is cost-minimizing for a continuum of the wage between two switch points.) Notice that only two wage curves intersect at each switch point on the frontier. The techniques that are cost-minimizing at each switch point differ in only one process. This is a non-fluke example, for these markups. For what it is worth, the switch point between the Delta and Gamma techniques exhibits capital-reversing.
Figure 2: The Wage Frontier Before a Four-Technique Pattern |
Table 3 shows the sequence of techniques that are cost-minimizing, along the wage frontier, at selected values of the markup for the iron industry. Figure 1, at the top of the post, illustrates the middle row. Presumably, two three-technique patterns have removed the Alpha and Gamma techniques from the frontier, for high values of the scale factor for the rate of profits. For the purposes of this post, I am not interested in those patterns. My point is focused on the switch point between the Eta and the Delta technique. Looking above at Table 2, one can see that, for these techniques, both processes in both the iron and corn industries are part of cost-minimizing techniques at the switch point. It follows that the Gamma and Theta techniques are cost-minimizing at this switch point, even though they do not appear on the wage frontier elsewhere. This is a fluke.
s_{1} | s_{2} | s_{3} | Techniques |
3/2 | 1 | 1 | Eta, Theta, Delta, Gamma, Alpha, Beta |
2.665 | 1 | 1 | Eta, Delta, Beta |
4 | 1 | 1 | Eta, Gamma, Delta, Beta |
Finally, Figure 3 shows the wage frontier at the last level of the markup in the iron industry that I want to consider. The sequence of cost-minimizing techniques of Eta, Theta, and Delta, for relatively low scale factors for the rate of profits, has been replaced by the sequence of Eta, Gamma, and Delta. This example shows one sequence for how the wage frontier can be varied by lasting changes in a markup in one industry.
Figure 3: The Wage Frontier After a Four-Technique Pattern |
5.0 Conclusion
Simple numerical examples are often presented in textbooks, such as Kurz and Salvadori's Theory of Production. They are often meant to illustrate phenomena that can appear in a more complicated example of a model. This post is an illustration of a fairly complicated example, where parts, in some sense, resemble simpler examples.
Monday, February 19, 2018
One Technique Replacing Another
Figure 1: One Way One Technique Can Replace Another |
The wage-rate of profits frontier (or wage frontier) is calculated with prices of production, given the techniques of production, available in the economy, for producing a given output. Suppose at one point in time, the techniques that lie along the wage frontier consist of the Alpha, Beta, and Gamma techniques, in order of an increasing rate of profits. As time passes, technical innovation alters coefficients of production, including for techniques that were not on the wage frontier at the initial point in time. Suppose at a later point in time, the techniques along the wage frontier now consist of the Alpha, Delta, and Gamma techniques. How did this replacement of the Beta technique by the Delta technique occur? What happened in the intervening time interval?
The pattern analysis I have been developing suggests answers to these questions. A pattern is a qualitative characterization of a part of the wage frontier associated with a change of switch points. And the patterns I have identified suggest three possibilities for the postulated change in techniques.
Figure 1, above, illustrates the first possibility. These illustrations are only schematic; the illustrated curves need not be straight lines. In the first pattern, two three-technique patterns succeed one another in time. A three-technique pattern arises when a switch point on the frontier is an intersection of the wage curves for three techniques. For the temporally first three-technique pattern, a switch point is replaced by two switch points, with a new technique being cost-minimizing for rates of profits between the two switch points. Four techniques, instead of three techniques, now lie on the wage frontier. For the later three-technique pattern, two switch points are replaced by one switch point. The middle technique at the original point in time is no longer cost-minimizing, for any rate of profits. The postulated initial sequence of techniques occurs before the first pattern, and the final sequence occurs after the second pattern.
A second possibility is that a reswitching pattern is followed by two three-technique patterns. (The shape of the curves are definitely off in Figure 2.) For the reswitching pattern, a new switch point occurs at which the wage curves for two techniques are tangent. For some time afterwards, this possibility is a case of reswitching. The two three-technique patterns remove the wage curve for the originally middle technique from the wage frontier. Once again, the postulated observations for the first and last point in time are consistent with this story.
Figure 2: A Second Way One Technique Can Replace Another |
A third possibility (Figure 3) also involves a sequence of two three-technique patterns. In this case, the temporally first three-technique pattern removes the wage-curve for the middle technique from the wage frontier. The wage frontier now has a succession of two cost-minimizing techniques along it. The second three-technique pattern puts a new technique in the middle of the wage frontier.
Figure 3: A Third Way One Technique Can Replace Another |
Two other stories can arise out of symmetries of, at least, the first and second possibility. And one might complicate the story by superimposing wage curves for other techniques somewhere in this story. In my exploration of numerical examples, I have usually found the switch points in reswitching examples disappearing with patterns over the axis for the rate of profits and the wage axis. That is not the case here.
I think that wage curves can be calculated from Leontief matrices, as derived from the National Income and Product Accounts (NIPA). Zonghie Han & Bertram Schefold and Stefano Zambelli have calculated wage frontiers from empirical data. But I think confirming these stories of technical change fit data is a challenge for those who know more about empirical research following on from Leontief's work. I suspect one would have to look at stylized facts, in some sense. I think I have been developing a perspective on technical innovation that would be worth exploring for empirical applications, even if I am not the one to do this.