tag:blogger.com,1999:blog-267065642017-06-26T15:02:36.149-04:00Thoughts On EconomicsRobert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.comBlogger1038125tag:blogger.com,1999:blog-26706564.post-1151835560707333482020-01-01T03:00:00.000-05:002017-01-03T06:51:08.056-05:00WelcomeI study economics as a hobby. My interests lie in Post Keynesianism, (Old) Institutionalism, and related paradigms. These seem to me to be approaches for understanding actually existing economies.<br /><br />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.<br /><br />In general, this blog is abstract, and I think I steer clear of commenting on practical politics of the day.<br /><br />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.<br /><br /><B>Comments Policy:</B> 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.Robert Vienneauhttp://www.blogger.com/profile/14748118392842775431noreply@blogger.com64tag:blogger.com,1999:blog-26706564.post-57146502584060544562017-06-24T13:03:00.001-04:002017-06-24T13:03:56.673-04:00Bifurcation Analysis in a Model of Oligopoly<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-_j9tTjtvDNI/WU6RbznnKSI/AAAAAAAAA38/5ToM22Jm5vwUjrt4Fwb7t-RBtVg4JJYnQCLcBGAs/s1600/OligopolyBifurcation.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-_j9tTjtvDNI/WU6RbznnKSI/AAAAAAAAA38/5ToM22Jm5vwUjrt4Fwb7t-RBtVg4JJYnQCLcBGAs/s320/OligopolyBifurcation.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 1: Bifurcation Diagram</b></td></tr></tbody></table><P>I have presented a <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2912181">model</A> of <A HREF="http://robertvienneau.blogspot.com/2017/02/a-reswitching-example-in-model-of.html">prices</A>of production in which the the rate of profits differs among industries. Such persistent differential rates of profits may be maintained because of perceptions by investors of different levels of risk among industries. Or they may reflect the ability of firms to maintain barriers to entry in different industries. In the latter case, the model is one of oligopoly. </P><P>This post is based on a specific numeric example for technology, namely, this <A HREF="http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.html">one</A>, in which labor and two commodities are used in the production of the same commodities. I am not going to reset out the model here. But I want to be able to refer to some notation. Managers know of two process processes for producing iron and one process for producing corn. Each process is specified by three coefficients of production. Hence, nine parameters specify the technology, and there is a choice between two techniques. In the model: </P><UL><LI>The rate of profits in the iron industry is <I>r</I><I>s</I><SUB>1</SUB>.</LI><LI>The rate of profits in the iron industry is <I>r</I><I>s</I><SUB>2</SUB>.</LI></UL><P>I call <I>r</I> the scale factor for the rates of profits. <I>s</I><SUB>1</SUB>is the markup for the rate rate of profits in the iron industry. And <I>s</I><SUB>2</SUB> is the markup for the rate of profits in the corn industry. So, with the two markups for the rates of profits, 11 parameters specify the model. </P><P>I suppose one could look at work by Edith Penrose, Michal Kalecki, Joseh Steindl, Paolo Sylos Labini, Alfred Eichner, or Robin Marris for a more concrete understanding of markups. </P><P>Anyways, a wage curve is associated with each technique. And that wage curve results in the wage being specified, in the system of equations for prices of production, given an exogenous specification of the scale factor for the rates of profits. Alternatively, the scale factor can be found, given the wage. Points in common (intersections) on the wage curves for the two techniques are switch points. </P><P>Depending on parameter values for the markups on the rates of profits, the example can have no, one, or two switch points. In the last case, the model is one of the reswitching of techniques. </P><P>A bifurcation diagram partitions the parameter space into regions where the model solutions, throughout a region, are topologically equivalent, in some sense. Theoretically, a bifurcation diagram for the example should be drawn in an eleven-dimensional space. I, however, take the technology as given and only vary the markups. Figure 1, is the resulting bifurcation diagram. </P><P>The model exhibits a certain invariance, manifested in the bifurcation diagram by the straight lines through the origin. Suppose each markup for the rates of profits were, say, doubled. Then, if the scale factor for the rates of profits were halved, the rates of profits in each industry would be unchanged. The wage and prices of production would also be unchanged. </P><P>So only the ratio between the markups matter for the model solution. In some sense, the two parameters for the markups can be reduced to one, the ratio between the rates of profits in the two industries. And this ratio is constant for each straight line in the bifurcation diagram. The reciprocal of the slopes of the lines labeled 2 and 4 in Figure 1 are approximately 0.392 and 0.938, respectively. These values are marked along the abscissa in the figure at the top of this <A HREF="http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.html">post</A>. </P><P>In the bifurcation diagram in Figure 1, I have numbered the regions and the loci constituting the boundaries between them. In a bifurcation diagram, one would like to know what a typical solution looks like in each region and how bifurcations occur. The point in this example is to understand changes in the relationships between the wage curves for the two techniques. And the wage curves for the techniques for the numbered regions and lines in Figure 1 look like (are topologically equivalent to) the corresponding numbered graphs in Figure 2 in this <A HREF="http://robertvienneau.blogspot.com/2017/06/continued-bifurcation-analysis-of.html">post</A></P><P>The model of oligopoly being analyzed here is open, insofar as the determinants of the functional distribution of income, of stable relative rates of profits among industries, and of the long run rate of growth have not been specified. Only comparisons of long run positions are referred to in talking about variations, in the solution to a model of prices of production, with variations in model parameters. That is, no claims are being made about transitions to long period equilibria. Nevertheless, the implications of the results in this paper for short period models, whether ones of classical gravitational processes, cross dual dynamics, <A HREF="http://robertvienneau.blogspot.com/2014/06/a-sophisticated-neoclassical-response.html">intertemporal equilibria, or temporary equilibria</A>, are well worth thinking about. </P><P>Mainstream economists frequently produce more complicated models, with conjectural variations, or game theory, or whatever, of firms operating in non-competitive markets. And they seem to think that models of competitive markets are more intuitive, with simple supply and demand properties and certain desirable properties. I think the Cambridge Capital Controversy raised fatal objections to this view long ago. Reswitching and capital reversing show that equilibrium prices are not scarcity indices, and the logic of comparisons of equilibrium positions, in competitive conditions does not conform to the principle of substitution. In the model of prices of production discussed here, there is a certain continuity between imperfections in competition and the case of free competition. The kind of dichotomy that I understand to exist in mainstream microeconomics just doesn't exist here. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-46929454737279056162017-06-20T18:23:00.000-04:002017-06-22T08:01:27.200-04:00Continued Bifurcation Analysis of a Reswitching Example<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-Pvn-jz3JFCE/WUuxS82BTrI/AAAAAAAAA3M/E8o4wvCWOJ0pKzicSUNqGc4-UduBQ3ikQCLcBGAs/s1600/TwoCoefficientsPerturbation.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-Pvn-jz3JFCE/WUuxS82BTrI/AAAAAAAAA3M/E8o4wvCWOJ0pKzicSUNqGc4-UduBQ3ikQCLcBGAs/s320/TwoCoefficientsPerturbation.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 1: Bifurcation Diagram</b></td></tr></tbody></table><P>This post is a continuation of the analysis in this <A HREF="http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.html">reswitching example</A>. That post presents an example of reswitching in a model of the production of commodities by means of commodities. The example is one of an economy in which two commodities, iron and corn, are produced. Managers of firms know of two processes for producing iron and one process for producing corn. The definition of technology results in a choice between two techniques of production. </P><P>The two-commodity model analyzed here is specified by nine parameters. Theoretically, a bifurcation diagram should be drawn in nine dimensions. But, being limited by the dimensions of the screen, I select two parameters. I take the inputs per unit output in the two processes for producing iron as given constants. I also take as given the amount of (seed) corn needed to produce a unit output of corn, in the one process known for producing corn. So the dimensions of my bifurcation diagram are the amount of labor required to produce a bushel corn and the amount of iron input required to produce a bushel corn. Both of these parameters must be non-negative. </P><P>I am interested in wage curves and, in particular, how many intersections they have. Figure 1, above, partitions the parameter space based on this rationale. I had to think some time about what this diagram implies for wage curves. In generating the points to interpolate, my Matlab/Octave code generated many graphs analogous to those in the linked post. I also generated Figure 2, which illustrates configurations of wage curves and switch points, for the number regions and loci in Figure 1. So I had some visualization help, from my code, in thinking about these implications. Anyways, I hope you can see that, from perturbations of one example, one can generate an infinite number of reswitching examples. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-DT3awQD5jis/WUuxLCpFh3I/AAAAAAAAA3I/WfBW0QKAPL4GCM8FhPOEEVPfwdh0Z9EnACLcBGAs/s1600/ManyWageCurves.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-DT3awQD5jis/WUuxLCpFh3I/AAAAAAAAA3I/WfBW0QKAPL4GCM8FhPOEEVPfwdh0Z9EnACLcBGAs/s320/ManyWageCurves.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 2: Some Wage Curves</b></td></tr></tbody></table><P>One can think of prices of production as (not necessarily stable) fixed points of short period dynamic processes. Economists have developed a number of dynamic processes with such fixed points. But I leave my analysis open to a choice of whatever dynamic process you like. In some sense, I am applying bifurcation analysis to the solution(s) of a system of algebraic equations. The closest analogue I know of in the literature is Rosser (1983), which is, more or less, a chapter in his well-known book. </P><P><B>Update (22 Jun 2017):</B> Added Figure 2, associated changes to Figure 1, and text. </P><B>References</B><UL><LI>J. Barkley Rosser (1983). Reswitching as a Cusp Catastrophe. <I>Journal of Economic Theory</I> V. 31: pp. 182-193.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-90180786222762690992017-06-15T14:44:00.000-04:002017-06-16T06:43:50.136-04:00Perfect Competition With An Uncountable Infinity Of Firms<B>1.0 Introduction</B><P>Consider a partial equilibrium model in which: </P><UL><LI>Consumers demand to buy a certain quantity of a commodity, given its price.</LI><LI>Firms produce (supply) a certain quantity of that commodity, given its price.</LI></UL><P>This is a model of perfect competition, since the consumers and producers take the price as given. In this post, I try to present a model of the supply curve in which the managers of firms do not make systematic mistakes. </P><P>This post is almost purely exposition. The exposition is concrete, in the sense that it is specialized for the economic model. I expect that many will read this as still plenty abstract. (I wish I had a better understanding of mathematical notation in HTML.) Maybe I will update this post with illustrations of approximations to integrals. </P><B>2.0 Firms Indexed on the Unit Interval</B><P>Suppose each firm is named (indexed) by a real number on the (closed) unit interval. That is, the set of firms, <B>X</B>, producing the given commodity is: </P><BLOCKQUOTE><B>X</B> = (0, 1) = {<I>x</I> | <I>x</I> is real and 0 < <I>x</I> < 1} </BLOCKQUOTE><P>Each firm produces a certain quantity, <I>q</I>, of the given quantity. I let the function, <I>f</I>, specify the quantity of the commodity that each firm produces. Formally, <I>f</I> is a function that maps the unit interval to the set of non-negative real numbers. So <I>q</I> is the quantity produced by the firm <I>x</I>, where: </P><BLOCKQUOTE><I>q</I> = <I>f</I>(<I>x</I>) </BLOCKQUOTE><B>2.1 The Number of Firms</B><P>How many firms are there? An infinite number of decimal numbers exist between zero and unity. So, obviously, an infinite number of firms exist in this model. </P><P>But this is not sufficient to specify the number of firms. Mathematicians have defined an <A HREF="http://robertvienneau.blogspot.com/2010/08/infinities-of-infinities.html">infinite number</A>of different size infinities. The smallest infinity is called <I>countable infinity</I>. The set of natural numbers, {0, 1, 2, ...}; the set of integers, {..., -2, -1, 0, 1, 2, ...}; and the set of rational numbers can all be be put into a one-to-one correspondence. Each of these sets contain a countable infinity of elements. </P><P>But the number of firms in the above model is more than that. The firms can be put into a one-to-one correspondence with the set of real numbers. So there exist, in the model, a uncountable infinity of firms. </P><B>2.2 To Know</B><P>Cantor's diagonalization argument, power sets, cardinal numbers. </P><B>3.0 The Quantity Supplied</B><P>Consider a set of firms, <B>E</B>, producing the specified commodity, not necessarily all of the firms. Given the amount produced by each firm, one would like to be able to say what is the total quantity supplied by these firms. So I introduce a notation to designate this quantity. Suppose <I>m</I>(<B>E</B>, <I>f</I>) is the quantity supplied by the firms in <B>E</B>, given that each firm in (0, 1) produces the quantity defined by the function <I>f</I>. </P><P>So, given the quantity supplied by each firm (as specified by the function <I>f</I>) and a set of firms <B>E</B>, the aggregate quantity supplied by those firms is given by the function <I>m</I>. And, if that set of firms is all firms, as indexed by the interval (0, 1), the function <I>m</I> yields the total quantity supplied on the market. </P><P>Below, I consider for which set of firms <I>m</I> is defined, conditions that might be reasonable to impose on <I>m</I>, a condition that is necessary for perfect competition, and two realizations of <I>m</I>, only one of is correct. </P><P>You might think that <I>m</I> should obviously be: </P><BLOCKQUOTE><I>m</I>(<B>E</B>, <I>f</I>) = ∫<SUB><B>E</B></SUB><I>f</I>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>and that the total quantity supplied by all firms is: </P><BLOCKQUOTE><I>Q</I> = <I>m</I>((0,1), <I>f</I>) = ∫<SUB>(0, 1)</SUB> <I>f</I>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>Whether or not this answer is correct depends on what you mean by an integral. Most introductory calculus classes, I gather, teach the Riemann integral. And, with that definition, the answer is wrong. But it takes quite a while to explain why. </P><B>3.1 A Sigma Algebra</B><P>One would like the function <I>m</I> to be defined for all subsets of (0, 1) and for all functions mapping the unit interval to the set of non-negative real numbers. Consider a "nice" function <I>f</I>, in some hand-waving sense. Let <I>m</I>be defined for a set of subsets of (0, 1) in which the following conditions are met: </P><UL><LI>The empty set is among the subsets of (0, 1) for which <I>m</I> is defined.</LI><LI><I>m</I> is defined for the interval (0, 1).</LI><LI>Suppose <I>m</I> is defined for <B>E</B>, where <B>E</B> is a subset of (0, 1). Let <B>E</B><SUP>c</SUP> be those elements of (0, 1) which are not in <B>E</B>. Then <I>m</I> is defined for <B>E</B><SUP>c</SUP>.</LI><LI>Suppose <I>m</I> is defined for <B>E</B><SUB>1</SUB> and <B>E</B><SUB>2</SUB>, both being subsets of (0, 1). Then <I>m</I> is defined for the union of <B>E</B><SUB>1</SUB> and <B>E</B><SUB>2</SUB>.</LI><LI>Suppose <I>m</I> is defined for <B>E</B><SUB>1</SUB> and <B>E</B><SUB>2</SUB>, both being subsets of (0, 1). Then <I>m</I> is defined for the intersection of <B>E</B><SUB>1</SUB> and <B>E</B><SUB>2</SUB>.</LI></UL><P>One might extend the last two conditions to a countable infinity of subsets of (0, 1). As I understand it, any set of subsets of (0, 1) that satisfy these conditions is a <A HREF="http://mathworld.wolfram.com/Sigma-Algebra.html">σ-<I>algebra</I></A>. A mathematical question arises: can one define the function <I>m</I> for the set of all subsets of (0, 1)? At any rate, one would like to define <I>m</I> for a maximal set of subsets of (0, 1), in some sense. I think this idea has something to do with <A HREF="http://mathworld.wolfram.com/BorelSet.html">Borel sets</A>. </P><B>3.2 A Measure</B><P>I now present some conditions on this function, <I>m</I>, that specifies the quantity supplied to the market by aggregating over sets of firms: </P><UL><LI>No output is produced by the empty set of firms:</LI><BLOCKQUOTE><I>m</I>(∅, <I>f</I>) = 0. </BLOCKQUOTE><LI>For any set of firms in the sigma algebra, market output is non-negative:</LI><BLOCKQUOTE><I>m</I>(<B>E</B>, <I>f</I>) ≥ 0. </BLOCKQUOTE><LI>For disjoint sets of firms in the sigma algebra, the market output of the union of firms is the sum of market outputs:</LI><BLOCKQUOTE>If <B>E</B><SUB>1</SUB> ∩ <B>E</B><SUB>1</SUB> = ∅, then <I>m</I>(<B>E</B><SUB>1</SUB> ∪ <B>E</B><SUB>1</SUB>, <I>f</I>) = <I>m</I>(<B>E</B><SUB>1</SUB>, <I>f</I>) + <I>m</I>(<B>E</B><SUB>2</SUB>, <I>f</I>) </BLOCKQUOTE></UL><P>The last condition can be extended to a countable set of disjoint sets in the sigma algebra. With this extension, the function <I>m</I> is a <A HREF="http://mathworld.wolfram.com/Measure.html">measure</A>. In other words, given firms indexed by the unit interval and a function specifying the quantity supplied by each firm, a function mapping from (certain) sets of firms to the total quantity supplied to a market by a set of firms is a measure, in this mathematical model. </P><P>One can specify a couple other conditions that seem reasonable to impose on this model of market supply. A set of firms indexed by an interval is a particularly simple set. And the aggregate quantity supplied to the market, when each of these firms produce the same amount is specified by the following condition: </P><BLOCKQUOTE><P>Let <B>I</B> = (<I>a</I>, <I>b</I>) be an interval in (0, 1). Suppose for all <I>x</I> in <B>I</B>: </P><BLOCKQUOTE><I>f</I>(<I>x</I>) = <I>c</I></BLOCKQUOTE><P>Then the quantity supplied to the market by the firms in this interval, <I>m</I>(<B>I</B>, <I>f</I>), is (<I>b</I> - <I>a</I>)<I>c</I>. </P></BLOCKQUOTE><P></P><B>3.3 Perfect Competition</B><P>Consider the following condition: </P><BLOCKQUOTE><P>Let <B>G</B> be a set of firms in the sigma algebra. Define the function <I>f</I><SUB><B>G</B></SUB>(<I>x</I>) to be <I>f</I>(<I>x</I>) when <I>x</I>is not an element of <B>G</B> and to be 1 + <I>f</I>(<I>x</I>) when <I>x</I>is in <B>G</B>. Suppose <B>G</B> has either a finite number of elements or a countable infinity number of elements. Then: </P><BLOCKQUOTE><I>m</I>((0,1), <I>f</I>) = <I>m</I>((0,1), <I>f</I><SUB><B>G</B></SUB>) </BLOCKQUOTE></BLOCKQUOTE><P>One case of this condition would be when <B>G</B> is a singleton. The above condition implies that when the single firm increases its output by a single unit, the total market supply is unchanged. </P><P>Another case would be when <B>G</B> is the set of firms indexed by the rational numbers in the interval (0, 1). If all these firms increased their individual supplies, the total market supply would still be unchanged. </P><P>Suppose the demand price for a commodity depends on the total quantity supplied to the market. Then the demand price would be unaffected by both one firm changing its output and up to a countably infinite number of firms changing their output. In other words, the above condition is a formalization of <I>perfect competition</I> in this model. </P><B>4.0 The Riemann Integral: An Incorrect Answer</B><P>I now try to describe why the usual introductory presentation of an integral cannot be used for this model of perfect competition. </P><P>Consider a special case of the model above. Suppose <I>f</I>(<I>x</I>) is zero for all <I>x</I>. And suppose that <B>G</B> is the set of rational numbers in (0, 1). So <I>f</I><SUB><B>G</B></SUB> is unity for all rational numbers in (0, 1) and zero otherwise. How could one define ∫<SUB>(0, 1)</SUB><I>f</I><SUB><B>G</B></SUB>(<I>x</I>) d<I>x</I>from a definition of the integral? </P><P>Define a <I>partition</I>, <I>P</I>, of (0, 1) to be a set {<I>x</I><SUB>0</SUB>, <I>x</I><SUB>1</SUB>, <I>x</I><SUB>2</SUB>, ..., <I>x</I><SUB><I>n</I></SUB>}, where: </P><BLOCKQUOTE>0 = <I>x</I><SUB>0</SUB> < <I>x</I><SUB>1</SUB> < <I>x</I><SUB>2</SUB> < ... < <I>x</I><SUB><I>n</I></SUB> = 1 </BLOCKQUOTE><P>The rational numbers are dense in the reals. This implies that, for any partition, each subinterval, [<I>x</I><SUB><I>i</I> - 1</SUB>, <I>x</I><SUB><I>i</I></SUB>] contains a rational number. Likewise, each subinterval contains an irrational real number. </P><P>Define, for <I>i</I> = 1, 2, ..., <I>n</I> the two following quantities: </P><BLOCKQUOTE><I>u</I><SUB><I>i</I></SUB> = supremum over [<I>x</I><SUB><I>i</I> - 1</SUB>, <I>x</I><SUB><I>i</I></SUB>] of <I>f</I><SUB><B>G</B></SUB>(<I>x</I>) </BLOCKQUOTE><P></P><BLOCKQUOTE><I>l</I><SUB><I>i</I></SUB> = infimum over [<I>x</I><SUB><I>i</I> - 1</SUB>, <I>x</I><SUB><I>i</I></SUB>] of <I>f</I><SUB><B>G</B></SUB>(<I>x</I>) </BLOCKQUOTE><P>For the function <I>f</I><SUB><B>G</B></SUB> defined above, <I>u</I><SUB><I>i</I></SUB> is always one, for all partitions and all subintervals. For this function, <I>l</I><SUB><I>i</I></SUB> is always zero. </P><P>A partition can be pictured as defining the bases of successive rectangles along the X axis. Each <I>u</I><SUB><I>i</I></SUB>specifies the height of a rectangle that just includes the function whose integral is being sought. For a smooth function (not our example), a nice picture could be drawn. The sum of the areas of these rectangles is an upper bound on the desired integral. Each partition yields a possibly different upper bound. The Riemann upper sum is the sum of the rectangles, for a given partition: </P><BLOCKQUOTE><I>U</I>(<I>f</I><SUB><B>G</B></SUB>, <I>P</I>) = (<I>x</I><SUB>1</SUB> - <I>x</I><SUB>0</SUB>) <I>u</I><SUB>1</SUB> + ... + (<I>x</I><SUB><I>n</I></SUB> - <I>x</I><SUB><I>n</I> - 1</SUB>) <I>u</I><SUB><I>n</I></SUB></BLOCKQUOTE><P>For the example, with a function that takes on unity for rational numbers, the Riemann upper sum is one for all partitions. The Riemann lower sum is the sum of another set of rectangles. </P><BLOCKQUOTE><I>L</I>(<I>f</I><SUB><B>G</B></SUB>, <I>P</I>) = (<I>x</I><SUB>1</SUB> - <I>x</I><SUB>0</SUB>) <I>l</I><SUB>1</SUB> + ... + (<I>x</I><SUB><I>n</I></SUB> - <I>x</I><SUB><I>n</I> - 1</SUB>) <I>l</I><SUB><I>n</I></SUB></BLOCKQUOTE><P>For the example, the Riemann lower sum is zero, whatever partition is taken. </P><P>The Riemann integral is defined in terms of the least upper bound and greatest lower bound on the integral, where the upper and lower bounds are given by Riemann upper and lower sums: </P><BLOCKQUOTE><P><B>Definition:</B> Suppose the infimum, over all partitions of (0, 1), of the set of Riemann upper sums is equal to the supremum, also over all partitions, of the set of Riemann lower sums. Let <I>Q</I> designate this common value. Then <I>Q</I> is the value of the <I>Riemann integral</I>: </P><BLOCKQUOTE><I>Q</I> = ∫<SUB>(0, 1)</SUB><I>f</I><SUB><B>G</B></SUB>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>If the infimum of Riemann upper sums is not equal to (exceeds) the supremum of the Riemann lower sums, then the Riemann integral of <I>f</I><SUB><B>G</B></SUB> is not defined. </P></BLOCKQUOTE><P>In the case of the example, the Riemann integral is not defined. One cannot use the Riemann integral to calculate the changed market supply from a countably infinite firms each increasing their output by one unit. </P><B>5.0 Lebesque Integration</B><P>The Riemann integral is based on partitioning the X axis. The Lebesque integral, on the other hand, is based on partitioning the Y axis, in some sense. Suppose one has some measure of the size of the set in the domain of a function where the function takes on some designated value. Then the contribution to the integral for that designated value can be seen as the product of that value and that size. The integral of a function can then be defined as the sum, over all possible values of the function, of such products. </P><B>5.1 Lebesque Outer Measure</B><P>Consider an interval, <B>I</B> = (<I>a</I>, <I>b</I>), in the real numbers. The (Lebesque) measure of that set is simply the length of the interval: </P><BLOCKQUOTE><I>m</I>*(<B>I</B>) = <I>b</I> - <I>a</I></BLOCKQUOTE><P>Let <B>E</B> be a set of real numbers. Let {<B>I</B><SUB><I>n</I></SUB>} be a set of an at most countable infinite number of open intervals such that </P><BLOCKQUOTE><B>E</B> is a subset of ∪ <B>I</B><SUB><I>n</I></SUB></BLOCKQUOTE><P>In other words, {<B>I</B><SUB><I>n</I></SUB>} is an <A HREF="http://mathworld.wolfram.com/OpenCover.html">open cover</A>of <B>E</B>. The <I>(Lebesque) measure</I> of <B>E</B> is defined to be: </P><BLOCKQUOTE><I>m</I>*(<B>E</B>) = inf [<I>m</I>*(<B>I</B><SUB>1</SUB>) + <I>m</I>*(<B>I</B><SUB>2</SUB>) + ...] </BLOCKQUOTE><P>where the infimum is taken over the set of countably infinite sets of intervals that cover <B>E</B>. </P><P>The Lebesque measure of any set that is at most countably infinite is zero. So the rational numbers is a set of Lebesque measure zero. So is a set containing a singleton. </P><P>A measurable set <B>E</B> can be used to decompose any other set <B>A</B> into those elements of that set that are also in <B>E</B> and those elements that are not. And the measure of <B>A</B> is the sum of the measures of those two set. </P><P>If a set is not measurable, there exists some set <B>A</B> where that sum does not hold. Given the <A HREF="http://mathworld.wolfram.com/AxiomofChoice.html">axiom of choice</A>non-measurable sets exist. As I understand it, the set of all measurable subsets of the real numbers is a sigma algebra. </P><B>5.2 Lebesque Integral for Simple Functions</B><P>Let <B>E</B> be a measurable subset of the real numbers. Define the characteristic function, χ<SUB><B>E</B></SUB>(<I>x</I>), for <B>E</B>, to be one, if <I>x</I> is an element of <B>E</B>, and zero, if <I>x</I> is not an element of <B>E</B>. </P><P>Suppose the function <I>g</I> takes on a finite number of values {<I>a</I><SUB>1</SUB>, <I>a</I><SUB>2</SUB>, ..., <I>a</I><SUB><I>n</I></SUB>}. Such a function is called a <I>simple function</I>. Let <B>A</B><SUB><I>i</I></SUB> be the set of real numbers where <I>g</I><SUB><I>i</I></SUB> = <I>a</I><SUB><I>i</I></SUB>. The function <I>g</I> can be represented as: </P><BLOCKQUOTE><I>g</I>(<I>x</I>) = <I>a</I><SUB>1</SUB> χ<SUB><B>A</B><SUB>1</SUB></SUB>(<I>x</I>) + ... + <I>a</I><SUB><I>n</I></SUB> χ<SUB><B>A</B><SUB><I>n</I></SUB></SUB>(<I>x</I>) </BLOCKQUOTE><P>The integral of such a simple function is: </P><BLOCKQUOTE>∫<I>g</I>(<I>x</I>) d<I>x</I> = <I>a</I><SUB>1</SUB> <I>m</I>*(<B>A</B><SUB>1</SUB>) + ... + <I>a</I><SUB><I>n</I></SUB> <I>m</I>*(<B>A</B><SUB><I>n</I></SUB>) </BLOCKQUOTE><P>This definition can be extended to non-simple functions by another limiting process. </P><B>5.3 Lebesque Upper and Lower Sums and the Integral</B><P>The Lebesque upper sum of a function <I>f</I> is: </P><BLOCKQUOTE><I>UL</I>(<B>E</B>, <I>f</I>) = sup over simple functions <I>g</I> ≥ <I>f</I> of ∫<SUB><B>E</B></SUB><I>g</I>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>One function is greater than or equal to another function if the value of the first function is greater than or equal to the value of the second function for all points in the common domain of the functions. The Lebesque lower sum is: </P><BLOCKQUOTE><I>LL</I>(<B>E</B>, <I>f</I>) = inf over simple functions <I>g</I> ≤ <I>f</I> of ∫<SUB><B>E</B></SUB><I>g</I>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>Suppose the Lebesque upper and lower sums are equal for a function. Denote that common quantity by <I>Q</I>. Then this is the value of the Lebesque integral of the function. </P><BLOCKQUOTE><I>Q</I> = ∫<SUB><B>E</B></SUB><I>f</I>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>When the Riemann integral exists for a function, the Lebesque integral takes on the same value. The Lebesque integral exists for more functions, however. The statement of the fundamental theorem of calculus is more complicated for the Lebesque integral than it is for the Riemann integral. Royden (1968) introduces the concept of a function of bounded variation in this context. </P><B>5.4 The Quantity Supplied to the Market</B><P>So the quantity supplied to the market by the firms indexed by the set <B>E</B>, when each firm produces the quantity specified by the function <I>f</I> is: </P><BLOCKQUOTE><I>m</I>(<B>E</B>, <I>f</I>) = ∫<SUB><B>E</B></SUB><I>f</I>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>where the integral is the Lebesque integral. In the special case, where the firms indexed by the rational numbers in the interval (0, 1) each supply one more unit of the commodity, the total quantity supplied to the market is unchanged: </P><BLOCKQUOTE><I>Q</I> = ∫<SUB>(0, 1)</SUB><I>f</I><SUB><B>G</B></SUB>(<I>x</I>) d<I>x</I> = ∫<SUB>(0, 1)</SUB><I>f</I>(<I>x</I>) d<I>x</I></BLOCKQUOTE><P>Here is a model of perfect competition, in which a countable infinity of firms can vary the quantity they produce and, yet, the total market supply is unchanged. </P><B>6.0 Conclusion</B><P>I am never sure about these sort of expositions. I suspect that most of those who have the patience to read through this have already seen this sort of thing. I learn something, probably, by setting them out. </P><P>I leave many questions above. In particular, I have not specified any process in which the above model of perfect competition is a limit of models with <I>n</I> firms. The above model certainly does not result from taking the limit at infinity of the number of firms in the Cournot model of systematically mistaken firms. That limit contains a countably infinite number of firms, each producing an infinitesimal quantity - a different model entirely. </P><P>I gather that economists have gone on from this sort of model. I think there are some models in which firms are indexed by the <A HREF="http://mathworld.wolfram.com/HyperrealNumber.html">hyperreals</A>. I do not know what theoretical problem inspired such models and have never studied non-standard analysis. </P><P>Another set of questions I have ignored arises in the philosophy of mathematics. I do not know how intuitionists would treat the multiplication of entities required to make sense of the above. Do considerations of computability apply, and, if so, how? </P><P>Some may be inclined to say that the above model has no empirical applicability to any possible actually existing market. The above mathematics is not specific to the economics model. It is very useful in understanding probability. For example, the probability density function for any continuous random variable is only defined up to a set of Lebesque measure zero. And probability theory is very useful empirically. </P><B>Appendix: Supremum and Infimum</B><P>I talk about the <A HREF="http://mathworld.wolfram.com/Supremum.html">supremum</A> and the <A HREF="http://mathworld.wolfram.com/Infimum.html">infimum</A>of a set above. These are sort of like the maximum and minimum of the set. </P><P>Let <B>S</B> be a subset of the real numbers. The supremum of <B>S</B>, written as sup <B>S</B>, is the least upper bound of <B>S</B>, if an upper bound exists. The infimum of <B>S</B> is written as inf <B>S</B>. It is the greatest lower bound of <B>S</B>, if a lower bound exists. </P><B>References</B><UL><LI>Robert Aumann (1964). Markets with a continuum of traders. <I>Econometrica</I>, V. 32, No. 1-2: pp. 39-50.</LI><LI>H. L. Royden (1968). <I>Real Analysis</I>, second edition.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com3tag:blogger.com,1999:blog-26706564.post-75614806920261477402017-06-11T15:59:00.000-04:002017-06-12T06:23:13.628-04:00Another Three-Commodity Example Of Price Wicksell Effects<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-NsxzAbp0KIE/WT2PukJkyyI/AAAAAAAAA2Y/McYy7iY4Wkc3biYfFx7SgYQdz-8kxqUZQCLcB/s1600/wicksellEffectsExample3.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-NsxzAbp0KIE/WT2PukJkyyI/AAAAAAAAA2Y/McYy7iY4Wkc3biYfFx7SgYQdz-8kxqUZQCLcB/s320/wicksellEffectsExample3.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 1: Price Wicksell Effects in Example</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post presents another example from my on-going simulation experiments. I am still focusing on simple models without the choice of technique. The example illustrates an economy in which price Wicksell effects are positive, for some ranges of the rate of profits, and negative for another range. </P><B>2.0 Technology</B><P>I used my implementation of the Monte-Carlo method to generate 20,000 viable, random economies in which three commodities are produced. For the 316 of these 20,000 economies in which price Wicksell effects are both negative and positive, the maximum vertical distance between the wage curve and an affine function is approximately 15% of the maximum wage. The example presented in this post is for that maximum. </P><P>The economy is specified by a process to produce each commodity and a commodity basket specifying the net output of the economy. Since the level of output is specified for each industry, no assumption is needed on returns to scale, I gather. But no harm will come from assuming Constant Returns to Scale (CRS). All capital is circulating capital; no fixed capital exists. All capital goods used as inputs in production are totally used up in producing the gross outputs. The capital goods must be replaced out of the harvest each year to allow production to continue on the same scale. The remaining commodities in the annual harvest constitute the given net national income. I assume the economy is in a stationary state. Workers advance their labor. They are paid wages out of the harvest at the end of the year. Net national income is taken as the numeraire. </P><P>Table 1 summarizes the technique in use in this example. The 3x3 matrix formed by the first three rows and columns is the Leontief input-output matrix. Each entry shows the physical quantity of the row commodity needed to produce one unit output of the column commodity. For example, 0.5955541 pigs are used each year to produce one bushel of corn. The last row shows labor coefficients, that is, the physical units of labor needed to produced one unit output of each commodity. The last column is net national income, in physical units of each commodity. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD><TD ALIGN="center"><B>Pigs<BR>Industry</B></TD><TD ALIGN="center"><B>Ale<BR>Industry</B></TD><TD ALIGN="center"><B>Net<BR>Output</B></TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0.0905726</TD><TD ALIGN="center">0.0021651</TD><TD ALIGN="center">0.0022885</TD><TD ALIGN="center">0.274545</TD></TR><TR><TD ALIGN="center">Pigs</TD><TD ALIGN="center">0.5955541</TD><TD ALIGN="center">0.2231379</TD><TD ALIGN="center">0.0054569</TD><TD ALIGN="center">0.097880</TD></TR><TR><TD ALIGN="center">Ale</TD><TD ALIGN="center">0.1202180</TD><TD ALIGN="center">0.6362278</TD><TD ALIGN="center">0.0232452</TD><TD ALIGN="center">0.804348</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">0.26273</TD><TD ALIGN="center">0.18555</TD><TD ALIGN="center">0.31306</TD><TD></TD></TR></TABLE><P></P><B>3.0 The Wage Curve</B><P>I now consider stationary prices such that the same rate of profits is made in each industry. The system of equations allow one to solve for the wage, as a function of a given rate of profits. The blue curve in Figure 2 is this wage curve. The maximum rate of profits, achieved when the wage is zero, is approximately 276.5%. The maximum wage, for a rate of profits of zero, is approximately 2.0278 numerate units per labor unit. As a contrast to the wage curve, I also draw a straight line, in Figure 2, connecting these maxima. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-vkAUgQxmEnY/WT2PoJN4-gI/AAAAAAAAA2U/FDaQZMSWG0EnleoP_PMktj8cs_34yr6nACLcB/s1600/wicksellEffectsExample1.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-vkAUgQxmEnY/WT2PoJN4-gI/AAAAAAAAA2U/FDaQZMSWG0EnleoP_PMktj8cs_34yr6nACLcB/s320/wicksellEffectsExample1.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curve in Example</b></td></tr></tbody></table><P>I do not think it is easy to see in the figure, but the wage curve is not of one convexity. The convexity changes at a rate of profits of approximately 25.35%, and I plot the point at which the convexity changes. </P><B>4.0 The Numeraire Value of Capital Goods</B><P>Since I have specified the net national product, the gross national product can be found from the Leontief input-output matrix. The gross national product is the sum of the capital goods, in a stationary state, and the net national product. The employed labor force can be found from labor coefficients and gross national product. </P><P>Given the rate of profits, one can find prices, as well as the wage. And one can use these prices to calculate the numeraire value of capital goods. Figure 1, at the top of this post, graphs the ratio of the value of capital goods to the employed labor force, as a function of the rate of profits. </P><P>A traditional, incorrect neoclassical idea is that a lower rate of profits incentivizes firms to increase the ratio of capital to labor. And a higher wage also incentivizes firms to increase the ratio of capital to labor. The region, for a low rate of profits, in which price Wicksell effects are positive already poses a problem for this vague neoclassical idea. </P><B>5.0 Conclusion</B><P>This example makes me feel better about my simulation approach. From some previous results, I was worried that I would have to rethink how I generate random coefficients. But, maybe if I generate enough economies, even with all coefficients, etc. confined to the unit interval, I will be able to find examples that approach visually interesting counter-examples to neoclassical economics. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-14579002617762019502017-06-08T07:45:00.000-04:002017-06-08T14:54:09.534-04:00Elsewhere<UL><LI>Ian Wright has had a <A HREF="https://ianwrightsite.wordpress.com">blog</A> for about six months.</LI><LI>Scott Carter <A HREF="http://sraffaarchive.org/2017/05/sraffas-notes-on-pcmc-are-online-now/">announces</A> that Sraffa's <A HREF="https://janus.lib.cam.ac.uk/db/node.xsp?id=EAD%2FGBR%2F0016%2FSRAFFA%2FD3%2F12">notes</A> for <I></I> are now available online. (The announcement is good for linking to Carter's paper explaining the arrangement of the notes.)</LI><LI>David Glasner has been thinking <A HREF="https://uneasymoney.com/2017/05/21/hayek-and-intertemporal-equilibrium/">about</A> <A HREF="https://uneasymoney.com/2017/05/28/correct-foresight-perfect-foresight-and-intertemporal-equilibrium/">intertemporal</A> <A HREF="https://uneasymoney.com/2017/06/04/roy-radner-and-the-equilibrium-of-plans-prices-and-price-expectations/">equilibrium</A>.</LI><LI>Brian Romanchuk <A HREF="http://www.bondeconomics.com/2017/05/the-horrifying-mathematics-of.html">questions</A> the use of models of infinitesimal agents in economics. (Some at <A HREF="https://www.econjobrumors.com/topic/the-horrifying-mathematics-of-infinitesimal-agents">ejmr</A> say he is totally wrong, but others cannot make any sense of such models, either. I am not sure if my use of a continuum of techniques <A HREF="http://robertvienneau.blogspot.com/2017/03/a-fluke-switch-point.html">here</A> can be justified as a limit.)</LI><LI>Miles Kimball <A HREF="https://blog.supplysideliberal.com/post/2017/5/29/there-is-no-such-thing-as-decreasing-returns-to-scale">argues</A> that there is no such thing as decreasing returns to scale.</LI></UL><P>Don't the last two bullets imply that the intermediate neoclassical microeconomic textbook treatment of perfect competition is balderdash, as Steve Keen says? </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-52807514703478700432017-06-06T08:47:00.000-04:002017-06-10T07:50:11.616-04:00Price Wicksell Effects in Random Economies<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-rGQKuBuds9g/WTL0J2iTZdI/AAAAAAAAA10/2-v7ZB5ICkEMpSLldow9e1kpKAvaB0lJQCLcB/s1600/VerticalDistanceDistribution.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-rGQKuBuds9g/WTL0J2iTZdI/AAAAAAAAA10/2-v7ZB5ICkEMpSLldow9e1kpKAvaB0lJQCLcB/s320/VerticalDistanceDistribution.png" width="320" height="271" data-original-width="1043" data-original-height="882" /></a></td></tr><tr><td align="center"><b>Figure 1: Blowup of Distribution of Maximum Distance of Frontier from Straight Line</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post is the third in a series. <A HREF="http://robertvienneau.blogspot.com/2017/05/i-just-simulated-6-billion-random.html">Here</A>is the first, and <A HREF="http://robertvienneau.blogspot.com/2017/05/distribution-of-maximum-rate-of-profits.html">here</A>is the second. </P><P>In this post, I am concerned with the probability that price Wicksell effects for a given technique are negative, positive, or both (for different rates of profits). A price Wicksell effect shows the change in the value of capital goods, for different rates of profits, for a technique. If a (non-zero) price Wicksell effect exists, for some range(s) of the rate of profits in which the technique is cost-minimizing, the rate of profits is unequal to the marginal product of capital, in the most straightforward sense. (This is the general case.) Furthermore, a positive price Wicksell effect shows that firms, in a comparison of stationary states, will want to employ more capital per person-hour at a higher rate of profits. The rate of profits is not a scarcity index, for some commodity called "capital", limited in supply. </P><P>My analysis is successful, in that I am able to calculate probabilities for the specific model of random economies. And I see that an appreciable probability exists that price Wicksell effects are positive. However, I wanted to find a visually appealing example of a wage frontier that exhibits both negative and positive Wicksell effects. The curve I end up with is close enough to an affine function that I doubt you can readily see the change in curvature. </P><P>Bertram Schefold has an explanation of this, based on the theory of random matrices. If the Leontief input-output matrix is random, in his sense (which matches my approach), the standard commodity will tend to contain all commodities in the same proportion, that is, proportional to a vector containing unity for all elements. And I randomly generate a numeraire (and net output vector) that will tend to be the same. So my random economies tend to deviate only slightly from standard proportions. And this deviation is smaller, the larger the number of commodities produced. So this post is, in some sense, an empirical validation of Schefold's findings. </P><B>2.0 Simulating Random Economies</B><P>The analysis in this post is based on an analysis, for economies that produce a specified number of commodities, of a specified sample size of random economies (Table 1). </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Number of Simulated Economies</B></CAPTION><TR><TD ALIGN="center"><B>Seed for<BR>Random<BR>Generator</B></TD><TD ALIGN="center"><B>Number of<BR>Commodities</B></TD><TD ALIGN="center"><B>Number of<BR>Economies</B></TD></TR><TR><TD ALIGN="center">66,965</TD><TD ALIGN="center">2</TD><TD ALIGN="center">2,020</TD></TR><TR><TD ALIGN="center">775,545</TD><TD ALIGN="center">3</TD><TD ALIGN="center">20,458</TD></TR><TR><TD ALIGN="center">586,658</TD><TD ALIGN="center">4</TD><TD ALIGN="center">2,747,934</TD></TR></TABLE><P>Each random economy is characterized by a Constant-Returns-to-Scale (CRS) technique, a numerate basket, and net output. The technique is specified by a: </P><UL><LI>A row vector of labor coefficients, where each element is the person-years of labor needed to a unit output of the numbered commodity.</LI><LI>A square Leontief input-output matrix, where each element is the units of input of the row commodity needed as input to produce a unit of the column commodity.</LI></UL><P>The numeraire and net output are column vectors. Net output is set to be the numeraire. The elements of the vector of labor coefficients, the Leontief matrix, and the numeraire are each realizations of independent and identically distributed random variables, uniformly distributed on the unit interval (from zero to one). Non-viable economies are discarded. So, as shown in the table above, more economies are randomly generated than the specified sample size (1,000). </P><P>I am treating both viability and the net output differently from Stefano Zambelli's approach. He bases net output on a given numeraire value of net output. Many vectors can result in the same value of net output in a Sraffa model. He chooses the vector for which the value of capital goods is minimized. This approach fits with Zambell's concentration on the aggregate production function. </P><B>3.0 Price Wicksell Effects</B><P>Table 2 shows my results. As I understand it, the probability that a wage curve for a random economy, in which more than one commodity is produced, will be a straight line is zero. And I find no cases of an affine function for the wage curve, in which the maximum wage (for a rate of profits of zero) and the maximum rate of profits (for a wage of zero) are connected by a straight line in the rate of profits-wage space. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: Price Wicksell Effects</B></CAPTION><TR><TD ALIGN="center"><B>Number<BR>of<BR>Industries</B></TD><TD ALIGN="center"><B>Number w/<BR>Negative<BR>Price<BR>Wicksell<BR>Effects</B></TD><TD ALIGN="center"><B>Number w/<BR>Positive<BR>Price<BR>Wicksell<BR>Effects</B></TD><TD ALIGN="center"><B>Number w/<BR>Both Price<BR>Wicksell<BR>Effects</B></TD></TR><TR><TD ALIGN="center">2</TD><TD ALIGN="center">548</TD><TD ALIGN="center">452</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">3</TD><TD ALIGN="center">603</TD><TD ALIGN="center">416</TD><TD ALIGN="center">19</TD></TR><TR><TD ALIGN="center">4</TD><TD ALIGN="center">679</TD><TD ALIGN="center">334</TD><TD ALIGN="center">13</TD></TR></TABLE><P>The wage curve in a two-commodity economy must be of a single curvature. So for a random economy in which two commodities are produced, price Wicksell effects are always negative or always positive, but never both. And that is what I find. I also find a small number of random economies, in which three or four commodities are produced, in which the wage curve has varying curvature through the economically-relevant range in the first quadrant. </P><B>4.0 Distribution of Displacement from Affine Frontier</B><P>I also measured how far, in some sense, these wage curves for random economies are from a straight line. I took the affine function, described above, connecting the intercepts, of the wage curve, with the rate of profits and the wage axes as a baseline. And I measured there absolute vertical distance between the wage curve and this affine function. (My code actually measures this distance at 600 points). I scale the maximum of this absolute distance by the maximum wage. Figure 1, above, graphs histograms of this scaled absolute vertical distance, expressed as a percentage. Tables 3 and 4 provide descriptive statistics for the empirical probability distribution. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: Parametric Statistics</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B></B></TD><TD ALIGN="center" COLSPAN="5"><B>Number of Produced Commodities</B></TD></TR><TR><TD ALIGN="center"><B>Two</B></TD><TD ALIGN="center"><B>Three</B></TD><TD ALIGN="center"><B>Four</B></TD></TR><TR><TD ALIGN="center">Sample Size</TD><TD ALIGN="center">1,000</TD><TD ALIGN="center">1,000</TD><TD ALIGN="center">1,000</TD></TR><TR><TD ALIGN="center">Mean</TD><TD ALIGN="center">1.962</TD><TD ALIGN="center">1.025</TD><TD ALIGN="center">0.498</TD></TR><TR><TD ALIGN="center">Std. Dev.</TD><TD ALIGN="center">3.428</TD><TD ALIGN="center">1.773</TD><TD ALIGN="center">0.772</TD></TR><TR><TD ALIGN="center">Skewness</TD><TD ALIGN="center">5.111</TD><TD ALIGN="center">4.837</TD><TD ALIGN="center">3.150</TD></TR><TR><TD ALIGN="center">Kurtosis</TD><TD ALIGN="center">48.467</TD><TD ALIGN="center">38.230</TD><TD ALIGN="center">12.492</TD></TR><TR><TD ALIGN="center">Coef. of Var.</TD><TD ALIGN="center">0.5724</TD><TD ALIGN="center">0.578</TD><TD ALIGN="center">0.645</TD></TR></TABLE><P></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: Nonparametric Statistics</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B></B></TD><TD ALIGN="center" COLSPAN="5"><B>Number of Produced Commodities</B></TD></TR><TR><TD ALIGN="center"><B>Two</B></TD><TD ALIGN="center"><B>Three</B></TD><TD ALIGN="center"><B>Four</B></TD></TR><TR><TD ALIGN="center">Minimum</TD><TD ALIGN="center">0.00018</TD><TD ALIGN="center">0.000251</TD><TD ALIGN="center">0.0000404</TD></TR><TR><TD ALIGN="center">1st Quartile</TD><TD ALIGN="center">0.179</TD><TD ALIGN="center">0.114</TD><TD ALIGN="center">0.0608</TD></TR><TR><TD ALIGN="center">Median</TD><TD ALIGN="center">0.653</TD><TD ALIGN="center">0.402</TD><TD ALIGN="center">0.203</TD></TR><TR><TD ALIGN="center">3rd Quartile</TD><TD ALIGN="center">2.211</TD><TD ALIGN="center">1.120</TD><TD ALIGN="center">0.583</TD></TR><TR><TD ALIGN="center">Maximum</TD><TD ALIGN="center">50.613</TD><TD ALIGN="center">23.374</TD><TD ALIGN="center">5.910</TD></TR><TR><TD ALIGN="center">IQR/Median</TD><TD ALIGN="center">3.110</TD><TD ALIGN="center">2.504</TD><TD ALIGN="center">2.574</TD></TR></TABLE><P>We see that the wage curves for these random economies tend not to deviate much from an affine function. And, as more commodities are produced, this deviation is less. </P><B>5.0 An Example</B><P>For three commodity economies, the maximum scaled displacement of the wage curve from a straight line I find is 23.4 percent. But, of those three-commodity economies with both negative and positive price Wicksell effects, the maximum displacement is only 0.736 percent. Table 5 provides the randomly generated parameters for this example. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 5: The Technology for a Three-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD><TD ALIGN="center"><B>Pigs<BR>Industry</B></TD><TD ALIGN="center"><B>Ale<BR>Industry</B></TD><TD ALIGN="center"><B>Net<BR>Output</B></TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0.5525152</TD><TD ALIGN="center">0.0024860</TD><TD ALIGN="center">0.2652761</TD><TD ALIGN="center">0.26077</TD></TR><TR><TD ALIGN="center">Pigs</TD><TD ALIGN="center">0.5164675</TD><TD ALIGN="center">0.7469286</TD><TD ALIGN="center">0.1128406</TD><TD ALIGN="center">0.42705</TD></TR><TR><TD ALIGN="center">Ale</TD><TD ALIGN="center">0.5636308</TD><TD ALIGN="center">0.0368399</TD><TD ALIGN="center">0.2110545</TD><TD ALIGN="center">0.98691</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">0.799364</TD><TD ALIGN="center">0.028111</TD><TD ALIGN="center">0.012866</TD><TD></TD></TR></TABLE><P>Figure 2 shows the wage curve for the example. This curve is not a straight line, no matter how close it may appear so to the eye. Figure 3 shows the distance between the wage curve and a straight line. Notice that the convexity towards the left of the curve in Figure 3 varies slightly from the convexity for the rest of the graph. This is a manifestation of price Wicksell effects in both directions. (I need to perform some more checks on my program.) </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-QiAI1UuP1jI/WTL0x-6QOtI/AAAAAAAAA14/L3yg2lId3mULm82Roa7a9DSRapNUL7HTwCLcB/s1600/wicksellEffectsExample1.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-QiAI1UuP1jI/WTL0x-6QOtI/AAAAAAAAA14/L3yg2lId3mULm82Roa7a9DSRapNUL7HTwCLcB/s320/wicksellEffectsExample1.jpg" width="320" height="240" data-original-width="1200" data-original-height="900" /></a></td></tr><tr><td align="center"><b>Figure 2: A Wage Frontier with Both Negative and Positive Price Wicksell Effects</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-dU4t-RUjI9U/WTL1KeoJNUI/AAAAAAAAA18/130ZPSxN_GsTaktZhjGpC5eqej1Kxo1XwCLcB/s1600/wicksellEffectsExample2.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-dU4t-RUjI9U/WTL1KeoJNUI/AAAAAAAAA18/130ZPSxN_GsTaktZhjGpC5eqej1Kxo1XwCLcB/s320/wicksellEffectsExample2.jpg" width="320" height="240" data-original-width="1200" data-original-height="900" /></a></td></tr><tr><td align="center"><b>Figure 3: Vertical Distance of Frontier from Straight Line</b></td></tr></tbody></table><P></P><B>6.0 Conclusion</B><P>I hope Bertram Schefold and Stefano Zambelli are aware of each other's work. </P><P><B>Postscript</B>: I had almost finished this post before Stefano Zambelli left this <A HREF="http://robertvienneau.blogspot.com/2015/05/paul-romer-confused-on-capital-theory.html?showComment=1496690057406#c68508124331593863">comment</A>. I'd like to hear from him at rvien@dreamscape.com. </P><B>References</B><UL><LI>Bertram Schefold (2013). Approximate Surrogate Production Functions. <I>Cambridge Journal of Economics</I>.</LI><LI>Stefano Zambelli (2004). The 40% neoclassical aggregate theory of production. <I>Cambridge Journal of Economics</I> 28(1): pp. 99-120.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-19019651010582947822017-05-27T13:41:00.000-04:002017-05-27T13:41:20.198-04:00Some Main Points of the Cambridge Capital Controversy<P>For the purposes of this very simplified and schematic post, I present the CCC as having two sides. </P><UL><LI>Views and achievements of Cambridge (UK) critics:</LI><UL><LI>Joan Robinson's argument for models set in historical time, not logical time.</LI><LI>Mathematical results in comparing long-run positions:</LI><UL><LI>Reswitching.</LI><LI><A HREF="http://robertvienneau.blogspot.com/2016/12/example-of-choice-of-technique.html">Capital reversing</A>.</LI><LI><A HREF="http://robertvienneau.blogspot.com/2006/05/empirical-evidence-exists-on-sraffa_16.html">Empirical results</A> and applications.</LI></UL><LI>Rediscovery of the logic of the Classical theory of value and distribution.</LI><LI>Arguments about the role a given quantity of capital in disaggregated neoclassical economic theory between 1870 and 1930.</LI><LI>Arguments that neoclassical models of intertemporal and temporary equilibrium do not escape the capital critique.</LI><LI>A critique of Keynes' marginal efficiency of capital and other aspects of <I>The General Theory</I>.</LI><LI>The recognition of precursors in Thorstein Veblen and earlier capital controversies in neoclassical economics.</LI></UL><LI>Views of neoclassical defenders:</LI><UL><LI>Paul Samuelson and Frank Hahn's, for example, acceptance and recognition of logical difficulties in aggregate production functions.</LI><LI>Recognition the equilibrium prices in disaggregate models are not scarcity indices; rejection of the principle of substitution.</LI><LI>Edwin Burmeister's championing of David Champerowne's <A HREF="http://robertvienneau.blogspot.com/2009/05/neoclassical-response-to-cambridge.html">chain index</A> measure of aggregate capital, useful for aggregate theory when, by happenstance, no positive real Wicksell effects exist.</LI><LI><A HREF="http://robertvienneau.blogspot.com/2014/06/a-sophisticated-neoclassical-response.html">Adoption</A> of models of inter temporal and temporary general equilibrium.</LI><LI>Assertion that such General Equilibrium models are not meant to be descriptive and, besides, have their own problems of stability, uniqueness, and determinateness, with no need for Cambridge critiques.</LI><LI>Samuel Hollander's argument for more continuity between classical and neoclassical economics than Sraffians see.</LI></UL></UL><P>I think I am still ignoring large aspects of the vast literature on the CCC. This post was inspired by Noah Smith's <A HREF="http://noahpinionblog.blogspot.com/2017/05/vast-literatures-as-mud-moats.html">anti-intellectualism</A>. Barkley Rosser brings up the CCC in his <A HREF="http://econospeak.blogspot.com/2017/05/on-virtues-of-good-review-essays.html">response</A> to Smith. I could list references for each point above. I am not sure I could even find a survey article that covered all those points, maybe not even a single <A HREF="http://robertvienneau.blogspot.com/2006/08/textbooks-for-teaching-non.html">book</A>. </P><P>So the CCC presents, to me, a convincing demonstration, through a counter-example to Smith's argument. In the comments to his post, Robert Waldmann <A HREF="http://noahpinionblog.blogspot.com/2017/05/vast-literatures-as-mud-moats.html?showComment=1495060863486#c6705582033485119000">brings up</A>old, paleo-Keynesian as an interesting rebuttal to a specific point. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-50648817979350651142017-05-25T08:01:00.000-04:002017-05-25T08:01:08.823-04:00Some Resources on Neoliberalism<P>Here are three:</P><UL><LI>Anthony Giddens, in <A HREF="https://www.amazon.com/Third-Way-Renewal-Social-Democracy/dp/0745622674"><I>The Third Way: The Renewal of Social Democracy</I></A> (1999), advocates a renewed social democracy. He contrasts what he is advocating with neoliberalism, which he summarizes as, basically, Margaret Thatcher's approach. Giddens recognizes that more flexible labor markets will not bring full employment and argues that unregulated globalism, including unregulated international financial markets, is a danger that must be addressed. He stresses the importance of environmental issues, on all levels from the personal to international. I wish he had something to say about labor unions, which I thought had an institutionalized role in the Labour Party, before Blair and Brown's "new labour" movement.</LI><LI>Charles Peters had a <A HREF="https://www.washingtonpost.com/archive/opinions/1982/09/05/a-neo-liberals-manifesto/21cf41ca-e60e-404e-9a66-124592c9f70d/?utm_term=.06ef0aae952b">A Neo-Liberal's Manifesto</A> in 1982. (See also <A HREF="http://coreyrobin.com/wp-content/uploads/2016/04/Charles-Peters-Neoliberalism.pdf">1983 piece</A> in <I>Washington Monthly</I>.) This was directed to the Democratic Party in the USA. It argues that they should reject the approach of the New Deal and the Great Society. Rather, they should put greater reliance on market solutions for progressive ends. I do not think Peters was aware that the term "neoliberalism" was already taken. Contrasting and comparing other uses with Peters' could occupy much time. </LI><LI>I have not got very far in reading Michel Foucault. <A HREF="https://www.amazon.com/Birth-Biopolitics-Lectures-Coll%C3%A8ge-1978-1979/dp/0312203411/"><I>The Birth of Biopolitics: Lectures at the Collège de France, 1978-1979</I></A>. Foucault focuses on German <A HREF="https://en.wikipedia.org/wiki/Ordoliberalism">ordoliberalism</A>and the Chicago school of economics. </LI></UL><P>Anyways, neoliberalism is something more specific than any centrist political philosophy, between socialist central planning and reactionary ethnic nationalism. George Monbiot has some short, <A HREF="https://www.theguardian.com/commentisfree/2016/nov/14/neoliberalsim-donald-trump-george-monbiot">popular</A><A HREF="https://www.theguardian.com/books/2016/apr/15/neoliberalism-ideology-problem-george-monbiot">accounts</A>. Read <A HREF="https://www.bloomberg.com/view/articles/2017-05-23/centrism-takes-on-the-extremes">Noah Smith</A>if you want confusion, incoherence, and ignorance, including ignorance of the literature. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com5tag:blogger.com,1999:blog-26706564.post-35051144505439889442017-05-19T08:31:00.000-04:002017-05-20T10:26:33.083-04:00Reversing Figure And Ground In Life-Like Celluar Automata<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD align="CENTER"><a href="https://2.bp.blogspot.com/-Rb8_P-5wHkw/V1Wkvx5r6CI/AAAAAAAAAow/DiGz4auRiu04vx_tnaB1L24_lTndx4o4wCLcB/s1600/LifeFlipLife100.JPG" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-Rb8_P-5wHkw/V1Wkvx5r6CI/AAAAAAAAAow/DiGz4auRiu04vx_tnaB1L24_lTndx4o4wCLcB/s320/LifeFlipLife100.JPG" /></a></td></tr><tr><td align="center"><b>Figure 1: Random Patterns in Life and Flip Life</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I <A HREF="http://robertvienneau.blogspot.com/2016/04/a-steam-experience-for-flash-mob.html">have</A><A HREF="http://robertvienneau.blogspot.com/2016/05/a-turing-machine-for-calculating.html">occasionally</A><A HREF="http://robertvienneau.blogspot.com/2016/05/a-turing-machine-for-binary-counter.html">posted</A><A HREF="http://robertvienneau.blogspot.com/2015/02/bad-math-in-good-math.html">about</A><A HREF="http://robertvienneau.blogspot.com/2012/12/anti-reductionism-example.html">automata</A>. A discussion with a colleague about Stephen Wolfram's <A HREF="http://www.wolframscience.com/nks/"><I>A New Kind of Science</I></A>reminded me that I had started this post some time last year. </P><P>This post has nothing to do with economics, albeit it does illustrate emergent behavior. And I have figures that are an eye test. I am subjectively original. But I assume somebody else has done this - that I am not objectively original. </P><P>This post is an exercise in combinatorics. There are 131,328 life-like Celluar Automata (CA), up to symmetry. </P><B>2.0 Conway's Game of Life</B><P><A HREF="https://en.wikipedia.org/wiki/John_Horton_Conway">John Conway</A>will probably ever be most famous for the Game of Life (GoL). I wish I understood <A HREF="http://www.brandonrayhaun.com/2015/07/19/moonshine-theory-i-symmetry-numbers-and-the-monster/">monstrous moonshine</A>. </P><P>The GoL is "played", if you can call it that, on an infinite plane divided into equally sized squares. The plane looks something like a chess board, extended forever. See the left side of Figure 1, above. Every square, at any moment in time, is in one of two states: alive or dead. Time is discrete. The rules of the game specify the state of each square at any moment in time, given the configuration at the previous instant. </P><P>The state of a square does not depend solely on its previous state. It also depends on the states of its neighbors. Two types of neighborhoods have been defined for a CA with a grid of square cells. The Von Neumann neighbors of a cell are the four cells above it, below it, and to the left and right. The Moore neighborhood (Figure 2) consists of the Von Neumann neighbors and the four cells diagonally adjacent to a given cell. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD align="CENTER"><a href="https://1.bp.blogspot.com/-jw5IB-PvJTM/V1WnrgV6I7I/AAAAAAAAApU/g2AN3UrEQFwkCUN2xEFPJUAZkvrqwXP0QCLcB/s1600/MooreNeighborhood.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-jw5IB-PvJTM/V1WnrgV6I7I/AAAAAAAAApU/g2AN3UrEQFwkCUN2xEFPJUAZkvrqwXP0QCLcB/s320/MooreNeighborhood.jpg" /></a></td></tr><tr><td align="center"><b>Figure 2: Moore Neighborhood of a Dead Cell</b></td></tr></tbody></table><P>The GoL is defined for Moore neighborhoods. State transition rules can be defined in terms of two cases: </P><UL><LI><B>Dead cells:</B> By default, a dead cell stays dead. If a cell was dead at the previous moment, it becomes (re)born at the next instant if the number of live cells in its Moore neighborhood at the previous moment was <I>x</I><SUB>1</SUB> or <I>x</I><SUB>2</SUB> or ... or <I>x</I><SUB><I>n</I></SUB>.</LI><LI><B>Alive Cells:</B> By default, a live cell becomes dead. If a cell was alive at the previous moment, it remains alive if the number of live cells in its Moore neighborhood at the previous moment was <I>y</I><SUB>1</SUB> or <I>y</I><SUB>2</SUB> or ... or <I>y</I><SUB><I>m</I></SUB>.</LI></UL><P>The state transition rules for the GoL can be specified by the notation B<I>x</I>/S<I>y</I>. Let <I>x</I> be the concatenation of the numbers <I>x</I><SUB>1</SUB>, <I>x</I><SUB>2</SUB>, ..., <I>x</I><SUB><I>n</I></SUB>. Let <I>y</I> be the concatenation of <I>y</I><SUB>1</SUB>, <I>y</I><SUB>2</SUB>, ..., <I>y</I><SUB><I>m</I></SUB>. The GoL is B3/S23. In other words, if exactly three of the neighbors of a dead cell are alive, it becomes alive for the next time step. If exactly two or or three of the neighbors of a live cell are alive, it remains alive at the next time step. Otherwise a dead cell remains dead, and a live cell becomes dead. </P><P>The GoL is an example of <A HREF="https://blogs.scientificamerican.com/guest-blog/the-top-10-martin-gardner-scientific-american-articles/">recreational mathematics</A>. Starting with random patterns, one can predict, roughly, the distributions of certain patterns when the CA settles down, in some sense. On the other hand, the specific patterns that emerge can only be found by iterating through the GoL, step by step. And one can engineer certain patterns. </P><B>3.0 Life-Like Celluar Automata</B><P>For the purposes of this post, a <I>life-like CA</I> is a CA defined with: </P><UL><LI>A two dimensional grid with square cells and discrete time</LI><LI>Two states for each cell</LI><LI>State transition rules specified for Moore neighborhoods</LI><LI>State transition rules that can be specified by the B<I>x</I>/S<I>y</I> notation.</LI></UL><P>How many life-like CA are there? This is the question that this post attempts to answer. </P><P>The Moore neighborhood of cell contains eight cells. Thus, for each of the digits 0, 1, 2, 3, 4, 5, 6, 7, and 8, they can appear in B<I>x</I>. For each digit, one has two choices. Either it appears in the birth rule or it does not. Thus, there are 2<SUP>9</SUP> birth rules. </P><P>The same logic applies to survival rules. There are 2<SUP>9</SUP> survival rules. </P><P>Each birth rule can be combined with any survival rule. So there are: </P><BLOCKQUOTE>2<SUP>9</SUP> 2<SUP>9</SUP> = 2<SUP>18</SUP></BLOCKQUOTE><P>life-like CA. But this number is too large. I am double counting, in some sense. </P><B>4.0 Reversing Figure and Ground</B><P>Figure 1 shows, side by side, grids from the GoL and from a CA called Flip Life. Flip Life is specified as B0123478/S01234678. Figure 3 shows a window from a computer program. In the window on the left, the rules for the GoL are specified. The window on the right is used to specify Flip Life. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD align="CENTER"><a href="https://4.bp.blogspot.com/-Sk9Eas9bZ8c/V1WlgLiS5MI/AAAAAAAAAo4/RbLetSFkmSAMSnMC_HkUut8vOI7YWea1ACLcB/s1600/LifeFlipLifeRules.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-Sk9Eas9bZ8c/V1WlgLiS5MI/AAAAAAAAAo4/RbLetSFkmSAMSnMC_HkUut8vOI7YWea1ACLcB/s320/LifeFlipLifeRules.jpg" /></a></td></tr><tr><td align="center"><b>Figure 3: Rules for Life and Flip Life</b></td></tr></tbody></table><P>Flip Life basically renames the states in the GoL. Cells that are called dead in the GoL are said to be alive in Flip Life. And cells that are alive in the GoL are dead in Flip Life. In counting the number of life-like CA, one should not count Flip Life separately from the GoL. In some sense, they are the same CA. </P><P>More generally, suppose B<I>x</I>/S<I>y</I> specifies a life-like CA, and let B<I>u</I>/S<I>v</I> be the life-like CA in which figure and ground are reversed. </P><UL><LI>For each digit <I>x</I><SUB><I>i</I></SUB> in <I>x</I>, 8 - <I>x</I><SUB><I>i</I></SUB>is not in <I>v</I>, and vice versa.</LI><LI>For each digit <I>y</I><SUB><I>j</I></SUB> in <I>y</I>, 8 - <I>y</I><SUB><I>j</I></SUB>is not in <I>u</I>, and vice versa.</LI></UL><P>So for any life-like CA, one can find another symmetrical CA in which dead cells become alive and vice versa. </P><B>5.0 Self Symmetrical CAs</B><P>One cannot just divide 2<SUP>18</SUP> by two to find the number of life-like CA, up to symmetry. Some rules define CA that are the same CA, when one reverses figure and ground. As an example, Figure 4 presents a screen snapshot for the CA called Day and Night, specified by the rule B1/S7. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD align="CENTER"><a href="https://2.bp.blogspot.com/-amlVObo9eMg/V1Wl95A8YOI/AAAAAAAAApE/do4IR8p5RIohfvDOrEhr0cAqnIeJeSwxACLcB/s1600/RocketsDayNight.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-amlVObo9eMg/V1Wl95A8YOI/AAAAAAAAApE/do4IR8p5RIohfvDOrEhr0cAqnIeJeSwxACLcB/s320/RocketsDayNight.jpg" /></a></td></tr><tr><td align="center"><b>Figure 4: Day and Night: An Example of a Self-Symmetrical Cellular Automaton</b></td></tr></tbody></table><P>Given rules for births, one can figure out what the rules must be for survival for the CA to be self-symmetrical. Thus, there are as many self-symmetrical life-like CAs as there are rules for births. </P><B>6.0 Combinatorics</B><P>I bring all of the above together in this section. Table 1 shows a tabulation of the number of life-like CAs, up to symmetry. </P><table align="CENTER" border=""><tbody><CAPTION><b>Table 1: Counting Life-Like Celluar Automata</b></CAPTION><TR align="CENTER"><TD><B></B></TD><TD><B>Number</B></TD></TR><TR><TD align="LEFT">Birth Rules</TD><TD align="CENTER">2<SUP>9</SUP></TD></TR><TR><TD align="LEFT">Survival Rules</TD><TD align="CENTER">2<SUP>9</SUP></TD></TR><TR><TD align="LEFT">Life-Like Rules</TD><TD align="CENTER">2<SUP>9</SUP> 2<SUP>9</SUP> = 262,144</TD></TR><TR><TD align="LEFT">Self-Symmetric Rules</TD><TD align="CENTER">2<SUP>9</SUP></TD></TR><TR><TD align="LEFT">Non-Self-Symmetric Rules</TD><TD align="CENTER">2<SUP>9</SUP>(2<SUP>9</SUP> - 1)</TD></TR><TR><TD align="LEFT">Without Symmetric Rules</TD><TD align="CENTER">2<SUP>8</SUP>(2<SUP>9</SUP> - 1)</TD></TR><TR><TD align="LEFT">With Self-Symmetric Rules Added Back</TD><TD align="CENTER">2<SUP>8</SUP>(2<SUP>9</SUP> + 1) = 131,328</TD></TR></tbody></table><P></P><B>7.0 Conclusion</B><P>How many of these 131,328 life-like CA are interesting? Answering this question requires some definition of what makes a CA interesting. It also requires some means of determining if some CA is in the set so defined. Some CAs are clearly not interesting. For example, consider a CA in which all cells eventually die off, leaving an empty grid. Or consider a CA in which, starting with a random grid, the grid remains random for all time, with no defined patterns ever forming. Somewhat more interesting would be a CA in which patterns grow like a crystal, repeating and duplicating. But perhaps an interesting definition of an interesting CA would be one that can simulate a Turing machine and thus may compute any computable function. The GoT happens to be Turing complete. </P><P><B>Acknowledgements:</B> I started with version 1.5 of Edwin Martin's implementation, in Java, of John Conway's Game of Life. I have modified this implementation in several ways. </P><B>References</B><UL><LI>David Eppstein (2010). <A HREF="http://arxiv.org/abs/0911.2890">Growth and Decay in Life-Like Celluar Automata</A></LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-20970327077072159222017-05-13T14:08:00.001-04:002017-05-13T14:12:21.785-04:00Innovation and Input-Output Matrices<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-So9XZIfZDqw/WRc0S-qMdaI/AAAAAAAAA1Q/oacTwiv-HAIln0aO_Hp5MXsQNnASzdWpwCLcB/s1600/IOTable.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-So9XZIfZDqw/WRc0S-qMdaI/AAAAAAAAA1Q/oacTwiv-HAIln0aO_Hp5MXsQNnASzdWpwCLcB/s320/IOTable.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: National Income and Product Accounts</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post contains some speculation about technical progress. </P><B>2.0 Non-Random Innovations and Almost Straight Wage Curves</B><P>The theory of the production of commodities by means of commodities imposes one restriction on wage-rate of profits curves: They should be downward-sloping. They can be of any convexity. They are high-order polynomials, where the order depends on the number of produced commodities. So no reason exists why they should not change convexity many times in the first quadrant, where the the rate of profits is positive and below the maximum range of profits. The theory of the choice of technique suggests that, if multiple processes are available for producing many commodities, many techniques will contribute to part of the wage-rate of profits frontier. </P><P>The empirical research does not show this. When I looked at <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2738605">all countries or regions</A>in the world, I found very little visual deviation from straight lines for most wage curves, for the ruling technique<SUP>1</SUP>. The exceptions tended to be undeveloped countries. Han and Schefold, in their empirical search for capital-theoretic paradoxes in OECD countries, also found mostly straight curves. And only a few techniques appeared on the frontier. </P><P>I have a qualitative explanation of this discrepancy between expectations from theory and empirical results. The theory I draw on above takes technology as given. It is as if economies are analyzed based on an instantaneous snapshot. But technology evolves as a dynamic process. The flows among industries and final demands have been built up over decades, if not centuries. </P><P>In advanced economies, technology does not change randomly. Large corporations have Research and Development departments, universities form extensive networks, and the government sponsors efforts to advance <A HREF="https://www.nasa.gov/directorates/heo/scan/engineering/technology/txt_accordion1.html">Technology Readiness Levels</A><SUP>2</SUP>. Sponsored research is not directed randomly. Technical feasibility is an issue, albeit that changes over time. Another concern is what is costly at the moment, with cost being defined widely. I suggest a constant effort to lower a reliance on high cost inputs in production process, over time, results in coefficients of production being lowered such that wage curves become more straight<SUP>3</SUP>. </P><P>The above story suggests that one should develop some mathematical theorems. I am aware of two areas of research in Sraffian economics that seem promising for further inquiry along these lines. First, consider Luigi Pasinetti's structural economic dynamics. I have an <A HREF="http://robertvienneau.blogspot.com/2013/07/trends-in-hardware-and-software-costs.html">analysis</A>of hardware and software costs in computer systems, which might be suggestive. Second, Bertram Schefold has been analyzing the relationship between the shape of wage curves; random matrices; and eigenvalues, including eigenvalues other than the Perron-Frobenius root. </P><B>3.0 Innovations Dividing Columns in Input-Output Table, Not Adjoining Completely New Ones</B><P>I have been moping during my day job how I cannot keep up with some of my fellow software developers. I return to, say, Java programming after a few years, and there is a whole new set of tools. And yet, much of what I have learned did not even exist when I received either of my college degrees. For example, creating an Android app in Android Studio or IntelliJ involves, minimally, XML, Java, and Virtual Machines for testing. Back in the 1980s, I saw some presentations from Marvin Zelkowitz for what might be described as an Integrated Development Environment (IDE). He had an editor that understood Pascal syntax, suggested statement completions, and, if I recall correctly, could be used to set breakpoints and examine states for executing code. I do not know how this work fed, for example, Eclipse. </P><P>Nowadays, you can specialize in developing web apps<SUP>4</SUP>. Some of my co-workers are Certified Information Systems Security Professionals (CISSPs). They know a lot of concepts that are sort of orthogonal to programming<SUP>5</SUP>. I also know people that work at Security Operations Centers (SOCs)<SUP>6</SUP>. And there are many other software specialities. </P><P>In short, software should no longer be considered a single industry. Glancing quickly at the <A HREF="https://www.bea.gov/industry/io_annual.htm">web site</A>for the Bureau of Economic Analysis, I note the following industries in the 2007 benchmark input-output tables: </P><UL><LI>Software publishers (511200)</LI><LI>Data processing, hosting, and related services (518200)</LI><LI>Internet publishing and broadcasting and Web search portals (518200)</LI><LI>Custom computer programming services (541511)</LI><LI>Computer systems design services (541512)</LI><LI>Other computer related services, including facilities management (54151A)</LI></UL><P>Coders, programmers, and software engineers definitely provide labor inputs in many other industries. Cybersecurity does not even appear above. </P><P>What would input-tables looked like, for software, in the 1970s? I speculate you might find industries for the manufacture of computers, telecommunication equipment, and satellites & space vehicles. And data processing would probably be an industry. </P><P>I am thinking that new industries come about, in modern economies, more by division and greater articulation of existing industries, not by suddenly creating completely new products. And this can be seen in divisions and movements in industries in National Income and Product Accounts (NIPA). One might explore innovation over the last half-century or so by looking at the evolution of industry taxonomies in the NIPA.<SUP>7</SUP>. </P><B>4.0 Conclusion</B><P>This post suggests some research directions<SUP>8</SUP>. At this point, I do not intend to pursue either. </P><B>Footnotes</B><OL><LI>Reviewers, several years ago, had three major objections to this paper. One was that I had to offer some suggestion why wage curves should be so straight. The other two were that I needed to offer a more comprehensive explanation of how to map from the raw data to the input-output tables I used and that I had to account for fixed capital and depreciation.</LI><LI>John Kenneth Galbraith's <I>The New Industrial State</I> is a somewhat dated analysis of these themes.</LI><LI>They also move outward.</LI><LI>The web is not old. Tools like Glassfish, Tomcat, and JBoss, and their commercial competitors are neat.</LI><LI>Such as Confidentiality, Integrity, and Availability; two-factor identification; Role-Based Access Control; taxonomies for vulnerabilities and intrusions; Public Key Infrastructure; symmetric and non-symmetric encryption; the Risk Management Framework (RMF) for Information Assurance (IA) Certification and Accreditation; and on and on.</LI><LI>A SOC differs from a Network Operations Center. Operators of a SOC have to know about host-based and network-based Intrusion Detection, Security Incident and Event Management (SIEM) systems, Situation Awareness, forensics, and so on.</LI><LI>One should be aware that part of the growth on the tracking of industries might be because computer technology has evolved. Von Neumann worried about numerical methods for calculating matrix inverses. Much bigger matrices are practical now.</LI><LI>I do not think my ideas in Section 3 are expressed well.</LI></OL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-74082388611596048032017-05-06T07:44:00.000-04:002017-05-20T10:28:05.749-04:00Distribution of Maximum Rate of Profits in Simulation<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-SkS0wzkAYTM/WQ2yTfr_DfI/AAAAAAAAA0c/Xz9mRDYot9cx5SCqYmlBFZTESJy-ZkdigCLcB/s1600/MaxRateDistribution1.png" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-SkS0wzkAYTM/WQ2yTfr_DfI/AAAAAAAAA0c/Xz9mRDYot9cx5SCqYmlBFZTESJy-ZkdigCLcB/s320/MaxRateDistribution1.png" width="320" height="269" /></a></td></tr><tr><td align="center"><b>Figure 1: Blowup of Distribution of Maximum Rate of Profits</b></td></tr></tbody></table><P>This post extends the results from my last <A HREF="http://robertvienneau.blogspot.com/2017/05/i-just-simulated-6-billion-random.html">post</A>. I think of the results presented here as providing information about the implementation of my simulation. I do not claim any implications about actually existing economies. I did not have any definite anticipations about what I would see. I suppose it could be of interest to regenerate these results where coefficients of production are randomly generated from some non-uniform distribution. </P><P>I continue to use a capability to generate a random economy, where such an economy is characterized by a single technique. A technique is specified by a row vector of labor coefficients and a corresponding square Leontief input-output matrix. The labor coefficients are randomly generated from a uniform distribution on (0.0, 1.0]. Each coefficient in the Leontief input-output matrix is randomly generated from a uniform distribution on [0.0, 1.0). The random number generator is as provided by the class java.util.Random, in the Java programming language. I am running Java version 1.8. </P><P>Each random economy is tested for viability. Non-viable economies are discarded. Table 1 shows how many economies needed to be generated, given the number of produced commodities, to end up with a sample size of 300 viable economies. The maximum rate of profits is calculated for each viable economy. The maximum rate of profits occurs when the wage is zero, and the workers live on air. Thus, labor coefficients do not matter for the calculation of the maximum rate of profits. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Number of Simulated Economies</B></CAPTION><TR><TD ALIGN="center"><B>Seed for<BR>Random<BR>Generator</B></TD><TD ALIGN="center"><B>Number of<BR>Commodities</B></TD><TD ALIGN="center"><B>Number of<BR>Economies</B></TD></TR><TR><TD ALIGN="center">368,424,234</TD><TD ALIGN="center">2</TD><TD ALIGN="center">610</TD></TR><TR><TD ALIGN="center">345,657</TD><TD ALIGN="center">3</TD><TD ALIGN="center">6,124</TD></TR><TR><TD ALIGN="center">4,566,843</TD><TD ALIGN="center">4</TD><TD ALIGN="center">826,471</TD></TR><TR><TD ALIGN="center">547,527</TD><TD ALIGN="center">5</TD><TD ALIGN="center">> 2<SUP>31</SUP> - 1</TD></TR></TABLE><P>I looked at the distribution of the maximum rate of profits, calculated as a percentage, in several ways. Figure 2 presents four histograms, superimposed on one another. Figure 1 expands the left tails of these histograms. I suppose Figure 2 is somewhat easier to make sense of than Figure 1, when you click on the image. Maybe the statistics in Tables 2 and 3 are clearer. One can see, for example, in random economies in which two commodities are produced, the mean of the maximum rate of profits is 43.9%. The minimum, in these 300 random economies, of the maximum rate of profits is about 0.03% and the maximum is 318%. If I wanted to be more thorough, I would have to review how skewness and kurtosis are calculated by default in the Java class org.apache.commons.math3.stat.descriptive.DescriptiveStatistics. The coefficient of variation is the ratio of the standard deviation to the mean. The nonparametric analogy, reported in the last row in Table 3, is the ratio of the Inter-Quartile Range to the median. Anyways, the distribution of the maximum rate of profits, in random viable economies generated by the simulation, is non-Gaussian and highly skewed, with a tail extending to the right. </P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-fZJ-UASFJsY/WQ2ydpRyTnI/AAAAAAAAA0g/4g2nQEh24lEXA8fw_XDmM4BvpL2CnkovACLcB/s1600/MaxRateDistribution.png" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-fZJ-UASFJsY/WQ2ydpRyTnI/AAAAAAAAA0g/4g2nQEh24lEXA8fw_XDmM4BvpL2CnkovACLcB/s320/MaxRateDistribution.png" width="320" height="269" /></a></td></tr><tr><td align="center"><b>Figure 2: Distribution of Maximum Rate of Profits</b></td></tr></tbody></table><P></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: Parametric Statistics</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B></B></TD><TD ALIGN="center" COLSPAN="5"><B>Number of Produced Commodities</B></TD></TR><TR><TD ALIGN="center"><B>Two</B></TD><TD ALIGN="center"><B>Three</B></TD><TD ALIGN="center"><B>Four</B></TD><TD ALIGN="center"><B>Five</B></TD></TR><TR><TD ALIGN="center">Sample Size</TD><TD ALIGN="center">300</TD><TD ALIGN="center">300</TD><TD ALIGN="center">300</TD><TD ALIGN="center">300</TD></TR><TR><TD ALIGN="center">Mean</TD><TD ALIGN="center">43.9</TD><TD ALIGN="center">15.7</TD><TD ALIGN="center">8.28</TD><TD ALIGN="center">4.95</TD></TR><TR><TD ALIGN="center">Std. Dev.</TD><TD ALIGN="center">50.2</TD><TD ALIGN="center">19.3</TD><TD ALIGN="center">7.53</TD><TD ALIGN="center">5.90</TD></TR><TR><TD ALIGN="center">Skewness</TD><TD ALIGN="center">2.10</TD><TD ALIGN="center">3.89</TD><TD ALIGN="center">1.22</TD><TD ALIGN="center">2.63</TD></TR><TR><TD ALIGN="center">Kurtosis</TD><TD ALIGN="center">5.14</TD><TD ALIGN="center">22.2</TD><TD ALIGN="center">0.882</TD><TD ALIGN="center">9.64</TD></TR><TR><TD ALIGN="center">Coef. of Var.</TD><TD ALIGN="center">0.875</TD><TD ALIGN="center">0.811</TD><TD ALIGN="center">1.10</TD><TD ALIGN="center">0.839</TD></TR></TABLE><P></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: Nonparametric Statistics</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B></B></TD><TD ALIGN="center" COLSPAN="5"><B>Number of Produced Commodities</B></TD></TR><TR><TD ALIGN="center"><B>Two</B></TD><TD ALIGN="center"><B>Three</B></TD><TD ALIGN="center"><B>Four</B></TD><TD ALIGN="center"><B>Five</B></TD></TR><TR><TD ALIGN="center">Minimum</TD><TD ALIGN="center">0.0327</TD><TD ALIGN="center">0.113</TD><TD ALIGN="center">0.0107</TD><TD ALIGN="center">0.00405</TD></TR><TR><TD ALIGN="center">1st Quartile</TD><TD ALIGN="center">9.35</TD><TD ALIGN="center">4.51</TD><TD ALIGN="center">2.52</TD><TD ALIGN="center">1.17</TD></TR><TR><TD ALIGN="center">Median</TD><TD ALIGN="center">25.3</TD><TD ALIGN="center">9.72</TD><TD ALIGN="center">5.70</TD><TD ALIGN="center">2.99</TD></TR><TR><TD ALIGN="center">3rd Quartile</TD><TD ALIGN="center">57.3</TD><TD ALIGN="center">19.9</TD><TD ALIGN="center">11.3</TD><TD ALIGN="center">6.27</TD></TR><TR><TD ALIGN="center">Maximum</TD><TD ALIGN="center">318</TD><TD ALIGN="center">168</TD><TD ALIGN="center">36.2</TD><TD ALIGN="center">44.2</TD></TR><TR><TD ALIGN="center">IQR/Median</TD><TD ALIGN="center">1.90</TD><TD ALIGN="center">1.58</TD><TD ALIGN="center">1.54</TD><TD ALIGN="center">1.70</TD></TR></TABLE><P>With the simulation, the maximum rate of profits tends to be smaller, the more commodities are produced. I wish I could extend these results to a lot more produced commodities. National Income and Product Accounts (NIPAs), at the grossest level of aggregation have on the order of 100 produced commodities. Even if results with the assumption of an arbitrary probability distribution for coefficients of production could be directly applied empirically, one would like confirmation that trends seen with a very small number of produced commodities continue. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-90646468891290641662017-05-03T08:17:00.000-04:002017-05-20T10:28:25.472-04:00I Just Simulated 6 Billion Random Economies<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-ev5mPY0xcZ4/WQnA5RfRJcI/AAAAAAAAAz8/Gnq-MO7Kr9wQBFOEzuWFUNgsI1a9jJTNgCLcB/s1600/ViabilityProbability.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-ev5mPY0xcZ4/WQnA5RfRJcI/AAAAAAAAAz8/Gnq-MO7Kr9wQBFOEzuWFUNgsI1a9jJTNgCLcB/s320/ViabilityProbability.jpg" width="320" height="211" /></a></td></tr><tr><td align="center"><b>Figure 1: Probability a Random Economy Will Be Viable</b></td></tr></tbody></table><P>I have begun working towards replicating certain simulation results reported by Stefano Zambelli's. </P><P>At this point, I have implemented a capability to generate a random economy, where such an economy is characterized by a single technique. A technique is specified by a row vector of labor coefficients and a corresponding square Leontief input-output matrix. The labor coefficients are randomly generated from a uniform distribution on (0.0, 1.0]. Each coefficient in the Leontief input-output matrix is randomly generated from a uniform distribution on [0.0, 1.0). The random number generator is as provided by the class java.util.Random, in the Java programming language. I am running Java version 1.8. </P><P>A Monte Carlo simulation, in the results reported here, tests each random economy for viability, where the technique, for each economy, is used to produce a specified number of commodities. A viable economy can reproduce the inputs used up in producing the outputs. If the economy is just viable, nothing is left over to pay the workers and the capitalists. The Hawkins-Simon condition can be used to check for viability. </P><P>Table 1 reports the results. The number of Monte Carlo runs, for each row, is 1,000,000,000. The seed is reported so I can replicate my results, if I want. I think I can provide a symmetry argument for why the probability for the first row should be 1/2. I reran the simulation for the last row with 2,000,000,000 runs and the same seed. I still found zero viable economies. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Simulation Results</B></CAPTION><TR><TD ALIGN="center"><B>Seed for<BR>Random<BR>Generator</B></TD><TD ALIGN="center"><B>Number of<BR>Commodities</B></TD><TD ALIGN="center"><B>Number of<BR>Viable<BR>Economies</B></TD><TD ALIGN="center"><B>Probability</B></TD></TR><TR><TD ALIGN="center">46,576,889</TD><TD ALIGN="center">2</TD><TD ALIGN="center">499,967,476</TD><TD ALIGN="center">49.9967476%</TD></TR><TR><TD ALIGN="center">89,058,538</TD><TD ALIGN="center">3</TD><TD ALIGN="center">50,198,690</TD><TD ALIGN="center">5.019869%</TD></TR><TR><TD ALIGN="center">7,586,338</TD><TD ALIGN="center">4</TD><TD ALIGN="center">372,339</TD><TD ALIGN="center">0.0372339</TD></TR><TR><TD ALIGN="center">784,054</TD><TD ALIGN="center">5</TD><TD ALIGN="center">99</TD><TD ALIGN="center">0.0000099%</TD></TR><TR><TD ALIGN="center">568,233,269</TD><TD ALIGN="center">6</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0%</TD></TR></TABLE><P>Zambelli suggests randomly specifying a rescaled output, in some sense, for the technology so as to ensure viability. I have a rough conceptual understanding of this step, but I need a better understanding to reduce it to source code. I think I'll go on to further analyses before revisiting the issue of viability. The above results certainly suggest that my analyses will be limited, in the mean time, to economies that produce only two, three, or maybe four commodities. </P><P>I think that Zambelli's approach is worthwhile for pursuing the results in which he is interested. One limitation arises with applying a probability distribution to one particular description of technology. In practice, coefficients of production evolve in a non-random manner. Pasinetti's structural dynamics is a good way of exploring <A HREF="http://robertvienneau.blogspot.com/2013/07/trends-in-hardware-and-software-costs.html">technical progress</A>in the tradition of Sraffa. </P><B>References</B><UL><LI>Stefano Zambelli (2004). The 40% neoclassical aggregate theory of production. <I>Cambridge Journal of Economics</I> 28(1): pp. 99-120.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-37492075750150383442017-04-20T16:30:00.000-04:002017-04-20T16:30:42.083-04:00Nonstandard Investments as a Challenge for Multiple Interest Rate Analysis?<B>1.0 Introduction</B><P>This post contains some musing on corporate finance and its relation to the theory of production. </P><B>2.0 Investments, the NPV, and the IRR</B><P>In finance, an <I>investment project</I> or, more shortly, an <I>investment</I>, is a sequence of dated cash flows. Consider an investment in which these cash flows take place at the end of <I>n</I> successive years. Let <I>C</I><SUB><I>t</I></SUB>; <I>t</I> = 0, 1, ..., <I>n</I> - 1; be the cash flow at the end of the <I>t</I>th year here, counting back from the last year in the investment. That is, <I>C</I><SUB><I>n</I> - 1</SUB> is the cash flow at the end of the first year in the investment, and <I>C</I><SUB>0</SUB> is the last cash flow. </P><P>The <I>Net Present Value</I> (NPV) of an investment is the sum of discounted cash flows in the investment. Let <I>r</I> be the interest rate used in time time discounting, and suppose all cash flows are discounted to the end of the first year in the investment. Then the NPV of the illustrative investment is: </P><BLOCKQUOTE>NPV<SUB>0</SUB>(<I>r</I>) = <I>C</I><SUB><I>n</I> - 1</SUB> + <I>C</I><SUB><I>n</I> - 2</SUB>/(1 + <I>r</I>) + ... + <I>C</I><SUB>0</SUB>/(1 + <I>r</I>)<SUP><I>n</I> - 1</SUP></BLOCKQUOTE><P>If the above expression is multiplied by (1 + <I>r</I>)<SUP><I>n</I> - 1</SUP>, one obtains the NPV of the investment with every cash flow discounted to the last year in the investment: </P><BLOCKQUOTE>NPV<SUB>1</SUB>(<I>r</I>) = <I>C</I><SUB><I>n</I> - 1</SUB>(1 + <I>r</I>)<SUP><I>n</I> - 1</SUP>+ <I>C</I><SUB><I>n</I> - 2</SUB>(1 + <I>r</I>)<SUP><I>n</I> - 2</SUP>+ ... + <I>C</I><SUB>0</SUB></BLOCKQUOTE><P>For the next step, I need some sign conventions. Let a positive cash flow designate revenues, and a negative cash flow be a cost. Suppose, for now, that the (temporally) first cash flow is a cost, that is negative. Then (-1/<I>C</I><SUB><I>n</I> - 1</SUB>) NPV<SUB>1</SUB>(<I>r</I>) is a polynomial in (1 + <I>r</I>), with unity as the coefficient for the highest-order term. All other terms are real. </P><P>Such a polynomial has <I>n</I> - 1 roots. These roots can be real numbers, either negative, zero, or positive. They can be complex. Since all coefficients of the polynomial are real, complex roots enter as conjugate pairs. Roots can be repeating. At any rate, the polynomial can be factored, as follows: </P><BLOCKQUOTE>NPV<SUB>1</SUB>(<I>r</I>) = (-<I>C</I><SUB><I>n</I> - 1</SUB>)(<I>r</I> - <I>r</I><SUB>0</SUB>) (<I>r</I> - <I>r</I><SUB>1</SUB>)... (<I>r</I> - <I>r</I><SUB><I>n</I> - 1</SUB>) </BLOCKQUOTE><P>where <I>r</I><SUB>0</SUB>, <I>r</I><SUB>1</SUB>, ..., <I>r</I><SUB><I>n</I> - 1</SUB> are the roots of the polynomial. Note that the interest rate appears only in terms in which the difference between the interest rate and one root is taken. And all roots appear on the Right Hand Side. I am going to call an specification of NPV with these properties an <I>Osborne expression</I> for NPV. </P><P>Suppose, for now, that at least one root is real and non-negative. The <I>Internal Rate of Return</I> (IRR) is the smallest real, non-negative root. For notational convenience, let <I>r</I><SUB>0</SUB> be the IRR. </P><B>3.0 Standard Investments in Selected Models of Production</B><P>A <I>standard investment</I> is one in which all negative cash flows precede all positive cash flows. Is there a theorem that an IRR exists for each standard investment? Perhaps this can be proven by discounting all cash flows to the end of the year in which the last outgoing cash flow occurs. Maybe one needs a clause that the undiscounted sum of the positive cash flows does not fall below the undiscounted sum of the negative cash flows. </P><P>At any rate, an Osborne expression for NPV has been calculated for standard investments characterizing two models of production. As I recall it, Osborne (2010) illustrates a more abstract discussion with a point-input, flow-output example. Consider a model in which a machine is first constructed, in a single year, from unassisted labor and land. That machine is then used to produce output over multiple years. Given certain assumptions on the pattern of the efficiency of the machine, this example is of a standard investment, with one initial negative cash flow followed by a finite sequence of positive cash flows. </P><P>On the other hand, I have <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058">presented</A>an example for a flow-input, point-output model. Techniques of production are represented as finite series of dated labor inputs, with output for sale on the market at a single point in the time. Each technique is characterized by a finite sequence of negative cash flows, followed by a single positive cash flow. </P><P>In each of these two examples, the NPV can be represented by an Osborne expression that combines information about all roots of a polynomial. Thus, basing an investment decision on the NPV uses more information than basing it on the IRR, which is a single root of the relevant polynomial. </P><B>4.0 Non-standard Investments and Pitfalls of the IRR</B><P>In a <I>non-standard investment</I> at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments can highlight three pitfalls in basing an investment decision on the IRR: </P><UL><LI>Multiple IRRs: The polynomial defining the IRR may have more than one real, non-negative root. What is the rationale for picking the smallest?</LI><LI>Inconsistency in recommendations based on IRR and NPV: The smallest real non-negative root may be positive (suggesting a good investment), with a negative NPV (suggesting a bad investment).</LI><LI>No IRR: All roots may be complex.</LI></UL><P>Berk and DeMarzo (2014) present the example in Table 1 as an illustration of the third pitfall. They imagine an author who receives an advance of $750 hundred thousands, sacrifices an income of $500 hundred thousand in each year of writing a book, and, finally, receives a royalty of one million dollars upon publication. The roots of the polynomial defining the NPV are -1.71196 + 0.78662 j, -1.71196 - 0.78662 j, 0.04529 + 0.30308 j, 0.04529 - 0.30308 j. All of these roots are complex; none satisfy the definition of the IRR. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: A Non-Standard Investment</B></CAPTION><TR><TD ALIGN="center"><B>Year</B></TD><TD ALIGN="center"><B>Revenue</B></TD></TR><TR><TD ALIGN="center">0</TD><TD ALIGN="center">750</TD></TR><TR><TD ALIGN="center">1</TD><TD ALIGN="center">-500</TD></TR><TR><TD ALIGN="center">2</TD><TD ALIGN="center">-500</TD></TR><TR><TD ALIGN="center">3</TD><TD ALIGN="center">-500</TD></TR><TR><TD ALIGN="center">4</TD><TD ALIGN="center">1,000</TD></TR></TABLE><P></P><B>5.0 Issues for Multiple Interest Rate Analysis</B><P>Osborne, in his 2014 book, extends his 2010 analysis of the NPV to consider the first and second pitfall above. Nowhere do I know of is an Osborne expression for the NPV derived for an example in which the third pitfall arises. </P><P>The idea that the pitfalls above for the use of the IRR might be a problem for multiple interest rate analysis was suggested to me anonymously. On even hours, I do not see this. Why should I care about how many roots there are in an Osborne expression for the NPV, their sign, or even if they are complex? </P><P>On the other hand, I wonder about how non-standard investments relate to the theory of production. I know that an example can be <A HREF="http://robertvienneau.blogspot.com/2017/01/reswitching-in-example-of-one-commodity.html">constructed</A>, in which the price of a used machine becomes negative before it becomes positive. Can the varying efficiency of the machine result in a non-standard investment? After all, the cash flow, in such an example of joint production, is the sum of the price of the conventional output of the machine and the price of the one-year older machine. Even when the latter is negative, the sum need not be negative. But, perhaps, it can be in some examples. </P><P>Not all techniques in models with joint production, of the production of commodities by means of commodities, can be represented as dated labor flows. I guess one can still talk about NPVs. Can one formulate an algorithm, based on NPVs, for the choice of technique? How would certain annoying possibilities, such as <A HREF="http://robertvienneau.blogspot.com/2009/08/some-issues-in-joint-production.html">cycling</A>be accounted for? Can one always formulate an Osborne expression for the NPV? Do properties of multiple interest rates have implications for, for example, a truncation rule in a model of fixed capital? Perhaps a non-standard investment, for a fixed capital example and one pitfall noted above, always has a cost-minimizing truncation in which the pitfall does not arise. Or perhaps the opposite is true. </P><P>Anyway, I think some issues could support further research relating models of production in economics and finance theory. Maybe one obtains, at least, a translation of terms. </P><B>Appendix: Technical Terminology</B><P>See body of post for definitions. </P><UL><LI>Flow Input, Point Output</LI><LI>Investment</LI><LI>Investment Project</LI><LI>Internal Rate of Return (IRR)</LI><LI>Net Present Value (NPV)</LI><LI>Non Standard Investment</LI><LI>Osborne Expression (for NPV)</LI><LI>Point-Input, Flow Output model</LI><LI>Standard Investment</LI></UL><B>References</B><UL><LI>Jonathan Berk and Peter DeMarzo (2014). <I>Corporate Finance</I>, 3rd edition. Boston: Pearson Education</LI><LI>Michael Osborne (2010). A resolution to the NPV-IRR debate? <I>Quarterly Review of Economics and Finance</I>, V. 50, Iss. 2: 234-239.</LI><LI>Michael Osborne (2014). <I>Multiple Interest Rate Analysis: Theory and Applications</I>. New York: Palgrave Macmillan</LI><LI>Robert Vienneau (2016). <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058"><I>The choice of technique with multiple and complex interest rates</I></A>, DRAFT.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-77463172424352089392017-04-13T15:43:00.000-04:002017-04-13T15:43:08.566-04:00Elsewhere<UL><LI>Beatrice Cherrier <A HREF="https://beatricecherrier.wordpress.com/2016/05/23/how-the-term-mainstream-economics-became-mainstream-a-speculation/">suggests</A> that Paul Samuelson originated the term "mainstream economics", in his textbook. (<B>h/t</B> I think I found this by Unlearning Economics's twitter feed.)</LI><LI>Jo Michell <A HREF="http://reteacheconomics.org/articles/review-econocracy/">reviews</A> <I>The Econocracy</I>, by Joe Earle, Cahal Moran, and Zach Ward-Perkins.</LI><LI>On twitter, Cameron Murray <A HREF="https://twitter.com/Rumplestatskin/status/833558739487567873">finds</A>, of the 46 who responded, 78% did not "learn about the Cambridge Capital Controversy at point in [their] degree".</LI><LI>In the <I>Guardian</I>, Kate Raworth <A HREF="https://www.theguardian.com/global-development-professionals-network/2017/apr/06/kate-raworth-doughnut-economics-new-economics">argues</A> that new economics is needed to replace the old economics and its foundation on false laws of physics.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-41069876809213737342017-04-01T08:41:00.000-04:002017-04-01T08:41:15.645-04:00Bifurcations With Variations In The Rate Of Growth<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-kuKKb2gCiXI/WN5BmGz7HTI/AAAAAAAAAzY/Vg_t-XRHAzgY5ZF-5eMFQLfOewT8wwu7QCLcB/s1600/LaborIntensity.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-kuKKb2gCiXI/WN5BmGz7HTI/AAAAAAAAAzY/Vg_t-XRHAzgY5ZF-5eMFQLfOewT8wwu7QCLcB/s320/LaborIntensity.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 1: Perversity and Non-Perversity in the Labor Market Varying with the Rate of Growth</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I have been considering how the existence and properties of switch points vary with parameters specifying numerical examples of models of the production of commodities by means of commodities. <A HREF="http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.html">Here</A><A HREF="http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.html">are</A><A HREF="http://robertvienneau.blogspot.com/2017/02/a-reswitching-example-in-model-of.html">some</A><A HREF="http://robertvienneau.blogspot.com/2016/12/perturbation-of-reswitching-example.html">examples</A><A HREF="http://robertvienneau.blogspot.com/2017/01/a-story-of-technical-innovation.html">of</A>such analyses of structural stability. This post adds to this series. </P><P>I consider a change in sign of real Wicksell effects to be a bifurcation. In the model in this post, the steady state rate of growth is an exogenous parameter. So a change of sign of real Wicksell effects, associated with a variation in the steady state rate of growth, is a bifurcation. </P><B>2.0 Technology</B><P>The technology for this example is as usual. Suppose two commodities, iron and corn, are produced in the example economy. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">305/494</TD><TD ALIGN="center">1</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD><TD ALIGN="center">2</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD><TD ALIGN="center">2/5</TD></TR></TABLE><P>For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the sole corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process. </P><P>Each technique is represented by a two-element row vector of labor coefficients and a 2x2 Leontief input-output matrix. For example, the vector of labor coefficients for the Beta technique, <B>a</B><SUB>0, β</SUB>, is: </P><BLOCKQUOTE><B>a</B><SUB>0, β</SUB> = [305/494, 1] </BLOCKQUOTE><P>The components of the Leontief matrix for the Beta technique, <B>A</B><SUB>β</SUB>, are: </P><BLOCKQUOTE><I>a</I><SUB>1,1, β</SUB> = 229/494 </BLOCKQUOTE><BLOCKQUOTE><I>a</I><SUB>1,2, β</SUB> = 2 </BLOCKQUOTE><BLOCKQUOTE><I>a</I><SUB>2,1, β</SUB> = 3/1976 </BLOCKQUOTE><BLOCKQUOTE><I>a</I><SUB>2,2, β</SUB> = 2/5 </BLOCKQUOTE><P>The labor coefficients for the Alpha technique, <B>a</B><SUB>0, α</SUB>, differ in the first element from those for the Beta technique. The Leontief matrix for the Alpha technique, <B>A</B><SUB>α</SUB>, differs from the Leontief matrix for the Beta technique in the first column. </P><P>(The mathematics in this post is set out in terms of linear algebra. I needed to remind myself of how to work out quantity flows with a positive rate of growth. I solved the example with Octave, the open-source equivalent of Matlab for the example. I haven't checked the graphs by also working them out by hand. You can click on the figures to see them somewhat larger.) </P><B>3.0 Prices and the Choice of Technique</B><P>Consider steady-state prices that repeat, year after year, as long as firms adopt the same technique. Let <B>a</B><SUB>0</SUB> and <B>A</B> be the labor coefficients and the Leontief matrix for that technique. Suppose labor is advanced and wages are paid out of the surplus at the end of the year. Then prices satisfy the following system of equations: </P><BLOCKQUOTE><B>p</B> <B>A</B> (1 + <I>r</I>) + <B>a</B><SUB>0</SUB> <I>w</I> = <B>p</B></BLOCKQUOTE><P>where <B>p</B> is a two-element row vector of prices, <I>w</I> is the wage, and <I>r</I> is the rate of profits. Let <B>e</B> be a column vector specifying the commodities constituting the numeraire. Then: </P><BLOCKQUOTE><B>p</B> <B>e</B> = 1 </BLOCKQUOTE><P>For the numerical example, a bushel corn is the numeraire, and <B>e</B>is the second column of the identity matrix. I think of the numeraire as in the proportions in which households consume commodities. </P><P>The system of equations for prices of production, including the equation for the numeraire, has one degree of freedom. Formally, one can solve for prices and the wage as functions of an externally given rate of profits. The first equation above can be rewritten as: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> <I>w</I> = <B>p</B> [<B>I</B> - (1 + <I>r</I>) <B>A</B>] </BLOCKQUOTE><P>Multiply through, on the right, by the inverse of the matrix in square brackets: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <I>w</I> = <B>p</B></BLOCKQUOTE><P>Multiply through, again on the right, by <B>e</B>: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B> <I>w</I> = <B>p</B> <B>e</B> = 1 </BLOCKQUOTE><P>Both sides of the above equation are scalars. The wage is: </P><BLOCKQUOTE><I>w</I> = 1/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>The above equation is called the <I>wage-rate of profits curve</I> or, more shortly, the <I>wage curve</I>. Prices of production are: </P><BLOCKQUOTE><B>p</B> = <B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP>/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>The above two equations solve the price system, in some sense. </P><P>Figure 2 plots the wage curves for the example. The downward-sloping blue and red curves show that, for each technique, a lower steady-state real wage is associated with a higher rate of profits. The two curves intersect at the two switch points, at rates of profits of 20% and 80%. For rates of profits between the switch points, the Alpha technique is cost-minimizing and its wage curve constitutes the outer envelope of the wage curves in this region. For feasible rates of profits outside that region, the Beta technique is cost-minimizing. (I talk more about this figure at least twice below.) </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-WRfMraI9O8Y/WN5CDkZI-SI/AAAAAAAAAzg/4MnfISsvuJMo6zy2K9B95ENZnfAINixzACLcB/s1600/WageCurves.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-WRfMraI9O8Y/WN5CDkZI-SI/AAAAAAAAAzg/4MnfISsvuJMo6zy2K9B95ENZnfAINixzACLcB/s320/WageCurves.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curves also Characterize Tradeoff Between Consumption per Worker and Steady State Rate of Growth</b></td></tr></tbody></table><P></P><B>4.0 Quantities</B><P>Suppose the steady-state rate of growth for this economy is 100 <I>g</I> percent. A system of equations, dual to the price equations, arises for quantity flows. Let <B>q</B> denote the column vector of gross quantities, per labor-year employed, produced in a given year. Let <B>y</B> be the column vector of net quantities, per labor-year. Net quantities constitute the surplus once the (circulating) capital goods advanced at the start of the year, for a given technique, are replaced: </P><BLOCKQUOTE><B>y</B> = <B>q</B> - <B>A</B> <B>q</B> = (<B>I</B> - <B>A</B>) <B>q</B></BLOCKQUOTE><P>Since quantities are defined per person-year, employment with these quantities is unity: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> <B>q</B> = 1 </BLOCKQUOTE><P>By hypothesis, net quantities are the sum of consumption and capital goods to accumulate at the steady state rate of profits: </P><BLOCKQUOTE><B>y</B> = <I>c</I> <B>e</B> + <I>g</I> <B>A</B> <B>q</B></BLOCKQUOTE><P>Substituting into the first equation in this section and re-arranging terms yields: </P><BLOCKQUOTE><I>c</I> <B>e</B> = [<B>I</B> - (1 + <I>g</I>) <B>A</B>] <B>q</B></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>c</I> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B> = <B>q</B></BLOCKQUOTE><P>Multiply through on the left by the row vector of labor coefficients: </P><BLOCKQUOTE><I>c</I> <B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B> = <B>a</B><SUB>0</SUB> <B>q</B> = 1 </BLOCKQUOTE><P>Consumption per person-year is: </P><BLOCKQUOTE><I>c</I> = 1/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>Gross quantities are: </P><BLOCKQUOTE><B>q</B> = [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>Interestingly enough, the relationship between consumption per worker and the rate of growth is identical to the relationship between the wage and the rate of profits. Thus, Figure 1 is also a graph of the trade-off, for the two technique, between steady-state consumption per worker and the rate of growth. One can think of the abscissa as relabeled the rate of growth and the ordinate as relabeled consumption per person-year. In the graph, the grey point illustrates consumption per worker at a rate of growth of 10% for the Beta technique. </P>The ordinate on this graph is consumption throughout the economy. If the rate of profits exceeds the rate of growth, both those obtaining income from wages and those obtaining income from profits will be consuming. When the rates of growth and profits are equal, all profits are accumulated. </P><B>5.0 Some Accounting Identities</B><P>The value of capital per worker is: </P><BLOCKQUOTE><I>k</I> = <B>p</B> <B>A</B> <B>q</B></BLOCKQUOTE><P>The value of net income per worker is: </P><BLOCKQUOTE><I>y</I> = <B>p</B> <B>y</B> = <B>p</B> (<B>I</B> - <B>A</B>) <B>q</B></BLOCKQUOTE><P>(I hope the distinction between the scalar <I>y</I> and the vector <B>y</B>is clear in this notation.) </P><P>The value of net income per worker can be expressed in terms of the sum of income categories. Rewrite the first equation in Section 3: </P><BLOCKQUOTE><B>p</B> (<B>I</B> - <B>A</B>) = <B>a</B><SUB>0</SUB> <I>w</I> + <B>p</B> <B>A</B> <I>r</I></BLOCKQUOTE><P>Multiply both sides by the vector of gross outputs: </P><BLOCKQUOTE><B>p</B> (<B>I</B> - <B>A</B>) <B>q</B> = <B>a</B><SUB>0</SUB> <B>q</B> <I>w</I> + <B>p</B> <B>A</B> <B>q</B> <I>r</I></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>y</I> = <I>w</I> + <I>k</I> <I>r</I></BLOCKQUOTE><P>In this model, net income per worker is the sum of wages and profits per worker. </P><P>Net income per worker can also be decomposed by how it is spent. For the third equation in Section 4, multiply both sides by the price vector: </P><BLOCKQUOTE><B>p</B> <B>y</B> = <I>c</I> <B>p</B> <B>e</B> + <I>g</I> <B>p</B> <B>A</B> <B>q</B></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>y</I> = <I>c</I> + <I>g</I> <I>k</I></BLOCKQUOTE><P>Net income per worker is the sum of consumption per worker and investment per worker. </P><P>Equating the two expressions for net income per worker allows one to derive an interesting graphical feature of Figure 1. This equation is: </P><BLOCKQUOTE><I>w</I> + <I>r</I> <I>k</I> = <I>c</I> + <I>g</I> <I>k</I></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE>(<I>r</I> - <I>g</I>) <I>k</I> = <I>c</I> - <I>w</I></BLOCKQUOTE><P>Or solving for the value of capital per worker: </P><BLOCKQUOTE> <I>k</I> = (<I>c</I> - <I>w</I>)/(<I>r</I> - <I>g</I>) </BLOCKQUOTE><P>Capital per worker, for a given technique, is the additive inverse of the slope of two points on the wage curve for that technique. Figure 1 illustrates for the Beta technique, with a rate of growth of 10% and a rate of profits of 80%, as at the upper switch point. </P><B>6.0 Real Wicksell Effects</b><P>This section and the next presents an analysis confined to prices at the switch point for a rate of profits of 80%. </P><P>For a rate of profits infinitesimally lower than 80%, the Alpha technique is cost-minimizing. And for a rate of profits infinitesimally higher, the Beta technique is cost minimizing. I have explained above how to calculate the value of capital per worker, for the two techniques, at any given rate of growth. </P><P>Abstract from any change in prices of production associated with a change in the rate of profits. The difference between capital per head for the Beta technique and capital per head for the Alpha technique, both calculated at the prices for the switch point, is the change in "real" capital around the switch point associated with an increase in the rate of profits. Figure 3 graphs this <I>real Wicksell effect</I> as a function of the rate of steady state growth. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-qJs_FGZkkBE/WN5B3eymRDI/AAAAAAAAAzc/HrYbT_geXh0qSE6mgxzquvK-mDTAYRsugCLcB/s1600/WicksellEffect.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-qJs_FGZkkBE/WN5B3eymRDI/AAAAAAAAAzc/HrYbT_geXh0qSE6mgxzquvK-mDTAYRsugCLcB/s320/WicksellEffect.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 3: Variation in Real Wicksell Effect with Steady State Rate of Growth</b></td></tr></tbody></table><P></P><P>Two regions are apparent in Figure 3. The intersection, at the left, of the downward-sloping graph with the axis for the change in the value of capital per worker shows that the real Wicksell effect is positive, for this switch point, in a stationary state. Around the given switch point, a higher rate of profits is associated, in a stationary state, with firms wanting to adopt a more capital-intensive technique. If a greater scarcity of capital caused the rate of profits to rise, so as to ration the supply of capital, such a logical possibility could not be demonstrated. </P><P>The real Wicksell effect, for the switch point at the higher rate of profits, is zero when the rate of growth is equal to the rate of profits at the other switch points. The value of capital per person-year is the same for the two techniques, in this case. Consider a line, in Figure 1, connecting the two switch points. It also connects the points on the wage curve for the Alpha technique for a rate of profits of 80% and a rate of growth of 20%. And the same goes for the wage curve for the Beta technique. </P><B>7.0 Real Wicksell Effects in the Labor Market</B><P>A variation in real Wicksell effects with the steady state rate of growth is also manifested in the labor market. I have echoed above some mathematics which shows that the value of national income is the dot product of a vector of prices with the vector of net quantity flows. The price vector depends, given the technique, on the rate of profits at which prices of production are found. The quantity vector depends on the steady state rate of growth. The reciprocal, (1/<I>y</I>), is the amount of labor firms want to hire, per numeraire unit of national income, for a given technique. The difference at a switch point between these reciprocals, for the two techniques, is another way of looking at real Wicksell effects. </P><P>Around the switch point at a rate of profits of 80%, a lower wage is associated with firms adopting the Beta technique. And a higher wage is associated with firms adopting the Alpha technique. The difference of the above reciprocals, between the Alpha and Beta techniques, is the increase in labor, per numeraire-unit net output, associated with an infinitesimal increase in wages, at the prices for the switch point. Figure 1 shows this difference, as a function of the steady state rate of growth, at the switch point with the higher rate of profits in the example. </P><P>Figure 1 qualitatively resembles Figure 3. For a stationary state, a higher wage is associated with firms wanting to employ more labor, per numeraire unit of net output. This effect is reversed for a high enough steady state rate of growth. The bifurcation, here too, occurs at the rate of growth for the switch point at 20%. </P><B>8.0 Conclusion</B><P>This post has illustrated a comparison among steady state growth paths at rates of profits associated with a switch point. And this switch point is "perverse" from the perspective of outdated neoclassical theory, at least at a low rate of growth. But the perversity of this switch point varies with the rate of growth. In the example, when the rate of growth is between the rate of profits at the two switch points, the second switch point becomes non-perverse. </P><P>And it can go the other way. Real Wicksell effects do not even need to be monotonic. I need to find an example with at least three commodities, two techniques, and three switch points. In such an example, the switch point with the largest rate of profits will have a negative real Wicksell effect for a stationary state, a positive real Wicksell effect for steady state rates of growth between the first two switch points, and a negative real Wicksell effect for higher rates of growth, between the second and third switch points. </P><P>(I want to look up Gandolfo (2008) in the light of <A HREF="http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.html">past</A> posts. Can I tell this <A HREF="http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.html">tale</A>in terms of increasing returns, instead of exogenous technical change?) </P><B>References</B><UL><LI>Giancarlo Gandolfo (2008). Comment on "C.E.S. production functions in the light of the Cambridge critique". <I>Journal of Macroeconomics</I>, V. 30, No. 2 (June): pp. 798-800.</LI><LI>Nell (1970). A note on Cambridge controversies in capital theory. <I>Journal of Economic Literature</I>V. 8, No. 1 (March): 41-44.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com1tag:blogger.com,1999:blog-26706564.post-26021696289018522692017-03-28T16:24:00.000-04:002017-03-29T06:05:45.095-04:00Niall Kishtainy's Pluralist, Popular History Of Economics<P>This post calls attention to <A HREF="https://www.amazon.com/Little-History-Economics-Histories/dp/0300206364">A Little History of Economics</A>, by Niall Kishtainy. Since I only skimmed this in a local bookstore, this post is not a proper review. </P><P>Kishtainy's book, in successive chapters, focuses on the lives of specific economists. The concluding chapter asks why one would want to be an economist, a question that is answered, I gather, by the preceding chapters. The book is in the same genre as Robert Heilbroner's <I>The Worldly Philosophers</I>. Since many more economists are covered in a short span, individual chapters are shorter. As I recall, economists (not in this order) covered include: </P><UL><LI>Augustine and Thomas Aquinas</LI><LI>Mercantilists</LI><LI>Francois Quesnay, Mirabeau, and other Physiocrats</LI><LI>Adam Smith</LI><LI>David Ricardo</LI><LI>Charles Fourier, Robert Owen, and other utopian socialists</LI><LI>Thomas Malthus</LI><LI>Friedrich List</LI><LI>Karl Marx</LI><LI>William Stanley and Alfred Marshall</LI><LI>German historical school, Austrian school, and the methodenstreit</LI><LI>Thorstein Veblen</LI><LI>Vladimir Lenin and John Hobson</LI><LI>Ludwig von Mises</LI><LI>Joan Robinson and Edward Chamberlin</LI><LI>John Maynard Keynes</LI><LI>Paul Samuelson, J. R. Hicks</LI><LI>Friedrich Hayek</LI><LI>Arthur Lewis, Paul Rosenstein-Rodan, and Raul Prebisch</LI><LI>Robert Solow, Trevor Swan, and Paul Romer</LI><LI>Joseph Schumpeter</LI><LI>Gary Becker</LI><LI>John Von Neumann and John Nash</LI><LI>Ken Arrow and Gerard Debreu</LI><LI>Fidel Castro, Che Guevara, and Andre Frank</LI><LI>James Buchanan</LI><LI>Milton Friedman</LI><LI>George Akerlof and Joseph Stiglitz</LI><LI>Hyman Minsky</LI><LI>John Muth, Eugene Fama, and Robert Lucas</LI><LI>Ed Prescott and Finn Kyland</LI><LI>Behavioral economics</LI><LI>Thomas Piketty</LI></UL><P>Doubtless, if I read the book in detail, I would have objections to specific details. The attempted coverage, however, seems quite impressive. This book extends from before to after most histories. And it covers a wider range of economists than most. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com2tag:blogger.com,1999:blog-26706564.post-86887325369647320132017-03-22T08:32:00.000-04:002017-03-22T08:32:05.011-04:00Krugman Confused On Trade, Capital Theory<P>Over on EconSpeak, Bruce Wilder provides <A HREF="http://econospeak.blogspot.com/2017/03/review-of-economism-bad-economics-and.html?showComment=1489869775705#c6656690971941685157">some</A><A HREF="http://econospeak.blogspot.com/2017/03/review-of-economism-bad-economics-and.html?showComment=1489870271977#c3201788875710212968">comments</A> on a post. He notes that economists wanting to criticize glib free-market ideology in the public discourse often seem unwilling to discard neoclassical economic theory. </P><P>Paul Krugman illustrates how theoretically conservative and neoclassical a liberal economist can be. (I use "liberal" in the sense of contemporary politics in the USA.) I refer to Krugman's <A HREF="https://krugman.blogs.nytimes.com/2017/03/20/robot-geometry-very-wonkish/?module=BlogPost-Title&version=Blog%20Main&contentCollection=Opinion&action=Click&pgtype=Blogs®ion=Body">post</A>from earlier this week, in which he adapts an analysis from the theory of international trade to consider technological innovation (e.g., robots). Krugman presents a diagram, in which endowments of capital and labor are measured along the two axes. Krugman does not seem to be aware that one cannot, in general, coherently talk about a quantity of capital, prior to and independently of prices. He goes on to talk about "capital-intensive" and "labor-intensive" techniques of production. </P><P>I point to my <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2237279">draft paper</A>, "On the loss from trade", to illustrate my point that one cannot meaningfully talk about the endowment of capital. </P><P>(I did submit this paper to a journal. A reviewer said it was not original enough. I emphasized that I was illustrating my points in a flow-input, point output model, with a one-way flow from factors of production to consumption goods, not a model of production of commodities by means of commodities. Steedman & Metcalfe (1979) also has a one-way model, albeit with a point-input, point-output model. So the reviewer's comments were fair. Embarrassingly, I cite other papers from that book. Apparently, I had forgotten that paper, if I ever read it. I suppose that, given the chance, I could have distinguished some of my points from those made in Steedman & Metcalfe (1979). Also, I close my model with utility-maximization; if I recall correctly, Steedman leaves such an exercise to the reader in papers in that book.) </P><B>Reference</B><UL><LI>Ian Steedman and J. S. Metcalfe (1979). 'On foreign trade'. In <I>Fundamental Issues in Trade Theory</I> (ed. by Ian Steedman).</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-62579035234011248362017-03-18T08:02:00.000-04:002017-03-18T08:02:10.727-04:00Reswitching Only Under Oligopoly<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-PU6HifxVWUo/WMkh365gGHI/AAAAAAAAAy4/dhEXkq9sPrQNU90tgUs6NGYBELhP9apjQCLcB/s1600/OligopolyPerturbation.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-PU6HifxVWUo/WMkh365gGHI/AAAAAAAAAy4/dhEXkq9sPrQNU90tgUs6NGYBELhP9apjQCLcB/s320/OligopolyPerturbation.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 1: Rates of Profits for Switch Points for Differential Rates of Profits</b></td></tr></tbody></table><B>1.0 Introduction</B><P>Suppose one knows the technology available to firms at a given point in time. That is, one knows the techniques among which managers of firms choose. And suppose one finds that reswitching cannot occur under this technology, given prices of production in which the same rate of profits prevails among all industries. But, perhaps, barriers to entry persist. If one analyzes the choice of technique for the given technology, under the assumption that prices of production reflect stable (non-unit) ratios of profits, differing among industries, reswitching may arise for the technology. The numerical example in this post demonstrates this logical possibility. </P><P>The numerical example follows a model of oligopoly I have previously <A HREF="http://robertvienneau.blogspot.com/2017/02/a-reswitching-example-in-model-of.html">outlined</A>. In some sense, the example is symmetrical to the example in this <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2912181">draft paper</A>. That example is of a reswitching example under pure competition, which becomes an example without reswitching and capital reversing, if the ratio of the rates of profits among industries differs enough. The example in this post, on the other hand, has no reswitching or capital reversing under pure competition. But if the ratios of the rates of profits becomes extreme enough, it becomes a reswitching example. </P><B>2.0 Technology</B><P>The technology for this example resembles many I have explained in past posts. Suppose two commodities, iron and corn, are produced in the example economy. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">305/494</TD><TD ALIGN="center">1</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD><TD ALIGN="center">11/10</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD><TD ALIGN="center">2/5</TD></TR></TABLE><P>For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the lone corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process. </P><B>3.0 Price Equations</B><P>The choice of technique is analyzed on the basis of cost-minimization, with prices of production. Suppose the Alpha technique is cost minimizing. Then the following system of equalities and inequalities hold: </P><BLOCKQUOTE>[(1/10)<I>p</I> + (1/40)](1 + <I>r</I><I>s</I><SUB>1</SUB>) + <I>w</I> = <I>p</I></BLOCKQUOTE><BLOCKQUOTE>[(229/494)<I>p</I> + (3/1976)](1 + <I>r</I><I>s</I><SUB>1</SUB>) + (305/494)<I>w</I> ≥ <I>p</I></BLOCKQUOTE><BLOCKQUOTE>[(11/10)<I>p</I> + (2/5)](1 + <I>r</I><I>s</I><SUB>2</SUB>) + <I>w</I> = 1 </BLOCKQUOTE><P>where <I>p</I> is the price of a unit of iron, and <I>w</I> is the wage. </P><P>The parameters <I>s</I><SUB>1</SUB> and <I>s</I><SUB>2</SUB> are given constants, such that <I>r</I><I>s</I><SUB>1</SUB>is the rate of profits in iron production and <I>r</I><I>s</I><SUB>2</SUB> is the rate of profits in corn production. The quotient <I>s</I><SUB>1</SUB>/<I>s</I><SUB>2</SUB> is the ratio, in this model, of the rate of profits in iron production to the rate of profits in corn production. Consider the special case: </P><BLOCKQUOTE><I>s</I><SUB>1</SUB> = <I>s</I><SUB>2</SUB> = 1 </BLOCKQUOTE><P>This is the case of free competition, with investors having no preference among industries. In this case, <I>r</I> is the rate of profits. I call <I>r</I> the scale factor for the rate of profits in the general case where <I>s</I><SUB>1</SUB> and <I>s</I><SUB>2</SUB>are unequal. </P><P>The above system of equations and inequalities embody the assumption that a unit corn is the numeraire. They also show labor as being advanced and wages as paid out of the surplus at the end of the period of production. If the second inequality is an equality, both the Alpha and the Beta techniques are cost-minimizing; this is a switch point. The Alpha technique is the unique cost-minimizing technique if it is a strict inequality. To create a system expressing that the Beta technique is cost-minimizing, the equality and inequality for iron production are interchanged. </P><B>4.0 Choice of Technique</B><P>The above system can be solved, given <I>s</I><SUB>1</SUB>, <I>s</I><SUB>2</SUB>, and the scale factor for the rate of profits. I record the solution for a couple of special cases, for completeness. Graphs of wage curves and a bifurcation diagram illustrate that stable (non-unitary) ratios of rates of profits can change the dynamics of markets. </P><B>4.1 Free Competition</B><P>Consider the special case of free competition. The wage curve for the Alpha technique is: </P><BLOCKQUOTE><I>w</I><SUB>α</SUB> = (41 - 38<I>r</I> + <I>r</I><SUP>2</SUP>)/[80(2 + <I>r</I>)] </BLOCKQUOTE><P>The price of iron, when the Alpha technique is cost-minimizing, is: </P><BLOCKQUOTE><I>p</I><SUB>α</SUB> = (5 - 3<I>r</I>)/[8(2 + <I>r</I>)] </BLOCKQUOTE><P>The wage curve for the Beta technique is: </P><BLOCKQUOTE><I>w</I><SUB>β</SUB> = (6,327 - 9,802<I>r</I> + 3,631<I>r</I><SUP>2</SUP>)/[20(1,201 + 213<I>r</I>)] </BLOCKQUOTE><P>When the Beta technique is cost-minimizing, the price of iron is: </P><BLOCKQUOTE><I>p</I><SUB>β</SUB> = [5(147 - 97<I>r</I>)]/[2(1,201 + 213<I>r</I>)] </BLOCKQUOTE><P>Figure 2 graphs the wage curves for the two techniques, under free competition and a uniform rate of profits among industries. The wage curves intersect at a single switch point, at a rate of profits of, approximately, 8.4%: </P><BLOCKQUOTE><I>r</I><SUB>switch</SUB> = (1/1,301)[799 - 24 (826<SUP>1/2</SUP>)] </BLOCKQUOTE><P>The wage curve for the Beta technique is on the outer envelope, of the wage curves, for rates of profits below the switch point. Thus, the Beta technique is cost-minimizing for low rates of profits. The Alpha technique is cost minimizing for feasible rates of profits above the switch point. Around the switch point, a higher rate of profits is associated with the adoption of a less capital-intensive technique. Under free competition, this is not a case of capital-reversing. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-OCbbq8y9zw8/WMkiEDD7WXI/AAAAAAAAAy8/7-w1YeHYVvYGOXwqWqk8l7PzmF3vgZ5sgCLcB/s1600/CompetitionWageCurves.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-OCbbq8y9zw8/WMkiEDD7WXI/AAAAAAAAAy8/7-w1YeHYVvYGOXwqWqk8l7PzmF3vgZ5sgCLcB/s320/CompetitionWageCurves.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curves for Free Competition</b></td></tr></tbody></table><P></P><B>4.2 A Case of Oligopoly</B><P>Now, I want to consider a case of oligopoly, in which firms in different industries are able to ensure long-lasting barriers to entry. These barriers manifest themselves with the following parameter values: </P><BLOCKQUOTE><I>s</I><SUB>1</SUB> = 4/5 </BLOCKQUOTE><BLOCKQUOTE><I>s</I><SUB>2</SUB> = 5/4 </BLOCKQUOTE><P>In this case, the wage curve for the Alpha technique is: </P><BLOCKQUOTE><I>w</I><SUB>α</SUB> = (4,100 - 4,435<I>r</I> + 100<I>r</I><SUP>2</SUP>)/[40(400 + 259<I>r</I>)] </BLOCKQUOTE><P>The price of iron, when the Alpha technique is cost-minimizing, is: </P><BLOCKQUOTE><I>p</I><SUB>α</SUB> = (125 - 96<I>r</I>)/(400 + 259<I>r</I>) </BLOCKQUOTE><P>The wage curve for the Beta technique is: </P><BLOCKQUOTE><I>w</I><SUB>β</SUB> = 8(126,540 - 195,289<I>r</I> + 72,620<I>r</I><SUP>2</SUP>)/[160(24,020 + 9,447<I>r</I>)] </BLOCKQUOTE><P>The price of iron, when the Beta technique is cost-minimizing, is: </P><BLOCKQUOTE><I>p</I><SUB>β</SUB> = 2(3,675 - 3,038<I>r</I>)/(24,020 + 9,447<I>r</I>) </BLOCKQUOTE><P>Figure 3 graphs the wage curves for the Alpha and Beta techniques, for the parameter values for this model of oligopoly. This is now an example of reswitching. The Beta technique is cost minimizing at low and high rates of profits. The Alpha technique is cost minimizing at intermediate rates. The switch points are at, approximately, a value of the scale factor for rates of profits of 12.07% and 77.66%, respectively. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-vxyeoTzy2Nk/WMkiQSVH8eI/AAAAAAAAAzA/qTWCV5A5Wkop6o4gbQaj5JqyHyBdY1h4ACLcB/s1600/OligopolyWageCurves.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-vxyeoTzy2Nk/WMkiQSVH8eI/AAAAAAAAAzA/qTWCV5A5Wkop6o4gbQaj5JqyHyBdY1h4ACLcB/s320/OligopolyWageCurves.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 3: Wage Curves for a Case of Oligopoly</b></td></tr></tbody></table><P></P><B>4.3 A Range of Ratios of Profit Rates</B><P>The above example of oligopoly can be generalized. I restrict myself to the case where the parameters expressing the ratio of rates of profits between industries satisfy: </P><BLOCKQUOTE><I>s</I><SUB>2</SUB> = 1/<I>s</I><SUB>1</SUB></BLOCKQUOTE><P>One can then consider how the shapes and locations of wage curves and switch points vary with continuous variation in <I>s</I><SUB>1</SUB>/<I>s</I><SUB>2</SUB>. Figure 1, at the top of this post, graphs the wage at switch points for a range of ratios of rates of profits. Since the Beta technique is cost-minimizing, in the graph, at all high feasible wages and low scale factor for the rates of profits, I only graph the maximum wage for the Beta technique. I do not graph the maximum wage for the Alpha technique. </P><P>As the ratio of the rate of profits in the iron industry to rate in the corn industry increases towards unity, the model changes from a region in which the Beta technique is dominant to a reswitching example to an example with only a single switch point. As expected, only one switch point exists when the rate of profits is uniform between industries. </P><B>5.0 Conclusion</B><P>So I have created and worked through an example where: </P><UL><LI>No reswitching or capital-reversing exists under pure competition, with all industries earning the same rate of profits.</LI><LI>Reswitching and capital-reversing can arise for oligopoly, with persistent differential rates of profits across industries.</LI></UL><P>No qualitative difference necessarily exists, in the long period theory of prices, between free competition and imperfections of competition. Doubtless, all sorts of complications of strategic behavior, asymmetric information, and so on are empirically important. But it seems confused to blame the failure of markets to clear or economic instability on such imperfections. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-26229930091321444462017-03-15T07:11:00.000-04:002017-03-15T07:32:01.777-04:00Bifurcations in a Reswitching Example<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-erBJxRNP0-Y/WMQY8GgGqvI/AAAAAAAAAyo/a5FM6mo9D1AIVJO4aE3audnUnr3k1htEwCLcB/s1600/SwitchPtsVsIronInputsAnnotated.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-erBJxRNP0-Y/WMQY8GgGqvI/AAAAAAAAAyo/a5FM6mo9D1AIVJO4aE3audnUnr3k1htEwCLcB/s320/SwitchPtsVsIronInputsAnnotated.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 1: Rates of Profits for Switch Points in One Dimension in Parameter Space</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post presents an example of structural variation in the qualitative behavior of a reswitching example, at different values for selected parameters. I know of few applications of bifurcation analysis to the Cambridge Capital Controversy. Most prominently, I think of Rosser (1983). I suppose I could also point to some of my <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2912181">draft</A> <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1307930">papers</A>. Although not presented this way, one could read Laibman and Nell (1977) as a bifurcation analysis, where the steady state rate of growth is the parameter being varied. </P><P>I guess one could read this post as a response to the empirical results in Han and Schefold (2006). Schefold has been developing a theoretical explanation, based on random matrices, of why capital-theoretic paradoxes might be empirically rare. I seem to have stumbled on an explanation of why such paradoxes might arise in practice, and yet might not be observable without more data. To fully address recent results from Schefold, on reswitching and random matrices, one should analyze the spectra of Leontief input-output matrices, which I do not do here. </P><B>2.0 Technology</B><P>Suppose two commodities, iron and corn, are produced in the economy in the numerical example. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">305/494</TD><TD ALIGN="center"><I>a</I><SUB>0,2</SUB></TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD><TD ALIGN="center"><I>a</I><SUB>1,2</SUB></TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD><TD ALIGN="center">2/5</TD></TR></TABLE><P>Assume <I>a</I><SUB>0,2</SUB> is non-negative, and that <I>a</I><SUB>1,2</SUB> is strictly positive. For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the lone corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process. </P><B>3.0 Choice of Technique</B><P>Managers of firms choose the technique to adopt based on cost-minimization. I take a bushel of corn as the numeraire. Assume that labor is advanced, and that wages are paid out of the surplus at the end of the year. For this post, I do not bother setting out equations for prices of production; I have done that many times in the past. </P><B>3.1 Reswitching for one Set of Parameter Values</B><P>Figure 2 illustrates that this is a reswitching example. This figure is drawn for the following values of the labor coefficient in the process for producing corn: </P><BLOCKQUOTE><I>a</I><SUB>0,2</SUB> = 1 </BLOCKQUOTE><P>The coefficient of production for iron in corn-production, in drawing Figure 2, is set to the following value: </P><BLOCKQUOTE><I>a</I><SUB>1,2</SUB> = 2 </BLOCKQUOTE><P>The economy exhibits capital-reversing around the switch point at an 80% rate of profits. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-9352QO3vLQU/WMQYgg_WAmI/AAAAAAAAAyg/SSrggNRiYDsevu9os2exXa_Q79nI3vHrQCLcB/s1600/WageCurvesAnnotated.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-9352QO3vLQU/WMQYgg_WAmI/AAAAAAAAAyg/SSrggNRiYDsevu9os2exXa_Q79nI3vHrQCLcB/s320/WageCurvesAnnotated.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage-Rate of Profits Curves</b></td></tr></tbody></table><P></P><B>3.2 Bifurcations with Variations in a Labor Coefficient</B><P>Wage-rate of profits curves are drawn for given coefficients of production. And they will be moved elsewhere for different levels of coefficients of production. Consequently, the existence and location of switch points differ, depending on the values for coefficients of production. </P><P>Accordingly, suppose all coefficients of production, except <I>a</I><SUB>0,2</SUB>, are as in the above reswitching example. Consider values of the labor coefficient for corn-production ranging from zero to three. The labor coefficient is plotted along the abscissa in Figure 3. The points on the blue locus in the figure show the rate of profits for the switch points, as a correspondence for the labor coefficient. The maximum rates of profits for the Alpha and Beta techniques are also graphed. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-35w9CICvgOY/WMQYqxRjPTI/AAAAAAAAAyk/c9BsEYuoTf0CeU_xtJhuFlRrWlCKnEBJQCLcB/s1600/SwitchPtsVsLaborInputAnnotated.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-35w9CICvgOY/WMQYqxRjPTI/AAAAAAAAAyk/c9BsEYuoTf0CeU_xtJhuFlRrWlCKnEBJQCLcB/s320/SwitchPtsVsLaborInputAnnotated.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 3: Rates of Profits for Switch Points as One Labor Coefficient Decreases</b></td></tr></tbody></table><P>Figure 3 shows a structural change in the example. Up to a value of <I>a</I><SUB>0,2</SUB> of approximately 2.74, this is a reswitching example. For parameter values strictly greater than that, no switch points exist. The maximum rates of profits for the two techniques are constant in Figure 3. The maximum rates of profits are found for a wage of zero, and they do not vary with the labor coefficient. In some sense, only the maximum rate of profits for the Beta technique is relevant in the figure. </P><B>3.3 Bifurcations with Variations in a Coefficient of Production for Iron</B><P>Figure 1, at the top of this post, also shows structural changes. The coefficient of production for iron in corn-production varies in the figure. <I>a</I><SUB>1,2</SUB>ranges from one to three. The other coefficients of production are as in the reswitching example in Section 3.1 above. And the blue locus shows the rate of profits at switch points. </P><P>The example can seen to have structural variations here, also, with three distinct regions for <I>a</I><SUB>1,2</SUB>, with the same qualitative behavior in each region. For a low enough value of the coefficient of production under consideration, only one switch point exists. The model remains a reswitching example for an intermediate range of this parameter. And for values of this coefficient of production strictly greater than approximately 2.53, the Beta technique is cost-minimizing for all feasible wages and rates of profits. </P><P>The maximum rates of profits, for the Alpha and Beta techniques, are also graphed in Figure 1. </P><B>4.0 A Story of Technological Process</B><P>Using the above example, one can tell a story of <A HREF="http://robertvienneau.blogspot.com/2017/01/a-story-of-technical-innovation.html">technological progress</A>. Suppose at the start of the story, corn production requires a relatively large input of direct labor and iron, per (gross) unit corn produced. Prices of production associated with this technology are such that only one technique is cost-minimizing. For all feasible wages and rates of profits, firms will want to adopt the Beta technique. </P><P>Suppose iron production is relatively stagnant, as compared to corn-production. Innovation in the corn industry reduces the labor and iron coefficients defining the single dominant corn-producing process. After some time, either or both coefficients will be reduced enough that the technology for this economy will have become a reswitching example. And around the switch point at the lower wage (and higher rate of profits), a higher wage is associated with the cost-minimizing technique requiring more labor to be hired, in the overall economy, per given bushel of corn produced (net). </P><P>But technological innovation continues to proceed apace. At a even lower coefficient of production for the iron input in the corn industry, the structural behavior of the economy changes again. Now a single switch point exists. And the results of the choice of technique around that switch point conforms to outdated neoclassical intuition. </P><B>5.0 Conclusion</B><P>This example has two properties that I think worth emphasizing. </P><P>The choice of technique in the example corresponds to a choice of a production process in the iron industry. As I have told the story, the technology is fixed in iron production. Innovation occurs in corn production. Thus, innovation in one industry can change the dynamics in another industry. </P><P>Second, suppose the technology is observed at a single point of time. Suppose the economy is more or less stationary, and that observation is taken at either the start or the end of the above story. Then neither reswitching nor capital reversing will be observed. Yet such phenomena might arise in the future or have arisen in the past. </P><B>References</B><UL><LI>David Laibman and Edward J. Nell (1977). Reswitching, Wicksell effects, and the neoclassical production function. <I>American Economic Review</I>. 67 (5): pp. 878-888.</LI><LI>Zonghie Han and Bertram Schefold (2006). <A HREF="https://academic.oup.com/cje/article-abstract/30/5/737/1683696/An-empirical-investigation-of-paradoxes?redirectedFrom=fulltext">An empirical investigation of paradoxes: reswitching and reverse capital deepening in capital theory</A>. <I>Cambridge Journal of Economics</I>. 30 (5): pp. 737-765.</LI><LI>J. Barkley Rosser, Jr. (1983). "Reswitching as a cusp catastrophe", <I>Journal of Economic Theory</I>. 31: pp. 182-193.</LI><LI>Bertram Schefold (2013). <A HREF="https://academic.oup.com/cje/article-abstract/37/5/1161/1678380/Approximate-surrogate-production-functions?redirectedFrom=fulltext">Approximate surrogate production functions</A>, <I>Cambridge Journal of Economics</I>. 37 (5): pp. 1161-1184.</LI><LI>Bertram Schefold (2016). <A HREF="https://academic.oup.com/cje/article-abstract/40/1/165/2604983/Profits-equal-surplus-value-on-average-and-the?redirectedFrom=fulltext">Profits equal surplus value on average and the significance of this result for the Marxian theory of accumulation: Being a new contribution to Engels' Prize Essay Competition, based on random matrices and on manuscripts recently published in the MEGA for the first time</A>. <I>Cambridge Journal of Economics</I>. 40 (1): pp. 165-199.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-54336200841219679762017-03-11T11:03:00.000-05:002017-03-11T11:03:00.496-05:00Here and Elsewhere<UL><LI>A commentator <A HREF="http://robertvienneau.blogspot.com/2010/08/when-adam-delved-and-eve-span-who-was.html">informs</A> me that the True Levelers revived some ideas put forth in the Peasants Revolt.</LI><LI>Another commentator <A HREF="http://robertvienneau.blogspot.com/2015/07/labor-reversing-without-capital-example.html?showComment=1487680214021#c7454708390002608422">points</A> me to Naoki Yoshihara's <A HREF="http://scholarworks.umass.edu/econ_workingpaper/222/">review</A> of Opocher and Steedman's recent book. Yoshihara has a point, but I think the practice of treating inputs and physically identical outputs as different dated commodities is less applicable in partial models, as opposed to full General Equilibrium. Accountants need guidelines that resist easy manipulation in calculating profits and losses.</LI><LI>Antonella Palumbo has a <A HREF="https://www.ineteconomics.org/perspectives/blog/can-it-happen-again-defining-the-battlefield-for-a-theoretical-revolution-in-economics">post</A>, "Can 'It' Happen Again? Defining the Battlefield for a Theoretical Revolution in Economics", at the Institute for New Economic Thinking. Palumbo argues that a revival of classical economics, without Say's law, can provide an alternative to neoclassical economics. And Keynes' macroeconomics can be usefully be combined with this revival.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-73038496066244036442017-03-08T17:20:00.001-05:002017-03-08T17:20:36.806-05:00A Fluke Switch Point<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-cs5md-XABPE/WF0cjnjFoDI/AAAAAAAAAu8/Lu4eYltVGycu3GELhuNjtHucO_DimzQKACLcB/s1600/SalvadoriSteedman1988Model2.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-cs5md-XABPE/WF0cjnjFoDI/AAAAAAAAAu8/Lu4eYltVGycu3GELhuNjtHucO_DimzQKACLcB/s320/SalvadoriSteedman1988Model2.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: The Choice of Technique in a Model with Four Techniques</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I think I may have an original criticism of (a good part of) neoclassical economics. For purposes of this post, I here define the use of continuously differential production functions as an essential element in the neoclassical theory of production. (This is a more restrictive characterization than I usually employ.) Consider this two-sector <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-two-good-model-cobb.html">example</A>, in which coefficients of production in both sectors varies continuously along the wage-rate of profits frontier. It would follow from this post, I guess, that neoclassical theory is a limit, in some sense, of an analysis in which all switch points are flukes. </P><P>I have presented many other, often unoriginal, examples with a continuum of techniques: </P><UL><LI><A HREF="http://robertvienneau.blogspot.com/2015/07/labor-reversing-without-capital-example.html">Labor reversing without capital</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2015/06/recurrence-of-capital-output-ratio.html">Recurrence of capital-output ratio without reswitching</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2015/04/an-example-with-heterogeneous-labor.html">An example with heterogeneous labor</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2014/08/labor-demand-in-fog.html">Labor demand in a fog</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2011/04/slope-of-demand-curve-varying-with.html">Slope of 'demand curve' varying with numeraire</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2009/12/negative-price-wicksell-effect-positive.html">Negative price Wicksell effect, positive real Wicksell effect</A></LI></UL><P>I have an <A HREF="http://robertvienneau.blogspot.com/2014/07/the-generality-of-sraffian-analysis-of.html">example</A> with an uncountably infinite number of techniques along the wage-rate of frontier, but discontinuities for (all?) marginal relationships. </P><B>2.0 Technology</B><P>I want to compare and contrast two models. The technology in the second model is an example in Salvadori and Steedman (1988). </P><P>Households consume a single commodity, called "corn", in both models. In both models, two processes are known for producing corn. And these processes require inputs of labor and a capital good to produce corn. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. Both models are models of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><B>2.1 First Model</B><P>The technology for the first model is shown in Table 1. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. The two processes for producing corn require inputs of distinct capital goods. One corn-producing process requires inputs of labor and iron, and the other requires inputs of labor and tin. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Industry Model</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Tin<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">(a)</TD><TD ALIGN="center">(b)</TD><TD ALIGN="center">(c)</TD><TD ALIGN="center">(d)</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Tin</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/2</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Two techniques, as shown in Table 2, are available for producing a net output of corn. A choice of a process for producing corn also entails a choice of which capital good is produced. When the processes are each operated on a appropriate scale, the gross output of the process producing the specific capital good exactly replaces the quantity of the capital good used up as an input, summed over both industries operated in the technique. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 2: Techniques in a Three-Commodity Model</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">b, d</TD></TR></TABLE><P></P><B>2.2 Second Model</B><P>The technology for the second model is shown in Table 3. Two processes are known for producing corn. Both corn-producing processes require inputs of labor and iron, but in different proportions. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">(a)</TD><TD ALIGN="center">(b)</TD><TD ALIGN="center">(c)</TD><TD ALIGN="center">(d)</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">1/2</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Table 4 lists the techniques available in the second model. The first two techniques superficially resemble the two techniques available in the first model. But, in this model, the first process for producing a capital good can be combined, in a technique, with the second corn-producing producing process. This combination of processes is called the Gamma technique. Likewise, the Delta technique combines the second process for producing a capital good with the first corn-producing processes. Nothing like the Gamma and Delta techniques are available in the first model. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 4: Techniques in a Two-Commodity Model</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">b, d</TD></TR><TR><TD ALIGN="center">Gamma</TD><TD ALIGN="center">a, d</TD></TR><TR><TD ALIGN="center">Delta</TD><TD ALIGN="center">b, c</TD></TR></TABLE><P></P><B>3.0 Prices of Production</B><P>Suppose the Alpha technique is cost-minimizing. Prices of production, which permit smooth reproduction of the economy, must satisfy the following system of two equations in three unknowns: </P><BLOCKQUOTE>(2/3)(1 + <I>r</I>) + <I>w</I><SUB>α</SUB> = <I>p</I><SUB>α</SUB></BLOCKQUOTE><BLOCKQUOTE>(2/3) <I>p</I><SUB>α</SUB>(1 + <I>r</I>) + <I>w</I><SUB>α</SUB> = 1 </BLOCKQUOTE><P>These equations are based on the assumption that labor is advanced, and wages are paid out of the surplus at the end of the year. The same rate of profits are generated in both industries. A unit quantity of corn is taken as the numeraire. </P><P>One of the variables in these equations can be taken as exogenous. The first row in Table 5 specifies the wage and the price of the appropriate capital good, as a function of the rate of profits. The equation in the second column is called the <I>wage-rate of profits curve</I>, also known as the <I>wage curve</I>, for the Alpha technique. Table 5 also shows solutions of the systems of equations for the prices of production for the other three techniques in the second model, above. I have deliberately chosen a notation such that the first two rows can be read as applying to either one of the two models. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 5: Wages and Prices by Technique</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Wage Curve</B></TD><TD ALIGN="center"><B>Prices</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center"><I>w</I><SUB>α</SUB> = (1 - 2 <I>r</I>)/3</TD><TD ALIGN="center"><I>p</I><SUB>α</SUB> = 1</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center"><I>w</I><SUB>β</SUB> = (1 - <I>r</I>)/4</TD><TD ALIGN="center"><I>p</I><SUB>β</SUB> = 1</TD></TR><TR><TD ALIGN="center">Gamma</TD><TD ALIGN="center"><I>w</I><SUB>γ</SUB> = 2(2 - 2<I>r</I> - <I>r</I><SUP>2</SUP>)<BR>/[3(5 + <I>r</I>)]</TD><TD ALIGN="center"><I>p</I><SUB>γ</SUB> = 2(7 + 4<I>r</I>)<BR>/[3(5 + <I>r</I>)]</TD></TR><TR><TD ALIGN="center">Delta</TD><TD ALIGN="center"><I>w</I><SUB>δ</SUB> = (2 - 2<I>r</I> - <I>r</I><SUP>2</SUP>)/(7 + 4<I>r</I>)</TD><TD ALIGN="center"><I>p</I><SUB>δ</SUB> = 3(5 + <I>r</I>)/[2(7 + 4<I>r</I>)]</TD></TR></TABLE><P>Figure 1, at the top of this post, graphs all four wage-curves. The wage curves for the Alpha and Beta techniques are straight lines. In the jargon, the processes comprising these techniques exhibit the same organic composition of capital. The wage curves for the Gamma and Delta techniques are not straight lines. All four wage-curves intersect at a single point, (<I>r</I>, <I>w</I>) = (20%, 1/5). (The wage curves for the Gamma and Delta techniques have the same intersection with the axis for the rate of profits.) </P><B>3.0 Choice of Technique</B><P>The cost-minimizing techniques form the outer envelope of the wage curves. For a given wage, the cost minimizing technique is the technique with the highest wage curve in Figure 1. A switch point is a point on the outer envelope at which more than one technique is cost-minimizing. All four wage curves intersect, in the figure, at the single switch point. </P><P>The Beta technique is cost-minimizing for wages to the left of the single switch point. The Alpha technique is cost-minimizing for all feasible wages greater than the wage at the switch point. Managers of firms replace both processes in the Alpha technique at the switch point with both processes in the Beta technique. </P><P>This is no problem for the first model above. The adoption of a new process for producing corn requires, if the economy is capable of self-replacement before and after the switch, that the process for producing iron or tin be replaced by the process for producing the other. </P><P>But consider the other model. For all processes in the Alpha technique to be replaced at a switch point, the wage curves for all techniques composed of all combinations of processes in the Alpha and Beta techniques. In other words, in the second model, wage curves for all four techniques must intersect at the switch point. The example in the second model is a fluke. </P><P>I have <A HREF="http://robertvienneau.blogspot.com/2016/12/tangency-of-wage-rate-of-profits-curves.html">previously</A>explained what makes a result a fluke, in the context of the analysis of the choice of technique. Qualitative properties, for generic results, continue to persist for some small variation in model parameters. </P><P>Consider a model with a discrete number of switch points. Consider the cost-minimizing techniques on both sides of a switch point. And suppose that same commodities are produced in both techniques, albeit in different proportions. Generically, only one process is replaced at such a switch point. All processes, except for that one, are common in both techniques. </P><B>5.0 A Generalization to An Uncountably Infinite Number of Processes in Each Industry</B><P>Consider a model with more than one industry, but a finite number. Suppose each industry has available an uncountably infinite number of processes. And, in each industry, the processes available for that industry can be described by a continuously differentiable production function. <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-two-good-model-cobb.html">Here</A>I present a two-commodity example with Cobb-Douglas production functions. </P><P>There are no switch points in such a model. The cost-minimizing technique varies continuously along the outer-envelope of wage curves. In fact, the processes in each industry, in the cost-minimizing technique varies continuously. Since there are no switch points at all, there is not a single switch point in which more than one process varies, as a fluke, with the cost-minimizing technique. </P><P>Nevertheless, cannot one see such "smooth" production functions as a limiting case? If so, it would be a generalization or extension of a discrete model, in which all switch points are flukes, to a continuum. From the perspective of the analysis of the choice of technique in discrete models, typical neoclassical models are nothing but flukes. </P><B>6.0 Conclusions</B><P>I actually found my negative conclusion surprising. I have tried to be conscious of the distinction between the structure of the two models in Section 2 above. I think at least some <A HREF="http://robertvienneau.blogspot.com/2009/12/negative-price-wicksell-effect-positive.html">examples</A> I have presented cannot be attacked by the above critique. They are examples of the first, not the second model. I tend to read Samuelson (1962) in the same way, as not sensitive to the critique in this post. </P><B>References</B><UL><LI>Neri Salvadori and Ian Steedman (1988). No reswitching? No switching! <I>Cambridge Journal of Economics</I>, V. 12: pp. 481-486.</LI><LI>Samuelson, P. A. (1962). Parable and Realism in Capital Theory: The Surrogate Production Function, <I></I> V. 29, No. 3: pp. 193-206.</LI><LI>J. E. Woods 1990. <I>The Production of Commodities: An Introduction to Sraffa</I>, Humanities Press International.</LI></UL>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-7121179218880918772017-03-04T08:59:00.000-05:002017-03-04T08:59:24.735-05:00Bifurcations Of Roots Of A Characteristic Equation<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-YgvQthZwB2M/WG-UAh1jZTI/AAAAAAAAAxk/J2mxoKQ8ufAre9ufa67CiANtatNgvGhWQCLcB/s1600/MIRABeta.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-YgvQthZwB2M/WG-UAh1jZTI/AAAAAAAAAxk/J2mxoKQ8ufAre9ufa67CiANtatNgvGhWQCLcB/s320/MIRABeta.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: Rates of Profits for Beta Technique</b></td></tr></tbody></table><P>I have previously considered all roots of a polynomial equation for the rate of profits in a model, of the choice of technique, in which each technique is specified by a finite series of dated labor inputs. One root is the traditional rate of profits, but there are uses for the other roots: <UL><LI>All roots appear in an equation defining the Net Present Value (NPV) for the technique, given the wage and the rate of profits.</LI><LI>All roots can be combined in an accounting identity for the difference between labor commanded and labor embodied, given the wage.</LI></UL>I thought it of interest to know whether these non-traditional roots are real or complex, as they <A HREF="http://robertvienneau.blogspot.com/2016/12/bifurcations-in-multiple-interest-rate.html">vary</A> with the wage. I am considering multiple roots in an attempt to build on and <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058">critique</A> Michael Osborne's approach to multiple interest rate analysis. </P><P>I also have considered examples of models of the production of commodities by means of commodities, in which at least one commodity is basic, in the sense of Sraffa. And I have attempted to <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2885821">apply</A> or <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882531">extend</A> my critique of multiple interest rate analysis to these models. The point of this post is to illustrate possibilities on the complex plane for multiple interest rates in these models. </P><P>A technique in models of the production of commodities by means of commodities, as least in the case when all capital is circulating capital, is specified as a vector of labor coefficients and a Leontief input-output matrix. In parallel with my approach to techniques specified by a finite sequence of dated labor inputs, consider wages as being advanced - that is, not paid at the end of the year out of the surplus - in such models. Given the wage and the numeraire, one can construct a square matrix in which each coefficient is the sum of the corresponding coefficient in the Leontief input-output matrix and the quantity of the commodity produced by that industry that is advanced to the workers, per unit output produced. I call this matrix the augmented input-output matrix. </P><P>A polynomial equation, called the <I>characteristic equation</I>, is solved to find eigenvalues of the augmented input-output matrix. The power of this polynomial is equal to the number of commodities produced by the technique. The number of roots for the polynomial is therefore equal to the number of commodities. A rate of profits corresponds to each root. Assume the Leontief input-output matrix is a <I>Sraffa matrix</I> and that the wage does not exceed a certain maximum. Under these conditions, the Perron-Frobenius theorem picks out the maximum eigenvalue of the augmented input-output matrix. The corresponding rate of profits is non-negative, and the prices of production of these commodities are positive at the given wage. I was not able to find an application for the other, non-traditional rates of profits. </P><P>I present a numerical example in this <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2885821">working paper</A>. This is a three-commodity example with two techniques. Figure 1 graphs the three roots, at different level of wages, for the Beta technique in that example. </P><P>In a previous <A HREF="http://robertvienneau.blogspot.com/2016/12/example-of-choice-of-technique.html">blog post</A>, I extend that example such that managers of firms have a choice of process for producing each of the three commodities. As a consequence, a choice among eight techniques arises. And one can draw a graph like Figure 1 for each technique in that example. Figure 2 shows the corresponding graph for the Delta technique. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-4LHXKjQF-B0/WG-UIPEfP5I/AAAAAAAAAxo/XPWRL6l8N20bTmY-pFSIwfu2PlVlt5DOACLcB/s1600/MIRADelta.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-4LHXKjQF-B0/WG-UIPEfP5I/AAAAAAAAAxo/XPWRL6l8N20bTmY-pFSIwfu2PlVlt5DOACLcB/s320/MIRADelta.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 2: Rates of Profits for Delta Technique</b></td></tr></tbody></table><P>In Figures 1 and 2, the rate of profits picked out by the Perron-Frobenius theorem and used to draw the wage-rate of profits curve for the technique lies along the line segment on the real axis on the left in the figure. A lower wage corresponds to a higher traditional rate of profits. Thus, points further to the right on this line segment correspond to a lower wage. A wage of zero leads to the right-most point on this line segment. The highest feasible wage corresponds to left-most point, at a rate of profits of zero, on this segment. </P><P>Two non-traditional rates of profits arise for the other two solutions of the characteristic equation. They are plotted to the right on the graphs in Figures 1 and 2. When complex, they are complex conjugates. I thought it of interest that, in Figure 2, they are purely real for two non-overlapping, distinct ranges of feasible levels of the wage. </P><P>I draw no practical, applied implications from the non-traditional rates of profits. I just think the graphs are curious. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-2349650651303581012017-03-02T10:15:00.000-05:002017-03-02T10:15:00.162-05:00Some Obituaries for Kenneth ArrowBill Black has one <A HREF="http://www.nakedcapitalism.com/2017/02/bill-black-kenneth-arrows-ignored-impossibility-theorem.html">here</A>, emphasizing Arrow's impossibility theorem. The blog, A Fine Theorem, has two of a planned four-part series. The <A HREF="https://afinetheorem.wordpress.com/2017/02/22/the-greatest-living-economist-has-passed-away-notes-on-kenneth-arrow-part-i/">first</A> is on the impossibility theorem. The <A HREF="https://afinetheorem.wordpress.com/2017/02/27/kenneth-arrow-part-ii-the-theory-of-general-equilibrium/">second</A> is about General Equilibrium. The two planned, I gather, are to be about learning-by-doing and health economics, respectively. <P>I have written several posts drawing on Arrow's work. This <A HREF="http://robertvienneau.blogspot.com/2014/06/a-sophisticated-neoclassical-response.html">one</A>, on a sophisticated neoclassical response to the Cambridge Capital Controversy, is among my most popular posts. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0tag:blogger.com,1999:blog-26706564.post-25555906417395596662017-02-09T08:17:00.000-05:002017-06-24T07:42:56.004-04:00A Reswitching Example in a Model of Oligopoly<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-f9VYOjtgjTg/WJsd3j8gTQI/AAAAAAAAAyE/s4XRFIPFgUgMwRYXmAHcBUyIxwFrY50DwCLcB/s1600/OligopolyRoots.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-f9VYOjtgjTg/WJsd3j8gTQI/AAAAAAAAAyE/s4XRFIPFgUgMwRYXmAHcBUyIxwFrY50DwCLcB/s320/OligopolyRoots.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>The Roots of a Cubic Polynomial Defining Switch Points</b></td></tr></tbody></table><P>I have a <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2912181">draft paper</A> up at SSRN. The abstract: </P><BLOCKQUOTE>This paper illustrates, through a numerical example of reswitching under oligopoly, the existence of implications from the Cambridge Capital Controversy for the theory of industrial organization. Oligopoly is modeled by given and persistent ratios in rates of profits among industries, as expressed in a system of equations for prices of production. The numerical example illustrates that this model of oligopoly is a pertubation of free competition. Some comparisons and contrasts are drawn to a model of free competition. </BLOCKQUOTE><P>In some sense, this paper shows a somewhat more comprehensive description of value through exogenous distribution than in Sraffa's book. The model can depict capitalists as squabbling over the division of the surplus that their class gets, as well as their struggle against the workers. I'd like to see an example of reswitching or capital reversing in this model, with all (price and real) Wicksell effects as negative in the example in the special case of free competition. I do not see why one cannot arise. Such an example would suggest that "perverse" examples can obtain empirically, even if they are not found in an analysis that presumes one common rate of profits among all industries. </P><P>The graph at the top of this post does not appear in the paper. In the model, the ratios of rates of profits among industries are given parameters. A cubic polynomial is defined for a given set of such ratios. Non-negative, real zeroes of that polynomial below a certain maximum define a scale factor for switch points. The location of the zeros varies with the ratios. I happen to be able to solve for the zeros. They are shown in the graph above. </P>Robert Vienneauhttp://www.blogger.com/profile/00872510108133281526noreply@blogger.com0