tag:blogger.com,1999:blog-26706564Fri, 21 Apr 2017 10:24:11 +0000Example in Mathematical EconomicsAustrian School Of EconomicsSraffa EffectsPaul KrugmanMethodology of EconomicsTowards Complex DynamicsKarl MarxInterpreting Classical EconomicsHistory vs. EquilibriumLabor MarketsCookbooks for Workshops of the FutureEmpirical Results - Distribution and MobilityInternational TradeEmpirical Input-Output AnalysisSteady State EconomicsMilton FriedmanTheory of ChoiceJoint ProductionProfile of an EconomistPolitical ScienceSteve KeenLudwig WittgensteinAnti-LibertarianismFull Cost PricesMaking Life More BrutishPrinciples of EconomicsEndogenous MoneyGame TheoryCriticisms of Sraffian EconomicsMultiple Interest RatesKaldor Business Cycle ModelNoam ChomskyAntonio GramsciFoucaultPolitical PhilosophyRecipes for Cookbooks for Workshops of the PresentVoices in the AirProtocolsZizekBiologyCorporate GovernanceEurolandVocabularyWeird ScienceMarkov ProcessThoughts On Economicshttp://robertvienneau.blogspot.com/noreply@blogger.com (Robert Vienneau)Blogger1026125tag: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-3749207575015038344Thu, 20 Apr 2017 20:30:00 +00002017-04-20T16:30:42.083-04:00Example in Mathematical EconomicsJoint ProductionMultiple Interest RatesNonstandard Investments as a Challenge for Multiple Interest Rate Analysis?<B>1.0 Introduction</B><P>This post contains some musing on corporate finance and its relation to the theory of production. </P><B>2.0 Investments, the NPV, and the IRR</B><P>In finance, an <I>investment project</I> or, more shortly, an <I>investment</I>, is a sequence of dated cash flows. Consider an investment in which these cash flows take place at the end of <I>n</I> successive years. Let <I>C</I><SUB><I>t</I></SUB>; <I>t</I> = 0, 1, ..., <I>n</I> - 1; be the cash flow at the end of the <I>t</I>th year here, counting back from the last year in the investment. That is, <I>C</I><SUB><I>n</I> - 1</SUB> is the cash flow at the end of the first year in the investment, and <I>C</I><SUB>0</SUB> is the last cash flow. </P><P>The <I>Net Present Value</I> (NPV) of an investment is the sum of discounted cash flows in the investment. Let <I>r</I> be the interest rate used in time time discounting, and suppose all cash flows are discounted to the end of the first year in the investment. Then the NPV of the illustrative investment is: </P><BLOCKQUOTE>NPV<SUB>0</SUB>(<I>r</I>) = <I>C</I><SUB><I>n</I> - 1</SUB> + <I>C</I><SUB><I>n</I> - 2</SUB>/(1 + <I>r</I>) + ... + <I>C</I><SUB>0</SUB>/(1 + <I>r</I>)<SUP><I>n</I> - 1</SUP></BLOCKQUOTE><P>If the above expression is multiplied by (1 + <I>r</I>)<SUP><I>n</I> - 1</SUP>, one obtains the NPV of the investment with every cash flow discounted to the last year in the investment: </P><BLOCKQUOTE>NPV<SUB>1</SUB>(<I>r</I>) = <I>C</I><SUB><I>n</I> - 1</SUB>(1 + <I>r</I>)<SUP><I>n</I> - 1</SUP>+ <I>C</I><SUB><I>n</I> - 2</SUB>(1 + <I>r</I>)<SUP><I>n</I> - 2</SUP>+ ... + <I>C</I><SUB>0</SUB></BLOCKQUOTE><P>For the next step, I need some sign conventions. Let a positive cash flow designate revenues, and a negative cash flow be a cost. Suppose, for now, that the (temporally) first cash flow is a cost, that is negative. Then (-1/<I>C</I><SUB><I>n</I> - 1</SUB>) NPV<SUB>1</SUB>(<I>r</I>) is a polynomial in (1 + <I>r</I>), with unity as the coefficient for the highest-order term. All other terms are real. </P><P>Such a polynomial has <I>n</I> - 1 roots. These roots can be real numbers, either negative, zero, or positive. They can be complex. Since all coefficients of the polynomial are real, complex roots enter as conjugate pairs. Roots can be repeating. At any rate, the polynomial can be factored, as follows: </P><BLOCKQUOTE>NPV<SUB>1</SUB>(<I>r</I>) = (-<I>C</I><SUB><I>n</I> - 1</SUB>)(<I>r</I> - <I>r</I><SUB>0</SUB>) (<I>r</I> - <I>r</I><SUB>1</SUB>)... (<I>r</I> - <I>r</I><SUB><I>n</I> - 1</SUB>) </BLOCKQUOTE><P>where <I>r</I><SUB>0</SUB>, <I>r</I><SUB>1</SUB>, ..., <I>r</I><SUB><I>n</I> - 1</SUB> are the roots of the polynomial. Note that the interest rate appears only in terms in which the difference between the interest rate and one root is taken. And all roots appear on the Right Hand Side. I am going to call an specification of NPV with these properties an <I>Osborne expression</I> for NPV. </P><P>Suppose, for now, that at least one root is real and non-negative. The <I>Internal Rate of Return</I> (IRR) is the smallest real, non-negative root. For notational convenience, let <I>r</I><SUB>0</SUB> be the IRR. </P><B>3.0 Standard Investments in Selected Models of Production</B><P>A <I>standard investment</I> is one in which all negative cash flows precede all positive cash flows. Is there a theorem that an IRR exists for each standard investment? Perhaps this can be proven by discounting all cash flows to the end of the year in which the last outgoing cash flow occurs. Maybe one needs a clause that the undiscounted sum of the positive cash flows does not fall below the undiscounted sum of the negative cash flows. </P><P>At any rate, an Osborne expression for NPV has been calculated for standard investments characterizing two models of production. As I recall it, Osborne (2010) illustrates a more abstract discussion with a point-input, flow-output example. Consider a model in which a machine is first constructed, in a single year, from unassisted labor and land. That machine is then used to produce output over multiple years. Given certain assumptions on the pattern of the efficiency of the machine, this example is of a standard investment, with one initial negative cash flow followed by a finite sequence of positive cash flows. </P><P>On the other hand, I have <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058">presented</A>an example for a flow-input, point-output model. Techniques of production are represented as finite series of dated labor inputs, with output for sale on the market at a single point in the time. Each technique is characterized by a finite sequence of negative cash flows, followed by a single positive cash flow. </P><P>In each of these two examples, the NPV can be represented by an Osborne expression that combines information about all roots of a polynomial. Thus, basing an investment decision on the NPV uses more information than basing it on the IRR, which is a single root of the relevant polynomial. </P><B>4.0 Non-standard Investments and Pitfalls of the IRR</B><P>In a <I>non-standard investment</I> at least one positive cash flow precedes a negative cash flow, and vice-versa. Non-standard investments can highlight three pitfalls in basing an investment decision on the IRR: </P><UL><LI>Multiple IRRs: The polynomial defining the IRR may have more than one real, non-negative root. What is the rationale for picking the smallest?</LI><LI>Inconsistency in recommendations based on IRR and NPV: The smallest real non-negative root may be positive (suggesting a good investment), with a negative NPV (suggesting a bad investment).</LI><LI>No IRR: All roots may be complex.</LI></UL><P>Berk and DeMarzo (2014) present the example in Table 1 as an illustration of the third pitfall. They imagine an author who receives an advance of $750 hundred thousands, sacrifices an income of $500 hundred thousand in each year of writing a book, and, finally, receives a royalty of one million dollars upon publication. The roots of the polynomial defining the NPV are -1.71196 + 0.78662 j, -1.71196 - 0.78662 j, 0.04529 + 0.30308 j, 0.04529 - 0.30308 j. All of these roots are complex; none satisfy the definition of the IRR. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: A Non-Standard Investment</B></CAPTION><TR><TD ALIGN="center"><B>Year</B></TD><TD ALIGN="center"><B>Revenue</B></TD></TR><TR><TD ALIGN="center">0</TD><TD ALIGN="center">750</TD></TR><TR><TD ALIGN="center">1</TD><TD ALIGN="center">-500</TD></TR><TR><TD ALIGN="center">2</TD><TD ALIGN="center">-500</TD></TR><TR><TD ALIGN="center">3</TD><TD ALIGN="center">-500</TD></TR><TR><TD ALIGN="center">4</TD><TD ALIGN="center">1,000</TD></TR></TABLE><P></P><B>5.0 Issues for Multiple Interest Rate Analysis</B><P>Osborne, in his 2014 book, extends his 2010 analysis of the NPV to consider the first and second pitfall above. Nowhere do I know of is an Osborne expression for the NPV derived for an example in which the third pitfall arises. </P><P>The idea that the pitfalls above for the use of the IRR might be a problem for multiple interest rate analysis was suggested to me anonymously. On even hours, I do not see this. Why should I care about how many roots there are in an Osborne expression for the NPV, their sign, or even if they are complex? </P><P>On the other hand, I wonder about how non-standard investments relate to the theory of production. I know that an example can be <A HREF="http://robertvienneau.blogspot.com/2017/01/reswitching-in-example-of-one-commodity.html">constructed</A>, in which the price of a used machine becomes negative before it becomes positive. Can the varying efficiency of the machine result in a non-standard investment? After all, the cash flow, in such an example of joint production, is the sum of the price of the conventional output of the machine and the price of the one-year older machine. Even when the latter is negative, the sum need not be negative. But, perhaps, it can be in some examples. </P><P>Not all techniques in models with joint production, of the production of commodities by means of commodities, can be represented as dated labor flows. I guess one can still talk about NPVs. Can one formulate an algorithm, based on NPVs, for the choice of technique? How would certain annoying possibilities, such as <A HREF="http://robertvienneau.blogspot.com/2009/08/some-issues-in-joint-production.html">cycling</A>be accounted for? Can one always formulate an Osborne expression for the NPV? Do properties of multiple interest rates have implications for, for example, a truncation rule in a model of fixed capital? Perhaps a non-standard investment, for a fixed capital example and one pitfall noted above, always has a cost-minimizing truncation in which the pitfall does not arise. Or perhaps the opposite is true. </P><P>Anyway, I think some issues could support further research relating models of production in economics and finance theory. Maybe one obtains, at least, a translation of terms. </P><B>Appendix: Technical Terminology</B><P>See body of post for definitions. </P><UL><LI>Flow Input, Point Output</LI><LI>Investment</LI><LI>Investment Project</LI><LI>Internal Rate of Return (IRR)</LI><LI>Net Present Value (NPV)</LI><LI>Non Standard Investment</LI><LI>Osborne Expression (for NPV)</LI><LI>Point-Input, Flow Output model</LI><LI>Standard Investment</LI></UL><B>References</B><UL><LI>Jonathan Berk and Peter DeMarzo (2014). <I>Corporate Finance</I>, 3rd edition. Boston: Pearson Education</LI><LI>Michael Osborne (2010). A resolution to the NPV-IRR debate? <I>Quarterly Review of Economics and Finance</I>, V. 50, Iss. 2: 234-239.</LI><LI>Michael Osborne (2014). <I>Multiple Interest Rate Analysis: Theory and Applications</I>. New York: Palgrave Macmillan</LI><LI>Robert Vienneau (2016). <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058"><I>The choice of technique with multiple and complex interest rates</I></A>, DRAFT.</LI></UL>http://robertvienneau.blogspot.com/2017/04/nonstandard-investments-as-challenge.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-7746317242435208939Thu, 13 Apr 2017 19:43:00 +00002017-04-13T15:43:08.566-04:00Elsewhere<UL><LI>Beatrice Cherrier <A HREF="https://beatricecherrier.wordpress.com/2016/05/23/how-the-term-mainstream-economics-became-mainstream-a-speculation/">suggests</A> that Paul Samuelson originated the term "mainstream economics", in his textbook. (<B>h/t</B> I think I found this by Unlearning Economics's twitter feed.)</LI><LI>Jo Michell <A HREF="http://reteacheconomics.org/articles/review-econocracy/">reviews</A> <I>The Econocracy</I>, by Joe Earle, Cahal Moran, and Zach Ward-Perkins.</LI><LI>On twitter, Cameron Murray <A HREF="https://twitter.com/Rumplestatskin/status/833558739487567873">finds</A>, of the 46 who responded, 78% did not "learn about the Cambridge Capital Controversy at point in [their] degree".</LI><LI>In the <I>Guardian</I>, Kate Raworth <A HREF="https://www.theguardian.com/global-development-professionals-network/2017/apr/06/kate-raworth-doughnut-economics-new-economics">argues</A> that new economics is needed to replace the old economics and its foundation on false laws of physics.</LI></UL>http://robertvienneau.blogspot.com/2017/04/elsewhere.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-4106987680921373734Sat, 01 Apr 2017 12:41:00 +00002017-04-01T08:41:15.645-04:00Example in Mathematical EconomicsSraffa EffectsSteady State EconomicsTowards Complex DynamicsBifurcations With Variations In The Rate Of Growth<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-kuKKb2gCiXI/WN5BmGz7HTI/AAAAAAAAAzY/Vg_t-XRHAzgY5ZF-5eMFQLfOewT8wwu7QCLcB/s1600/LaborIntensity.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-kuKKb2gCiXI/WN5BmGz7HTI/AAAAAAAAAzY/Vg_t-XRHAzgY5ZF-5eMFQLfOewT8wwu7QCLcB/s320/LaborIntensity.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 1: Perversity and Non-Perversity in the Labor Market Varying with the Rate of Growth</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I have been considering how the existence and properties of switch points vary with parameters specifying numerical examples of models of the production of commodities by means of commodities. <A HREF="http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.html">Here</A><A HREF="http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.html">are</A><A HREF="http://robertvienneau.blogspot.com/2017/02/a-reswitching-example-in-model-of.html">some</A><A HREF="http://robertvienneau.blogspot.com/2016/12/perturbation-of-reswitching-example.html">examples</A><A HREF="http://robertvienneau.blogspot.com/2017/01/a-story-of-technical-innovation.html">of</A>such analyses of structural stability. This post adds to this series. </P><P>I consider a change in sign of real Wicksell effects to be a bifurcation. In the model in this post, the steady state rate of growth is an exogenous parameter. So a change of sign of real Wicksell effects, associated with a variation in the steady state rate of growth, is a bifurcation. </P><B>2.0 Technology</B><P>The technology for this example is as usual. Suppose two commodities, iron and corn, are produced in the example economy. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">305/494</TD><TD ALIGN="center">1</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD><TD ALIGN="center">2</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD><TD ALIGN="center">2/5</TD></TR></TABLE><P>For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the sole corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process. </P><P>Each technique is represented by a two-element row vector of labor coefficients and a 2x2 Leontief input-output matrix. For example, the vector of labor coefficients for the Beta technique, <B>a</B><SUB>0, β</SUB>, is: </P><BLOCKQUOTE><B>a</B><SUB>0, β</SUB> = [305/494, 1] </BLOCKQUOTE><P>The components of the Leontief matrix for the Beta technique, <B>A</B><SUB>β</SUB>, are: </P><BLOCKQUOTE><I>a</I><SUB>1,1, β</SUB> = 229/494 </BLOCKQUOTE><BLOCKQUOTE><I>a</I><SUB>1,2, β</SUB> = 2 </BLOCKQUOTE><BLOCKQUOTE><I>a</I><SUB>2,1, β</SUB> = 3/1976 </BLOCKQUOTE><BLOCKQUOTE><I>a</I><SUB>2,2, β</SUB> = 2/5 </BLOCKQUOTE><P>The labor coefficients for the Alpha technique, <B>a</B><SUB>0, α</SUB>, differ in the first element from those for the Beta technique. The Leontief matrix for the Alpha technique, <B>A</B><SUB>α</SUB>, differs from the Leontief matrix for the Beta technique in the first column. </P><P>(The mathematics in this post is set out in terms of linear algebra. I needed to remind myself of how to work out quantity flows with a positive rate of growth. I solved the example with Octave, the open-source equivalent of Matlab for the example. I haven't checked the graphs by also working them out by hand. You can click on the figures to see them somewhat larger.) </P><B>3.0 Prices and the Choice of Technique</B><P>Consider steady-state prices that repeat, year after year, as long as firms adopt the same technique. Let <B>a</B><SUB>0</SUB> and <B>A</B> be the labor coefficients and the Leontief matrix for that technique. Suppose labor is advanced and wages are paid out of the surplus at the end of the year. Then prices satisfy the following system of equations: </P><BLOCKQUOTE><B>p</B> <B>A</B> (1 + <I>r</I>) + <B>a</B><SUB>0</SUB> <I>w</I> = <B>p</B></BLOCKQUOTE><P>where <B>p</B> is a two-element row vector of prices, <I>w</I> is the wage, and <I>r</I> is the rate of profits. Let <B>e</B> be a column vector specifying the commodities constituting the numeraire. Then: </P><BLOCKQUOTE><B>p</B> <B>e</B> = 1 </BLOCKQUOTE><P>For the numerical example, a bushel corn is the numeraire, and <B>e</B>is the second column of the identity matrix. I think of the numeraire as in the proportions in which households consume commodities. </P><P>The system of equations for prices of production, including the equation for the numeraire, has one degree of freedom. Formally, one can solve for prices and the wage as functions of an externally given rate of profits. The first equation above can be rewritten as: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> <I>w</I> = <B>p</B> [<B>I</B> - (1 + <I>r</I>) <B>A</B>] </BLOCKQUOTE><P>Multiply through, on the right, by the inverse of the matrix in square brackets: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <I>w</I> = <B>p</B></BLOCKQUOTE><P>Multiply through, again on the right, by <B>e</B>: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B> <I>w</I> = <B>p</B> <B>e</B> = 1 </BLOCKQUOTE><P>Both sides of the above equation are scalars. The wage is: </P><BLOCKQUOTE><I>w</I> = 1/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>The above equation is called the <I>wage-rate of profits curve</I> or, more shortly, the <I>wage curve</I>. Prices of production are: </P><BLOCKQUOTE><B>p</B> = <B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP>/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>r</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>The above two equations solve the price system, in some sense. </P><P>Figure 2 plots the wage curves for the example. The downward-sloping blue and red curves show that, for each technique, a lower steady-state real wage is associated with a higher rate of profits. The two curves intersect at the two switch points, at rates of profits of 20% and 80%. For rates of profits between the switch points, the Alpha technique is cost-minimizing and its wage curve constitutes the outer envelope of the wage curves in this region. For feasible rates of profits outside that region, the Beta technique is cost-minimizing. (I talk more about this figure at least twice below.) </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-WRfMraI9O8Y/WN5CDkZI-SI/AAAAAAAAAzg/4MnfISsvuJMo6zy2K9B95ENZnfAINixzACLcB/s1600/WageCurves.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-WRfMraI9O8Y/WN5CDkZI-SI/AAAAAAAAAzg/4MnfISsvuJMo6zy2K9B95ENZnfAINixzACLcB/s320/WageCurves.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curves also Characterize Tradeoff Between Consumption per Worker and Steady State Rate of Growth</b></td></tr></tbody></table><P></P><B>4.0 Quantities</B><P>Suppose the steady-state rate of growth for this economy is 100 <I>g</I> percent. A system of equations, dual to the price equations, arises for quantity flows. Let <B>q</B> denote the column vector of gross quantities, per labor-year employed, produced in a given year. Let <B>y</B> be the column vector of net quantities, per labor-year. Net quantities constitute the surplus once the (circulating) capital goods advanced at the start of the year, for a given technique, are replaced: </P><BLOCKQUOTE><B>y</B> = <B>q</B> - <B>A</B> <B>q</B> = (<B>I</B> - <B>A</B>) <B>q</B></BLOCKQUOTE><P>Since quantities are defined per person-year, employment with these quantities is unity: </P><BLOCKQUOTE><B>a</B><SUB>0</SUB> <B>q</B> = 1 </BLOCKQUOTE><P>By hypothesis, net quantities are the sum of consumption and capital goods to accumulate at the steady state rate of profits: </P><BLOCKQUOTE><B>y</B> = <I>c</I> <B>e</B> + <I>g</I> <B>A</B> <B>q</B></BLOCKQUOTE><P>Substituting into the first equation in this section and re-arranging terms yields: </P><BLOCKQUOTE><I>c</I> <B>e</B> = [<B>I</B> - (1 + <I>g</I>) <B>A</B>] <B>q</B></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>c</I> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B> = <B>q</B></BLOCKQUOTE><P>Multiply through on the left by the row vector of labor coefficients: </P><BLOCKQUOTE><I>c</I> <B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B> = <B>a</B><SUB>0</SUB> <B>q</B> = 1 </BLOCKQUOTE><P>Consumption per person-year is: </P><BLOCKQUOTE><I>c</I> = 1/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>Gross quantities are: </P><BLOCKQUOTE><B>q</B> = [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>/{<B>a</B><SUB>0</SUB> [<B>I</B> - (1 + <I>g</I>) <B>A</B>]<SUP>-1</SUP> <B>e</B>} </BLOCKQUOTE><P>Interestingly enough, the relationship between consumption per worker and the rate of growth is identical to the relationship between the wage and the rate of profits. Thus, Figure 1 is also a graph of the trade-off, for the two technique, between steady-state consumption per worker and the rate of growth. One can think of the abscissa as relabeled the rate of growth and the ordinate as relabeled consumption per person-year. In the graph, the grey point illustrates consumption per worker at a rate of growth of 10% for the Beta technique. </P>The ordinate on this graph is consumption throughout the economy. If the rate of profits exceeds the rate of growth, both those obtaining income from wages and those obtaining income from profits will be consuming. When the rates of growth and profits are equal, all profits are accumulated. </P><B>5.0 Some Accounting Identities</B><P>The value of capital per worker is: </P><BLOCKQUOTE><I>k</I> = <B>p</B> <B>A</B> <B>q</B></BLOCKQUOTE><P>The value of net income per worker is: </P><BLOCKQUOTE><I>y</I> = <B>p</B> <B>y</B> = <B>p</B> (<B>I</B> - <B>A</B>) <B>q</B></BLOCKQUOTE><P>(I hope the distinction between the scalar <I>y</I> and the vector <B>y</B>is clear in this notation.) </P><P>The value of net income per worker can be expressed in terms of the sum of income categories. Rewrite the first equation in Section 3: </P><BLOCKQUOTE><B>p</B> (<B>I</B> - <B>A</B>) = <B>a</B><SUB>0</SUB> <I>w</I> + <B>p</B> <B>A</B> <I>r</I></BLOCKQUOTE><P>Multiply both sides by the vector of gross outputs: </P><BLOCKQUOTE><B>p</B> (<B>I</B> - <B>A</B>) <B>q</B> = <B>a</B><SUB>0</SUB> <B>q</B> <I>w</I> + <B>p</B> <B>A</B> <B>q</B> <I>r</I></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>y</I> = <I>w</I> + <I>k</I> <I>r</I></BLOCKQUOTE><P>In this model, net income per worker is the sum of wages and profits per worker. </P><P>Net income per worker can also be decomposed by how it is spent. For the third equation in Section 4, multiply both sides by the price vector: </P><BLOCKQUOTE><B>p</B> <B>y</B> = <I>c</I> <B>p</B> <B>e</B> + <I>g</I> <B>p</B> <B>A</B> <B>q</B></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE><I>y</I> = <I>c</I> + <I>g</I> <I>k</I></BLOCKQUOTE><P>Net income per worker is the sum of consumption per worker and investment per worker. </P><P>Equating the two expressions for net income per worker allows one to derive an interesting graphical feature of Figure 1. This equation is: </P><BLOCKQUOTE><I>w</I> + <I>r</I> <I>k</I> = <I>c</I> + <I>g</I> <I>k</I></BLOCKQUOTE><P>Or: </P><BLOCKQUOTE>(<I>r</I> - <I>g</I>) <I>k</I> = <I>c</I> - <I>w</I></BLOCKQUOTE><P>Or solving for the value of capital per worker: </P><BLOCKQUOTE> <I>k</I> = (<I>c</I> - <I>w</I>)/(<I>r</I> - <I>g</I>) </BLOCKQUOTE><P>Capital per worker, for a given technique, is the additive inverse of the slope of two points on the wage curve for that technique. Figure 1 illustrates for the Beta technique, with a rate of growth of 10% and a rate of profits of 80%, as at the upper switch point. </P><B>6.0 Real Wicksell Effects</b><P>This section and the next presents an analysis confined to prices at the switch point for a rate of profits of 80%. </P><P>For a rate of profits infinitesimally lower than 80%, the Alpha technique is cost-minimizing. And for a rate of profits infinitesimally higher, the Beta technique is cost minimizing. I have explained above how to calculate the value of capital per worker, for the two techniques, at any given rate of growth. </P><P>Abstract from any change in prices of production associated with a change in the rate of profits. The difference between capital per head for the Beta technique and capital per head for the Alpha technique, both calculated at the prices for the switch point, is the change in "real" capital around the switch point associated with an increase in the rate of profits. Figure 3 graphs this <I>real Wicksell effect</I> as a function of the rate of steady state growth. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-qJs_FGZkkBE/WN5B3eymRDI/AAAAAAAAAzc/HrYbT_geXh0qSE6mgxzquvK-mDTAYRsugCLcB/s1600/WicksellEffect.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-qJs_FGZkkBE/WN5B3eymRDI/AAAAAAAAAzc/HrYbT_geXh0qSE6mgxzquvK-mDTAYRsugCLcB/s320/WicksellEffect.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 3: Variation in Real Wicksell Effect with Steady State Rate of Growth</b></td></tr></tbody></table><P></P><P>Two regions are apparent in Figure 3. The intersection, at the left, of the downward-sloping graph with the axis for the change in the value of capital per worker shows that the real Wicksell effect is positive, for this switch point, in a stationary state. Around the given switch point, a higher rate of profits is associated, in a stationary state, with firms wanting to adopt a more capital-intensive technique. If a greater scarcity of capital caused the rate of profits to rise, so as to ration the supply of capital, such a logical possibility could not be demonstrated. </P><P>The real Wicksell effect, for the switch point at the higher rate of profits, is zero when the rate of growth is equal to the rate of profits at the other switch points. The value of capital per person-year is the same for the two techniques, in this case. Consider a line, in Figure 1, connecting the two switch points. It also connects the points on the wage curve for the Alpha technique for a rate of profits of 80% and a rate of growth of 20%. And the same goes for the wage curve for the Beta technique. </P><B>7.0 Real Wicksell Effects in the Labor Market</B><P>A variation in real Wicksell effects with the steady state rate of growth is also manifested in the labor market. I have echoed above some mathematics which shows that the value of national income is the dot product of a vector of prices with the vector of net quantity flows. The price vector depends, given the technique, on the rate of profits at which prices of production are found. The quantity vector depends on the steady state rate of growth. The reciprocal, (1/<I>y</I>), is the amount of labor firms want to hire, per numeraire unit of national income, for a given technique. The difference at a switch point between these reciprocals, for the two techniques, is another way of looking at real Wicksell effects. </P><P>Around the switch point at a rate of profits of 80%, a lower wage is associated with firms adopting the Beta technique. And a higher wage is associated with firms adopting the Alpha technique. The difference of the above reciprocals, between the Alpha and Beta techniques, is the increase in labor, per numeraire-unit net output, associated with an infinitesimal increase in wages, at the prices for the switch point. Figure 1 shows this difference, as a function of the steady state rate of growth, at the switch point with the higher rate of profits in the example. </P><P>Figure 1 qualitatively resembles Figure 3. For a stationary state, a higher wage is associated with firms wanting to employ more labor, per numeraire unit of net output. This effect is reversed for a high enough steady state rate of growth. The bifurcation, here too, occurs at the rate of growth for the switch point at 20%. </P><B>8.0 Conclusion</B><P>This post has illustrated a comparison among steady state growth paths at rates of profits associated with a switch point. And this switch point is "perverse" from the perspective of outdated neoclassical theory, at least at a low rate of growth. But the perversity of this switch point varies with the rate of growth. In the example, when the rate of growth is between the rate of profits at the two switch points, the second switch point becomes non-perverse. </P><P>And it can go the other way. Real Wicksell effects do not even need to be monotonic. I need to find an example with at least three commodities, two techniques, and three switch points. In such an example, the switch point with the largest rate of profits will have a negative real Wicksell effect for a stationary state, a positive real Wicksell effect for steady state rates of growth between the first two switch points, and a negative real Wicksell effect for higher rates of growth, between the second and third switch points. </P><P>(I want to look up Gandolfo (2008) in the light of <A HREF="http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.html">past</A> posts. Can I tell this <A HREF="http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.html">tale</A>in terms of increasing returns, instead of exogenous technical change?) </P><B>References</B><UL><LI>Giancarlo Gandolfo (2008). Comment on "C.E.S. production functions in the light of the Cambridge critique". <I>Journal of Macroeconomics</I>, V. 30, No. 2 (June): pp. 798-800.</LI><LI>Nell (1970). A note on Cambridge controversies in capital theory. <I>Journal of Economic Literature</I>V. 8, No. 1 (March): 41-44.</LI></UL>http://robertvienneau.blogspot.com/2017/04/bifurcations-with-variations-in-rate-of.htmlnoreply@blogger.com (Robert Vienneau)1tag:blogger.com,1999:blog-26706564.post-2602169628901852269Tue, 28 Mar 2017 20:24:00 +00002017-03-29T06:05:45.095-04:00Niall Kishtainy's Pluralist, Popular History Of Economics<P>This post calls attention to <A HREF="https://www.amazon.com/Little-History-Economics-Histories/dp/0300206364">A Little History of Economics</A>, by Niall Kishtainy. Since I only skimmed this in a local bookstore, this post is not a proper review. </P><P>Kishtainy's book, in successive chapters, focuses on the lives of specific economists. The concluding chapter asks why one would want to be an economist, a question that is answered, I gather, by the preceding chapters. The book is in the same genre as Robert Heilbroner's <I>The Worldly Philosophers</I>. Since many more economists are covered in a short span, individual chapters are shorter. As I recall, economists (not in this order) covered include: </P><UL><LI>Augustine and Thomas Aquinas</LI><LI>Mercantilists</LI><LI>Francois Quesnay, Mirabeau, and other Physiocrats</LI><LI>Adam Smith</LI><LI>David Ricardo</LI><LI>Charles Fourier, Robert Owen, and other utopian socialists</LI><LI>Thomas Malthus</LI><LI>Friedrich List</LI><LI>Karl Marx</LI><LI>William Stanley and Alfred Marshall</LI><LI>German historical school, Austrian school, and the methodenstreit</LI><LI>Thorstein Veblen</LI><LI>Vladimir Lenin and John Hobson</LI><LI>Ludwig von Mises</LI><LI>Joan Robinson and Edward Chamberlin</LI><LI>John Maynard Keynes</LI><LI>Paul Samuelson, J. R. Hicks</LI><LI>Friedrich Hayek</LI><LI>Arthur Lewis, Paul Rosenstein-Rodan, and Raul Prebisch</LI><LI>Robert Solow, Trevor Swan, and Paul Romer</LI><LI>Joseph Schumpeter</LI><LI>Gary Becker</LI><LI>John Von Neumann and John Nash</LI><LI>Ken Arrow and Gerard Debreu</LI><LI>Fidel Castro, Che Guevara, and Andre Frank</LI><LI>James Buchanan</LI><LI>Milton Friedman</LI><LI>George Akerlof and Joseph Stiglitz</LI><LI>Hyman Minsky</LI><LI>John Muth, Eugene Fama, and Robert Lucas</LI><LI>Ed Prescott and Finn Kyland</LI><LI>Behavioral economics</LI><LI>Thomas Piketty</LI></UL><P>Doubtless, if I read the book in detail, I would have objections to specific details. The attempted coverage, however, seems quite impressive. This book extends from before to after most histories. And it covers a wider range of economists than most. </P>http://robertvienneau.blogspot.com/2017/03/niall-kishtainys-pluralist-populst.htmlnoreply@blogger.com (Robert Vienneau)2tag:blogger.com,1999:blog-26706564.post-8688732536964732013Wed, 22 Mar 2017 12:32:00 +00002017-03-22T08:32:05.011-04:00International TradePaul KrugmanSraffa EffectsKrugman Confused On Trade, Capital Theory<P>Over on EconSpeak, Bruce Wilder provides <A HREF="http://econospeak.blogspot.com/2017/03/review-of-economism-bad-economics-and.html?showComment=1489869775705#c6656690971941685157">some</A><A HREF="http://econospeak.blogspot.com/2017/03/review-of-economism-bad-economics-and.html?showComment=1489870271977#c3201788875710212968">comments</A> on a post. He notes that economists wanting to criticize glib free-market ideology in the public discourse often seem unwilling to discard neoclassical economic theory. </P><P>Paul Krugman illustrates how theoretically conservative and neoclassical a liberal economist can be. (I use "liberal" in the sense of contemporary politics in the USA.) I refer to Krugman's <A HREF="https://krugman.blogs.nytimes.com/2017/03/20/robot-geometry-very-wonkish/?module=BlogPost-Title&version=Blog%20Main&contentCollection=Opinion&action=Click&pgtype=Blogs®ion=Body">post</A>from earlier this week, in which he adapts an analysis from the theory of international trade to consider technological innovation (e.g., robots). Krugman presents a diagram, in which endowments of capital and labor are measured along the two axes. Krugman does not seem to be aware that one cannot, in general, coherently talk about a quantity of capital, prior to and independently of prices. He goes on to talk about "capital-intensive" and "labor-intensive" techniques of production. </P><P>I point to my <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2237279">draft paper</A>, "On the loss from trade", to illustrate my point that one cannot meaningfully talk about the endowment of capital. </P><P>(I did submit this paper to a journal. A reviewer said it was not original enough. I emphasized that I was illustrating my points in a flow-input, point output model, with a one-way flow from factors of production to consumption goods, not a model of production of commodities by means of commodities. Steedman & Metcalfe (1979) also has a one-way model, albeit with a point-input, point-output model. So the reviewer's comments were fair. Embarrassingly, I cite other papers from that book. Apparently, I had forgotten that paper, if I ever read it. I suppose that, given the chance, I could have distinguished some of my points from those made in Steedman & Metcalfe (1979). Also, I close my model with utility-maximization; if I recall correctly, Steedman leaves such an exercise to the reader in papers in that book.) </P><B>Reference</B><UL><LI>Ian Steedman and J. S. Metcalfe (1979). 'On foreign trade'. In <I>Fundamental Issues in Trade Theory</I> (ed. by Ian Steedman).</LI></UL>http://robertvienneau.blogspot.com/2017/03/krugman-confused-on-trade-capital-theory.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-6257903523401124836Sat, 18 Mar 2017 12:02:00 +00002017-03-18T08:02:10.727-04:00Example in Mathematical EconomicsFull Cost PricesSraffa EffectsTowards Complex DynamicsReswitching Only Under Oligopoly<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-PU6HifxVWUo/WMkh365gGHI/AAAAAAAAAy4/dhEXkq9sPrQNU90tgUs6NGYBELhP9apjQCLcB/s1600/OligopolyPerturbation.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-PU6HifxVWUo/WMkh365gGHI/AAAAAAAAAy4/dhEXkq9sPrQNU90tgUs6NGYBELhP9apjQCLcB/s320/OligopolyPerturbation.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 1: Rates of Profits for Switch Points for Differential Rates of Profits</b></td></tr></tbody></table><B>1.0 Introduction</B><P>Suppose one knows the technology available to firms at a given point in time. That is, one knows the techniques among which managers of firms choose. And suppose one finds that reswitching cannot occur under this technology, given prices of production in which the same rate of profits prevails among all industries. But, perhaps, barriers to entry persist. If one analyzes the choice of technique for the given technology, under the assumption that prices of production reflect stable (non-unit) ratios of profits, differing among industries, reswitching may arise for the technology. The numerical example in this post demonstrates this logical possibility. </P><P>The numerical example follows a model of oligopoly I have previously <A HREF="http://robertvienneau.blogspot.com/2017/02/a-reswitching-example-in-model-of.html">outlined</A>. In some sense, the example is symmetrical to the example in this <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2912181">draft paper</A>. That example is of a reswitching example under pure competition, which becomes an example without reswitching and capital reversing, if the ratio of the rates of profits among industries differs enough. The example in this post, on the other hand, has no reswitching or capital reversing under pure competition. But if the ratios of the rates of profits becomes extreme enough, it becomes a reswitching example. </P><B>2.0 Technology</B><P>The technology for this example resembles many I have explained in past posts. Suppose two commodities, iron and corn, are produced in the example economy. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">305/494</TD><TD ALIGN="center">1</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD><TD ALIGN="center">11/10</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD><TD ALIGN="center">2/5</TD></TR></TABLE><P>For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the lone corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process. </P><B>3.0 Price Equations</B><P>The choice of technique is analyzed on the basis of cost-minimization, with prices of production. Suppose the Alpha technique is cost minimizing. Then the following system of equalities and inequalities hold: </P><BLOCKQUOTE>[(1/10)<I>p</I> + (1/40)](1 + <I>r</I><I>s</I><SUB>1</SUB>) + <I>w</I> = <I>p</I></BLOCKQUOTE><BLOCKQUOTE>[(229/494)<I>p</I> + (3/1976)](1 + <I>r</I><I>s</I><SUB>1</SUB>) + (305/494)<I>w</I> ≥ <I>p</I></BLOCKQUOTE><BLOCKQUOTE>[(11/10)<I>p</I> + (2/5)](1 + <I>r</I><I>s</I><SUB>2</SUB>) + <I>w</I> = 1 </BLOCKQUOTE><P>where <I>p</I> is the price of a unit of iron, and <I>w</I> is the wage. </P><P>The parameters <I>s</I><SUB>1</SUB> and <I>s</I><SUB>2</SUB> are given constants, such that <I>r</I><I>s</I><SUB>1</SUB>is the rate of profits in iron production and <I>r</I><I>s</I><SUB>2</SUB> is the rate of profits in corn production. The quotient <I>s</I><SUB>1</SUB>/<I>s</I><SUB>2</SUB> is the ratio, in this model, of the rate of profits in iron production to the rate of profits in corn production. Consider the special case: </P><BLOCKQUOTE><I>s</I><SUB>1</SUB> = <I>s</I><SUB>2</SUB> = 1 </BLOCKQUOTE><P>This is the case of free competition, with investors having no preference among industries. In this case, <I>r</I> is the rate of profits. I call <I>r</I> the scale factor for the rate of profits in the general case where <I>s</I><SUB>1</SUB> and <I>s</I><SUB>2</SUB>are unequal. </P><P>The above system of equations and inequalities embody the assumption that a unit corn is the numeraire. They also show labor as being advanced and wages as paid out of the surplus at the end of the period of production. If the second inequality is an equality, both the Alpha and the Beta techniques are cost-minimizing; this is a switch point. The Alpha technique is the unique cost-minimizing technique if it is a strict inequality. To create a system expressing that the Beta technique is cost-minimizing, the equality and inequality for iron production are interchanged. </P><B>4.0 Choice of Technique</B><P>The above system can be solved, given <I>s</I><SUB>1</SUB>, <I>s</I><SUB>2</SUB>, and the scale factor for the rate of profits. I record the solution for a couple of special cases, for completeness. Graphs of wage curves and a bifurcation diagram illustrate that stable (non-unitary) ratios of rates of profits can change the dynamics of markets. </P><B>4.1 Free Competition</B><P>Consider the special case of free competition. The wage curve for the Alpha technique is: </P><BLOCKQUOTE><I>w</I><SUB>α</SUB> = (41 - 38<I>r</I> + <I>r</I><SUP>2</SUP>)/[80(2 + <I>r</I>)] </BLOCKQUOTE><P>The price of iron, when the Alpha technique is cost-minimizing, is: </P><BLOCKQUOTE><I>p</I><SUB>α</SUB> = (5 - 3<I>r</I>)/[8(2 + <I>r</I>)] </BLOCKQUOTE><P>The wage curve for the Beta technique is: </P><BLOCKQUOTE><I>w</I><SUB>β</SUB> = (6,327 - 9,802<I>r</I> + 3,631<I>r</I><SUP>2</SUP>)/[20(1,201 + 213<I>r</I>)] </BLOCKQUOTE><P>When the Beta technique is cost-minimizing, the price of iron is: </P><BLOCKQUOTE><I>p</I><SUB>β</SUB> = [5(147 - 97<I>r</I>)]/[2(1,201 + 213<I>r</I>)] </BLOCKQUOTE><P>Figure 2 graphs the wage curves for the two techniques, under free competition and a uniform rate of profits among industries. The wage curves intersect at a single switch point, at a rate of profits of, approximately, 8.4%: </P><BLOCKQUOTE><I>r</I><SUB>switch</SUB> = (1/1,301)[799 - 24 (826<SUP>1/2</SUP>)] </BLOCKQUOTE><P>The wage curve for the Beta technique is on the outer envelope, of the wage curves, for rates of profits below the switch point. Thus, the Beta technique is cost-minimizing for low rates of profits. The Alpha technique is cost minimizing for feasible rates of profits above the switch point. Around the switch point, a higher rate of profits is associated with the adoption of a less capital-intensive technique. Under free competition, this is not a case of capital-reversing. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-OCbbq8y9zw8/WMkiEDD7WXI/AAAAAAAAAy8/7-w1YeHYVvYGOXwqWqk8l7PzmF3vgZ5sgCLcB/s1600/CompetitionWageCurves.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-OCbbq8y9zw8/WMkiEDD7WXI/AAAAAAAAAy8/7-w1YeHYVvYGOXwqWqk8l7PzmF3vgZ5sgCLcB/s320/CompetitionWageCurves.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage Curves for Free Competition</b></td></tr></tbody></table><P></P><B>4.2 A Case of Oligopoly</B><P>Now, I want to consider a case of oligopoly, in which firms in different industries are able to ensure long-lasting barriers to entry. These barriers manifest themselves with the following parameter values: </P><BLOCKQUOTE><I>s</I><SUB>1</SUB> = 4/5 </BLOCKQUOTE><BLOCKQUOTE><I>s</I><SUB>2</SUB> = 5/4 </BLOCKQUOTE><P>In this case, the wage curve for the Alpha technique is: </P><BLOCKQUOTE><I>w</I><SUB>α</SUB> = (4,100 - 4,435<I>r</I> + 100<I>r</I><SUP>2</SUP>)/[40(400 + 259<I>r</I>)] </BLOCKQUOTE><P>The price of iron, when the Alpha technique is cost-minimizing, is: </P><BLOCKQUOTE><I>p</I><SUB>α</SUB> = (125 - 96<I>r</I>)/(400 + 259<I>r</I>) </BLOCKQUOTE><P>The wage curve for the Beta technique is: </P><BLOCKQUOTE><I>w</I><SUB>β</SUB> = 8(126,540 - 195,289<I>r</I> + 72,620<I>r</I><SUP>2</SUP>)/[160(24,020 + 9,447<I>r</I>)] </BLOCKQUOTE><P>The price of iron, when the Beta technique is cost-minimizing, is: </P><BLOCKQUOTE><I>p</I><SUB>β</SUB> = 2(3,675 - 3,038<I>r</I>)/(24,020 + 9,447<I>r</I>) </BLOCKQUOTE><P>Figure 3 graphs the wage curves for the Alpha and Beta techniques, for the parameter values for this model of oligopoly. This is now an example of reswitching. The Beta technique is cost minimizing at low and high rates of profits. The Alpha technique is cost minimizing at intermediate rates. The switch points are at, approximately, a value of the scale factor for rates of profits of 12.07% and 77.66%, respectively. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-vxyeoTzy2Nk/WMkiQSVH8eI/AAAAAAAAAzA/qTWCV5A5Wkop6o4gbQaj5JqyHyBdY1h4ACLcB/s1600/OligopolyWageCurves.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-vxyeoTzy2Nk/WMkiQSVH8eI/AAAAAAAAAzA/qTWCV5A5Wkop6o4gbQaj5JqyHyBdY1h4ACLcB/s320/OligopolyWageCurves.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 3: Wage Curves for a Case of Oligopoly</b></td></tr></tbody></table><P></P><B>4.3 A Range of Ratios of Profit Rates</B><P>The above example of oligopoly can be generalized. I restrict myself to the case where the parameters expressing the ratio of rates of profits between industries satisfy: </P><BLOCKQUOTE><I>s</I><SUB>2</SUB> = 1/<I>s</I><SUB>1</SUB></BLOCKQUOTE><P>One can then consider how the shapes and locations of wage curves and switch points vary with continuous variation in <I>s</I><SUB>1</SUB>/<I>s</I><SUB>2</SUB>. Figure 1, at the top of this post, graphs the wage at switch points for a range of ratios of rates of profits. Since the Beta technique is cost-minimizing, in the graph, at all high feasible wages and low scale factor for the rates of profits, I only graph the maximum wage for the Beta technique. I do not graph the maximum wage for the Alpha technique. </P><P>As the ratio of the rate of profits in the iron industry to rate in the corn industry increases towards unity, the model changes from a region in which the Beta technique is dominant to a reswitching example to an example with only a single switch point. As expected, only one switch point exists when the rate of profits is uniform between industries. </P><B>5.0 Conclusion</B><P>So I have created and worked through an example where: </P><UL><LI>No reswitching or capital-reversing exists under pure competition, with all industries earning the same rate of profits.</LI><LI>Reswitching and capital-reversing can arise for oligopoly, with persistent differential rates of profits across industries.</LI></UL><P>No qualitative difference necessarily exists, in the long period theory of prices, between free competition and imperfections of competition. Doubtless, all sorts of complications of strategic behavior, asymmetric information, and so on are empirically important. But it seems confused to blame the failure of markets to clear or economic instability on such imperfections. </P>http://robertvienneau.blogspot.com/2017/03/reswitching-only-under-oligopoly.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-2622993009132144446Wed, 15 Mar 2017 11:11:00 +00002017-03-15T07:32:01.777-04:00Example in Mathematical EconomicsSraffa EffectsTowards Complex DynamicsBifurcations in a Reswitching Example<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-erBJxRNP0-Y/WMQY8GgGqvI/AAAAAAAAAyo/a5FM6mo9D1AIVJO4aE3audnUnr3k1htEwCLcB/s1600/SwitchPtsVsIronInputsAnnotated.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-erBJxRNP0-Y/WMQY8GgGqvI/AAAAAAAAAyo/a5FM6mo9D1AIVJO4aE3audnUnr3k1htEwCLcB/s320/SwitchPtsVsIronInputsAnnotated.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 1: Rates of Profits for Switch Points in One Dimension in Parameter Space</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post presents an example of structural variation in the qualitative behavior of a reswitching example, at different values for selected parameters. I know of few applications of bifurcation analysis to the Cambridge Capital Controversy. Most prominently, I think of Rosser (1983). I suppose I could also point to some of my <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2912181">draft</A> <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1307930">papers</A>. Although not presented this way, one could read Laibman and Nell (1977) as a bifurcation analysis, where the steady state rate of growth is the parameter being varied. </P><P>I guess one could read this post as a response to the empirical results in Han and Schefold (2006). Schefold has been developing a theoretical explanation, based on random matrices, of why capital-theoretic paradoxes might be empirically rare. I seem to have stumbled on an explanation of why such paradoxes might arise in practice, and yet might not be observable without more data. To fully address recent results from Schefold, on reswitching and random matrices, one should analyze the spectra of Leontief input-output matrices, which I do not do here. </P><B>2.0 Technology</B><P>Suppose two commodities, iron and corn, are produced in the economy in the numerical example. As shown in Table 1, two processes are known for producing iron, and one corn-producing process is known. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. This is an example of a model of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">305/494</TD><TD ALIGN="center"><I>a</I><SUB>0,2</SUB></TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">229/494</TD><TD ALIGN="center"><I>a</I><SUB>1,2</SUB></TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">1/40</TD><TD ALIGN="center">3/1976</TD><TD ALIGN="center">2/5</TD></TR></TABLE><P>Assume <I>a</I><SUB>0,2</SUB> is non-negative, and that <I>a</I><SUB>1,2</SUB> is strictly positive. For the economy to be self-reproducing, both iron and corn must be produced each year. Two techniques of production are available. The Alpha technique consists of the first iron-producing process and the lone corn-producing process. The Beta technique consists of the remaining iron-producing process and the corn-producing process. </P><B>3.0 Choice of Technique</B><P>Managers of firms choose the technique to adopt based on cost-minimization. I take a bushel of corn as the numeraire. Assume that labor is advanced, and that wages are paid out of the surplus at the end of the year. For this post, I do not bother setting out equations for prices of production; I have done that many times in the past. </P><B>3.1 Reswitching for one Set of Parameter Values</B><P>Figure 2 illustrates that this is a reswitching example. This figure is drawn for the following values of the labor coefficient in the process for producing corn: </P><BLOCKQUOTE><I>a</I><SUB>0,2</SUB> = 1 </BLOCKQUOTE><P>The coefficient of production for iron in corn-production, in drawing Figure 2, is set to the following value: </P><BLOCKQUOTE><I>a</I><SUB>1,2</SUB> = 2 </BLOCKQUOTE><P>The economy exhibits capital-reversing around the switch point at an 80% rate of profits. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-9352QO3vLQU/WMQYgg_WAmI/AAAAAAAAAyg/SSrggNRiYDsevu9os2exXa_Q79nI3vHrQCLcB/s1600/WageCurvesAnnotated.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-9352QO3vLQU/WMQYgg_WAmI/AAAAAAAAAyg/SSrggNRiYDsevu9os2exXa_Q79nI3vHrQCLcB/s320/WageCurvesAnnotated.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage-Rate of Profits Curves</b></td></tr></tbody></table><P></P><B>3.2 Bifurcations with Variations in a Labor Coefficient</B><P>Wage-rate of profits curves are drawn for given coefficients of production. And they will be moved elsewhere for different levels of coefficients of production. Consequently, the existence and location of switch points differ, depending on the values for coefficients of production. </P><P>Accordingly, suppose all coefficients of production, except <I>a</I><SUB>0,2</SUB>, are as in the above reswitching example. Consider values of the labor coefficient for corn-production ranging from zero to three. The labor coefficient is plotted along the abscissa in Figure 3. The points on the blue locus in the figure show the rate of profits for the switch points, as a correspondence for the labor coefficient. The maximum rates of profits for the Alpha and Beta techniques are also graphed. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-35w9CICvgOY/WMQYqxRjPTI/AAAAAAAAAyk/c9BsEYuoTf0CeU_xtJhuFlRrWlCKnEBJQCLcB/s1600/SwitchPtsVsLaborInputAnnotated.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-35w9CICvgOY/WMQYqxRjPTI/AAAAAAAAAyk/c9BsEYuoTf0CeU_xtJhuFlRrWlCKnEBJQCLcB/s320/SwitchPtsVsLaborInputAnnotated.jpg" width="320" height="180" /></a></td></tr><tr><td align="center"><b>Figure 3: Rates of Profits for Switch Points as One Labor Coefficient Decreases</b></td></tr></tbody></table><P>Figure 3 shows a structural change in the example. Up to a value of <I>a</I><SUB>0,2</SUB> of approximately 2.74, this is a reswitching example. For parameter values strictly greater than that, no switch points exist. The maximum rates of profits for the two techniques are constant in Figure 3. The maximum rates of profits are found for a wage of zero, and they do not vary with the labor coefficient. In some sense, only the maximum rate of profits for the Beta technique is relevant in the figure. </P><B>3.3 Bifurcations with Variations in a Coefficient of Production for Iron</B><P>Figure 1, at the top of this post, also shows structural changes. The coefficient of production for iron in corn-production varies in the figure. <I>a</I><SUB>1,2</SUB>ranges from one to three. The other coefficients of production are as in the reswitching example in Section 3.1 above. And the blue locus shows the rate of profits at switch points. </P><P>The example can seen to have structural variations here, also, with three distinct regions for <I>a</I><SUB>1,2</SUB>, with the same qualitative behavior in each region. For a low enough value of the coefficient of production under consideration, only one switch point exists. The model remains a reswitching example for an intermediate range of this parameter. And for values of this coefficient of production strictly greater than approximately 2.53, the Beta technique is cost-minimizing for all feasible wages and rates of profits. </P><P>The maximum rates of profits, for the Alpha and Beta techniques, are also graphed in Figure 1. </P><B>4.0 A Story of Technological Process</B><P>Using the above example, one can tell a story of <A HREF="http://robertvienneau.blogspot.com/2017/01/a-story-of-technical-innovation.html">technological progress</A>. Suppose at the start of the story, corn production requires a relatively large input of direct labor and iron, per (gross) unit corn produced. Prices of production associated with this technology are such that only one technique is cost-minimizing. For all feasible wages and rates of profits, firms will want to adopt the Beta technique. </P><P>Suppose iron production is relatively stagnant, as compared to corn-production. Innovation in the corn industry reduces the labor and iron coefficients defining the single dominant corn-producing process. After some time, either or both coefficients will be reduced enough that the technology for this economy will have become a reswitching example. And around the switch point at the lower wage (and higher rate of profits), a higher wage is associated with the cost-minimizing technique requiring more labor to be hired, in the overall economy, per given bushel of corn produced (net). </P><P>But technological innovation continues to proceed apace. At a even lower coefficient of production for the iron input in the corn industry, the structural behavior of the economy changes again. Now a single switch point exists. And the results of the choice of technique around that switch point conforms to outdated neoclassical intuition. </P><B>5.0 Conclusion</B><P>This example has two properties that I think worth emphasizing. </P><P>The choice of technique in the example corresponds to a choice of a production process in the iron industry. As I have told the story, the technology is fixed in iron production. Innovation occurs in corn production. Thus, innovation in one industry can change the dynamics in another industry. </P><P>Second, suppose the technology is observed at a single point of time. Suppose the economy is more or less stationary, and that observation is taken at either the start or the end of the above story. Then neither reswitching nor capital reversing will be observed. Yet such phenomena might arise in the future or have arisen in the past. </P><B>References</B><UL><LI>David Laibman and Edward J. Nell (1977). Reswitching, Wicksell effects, and the neoclassical production function. <I>American Economic Review</I>. 67 (5): pp. 878-888.</LI><LI>Zonghie Han and Bertram Schefold (2006). <A HREF="https://academic.oup.com/cje/article-abstract/30/5/737/1683696/An-empirical-investigation-of-paradoxes?redirectedFrom=fulltext">An empirical investigation of paradoxes: reswitching and reverse capital deepening in capital theory</A>. <I>Cambridge Journal of Economics</I>. 30 (5): pp. 737-765.</LI><LI>J. Barkley Rosser, Jr. (1983). "Reswitching as a cusp catastrophe", <I>Journal of Economic Theory</I>. 31: pp. 182-193.</LI><LI>Bertram Schefold (2013). <A HREF="https://academic.oup.com/cje/article-abstract/37/5/1161/1678380/Approximate-surrogate-production-functions?redirectedFrom=fulltext">Approximate surrogate production functions</A>, <I>Cambridge Journal of Economics</I>. 37 (5): pp. 1161-1184.</LI><LI>Bertram Schefold (2016). <A HREF="https://academic.oup.com/cje/article-abstract/40/1/165/2604983/Profits-equal-surplus-value-on-average-and-the?redirectedFrom=fulltext">Profits equal surplus value on average and the significance of this result for the Marxian theory of accumulation: Being a new contribution to Engels' Prize Essay Competition, based on random matrices and on manuscripts recently published in the MEGA for the first time</A>. <I>Cambridge Journal of Economics</I>. 40 (1): pp. 165-199.</LI></UL>http://robertvienneau.blogspot.com/2017/03/bifurcations-in-reswitching-example.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-5433620084121967976Sat, 11 Mar 2017 16:03:00 +00002017-03-11T11:03:00.496-05:00Here and Elsewhere<UL><LI>A commentator <A HREF="http://robertvienneau.blogspot.com/2010/08/when-adam-delved-and-eve-span-who-was.html">informs</A> me that the True Levelers revived some ideas put forth in the Peasants Revolt.</LI><LI>Another commentator <A HREF="http://robertvienneau.blogspot.com/2015/07/labor-reversing-without-capital-example.html?showComment=1487680214021#c7454708390002608422">points</A> me to Naoki Yoshihara's <A HREF="http://scholarworks.umass.edu/econ_workingpaper/222/">review</A> of Opocher and Steedman's recent book. Yoshihara has a point, but I think the practice of treating inputs and physically identical outputs as different dated commodities is less applicable in partial models, as opposed to full General Equilibrium. Accountants need guidelines that resist easy manipulation in calculating profits and losses.</LI><LI>Antonella Palumbo has a <A HREF="https://www.ineteconomics.org/perspectives/blog/can-it-happen-again-defining-the-battlefield-for-a-theoretical-revolution-in-economics">post</A>, "Can 'It' Happen Again? Defining the Battlefield for a Theoretical Revolution in Economics", at the Institute for New Economic Thinking. Palumbo argues that a revival of classical economics, without Say's law, can provide an alternative to neoclassical economics. And Keynes' macroeconomics can be usefully be combined with this revival.</LI></UL>http://robertvienneau.blogspot.com/2017/03/here-and-elsewhere.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-7303849606624403644Wed, 08 Mar 2017 22:20:00 +00002017-03-08T17:20:36.806-05:00Example in Mathematical EconomicsSraffa EffectsA Fluke Switch Point<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-cs5md-XABPE/WF0cjnjFoDI/AAAAAAAAAu8/Lu4eYltVGycu3GELhuNjtHucO_DimzQKACLcB/s1600/SalvadoriSteedman1988Model2.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-cs5md-XABPE/WF0cjnjFoDI/AAAAAAAAAu8/Lu4eYltVGycu3GELhuNjtHucO_DimzQKACLcB/s320/SalvadoriSteedman1988Model2.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: The Choice of Technique in a Model with Four Techniques</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I think I may have an original criticism of (a good part of) neoclassical economics. For purposes of this post, I here define the use of continuously differential production functions as an essential element in the neoclassical theory of production. (This is a more restrictive characterization than I usually employ.) Consider this two-sector <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-two-good-model-cobb.html">example</A>, in which coefficients of production in both sectors varies continuously along the wage-rate of profits frontier. It would follow from this post, I guess, that neoclassical theory is a limit, in some sense, of an analysis in which all switch points are flukes. </P><P>I have presented many other, often unoriginal, examples with a continuum of techniques: </P><UL><LI><A HREF="http://robertvienneau.blogspot.com/2015/07/labor-reversing-without-capital-example.html">Labor reversing without capital</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2015/06/recurrence-of-capital-output-ratio.html">Recurrence of capital-output ratio without reswitching</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2015/04/an-example-with-heterogeneous-labor.html">An example with heterogeneous labor</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2014/08/labor-demand-in-fog.html">Labor demand in a fog</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2011/04/slope-of-demand-curve-varying-with.html">Slope of 'demand curve' varying with numeraire</A></LI><LI><A HREF="http://robertvienneau.blogspot.com/2009/12/negative-price-wicksell-effect-positive.html">Negative price Wicksell effect, positive real Wicksell effect</A></LI></UL><P>I have an <A HREF="http://robertvienneau.blogspot.com/2014/07/the-generality-of-sraffian-analysis-of.html">example</A> with an uncountably infinite number of techniques along the wage-rate of frontier, but discontinuities for (all?) marginal relationships. </P><B>2.0 Technology</B><P>I want to compare and contrast two models. The technology in the second model is an example in Salvadori and Steedman (1988). </P><P>Households consume a single commodity, called "corn", in both models. In both models, two processes are known for producing corn. And these processes require inputs of labor and a capital good to produce corn. All processes exhibit Constant Returns to Scale, and all processes require services of inputs over a year to produce output of a single commodity available at the end of the year. Both models are models of circulating capital. Nothing remains at the end of the year of the capital goods whose services are used by firms during the production processes. </P><B>2.1 First Model</B><P>The technology for the first model is shown in Table 1. Each column lists the inputs, in physical units, needed to produce one physical unit of the output for the industry for that column. The two processes for producing corn require inputs of distinct capital goods. One corn-producing process requires inputs of labor and iron, and the other requires inputs of labor and tin. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Industry Model</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Tin<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">(a)</TD><TD ALIGN="center">(b)</TD><TD ALIGN="center">(c)</TD><TD ALIGN="center">(d)</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Tin</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/2</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Two techniques, as shown in Table 2, are available for producing a net output of corn. A choice of a process for producing corn also entails a choice of which capital good is produced. When the processes are each operated on a appropriate scale, the gross output of the process producing the specific capital good exactly replaces the quantity of the capital good used up as an input, summed over both industries operated in the technique. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 2: Techniques in a Three-Commodity Model</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">b, d</TD></TR></TABLE><P></P><B>2.2 Second Model</B><P>The technology for the second model is shown in Table 3. Two processes are known for producing corn. Both corn-producing processes require inputs of labor and iron, but in different proportions. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 3: The Technology for a Two-Industry Model</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">(a)</TD><TD ALIGN="center">(b)</TD><TD ALIGN="center">(c)</TD><TD ALIGN="center">(d)</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">1/2</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">2/3</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Table 4 lists the techniques available in the second model. The first two techniques superficially resemble the two techniques available in the first model. But, in this model, the first process for producing a capital good can be combined, in a technique, with the second corn-producing producing process. This combination of processes is called the Gamma technique. Likewise, the Delta technique combines the second process for producing a capital good with the first corn-producing processes. Nothing like the Gamma and Delta techniques are available in the first model. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 4: Techniques in a Two-Commodity Model</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">b, d</TD></TR><TR><TD ALIGN="center">Gamma</TD><TD ALIGN="center">a, d</TD></TR><TR><TD ALIGN="center">Delta</TD><TD ALIGN="center">b, c</TD></TR></TABLE><P></P><B>3.0 Prices of Production</B><P>Suppose the Alpha technique is cost-minimizing. Prices of production, which permit smooth reproduction of the economy, must satisfy the following system of two equations in three unknowns: </P><BLOCKQUOTE>(2/3)(1 + <I>r</I>) + <I>w</I><SUB>α</SUB> = <I>p</I><SUB>α</SUB></BLOCKQUOTE><BLOCKQUOTE>(2/3) <I>p</I><SUB>α</SUB>(1 + <I>r</I>) + <I>w</I><SUB>α</SUB> = 1 </BLOCKQUOTE><P>These equations are based on the assumption that labor is advanced, and wages are paid out of the surplus at the end of the year. The same rate of profits are generated in both industries. A unit quantity of corn is taken as the numeraire. </P><P>One of the variables in these equations can be taken as exogenous. The first row in Table 5 specifies the wage and the price of the appropriate capital good, as a function of the rate of profits. The equation in the second column is called the <I>wage-rate of profits curve</I>, also known as the <I>wage curve</I>, for the Alpha technique. Table 5 also shows solutions of the systems of equations for the prices of production for the other three techniques in the second model, above. I have deliberately chosen a notation such that the first two rows can be read as applying to either one of the two models. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 5: Wages and Prices by Technique</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Wage Curve</B></TD><TD ALIGN="center"><B>Prices</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center"><I>w</I><SUB>α</SUB> = (1 - 2 <I>r</I>)/3</TD><TD ALIGN="center"><I>p</I><SUB>α</SUB> = 1</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center"><I>w</I><SUB>β</SUB> = (1 - <I>r</I>)/4</TD><TD ALIGN="center"><I>p</I><SUB>β</SUB> = 1</TD></TR><TR><TD ALIGN="center">Gamma</TD><TD ALIGN="center"><I>w</I><SUB>γ</SUB> = 2(2 - 2<I>r</I> - <I>r</I><SUP>2</SUP>)<BR>/[3(5 + <I>r</I>)]</TD><TD ALIGN="center"><I>p</I><SUB>γ</SUB> = 2(7 + 4<I>r</I>)<BR>/[3(5 + <I>r</I>)]</TD></TR><TR><TD ALIGN="center">Delta</TD><TD ALIGN="center"><I>w</I><SUB>δ</SUB> = (2 - 2<I>r</I> - <I>r</I><SUP>2</SUP>)/(7 + 4<I>r</I>)</TD><TD ALIGN="center"><I>p</I><SUB>δ</SUB> = 3(5 + <I>r</I>)/[2(7 + 4<I>r</I>)]</TD></TR></TABLE><P>Figure 1, at the top of this post, graphs all four wage-curves. The wage curves for the Alpha and Beta techniques are straight lines. In the jargon, the processes comprising these techniques exhibit the same organic composition of capital. The wage curves for the Gamma and Delta techniques are not straight lines. All four wage-curves intersect at a single point, (<I>r</I>, <I>w</I>) = (20%, 1/5). (The wage curves for the Gamma and Delta techniques have the same intersection with the axis for the rate of profits.) </P><B>3.0 Choice of Technique</B><P>The cost-minimizing techniques form the outer envelope of the wage curves. For a given wage, the cost minimizing technique is the technique with the highest wage curve in Figure 1. A switch point is a point on the outer envelope at which more than one technique is cost-minimizing. All four wage curves intersect, in the figure, at the single switch point. </P><P>The Beta technique is cost-minimizing for wages to the left of the single switch point. The Alpha technique is cost-minimizing for all feasible wages greater than the wage at the switch point. Managers of firms replace both processes in the Alpha technique at the switch point with both processes in the Beta technique. </P><P>This is no problem for the first model above. The adoption of a new process for producing corn requires, if the economy is capable of self-replacement before and after the switch, that the process for producing iron or tin be replaced by the process for producing the other. </P><P>But consider the other model. For all processes in the Alpha technique to be replaced at a switch point, the wage curves for all techniques composed of all combinations of processes in the Alpha and Beta techniques. In other words, in the second model, wage curves for all four techniques must intersect at the switch point. The example in the second model is a fluke. </P><P>I have <A HREF="http://robertvienneau.blogspot.com/2016/12/tangency-of-wage-rate-of-profits-curves.html">previously</A>explained what makes a result a fluke, in the context of the analysis of the choice of technique. Qualitative properties, for generic results, continue to persist for some small variation in model parameters. </P><P>Consider a model with a discrete number of switch points. Consider the cost-minimizing techniques on both sides of a switch point. And suppose that same commodities are produced in both techniques, albeit in different proportions. Generically, only one process is replaced at such a switch point. All processes, except for that one, are common in both techniques. </P><B>5.0 A Generalization to An Uncountably Infinite Number of Processes in Each Industry</B><P>Consider a model with more than one industry, but a finite number. Suppose each industry has available an uncountably infinite number of processes. And, in each industry, the processes available for that industry can be described by a continuously differentiable production function. <A HREF="http://robertvienneau.blogspot.com/2013/04/choice-of-technique-two-good-model-cobb.html">Here</A>I present a two-commodity example with Cobb-Douglas production functions. </P><P>There are no switch points in such a model. The cost-minimizing technique varies continuously along the outer-envelope of wage curves. In fact, the processes in each industry, in the cost-minimizing technique varies continuously. Since there are no switch points at all, there is not a single switch point in which more than one process varies, as a fluke, with the cost-minimizing technique. </P><P>Nevertheless, cannot one see such "smooth" production functions as a limiting case? If so, it would be a generalization or extension of a discrete model, in which all switch points are flukes, to a continuum. From the perspective of the analysis of the choice of technique in discrete models, typical neoclassical models are nothing but flukes. </P><B>6.0 Conclusions</B><P>I actually found my negative conclusion surprising. I have tried to be conscious of the distinction between the structure of the two models in Section 2 above. I think at least some <A HREF="http://robertvienneau.blogspot.com/2009/12/negative-price-wicksell-effect-positive.html">examples</A> I have presented cannot be attacked by the above critique. They are examples of the first, not the second model. I tend to read Samuelson (1962) in the same way, as not sensitive to the critique in this post. </P><B>References</B><UL><LI>Neri Salvadori and Ian Steedman (1988). No reswitching? No switching! <I>Cambridge Journal of Economics</I>, V. 12: pp. 481-486.</LI><LI>Samuelson, P. A. (1962). Parable and Realism in Capital Theory: The Surrogate Production Function, <I></I> V. 29, No. 3: pp. 193-206.</LI><LI>J. E. Woods 1990. <I>The Production of Commodities: An Introduction to Sraffa</I>, Humanities Press International.</LI></UL>http://robertvienneau.blogspot.com/2017/03/a-fluke-switch-point.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-712117921888091877Sat, 04 Mar 2017 13:59:00 +00002017-03-04T08:59:24.735-05:00Example in Mathematical EconomicsMultiple Interest RatesBifurcations Of Roots Of A Characteristic Equation<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-YgvQthZwB2M/WG-UAh1jZTI/AAAAAAAAAxk/J2mxoKQ8ufAre9ufa67CiANtatNgvGhWQCLcB/s1600/MIRABeta.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-YgvQthZwB2M/WG-UAh1jZTI/AAAAAAAAAxk/J2mxoKQ8ufAre9ufa67CiANtatNgvGhWQCLcB/s320/MIRABeta.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: Rates of Profits for Beta Technique</b></td></tr></tbody></table><P>I have previously considered all roots of a polynomial equation for the rate of profits in a model, of the choice of technique, in which each technique is specified by a finite series of dated labor inputs. One root is the traditional rate of profits, but there are uses for the other roots: <UL><LI>All roots appear in an equation defining the Net Present Value (NPV) for the technique, given the wage and the rate of profits.</LI><LI>All roots can be combined in an accounting identity for the difference between labor commanded and labor embodied, given the wage.</LI></UL>I thought it of interest to know whether these non-traditional roots are real or complex, as they <A HREF="http://robertvienneau.blogspot.com/2016/12/bifurcations-in-multiple-interest-rate.html">vary</A> with the wage. I am considering multiple roots in an attempt to build on and <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058">critique</A> Michael Osborne's approach to multiple interest rate analysis. </P><P>I also have considered examples of models of the production of commodities by means of commodities, in which at least one commodity is basic, in the sense of Sraffa. And I have attempted to <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2885821">apply</A> or <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882531">extend</A> my critique of multiple interest rate analysis to these models. The point of this post is to illustrate possibilities on the complex plane for multiple interest rates in these models. </P><P>A technique in models of the production of commodities by means of commodities, as least in the case when all capital is circulating capital, is specified as a vector of labor coefficients and a Leontief input-output matrix. In parallel with my approach to techniques specified by a finite sequence of dated labor inputs, consider wages as being advanced - that is, not paid at the end of the year out of the surplus - in such models. Given the wage and the numeraire, one can construct a square matrix in which each coefficient is the sum of the corresponding coefficient in the Leontief input-output matrix and the quantity of the commodity produced by that industry that is advanced to the workers, per unit output produced. I call this matrix the augmented input-output matrix. </P><P>A polynomial equation, called the <I>characteristic equation</I>, is solved to find eigenvalues of the augmented input-output matrix. The power of this polynomial is equal to the number of commodities produced by the technique. The number of roots for the polynomial is therefore equal to the number of commodities. A rate of profits corresponds to each root. Assume the Leontief input-output matrix is a <I>Sraffa matrix</I> and that the wage does not exceed a certain maximum. Under these conditions, the Perron-Frobenius theorem picks out the maximum eigenvalue of the augmented input-output matrix. The corresponding rate of profits is non-negative, and the prices of production of these commodities are positive at the given wage. I was not able to find an application for the other, non-traditional rates of profits. </P><P>I present a numerical example in this <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2885821">working paper</A>. This is a three-commodity example with two techniques. Figure 1 graphs the three roots, at different level of wages, for the Beta technique in that example. </P><P>In a previous <A HREF="http://robertvienneau.blogspot.com/2016/12/example-of-choice-of-technique.html">blog post</A>, I extend that example such that managers of firms have a choice of process for producing each of the three commodities. As a consequence, a choice among eight techniques arises. And one can draw a graph like Figure 1 for each technique in that example. Figure 2 shows the corresponding graph for the Delta technique. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-4LHXKjQF-B0/WG-UIPEfP5I/AAAAAAAAAxo/XPWRL6l8N20bTmY-pFSIwfu2PlVlt5DOACLcB/s1600/MIRADelta.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-4LHXKjQF-B0/WG-UIPEfP5I/AAAAAAAAAxo/XPWRL6l8N20bTmY-pFSIwfu2PlVlt5DOACLcB/s320/MIRADelta.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 2: Rates of Profits for Delta Technique</b></td></tr></tbody></table><P>In Figures 1 and 2, the rate of profits picked out by the Perron-Frobenius theorem and used to draw the wage-rate of profits curve for the technique lies along the line segment on the real axis on the left in the figure. A lower wage corresponds to a higher traditional rate of profits. Thus, points further to the right on this line segment correspond to a lower wage. A wage of zero leads to the right-most point on this line segment. The highest feasible wage corresponds to left-most point, at a rate of profits of zero, on this segment. </P><P>Two non-traditional rates of profits arise for the other two solutions of the characteristic equation. They are plotted to the right on the graphs in Figures 1 and 2. When complex, they are complex conjugates. I thought it of interest that, in Figure 2, they are purely real for two non-overlapping, distinct ranges of feasible levels of the wage. </P><P>I draw no practical, applied implications from the non-traditional rates of profits. I just think the graphs are curious. </P>http://robertvienneau.blogspot.com/2017/03/bifurcations-of-roots-of-characteristic.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-234965065130358101Thu, 02 Mar 2017 15:15:00 +00002017-03-02T10:15:00.162-05:00Profile of an EconomistSome Obituaries for Kenneth ArrowBill Black has one <A HREF="http://www.nakedcapitalism.com/2017/02/bill-black-kenneth-arrows-ignored-impossibility-theorem.html">here</A>, emphasizing Arrow's impossibility theorem. The blog, A Fine Theorem, has two of a planned four-part series. The <A HREF="https://afinetheorem.wordpress.com/2017/02/22/the-greatest-living-economist-has-passed-away-notes-on-kenneth-arrow-part-i/">first</A> is on the impossibility theorem. The <A HREF="https://afinetheorem.wordpress.com/2017/02/27/kenneth-arrow-part-ii-the-theory-of-general-equilibrium/">second</A> is about General Equilibrium. The two planned, I gather, are to be about learning-by-doing and health economics, respectively. <P>I have written several posts drawing on Arrow's work. This <A HREF="http://robertvienneau.blogspot.com/2014/06/a-sophisticated-neoclassical-response.html">one</A>, on a sophisticated neoclassical response to the Cambridge Capital Controversy, is among my most popular posts. </P>http://robertvienneau.blogspot.com/2017/03/some-obituaries-for-kenneth-arrow.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-2555590641739559666Thu, 09 Feb 2017 13:17:00 +00002017-03-14T08:51:52.683-04:00Example in Mathematical EconomicsFull Cost PricesMultiple Interest RatesSraffa EffectsA Reswitching Example in a Model of Oligopoly<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-f9VYOjtgjTg/WJsd3j8gTQI/AAAAAAAAAyE/s4XRFIPFgUgMwRYXmAHcBUyIxwFrY50DwCLcB/s1600/OligopolyRoots.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-f9VYOjtgjTg/WJsd3j8gTQI/AAAAAAAAAyE/s4XRFIPFgUgMwRYXmAHcBUyIxwFrY50DwCLcB/s320/OligopolyRoots.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>The Roots of a Cubic Polynomial Defining Switch Points</b></td></tr></tbody></table><P>I have a <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2912181">draft paper</A> up at SSRN. The abstract: </P><BLOCKQUOTE>This paper illustrates, through a numerical example of reswitching under oligopoly, the existence of implications from the Cambridge Capital Controversy for the theory of industrial organization. Oligopoly is modeled by given and persistent ratios in rates of profits among industries, as expressed in a system of equations for prices of production. The numerical example illustrates that this model of oligopoly is a pertubation of free competition. Some comparisons and contrasts are drawn to a model of free competition. </BLOCKQUOTE><P>In some sense, this paper shows a somewhat more comprehensive description of value through exogenous distribution than in Sraffa's book. The model can depict capitalists as squabbling over the division of the surplus that their class gets, as well as their struggle against the workers. I'd like to see an example of reswitching or capital reversing in this model, with all (price and real) Wicksell effects as negative in the example in the special case of free competition. I do not see why one cannot arise. Such an example would suggest that "perverse" examples can obtain empirically, even if they are not found in an analysis that presumes one common rate of profits among all industries. </P><P>The graph at the top of this post does not appear in the paper. In the model, the ratios of rates of profits among industries are given parameters. A cubic polynomial is defined for a given set of such ratios. Non-negative, real zeroes of that polynomial below a certain maximum define a scale factor for switch points. The location of the zeros varies with the ratios. I happen to be able to solve for the zeros. They are shown in the graph above. </P>http://robertvienneau.blogspot.com/2017/02/a-reswitching-example-in-model-of.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-1394742289641934939Mon, 23 Jan 2017 13:01:00 +00002017-01-23T08:01:24.047-05:00Festschrifts for Sraffians<P>We are on, maybe, the third generations of Sraffians. An annoyance and a delight of publicly taking up a topic is that one must continually read advances, whether large or small, in your topic. I've read some festschrifts over the last several decades: </P><UL><LI><I>Competing Economic Theories: Essays in memory of Giovanni Carvale</I>, edited by Serio Nisticó and Domenico Tosato.</LI><LI><I>Value, Distribution and Capital: Essays in honour of Pierangelo Garegnani</I>, edited by Gary Mongiovi and Fabio Petri.</LI><LI><I>Economic Theory and Economic Thought: Essays in Honour of Ian Steedman</I>, edited by John Vint, J. Stanley Metcalfe, Heinz D. Kurz, Neri Salvadori, and Paul A. Samuelson.</LI></UL><P>I'm aware of some I have not read: </P><UL><LI><I>Social Fairness and Economics: Essays in the spirit of Duncan Foley</I>, edited by Lance Taylor, Armon Rezai, and Thomas Michl.</LI><LI><I>Keynes, Sraffa and the Criticism of Neoclassical Theory: Essays in honour of Heinz Kurz</I>, edited by Neri Salvadori and Christian Gehrke.</LI><LI><I>Classical Political Economy and Modern Theory: Essays in honour of Heinz Kurz</I>, edited by Christian Gehrke, Neri Salvadori, Ian Steedman and Richard Sturn.</LI><LI><I>Economic Theory and its History</I>, edited by Guiseppe Freni, Heinz D. Kurz, Andrea Mario Lavezzi, and Rudolfo Signorino. (This apparently is a celebration of Neri Salvadori's work.)</LI><LI><I>Production, Distribution and Trade: Essays in honour of Sergio Parrinello</I>, edited by Adriano Birolo, Duncan K. Foley, Heinz D. Kurz, Bertram Schefold and Ian Steedman</LI><LI><I>The Evolution of Economic Theory: Essays in honour of Bertram Schefold</I>, edited by Volker Caspari.</LI></UL><P>This post provides another demonstration that at least one school of heterodox economists looks, from the outside, like any other group of academics with common research interests. I have posted about <A HREF="http://robertvienneau.blogspot.com/2016/03/post-keynesianism-from-outside.html">Post Keynesianism</A> in this respect. Likewise, I once listed <A HREF="http://robertvienneau.blogspot.com/2006/08/textbooks-for-teaching-non.html">textbooks</A>. </P>http://robertvienneau.blogspot.com/2017/01/festschrifts-for-sraffians.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-1187812412425504798Mon, 16 Jan 2017 15:22:00 +00002017-01-16T10:22:56.914-05:00Example in Mathematical EconomicsSraffa EffectsA Story Of Technical Innovation<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-E9lkHMoshxo/WF0eNZSrbWI/AAAAAAAAAvI/jh1QrfH8GxoMgnIdA-mSBZy-uyYu6886wCLcB/s1600/Fujomoto1983.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-E9lkHMoshxo/WF0eNZSrbWI/AAAAAAAAAvI/jh1QrfH8GxoMgnIdA-mSBZy-uyYu6886wCLcB/s320/Fujomoto1983.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: The Choice of Technique in a Model with Four Techniques</b></td></tr></tbody></table><B>1.0 Introduction</B><P>I often present examples of the choice of technique as an internal critique of neoclassical economics. The example in this post, however, is closer to how I think techniques evolve in actually existing capitalist economies. Managers of firms know a limited number of processes in each industry, sometimes only the one in use. Accounting techniques specify a set of prices. An innovation provides a new process in a given industry. The first firm to adopt that process may obtain supernormal profits, whatever the wage or normal rate of profits. Other firms will strive to move into the industry with supernormal profits and to use the new process. Prices associated with the new technique result in the wage-rate of profits frontier being moved outward, perhaps along its full extent. </P><P>This example was introduced by Fujimoto (1983). I know it most recently from problem 22 in Woods (1990: p. 126). It is also a problem in Kurz and Salvadori (1995). Fujimoto probably labels it a curiosum because of details more specific than the above overview of how Sraffians might treat technical change. </P><B>2.0 Technology</B><P>This example is a two-commodity model, in which both commodities, called <I>iron</I> and <I>corn</I>, are basic. Suppose iron is used exclusively as a capital good, and corn is used for both consumption and as a capital good. Consider the processes shown in Table 1. Each process exhibits Constant Returns to Scale. The coefficients in each column show required inputs, per unit output, in each industry for each process. Each process requires a year to complete, and outputs become available at the end of the year. This is a circulating capital model. All commodity inputs are totally used up in the year by providing their services during the course of the year. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">(a)</TD><TD ALIGN="center">(b)</TD><TD ALIGN="center">(c)</TD><TD ALIGN="center">(d)</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">3/5</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">2/5</TD><TD ALIGN="center">1/5</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">2/5</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>For this economy to be reproduced, both iron and corn must be (re)produced. A technique consists of an iron-producing and a corn-producing process. Table 2 lists the four techniques that can be formed from the processes listed in Table 1. In this example, not all processes or techniques are known at the start of the dynamic process under consideration. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 2: Techniques</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">a, d</TD></TR><TR><TD ALIGN="center">Gamma</TD><TD ALIGN="center">b, c</TD></TR><TR><TD ALIGN="center">Delta</TD><TD ALIGN="center">b, d</TD></TR></TABLE><P></P><B>3.0 Price Systems</B><P>A system of prices of production characterize smooth reproduction with a given technique. Suppose a unit of corn is the numeraire. Let <I>w</I> be the wage, <I>r</I> be the (normal) rate of profits, and <I>p</I> be the price of a unit of iron. Suppose labor is advanced, and wages are paid out of the surplus. If the Alpha technique is in use, prices of production satisfy the following system of two equations in three unknowns: </P><BLOCKQUOTE>(2/5)(1 + <I>r</I>) + (1/2) <I>w</I> = <I>p</I></BLOCKQUOTE><BLOCKQUOTE>(2/5) <I>p</I> (1 + <I>r</I>) + (1/2) <I>w</I> = 1 </BLOCKQUOTE><P>A non-negative price of iron and wage can be found for all rates of profit between zero and a maximum associated with the technique. Figure 1 illustrates one way of depicting this single degree of freedom, for each technique. </P><B>4.0 Innovations</B><P>I use the above model to tell a story of technological progress. Suppose at the start, managers of firms only know one process for producing iron and one process for producing corn. Let these be the processes comprising the Alpha technique. In this story, the rate of profits is exogenous, at a level below the rate of profits associated with the switch point between the Gamma and Delta technique, not that that switch point is relevant at the start of this story. </P><P>Somehow or other, prices of production provide a reference for market prices. For such prices, the economy is on the wage-rate of profits curve for the Alpha technique in Figure 1. This curve is closest to the origin in the figure. </P><P>Suppose researchers in the corn industry discover a new process for producing corn, namely process (d). A choice of technique arises. Corn producers see that they can earn extra profits by adopting this technique at Alpha prices. The Beta technique becomes dominant. Eventually, the extra profits are competed away, and the economy lies on the wage-rate of profits curve for the Beta technique. Under the assumption of an externally specified rate of profits, the wage has increased. </P><P>Next, an innovation occurs in the iron industry. Firms discover process (b). At Beta prices, it pays for iron-producing firms to adopt this new process. The wage-rate of profits curve for the Delta technique lies outside the wage-rate of profits curve for the Beta technique. Thus, the Delta technique dominates the Beta technique. But prices of production associated with the Delta technique cannot rule. If the Delta technique were prevailing, corn-producing firms would find they can earn extra profits by discarding process (d) and reverting to process (c). The Gamma technique is dominant at the given rate of profits, and workers will end up earning a still higher wage. </P><P>I guess this story does not apply to the United States these days. In the struggle over the increased surplus provided by technological innovation, workers do not seem to be gaining much. At any rate, Table 3 summarizes the temporal sequence of the dominant technique in this story. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 3: A Temporal Series of Innovations</B></CAPTION><TR><TD ALIGN="center"><B>Events</B></TD><TD ALIGN="center"><B>Dominant<BR>Technique</B></TD><TD ALIGN="center"><B>Processes<BR>in Use</B></TD></TR><TR><TD ALIGN="center">Processes (a) and (c) known</TD><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c</TD></TR><TR><TD ALIGN="center">Processes (d) introduced</TD><TD ALIGN="center">Beta</TD><TD ALIGN="center">a, d</TD></TR><TR><TD ALIGN="center">Process (b) introduced</TD><TD ALIGN="center">Gamma</TD><TD ALIGN="center">b, c</TD></TR></TABLE><P>I do not see why one could not create an example with a single switch point between the Gamma and Delta techniques, where that switch point is at a wage below the maximum wage for the Alpha technique. For such a postulated example, one could tell story, like the above, with a given wage. The capitalists would end up with all the benefits from technological progress. </P><B>5.0 Conclusion</B><P>This example illustrates that innovation in one industry (that is, the production of iron) can result in the managers of firms in another industry (corn-production) discarding a previously introduced innovation and reverting to an old process of production. </P><B>References</B><UL><LI>T. Fujimoto 1983. Inventions and Technical Change: A Curiosum, <I>Manchester School</I>, V. 51: pp. 16-20.</LI><LI>Heinz D. Kurz and Neri Salvadori 1995. <I>Theory of Production: A Long Period Analysis</I>, Cambridge University Press.</LI><LI>J. E. Woods 1990. <I>The Production of Commodities: An Introduction to Sraffa</I>, Humanities Press International.</LI></UL>http://robertvienneau.blogspot.com/2017/01/a-story-of-technical-innovation.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-6808891786080858040Sat, 14 Jan 2017 17:20:00 +00002017-03-15T07:33:07.663-04:00Example in Mathematical EconomicsFull Cost PricesSraffa EffectsA Model Of Oligopoly<B>1.0 Introduction</B><P>Suppose barriers to entry exist in an economy. Entrepreneurs and capitalists find that they cannot freely enter or exit some industries. And these barriers are manifested by stable ratios of rates of profits among industries. This post presents equations for prices of production under these assumptions. </P><P>I suggest that the model presented here fits into the tradition of Old Industrial Organization, as formulated by Joe Bain and Paolo Sylos Labini. As I understand it, Sylos Labini may have once written down equations like these, but never presented them or published them. I suppose this model is also related to work Piero Sraffa <A HREF="http://robertvienneau.blogspot.com/2014/01/impact-of-piero-sraffa-on-industrial.html">published</A> in the 1920s. </P><B>2.0 The Model</B><P>Consider an economy consisting of <I>n</I> industries. Suppose the rate of profits in the <I>j</I>th industry is (<I>s</I><SUB><I>j</I></SUB> <I>r</I>), where <I>r</I> is the base rate of profits, <I>s</I><SUB><I>j</I></SUB> is positive, and: </P><BLOCKQUOTE><I>s</I><SUB>1</SUB> + <I>s</I><SUB>2</SUB> + ... + <I>s</I><SUB><I>n</I></SUB> = 1 </BLOCKQUOTE><P>For simplicity, I limit my attention to a circulating capital model of the production of commodities by means of commodities. For the technique in use, let <I>a</I><SUB><I>i</I>, <I>j</I></SUB> be the quantity of the <I>i</I>th commodity used to produce a unit of output in the <I>j</I>th industry. Homogeneous labor is the only unproduced input in each industry. Let <I>a</I><SUB>0, <I>j</I></SUB> be the person years of labor used to produce a unit output in the <I>j</I>th industry. I assume labor is advanced, and wages are paid out of the surplus at the end of production period, say, a year. Then prices of production, which ensure a smooth reproduction of the economy, satisfy the following system of equations: </P><BLOCKQUOTE>(<I>a</I><SUB>1, 1</SUB> <I>p</I><SUB>1</SUB> + <I>a</I><SUB>2, 1</SUB> <I>p</I><SUB>2</SUB> + ... + <I>a</I><SUB><I>n</I>, 1</SUB> <I>p</I><SUB><I>n</I></SUB>)(1 + <I>s</I><SUB>1</SUB> <I>r</I>) + <I>w</I> <I>a</I><SUB>0, 1</SUB> = <I>p</I><SUB>1</SUB></BLOCKQUOTE><BLOCKQUOTE>(<I>a</I><SUB>1, 2</SUB> <I>p</I><SUB>1</SUB> + <I>a</I><SUB>2, 2</SUB> <I>p</I><SUB>2</SUB> + ... + <I>a</I><SUB><I>n</I>, 2</SUB> <I>p</I><SUB><I>n</I></SUB>)(1 + <I>s</I><SUB>2</SUB> <I>r</I>) + <I>w</I> <I>a</I><SUB>0, 2</SUB> = <I>p</I><SUB>2</SUB></BLOCKQUOTE><BLOCKQUOTE> . . . </BLOCKQUOTE><BLOCKQUOTE>(<I>a</I><SUB>1, <I>n</I></SUB> <I>p</I><SUB>1</SUB> + <I>a</I><SUB>2, <I>n</I></SUB> <I>p</I><SUB>2</SUB> + ... + <I>a</I><SUB><I>n</I>, <I>n</I></SUB> <I>p</I><SUB><I>n</I></SUB>)(1 + <I>s</I><SUB><I>n</I></SUB> <I>r</I>) + <I>w</I> <I>a</I><SUB>0, <I>n</I></SUB> = <I>p</I><SUB><I>n</I></SUB></BLOCKQUOTE><P>The coefficients of production, including labor coefficients, and the ratios of the rate of profits are given parameters in the above system of equations. The unknowns are the prices, the wage, and the base rate of profits. Since only relative prices matter in this model, one degree of freedom is eliminated by choosing a numeraire: </P><BLOCKQUOTE><I>p</I><SUB>1</SUB> <I>q</I><SUP>*</SUP><SUB>1</SUB> + ... + <I>p</I><SUB><I>n</I></SUB> <I>q</I><SUP>*</SUP><SUB><I>n</I></SUB> = 1 </BLOCKQUOTE><P>Since there are <I>n</I> price equations, appending the above equation for the specified numeraire yields a model with (<I>n</I> + 1) equations and (<I>n</I> + 2) unknowns. One degree of freedom remains. </P><B>3.0 In Matrix Form</B><P>The above model can be expressed more concisely in matrix form. Define: </P><UL><LI><B>I</B> is the identity matrix.</LI><LI><B>e</B> is a column vector in which each element is 1.</LI><LI><B>S</B> is a diagonal matrix, with <I>s</I><SUB>1</SUB>, <I>s</I><SUB>2</SUB>, ..., <I>s</I><SUB><I>n</I></SUB> along the principal diagonal.</LI><LI><B>p</B> is a row vector of prices.</LI><LI><B>q<SUP>*</SUP></B> is the column vector representing the numeraire.</LI><LI><B>A</B> is the Leontief input-output matrix, representing the technique in use.</LI><LI><B>a<SUB>0</SUB></B> is the row vector of labor coefficients for the technique.</LI></UL><P>The model consists of the following equations: </P><BLOCKQUOTE><B>e</B><SUP>T</SUP> <B>S</B> <B>e</B> = 1 </BLOCKQUOTE><BLOCKQUOTE><B>p</B> <B>A</B> (<B>I</B> + <I>r</I> <B>S</B>) + <I>w</I> <B>a<SUB>0</SUB></B> = <B>p</B></BLOCKQUOTE><BLOCKQUOTE><B>p</B> <B>q<SUP>*</SUP></B> = 1 </BLOCKQUOTE><P></P><B>4.0 Conclusion</B><P>One could develop the above model in various directions. For example, one could plot the wage-base rate of profits curve for the technique in use. Of interest to me would be presenting examples of the choice of technique, including reswitching and capital-reversing. The Sraffian critique of neoclassical economics is not confined to the theory of perfect competition. </P><B>Update (16 January 2017):</B> I find I have outlined this model <A HREF="http://robertvienneau.blogspot.com/2013/05/kalecki-and-sraffa-compatible.html">before</A>.http://robertvienneau.blogspot.com/2017/01/a-model-of-oligopoly.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-3672563238986995335Sun, 01 Jan 2017 15:48:00 +00002017-01-03T06:53:34.842-05:00Austrian School Of EconomicsExample in Mathematical EconomicsJoint ProductionSraffa EffectsReswitching In An Example Of A One-Commodity Model<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-FYPuIpYrrxg/WGZQ8H5yMTI/AAAAAAAAAwM/R7m-tuLE1tgrZh5Vw-J4Ba-9sAGOx6AdACLcB/s1600/OneGoodFrontier.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-FYPuIpYrrxg/WGZQ8H5yMTI/AAAAAAAAAwM/R7m-tuLE1tgrZh5Vw-J4Ba-9sAGOx6AdACLcB/s320/OneGoodFrontier.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: The Choice of Technique in a Model with One Commodity</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post presents a reswitching example in a one-good model. The single produced commodity in the example can be used as both a consumption and a capital good. It is produced by expenditure of labor with its services. It lasts for three production periods, and its technical efficiency varies over the course of its lifetime, when used as a capital good. </P><P>I do not remember any comparable numeric example in the literature. If I recall Ian Steedman's 1994 article, he gives instructions for constructing a one-good example, but does not present one. Maybe if I reread it now, I will find it clearer. I have previously worked through a <A HREF="http://robertvienneau.blogspot.com/2008/04/reswitching-example-with-fixed-capital.html">reswitching example</A>, with fixed capital, from J. E. Woods. But that is a multi-commodity model. I have also once <A HREF="http://robertvienneau.blogspot.com/2007/03/depreciation-of-one-hoss-shay.html">echoed</A> Sraffa's analysis of depreciation charges, in a case with constant efficiency. </P><B>2.0 Technology</B><P>This is an example of fixed capital, a kind of joint production. Three production processes are known by the manager of firms, and they each exhibit constant returns to scale. Each process requires a year to complete, and each process produces new widgets. Table 1 shows the inputs for each process, when operated at a unit level, and Table 2 shows the outputs. For example, process I requires inputs of labor and new widgets. The outputs of process I consist of new widgets and the widgets which provided their services throughout the year it is under operation. Those leftover widgets are one year older, though. Consumers consume new widgets during the year following on their purchase. The physical life of widgets, when providing services for production, is three years. Thus, three processes can be operated in production. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Inputs for The Technology</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">(I)</TD><TD ALIGN="center">(II)</TD><TD ALIGN="center">(III)</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">10</TD><TD ALIGN="center">60</TD><TD ALIGN="center">13/2</TD></TR><TR><TD ALIGN="center">New Widgets</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">One-Year Old Widgets</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Two-Year Old Widgets</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/3</TD></TR></TABLE><P></P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Outputs for The Technology</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Output</B></TD><TD ALIGN="center" COLSPAN="3"><B>Process</B></TD></TR><TR><TD ALIGN="center">(I)</TD><TD ALIGN="center">(II)</TD><TD ALIGN="center">(III)</TD></TR><TR><TD ALIGN="center">New Widgets</TD><TD ALIGN="center">1</TD><TD ALIGN="center">7/12</TD><TD ALIGN="center">79/20</TD></TR><TR><TD ALIGN="center">One-Year Old Widgets</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Two-Year Old Widgets</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">0</TD></TR></TABLE><P>Firms are not required to operate all three processes. They can truncate the use of widgets after one or two years. Assume free disposal, that is, that discarding widgets does not incur a cost. Under these assumptions, three techniques are available to produce new widgets, as shown in Table 3. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 3: Techniques</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">I</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">I, II</TD></TR><TR><TD ALIGN="center">Gamma</TD><TD ALIGN="center">I, II, II</TD></TR></TABLE><P></P><B>3.0 Price Equations</B><P>Managers of widget-producing firms choose the technique, that is, the truncation period, on the basis of cost. As usual, consider a competitive economy, in which workers and firms are free to seek out higher wages and profits, respectively. Revenues and costs are calculated on the basis of a set of prices in which workers and firms have no incentive to move out of one process and into another. Workers receive a common wage of <I>w</I> new widgets per person-year. Assume workers are paid out of the surplus at the end of each year. Firms receive a rate of profits of 100 <I>r</I> percent in operating each process in use. For notational convenience, define <I>R</I>: </P><BLOCKQUOTE><I>R</I> = 1 + <I>r</I></BLOCKQUOTE><P>Let <I>p</I><SUB>1</SUB> be the price of a one-year old widget and <I>p</I><SUB>2</SUB> the price of a two-year old widget. </P><P>I confine the systems of price equations for the Alpha and Beta techniques to an appendix. Accordingly, assume the Gamma technique is in use. The wage, prices, and the rate of profits must satisfy a system of three equations: </P><BLOCKQUOTE>(1/3)<I>R</I> + 10 <I>w</I> = 1 + (1/3) <I>p</I><SUB>1</SUB></BLOCKQUOTE><BLOCKQUOTE>(1/3) <I>p</I><SUB>1</SUB> <I>R</I> + 60 <I>w</I> = 7/12 + (1/3) <I>p</I><SUB>2</SUB></BLOCKQUOTE><BLOCKQUOTE>(1/3) <I>p</I><SUB>2</SUB> <I>R</I> + (13/2) <I>w</I> = 79/20 </BLOCKQUOTE><P>Given the rate of profits below some maximum, one can solve for the wage: </P><BLOCKQUOTE><I>w</I><SUB>γ</SUB> = (60 <I>R</I><SUP>2</SUP> + 35 <I>R</I> + 237 - 20 <I>R</I><SUP>3</SUP>)/(10 (60 <I>R</I><SUP>2</SUP> + 360 <I>R</I> + 39)) </BLOCKQUOTE><P>The price of two-year old and one-year old widgets fall out: </P><BLOCKQUOTE><I>p</I><SUB>2, γ</SUB> = (237 - 390 <I>w</I><SUB>γ</SUB>)/(20 <I>R</I>) </BLOCKQUOTE><BLOCKQUOTE><I>p</I><SUB>1, γ</SUB> = (7 + 4 <I>p</I><SUB>2, γ</SUB> - 720 <I>w</I><SUB>γ</SUB>)/(4 <I>R</I>) </BLOCKQUOTE><P>If you feel like it, you can substitute on the Right Hand Sides of the above two equations so as to express prices as functions exclusively of the rate of profits. </P><B>4.0 Choice of Technique</B><P>I have explained above how to find the wage, as a function of the rate of profits, when the Gamma technique is in use. The wage-rate of profits curves for the Alpha and Beta technique are, respectively: </P><BLOCKQUOTE><I>w</I><SUB>α</SUB> = (3 - <I>R</I>)/30 </BLOCKQUOTE><BLOCKQUOTE><I>w</I><SUB>β</SUB> = (12 <I>R</I> + 7 - 4 <I>R</I><SUP>2</SUP>)/(120 (<I>R</I> + 6)) </BLOCKQUOTE><P>Figure 1 graphs all three wage curves. The cost-minimizing technique, at any given rate of profits, is the technique on the outer envelope of the wage curves. The switch points between the Alpha and Gamma techniques are at rates of profits of 10% and 50%. Below 10% and above 50%, the Gamma technique is cost-minimizing. Widgets are used in production processes to the extent of their physical life. Between these rate of profits, the Alpha technique is cost minimizing. The use of widgets, as capital goods, is truncated after one year. For what it is worth, the switch point between the Alpha and Beta techniques, within the outer envelope, is at a rate of profits of 41/24, approximately 171%. </P><B>4.1 A Direct Method with Alpha Prices</B><P>In the general theory of joint production, an analysis of the choice of technique cannot generally be based on wage-rate of profits curves. Such an analysis does work in this model of fixed capital. But I checked it with a more direct method of analysis. </P><P>Suppose the Alpha technique is in use. The prices of one-year old and two-year old widgets are zero. The wage is as found from the system of price equations associated with the Alpha technique. Would it pay to produce new widgets with one-year old or two-year old widgets? Figure 2 shows calculations to determine if supernormal profits can be earned with the Beta or Gamma technique. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-KskAeYyZIVM/WGZlKs94AKI/AAAAAAAAAwc/NX4FzPjv_vEcwC1An-SGHKzJje9Pm9y5wCLcB/s1600/OneGoodAlphaPrices.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-KskAeYyZIVM/WGZlKs94AKI/AAAAAAAAAwc/NX4FzPjv_vEcwC1An-SGHKzJje9Pm9y5wCLcB/s320/OneGoodAlphaPrices.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 2: Supernormal Profits at Alpha Prices</b></td></tr></tbody></table><P>Since the Alpha technique is in use, its net present value is zero. Extra profits are assumed to have been competed away. </P><P>If the Beta technique is operated, no extra profits or losses are earned in operating process I under the Beta technique. In operating process II, the services of old widgets are free, and no revenues are received for the two-year old widgets disposed of at the end of the year. At low rates of profits and high wages, the revenues received for new widgets produced with process II do not cover labor costs. At high rates of profits, the opposite is the case. These prices, when wages are low, signal to firms that they can earn extra profits by extending the truncation period one year. </P><P>The analysis for the Gamma technique is more cumbersome. Firms can not adopt process III without also operating process II. Accordingly, the net present value for the Gamma technique is found by accumulating all costs and revenues for all three processes to the end of the third year. In such a weighted sum, the revenues for process I are multiplied by (1 + <I>r</I>)<SUP>2</SUP>, and the revenues for process II are multiplied by (1 + <I>r</I>). At a rate of profits below 10% and above 50%, firms will want to adopt the Gamma technique and produce with old widgets to the end of their physical life. </P><B>4.2 A Direct Method with Beta Prices</B><P>I find of interest some complications that arise in applying this direct method with wages and prices, as calculated for the Beta price system. Figure 3 shows the net present value for the processes comprising each of the three techniques. At all rates of profits, these prices signal that firms should extend the truncation period, from two years, to the three years specified by the Gamma technique. This is so, even for rates of profits between 10% and 50%. If firms start at a two-year truncation period, they will only find that they need to truncate to one year after first extending production to three years. (See A.3 in the Appendix for graphs associated with the Gamma price system.) </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-ekcjn5Q8ft4/WGZnxueLxKI/AAAAAAAAAwo/fPEx-eVWTMokSWdsqwKgW5VmxuPIzN4sACLcB/s1600/OneGoodBetaPrices.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-ekcjn5Q8ft4/WGZnxueLxKI/AAAAAAAAAwo/fPEx-eVWTMokSWdsqwKgW5VmxuPIzN4sACLcB/s320/OneGoodBetaPrices.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 3: Supernormal Profits at Beta Prices</b></td></tr></tbody></table><P>When the Beta technique is operated, a price must be assigned to the price of a one-year old widget. In the theory of joint production, prices can be negative when calculated for rates of profits below the maximum. (This is not so for pure circular capital models without joint production.) Figure 4 graphs the price of one-year old widgets, under the system of prices associated with the Beta technique. This price is negative for rates of profits of approximately 171%. Confine your attention to the Alpha and Beta techniques for a second. The negative price of one-year old widgets signals firms that it is profitable to truncate production from two years to one year. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-iPxFeLOcMbk/WGe8PS-CAhI/AAAAAAAAAxI/4pCywcj188EUWTMOvVvg0L1AEynttI6PACLcB/s1600/OneGoodBetaPrices2.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-iPxFeLOcMbk/WGe8PS-CAhI/AAAAAAAAAxI/4pCywcj188EUWTMOvVvg0L1AEynttI6PACLcB/s320/OneGoodBetaPrices2.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 4: Price of One-Year Widget with Beta Technique</b></td></tr></tbody></table><P>The application of a direct method for comparing costs and revenues for techniques of production confirms, in the context of this model of fixed capital, the results of constructing the outer envelope curve from the wage-rate of profits curves for the techniques. </P><B>5.0 Conclusion</B><P>The literature on macroeconomics contains many models with aggregate production functions and in which the one produced commodity can be either consumed by households or used as a capital good in further production. And, in many of these models, this capital good depreciates over many periods. These models are one-good models, in the sense of this post. Some mainstream economists ignorantly assert that the Cambridge Capital Controversy was exclusively about problems in aggregating capital goods. Since mainstream economists are aware of aggregation issues, they somehow conclude they are justified in ignoring the controversy. I usually refute this rot by pointing out consequences for microeconomics of the analysis of the choice of technique. This post takes an alternative approach. It examines a highly aggregated model. And issues related to Sraffa effects arise in the one-good model, too. </P><P>This example also has a bearing on a misunderstanding common among the Austrian school of economics. Böhm Bawerk, at least, thought of production processes taking a longer amount of time as being more capital intensive and, therefore, more productive, in some sense. Firms are supposedly restricted in how willing they are to temporally extend production processes because of the scarcity of capital, as reflected in the interest rate. If households were less impatient and more willing to save, the interest rate would fall and firms would adopt longer processes. One can find many Austrian school economists (for example, Hayek in the 1930s) rejecting the idea that there exists a meaningful quantitative measure of roundaboutness or the period of production, whether independent of prices or not. But Austrian school economists generally retain a sense that the theory is insightful and somehow qualitatively true. The numeric example challenges this idea. One would think that a truncation of the production process, with the capital good not reaching its physical lifetime, is unambiguously less roundabout. As noted in Figure 1, for one switch point in the example, such a truncation can be associated with a lower interest rate. </P><B> Appendix A</B><P>I confine various mathematical details to this appendix. </P><B>A.1 Alpha Price Equations</B><P>If the Alpha technique is in use, the prices of one-year old and two-year old widgets is zero: </P><BLOCKQUOTE><I>p</I><SUB>1, α</SUB> = <I>p</I><SUB>2, α</SUB> = 0 </BLOCKQUOTE>The wage and the rate of profits are related by the coefficients of production for process I: <BLOCKQUOTE>(1/3)<I>R</I> + 10 <I>w</I> = 1 </BLOCKQUOTE><P>The wage, under the Alpha technique, can be expressed as a function of the rate of profits, as illustrated in Figure 1. </P><B>A.2 Beta Price Equations</B><P>If the Beta technique is in use, the price two-year old widgets is zero: </P><BLOCKQUOTE><I>p</I><SUB>2, β</SUB> = 0 </BLOCKQUOTE><P>A system of two equations arises for the Beta technique: </P><BLOCKQUOTE>(1/3)<I>R</I> + 10 <I>w</I> = 1 + (1/3) <I>p</I><SUB>1</SUB></BLOCKQUOTE><BLOCKQUOTE>(1/3) <I>p</I><SUB>1</SUB> <I>R</I> + 60 <I>w</I> = 7/12 </BLOCKQUOTE><P>The wage as a function of the rate of profits, for the Beta technique, is also illustrated above. The price of one-year old widgets is: </P><BLOCKQUOTE><I>p</I><SUB>1, β</SUB> = <I>R</I> + 30 <I>w</I><SUB>β</SUB> - 3 </BLOCKQUOTE><B>A.3 Direct Method with Gamma Prices</B><P>This section presents two graphs with wages and the rate of profits found from the system of prices for the Gamma technique. Figure 5 shows the net present value of truncating the use of widgets after one, two, or three years. For all techniques, revenues and costs are accumulated, at the going rate of profits to the end of the last year in which widgets are produced with the technique. The net present value for operating the Alpha technique (truncating after one year) is positive between the switch points at rates of profits of 10% and 50%. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-qx2zQqY_mP4/WGZptlgDPgI/AAAAAAAAAw4/YkAuEpT2RgM5RcrnM9mi71yEsWpC4dG7QCLcB/s1600/OneGoodGammaPrices.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-qx2zQqY_mP4/WGZptlgDPgI/AAAAAAAAAw4/YkAuEpT2RgM5RcrnM9mi71yEsWpC4dG7QCLcB/s320/OneGoodGammaPrices.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 5: Supernormal Profits at Gamma Prices</b></td></tr></tbody></table><P>Figure 6 shows the prices of one-year old and two-year old widgets for the solution to the Gamma price system. Although not very easy to see in the graph, the price of one-year old widgets is negative at rates of profits between 10% and 50%. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-ABEw0deHkCc/WGe8l5uJtqI/AAAAAAAAAxM/id75-R8juzgv7JVwNxK2scpvRn7YcsK6QCLcB/s1600/OneGoodGammaPrices2.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-ABEw0deHkCc/WGe8l5uJtqI/AAAAAAAAAxM/id75-R8juzgv7JVwNxK2scpvRn7YcsK6QCLcB/s320/OneGoodGammaPrices2.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 6: Price of Widgets with Gamma Technique</b></td></tr></tbody></table><P></P><B>References</B><UL><LI>Ian Steedman. 1994. 'Perverse' Behaviour in a 'One Commodity' Model. <I>Cambridge Journal of Economics</I>, V. 18, No. 3: pp. 299-311.</LI></UL>http://robertvienneau.blogspot.com/2017/01/reswitching-in-example-of-one-commodity.htmlnoreply@blogger.com (Robert Vienneau)6tag:blogger.com,1999:blog-26706564.post-6276993000011922917Fri, 30 Dec 2016 16:31:00 +00002016-12-30T11:31:34.983-05:00On Ajit Sinha On Sraffa<P>Over at the Institute for New Economic Theory (INET), Ajit Sinha <A HREF="https://www.ineteconomics.org/perspectives/blog/sraffas-revolution-in-economic-theory">discusses</A> the Sraffian revolution. Scott Carter <A HREF="http://sraffaarchive.org/2016/12/no-scholar-this-site-included-knows-unequivocally-what-sraffa-really-meant/">cautions</A> that, in interpreting Sraffa's thought, his archives have barely been touched. </P><P>Sinha's article has this blurb, with which I entirely agree: </P><BLOCKQUOTE>"The prominence of the debate over 'reswitching' has obscured the importance of Piero Sraffa's profound contribution to economics. It's time to revisit and build on that body of work." </BLOCKQUOTE><P>One can agree with the above without following Sinha very far. In analyzing the choice of technique, I <A HREF="http://robertvienneau.blogspot.com/search/label/Sraffa%20Effects">often</A> point out more than reswitching. I try to find effects in other markets than the capital markets and go in other directions. Since my motivation for working through these examples is frequently an internal criticism of neoclassical economics, I am frequently willing to assume Constant Returns to Scale and perfect competition, in the sense that firms take prices as given. One might argue that this misses Sraffa's point. Besides one can use 'reswitching' as a synecdoche for such analyses of the choice of technique. </P><P>How do I know that <I>Production of Commodities by Means of Commodities: Prelude to a Critique of Economic Theory</I> was about more than Sraffa effects, as seen in the analysis of the choice of technique? Only the last chapter in the book deals with the choice of technique. (Maybe, earlier chapters on joint production, rent, and fixed capital might have been clearer if they came after this chapter.) Sraffa doesn't present this one-chapter, final part of his book as a climax that all before is leading up to. In fact, he explicitly says, in the first paragraph, that the status of that chapter is somewhat different from the rest of the book: </P><BLOCKQUOTE>"Anyone accustomed to think in terms of the equilibrium of demand and supply may be inclined, on reading these pages, to suppose that the argument rests on a tacit assumption of constant returns in all industries. If such a supposition is found helpful, there is no harm in the reader's adopting it as a temporary working hypothesis. In fact, however, no such assumption is made. No changes in output and (at any rate in Parts I and II [Part III presents switches in methods of production - RLV]) no changes in the proportions in which different means of production are used by an industry are considered, so that no question arises as to the variation or constancy of returns. The investigation is concerned exclusively with such properties of an economic system as do not depend on changes in the scale of production or in the proportions of 'factors'." </BLOCKQUOTE><P>The analysis of the choice of technique shows that much neoclassical teaching and "practical" applications is humbug. But that does not exhaust Sraffa's point. Turning to the first sentence of the next paragraph in the preface can help: </P><BLOCKQUOTE>"This standpoint, which is that of the old classical economists from Adam Smith to Ricardo, has been submerged and forgotten since the advent of the 'marginal' method." </BLOCKQUOTE><P>A second major emphasis of Sraffa's scholarship, including his 1960 book, is the rediscovery of the logic of the <A HREF="http://robertvienneau.blogspot.com/search/label/Interpreting%20Classical%20Economics">classical theory</A> of value and distribution. Sraffians can claim to have a theory that can serve as an alternative to neoclassical theory and that is empirically applicable (for example, by Leontief and those aware of the National Income and Product Accounts (NIPA).) This rediscovery provides an external critique of neoclassical theory. </P><P>By the way, the development of this external critique provides, for example, Pierangelo Garegnani for a defense of the claim that the analysis of the attraction of market prices to prices of production is building on Sraffa's work. Sraffa's book does not discuss market processes or the classical theory of competition: </P><BLOCKQUOTE>"A less one-sided description than cost of production seems therefore required. Such classical terms as 'necessary price', 'natural price' or 'price of production' would meet the case, but value and price have been preferred as being shorter and in the present context (which contains no reference to market prices) no more ambiguous." (PoCbMoC, p. 9) </BLOCKQUOTE><P>One could read Sraffa as being able to take many aspects of classical political economy as given, including analyses of market prices. How should ideas that Sraffa explicitly choose to include in his archives, but not publish in his lifetime, influence our interpretation? </P><P>None of this gets to Sinha's point. He thinks, as I understand it, that Sraffa offers more than a rediscovery of classical political economy. Sraffa offers innovations in our understanding of prices and distribution, and these innovations can help us better understand actually existing capitalist economies. (Some of these innovations might be Wittgenstein-like in that they allow us to improve by discarding lots of rubbish.) I daresay Scott Carter agrees with that claim, even though he might disagree with details of Sinha's understanding of the Standard Commodity. </P>http://robertvienneau.blogspot.com/2016/12/on-ajit-sinha-on-sraffa.htmlnoreply@blogger.com (Robert Vienneau)4tag:blogger.com,1999:blog-26706564.post-5058425994538569414Tue, 27 Dec 2016 15:27:00 +00002016-12-29T13:07:32.872-05:00Example in Mathematical EconomicsSraffa EffectsTangency of Wage-Rate of Profits Curves<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-jzlP-29Z7UM/WGJtzzK4iFI/AAAAAAAAAvw/MsuzPmUSRoAzAWBlQ2XM4O5W7IDmXKTZACLcB/s1600/TangencyFrontier.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-jzlP-29Z7UM/WGJtzzK4iFI/AAAAAAAAAvw/MsuzPmUSRoAzAWBlQ2XM4O5W7IDmXKTZACLcB/s320/TangencyFrontier.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: The Choice of Technique in a Model with One Switch Point</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This post presents an example in which vertically-integrated firms producing a consumption good have a choice between two techniques. The wage-rate of profits curves for the techniques have a single switch point, at which they are tangent. I, and, I dare say, most economists who are aware of the illustrated possibility, consider this a fluke, a possibility that cannot be expected to arise in practice. </P><B>2.0 Technology</B><P>The technology in this example has the structure of Garegnani's generalization of Samuelson's surrogate production function. One commodity, corn, can be produced from inputs of labor and a single capital good. Two processes are known for producing corn, and each process requires a different capital good, called "iron" and "copper". Each capital good is produced, if at all, by a process that requires inputs of labor and that capital good. Each process requires a year to complete, and the services of the capital good fully consume that capital good during the course of the year. No stock of iron or copper remains at the end of the year to carry over into the next year. </P><P>Constant Returns to Scale are assumed for each process. Table 1 shows the coefficients of production for the four processes specified by the technology. Each column corresponds to a process. The coefficients of production specify the input of the row commodity that is needed to produce a unit output of the commodity for the column. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology for a Three-Industry Model</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center"><B>Iron<BR>Industry</B></TD><TD ALIGN="center"><B>Copper<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">(a)</TD><TD ALIGN="center">(b)</TD><TD ALIGN="center">(c)</TD><TD ALIGN="center">(d)</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1</TD><TD ALIGN="center">17,328/8,281</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>Two techniques are available for producing corn (Table 2). The Alpha technique consists of the process for producing iron and the corn-producing process that requires an input of iron. The Beta technique consists of the copper-producing process and the corn-producing process using services provided by copper. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 2: Techniques</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">b, d</TD></TR></TABLE><P></P><B>3.0 Prices and the Choice of Technique</B><P>The technique, as usual, is chosen by managers of firms to minimize costs. Corn is taken as the numeraire, and wages are paid at the end of the year. Prices of production, in which all extra profits above a common rate have been competed away, are used to calculate costs. For analytical convenience, in this pose I take the rate of profits as given. </P><P>Suppose the Alpha technique is chosen. Under the above assumptions, the price of iron and the wage must satisfy the following system of two equations: </P><BLOCKQUOTE>(1/2) <I>p</I><SUB>iron</SUB> (1 + <I>r</I>) + <I>w</I><SUB>α</SUB> = <I>p</I><SUB>iron</SUB></BLOCKQUOTE><BLOCKQUOTE>3 <I>p</I><SUB>iron</SUB> (1 + <I>r</I>) + <I>w</I><SUB>α</SUB> = 1 </BLOCKQUOTE><P>A similar system arises for the Beta technique, but as applied to the price of copper and the coefficients of production for the Beta technique. </P><P>For a non-negative rate of profits, up to a certain maximum rate that depends on the technique, one can solve each system of equations for the wage and the price of the relevant capital good. The resulting wage-rate of profits curve for the Alpha technique is: </P><BLOCKQUOTE><I>w</I><SUB>α</SUB> = (1 - <I>r</I>)/(7 + 5 <I>r</I>) </BLOCKQUOTE><P>The maximum wage for the Alpha technique, 1/7 bushels per person-year arises for a rate of profits of zero in the above equation. The maximum rate of profits, for the Alpha technique, is 100% and occurs when the wage is zero. </P><P>The wage-rate of profits curve for the Beta technique is: </P><BLOCKQUOTE><I>w</I><SUB>β</SUB> = (43 - 48 <I>r</I>)/361 </BLOCKQUOTE><P>For what it is worth, the maximum wage for the Beta technique is 43/361 bushels per person-year. The maximum rate of profits is 43/48, approximately 90%. Both the maximum wage and the maximum rate of profits for the Beta technique are dominated by the corresponding values for the Alpha technique. </P><P>Figure 1, at the top of this post, graphs the wage-rate of profits curves for both techniques. Since the coefficients of production in copper-production are a constant multiple (48/91) of the coefficients of production in the process for producing corn from copper, the wage-rate of profits curve for the Beta technique is a straight line. The wage-rate of profits curve for the cost-minimizing technique forms the outer envelope in Figure 1. The Alpha technique minimizes costs for all feasible rates of profits and wages. </P><P>One switch point arises in this example. It is at 50%, half the maximum rate of profits for the Alpha technique. The wage is 1/19 bushels per person-year at the switch point, and the slope of both wage-rate of profits curves has a value -48/361 at the switch point. One can find the rate of profits for the switch point by equating the functions for <I>w</I><SUB>α</SUB> and <I>w</I><SUB>β</SUB>. A quadratic equation arises for the rate of profits, and 50% is a repeated root for this polynomial. Both the Alpha and Beta techniques are cost-minimizing at the switch point. </P><B>4.0 The Market for "Capital"</B><P>One can find gross outputs of each process needed to produce a bushel of corn. If the Alpha technique is used, gross outputs consist of two tons iron and 1 bushel corn. For the Beta technique, gross outputs consist of 91/43 tons copper and one bushel corn. The quantity of the capital good, in physical units, needed to produce a net output of one unit of the numeraire good is immediately obvious in this technology. The total quantity of labor, over all processes in a technique, for producing a net output of corn is vector dot product of the labor coefficients, for the technique, and the gross outputs. </P><P>The quantity of the capital good must be evaluated with prices so as to graph, say, the amount of capital per person-year for each technique in one space. Since the wage-rate of profits curve for the Alpha technique has some non-zero convexity, the price of iron varies with the given rate of profits: </P><BLOCKQUOTE><I>p</I><SUB>iron</SUB> = 2/(7 + 5 <I>r</I>) </BLOCKQUOTE><P>The price of copper is a constant 48/91 bushels per ton. </P><P>Table 2 brings these calculations together. It shows the ratio of the value of the capital good to labor inputs. The horizontal line shows the real Wicksell effect at the switch point. If one wanted, one could remove the price Wicksell effects with Champernowne's <A HREF="http://robertvienneau.blogspot.com/2009/05/neoclassical-response-to-cambridge.html">chain index</A> measure of capital. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-dPOxHu4Q5-U/WGJtr0Ttv0I/AAAAAAAAAvs/PR-KRAJzcsQp3YKaifbwRHnKyHP43q2NwCLcB/s1600/TangencyCapitalMarket.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-dPOxHu4Q5-U/WGJtr0Ttv0I/AAAAAAAAAvs/PR-KRAJzcsQp3YKaifbwRHnKyHP43q2NwCLcB/s320/TangencyCapitalMarket.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 2: Capital per Worker versus Rate of Profits</b></td></tr></tbody></table><B>5.0 The Labor Market</B><P>For completeness, Figure 3 graphs the wage against the amount of labor hired, across all industries, to produce a net output of corn with cost-minimizing techniques. A linear combination of the techniques at the switch point is shown here, also, by a horizontal line. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-tsys0TjZNlY/WGJtXpJtKYI/AAAAAAAAAvo/rF9kYxMcaDcD5jdk-IS_j69ujdifpXBOQCLcB/s1600/TangencyLaborMarket.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-tsys0TjZNlY/WGJtXpJtKYI/AAAAAAAAAvo/rF9kYxMcaDcD5jdk-IS_j69ujdifpXBOQCLcB/s320/TangencyLaborMarket.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 3: Labor per Unit Output versus Wage</b></td></tr></tbody></table><B>6.0 Why This Example is a Fluke</B><P>Generic results show a certain structural stability. Qualitative properties, for generic results, continue to persist for some small variation in model parameters. This is not the case for the example. Small variations will lead to either two switch points (that is, reswitching) or no switch points. In the latter case, the Beta technique would be dominated and never cost-minimizing. </P><P>I look to the mathematics of dynamical systems for an analogy. One can look at prices of production as fixed points in some dynamical system. For example, consider a classical view of competition, in which firms and investors are able to shift from the production in one industry to production in another. (Literature on such dynamical processes can be found under the keyword of "cross-dual dynamics".) Neoclassical economists might look at prices of production as a special case of an intertemporal equilibrium, in which initial endowments just happen to be such that relative spot prices do not vary with time. Or one can consider prices of production as partially characterizing a fixed point, in a limiting process, as time grows without bound in <A HREF="http://robertvienneau.blogspot.com/2014/06/a-sophisticated-neoclassical-response.html">neoclassical</A> models of intertemporal or temporary equilibria. </P><P>At any rate, hyperbolic points are considered generic in dynamical systems. In discrete time, no eigenvalues of the linearization around a hyperbolic point lie on the unit circle. Continuing in the jargon, no center manifold exists for a hyperbolic point. Non-hyperbolic fixed points are important in that they indicate a bifurcation. </P>http://robertvienneau.blogspot.com/2016/12/tangency-of-wage-rate-of-profits-curves.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-3709085281062613526Wed, 21 Dec 2016 14:29:00 +00002016-12-30T07:17:15.143-05:00Example in Mathematical EconomicsLabor MarketsSraffa EffectsExample Of The Choice Of Technique<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-BQ7fDq6jYW8/WFfc8IHtZuI/AAAAAAAAAs0/hVu3R_2NId0nOWliUwuoJ1lcMW4i0KMQgCLcB/s1600/AggregateProductionFunction.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-BQ7fDq6jYW8/WFfc8IHtZuI/AAAAAAAAAs0/hVu3R_2NId0nOWliUwuoJ1lcMW4i0KMQgCLcB/s320/AggregateProductionFunction.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: Aggregate Production Function</b></td></tr></tbody></table><B>1.0 Introduction</B><P>This long post presents an analysis of the choice of technique in a three-commodity example. This example extends a <A HREF="http://robertvienneau.blogspot.com/2016/12/perturbation-of-reswitching-example.html">previous post</A>. Two processes are known for producing each commodity. The example is simple in that it is of a model only of circulating capital. No fixed capital - that is, machines that last more than one period - exists in the model. Homogeneous labor is the only non-produced input used in production. </P><P>Despite these simplifications, many readers may prefer that I revert to examples with fewer commodities and processes. Eight techniques arise for analysis. All three commodities are basic in all techniques. I end up with 34 switch points. Even so, various possibilities in the theory are not illustrated by the example. (Heinz Kurz and Neri Salvadori probably have better examples. I also like J. E. Woods for exploring possibilities in linear models of production.) The example does suggest, however, that the exposed errors taught, around the world, to students of microeconomics and macroeconomics cannot be justified by the use of continuously differentiable, microeconomic production functions. </P><B>2.0 Technology</B><P>This economy produces a single consumption good, called corn. Corn is also a capital good, that is, a produced commodity used in the production of other commodities. In fact, iron, steel, and corn are capital goods in this example. So three industries exist. One produces iron, another produces steel, and the last produces corn. Two processes exist in each industry for producing the output of that industry. Each process exhibits Constant Returns to Scale (CRS) and is characterized by coefficients of production. Coefficients of production (Table 1) specify the physical quantities of inputs required to produce a unit output in the specified industry. All processes require a year to complete, and the inputs of iron, steel, and corn are all consumed over the year in providing their services so as to yield output at the end of the year. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center" ROWSPAN="2"><B>Input</B></TD><TD ALIGN="center" COLSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Steel<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn<BR>Industry</B></TD></TR><TR><TD ALIGN="center">a</TD><TD ALIGN="center">b</TD><TD ALIGN="center">c</TD><TD ALIGN="center">d</TD><TD ALIGN="center">e</TD><TD ALIGN="center">f</TD></TR><TR><TD ALIGN="center">Labor</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">1/10</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">7/20</TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD ALIGN="center">Iron</TD><TD ALIGN="center">1/6</TD><TD ALIGN="center">2/5</TD><TD ALIGN="center">1/200</TD><TD ALIGN="center">1/100</TD><TD ALIGN="center">1</TD><TD ALIGN="center">0</TD></TR><TR><TD ALIGN="center">Steel</TD><TD ALIGN="center">1/200</TD><TD ALIGN="center">1/400</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">3/10</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/4</TD></TR><TR><TD ALIGN="center">Corn</TD><TD ALIGN="center">1/300</TD><TD ALIGN="center">1/300</TD><TD ALIGN="center">1/300</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR></TABLE><P>A technique consists of a process in each industry. Table 2 specifies the eight techniques that can be formed from the processes specified by the technology. If you work through this example, you will find that to produce a net output of one bushel corn, inputs of iron, steel, and corn all need to be produced to reproduce the capital goods used up in producing that bushel. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 2: Techniques</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c, e</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">a, c, f</TD></TR><TR><TD ALIGN="center">Gamma</TD><TD ALIGN="center">a, d, e</TD></TR><TR><TD ALIGN="center">Delta</TD><TD ALIGN="center">a, d, f</TD></TR><TR><TD ALIGN="center">Epsilon</TD><TD ALIGN="center">b, c, e</TD></TR><TR><TD ALIGN="center">Zeta</TD><TD ALIGN="center">b, c, f</TD></TR><TR><TD ALIGN="center">Eta</TD><TD ALIGN="center">b, d, e</TD></TR><TR><TD ALIGN="center">Theta</TD><TD ALIGN="center">b, d, f</TD></TR></TABLE><P></P><B>3.0 Choice of Technique</B><P>Managers of firms choose processes in their industry to minimize costs. So one must consider prices in analyzing the choice of technique. Assume that corn is the numeraire. In other words, the price of a bushel corn is one monetary unit. I assume that labor is advanced, and that wages are paid out of the surplus at the end of the year. </P><P>These conditions specify a system of three equations that must be satisfied if a technique is to be chosen. For example, suppose the Alpha technique is in use. Let <I>w</I><SUB>α</SUB> be the wage and <I>r</I><SUB>α</SUB> the rate of profits. Let <I>p</I><SUB>1</SUB> be the price of iron and <I>p</I><SUB>2</SUB> the price of steel. If managers are willing to continue producing iron, steel, and corn with the Alpha technique, the following three equations apply: </P><BLOCKQUOTE>((1/6)<I>p</I><SUB>1</SUB> + (1/200)<I>p</I><SUB>2</SUB> + (1/300))(1 + <I>r</I><SUB>α</SUB>) + (1/3)<I>w</I><SUB>α</SUB> = <I>p</I><SUB>1</SUB> </BLOCKQUOTE><BLOCKQUOTE>((1/200)<I>p</I><SUB>1</SUB> + (1/4)<I>p</I><SUB>2</SUB> + (1/300))(1 + <I>r</I><SUB>α</SUB>) + (1/2)<I>w</I><SUB>α</SUB> = <I>p</I><SUB>2</SUB> </BLOCKQUOTE><BLOCKQUOTE>(<I>p</I><SUB>1</SUB>)(1 + <I>r</I><SUB>α</SUB>) + <I>w</I><SUB>α</SUB> = 1 </BLOCKQUOTE><P>These equations apply to iron, steel, and corn production, respectively. They show the same rate of profits being earned in each industry. Confining one's attention to the three processes comprising the Alpha technique, they show the same rate of profits being earned in each industry. Managers will not want to disinvest in one industry and invest in another, at least, with these three processes available. </P><P>Suppose the wage is given, is non-negative, and does not exceed a certain maximum specified, for a technique, by a zero rate of profits. Then, for each technique, one can find the rate of profits and prices of commodities. The function relating the rate of profits to the wage for a technique is known as the wage-rate of profits curve, or, more shortly, the wage curve for the technique. Figure 2 graphs the wage-rate of profits curves for the eight techniques in the example. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://2.bp.blogspot.com/-_GGOZOkzows/WFqEC6Epd0I/AAAAAAAAAuk/AahGp49BJWEDBOcYBtLWx_eBGCjYB5nBwCLcB/s1600/EightTechniqueFrontier.jpg" imageanchor="1" ><img border="0" src="https://2.bp.blogspot.com/-_GGOZOkzows/WFqEC6Epd0I/AAAAAAAAAuk/AahGp49BJWEDBOcYBtLWx_eBGCjYB5nBwCLcB/s320/EightTechniqueFrontier.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 2: Wage-Rate of Profits Curves</b></td></tr></tbody></table><P>The cost minimizing technique, at a given wage, maximizes the rate of profits. That is, wage curves for cost minimizing techniques form the outer envelope of the wage curves graphed in Figure 2. Table 3 lists the cost minimizing techniques for the example, from a wage of zero to the maximum wage. The switch points on the frontier are pointed out in Figure 2. This is <I>not</I> an example of reswitching or of the recurrence of techniques. No technique is repeated in Table 3. It is an example of process recurrence. The corn-producing process labeled "e", repeats in Table 3. I label the switch point between Alpha and Beta as "perverse" just to emphasize that results arise for it that violate the beliefs of outdated and erroneous neoclassical economists. From the standpoint of current theory, it is not any more surprising than non-perverse switch points. </P><TABLE ALIGN="center" BORDER="1" CELLPADDING="1" CELLSPACING="1"><CAPTION><B>Table 3: Techniques on Frontier</B></CAPTION><TR><TD ALIGN="center"><B>Technique</B></TD><TD ALIGN="center"><B>Processes</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">a, c, e</TD></TR><TR><TD ALIGN="center">Beta</TD><TD ALIGN="center">a, c, f</TD></TR><TR><TD ALIGN="center">Delta</TD><TD ALIGN="center">a, d, f</TD></TR><TR><TD ALIGN="center">Theta</TD><TD ALIGN="center">b, d, f</TD></TR><TR><TD ALIGN="center">Eta</TD><TD ALIGN="center">b, d, e</TD></TR></TABLE><P></P><P>I experimented, somewhat, with the coefficients of production for alternative processes in the various industries, but not all that much. Thirty four switch points exist in the example, including switch points (some "perverse") inside the frontier. No techniques have three switch points, even though in a model with three basic commodities, such can happen. As noted above, no reswitching occurs on the frontier. But consider switch points for each pair of techniques, including within the frontier. Under this way of looking at the example, reswitching arises for the following pair of techniques: </P><UL><LI>Alpha and Beta: Vary in corn-producing process.</LI><LI>Alpha and Delta: Vary in steel-producing process.</LI><LI>Alpha and Zeta: Vary in iron-producing and corn producing processes.</LI><LI>Alpha and Theta: No processes in common.</LI><LI>Gamma and Delta: Vary in corn-producing processes.</LI><LI>Gamma and Zeta: No processes in common.</LI><LI>Gamma and Theta: Vary in iron-producing and corn producing processes.</LI></UL><P>Generically, in models with all commodities basic, techniques that switch on the frontier differ in one process. So one could form a reswitching example with two technique out of the processes comprising, for example, the Alpha and Delta techniques. </P><P>Figure 2 is complicated, and some properties of the wage curves are hard to see, no matter how close you look. All wage curves slope downward, as must be the case. The wage curve for, for example, the Alpha technique varies in convexity, depending at what wage you find its second derivative. For high wages, the wage curve for Alpha lies just below Gamma's, the wage curve for Beta is just below Zeta's, and the wage curve for Delta is just below Theta's. (By "high wages", I mean wages larger than the wage for the single switch point for the given pair of techniques.) The wage curves for Epsilon and Eta are visually indistinguishable in the figure. They have a single switch point at a fairly low wage, and above that, the wage curve for Epsilon lies below Eta's. I wonder how much variations in the parameters specifying the technology result in variation in the location of wage curves. </P><B>4.0 The Capital "Market" and Aggregate Production Function</B><P>The example illustrates certain results that I find of interest. Suppose the economy produces a net output of corn. Given the wage, one can identify the cost minimizing technique. By use of the Leontief inverse for that technique, one can calculate the level of outputs in the iron, steel, and corn industries needed to replace the capital goods used up in producing a given net output of corn. In a standard notation, used in previous posts: </P><BLOCKQUOTE><B>q</B> = (<B>I</B> - <B>A</B>)<SUP>-1</SUP> (<I>c</I> <B>e</B><SUB>3</SUB>) </BLOCKQUOTE><P>where <B>I</B> is the identity matrix, <B>e</B><SUB>3</SUB> is the third column of the identity matrix, <I>c</I> is the quantity of corn produced for the net output, <B>A</B> is the Leontief matrix for the cost minimizing technique at the given wage, and <B>q</B> is the column vector of gross outputs of iron, steel, and corn. (This relationship can be extended to a steady state, positive rate of growth, up to a maximum rate of growth.) </P><P>For the given wage, one can find prices that are consistent with the adoption of the cost minimizing technique. Let <B>p</B> be the three-element row vector for these prices. (Since corn is the numeraire, <I>p</I><SUB>3</SUB> is unity.) Consider the production of a net output of corn. The column vector of capital goods needed to produce this net output is (<B>A</B> <B>q</B>). The value of these capital goods is: </P><BLOCKQUOTE><I>K</I> = <B>p</B> <B>A</B> <B>q</B></BLOCKQUOTE><P>Let <B>a</B><SUB>0</SUB> be the row vector of labor coefficients for the cost minimizing technique. </P><BLOCKQUOTE><I>L</I> = <B>a</B><SUB>0</SUB> <B>q</B> = <B>a</B><SUB>0</SUB> (<B>I</B> - <B>A</B>)<SUP>-1</SUP> (<I>c</I> <B>e</B><SUB>3</SUB>) </BLOCKQUOTE><P>Net output per worker is easily found: </P><BLOCKQUOTE><I>y</I> = <I>c</I>/<I>L</I></BLOCKQUOTE><P>Likewise, capital per worker is: </P><BLOCKQUOTE><I>k</I> = <I>K</I>/<I>L</I></BLOCKQUOTE><P>This algebra allows one to draw certain graphs for the example. Figure 3 shows the value of the capital goods the managers of firms want to employ per worker as a function of the rate of profits. As is typical in the Marshallian tradition for graphing supposedly downward-sloping demand functions, the "quantity" variable - that is, the value of the capital goods - is on the abscissa. The "price" variable - that is, the rate of profits - is graphed on the ordinate. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-SmY_lkikMKA/WFfnEtqqK5I/AAAAAAAAAtg/uDlJXIMDcQQl7M8VLsiA-U3tF09pL_YrACLcB/s1600/CapitalPerHead.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-SmY_lkikMKA/WFfnEtqqK5I/AAAAAAAAAtg/uDlJXIMDcQQl7M8VLsiA-U3tF09pL_YrACLcB/s320/CapitalPerHead.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 3: Value of Capital Hired at Different Rates of Profit</b></td></tr></tbody></table><P>Switch points appear in Figure 3 as horizontal lines. They result from varying linear combinations of techniques, at a given price system. The curves, that are not quite vertical, between the switch points result from variations in prices and the rate of profits, for a given cost minimizing technique, with the wage along the wage-rate of profits frontier. Variation from the vertical for these curves is known as a <I>price Wicksell effect</I>. </P><P>While it is not obvious from the figure, the sign of the slope of the curve above the switch point between the Alpha and Beta technique changes over the range in which Alpha is the cost-minimizing technique. (This change in the direction of the price Wicksell effect is equivalent to a change in the convexity of the wage curve for the Alpha technique in the region where it lies on the outer frontier in Figure 2.) In the lower part of this uppermost locus, a lower rate of profits is associated with a greater value of capital goods per worker. This is a negative price Wicksell effect. Elsewhere, in the graph, price Wicksell effects are positive. It is not clear to me that neoclassical economists, at least after the Cambridge Capital Controversy, have any definite <A HREF="http://robertvienneau.blogspot.com/2009/05/neoclassical-response-to-cambridge.html">beliefs</A> about the direction of price Wicksell effects. </P><P>The direction of real Wicksell effects cannot be reconciled with traditional neoclassical theory. Consider, first, the switch point between the Theta and Eta techniques. Compare the value of capital per worker at a rate of profits slightly higher than the rate of profits at the switch point with capital per worker at a rate of profits slightly lower. Notice that with this notional variation, a higher value of capital per worker is associated with a lower rate of profits. This is a negative real Wicksell effect. If capital were a factor of production, a lower equilibrium rate of profits would indicate it is less scarce, and firms would be induced to adopt a more capital-intensive technique of production. Thus, a negative real Wicksell effect illustrates traditional, mistaken neoclassical theory. But, in the example, the real Wicksell effect is positive at the switch point between the Alpha and Beta techniques. </P><P>I have above outlined how to calculated the value of output per worker and capital per worker as the wage or the rate of profits parametrically varies. Figure 1, at the head of the post, graphs the value of output per worker versus capital per worker. The scribble at the top is the production function, as in, for example, Solow's growth model. </P><P>Before considering the details of this function in the example, note that the production function is not a technological relationship, showing the quantity of a physical output that can be produced from physical inputs. Prices must be determined before it can be drawn. In particular, either the wage or the rate of profits must be given to determine a particular point on the production function. Suppose all real Wicksell effects happen to be negative, and the slope of the production function, for some index of capital intensity, happens to be equal to the rate of profits (at each switch point). Since one had to start with the wage or the rate of profits, even then one could not use the production function to determine distribution. Deriving such a marginal productivity relationship seems to be besides the point when it comes to defending neoclassical theory. </P><P>Now to details. Between switch points, a single technique lies on the frontier in Figure 2. Given the technique and net output, a certain constant output per worker results, no matter what the wage and the rate of profits in the region where that wage curve lies on the frontier. Thus, the horizontal lines in the graph of the production function reflect a region in which a switch of techniques does not occur. The downward-sloping and upward-sloping lines, in the production function, illustrate switch points. At each switch point, a linear combination of techniques minimizes costs. The perverse switch point is reflected in the production function by an upward slope at the switch point, as the rate of profits parametrically increases. I gather it is a theorem that greater capital per worker is associated with more output per worker. But in the "perverse" case, greater capital per worker is associated with a greater rate of profits. </P><B>5.0 The Labor "Market"</B><P>So much for neoclassical macroeconomics. Next, consider how much labor, firms want to hire over all three industries, to produce a given net output of corn (Figure 4). (I still follow the Marshallian tradition of putting the price variable on the Y-axis and the quantity variable on the X-axis.) Around the switch point between the Alpha and Beta technique, a slightly higher wage is associated with firms wanting to employ more labor, given net output. In the traditional neoclassical theory, a higher wage would indicate to firms that labor is scarcer, and firms would be induced to adopt less labor-intensive techniques of production. The example shows that this theory is logically invalid. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://1.bp.blogspot.com/-lQnKpa7EuA8/WFqD5V-WeeI/AAAAAAAAAug/rU11lvfodoU9k_Ygk75BDOWs2AtPoPZdACLcB/s1600/VerticalIntegration.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-lQnKpa7EuA8/WFqD5V-WeeI/AAAAAAAAAug/rU11lvfodoU9k_Ygk75BDOWs2AtPoPZdACLcB/s320/VerticalIntegration.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 4: Labor Employed at Different Wages</b></td></tr></tbody></table><P></P><B>6.0 Labor Employed Directly in Corn Production</B><P>Although this is not an example of reswitching, it is an example of process recurrence. (I was pleased to see that each of the six production processes is part of at least one technique with a wage curve on the frontier.) Since two processes are available for producing corn, the amount of labor that corn-producing firms want to produce, at a non-switch point, is either 1.0 or 1.5 person-years per gross output of the corn industry. These are the labor coefficients, for processes "e" and "f", in Table 1. The labor coefficients account for the locations of the vertical lines in Figure 5. Once again, a linear combination of techniques is possible at switch points. If the pair of techniques that are cost minimizing at a switch point differ in the corn-producing process, a horizontal line is shown in Table 5. The analysis of the choice of technique is needed to locate these horizontal lines. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://3.bp.blogspot.com/-WEDdEj0IMAM/WFqDyE9c7XI/AAAAAAAAAuc/Yabl39HDd50-8KIq5scoMeUL7ERHAGhuQCLcB/s1600/ProcessReccurence.jpg" imageanchor="1" ><img border="0" src="https://3.bp.blogspot.com/-WEDdEj0IMAM/WFqDyE9c7XI/AAAAAAAAAuc/Yabl39HDd50-8KIq5scoMeUL7ERHAGhuQCLcB/s320/ProcessReccurence.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 5: Labor Directly Employed in Producing Corn</b></td></tr></tbody></table><P>Figure 5 shows, that around the switch point for the Alpha and Beta techniques, a higher wage is associated with corn-producing firms wanting to hire more labor for direct employment in producing corn. So much for microeconomics. Those exploring the theory of production have found other results that contradict neoclassical microeconomics. </P><B>7.0 Conclusion</B><P>The above has presented an example in which, in each industry, firms have some capability to trade off inputs, in some sense. For producing a unit output of iron or steel, they might be able to lower labor inputs at the expense of needing to hire more commodities used directly in producing that output. As I understand it, if possibilities of substitution are increased without end, traditional mistaken parables, preached by mainstream economists, are not restored. Suppose the cost-minimizing technique varied continuously along the wage-rate of profits frontier. A specific coefficient of production, as a process varied in some industry, would not necessarily vary continuously. The stories of marginal adjustments that many mainstream economists have been telling for over a century seem to be contradicted by the theory of production. </P><P>I have highlighted three results, at least, for the example: </P><UL><LI>Around a so-called perverse switch point, a lower rate of profits is associated with firms wanting to adopt a technique in which the value of capital goods, per worker, is <I>less</I> than at a higher rate of profits.</LI><LI>In the labor market for the economy as a whole, a higher wage can be associated with firms wanting to employ <I>more</I> workers to produce a given (net) output.</LI><LI>In a given industry, a higher wage can be associated with firms in that industry wanting to employ <I>more</I> workers to produce a given (gross) output.</LI></UL><P>The last result, at least, is independent of the first. For instance, examples exist of non-perverse switch points in which this result arises. </P><P>The theory of supply and demand has been lying in tatters, destroyed for about half a century. Many economists seem to be ignorant of this, though. </P>http://robertvienneau.blogspot.com/2016/12/example-of-choice-of-technique.htmlnoreply@blogger.com (Robert Vienneau)2tag:blogger.com,1999:blog-26706564.post-7903701427870581239Tue, 20 Dec 2016 06:42:00 +00002016-12-20T01:42:03.661-05:00Multiple Interest RatesThe Production of Commodities and Multiple Interest Rate Analysis<P>I've rewritten my analysis of the application of multiple interest rate analysis to models of the production of commodities by means of commodities. (This analysis is limited to circulating capital models, in which there exists no land or long-lasting machines.) I like to think this newer <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2885821">paper</A> is more focused than my earlier <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882531">paper</A>. For example, I do not have an aside, with graphs, about bifurcation theory, as applied to polynomial equations. I also have an example which I think provides more easily visualizable graphs. I still think these papers are better at raising questions than reaching conclusions.</P>http://robertvienneau.blogspot.com/2016/12/the-production-of-commodities-and.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-7576282914999864696Fri, 16 Dec 2016 14:00:00 +00002016-12-16T09:00:02.884-05:00Example in Mathematical EconomicsSraffa EffectsPerturbation Of A Reswitching Example<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><TR><TD><a href="https://4.bp.blogspot.com/-GZ04-hTvD3w/WFKlW6cYkeI/AAAAAAAAAsc/clJaO0LfctA4HsSsBO33PYWdAwdyE6eJgCLcB/s1600/ThreeGoodsFrontier.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-GZ04-hTvD3w/WFKlW6cYkeI/AAAAAAAAAsc/clJaO0LfctA4HsSsBO33PYWdAwdyE6eJgCLcB/s320/ThreeGoodsFrontier.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: Wage-Rate of Profits Curve for Two Techniques</b></td></tr></tbody></table><B>1.0 Introduction</B><P>In this post, I consider a perturbation of the data on technology in this <A HREF="http://robertvienneau.blogspot.com/2009/04/yet-another-cambridge-controversy-yacc.html">example</A> of the production of commodities by means of commodities. This example is of the choice of technique from two techniques. Each technique can be used to produced a commodity, corn, used for consumption and as the numeraire. The perturbations considered here drastically changes the qualitative characterization of the technology. And they only slightly change the location of switch points and the maximum wages, for the two techniques. These perturbations also only slightly change the maximum rate of profits for one technique. They do, however, drastically lower the maximum rate of profits for the other technique. </P><B>2.0 Two Techniques With Two Perturbations</B><P>Table 1 displays the technology available to the firms in this example. (I have renamed the industries and commodities.) Each column defines the coefficients of production for a process for producing the output of an industry. Only one process is known for producing iron, and only one process is available for producing steel. Two processes are known for producing corn. Coefficients of production show how much of each input must be available, to provide flows of services of that input over the year, per unit output produced and available at the end of the year. The parameters δ and ε must both be nonnegative for a given technology. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ROWSPAN="2"><B>Inputs</B></TD><TD ALIGN="center" ROWSPAN="2"><B>Iron<BR>Industry</B></TD><TD ALIGN="center" ROWSPAN="2"><B>Steel<BR>Industry</B></TD><TD ALIGN="center" COLSPAN="2"><B>Corn Industry</B></TD></TR><TR><TD ALIGN="center">Alpha</TD><TD ALIGN="center">Beta</TD></TR><TR><TD>Labor (Person-Years):</TD><TD ALIGN="center">1/3</TD><TD ALIGN="center">1/2</TD><TD ALIGN="center">1</TD><TD ALIGN="center">3/2</TD></TR><TR><TD>Iron (Tons):</TD><TD ALIGN="center">1/6</TD><TD ALIGN="center">ε</TD><TD ALIGN="center">1</TD><TD ALIGN="center">0</TD></TR><TR><TD>Steel (Tons):</TD><TD ALIGN="center">ε</TD><TD ALIGN="center">1/4</TD><TD ALIGN="center">0</TD><TD ALIGN="center">1/4</TD></TR><TR><TD>Corn (Bushels):</TD><TD ALIGN="center">δ</TD><TD ALIGN="center">δ</TD><TD ALIGN="center">0</TD><TD ALIGN="center">0</TD></TR><TR><TD><B>Output (Various):</B></TD><TD ALIGN="center">1</TD><TD ALIGN="center">1</TD><TD ALIGN="center">1</TD><TD ALIGN="center">1</TD></TR></TABLE><P>Two techniques are defined, in this technology, for producing a net output of corn. Each technique consists of a single process for producing corn and whichever of the iron-producing and steel-producing processes (sometimes both) is needed to reproduce the capital goods used up in producing a net output of corn. </P><B>2.1 No Basic Commodities</B><P>Consider the special case where: </P><BLOCKQUOTE>δ = ε = 0 </BLOCKQUOTE><P>In this case, one can say that in both techniques, no commodity is basic. Or one might say that, in each technique, one commodity is basic, and that which commodity is basic varies with the technique. It depends on how you look at it. </P><P>In the Alpha technique, corn is produced with the process labeled Alpha. Iron is used as an input in producing iron and in producing corn. Corn is not an input in any process, and steel is not produced. If one disregarded the non-produced commodity, steel, one could say iron is the single basic commodity. On the other hand, if one included steel as a possible commodity, iron would not be basic, since it does not enter into the production of steel, either directly or indirectly. </P><P>The same paragraph could be written about the Beta technique, with the role of iron and steel reversed. </P><B>2.2 Three Basic Commodities</B><P>Cosider a case in which both the δ and ε parameters are (small) positive numbers. I worked out the following case: </P><BLOCKQUOTE>δ = 1/300 </BLOCKQUOTE><BLOCKQUOTE>ε = 1/200 </BLOCKQUOTE><P>In this case, the Alpha technique consists of the iron-producing process, the steel-producing process, and the corn-producing process labeled Alpha. All three commodities are basic. Corn enters indirectly into the production of corn through both iron and steel. Similarly, all three commodities are basic in the Beta technique. </P><P>So one sees that the structure of production, in both techniques, is qualitatively different in these two cases. This difference is seen in which commodities are basic, and which are not. </P><B>3.0 Wage-Rate of Profits Curves</B><P>The managers of firms choose the processes comprising the technique so as to minimize cost. Let a bushel of corn be the numeraire. Suppose labor is advanced, and wages are paid out of the output available at the end of the year. </P><P>For each technique, these assumptions are such that a relation between the wage and the rate of of profits arises. Both the wage and the rate of profits range between zero and a finite maximum wage or rate of profits. The higher the rate of profits, the lower the wage and vice versa. You can see these wage-rate of profits curves graphed in the first figure <A HREF="http://robertvienneau.blogspot.com/2009/04/yet-another-cambridge-controversy-yacc.html">here</A> for the special case in which δ = ε = 0. Figure 1, at the top of this post, graphs these wage curves for the specific positive values of δ and ε graphed above. </P><P>The choice of technique can be analyzed based on the outer frontier of the wage-rate of profits curves for the technique. For a given rate of profits, the cost-minimizing technique is the one with the highest wage at that rate of profits. At switch points, more than one technique is cost-minimizing. Firms can adopt a linear combination of the techniques on the outer frontier at switch points. </P><P>This is a reswitching example for the perturbations considered here. The Alpha technique is cost-minimizing at low and high rates of profits, with the Beta technique cost-minimizing at intermediate rates of profits. Tables 1 and 2 specify the location of the switch points, as well as the maximum wages and rates of profits for the two techniques. These solution values can be found as easy-to-calculate rational numbers for the original case, as shown in Table 1. Table 2 lists approximate values. </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Model with δ = ε = 0</B></CAPTION><TR><TD ALIGN="center"><B>Variable</B></TD><TD ALIGN="center"><B>Alpha Technique</B></TD><TD ALIGN="center"><B>Beta Technique</B></TD></TR><TR><TD ALIGN="center">Maximum Wage</TD><TD ALIGN="center">5/7 = 0.7143</TD><TD ALIGN="center">3/5 = 0.6</TD></TR><TR><TD ALIGN="center">Maximum Rate of Profits</TD><TD ALIGN="center">500%</TD><TD ALIGN="center">300%</TD></TR><TR><TD ALIGN="center" COLSPAN="3">First Switch Point</TD></TR><TR><TD ALIGN="center">Wage</TD><TD ALIGN="center" COLSPAN="2">1/2 = 0.5</TD></TR><TR><TD ALIGN="center">Rate of Profits</TD><TD ALIGN="center" COLSPAN="2">100%</TD></TR><TR><TD ALIGN="center" COLSPAN="3">Second Switch Point</TD></TR><TR><TD ALIGN="center">Wage</TD><TD ALIGN="center" COLSPAN="2">1/3 = 0.3333</TD></TR><TR><TD ALIGN="center">Rate of Profits</TD><TD ALIGN="center" COLSPAN="2">200%</TD></TR></TABLE><P> </P><TABLE CELLSPACING="1" CELLPADDING="1" BORDER="1" ALIGN="center"><CAPTION><B>Table 1: Perturbed Model</B></CAPTION><TR><TD ALIGN="center"><B>Variable</B></TD><TD ALIGN="center"><B>Alpha Technique</B></TD><TD ALIGN="center"><B>Beta Technique</B></TD></TR><TR><TD ALIGN="center">Maximum Wage</TD><TD ALIGN="center">0.7094</TD><TD ALIGN="center">0.5991</TD></TR><TR><TD ALIGN="center">Maximum Rate of Profits</TD><TD ALIGN="center">298.0%</TD><TD ALIGN="center">294.1%</TD></TR><TR><TD ALIGN="center" COLSPAN="3">First Switch Point</TD></TR><TR><TD ALIGN="center">Wage</TD><TD ALIGN="center" COLSPAN="2">0.5153</TD></TR><TR><TD ALIGN="center">Rate of Profits</TD><TD ALIGN="center" COLSPAN="2">84.15%</TD></TR><TR><TD ALIGN="center" COLSPAN="3">Second Switch Point</TD></TR><TR><TD ALIGN="center">Wage</TD><TD ALIGN="center" COLSPAN="2">0.2376</TD></TR><TR><TD ALIGN="center">Rate of Profits</TD><TD ALIGN="center" COLSPAN="2">231.2%</TD></TR></TABLE><P>Small variations in the data defining the technology results in small variations in, for example, the maximum wages and the location of switch points. Decreased requirements for commodity inputs in production processes results in an outward movement of the wage-rate of profits curves and the outer frontier. But some changes resulting from these perturbations of the data are discontinuous. The maximum rate of profits is the most noticeable in this example. When iron is the only input in the iron-producing process, the maximum rate of profits for Alpha is 500%. (This maximum depends only on how much iron is required to produce a unit output of iron.) A perturbation that results in all three commodities being basic in both techniques abruptly lowers this maximum rate of profits to below 300%, the maximum rate of profits in the Beta technique in the original example. I also like that the perturbed model, with three basic commodities, removes the necessity for the convexity of a wage curve to be fixed in one direction for the entire curve. </P><B>4.0 Conclusion</B><P>This example has illustrated the transformation of a simple reswitching example, through perturbations, to another example, in which all commodities are basic. In this three-commodity example, with all commodities basic, the wage-rate of profits curve for the Alpha technique varies in convexity along its extent. Such a variation in convexity is a general property of multicommodity models of the production of commodities by means of commodity, but cannot be seen in two-commodity examples. </P>http://robertvienneau.blogspot.com/2016/12/perturbation-of-reswitching-example.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-16464628146087068Mon, 12 Dec 2016 18:43:00 +00002016-12-12T13:43:55.818-05:00Multiple Interest RatesTrivial Application of Multiple Interest Rate Analysis<P>I should have put the following in my <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882531">working paper</A>, on <I>Basic Commodities and Multiple Interest Rate Analysis</I>. This would go somewhere after Equation 10. </P><P>Let a technique of production be specified by a row vector, <B>a</B><SUB>0</SUB>, of labor coefficients and a square Leontief input-output matrix, <B>A</B>. The <I>j</I>th labor coefficient, <I>a</I><SUB>0,<I>j</I></SUB>, and the <I>j</I>th column, <B>a</B><SUB>.,<I>j</I></SUB>, of <B>A</B> represent the process for producing the <I>j</I> commodity when this technique is in use. </P><P>Consider a firm producing the <I>j</I>th commodity with this process. Suppose the firm faces prices of inputs and outputs, as represented by the row vector <B>p</B>. Let <I>w</I> be the given wage and <I>r</I> be the given rate of profits. Then the Net Present Value (NPV) for using this process, per unit output of the <I>j</I> commodity is: </P><BLOCKQUOTE>NPV<SUB><I>j</I></SUB>(<I>r</I>) = <I>p</I><SUB><I>j</I></SUB> - (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)(1 + <I>r</I>) </BLOCKQUOTE><P>Let <I>r</I><SUB>1</SUB> be the Internal Rate of Return (IRR) for this process. By definition, the NPV, evaluated for the IRR, is zero: </P><BLOCKQUOTE>NPV<SUB><I>j</I></SUB>(<I>r</I><SUB>1</SUB>) = 0 </BLOCKQUOTE><P>As the appendix proves, one can derive: </P><BLOCKQUOTE>NPV<SUB><I>j</I></SUB>(<I>r</I>) = - (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)(<I>r</I> - <I>r</I><SUB>1</SUB>) </BLOCKQUOTE><P>In words, when an investment project consists of one payout and one expenditure, with the payout coming one period after the expenditure, the Net Present Value of the investment is the additive inverse of the (first) expenditure, accumulated for one period at the difference between the given rate of profits and the Internal Rate of Return for the investment. Notice that NPV is only positive if the rate of profits used for accumulating costs falls below the internal rate of returns. </P><P>This is a trivial application of multiple interest rate analysis because it applies when the multiplicity is one. The above formulation of NPV was suggested to me, however, by first considering a non-trivial <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2869058">application</A>. </P><B>Appendix</B><P>By the definition of the IRR: </P><BLOCKQUOTE><I>r</I><SUB>1</SUB> = [<I>p</I><SUB><I>j</I></SUB>/(<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)] - 1 </BLOCKQUOTE><P>Substitute: </P><BLOCKQUOTE>- (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)(<I>r</I> - <I>r</I><SUB>1</SUB>) = - (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)<I>r</I> + <I>p</I><SUB><I>j</I></SUB> - (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>) </BLOCKQUOTE><P>Or:</P><BLOCKQUOTE>- (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)(<I>r</I> - <I>r</I><SUB>1</SUB>) = -(<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)(<I>r</I> + 1) + <I>p</I><SUB><I>j</I></SUB></BLOCKQUOTE><P>Which is to say: </P><BLOCKQUOTE>- (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)(<I>r</I> - <I>r</I><SUB>1</SUB>) = <I>p</I><SUB><I>j</I></SUB> - (<B>p</B> <B>a</B><SUB>.,<I>j</I></SUB> + <I>w</I> <I>a</I><SUB>0,<I>j</I></SUB>)(1 + <I>r</I>) </BLOCKQUOTE><P>But the term on the right is the definition of NPV. So the two expressions for NPV in the main text are equivalent. </P>http://robertvienneau.blogspot.com/2016/12/trivial-application-of-multiple.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-1441787396786203519Fri, 09 Dec 2016 12:23:00 +00002016-12-10T13:29:25.688-05:00Multiple Interest RatesBasic Commodities and Multiple Interest Rate AnalysisI have a new <A HREF="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2882531">working paper</A> on the Social Science Research Network: <BLOCKQUOTE><B>Abstract:</B> This paper considers the application of multiple interest rate analysis to a model of the production of commodities by means of commodities. A polynomial, for the characteristic equation of the augmented input-output matrix, is used in defining the rate of profits in such a model. Only one root is found to be economically meaningful. No non-trivial application of multiple interest rate analysis is found in the analysis of the choice of technique. On the other hand, multiple interest rate analysis can be used in defining Net Present Value in an approximate model, in which techniques are represented as finite series of dated labor inputs. The product of the quantity of the first labor input and the composite interest rate approaches, in the limit, the difference between the labor commanded by and the labor embodied in final output in the full model. </BLOCKQUOTE><P>I am proud of some observations in this paper. Nevertheless, I think it tries to go in too many directions at once. It is also longer than I like. It may seem, at first glance, to be longer than it is. I have ten graphs scattered throughout. </P><P>Michael Osborne cannot deny that I have taken his research seriously. He needs somebody with more academic credibility than me to write on his topic, though. </P><P>This is one paper where I would not mind being shown to be wrong. I did not find any use for more than one eigenvalue of what I am calling the augmented input-output matrix. If somebody can find something useful, along the line of multiple interest rate analysis, to say about all eigenvalues, I would be interested to hear of it. </P><P><B>Update:</B> I accidentally first posted without a "not" in the first sentence of the last paragraph. (I normally silently update typographic errors, but that one changes the meaning.) </P>http://robertvienneau.blogspot.com/2016/12/basic-commodities-and-multiple-interest.htmlnoreply@blogger.com (Robert Vienneau)0tag:blogger.com,1999:blog-26706564.post-4608848191342504323Tue, 06 Dec 2016 15:02:00 +00002016-12-06T10:02:09.479-05:00Example in Mathematical EconomicsMultiple Interest RatesBifurcations In Multiple Interest Rate Analysis<table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><a href="https://4.bp.blogspot.com/-qYzQv2Y7oKc/WEbNxSgu2xI/AAAAAAAAAsE/HYrjB-ff6PYoPEySBlwuTZ23tLkNi8VngCLcB/s1600/ThreePolynomials.jpg" imageanchor="1" ><img border="0" src="https://4.bp.blogspot.com/-qYzQv2Y7oKc/WEbNxSgu2xI/AAAAAAAAAsE/HYrjB-ff6PYoPEySBlwuTZ23tLkNi8VngCLcB/s320/ThreePolynomials.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 1: Three Trinomials</b></td></tr></tbody></table><B>1.0 Introduction</B><P>Typically, in calculating the Internal Rate of Return (IRR), a polynomial function arises. The IRR is the smallest, non-negative rate of profits, as calculated from a root of this function. The other roots are almost always ignored as having no economic meaning. </P><P>Michael Osborne, as I understand it, is pursuing a research project of investigating the use of all the roots of such polynomial functions that arise in financial analysis. A polynomial of degree <I>n</I> has <I>n</I> roots in the complex plane. I have noticed that the roots, other than the IRR, for examples that might arise in practice, can vary in whether they are real, repeating, or complex. </P><P>Bifurcation analysis, as developed for the study of dynamic systems might therefore have an application in multiple interest rate analysis. (This post is not about a dynamic system. I do not know how many of these <A HREF="https://www.google.com/?gws_rd=ssl#q=bifurcation+%22theory+of+equations%22">results</A> are about the theory of equations, independently of dynamical systems.) On the other hand, Osborne typically presents his analyses in terms of complex numbers. So I am not sure that he need care about these details. </P><B>2.0 An Example</B><P>Table 1 specifies the technology to be analyzed in this post. This technology produces an output of corn at the end of one specified year. The production of corn requires inputs of flows of labor in each of the three preceding years (and no other inputs). The labor inputs, per unit corn output, are listed in the table. </P><TABLE BORDER="1" ALIGN="center"><CAPTION><B>Table 1: The Technology</B></CAPTION><TR><TD ALIGN="center"><B>Year<BR>Before<BR>Output</B></TD><TD ALIGN="center" COLSPAN="2"><B>Labor Hired<BR>for Each Technique</B></TD></TR><TR><TD ALIGN="center"><B>1</B></TD><TD ALIGN="center"><I>L</I><SUB>1</SUB> = 0.18 Person-Years</TD></TR><TR><TD ALIGN="center"><B>2</B></TD><TD ALIGN="center"><I>L</I><SUB>2</SUB> = 4.468 Person-Years</TD></TR><TR><TD ALIGN="center"><B>3</B></TD><TD ALIGN="center"><I>L</I><SUB>3</SUB> = 0.527438298 Person-Years</TD></TR></TABLE><P>Let a unit of corn be the numeraire. Suppose firms face a wage of <I>w</I> and a rate of profits, <I>r</I>, to be used for time discounting. Wages are assumed to be advanced. That is, workers are paid at the start of the year for each year in which they supply flows of labor. Accumulate all costs to the end of the year in which the harvest occurs. Then the Net Present Value for this technology is: </P><BLOCKQUOTE>NPV(<I>r</I>) = 1 - <I>w</I>[<I>L</I><SUB>1</SUB>(1 + <I>r</I>) + <I>L</I><SUB>2</SUB>(1 + <I>r</I>)<SUP>2</SUP> + <I>L</I><SUB>3</SUB>(1 + <I>r</I>)<SUP>3</SUP>] </BLOCKQUOTE><P>The NPV is a third-degree polynomial. The wage can be considered a parameter. Figure 1, above, graphs this polynomial for three specific values of this parameters. In decreasing order, wages are 11/250, 11/500, and 2/250 bushels per person-years for these graphs. </P><P>Given the wage, the IRR is the intersection of the appropriate polynomial with the positive real axis in Figure 1. These IRRs are approximately 101.1%, 175.5%, and 329.5%, respectively. Suppose the economy were competitive, in the sense that capitalists can freely invest and disinvest in any industry. No barriers to entry exist. Then, if this technology is actually in use in producing corn and the wage were the independent variable, the rate of profits would tend to the IRR found for the wage. Profits and losses other than those earned at this rate of profits would be competed away. </P><P>The above graph suggests that, perhaps, the NPV for all wages intersects in two points, one of which is a local maximum. I do not know if this is so. Nor have I thought about why this might be. I guess it is fairly obvious that the local maximum is always at the same rate of profits. The wage drops out of the equation formed by setting the derivative of the NPV, with respect to the rate of profits, to zero. </P><P>I want to focus on the number of crossings of the real axis in the above graph. Figure 2 shows all roots of the polynomial equation defining the NPV. For a maximum wage, the IRR is zero, and it is greater to the right, along the real axis, for a smaller wage. The corresponding real roots, for the maximum wage, are the greatest and least negative rate of profits along the two loci shown in the left half of Figure 2. For smaller wages, these two real roots lie closer together, until around the middle wage used in constructing Figure 1, only one negative, repeated root exists. For any lower wage, the two roots that are not the IRR are complex conjugates. When the wage approaches zero, the workers live on air and all three roots go to (positive or negative) infinity. </P><table align="center" border="0" cellpadding="1" cellspacing="1"><tbody><a href="https://1.bp.blogspot.com/-v81qA6QD5xY/WEa-jzFvoyI/AAAAAAAAArc/qVwpc5ueldAtXJzGCOfh715gpzdDZNq2ACLcB/s1600/ThreeRoots.jpg" imageanchor="1" ><img border="0" src="https://1.bp.blogspot.com/-v81qA6QD5xY/WEa-jzFvoyI/AAAAAAAAArc/qVwpc5ueldAtXJzGCOfh715gpzdDZNq2ACLcB/s320/ThreeRoots.jpg" width="320" height="240" /></a></td></tr><tr><td align="center"><b>Figure 2: Multiple Rates of Profit for The Technique</b></td></tr></tbody></table><P>This post has presented an example for thinking about multiple interest rate analysis. It is mainly a matter of raising questions. I do not know how the mathematics for investigating these questions impacts practical applications of multiple interest rate analysis. </P>http://robertvienneau.blogspot.com/2016/12/bifurcations-in-multiple-interest-rate.htmlnoreply@blogger.com (Robert Vienneau)2