tag:blogger.com,1999:blog-26706564Wed, 20 Sep 2017 12:13:41 +0000Example in Mathematical EconomicsAustrian School Of EconomicsSraffa EffectsPaul KrugmanTowards Complex DynamicsMethodology of EconomicsKarl MarxInterpreting Classical EconomicsLabor MarketsHistory vs. EquilibriumCookbooks for Workshops of the FutureEmpirical Results - Distribution and MobilityInternational TradeEmpirical Input-Output AnalysisSteady State EconomicsMilton FriedmanTheory of ChoiceJoint ProductionProfile of an EconomistPolitical ScienceSteve KeenFull Cost PricesLudwig WittgensteinAnti-LibertarianismMaking Life More BrutishMultiple Interest RatesPrinciples of EconomicsEndogenous MoneyGame TheoryCriticisms of Sraffian EconomicsKaldor Business Cycle ModelNoam ChomskyPolitical PhilosophyAntonio GramsciFoucaultRecipes for Cookbooks for Workshops of the PresentProtocolsVoices in the AirWeird ScienceZizekBiologyCorporate GovernanceEurolandVocabularyMarkov ProcessThoughts On Economicshttp://robertvienneau.blogspot.com/noreply@blogger.com (Robert Vienneau)Blogger1059125tag:blogger.com,1999:blog-26706564.post-115183556070733348Wed, 01 Jan 2020 08:00:00 +00002017-01-03T06:51:08.056-05:00WelcomeI 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.http://robertvienneau.blogspot.com/2007/12/welcome.htmlnoreply@blogger.com (Robert Vienneau)64tag:blogger.com,1999:blog-26706564.post-2641089946211750416Mon, 18 Sep 2017 11:19:00 +00002017-09-20T08:13:41.389-04:00Example in Mathematical EconomicsSraffa EffectsAnother Example Of A Real Wicksell Effect Of Zero<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-8QBEotZprSk/WcJZaAJ8cTI/AAAAAAAABBY/ll-tcKCFQjUusgFkezQeT2z0W7L6tbNQACLcBGAs/s1600/WageCurves2.gif" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-8QBEotZprSk/WcJZaAJ8cTI/AAAAAAAABBY/ll-tcKCFQjUusgFkezQeT2z0W7L6tbNQACLcBGAs/s320/WageCurves2.gif" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: A Reswitching Example with a Fluke Switch Point</b></td></tr></tbody></table><B>1.0 Introduction</B><P>A switch point that occurs at a rate of profits of zero is a fluke. This post presents a two-commodity example with a choice between two techniques, in which restitching occurs, and one switch point is such a fluke. Total employment per unit of net output is unaffected by the choice of technique. Furthermore, the numeraire-value of capital per unit net output is also unaffected by the mix of techniques adopted at a switch point with a positive rate of profits. This is not the first example I present in a draft <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3032428">paper</A>. </P><B>2.0 Technology</B><P>Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. (The coefficients were "nicer" fractions before I started perturbing it. Octave code was useful.) </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="3"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Industry</B></TD></TR><TR><TD ALIGN="center" ROWSPAN="2"><B>Iron</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TD><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">5,191/5,770</TD><TD ALIGN="center">305/494</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">9/20</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">2</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD></TR></TABLE><P>This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta. </P><B>3.0 Quantity Flows</B><P>Quantity flows can be analyzed independently of prices. Suppose the economy is in a self-replacing state, with a net output consisting only of corn. Table 2 displays quantity flows for the Alpha technique, when the net output consists of a bushel of corn. The last row shows gross outputs, for each industry. The entries in the three previous rows are found by scaling the coefficients of production in Table 1 by these gross outputs. Table 3 displays corresponding quantity flows for the Beta technique. </P><P>Consider the quantity flows for the Alpha technique. The row for iron shows that each year, the sum (9/356) + (11/356) = 5/89 tons are used as iron inputs in the iron and corn industries. These inputs are replaced at the end of the year by the output of the iron industry, with no surplus iron left over. In the corn industry, the sum 10/89 + 11/89 = 21/89 bushels are used as corn inputs in the two industries. When these inputs are replaced out of the output of the corn industry, a surplus of one bushel of corn remains. The net output of the economy, when these processes are operated in these proportions, is one bushel corn. The table allows one to calculate, for each technique, the labor aggregated over all industries per net unit output of the corn industry. Likewise, one can find the aggregate physical quantities of capital goods per net unit output of corn. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: Quantity Flows for Alpha Technique</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Industry</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Corn</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">5/89 ≈ 0.0562 Person-Yrs.</TD><TD ALIGN="center">57,101/51,353 ≈ 1.11 Person-Yrs.</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">9/356 ≈ 0.0253 Tons</TD><TD ALIGN="center">11/356 ≈ 0.0309 Tons</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">10/89 ≈ 0.112 Bushels</TD><TD ALIGN="center">11/89 ≈ 0.124 Bushels</TD></TR><TR><TD ALIGN="center"><B>Output</B></TD><TD ALIGN="center">5/89 ≈ 0.0562 Tons</TD><TD ALIGN="center">110/89 ≈ 1.24 Bushels</TD></TR></TABLE><P></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: Quantity Flows for Beta Technique</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Industry</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Corn</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">3/577 ≈ 0.00520 Person-Yrs.</TD><TD ALIGN="center">671/577 ≈ 1.16 Person-Yrs.</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">27/11,540 ≈ 0.00234 Tons</TD><TD ALIGN="center">33/11,540 ≈ 0.00286 Tons</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">6/577 ≈ 0.0104 Bushels</TD><TD ALIGN="center">2,519/2885 ≈ 0.873 Bushels</TD></TR><TR><TD ALIGN="center"><B>Output</B></TD><TD ALIGN="center">3/577 ≈ 0.00520 Tons</TD><TD ALIGN="center">5,434/2,885 ≈ 1.88 Bushels</TD></TR></TABLE><P></P><B>4.0 Prices and the Choice of Technique</B><P>The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage curve for the two techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. In the example, the Beta technique is cost minimizing for high rates of profits, while the Alpha technique is cost-minimizing between the two switch points. At the switch points, any linear combination of the two techniques is cost-minimizing. </P><P>One switch point is a fluke; it occurs for a rate of profits of zero. Any infinitesimal variation in the coefficients of production would result in the switch point no longer being on the wage axis. This intersection between the wage curves would then either occur at a negative or positive rate of profits. In the former case, the example would be one with a single switch point with a non-negative, feasible rate of profits, and the real Wicksell effect would be negative at that switch point. In the latter case, it would be a reswitching example, with the Beta technique uniquely cost-minimizing for low and high rates of profits. The real Wicksell effect would be negative at the first switch point and positive at the second. </P><B>5.0 Aggregates</B><P>In calculating wage curves, one can also find prices for each rate of profits. Table 5 shows certain aggregates, as obtained from Tables 2 and 3 and the price of iron at the switch point with a positive rate of profits. (Table 4 shows this price.) The numeraire value of capital per person-year, for a given technique and a given rate of profits, is the additive inverse of the slope of a line joining the intercept of the technique's wage curve with the wage axis to a point on the wage curve at the specified rate of profits. The capital-labor ratio, for a given technique, varies with the rate of profits, unless the wage curve is a straight line. Since a switch point occurs on the wage axis, the capital-labor ratio for both techniques at the other switch point is identical. As seen in Table 5, it does not vary among the two cost-minimizing techniques at the switch point with a positive rate of profits. The real Wicksell effect is zero at this switch point. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 4: Price Variables at Switch Point with Real Wicksell Effect of Zero</B></CAPTION><TR><TD ALIGN="center"><B>Variable</B></TD><TD ALIGN="center"><B>Value</B></TD></TR><TR><TD ALIGN="center">Rate of Profits</TD><TD ALIGN="center">125,483/209,727 ≈ 59.8 Percent</TD></TR><TR><TD ALIGN="center">Wage</TD><TD ALIGN="center">9,226,807/24,957,513 ≈ 0.370 Bushels per Person-Yr.</TD></TR><TR><TD ALIGN="center">Wage</TD><TD ALIGN="center">7,558/595 ≈ 12.7 Bushels per Ton</TD></TR></TABLE><P></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 5: Aggregates at Switch Point with Real Wicksell Effect of Zero</B></CAPTION><TR><TD ALIGN="center"></TD><TD ALIGN="center" COLSPAN="2"><B>Technique</B></TD></TR><TR><TD ALIGN="center"></TD><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TR><TR><TD ALIGN="center">Net Output</TD><TD ALIGN="center" COLSPAN="2">1 Bushel Corn</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center" COLSPAN="2">674/577 ≈ 1.17 Person-Years</TD></TR><TR><TD ALIGN="center" ROWSPAN="2">Physical Capital</TD><TD ALIGN="center">5/89 Tons Iron</TD><TD ALIGN="center">3/577 Tons Iron</TD></TR><TR><TD ALIGN="center">21/89 Bushels Corn</TD><TD ALIGN="center">2,549/2,885 Bushels Corn</TD></TR><TR><TD ALIGN="center">Financial Capitl</TD><TD ALIGN="center" COLSPAN="2">113/119 ≈ 0.945 Bushels Corn</TD></TR><TR><TD ALIGN="center">Capital-Labor Ratio</TD><TD ALIGN="center" COLSPAN="2">65,201/80,206 ≈ 0.813 Bushels per Person-Yr.</TD></TR></TABLE><P></P><B>6.0 Implications</B><P>A certain sort of indeterminacy arises in the example. For a given quantity of corn produced net, the ratio of labor employed in corn production to labor employed in iron production varies, at the switch point with a positive rate of profits, from around 1/5 to just over 223 to one. A change in technique leaves total employment unchanged, given net output, even as it alters the allocation of labor among industries. At the switch point, a change in technique, given net output, leaves the total value of capital unchanged, while, once again, altering its allocation among industries. </P><P>Suppose the economy is in a stationary state with the wage slightly below the wage at the switch point with a real Wicksell effect of zero. The Beta technique is in use. Consider what happens if a positive shock to wages result in a wage permanently higher than the wage at the switch point. The shock might be, for example, from an unanticipated increase in the minimum wage. Prices and outputs will be out of proportion, and a perhaps long disequilibrium adjustment process begins. Suppose that, eventually, after all this folderol, the economy, once more, attains another stationary state. The Alpha technique will now be in use. Labor hired per unit net output will be unchanged. The only variation in the value of capital goods per unit labor is a result of price changes, independent of the change in technique. </P>http://robertvienneau.blogspot.com/2017/09/another-example-of-real-wicksell-effect.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-7885380600456315186Thu, 14 Sep 2017 11:47:00 +00002017-09-14T07:47:46.304-04:00Example in Mathematical EconomicsSraffa EffectsBifurcation Diagram for Fluke Switch Point<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-ZyQsjhSgX-Y/WbVvrVILd2I/AAAAAAAABAc/7KcV9Axq3mUrscAWUJGvRVmd5zIoL8pMACLcBGAs/s1600/Coefficients.gif" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-ZyQsjhSgX-Y/WbVvrVILd2I/AAAAAAAABAc/7KcV9Axq3mUrscAWUJGvRVmd5zIoL8pMACLcBGAs/s320/Coefficients.gif" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: A Bifurcation Diagram</b></td></tr></tbody></table><P>I have previously <A HREF="http://robertvienneau.blogspot.com/2017/08/a-fluke-switch-point-with-real-wicksell.html">illustrated</A>a case in which real Wicksell effects are zero. I wrote this post to present an argument that that example is not a matter of round-off error confusing me. </P><P>Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="3"><B>Input</B></TD></TD><TD ALIGN="center" COLSPAN="3"><B>Industry</B></TD></TR><TR><TD ALIGN="center" ROWSPAN="2"><B>Iron</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TD><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center"><I>a</I><SUB>0,2</SUB><SUP>α</SUP></TD><TD ALIGN="center">305/494</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">9/20</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center"><I>a</I><SUB>2,1</SUB></TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD></TR></TABLE><P>Figure 1 shows two loci in the parameter space defined by the two coefficients of production <I>a</I><SUB>0,2</SUB><SUP>α</SUP> and <I>a</I><SUB>2,1</SUB>. The solid line represents coefficients of production for which the wage curves for the two techniques are tangent at a point of intersection. The dashed line represents parameters for which a switch point exists on the wage axis. The point at which these two loci are tangent specifies the parameters for this <A HREF="http://robertvienneau.blogspot.com/2017/08/a-fluke-switch-point-with-real-wicksell.html">example</A>. Figure 2 repeats the graph of the wage curves for that example. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-n6d2X0d3NH4/WaMWguoBpSI/AAAAAAAAA_o/YZ6enqLv8RcG551uNcfAlAIShEykccqMgCLcBGAs/s1600/WageCurves.gif" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-n6d2X0d3NH4/WaMWguoBpSI/AAAAAAAAA_o/YZ6enqLv8RcG551uNcfAlAIShEykccqMgCLcBGAs/s320/WageCurves.gif" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 2: A Fluke Switch Point</b></td></tr></tbody></table><P>Suppose coefficients are as in the example in the main text, but <I>a</I><SUB>0,2</SUB><SUP>α</SUP> is somewhat greater. Then the wage curve for the Alpha technique lies below the wage curve for Beta for all non-negative rates of profits not exceeding the maximum rate of profits. For all feasible rate of profits, Beta is cost-minimizing. On the other hand, if <I>a</I><SUB>0,2</SUB><SUP>α</SUP> is somewhat less than in the example, the wage curve for Alpha is somewhat higher than in Figure 2. The wage curve for Alpha will intersect the wage curve for Beta at two points, one with a negative rate of profits exceeding one hundred percent and one for a switch point with a positive rate of profit. As indicated in Figure 1, this combination of parameters is an example of the reserve substitution of labor </P><P>In the region graphed in Figure 1, if the coefficient of production <I>a</I><SUB>0,2</SUB><SUP>α</SUP> falls below the loci at which the two wage curves are tangent, the wage curves will have two intersections. Suppose <I>a</I><SUB>2,1</SUB> is greater than in the example in the main text. In the corresponding region between the two loci in Figure 1, the rate of profits at both intersections of the wage curves are negative. In this region of the parameter space, Beta remains cost-minimizing for all feasible non-negative rates of profits. If <I>a</I><SUB>2,1</SUB> is less than in the example, the rate of profits for both intersections are positive in the region between the two loci. The example is one of reswitching. In effect, which intersection of the wage curves is a switch point on the wage axis changes along the locus for the switch point on the wage axis. </P><P>Consider the rate of profits at which the wage curves have a repeated intersection, that is, are tangent, for the corresponding locus in Figure 1. Toward the left of the figure, this rate of profits is positive, while it is negative toward the right. By continuity, this rate of profits is zero for a single point in the graphed part of the parameter space. The two loci must be tangent for this set of parameters. The appearance of a switch point with a real Wicksell effect of zero in this <A HREF="http://robertvienneau.blogspot.com/2017/08/a-fluke-switch-point-with-real-wicksell.html">post</A>is not a result of round-off error or finite precision arithmetic. Such a point exists for exactly specified coefficients of production. </P>http://robertvienneau.blogspot.com/2017/09/bifurcation-diagram-for-fluke-switch.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-5749471530208207768Thu, 07 Sep 2017 11:55:00 +00002017-09-07T07:55:46.073-04:00Fluke Switch Points and a Real Wicksell Effect of Zero<P>I have put up a <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3032428">draft paper</A>with the post title on my SSRN site. </P><BLOCKQUOTE><B>Abstract:</B> This note presents two numerical examples, in a model with two techniques of production, of a switch point with a real Wicksell effect of zero. The variation in the technique adopted, at the switch point, leaves employment and the value of capital per unit net output unchanged. This invariant generalizes to switch points with a real Wicksell effect of zero for steady states with a positive rate of growth. </BLOCKQUOTE>http://robertvienneau.blogspot.com/2017/09/fluke-switch-points-and-real-wicksell.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-2994241785290815293Thu, 31 Aug 2017 11:56:00 +00002017-09-10T12:57:31.547-04:00Example in Mathematical EconomicsSraffa EffectsA Fluke Switch Point With A Real Wicksell Effect Of Zero<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-n6d2X0d3NH4/WaMWguoBpSI/AAAAAAAAA_o/YZ6enqLv8RcG551uNcfAlAIShEykccqMgCLcBGAs/s1600/WageCurves.gif" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-n6d2X0d3NH4/WaMWguoBpSI/AAAAAAAAA_o/YZ6enqLv8RcG551uNcfAlAIShEykccqMgCLcBGAs/s320/WageCurves.gif" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: A Fluke Switch Point</b></td></tr></tbody></table><B>1.0 Introduction</B><P>A switch point in which the wage curves for two techniques are tangent to one another at the switch point is a fluke. Likewise, a switch point that occurs at a rate of profits of zero is a fluke. This post presents a two-commodity example with a choice between two techniques, in which the single switch point is simultaneously both types of flukes. The wage curves are tangent at the switch point, and the switch point occurs at a rate of profits of zero. </P><B>2.0 Technology</B><P>Consider the technology illustrated in Table 1. The managers of firms know of three processes of production. These processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. (The coefficients were found by first creating an example with two wage curves tangent at a switch point. Selected coefficients were then varied to move the switch point to the wage axis. A binary search improved the approximation. Octave code was useful.) </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="3"><B>Input</B></TD></TD><TD ALIGN="center" COLSPAN="3"><B>Industry</B></TD></TR><TR><TD ALIGN="center" ROWSPAN="2"><B>Iron</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TD><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">0.802403</TD><TD ALIGN="center">305/494</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">9/20</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">3.9973702</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD></TR></TABLE><P>This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the iron-producing process and the corn-producing process labeled Beta. </P><B>3.0 Quantity Flows</B><P>Quantity flows can be analyzed independently of prices. Suppose the economy is in a self-replacing state, with a net output consisting only of corn. Table 2 displays (approximate) quantity flows for the Alpha technique, when the net output consists of a bushel of corn. The last row shows gross outputs, for each industry. The entries in the three previous rows are found by scaling the coefficients of production in Table 1 for Alpha by these gross outputs. The row for iron shows that each year, the sum 0.02848 + 0.3480 = 0.6328 tons are used as inputs in the iron and corn industries. These inputs are replaced at the end of the year by the output of the iron industry, with no surplus iron left over. Similarly, the output of the corn industry replaces the inputs of corn for the two industries, leaving a net output of one bushel corn. </P><P></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: Quantity Flows for Alpha Technique</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Industries</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Corn</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">0.06328</TD><TD ALIGN="center">1.11708</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">0.02848</TD><TD ALIGN="center">0.03480</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0.25296</TD><TD ALIGN="center">0.13922</TD></TR><TR><TD ALIGN="center"><B>Outputs</B></TD><TD ALIGN="center">0.06328</TD><TD ALIGN="center">1.39217</TD></TR></TABLE><P>Table 3 shows corresponding quantity flows for the Beta technique. As above, the net output is one bushel corn. These tables allow one to calculate, for each technique, the labor aggregated over all industries per net unit output of the corn industry. Likewise, one can find the aggregate physical quantities of capital goods per net unit output of corn. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: Quantity Flows for Beta Technique</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Industries</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Corn</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">0.00525</TD><TD ALIGN="center">1.17512</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">0.00236</TD><TD ALIGN="center">0.00289</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0.02100</TD><TD ALIGN="center">0.88230</TD></TR><TR><TD ALIGN="center"><B>Outputs</B></TD><TD ALIGN="center">0.00525</TD><TD ALIGN="center">1.90330</TD></TR></TABLE><P></P><B>4.0 Prices</B><P>The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage-rate of profits for the two techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. The Beta technique is cost-minimizing at all feasible rates of profits. At the switch point, the Alpha technique is also cost-minimizing. Furthermore, at the switch point, any linear combination of the techniques is cost-minimizing. </P><P>In calculating wage curves, one can also find prices for each rate of profits. Table 4 shows certain aggregates, as obtained from Tables 2 and 3 and the price of iron at the switch point. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 4: Aggregates at the Switch Point</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Aggregate</B></TD><TD ALIGN="center" COLSPAN="2"><B>Technique</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TR><TR><TD ALIGN="center">Net Output</TD><TD ALIGN="center" COLSPAN="2">1 Bushel Corn</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center" COLSPAN="2">1.18036 Person-Years</TD></TR><TR><TD ALIGN="center">Physical Capital</TD><TD ALIGN="center">0.06328 Tons<BR>0.39217 Bushels</TD><TD ALIGN="center">0.00525 Tons,<BR>0.90330 Bushels</TD></TR><TR><TD ALIGN="center">Financial Capital</TD><TD ALIGN="center" COLSPAN="2">0.94957 Bushels</TD></TR></TABLE><P>A certain sort of indeterminancy arises in the example. For a given quantity of corn produced net, the ratio of labor employed in corn production to labor employed in iron production varies at the switch point from approximately 17.7 to 223.7. A change in technique leaves total employment unchanged, given net output, even as it alters the allocation of labor between industries. It is also the case that, at the switch point, a change in technique, given net output, leaves the total value of capital unchanged, while, once again, altering its allocation between industries. </P><P>For non-fluke switch points, aggregate employment and the aggregate value of capital, per unit net output, vary with the technique. If the technique that is cost minimizing at an infinitesimally greater rate of profits than associated with the switch point has a greater value of capital per net output at the switch point, the real Wicksell effect is positive. If that technique has a smaller value of capital per net output, still using the prices at the switch point to value capital goods, is negative. (Edwin Burmeister argues that a negative real Wicksell effect is the appropriate formalization of the neoclassical idea of capital-deepening.) The fluke switch point presented here has a zero real Wicksell effect. </P><P>The indeterminacy at the switch point is related to both fluke properties of the switch point. Net output per worker, for a given technique, is shown by the intersection of the wage curve for the technique with the wage axis. Since both curves intersect the wage axis at the same point, they produce the same net output per worker. Thus, both techniques result in the same overall employment, per bushel corn produced net. </P><P>The wage curve also shows the value of capital per worker. For a given technique and rate of profits, the numeraire value of capital per person-year is the absolute value of the slope of the secant connecting the point on the wage curve specified by the rate of profits and the intercept with the wage axis. In the limit, when the rate of profits is zero, the value of capital per person-year is the absolute value of the slope of the tangent. The tangency of the wage curves at the switch point on the wage axis implies that both techniques have the same value of capital per person-year. </P><P><B>Update (10 Sept. 2017):</B> Fixed transcription error in coefficients of production. </P>http://robertvienneau.blogspot.com/2017/08/a-fluke-switch-point-with-real-wicksell.htmlnoreply@blogger.com (Robert Vienneau)2tag:blogger.com,1999:blog-26706564.post-2537386468159998739Sun, 27 Aug 2017 18:51:00 +00002017-08-27T14:51:57.638-04:00Example in Mathematical EconomicsSraffa EffectsExample With Four Normal Forms For Bifurcations Of Switch Points<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-42fNOmXNIuw/WZCnyCwK5tI/AAAAAAAAA-s/ebPxqZDKhRAn4XDouANQ7G8GO0QHGviuQCLcBGAs/s1600/BifurcationBlowup.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-42fNOmXNIuw/WZCnyCwK5tI/AAAAAAAAA-s/ebPxqZDKhRAn4XDouANQ7G8GO0QHGviuQCLcBGAs/s320/BifurcationBlowup.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: A Blowup of a Bifurcation Diagram</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I have been working on an <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3011850">analysis</A> of structural economic dynamics with a choice of technique. Technical progress can result in a variation in the switch points and the succession of techniques with wage curves on the outer wage frontier. I call such a variation a bifurcation, and I have identified normal forms for four generic bifurcations. This post prevents an example in which all four generic bifurcations appear. </P><B>2.0 Technology</B><P>The example in is one of an economy in which four commodities can be produced. These commodities are called iron, copper, uranium, and corn. The managers of firms know of one process for producing each of the first three commodities. They know of three processes for producing corn. Table 1 specifies the inputs required for a unit output for each of these six processes. Each column specifies the inputs needed for the process to produce a unit output of the designated industry. Variations in the parameters <I>a</I><SUB>11, β</SUB> and <I>a</I><SUB>11, γ</SUB>can result in different switch points appearing on the frontier. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for Three of Four Industries</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Industry</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Copper</B></TD><TD ALIGN="center"><B>Uranium</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">17,328/8,281</TD><TD ALIGN="center">1</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center"><I>a</I><SUB>11, β</SUB></TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Uranium</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center"><I>a</I><SUB>11, γ</SUB></TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Three processes are known for producing corn (Table 2). As usual, these processes exhibit CRS. And each column specifies inputs needed to produce a unit of corn with that process. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: The Technology the Corn Industry</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Process</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD><TD ALIGN="center"><B>Gamma</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">361/91</TD><TD ALIGN="center">3.63505</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">3</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Uranium</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1.95561</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P></P><B>3.0 Technical Progress</B><P></P><B>3.1 Progress in Copper Production</B><P>Consider the variation in the number and location of switch points as the coefficient of production for the input of copper per unit copper produced, <I>a</I><SUB>11, β</SUB>, falls from over 48/91 to around 1/4. In this analysis, the coefficient of production for the input of uranium per unit uranium produced, <I>a</I><SUB>11, γ</SUB>, is set to 3/5. This variation in <I>a</I><SUB>11, β</SUB>, while all other coefficients of production are fixed, describes a type of technical progress in the copper industry. </P><P>Figure 2 shows the configuration of wage curves near the start of this story. The Gamma technique is never cost-minimizing. For all feasible rates of profits, the wage curve for the Gamma technique falls within the wage frontier. For a parameter value of <I>a</I><SUB>11, β</SUB> of 48/91, the Alpha technique is always cost-minimizing. A single switch point exists, at which the wage curve for the Beta technique is tangent to the wage curve for the Alpha technique, and the Beta technique is also cost-minimizing. I call a configuration of wage curves like that in Figure 2 a <I>reswitching bifurcation</I>. For a slightly lower value of <I>a</I><SUB>11, β</SUB>, two switch points would emerge. The Alpha technique would be cost-minimizing for low and high rates of profits, with the Beta technique cost-minimizing at intermediate rates of profits. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-o4N3lRo4LeM/WYhauXRcIEI/AAAAAAAAA94/OuwH9xGy3GEJdTBZmPdLhplhJqVrzlGfgCLcBGAs/s1600/Slide2.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-o4N3lRo4LeM/WYhauXRcIEI/AAAAAAAAA94/OuwH9xGy3GEJdTBZmPdLhplhJqVrzlGfgCLcBGAs/s320/Slide2.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 2: A Reswitching Bifurcation</b></td></tr></tbody></table><P>Figure 3 shows the configuration of wage curves when <I>a</I><SUB>11, β</SUB> has fallen to one half. The interval with high rates of profits where the Alpha technique is uniquely cost-minimizing has vanished. The switch point between Alpha and Beta at high rates of profits occurs at a wage of zero. I call Figure 3 an example of a <I>bifurcation around the axis for the rate of profits</I>. For a slightly smaller value of <I>a</I><SUB>11, β</SUB>, the switch point on the axis would vanish, and only one switch point would exist, in this example, for a non-negative wage. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-hMsfDNF3u7A/WYhao5QW-vI/AAAAAAAAA90/G4yhynU-kUIwk9xdvEN2c8sYI-98XEJBACLcBGAs/s1600/Slide3.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-hMsfDNF3u7A/WYhao5QW-vI/AAAAAAAAA90/G4yhynU-kUIwk9xdvEN2c8sYI-98XEJBACLcBGAs/s320/Slide3.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 3: A Bifurcation around the Axis for the Rate of Profits</b></td></tr></tbody></table><P>Suppose the coefficient of production <I>a</I><SUB>11, β</SUB> were to fall to approximately 0.31008. Figure 4 shows the resulting configuration of wage curves. The Beta technique is cost-minimizing for all feasible positive rates of profit. A single switch point exists, between Alpha and Beta, on the wage axis. If <I>a</I><SUB>11, β</SUB> were to fall even further, no switch points would exist, and Beta would also be cost-minimizing for a rate of profits of zero. I call this an example of a <I>bifurcation around the wage axis</I>. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-5O4580Ggyew/WYhajDWiw3I/AAAAAAAAA9w/3TjfWzukDrQlQH8EhaLhM5YrEebMmce6gCLcBGAs/s1600/Slide4.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-5O4580Ggyew/WYhajDWiw3I/AAAAAAAAA9w/3TjfWzukDrQlQH8EhaLhM5YrEebMmce6gCLcBGAs/s320/Slide4.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 4: A Bifurcation around the Wage Axis</b></td></tr></tbody></table><P>Figures 5 and 6 summarize the above discussion. The coefficient of production <I>a</I><SUB>11, β</SUB> is plotted on the abscissa in each figure. The rates of profits and the wage, respectively, are plotted on the ordinate. Switch points are graphed. The maximum rates of profits for the Alpha and Beta technique are plotted in Figure 5. In Figure 6, the maximum wages for Alpha and Beta are plotted. Each of the three bifurcations in Figure 2, 3, and 4 is shown as a thin vertical line in Figures 5 and 6. The wage curve for the Beta techniques moves outward as one passes from the right to the left in the figures. One can see the single switch point becoming two, and the distance between the two, in terms of either the rate of profits of the wage, becoming greater. The rate of profits for one switch point eventually exceeds the maximum rate of profits and disappears. The rate of profits for the other switch point falls below zero, leaving Beta cost-minimizing for all feasible rates of profits and wages. In short, structural economic dynamics, for the case examined here, can be summarized in either one of these two graphs. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-eXuxTmNFXrk/WYhaW_zIqWI/AAAAAAAAA9s/BkTB4Jd0Ep4SL3H0AzHU8pOXO6j2ZxgoQCLcBGAs/s1600/Slide5.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-eXuxTmNFXrk/WYhaW_zIqWI/AAAAAAAAA9s/BkTB4Jd0Ep4SL3H0AzHU8pOXO6j2ZxgoQCLcBGAs/s320/Slide5.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 5: A Bifurcation Diagram for Technical Progress in the Copper Industry</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-XcPizWH9fZE/WYhaHsbfiSI/AAAAAAAAA9o/99ysT5Ka93cimCPE5UyMIxVxTiBIeCx1QCLcBGAs/s1600/Slide6.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-XcPizWH9fZE/WYhaHsbfiSI/AAAAAAAAA9o/99ysT5Ka93cimCPE5UyMIxVxTiBIeCx1QCLcBGAs/s320/Slide6.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 6: A Bifurcation Diagram for Technical Progress in the Copper Industry</b></td></tr></tbody></table><P></P><B>3.2 Progress in Uranium Production</B><P>An analysis of technical progress in the uranium industry illustrates another type of bifurcation. Let <I>a</I><SUB>11, β</SUB> be set to 51/100, and let the coefficient of production for the input of uranium per unit uranium produced, <I>a</I><SUB>11, γ</SUB>, fall from around 0.55 to 0.4. Figure 7 shows the configuration of wage curves when <I>a</I><SUB>11, γ</SUB> is approximately 0.537986. The wage curves for Alpha and Beta exhibit reswitching. The wage curve for the Gamma technique also intersects the switch point at the lower rate of profits. I call such a configuration of wage curves a <I>three-technique bifurcation</I>. Aside from the switch point, the Gamma technique is never cost-minimizing. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-VrIKvDkNNZA/WYhZ6r2mPPI/AAAAAAAAA9k/KfqiA8DJ9JExd-B4uDoHxMi5TWj2Bq5dACLcBGAs/s1600/Slide7.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-VrIKvDkNNZA/WYhZ6r2mPPI/AAAAAAAAA9k/KfqiA8DJ9JExd-B4uDoHxMi5TWj2Bq5dACLcBGAs/s320/Slide7.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 7: A Three Technique Bifurcation</b></td></tr></tbody></table><P>As <I>a</I><SUB>11, γ</SUB> decreases, the wage curve for the Gamma technique moves outward. At an intermediate value, the wage curve for Gamma intersects the wage curves for Alpha and Beta at different switch points. The reswitching example is transformed into one of capital reversing without reswitching. </P><P>Figure 8 displays a case where the wage curve for Gamma has moved outwards until it intersects the other switch point for the reswitching example. Other than at the switch point, the Beta technique is not cost minimizing for any feasible rate of profits. Figure 8 is also a case of a three-technique bifurcation. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-1gfpmRLsktM/WYhZxQyGguI/AAAAAAAAA9g/oSuI4YQLegQKsCNBMGIjEuiqRZQ402DiQCLcBGAs/s1600/Slide8.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-1gfpmRLsktM/WYhZxQyGguI/AAAAAAAAA9g/oSuI4YQLegQKsCNBMGIjEuiqRZQ402DiQCLcBGAs/s320/Slide8.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 8: Another Three Technique Bifurcation</b></td></tr></tbody></table><P>Figure 9 is a bifurcation diagram illustrating this analysis of technical progress in the uranium industry. It graphs the rate of profits against the coefficient of production <I>a</I><SUB>11, γ</SUB>. Switch points on the wage frontier, as well as the maximum rates of profits for the Alpha and Gamma technique, are graphed. The two thin vertical lines toward the right side of the graph are the two three-technique bifurcations. For a slightly lower value of <I>a</I><SUB>11, γ</SUB> than used in Figure 8, this is a reswitching example between Alpha and Gamma. As <I>a</I><SUB>11, γ</SUB> falls even lower, both switch points disappear over the axis for the rate of profits and the wage, respectively, in a graph of wage curves. That is, this example exhibits another illustration of both a bifurcation around the axis for the rate of profits and a bifurcation around the wage axis. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-txYi4bhkWLc/WYhWLbH1K_I/AAAAAAAAA9M/d27adJ6Q0GQOLqo3o7aECUmihWX0_dy6wCLcBGAs/s1600/Slide9.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-txYi4bhkWLc/WYhWLbH1K_I/AAAAAAAAA9M/d27adJ6Q0GQOLqo3o7aECUmihWX0_dy6wCLcBGAs/s320/Slide9.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 9: A Bifurcation Diagram for Technical Progress in the Uranium Industry</b></td></tr></tbody></table><P></P><B>3.3 Another Bifurcation Diagram</B><P>Sections 3.1 and 3.2 each graph switch points against a parameter in the numerical example. A more comprehensive analysis would consider all possible combinations of valid parameter values. One would need to draw a twelve-dimensional space. A part of the space defined by feasible combinations of positive values of <I>a</I><SUB>11, β</SUB> and <I>a</I><SUB>11, γ</SUB> is illustrated in Figure 10, instead Eleven regions are numbered in the figure. Figure 1 enlarges part of Figure 10 and labels the loci dividing regions with specific types of bifurcations. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-l7mL3ApQ198/WZCn53MGdPI/AAAAAAAAA-w/SSN9nO3FQH8WpXzF3_Bs5jioyV0sxmfBACLcBGAs/s1600/BifurcationDiagram.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-l7mL3ApQ198/WZCn53MGdPI/AAAAAAAAA-w/SSN9nO3FQH8WpXzF3_Bs5jioyV0sxmfBACLcBGAs/s320/BifurcationDiagram.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 10: A Bifurcation Diagram for the Parameter Space</b></td></tr></tbody></table><P>Each numbered region contains an interior. For points in the interior of a region, a sufficiently small perturbation of the coefficients of production <I>a</I><SUB>11, β</SUB> and <I>a</I><SUB>11, γ</SUB> leaves unchanged the number and pattern of switch points. The sequence of cost-minimizing techniques along the wage frontier between switch points is also invariant within regions. Accordingly, Table 3 lists switch points and cost-minimizing techniques for each region. The techniques are specified in order, from a rate of profits of zero to the maximum rate of profits. In several regions, such as region 2, the same technique is listed more than once, since it appears on the wage frontier in two disjoint intervals. Each locus dividing a pair of regions is a bifurcation. The reader can check that the labels for bifurcations in Figure 1 are consistent with Table 3. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: Techniques on the Wage Frontier</B></CAPTION><TR><TD ALIGN="center"><B>Region</B></TD><TD ALIGN="center" COLSPAN="3"><B>Techniques</B></TD></TR><TR><TD ALIGN="center">1</TD><TD ALIGN="center">Alpha throughout</TD></TR><TR><TD ALIGN="center">2</TD><TD ALIGN="center">Alpha, Beta, Alpha</TD></TR><TR><TD ALIGN="center">3</TD><TD ALIGN="center">Alpha, Beta</TD></TR><TR><TD ALIGN="center">4</TD><TD ALIGN="center">Beta throughout</TD></TR><TR><TD ALIGN="center">5</TD><TD ALIGN="center">Alpha, Gamma, Alpha</TD></TR><TR><TD ALIGN="center">6</TD><TD ALIGN="center">Alpha, Gamma, Alpha, Beta, Alpha</TD></TR><TR><TD ALIGN="center">7</TD><TD ALIGN="center">Alpha, Gamma, Beta, Alpha</TD></TR><TR><TD ALIGN="center">8</TD><TD ALIGN="center">Alpha, Gamma, Beta</TD></TR><TR><TD ALIGN="center">9</TD><TD ALIGN="center">Alpha, Gamma</TD></TR><TR><TD ALIGN="center">10</TD><TD ALIGN="center">Gamma</TD></TR><TR><TD ALIGN="center">11</TD><TD ALIGN="center">Gamma, Beta</TD></TR></TABLE><P>To aid in visualization, Figures 11, 12, and 13 graph wage curves and switch points on the wage frontier for each of the eleven regions. Within a region, the number of and characteristics of intersections of wage curves not on the frontier can vary. For example, the graph for region 8 in the lower right of Figure 12 shows an intersection between the wage curves for the Alpha and Gamma techniques at a high rate of profits. That second intersection between these wage curves can disappear over the axis for the rate of profits while leaving the sequence, if not the location, of cost-minimizing techniques and switch points on the frontier unchanged. </P><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-37I2eQULV5A/WZWQ2MvPLKI/AAAAAAAAA_Q/cgAnYh8NZhMcGpjK1x-5aZ56q64_7TdtQCLcBGAs/s1600/TypicalWageCurves14.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-37I2eQULV5A/WZWQ2MvPLKI/AAAAAAAAA_Q/cgAnYh8NZhMcGpjK1x-5aZ56q64_7TdtQCLcBGAs/s320/TypicalWageCurves14.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 11: Wage Curves for Regions 1 through 4</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-Lv9U-vsWHsE/WZWQwc_BX8I/AAAAAAAAA_M/MW_EMp6amo8uUaAUvIXfS4ePxuhg8P3mwCLcBGAs/s1600/TypicalWageCurves58.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-Lv9U-vsWHsE/WZWQwc_BX8I/AAAAAAAAA_M/MW_EMp6amo8uUaAUvIXfS4ePxuhg8P3mwCLcBGAs/s320/TypicalWageCurves58.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 12: Wage Curves for Regions 5 through 8</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-xtnOsS126OQ/WZWQoGLESzI/AAAAAAAAA_I/J4D6bLXvLhsiqfnqHoLqpVxOXnY0qvjqgCLcBGAs/s1600/TypicalWageCurve911.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-xtnOsS126OQ/WZWQoGLESzI/AAAAAAAAA_I/J4D6bLXvLhsiqfnqHoLqpVxOXnY0qvjqgCLcBGAs/s320/TypicalWageCurve911.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 13: Wage Curves for Regions 9 through 11</b></td></tr></tbody></table><P>The numerical example is an instance of the Samuelson-Garegnani model. Variations in the two coefficients of production for the copper industry have no effect on the location of intersections between wage curves for Alpha and Gamma. Thus, one obtains the horizontal lines in Figures 1 and 10. Likewise, variations in <I>a</I><SUB>11, γ</SUB> do not affect intersections between the wages curves for Alpha and Beta. This property results in the vertical lines in the bifurcation diagram. Bifurcations in which wage curves for both Beta and Gamma are involved result in the more or less diagonal curves in Figures 1 and 10. </P><P>Section 3.1 tells a tale of technical progress in the copper industry. This story is illustrated by the bifurcation diagrams in Figures 1 and 10. The chosen values for <I>a</I><SUB>11, β</SUB> divide regions 1, 2, 3, and 4. Figure 2 lies along the vertical line dividing regions 1 and 2. Figure 3 illustrates the division between regions 2 and 3, and Figure 4 illustrates the corresponding division between regions 3 and 4. The vertical line towards the left side of Figure 10 is a bifurcation across the wage axis. </P><P>Similarly, Section 3.2 illustrates bifurcations along a movement downward in Figures 1 and 10. Such a downward movement would pass through regions 2, 7, 5, 9, and 10. Figure 7 illustrates parameters on the locus dividing regions 2 and 7. Figure 8 illustrates the division between regions 7 and 5. The line dividing regions 5 and 9 is a bifurcation around the axis for the rate of profits, and the line dividing regions 9 and 10 is a bifurcation around the wage axis. All four bifurcations are illustrated in Figure 9. </P><P>The above partitioning of the parameter space formed by coefficients of production suggests the existence of bifurcations not yet illustrated. For example, a three-technique bifurcation is located anywhere along the locus dividing regions 6 and 7. This bifurcation differs from the three-technique bifurcations illustrated by Figures 7 and 8. Or consider the point that separates regions 1, 2, 5, and 6. The Alpha technique is cost minimizing for all feasible rates of profits for these coefficients of production. Two switch points exist, and at each one of these switch points another technique is tied with the Alpha technique. The wage curve for the Gamma technique is tangent to the wage curve for the Alpha technique at the switch point with the lower rate of profits. The wage curve for the Beta technique is tangent to the wage curve for the alpha technique at the other switch point. The point on the intersection between the loci dividing regions 2, 6, and 7 is interesting. The coefficients of production specified by this point characterize a three-technique bifurcation in which the wage curves for the Alpha and Gamma techniques are tangent at the appropriate switch point. This discussion has not exhausted the possibilities. </P>http://robertvienneau.blogspot.com/2017/08/example-with-four-normal-forms-for.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-8265771729921047871Tue, 22 Aug 2017 13:30:00 +00002017-08-22T09:30:17.599-04:00Karl MarxThe Concept Of Totality<P>This post is inspired by <A HREF="https://www.nasa.gov/feature/2017-solar-eclipse-highlights">current events</A></P><BLOCKQUOTE><P>"It is not the primacy of economic motives in historical explanation that constitutes the decisive difference between Marxism and bourgeois thought, but the point of view of totality. The category of totality, the all-pervasive supremacy of the whole over the parts is the essence of the method which Marx took over from Hegel and brilliantly transformed into the foundations of a wholly new science. The capitalist separation of the producer from the total process of production, the division of the process of labour into parts at the cost of the individual humanity of the worker, the atomisation of society into individuals who simply go on producing without rhyme or reason, must all have a profound influence on the thought, the science and the philosophy of capitalism. Proletarian science is revolutionary not just by virtue of its revolutionary ideas which it opposes to bourgeois society, but above all because of its method. <I>The primacy of the category of totality is the bearer of the principle of revolution in science</I>. </P><P>The revolutionary nature of Hegelian dialectics had often been recognised as such before Marx, notwithstanding Hegel's own conservative applications of the method. But no one had converted this knowledge into a science of revolution. It was Marx who transformed the Hegelian method into what Herzen described as the 'algebra of revolution'. It was not enough, however, to give it a materialist twist. The revolutionary principle inherent in Hegel's dialectic was able to come to the surface less because of that than because of the validity of the method itself, viz. the concept of totality, the subordination of every part to the whole unity of history and thought. In Marx the dialectical method aims at understanding society as a whole. Bourgeois thought concerns itself with objects the arise either from the process of studying phenomena in isolation, or from the division of labour and specialisation in the different disciplines. It holds abstractions to 'real' if it is naively realistic, and 'autonomous' if it is critical." </P>-- Georg Lukács, <I>History and Class Consciousness</I> (trans. by Rodney Livingstone), MIT Press (1971): pp. 27-28. </BLOCKQUOTE>http://robertvienneau.blogspot.com/2017/08/the-concept-of-totality.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-3757094239410824327Sun, 20 Aug 2017 19:33:00 +00002017-08-20T15:33:10.178-04:00Example in Mathematical EconomicsSraffa EffectsA Reswitching Bifurcation, Reflected<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-N5BknkS80o4/WX48qYaRWnI/AAAAAAAAA8Q/BIcAY8MUCtMQO-US9kq9d89pdg_ZaetuQCLcBGAs/s1600/ReswitchingBifurcations.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-N5BknkS80o4/WX48qYaRWnI/AAAAAAAAA8Q/BIcAY8MUCtMQO-US9kq9d89pdg_ZaetuQCLcBGAs/s320/ReswitchingBifurcations.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: Two Bifurcation Diagrams Horizontally Reflecting</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post continues my investigation of structural economic dynamics. I am interested in how technological progress can change the analysis of the choice of technique. I have <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3011850">four normal forms</A>for how switch points can appear on or disappear from the wage frontier, as a result of changes in coefficients of production. This post concentrates on what I call a <A HREF="https://robertvienneau.blogspot.com/2016/12/tangency-of-wage-rate-of-profits-curves.html">reswitching bifurcation</A>. </P><P>Each bifurcation can be described by how wages curves look around the bifurcation before, at, and after the bifurcation. I claim that, in some sense, order does not matter. For each normal form, bifurcations can exist in either order. I have proven this, for three of the bifurcations, by constructing the normal forms in both orders. This post completes the proof by exhibiting both orders for the reswitching bifurcation. </P><B>2.0 Technology</B><P>Consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. Technology is defined in terms of two parameters, <I>u</I> and <I>v</I>. <I>u</I> denotes the quantity of labor needed to produce a unit iron in the iron industry. <I>v</I> is the quantity of labor needed to produce a unit copper. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Commodity Example</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="3"><B>Input</B></TD><TD ALIGN="center" COLSPAN="4"><B>Industry</B></TD></TR><TR><TD ALIGN="center" ROWSPAN="2"><B>Iron</B></TD><TD ALIGN="center" ROWSPAN="2"><B>Copper</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center"><I>u</I></TD><TD ALIGN="center"><I>v</I></TD><TD ALIGN="center">1</TD><TD ALIGN="center">361/91</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">0</TD><TD ALIGN="center">3</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">48/91</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>As usual, this technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta. </P><B>3.0 Selected Configurations of Wage Curves</B><P></P><B>3.1 A Reswitching Bifurcation</B><P>Consider certain specified parameter values for the coefficients of production denoting the amount of labor needed to produce one unit of iron and one unit of copper. In particular, let <I>u</I> be 1, and let <I>v</I> be 17,328/8,281. Figure 2 graphs the wage curves for the two techniques in this case. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-PueU0IaB3BM/WX48fmVycjI/AAAAAAAAA8M/UTgo9yaj1LglakbW3jEAZRR3A3cCEjo8wCLcBGAs/s1600/WageCurvesAtBifurcation.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-PueU0IaB3BM/WX48fmVycjI/AAAAAAAAA8M/UTgo9yaj1LglakbW3jEAZRR3A3cCEjo8wCLcBGAs/s320/WageCurvesAtBifurcation.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curves at the Bifurcation</b></td></tr></tbody></table><P>I call this case a reswitching bifurcation. Like all bifurcations, it is a fluke case. </P><B>3.2 Improvements in Iron Production Around The Reswitching Bifurcation</B><P>Consider variations in <I>u</I>, from some parameter larger than its value in the above reswitching bifurcation to some lower value. In this part of the story, the value of <I>v</I> is assumed to be fixed at its value for the bifurcation. The right half of Figure 1, at the top of this post, illustrates this story. </P><P>For a high value of <I>u</I>, to the right of the right of Figure 1, the wage curve for Alpha is moved inside its location in Figure 2. The wage curves for the Alpha and Beta techniques intersect at two points. It is a reswitching example. A fall in <I>u</I> is illustrated by a movement to the left on the right side of Figure 1. The two switch points become closer and closer along the wage frontier. The reswitching bifurcation is illustrated by the thin vertical line in Figure 1. For any smaller value of <I>u</I>, the Alpha technique is cost minimizing for all feasible rates of profits or wages. </P><B>3.3 Improvements in Copper Production Around The Reswitching Bifurcation</B><P>Now consider variations in <I>v</I>, with <I>u</I> fixed at the value for the bifurcation illustrated in Figure 2. Technical progress in the copper industry is illustrated by a movement to the left on the left side of Figure 1. For a high value of <I>v</I>, the wage curve for the Beta technique is inside the wage curve for the Alpha technique. The Alpha technique is cost-minimizing for all feasible rates of profits. As <I>v</I> decreases, the wage curve for the Beta technique moves outward, until it reaches the reswitching bifurcation. For smaller values of <I>v</I>, the example becomes, once again, a reswitching example. A second bifurcation is illustrated on the left side of Figure 1, when the switch point at the higher rate of profits moves across the axis for the wage. The labor input for copper has become so small that the Beta technique is cost-minimizing for any sufficiently large enough wage and small rate of profits. </P><B>4.0 Conclusion</B><P>The bifurcation depends on a certain relative configuration of wage curves, in which one is tangent to the other at a switch point. Whether technical progress around the bifurcation results in reswitching appearing or disappearing depends on which wage curve is moving outwards faster around the switch point(s). Either order is possible. </P>http://robertvienneau.blogspot.com/2017/08/a-reswitching-bifurcation-reflected.htmlnoreply@blogger.com (Robert Vienneau)2tag:blogger.com,1999:blog-26706564.post-6801146111260483324Tue, 15 Aug 2017 12:20:00 +00002017-08-15T08:20:35.062-04:00Elsewhere<UL><LI>Nick Hanauer <A HREF="http://www.politico.com/magazine/story/2017/07/18/to-my-fellow-plutocrats-you-can-cure-trumpism-215347">argues</A> for some policies that postulate:</LI><UL><LI>Income distribution is not a matter of supply and demand or any other sort of economic natural laws.</LI><LI>That a more egalitarian distribution of income leads to an increased demand and generalized shared prosperity.</LI></UL><LI>Tom Palley <A HREF="http://fpif.org/from_keynesianism_to_neoliberalism_shifting_paradigms_in_economics/">contrasts</A>neoliberalism with an economic theory with an approach with another "theory of income distribution and its theory of aggregate employment determination".</LI><LI>Elizabeth Bruenig <A HREF="https://medium.com/@ebruenig/understanding-liberals-versus-the-left-5cff7ea41fd8">contrasts</A> liberalism with the the left.</LI><LI>Paul Blest <A HREF="https://theoutline.com/post/1925/why-are-neoliberals-such-big-babies">laughs</A> at whining neoliberals</LI><LI>Chris Lehmann <A HREF="https://thebaffler.com/the-jaundiced-eyeball/sail-trimmers">considers</A>how the turn of the US's Democratic Party to neoliberalism lowers its electoral prospects.</LI></UL><P>Is the distinction between democratic socialism and social democracy of no practical importance at the moment in any nation's politics? I think of the difference in two ways. First, in the United States in the 1970s, leftists had an argument. Self-defined social democrats became Neoconservatives, while democratic socialists found the Democratic Socialists of America (DSA). Second, both are reformists approaches to capitalism, advocating tweaks to, as Karl Popper argued for, prevent unnecessary pain. But social democrats have no ultimate goal of replacing capitalism, while democratic socialists want to end up with a transformed system. </P>http://robertvienneau.blogspot.com/2017/08/elsewhere.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-2063059658418968620Sat, 12 Aug 2017 16:54:00 +00002017-08-14T07:39:29.321-04:00A Fluke Of A Fluke Switch Point<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-iyPam6EQJ4o/WY8h6WCIQaI/AAAAAAAAA-U/j1Lk0HveZYMCBP_6lTwYVbuDB-nrhtghwCLcBGAs/s1600/FlukeWageCurves.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-iyPam6EQJ4o/WY8h6WCIQaI/AAAAAAAAA-U/j1Lk0HveZYMCBP_6lTwYVbuDB-nrhtghwCLcBGAs/s320/FlukeWageCurves.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: Wage Curves</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post presents an example of the analysis of the choice of technique in competitive markets. The example is one with three techniques and two switch points. The wage curves for the Alpha and Beta techniques are tangent at one of the switch points. This is a <A HREF="http://robertvienneau.blogspot.com/2016/12/tangency-of-wage-rate-of-profits-curves.html">fluke</A>. And the wage curves for all three techniques all pass through that same switch point. This, too, is a fluke. </P><P>I suppose that the example is one of reswitching and capital-reversing is the least interesting property of the example. Paul Samuelson was simply wrong in labeling such phenomena as perverse. A non-generic bifurcation, like the illustrated one, falls out of a comprehensive analysis of possible configurations of wage curves. </P><B>2.0 Technology</B><P>The technology in the example has a particularly simple structure. Firms can produce one of three capital goods, which I am arbitrarily labeling iron, copper, and uranium. Table 1 shows the production processes known for producing each metal. One process is known for producing each, and each metal is produced out of inputs of labor and that metal. Each process requires a year to complete, uses up all its material inputs, and exhibits Constant Returns to Scale. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for Three of Four Industries</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Industry</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Copper</B></TD><TD ALIGN="center"><B>Uranium</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">17,328/8,281</TD><TD ALIGN="center">1</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">48/91</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Uranium</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0.53939</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Three processes are known for producing corn (Table 2), which is the consumption good. This economy can be sustained by adopting one of three techniques. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta. Finally, the Gamma technique consists of the remaining two processes. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: The Technology the Corn Industry</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Process</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD><TD ALIGN="center"><B>Gamma</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">361/91</TD><TD ALIGN="center">3.63505</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">3</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Uranium</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1.95561</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><B>3.0 The Choice of Technique</B><P>The choice of technique is analyzed based on prices of production and cost-minimization. Labor is assumed to be advanced, and wages are paid out of the surplus product at the end of the year. Corn is taken as the numeraire. Figure 1 graphs the wage-rate of profits for the three techniques. The cost-minimizing technique, at a given rate of profits, maximizes the wage. That is, the cost-minimizing techniques form the outer envelope, also known as, the wage frontier, from the wage curves. Aside from switch points, the Alpha technique is cost-minimizing at low and high rates of profits, with the Gamma technique cost-minimizing between the switch points. At switch points, any linear combination of the techniques with wage curves going through that switch point are cost-minimizing. </P><P>The wage curve for the Beta technique is a straight line. This affine property results from the Organic Composition of Capital being the same in copper production and in corn production, when the Beta technique is adopted. To help visualization, I also graph the difference between the wage curves (Figure 2). The Beta technique is only cost-minimizing at the switch point at the higher rate of profits. The tangency of the wage curves for the Alpha and Beta techniques is manifested in Figure 2 by the non-negativity of the difference in these curves. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-v4F3AJuNhy4/WY8h0WMgNBI/AAAAAAAAA-Q/H_JrI5WSz7k5KKgz9skYdwNnNerqJqrFQCLcBGAs/s1600/FlukeDifferences.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-v4F3AJuNhy4/WY8h0WMgNBI/AAAAAAAAA-Q/H_JrI5WSz7k5KKgz9skYdwNnNerqJqrFQCLcBGAs/s320/FlukeDifferences.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 2: Distance Between Wage Curves</b></td></tr></tbody></table><P></P><B>4. Conclusion</B><P>I'm sort of proud of this example. I suppose I could, at least, submit it for publication somewhere. But it is only a side effect of a larger <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3011850">project</A> I guess I am pursuing. </P><P>I want to introduce a distinction among fluke switch points. Every bifurcation (that is, a change in the sequence of switch points and cost-minimizing techniques along the wage frontier) is a fluke. Some perturbation of a coefficient of production from a bifurcation value will change that sequence. Suppose a perturbation of a coefficient of production not involved in a bifurcation, in some sense, leaves the qualitative story unchanged. One can use the same bifurcation to tell a story about, say, technological progress. This is a generic bifurcation. </P><P>Accept, for the sake of argument, that prices of production tell us something about actual prices. The economy is never in an equilibrium, but owners of firms are always interested in increasing their profits. One can never expect observed technology to meet the fluke conditions of a generic bifurcation. But it can tell us something about how the dynamics of income distribution, for example, vary with technological progress. </P><P>Suppose one perturbs, in the example, the coefficient of production for the amount of iron needed to produce iron. (I denote this coefficient, in a fairly standard notation, as <I>a</I><SUB>1,1</SUB><SUP>β</SUP>.) Then, either the wage curves for the. Alpha and Beta techniques will not intersect at all or they will intersect twice. In the latter case, one can vary <I>a</I><SUB>1,1</SUB><SUP>γ</SUP> to find an example in which all three wage curves intersect at one or another of the switch points. But the tangency will be lost. So I consider the fluke point illustrated to be a non-generic bifurcation. </P><P>Non-generic bifurcations arise in a complete bifurcation analysis. The model illustrated remains open. Income distribution is not specified. Nevertheless, I think this theoretical analysis can say something to those who are attempting to empirically apply the Leontief-Sraffa model. </P>http://robertvienneau.blogspot.com/2017/08/a-fluke-of-fluke-switch-point.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-974291584306486143Mon, 07 Aug 2017 22:41:00 +00002017-08-08T06:35:08.898-04:00Multiple Interest RatesSome Unresolved Issues In Multiple Interest Rate Analysis<B>1.0 Introduction</B><P>Come October, as I understand it, the <I>Review of Political Economy</I> will publish, in hardcopy, my article <A HREF="http://www.tandfonline.com/doi/abs/10.1080/09538259.2017.1346039">The Choice of Technique with Multiple and Complex Interest Rates</A>. I discuss in this post questions I do not understand. </P><B>2.0 Non-Standard Investments and Fixed Capital</B><P>Consider a point-input, flow-output model. In the first year, unassisted labor produces a long-lived machine. In successive years, labor and a machine of a specific history are used to produce outputs of a consumption good and a one-year older machine. The efficiency of the machine may vary over the course of its physical lifetime. When the machine should be junked is a choice variable in some economic models. </P><P>I am aware that in this, or closely related models, the price of a machine of a specific date can be <A HREF="http://robertvienneau.blogspot.com/2017/01/reswitching-in-example-of-one-commodity.html">negative</A>. The total value of outputs at such a year in which the price of the machine of the machine is negative, however, is the difference between the sum of the price of the machine & the consumption good and the price of any inputs, like labor, that are hired in that year. Can one create a numerical example of such a case in which the net value, in a given year, is negative, where that negative value is preceded and followed by years with a positive net value? </P><P>If so, this would an example of a <I>non-standard investment</I>. A standard investment is one in which all negative cash flows precede all positive cash flows. In a non-standard investment at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments create the possibility that all roots of the polynomial used to define the Internal Rate of Return (IRR) are complex. Can one create an example with fixed capital or, more generally, joint production in which this possibility arises? </P><P>Does corporate finance theory reach the same conclusions about the economic life of a machine as Sraffian analysis in such a case? Can one express Net Present Value (NPV) as a function combining the difference between the interest rate and each IRR in this case, even though all IRRs are complex? (I call such a function an <A HREF="http://robertvienneau.blogspot.com/2017/04/nonstandard-investments-as-challenge.html">Osborne expression</A>for the NPV.) </P><B>3.0 Generalizing the Composite Interest Rate to the Production of Commodities by Means of Commodities</B><P>In my article, I follow Michael Osborne in deriving what he calls a composite interest rate, that combines all roots of the polynomial defining the IRR. I disagree with him, in that I do not think this composite interest rate is useful in analyzing the choice of technique. But we both obtain, in a flow-input, point output model, an equation I find interesting. </P><P>This equation states that the difference between the labor commanded by a commodity and the labor embodied in that commodity is the product of the first input of labor per unit output and the composite interest rate. Can you give an intuitive, theoretical explanation of this result? (I am aware that Osborne and Davidson give an explanation, that I can sort of understand when concentrating, in terms of the Austrian average period of production.) </P><P>A model of the production of commodities by means of commodities can be approximated by a model of a finite sequence of labor inputs. The model becomes exact as the number of dated labor inputs increases without bound. In the limit, the labor command by each commodity is a finite value. So is the labor embodied. And the quantity for the first labor input decrease to zero. Thus, the composite interest rate increases without bound. How, then, can the concept of the composite interest rate be extended to a model of the production of commodities by means of commodities? </P><B>4.0 Further Comments on Multiple Interest Rates with the Production of Commodities by Means of Commodities</B><P>In models of the production of commodities by means of commodities, various polynomials arise in which one root is the rate of profits. I have <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2885821">considered</A>, for example, the characteristic equation for a certain matrix related to real wages, labor inputs, and the Leontief input-output matrix associated with a technique of production. Are all roots of such polynomials useful for some analysis? How so? </P><P>Luigi Pasinetti, in the context of a theory of <A HREF="https://www.cambridge.org/core/books/structural-economic-dynamics/913CC279161B6C13E4819FB9DF05D4D3"><I>Structural Economic Dynamics</I></A>, has described what he calls the <I>natural system</I>. In the price system associated with the natural system, multiple interest rates arise, one for each produced commodity. Can these multiple interest rates be connected to Osborne's <STRIKE>natural</STRIKE> multiple interest rates? </P><B>5.0 Conclusion</B><P>I would not mind reading attempts to answer the above questions. </P>http://robertvienneau.blogspot.com/2017/08/some-unresolved-issues-in-multiple.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-5527684832199987863Fri, 04 Aug 2017 17:11:00 +00002017-08-04T13:11:42.087-04:00Example in Mathematical EconomicsSraffa EffectsSwitch Points and Normal Forms for Bifurcations<P>I have put up a <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3011850">working paper</A>, with the post title, on my Social Sciences Research Network (SSRN) site. </P><BLOCKQUOTE><B>Abstract:</B> The choice of technique can be analyzed, in a circulating capital model of prices of production, by constructing the wage frontier. Switch points arise when more than one technique is cost-minimizing for a specified rate of profits. This article defines four normal forms for structural bifurcations, in which the number and sequence of switch points varies with a variation in one model parameter, such as a coefficient of production. The 'perversity' of switch points that appear on and disappear from the wage frontier is analyzed. The conjecture is made that no other normal forms exist of codimension one. </BLOCKQUOTE>http://robertvienneau.blogspot.com/2017/08/switch-points-and-normal-forms-for.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-8525252850275771338Tue, 01 Aug 2017 12:27:00 +00002017-08-01T08:27:41.804-04:00Example in Mathematical EconomicsSraffa EffectsSwitch Points Disappearing Or Appearing Over The Axis For The Rate Of Profits<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-XO58F4cKxqA/WXy_vteMcaI/AAAAAAAAA7Q/f1rVBOG_MMAltGDM_RVkLn46Xl3r-qQPwCLcBGAs/s1600/Slide1.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-XO58F4cKxqA/WXy_vteMcaI/AAAAAAAAA7Q/f1rVBOG_MMAltGDM_RVkLn46Xl3r-qQPwCLcBGAs/s320/Slide1.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: Two Bifurcation Diagrams Horizontally Reflecting</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post continues my investigation of structural economic dynamics. I am interested in how technological progress can change the analysis of the choice of technique. In this case, I explore how a decrease in a coefficient of production can cause a switch point to appear or disappear over the axis for the rate of profits. </P><B>2.0 Technology</B><P>Consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. Technology is defined in terms of two parameters, <I>u</I> and <I>v</I>. <I>u</I> denotes the quantity of iron needed to produce a unit iron in the iron industry. <I>v</I> is the quantity of copper needed to produce a unit copper. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Commodity Example</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="3"><B>Input</B></TD><TD ALIGN="center" COLSPAN="4"><B>Industry</B></TD></TR><TR><TD ALIGN="center" ROWSPAN="2"><B>Iron</B></TD><TD ALIGN="center" ROWSPAN="2"><B>Copper</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2/3</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center"><I>u</I></TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center"><I>v</I></TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/3</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>As usual, this technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta. </P><P>I make all my standard assumptions. The choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Hence, wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves. </P><B>3.0 Innovations</B><P>I have two stories of technical innovation. In one, improvements are made in the process for producing copper. As a consequence, the wage curve for the Beta technique moves outward. In the other story, improvements are made in the iron industry, and the wage curve for the Alpha technique moves outwards. The bifurcations that occur in the two stories are mirror reflections of one another, in some sense. </P><B>3.1 Improvements in Copper Production</B><P>Let <I>u</I> be fixed at 1/3 tons per ton. The wage curve for the Alpha technique is a downward sloping straight line. Let <I>v</I> decrease from 1/2 to 3/10. When <I>v</I> is 1/3, the wage curve for the Beta technique is also a straight line. I created the example to have linear (actually, affine) wage curves at the bifurcation for convenience. The bifurcation does not require such. </P><P>Figure 2 shows the wage curves when the copper coefficient for copper production is a high value, in the range under consideration. A single switch point exists, and the Alpha technique is cost-minimizing if the rate of profits is high. As <I>v</I> decreases, the switch point moves to a higher and higher rate of profits. (These statements are about the shapes of mathematical functions. They are not about historical processes set in time.) Figure 3 shows the wage curves when <I>v</I> is 1/3. The switch point is now on the axis for the rate of profits. For any non-negative rate of profits below the maximum, the Beta technique is cost-minimizing. Finally, Figure 4 shows the wage curves for an even lower copper coefficient in copper production. Now, there is no switch point, and the Beta technique is always cost-minimizing, for all possible prices of production. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-rqnCE2JoFLg/WXy_5E0-QWI/AAAAAAAAA7U/ruf-rjNGHX86fv4lw_4KqWR9O1QbMAhJgCLcBGAs/s1600/Slide2.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-rqnCE2JoFLg/WXy_5E0-QWI/AAAAAAAAA7U/ruf-rjNGHX86fv4lw_4KqWR9O1QbMAhJgCLcBGAs/s320/Slide2.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curves Without Improvement in Copper Production</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/-sB54DM-D_9g/WXy_9lrf3NI/AAAAAAAAA7Y/GuFr4rIhWLYc3U1EJdhc_mh0vCpOpMUYwCLcBGAs/s1600/Slide3.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-sB54DM-D_9g/WXy_9lrf3NI/AAAAAAAAA7Y/GuFr4rIhWLYc3U1EJdhc_mh0vCpOpMUYwCLcBGAs/s320/Slide3.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 3: Wage Curves For A Bifurcation</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://4.bp.blogspot.com/-zLrRZJEoIvo/WXzACRpufXI/AAAAAAAAA7c/1r0113dH3BMA5WzIt-RNIxiCKJWjtWengCLcBGAs/s1600/Slide4.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-zLrRZJEoIvo/WXzACRpufXI/AAAAAAAAA7c/1r0113dH3BMA5WzIt-RNIxiCKJWjtWengCLcBGAs/s320/Slide4.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 4: Wage Curves After Improvements in Copper Production</b></td></tr></tbody></table><P></P><B>3.2 Improvements in Iron Production</B><P>Now let <I>v</I> be set at 1/3. Let <I>u</I> decrease from 1/2 to 3/10. Figure 5 shows the wage curves at the high end for the iron coefficient in iron production. No switch point exists, and the Beta technique is always cost-minimizing. I thought about repeating Figure 3, for <I>v</I> decreased to 1/3. The same configuration of wage curves, with a bifurcation, appears in this story. Figure 6, shows that the switch point appears for an even lower value of the iron coefficient. </P><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-_mObjYdOP4Q/WXzANPoXqXI/AAAAAAAAA7g/Q6JY5O1mLC8VoW-my3WGQ6plYXFjPpKTACLcBGAs/s1600/Slide6.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-_mObjYdOP4Q/WXzANPoXqXI/AAAAAAAAA7g/Q6JY5O1mLC8VoW-my3WGQ6plYXFjPpKTACLcBGAs/s320/Slide6.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 5: Wage Curves Without Improvement in Iron Production</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-Fx-1LJ1JN1g/WXzAUWzyA2I/AAAAAAAAA7k/Yl3e-Vtgw90VOf-VCbJ8KOFJN0iIkPyhQCLcBGAs/s1600/Slide7.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-Fx-1LJ1JN1g/WXzAUWzyA2I/AAAAAAAAA7k/Yl3e-Vtgw90VOf-VCbJ8KOFJN0iIkPyhQCLcBGAs/s320/Slide7.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 6: Wage Curves After Improvements in Iron Production</b></td></tr></tbody></table><P></P><B>3.3 Improvements in Both Iron and Copper Industries</B><P>I might as well graph (Figure 7) the copper coefficient in copper production against the iron coefficient in iron production. The bifurcation occurs when the maximum rates of profits are identical in the Alpha and Beta technique. In a model with the simple structure of the example, this occurs when <I>u</I> = <I>v</I>. Representative illustrations of wage curves are shown in the regions in the parameter space. A switch point below the maximum rate of profits exists only above the line in parameter space representing the bifurcation. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-xj1ono_l6NY/WXzQb80p7zI/AAAAAAAAA74/S6XyulDd09IBhqom4q22toxwvXKqmrhAACLcBGAs/s1600/RateOfProfitsBifurcation.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-xj1ono_l6NY/WXzQb80p7zI/AAAAAAAAA74/S6XyulDd09IBhqom4q22toxwvXKqmrhAACLcBGAs/s320/RateOfProfitsBifurcation.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 7: Bifurcation Diagram for Two Coefficients of Production</b></td></tr></tbody></table><P>The story in Section 3.1 corresponds to moving downwards on a vertical line in Figure 7. The left-hand side of Figure 1, at the top of this post, is another way of illustrating this story. On the other hand, Section 3.2 tells a story of moving leftwards on a horizontal line in Figure 7. The right-hand side of Figure 1 illustrates this story. </P><P>Focus on the intersections, in the two sides of Figure 1 of the blue, red, and purple loci. Can you see that, in some sense, they are reflections, up to a topological equivalence? </P><B>4.0 Discussion</B><P>I have a <A HREF="http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.html">reswitching example</A>with a switch point disappearing over the axis for the rate of profits. In that example, the disappearing switch point is 'perverse', that is, it has a positive real Wicksell effect. In the examples in Section 3 above, the disappearing or appearing switch point is 'normal', with a negative real Wicksell effect. </P><P></P>http://robertvienneau.blogspot.com/2017/08/switch-points-disappearing-or-appearing.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-3459786001531218013Fri, 28 Jul 2017 12:13:00 +00002017-07-28T08:13:22.244-04:00Example in Mathematical EconomicsSraffa EffectsBifurcations Along Wage Frontier, Reflected<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-Lg-UtFCXgmQ/WXnWFp6roJI/AAAAAAAAA6k/vcNe1eIMvPIkWiJcLK54pN3nNzGLN-opACLcBGAs/s1600/Slide5.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-Lg-UtFCXgmQ/WXnWFp6roJI/AAAAAAAAA6k/vcNe1eIMvPIkWiJcLK54pN3nNzGLN-opACLcBGAs/s320/Slide5.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: Bifurcation Diagram</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post continues a series investigating structural economic dynamics. I think most of those who understand prices of production - say, after working through Kurz and Salvadori (1995) - understand that technical innovation can change the appearance of the wage frontier. (The wage frontier is also called the wage-rate of profits frontier and the factor-price frontier.) Changes in coefficients of production can create or destroy a reswitching example. But, as far as I know, nobody has systematically explored how this happens in theory. </P><P>I claim that when switch points appear on or disappear off of the wage frontier, these bifurcations follow a few normal forms. I have been describing each normal form as a story of a coefficient of production being reduced by technical innovation. I further claim that, in some sense, the order of changes along the wage frontier is not specified. One can find an example with a decreasing coefficient of production in which the order is the opposite of some other example of technical innovation with the same normal form. </P><P>This post is one of a series providing the proof that order does not matter. The example in this post relates to this previous <A HREF="http://robertvienneau.blogspot.com/2017/07/bifurcations-along-wage-frontier.html">example</A>. </P><B>2.0 Technology</B><P>The example in this post is one of an economy in which four commodities can be produced. These commodities are called iron, steel, copper, and corn. The managers of firms know (Table 1) of one process for producing each of the first three commodities. These processes exhibit Constant Returns to Scale. Each column specifies the inputs required to produce a unit output for the corresponding industry. Variations in the parameter <I>v</I>can result in different switch points appearing on the frontier. Each process requires a year to complete and uses up all of its inputs in that year. There is no fixed capital. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for Three of Four Industries</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Industry</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Steel</B></TD><TD ALIGN="center"><B>Copper</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">53/180</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Steel</TD><TD ALIGN="center">0</TD><TD ALIGN="center"><I>v</I></TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Three processes are known for producing corn (Table 2). As usual, these processes exhibit CRS. And each column specifies inputs needed to produce a unit of corn with that process. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: The Technology the Corn Industry</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Process</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD><TD ALIGN="center"><B>Gamma</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Steel</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Four techniques are available for producing a net output of a unit of corn. The Alpha technique consists of the iron-producing process and the Alpha corn-producing process. The Beta technique consists of the steel-producing process and the Beta process. The Gamma technique consists of the remaining two processes. </P><P>As usual, the choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves. </P><B>3.0 Technical Progress</B><P>Figures 2 through 5 illustrate wage curves for different levels of the coefficient of production denoted <I>v</I> in the table. Figure 2 shows that for a relatively high parameter value, the switch point between the Alpha and Gamma techniques is the only switch point on the outer frontier. For continuously lower parameter values of <I>v</I>, the wage curve moves outward. Figure 3 illustration the bifurcation value, a fluke case in which the wage curves for all three techniques intersect in a single switch point. Other than at the switch point, the wage curve for the Beta technique is not on the frontier. But, for a slightly lower parameter value (Figure 4), the wage curve for the Beta technique, along with switch points between the Alpha and the Beta techniques and between the Beta and Gamma techniques, is on the frontier. The intersection between the wage curves for the Alpha and Gamma techniques is no longer on the frontier. Figure 5 illustrates another bifurcation in the example. The focus of this post is not on this bifurcation, in which a switch point disappears over the axis for the rate of profits. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-IXCXNQ-Of0o/WXnW3dJTVmI/AAAAAAAAA6s/sJMdtQwiexgYtDQt6FtpmjJofWgo1QfPACLcBGAs/s1600/Slide1.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-IXCXNQ-Of0o/WXnW3dJTVmI/AAAAAAAAA6s/sJMdtQwiexgYtDQt6FtpmjJofWgo1QfPACLcBGAs/s320/Slide1.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curves with High Steel Inputs in Steel Production</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-lyvbP22THBY/WXnXAiq_OVI/AAAAAAAAA6w/wFt0k3HB0JAC9QvuceWyEXBfIjgmmzpvQCLcBGAs/s1600/Slide2.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-lyvbP22THBY/WXnXAiq_OVI/AAAAAAAAA6w/wFt0k3HB0JAC9QvuceWyEXBfIjgmmzpvQCLcBGAs/s320/Slide2.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 3: Wage Curves with Medium Steel Inputs in Steel Production</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/-pNZ1JjcKzXE/WXnXLnCMCQI/AAAAAAAAA60/rGM31DaJ8DQx0KhDPJlZxDPn09mRAL3xACLcBGAs/s1600/Slide3.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-pNZ1JjcKzXE/WXnXLnCMCQI/AAAAAAAAA60/rGM31DaJ8DQx0KhDPJlZxDPn09mRAL3xACLcBGAs/s320/Slide3.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 4: Wage Curves with Low Steel Inputs in Steel Production</b></td></tr></tbody></table><P></P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-Wm4_0ozNm9c/WXnXUdbsodI/AAAAAAAAA64/tAzvhOrNKUgGPHTanb18ql7dLFvJ_D3OACLcBGAs/s1600/Slide4.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-Wm4_0ozNm9c/WXnXUdbsodI/AAAAAAAAA64/tAzvhOrNKUgGPHTanb18ql7dLFvJ_D3OACLcBGAs/s320/Slide4.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 5: Wage Curves with Lowest Steel Inputs in Steel Production</b></td></tr></tbody></table><P></P><B>4.0 Discussion</B><P>The bifurcation on the right in Figure 1, at the top of this post, is topologically equivalent to the horizontal reflection of the bifurcation on the right in the equivalent figure in this previous <A HREF="http://robertvienneau.blogspot.com/2017/07/bifurcations-along-wage-frontier.html">post</A>. (On the other hand, the bifurcations on the upper left in both diagrams are the same normal form, in the same order.) </P><P>The bifurcation described in this post is a local bifurcation. To characterize this bifurcation, one need only look at small range of rates of profits and coefficients of production around a critical value. Accordingly, then wage curves involved in the bifurcations could intercept any number of times, in some other example of this normal form, at positive rates of profits. Each of the three switch points involved in the bifurcation could have any direction for real Wicksell effects, positive or negative. </P><P>The bifurcation, as depicted in this post, replaces one switch point on the wage curve with two switch points. It could be that the switch point disappearing exhibits capital-reversing, and both of the two new switch points appearing also exhibit capital-reversing. But any of five other other combinations are possible. </P>http://robertvienneau.blogspot.com/2017/07/bifurcations-along-wage-frontier_28.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-8417176365479286662Wed, 26 Jul 2017 12:12:00 +00002017-07-26T08:12:54.356-04:00Example in Mathematical EconomicsMultiple Interest RatesSraffa EffectsThe Choice Of Technique With Multiple And Complex Interest Rates<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-GeTgo7HCmDM/WW5ekc668iI/AAAAAAAAA5w/SG87ymluykkXwsCzp7LHafiZI3512On3gCLcBGAs/s1600/WordCountAll.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-GeTgo7HCmDM/WW5ekc668iI/AAAAAAAAA5w/SG87ymluykkXwsCzp7LHafiZI3512On3gCLcBGAs/s320/WordCountAll.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr></tbody></table><P>My article with the post title is now <A HREF="http://www.tandfonline.com/doi/full/10.1080/09538259.2017.1346039">available</A> on the website for the <I>Review of Political Economy</I>. It will be, I gather, in the October 2017 hardcopy issue. The abstract follows. </P><P><B>Abstract:</B>This article clarifies the relations between internal rates of return (IRR), net present value (NPV), and the analysis of the choice of technique in models of production analyzed during the Cambridge capital controversy. Multiple and possibly complex roots of polynomial equations defining the IRR are considered. An algorithm, using these multiple roots to calculate the NPV, justifies the traditional analysis of reswitching. </P>http://robertvienneau.blogspot.com/2017/07/the-choice-of-technique-with-multiple.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-1419926844955102412Sun, 23 Jul 2017 19:35:00 +00002017-07-23T15:35:55.744-04:00Example in Mathematical EconomicsLabor MarketsSraffa EffectsA Switch Point Disappearing Over The Wage Axis<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-dHecGTK9hpg/WXTxR6Uh3hI/AAAAAAAAA6M/6UAQ9DHTdMwhyw2o3iXtC-DfjGGM4l2dQCLcBGAs/s1600/BifurcationOnWageAxis4.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-dHecGTK9hpg/WXTxR6Uh3hI/AAAAAAAAA6M/6UAQ9DHTdMwhyw2o3iXtC-DfjGGM4l2dQCLcBGAs/s320/BifurcationOnWageAxis4.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: Bifurcation Diagram</b></td></tr></tbody></table><B>1.0 Introduction</B><P>In a series of posts, I have been exploring structural economic dynamics. Innovation reduces coefficients of production. Such reductions can vary the number and sequence of switch points on the wage frontier. I call such a variation a <I>bifurcation</I>. And I think such bifurcations, at least if only one coefficient decreases, fall into a small number of normal forms. </P><P>One possibility is that a decrease in a coefficient of production results in a switch point appearing over the wage axis, as illustrated <A HREF="http://robertvienneau.blogspot.com/2017/07/a-switch-point-on-wage-axis.html">here</A>. This post modifies that example such that the switch point disappears over the wage axis with a decrease in a coefficient of production. </P><B>2.0 Technology</B><P>Accordingly, consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. In this post, I consider how variations in the parameter <I>u</I> affect the number of switch points. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Commodity Example</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="3"><B>Input</B></TD><TD ALIGN="center" COLSPAN="4"><B>Industry</B></TD></TR><TR><TD ALIGN="center" ROWSPAN="2"><B>Iron</B></TD><TD ALIGN="center" ROWSPAN="2"><B>Copper</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">21/8</TD><TD ALIGN="center"><I>u</I></TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta. </P><P>The choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Hence, wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves. </P><B>3.0 Results</B><P>Consider variations in <I>u</I>, the input of labor in the copper industry, per unit copper produced. Figure 1 shows the effects of such variations. For a high value of this coefficient, a single switch point exists. The Alpha technique is cost-minimizing at high wages (or low rates of profits). The Beta technique is cost-minimizing at low wages (or high rates of profits). </P><P>Suppose that technical innovations reduce <I>u</I> to 3/2. Then the switch point occurs at the maximum wage. For all positive rates of profits (not exceeding the maximum), the Beta technique is cost-minimizing. At a rate of profits of zero, both techniques (or any linear combination of them) are eligible for adoption by cost-minimizing firms. </P><P>A third regime arises when technical innovations reduce <I>u</I> even more. The a technique is cost-minimizing for all feasible rates of profits, including a rate of profits of zero. </P><B>4.0 Discussion</B><P>So this example has illustrated that the bifurcation diagram at the top of this <A HREF="http://robertvienneau.blogspot.com/2017/07/a-switch-point-on-wage-axis.html">previous post</A>can be reflected across a vertical line where the bifurcation occurs. An abstract description of a bifurcation in which a switch point crosses the wage axis does not have a direction, in some sense. Either direction is possible. </P><P>The illustrated bifurcation is, in some sense, local. The illustrated phenomenon might occur in what is originally a reswitching example. That is, the bifurcation concerns only what happens around a small rate of profit (or near the maximum wage). It is compatible with wage curves that have a second intersection on the frontier at a higher rate of profits. In such a case, the switch point at the higher rate of profits will remain. But the bifurcation will transform it from a 'perverse' switch point to a 'normal' one. </P><P>As I understand it, such a bifurcation of a reswitching will be manifested in the labor market with 'paradoxical' behavior. Suppose the first switch point disappears over the wage axis. Around the second switch point, a comparison of long period (stationary) positions will find a higher wage associated with the adoption of a technique that requires less labor per (net) unit output, for the economy as a whole. But, in the corn industry, a higher wage will be associated with the adoption of a technique that requires more labor per (gross) unit corn produced. </P><P>This is just one of those possibilities that demonstrates the Cambridge Capital Controversy is not merely a critique of aggregation, macroeconomics, and the aggregate production function. It has implications for microeconomics, too. </P>http://robertvienneau.blogspot.com/2017/07/a-switch-point-disappearing-over-wage.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-8846269347261073003Thu, 20 Jul 2017 12:31:00 +00002017-07-20T12:31:13.568-04:00Piers Anthony, Neoliberal<P><A HREF="https://www.amazon.com/Spell-Chameleon-Xanth-Book/dp/0345347536/"><I>A Spell for Chameleon</I></A>, the first book of the <I>Xanth</I> series, shows that Piers Anthony is a neoliberal<SUP>1</SUP>. Magicians are important characters in Xanth, and <I>A Spell</I> introduces us to at least two, Humphrey<SUP>2</SUP> and Evil Magician Trent. </P><P>We find that "Evil" is just what Trent is called. We are not supposed to regard him as such. And he bases his life entirely on market transactions, even though the setting is a feudal society. Everything is an agreement to a contract, or not, for mutual advantage. An upright person adheres to the spirit of his deals, even when unforeseen circumstances make it unclear what his promises entail in this new situation. </P><P>Humphrey is also all about deals. He doesn't like to answer questions, so he always sets the questioner three challenges. Some of these challenges require the questioner to do something for him. </P><P>For both Humphrey and Trent, quid pro quo agreements can extend to the most intimate relationships<SUP>3</SUP>. </P><P>I was prompted to think about neoliberalism by this Mike Konczal <A HREF="https://www.vox.com/the-big-idea/2017/7/18/15992226/neoliberalism-chait-austerity-democratic-party-sanders-clinton">article</A> in <I>Vox</I>. </P><B>Footnotes</B><OL><LI>One can argue that I am conflating the views of the author with the views of his characters. I think the novels portray both magician Humphrey and Trent in a positive light, but am willing to entertain argument.</LI><LI>Humphrey, since he has access to the fountain of youth, as I recall, is an important character throughout the series. I have read hardly any after the first five or ten.</LI><LI>Feminists might have something to say about this light reading. The hero, Bink, finds his perfect mate gives him variety, with the young woman's cycle combining certain stereotypical attributes.</LI></OL>http://robertvienneau.blogspot.com/2017/07/piers-anthony-neoliberal.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-3451227381323213178Sun, 16 Jul 2017 20:54:00 +00002017-07-16T16:54:12.833-04:00Example in Mathematical EconomicsSraffa EffectsBifurcations Along Wage Frontier<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-uLNIYkLvskQ/WWNodaGwq_I/AAAAAAAAA4U/4gUe9XomoTsN9gmCnLo3xB6V1SjCwHmagCLcBGAs/s1600/BifurcationsOnFrontier.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-uLNIYkLvskQ/WWNodaGwq_I/AAAAAAAAA4U/4gUe9XomoTsN9gmCnLo3xB6V1SjCwHmagCLcBGAs/s320/BifurcationsOnFrontier.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: Bifurcation Diagram</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post continues my <A HREF="https://robertvienneau.blogspot.com/2017/07/generic-bifurcations-and-switch-points.html">exploration</A> of the variation in the number and "perversity" of switch points in a model of prices of production. This post presents a case in which one switch point replaces two switch points on the wage frontier. </P><P></P><B>2.0 Technology</B><P>The example in this post is one of an economy in which four commodities can be produced. These commodities are called iron, steel, copper, and corn. The managers of firms know (Table 1) of one process for producing each of the first three commodities. These processes exhibit Constant Returns to Scale. Each column specifies the inputs required to produce a unit output for the corresponding industry. Variations in the parameter <I>d</I>can result in different switch points appearing on the frontier. Each process requires a year to complete and uses up all of its inputs in that year. There is no fixed capital. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for Three of Four Industries</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Industry</B></TD></TR><TR><TD ALIGN="center"><B>Iron</B></TD><TD ALIGN="center"><B>Steel</B></TD><TD ALIGN="center"><B>Copper</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center"><I>d</I></TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Steel</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Three processes are known for producing corn (Table 2). As usual, these processes exhibit CRS. And each column specifies inputs needed to produce a unit of corn with that process. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 2: The Technology the Corn Industry</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="3"><B>Process</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD><TD ALIGN="center"><B>Gamma</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Steel</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Four techniques are available for producing a net output of a unit of corn. The Alpha technique consists of the iron-producing process and the Alpha corn-producing process. The Beta technique consists of the steel-producing process and the Beta process. The Gamma technique consists of the remaining two processes. </P><P>As usual, the choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves. </P><B>3.0 Result of Technical Progress</B><P>Figure 2 shows wage curves when <I>d</I> is 1/3, a fairly high value in this analysis. The wage curves for all three techniques are on the frontier. For certain ranges of the rate of profits, each technique is cost-minimizing. The switch point between the Alpha and Gamma techniques is not on the frontier. No infinitesimal variation in the rate of profits will result in a transition from a position in which the Alpha technique is cost-minimizing in the long period to one in which the Gamma technique is cost-minimizing. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-YOYZzwbYKF4/WWNriO8lEMI/AAAAAAAAA4w/eMMqNSSCsk0US7dxehtfxZothaUJIWdzgCLcBGAs/s1600/TwoSwitchPoints.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-YOYZzwbYKF4/WWNriO8lEMI/AAAAAAAAA4w/eMMqNSSCsk0US7dxehtfxZothaUJIWdzgCLcBGAs/s320/TwoSwitchPoints.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 2: Two Switch Points on Frontier</b></td></tr></tbody></table><P>Suppose technical progress reduces <I>d</I> to 53/180. Figure 3 shows the resulting configuration of the wage curves. There is a single switch point, in which all three wage curves intersect. Aside from the switch point, the Beta technique is no longer cost-minimizing for any other rate of profits. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-DKcN4fm1VdI/WWNrKwO8S3I/AAAAAAAAA4o/fCIVMSzp13U5IQBiakJbiiI45wLemNulQCLcBGAs/s1600/OneSwitchPoint1.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-DKcN4fm1VdI/WWNrKwO8S3I/AAAAAAAAA4o/fCIVMSzp13U5IQBiakJbiiI45wLemNulQCLcBGAs/s320/OneSwitchPoint1.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 3: One Switch Point on Frontier</b></td></tr></tbody></table><P>Figure 4 shows the wage curves when the parameter <I>d</I> has been reduced to 1/5. For <I>d</I> between 53/180 and 1/5, the wage frontier is constructed from the wage curves for the Alpha and Gamma techniques. The Beta technique is never cost minimizing, and the switch point between the Beta and Gamma techniques does not lie on the frontier. The wage curves for the Alpha and Beta techniques have an intersection in the first quadrant only for part of that range for the parameter <I>d</I>. That intersection, however, is never on the frontier for that range. For a value of <I>d</I>less than 1/5, the Alpha technique is dominant. The Beta and Gamma techniques are no longer cost minimizing for any rate of profits. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-G8WsuvireuU/WWNrQWFmdeI/AAAAAAAAA4s/uEBatoRWFlY55xKRTwNyHnbJzWjO2kDEwCLcBGAs/s1600/OneSwitchPoint2.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-G8WsuvireuU/WWNrQWFmdeI/AAAAAAAAA4s/uEBatoRWFlY55xKRTwNyHnbJzWjO2kDEwCLcBGAs/s320/OneSwitchPoint2.jpg" width="320" height="180" data-original-width="720" data-original-height="405" /></a></td></tr><tr><td align="center"><b>Figure 4: Bifurcation in which Switch Point on Frontier Disappears</b></td></tr></tbody></table><B>4.0 Conclusion</B><P>Figure 1, at the top of the post, summarizes the example. Technical progress can result in a change of the number of switch points, where those switch points disappear and appear along the inside of the wage frontier. Bifurcations need not be across the axes for the wage or the rate of profits. </P>http://robertvienneau.blogspot.com/2017/07/bifurcations-along-wage-frontier.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-2562294352860846383Wed, 12 Jul 2017 00:07:00 +00002017-07-13T13:29:22.860-04:00Example in Mathematical EconomicsLabor MarketsSraffa EffectsA Switch Point on the Wage Axis<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://1.bp.blogspot.com/-3G0lreXdQ5g/WWVgPEahaBI/AAAAAAAAA5U/2EWnLLuHT1INoJOXpz3pqLw0Wu5GHkpNwCLcBGAs/s1600/BifurcationOnWageAxis3.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-3G0lreXdQ5g/WWVgPEahaBI/AAAAAAAAA5U/2EWnLLuHT1INoJOXpz3pqLw0Wu5GHkpNwCLcBGAs/s320/BifurcationOnWageAxis3.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 1: Bifurcation Diagram</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I have been exploring the variation in the number and "perversity" of switch points in a model of prices of production. I <A HREF="https://robertvienneau.blogspot.com/2017/07/generic-bifurcations-and-switch-points.html">conjecture</A>that generic changes in the number of switch points with variations in model parameters can be classified into a few types of bifurcations. (This conjecture needs a more precise statement.) This post fills a lacuna in this conjecture. I give an example of a case that I have not previously illustrated. </P><B>2.0 Technology</B><P>Consider the technology illustrated in Table 1. The managers of firms know of four processes of production. And these processes exhibit Constant Returns to Scale. The column for the iron industry specifies the inputs needed to produce a ton of iron. In this post, I consider how variations in the parameter <I>e</I> affect the number of switch points. The column for the copper industry likewise specifies the inputs needed to produce a ton of copper. Two processes are known for producing corn, and their coefficients of production are specified in the last two columns in the table. Each process is assumed to require a year to complete and uses up all of its commodity inputs. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Commodity Example</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="3"><B>Input</B></TD><TD ALIGN="center" COLSPAN="4"><B>Industry</B></TD></TR><TR><TD ALIGN="center" ROWSPAN="2"><B>Iron</B></TD><TD ALIGN="center" ROWSPAN="2"><B>Copper</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn</B></TD></TR><TR><TD ALIGN="center"><B>Alpha</B></TD><TD ALIGN="center"><B>Beta</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center"><I>e</I></TD><TD ALIGN="center">3/2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Copper</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/5</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>This technology presents a problem of the choice of technique. The Alpha technique consists of the iron-producing process and the corn-producing process labeled Alpha. Similarly, the Beta technique consists of the copper-producing process and the corn-producing process labeled Beta. </P><P>The choice of technique is based on cost-minimization. Consider prices of production, which are stationary prices that allow the economy to be reproduced. A wage curve, showing a tradeoff between wages and the rate of profits, is associated with each technique. In drawing such a curve, I assume that a bushel corn is the numeraire and that labor is advanced. Hence, wages are paid out of the surplus product at the end of the year. The chosen technique, for, say, a given wage, maximizes the rate of profits. The wage curve for the cost-minizing technique at the given wage lies on the outer frontier formed from all wage curves. </P><B>3.0 A Result of Technical Progress</B><P>For a high value of the parameter <I>e</I>, the Beta technique minimizes costs, for all feasible wages and rates of profits. Figure 2 illustrates wage curves when <I>e</I> is equal to 21/8. For any wage below the maximum, the Beta technique is cost minimizing. But at a rate of profits of zero, a switch point arises. Both techniques are cost-minimizing. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://3.bp.blogspot.com/-XPwrKROLX0s/WWVgGceix8I/AAAAAAAAA5Q/MkdmU5mPVbsPmVQ26RsMAs48_RA6pPAXwCLcBGAs/s1600/BifurcationOnWageAxis1.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-XPwrKROLX0s/WWVgGceix8I/AAAAAAAAA5Q/MkdmU5mPVbsPmVQ26RsMAs48_RA6pPAXwCLcBGAs/s320/BifurcationOnWageAxis1.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 2: A Switch Point on the Wage Axis</b></td></tr></tbody></table><P>Suppose technical progress further decreases the person-years needed as input for each ton iron produced. Figure 3 illustrates wage curves when <I>e</I> has fallen to one. For low wages, the Beta technique is cost-minimizing. For high wages, the Alpha technique is preferred. As a result of the structural variation under consideration, the switch point is on the frontier within the first quadrant. It is no longer an intersection of two wage curves with the wage axis. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD ALIGN="center"><a href="https://2.bp.blogspot.com/-Ksto8r05t3U/WWVgAWmTgJI/AAAAAAAAA5M/z2a19-Zo6_cpXWKrBKtjmd1qGNpjZNvZQCLcBGAs/s1600/BifurcationOnWageAxis2.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-Ksto8r05t3U/WWVgAWmTgJI/AAAAAAAAA5M/z2a19-Zo6_cpXWKrBKtjmd1qGNpjZNvZQCLcBGAs/s320/BifurcationOnWageAxis2.jpg" width="320" height="180" data-original-width="1440" data-original-height="810" /></a></td></tr><tr><td align="center"><b>Figure 3: A Perturbation of the Switch Point on the Wage Axis</b></td></tr></tbody></table><P>By the way, this switch point conforms to outdated neoclassical mumbo jumbo. In a comparison of stationary states, a lower wage around the switch point is associated with the adoption of a more labor-intensive technique. When analyzing switch points, this is a special case with no claim to logical necessity. <A HREF="http://johnhcochrane.blogspot.com/2017/07/whats-good-about-economics-sometimes.html">John Cochrane</A>and <A HREF="http://econlog.econlib.org/archives/2017/07/minimally_convi.html">Bryan Caplan</A>are ignorant of price theory. Contrast with <A HREF="https://academic.oup.com/cje/article-abstract/38/5/1087/1683221/Do-labour-supply-and-demand-curves-exist?redirectedFrom=fulltext">Steve Fleetwood</A>. </P><B>4.0 Conclusion</B><P>Technical progress can result in a new switch point appearing over the axis for the wage. Given a stationary state, this switch point is "non-perverse" until the occurrence of another structural bifurcation. </P>http://robertvienneau.blogspot.com/2017/07/a-switch-point-on-wage-axis.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-5474754967565458932Sat, 08 Jul 2017 12:18:00 +00002017-07-08T08:18:53.742-04:00Example in Mathematical EconomicsSraffa EffectsTowards Complex DynamicsGeneric Bifurcations and Switch Points<P>This post states a mathematical conjecture. </P><P>Consider a model of prices of production in which a choice of technique exists. The parameters of model consist of coefficients of production for each technique and given ratios for the rates of profits among industries. The choice of technique can be analyzed based on wage curves. A point that lies simultaneously on the outer envelope of all wage curves and the wage curves for two techniques (for non-negative wages and rates of profits not exceeding the maximum rates of profits for both techniques) is a <I>switch point</I>. </P><P><B>Conjecture:</B> The number of switch points is a function of the parameters of the model. The number of switch points varies with variations in the parameters. </P><UL><LI>A pair of switch points can arise if:</LI><UL><LI>One wage curve dominates another for one set of parameter values.</LI><LI>The wage curves become tangent at a single switch point, for a change in one parameter.</LI><LI>The point of tangency breaks up into two switch points (reswitching) as that parameter continues in the same direction.</LI></UL><LI>A switch point can disappear (for an economically relevant ranges of wages) if:</LI><UL><LI>A switch point exists for some set of parameter values.</LI><LI>For some variation of a parameter, that switch point becomes the intersection of both wage curves with one of the axes (the wage or the rate of profits).</LI><LI>A further variation of the parameter in the same direction leads to the point of intersection of the wage curves falling out of the first quadrant.</LI></UL><LI>Like the above, but a switch point can disappear if a variation in a parameter results in that intersection of two wage curves falling off the outer envelope. (A third wage curve becomes dominant for the wage at which the intersection occurs.)</LI></UL><P>The above three possibilities are the only generic bifurcations in which the number of switch points can change with model parameters. </P><P><B>Proof:</B> By incredibility. How could it be otherwise? </P><P>I claim that the above conjecture applies to a model with <I>n</I> commodities, not just the two-commodity example I have <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2993913">previously</A><A HREF="https://robertvienneau.blogspot.com/2017/06/bifurcations-and-switchpoints.html">analyzed</A>. It applies to a choice among as many finite techniques as you please. Different techniques may require different capital goods as inputs. Not all commodities need be basic. </P><P>In actuality, I do not know how to prove this. I am not sure what it means for a bifurcation to be <I>generic</I> in the above conjecture, but I want to allow for a combination of, say, two of the three possibilities. For example, the point of tangency for two wage curves (in the first case) may simultaneously be the intersection of both wage curves with the axis for the rate of profits. In this case, only one switch point arises with continuous variation of model parameters; the other falls below the axis for the rate of profits. I want to say such a bifurcation is non-generic, in some sense. </P><P>This post needs pictures. I assume the third possibility can arise for some parameter in at least one of <A HREF="https://robertvienneau.blogspot.com/2017/03/a-fluke-switch-point.html">these</A><A HREF="https://robertvienneau.blogspot.com/2016/12/example-of-choice-of-technique.html">examples</A>. (Maybe I need to think harder to be sure that the number of switch points changes. What do I want to say is non-generic here?) I have an example in which a switch point disappears by falling below the axis for the rate of profits, but I do not have an example of a switch point disappearing by crossing the wage axis. </P>http://robertvienneau.blogspot.com/2017/07/generic-bifurcations-and-switch-points.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-1929976725492994241Tue, 04 Jul 2017 12:06:00 +00002017-07-04T08:06:01.687-04:00Example in Mathematical EconomicsPolitical ScienceVoting Efficiency Gap: A Performative Theory?<TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Distribution of Votes Among Parties and Districts</B></CAPTION><TR><TD ALIGN="center"><B>District</B></TD><TD ALIGN="center"><B>Tories</B></TD><TD ALIGN="center"><B>Whigs</B></TD><TD ALIGN="center"><B>Total</B></TD></TR><TR><TD ALIGN="center">I</TD><TD ALIGN="center">51</TD><TD ALIGN="center">49</TD><TD ALIGN="center">100</TD></TR><TR><TD ALIGN="center">II</TD><TD ALIGN="center">51</TD><TD ALIGN="center">49</TD><TD ALIGN="center">100</TD></TR><TR><TD ALIGN="center">III</TD><TD ALIGN="center">33</TD><TD ALIGN="center">67</TD><TD ALIGN="center">100</TD></TR><TR><TD ALIGN="center"><B>Total</B></TD><TD ALIGN="center">135</TD><TD ALIGN="center">165</TD><TD ALIGN="center">300</TD></TR></TABLE><B>1.0 Introduction</B><P>This post, amazingly enough, is on current events. Stephanopoulos and McGhee have developed a formula, the efficiency gap, that measures the partisanship of the lines drawn for legislative districts. In this post, I present a numerical illustration of this formula and connect it to current events. I conclude with some questions. </P><B>2.0 Numerical Example</B><P>Consider a population of 300 voters divided between two parties. The Whigs are in the majority, with 55% of the electorate. Suppose the government has a three-member council, with each member elected from a district. And each district contains 100 voters. </P><B>2.1 Drawing Districts</B><P>The Tories, despite being the minority party have drawn the districts. The votes in the last election are as in Table 1. The Tories are in the minority of the population, but hold two out of three council seats. </P><P>The Tories, in this example, cannot win all seats. In the seats they lose, they want to pack as many Whigs as possible. So where the Whigs win, they win overdominatingly. Many of the Whig votes in that single district are wasted on running up a victory more than necessary. On the other hand, the Tories try to draw their winning districts to win as narrowly as possible. The Whig votes in the districts in which the Whigs lose are said to be cracked. </P><P>This is an extreme example, sensitive to small variations in the districts in which the Tories win. They would probably want safer majorities in those districts. </P><P>As far as I can see, the drawing of odd-shaped district lines is not necessary for gerrymandering. Consider a city surrounded by suburbs and a rural area. Suppose, that downtown tends to vote differently than the suburbs and rural areas. One could imagine district lines drawn outward from the central city. Depending on relative populations, that might distribute the urban voters such that they predominate in all districts. On the other hand, one might create a few compact districts in the center to pack many urban voters, with the ones remaining in cropped pizza slices having their votes cracked. </P><B>2.2 Wasted Votes</B><P>Define a vote to be wasted if either it is for a losing candidate in your district or it is for a winning candidate, but it exceeds the number needed for a majority in that district. The number of wasted votes for each party in the numerical example is: </P><UL><LI>The Tories have 33 wasted votes.</LI><LI>The Whigs have 49 + 49 + (67 - 51) = 114 wasted votes.</LI></UL><P>The efficiency gap is a single number that combines the number of wasted votes in both parties. An invariance property arises here. As I have defined it, the number of wasted votes, summed across parties, in each district is 49. Forty nine is one less than half the number of votes in a district. This is no accident. </P><B>2.3 Arithmetic</B><P>In calculating the efficiency gap, one takes the absolute value of the difference between the parties in the number of wasted votes. In the example, this number is | 33 - 114 | = 81. </P><P>The efficiency gap is the ratio of this positive difference to the number of voters. So the efficiency gap in the example is 81/300 = 27%. </P><B>3.0 Contemporary Relevance in the United States</B><P>The United States Supreme Court has decided, in a number of cases over the last decades, that gerrymandering might be something they can rule on. Partisan redistricting is not purely a political issue that they do not want to get involved in. Apparently, however, they have never found a clear example. </P><P>But what is gerrymandering? Can they define some sort of rule that lower courts can use? How would politicians drawing up district lines know whether or not their decisions will withstand challenges in court? Apparently, Justice Kennedy, among others expressed a hankering for some such rule in his decision in <A HREF="https://www.law.cornell.edu/supct/html/05-204.ZS.html">League of United Latin American Citizens (LULAC) vs. Perry</A> (2006). </P><P><A HREF="https://www.brennancenter.org/legal-work/whitford-v-gill">Gill vs. Whitford</A> is a current case on the Supreme Court docket. And the efficiency gap, which is relatively new mathematics, may be discussed in the pleadings, at least, in this case. </P><P>So the creation of the mathematical formula illustrated above might affect the law in the United States. If so, it will impact how districts are drawn and what some consider fair. It is interesting that I can now raise the issue of the performativity of mathematics in a non-historical context, while the mathematics is, perhaps, performing. </P><B>4.0 Questions</B><P>I am working on reading two of the three references below. (Articles in law reviews seem to be consistently lengthy.) I have some questions and comments. </P><P>Berstein and Duchin (2017) seems to raise some severe objections. Suppose the election in a district with 100 voters is decided either 75 to 25 or 76 to 24. The way I have defined it, the difference in wasted votes in this district is (24 - 25) or (25 - 24). That is, this district contributes one vote to the difference in wasted votes. So the definition of the efficiency gap privileges races that are won with 75% of the vote. </P><P>Consider a case in which one party has support from 75 percent of the voters. Suppose the districts are drawn such that each district casts 75% of their votes for that party. So this party wins 100% of the seats and the efficiency gap is minimized. Do we want to say this is not an example of gerrymandering? </P><P>Is the efficiency gap related to <A HREF="http://robertvienneau.blogspot.com/2016/04/math-is-power.html">power</A> <A HREF="http://robertvienneau.blogspot.com/2016/06/getting-greater-weight-for-your-vote.html">indices</A> somehow or other? How should the efficiency gap be calculated if more than two parties are contesting an election? Mayhaps, one should calculate the efficiency gap for each pair of parties. This loses the simplicity of a single number. Also, sometimes clever Republican strategists might try to help themselves by helping the Green Party, at the expense of the Democratic Party. How does this measure compare and contrast with other measures? As I understand it, a measure of partisan swing, for example, relies on counterfactuals, while the efficiency gap is not counterfactual. </P><B>References</B><UL><LI>Mira Bernstein and Moon Duchin (2017). <A HREF="https://arxiv.org/pdf/1705.10812.pdf">A Formula Goes to Court: Partisan Gerrymandering and the Efficiency Gap</A>.</LI><LI><A HREF="http://sites.tufts.edu/gerrymandr/about-the-august-workshop/">Geometry of Redistricting</A>, Tufts University, to be held 7-11 August 2017</LI><LI>Nicholas Stephanopoulos and Eric McGhee (2015). <A HREF="http://chicagounbound.uchicago.edu/uclrev/vol82/iss2/4/">Partisan Gerrymandering and the Efficiency Gap</A>, <I>University of Chicago Law Review</I>.</LI></UL>http://robertvienneau.blogspot.com/2017/07/voting-efficiency-gap-performative.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-2213279239959305510Fri, 30 Jun 2017 11:48:00 +00002017-07-01T12:00:33.760-04:00Example in Mathematical EconomicsFull Cost PricesSraffa EffectsTowards Complex DynamicsBifurcations And Switchpoints<P>I have organized a series of my posts together into a working paper, titled <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2993913"><I>Bifurcations and Switch Points</I></A>. Here is the abstract: </P><BLOCKQUOTE>This article analyzes structural instabilities, in a model of prices of production, associated with variations in coefficients of production, in industrial organization, and in the steady-state rate of growth. Numerical examples are provided, with illustrations, demonstrating that technological improvements or the creation of differential rates of profits can create a reswitching example. Variations in the rate of growth can change a "perverse" switch point into a normal one or vice versa. These results seem to have implications for the stability of short-period dynamics and suggest an approach to sensitivity analysis for certain empirical results regarding the presence of Sraffa effects. </BLOCKQUOTE><P>Here are links to previous expositions of parts of this analysis: </P><UL><LI>Variation in selected coefficients of production:</LI><UL><LI><A HREF="http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.html">Variation of switch points</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2017/06/continued-bifurcation-analysis-of.html">A bifurcation diagram</A></LI></UL><LI>A model of prices of production under oligopoly:</LI><UL><LI><A HREF="http://robertvienneau.blogspot.com/2017/02/a-reswitching-example-in-model-of.html">The model</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.html">Variation of switch points with markups</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2017/06/bifurcation-analysis-in-model-of.html">A bifurcation diagram</A></LI></UL><LI><A HREF="http://robertvienneau.blogspot.com/2017/04/bifurcations-with-variations-in-rate-of.html">Bifurcations with variations in the rate of growth</A></LI></UL><P>In <A HREF="https://www.blogger.com/comment.g?blogID=26706564&postID=5714650258406054456">comments</A>, Sturai suggests additional research with the model of oligopoly. One could take the standard commodity as such that it has no markup. What I am calling the scale factor for the rate of profits would be the rate of profits made in the production of the standard commodity. Markups for individual industries would be based on this. I have identified a problem, much like the transformation problem, in comparing and contrasting free competition and oligopoly. I would have to think about this. </P>http://robertvienneau.blogspot.com/2017/06/bifurcations-and-switchpoints.htmlnoreply@blogger.com (Robert Vienneau)2tag:blogger.com,1999:blog-26706564.post-5714650258406054456Sat, 24 Jun 2017 17:03:00 +00002017-06-27T06:13:28.565-04:00Example in Mathematical EconomicsFull Cost PricesSraffa EffectsTowards Complex DynamicsBifurcation 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 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 corn 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>http://robertvienneau.blogspot.com/2017/06/bifurcation-analysis-in-model-of.htmlnoreply@blogger.com (Robert Vienneau)4tag:blogger.com,1999:blog-26706564.post-4692945473727905616Tue, 20 Jun 2017 22:23:00 +00002017-06-22T08:01:27.200-04:00Example in Mathematical EconomicsSraffa EffectsTowards Complex DynamicsContinued 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>http://robertvienneau.blogspot.com/2017/06/continued-bifurcation-analysis-of.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-9018078622276269099Thu, 15 Jun 2017 18:44:00 +00002017-06-16T06:43:50.136-04:00Example in Mathematical EconomicsPerfect 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>http://robertvienneau.blogspot.com/2017/06/perfect-competition-with-uncountable.htmlnoreply@blogger.com (Robert Vienneau)3