From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Tue Aug 21 2007 - 08:45:13 EDT
You elided the key passagea in your quote. Here is the full text : The standard method of solving simultaneous equations is Gaussian elimination. 4 It is equivalent to the school textbook method. This method yields an exact solution in a running time proportional to the cube of the number of equations (see Sedgewick, 1983, chapter 5).5 Let us suppose that the number of distinct types of output in the economy to be planned is of the order of a million (106). In that case the Gaussian elimination method applied to the input–output table would require 106 cubed or 1018 (a million million million) iterations, each of which might contain ten primitive computer instructions. Suppose we can run the problem on a modern Japanese supercomputer such as the Fujitsu VP200 or the Hitachi S810/20, then how long will it take? These machines are capable of performing around 200 million arithmetic operations per second when working on large volumes of data (see Lubeck et al., 1985).6 So the time taken to compute all of the labour values of the economy would be on the order of 50 billion seconds or 16 thousand years. Rather obviously, this is far too slow. When one runs into a scale problem like this it is often convenient to reformulate the task in different terms. The input–output table for an economy is in practice likely to be mostly blanks. In reality each product has on average only a few tens or at most hundreds of inputs to its production rather than a million. This makes it more economical to represent the system in terms of a vector of lists rather than a matrix. In consequence, there are short-cuts which can be taken to arrive at a result. We can use another approach, that of successive approximation. The idea here is that as a first approximation we ignore all inputs to the production process apart from directly expended labour. This gives us a first, approximate estimate of each product’s labour value. It will be an underestimate because it ignores the non-labour inputs to the production process. To arrive at our second approximation we add in the non-labour inputs valued on the basis of the labour values computed in the first phase. This will get us one step closer to the true labour values. Repeated application of this process will give us the answer to the desired degree of accuracy. If about half the value of an average product is derived from direct labour inputs then each iteration round our approximation process will add one binary digit of significance to our answer. An answer correct to four significant decimal digits (which is better than the market can achieve) would require about 15 iterations round our approximation process. The time order complexity of this algorithm7 is proportional to the number of products times the average number of inputs per product, times the desired accuracy of the result in digits. On our previous assumptions this could be computed on a supercomputer in a few minutes, rather than the thousands of years required for Gaussian elimination.8 We say that Guassian elimination is inefficient for large sparse economic matrices, and that iterative solutions are to be preferred. 4We start off with n equations in n unknowns. These can be reduced to n − 1 equations in n − 1 unknowns by adding appropriate multiples of the nth equation to each of the first n−1 equations. This step is then iterated until eventually we have 1 equation in 1 unknown. This is immediately soluble. We then back-substitute this result in the immediately preceding system of 2 equations in 2 unknowns, and so on. 5The intuition behind this is simple. For each of the variables eliminated we must perform n(n − 1) multiplications. There are n variables to eliminate, hence the complexity of the problem is of order n3. 6It should be borne in mind that computer technology has advanced considerably since the mid ’80s. By the mid 1990s manufacturers hope to deliver machines capable of about 1 million million operations per second. From: OPE-L [mailto:OPE-L@SUS.CSUCHICO.EDU] On Behalf Of Alejandro Agafonow Sent: 20 August 2007 23:09 To: OPE-L@SUS.CSUCHICO.EDU Subject: Re: [OPE-L] A startling quotation from Engels So I have misread your Towards a New Socialism, Paul. But what did you try to say below?: «The conditions of production can be represented as an input–output table, and from this table a set of equations can be derived, as in the examples above. In principle, these are clearly solvable —we have the same number of equations as we have unknown labour values to solve for. The question is whether the system is practically solvable. […] The standard method of solving simultaneous equations is Gaussian elimination. It is equivalent to the school textbook method. This method yields an exact solution in a running time proportional to the cube of the number of equations. […] An answer correct to four significant decimal digits (which is better than the market can achieve) would require about 15 iterations round our approximation process. […] On our previous assumptions this could be computed on a supercomputer in a few minutes, rather than the thousands of years required for Gaussian elimination.» (49-50) Alejandro Agafonow ----- Mensaje original ---- De: Paul Cockshott <wpc@DCS.GLA.AC.UK> Para: OPE-L@SUS.CSUCHICO.EDU Enviado: lunes, 20 de agosto, 2007 22:46:54 Asunto: Re: [OPE-L] A startling quotation from Engels Alejandro wrote: ---------------- Kantorovich so willing to these arrangements wrote: «The computation of optimal solution has its difficulties as well. In spite of the presence of efficient algorithms and codes practical linear programmes are not too simple since they are very large. The difficulties grow significantly when the linear model is modified by any of its generalization.» (pp. 21 in Leonid Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", The American Economic Review, Vol. 79, Nº6.) C&C have proposed to use the method of Gaussian elimination to deal with the empty boxes in a Leontief matrix, -------------------- This is a bit of confusion Alejandro. We argue against the use of Guassian elimination for computing labour values at the economy wide level since Guassian elimination has a computational time that is proportional to the cube of the number of commodities. Instead what we propose is an iterative approximation technique, which is what we have used in all our empirical research. This is much faster for really large problems. With respect to Kantorovich, he was certainly right at the time he wrote, since the best method then known of doing linear optimisation - the simplex approach, was in the worst case exponential in complexity. IN 1984 Karkamar developed a new interior point method for linear programming which markedly improved performance. We were unaware of Karkamars work when we wrote our book in the late 80s but we proposed instead another interior point method of our own for plan optimisation which also appears to be better than the simplex method. I will be visiting Dr Gondzio at Edinburgh University tomorrow to discuss the work he has been doing on solving linear programming problems with one billion variables with novel algorithms. So this area is experiencing very significant advances in the last few years. Paul Cockshott www.dcs.gla.ac.uk/~wpc -----Original Message----- From: OPE-L on behalf of Alejandro Agafonow Sent: Mon 8/20/2007 5:55 PM To: OPE-L@SUS.CSUCHICO.EDU Subject: [OPE-L] A startling quotation from Engels Dear Prof. Claus, Bullock and Bendien: If you equate «exchange value» with «self-interested production aimed by profit», then your intent to liquidate the law of value has sense, but I'm not sure you are meaning this. It seems to me that you equate «exchange value» with a social organization of production that allows social division of labour based on decentralization and pricing. If your liquidation of the law of value intends to erase decentralization and pricing, your model of socialism only could function with self-sufficient small production units -and its feasibility has a very limited sense. Prof. Claus is right when he said that the acid test of value (use value) is the marketability of the product of labour: «If the product of labor is unable to be sold, this means that the society has not needed it, and consequently the labor spent in its production is not social labor. And to the extent that value is the expression of social labor, such a product has no value.» But how can you guarantee a proper scale of marketability in the framework of a self-sufficient small production units? I think Prof. Claus are thinking in this kind of self-sufficient units when wrote: «If there is no exchange, every production unit or community has to produce its essential means of consumption and production under the conditions in which it produces [.] If the production is for self-consumption, the product is usefull whatever the time required to produce it [.]» I think that all you have the temptation to equate «exchange value» with the property «marketability» of labour's products. That temptation comes from the atavism of the round-about argument in Marx & Engels. Due to the production scale of the nowadays industrial economies any decision of the usefulness of products must be done in the framework of a consumption market, but is precisely this link in the production process of a large socialist economy that is missing in Marx, Engels and Bullock, when the last one wrote: «With socialism and then communism [.] A choice will be made about the usefulness of products, and the quantity to be produced, not their exchange value, which will not exist.» Fortunately, Cockshott & Cottrell are very clear in this respect. They allow a full consumer market mechanism. Finally, unless you adopt the surprisingly unorthodox interpretation of Engels concerning the depreciation of labour content if market doesn't reach the point where commodities are fully useful, you'll incur in a «dichotomy» saying that a product has value (labour) because a human being produced it, but also because it is of value (usefulness). Bendien incurred in this dichotomy when he wrote: «[.] "products of labour" qua use-values have values, to be precise, commodities have values ONLY BECAUSE they are products of general human labour which, therefore, have values. You can of course now turn all this around, and argue that products of labour have values ONLY IF they are commodities, but that is not Marx's argument. His argument here is clearly that products (use-values) have value, because they are products of human labour as such, human labour in general.» Bullock: «[.] the use of computers to measure the time it took to produce goods and services, was well discussed eg by the late 1960's the Institute of Workers Control ( Nottingham UK) was already proposing such technical solutions to the problems of production and distribution by state owned operations.» Are there any publication concerning these works? I didn't know it. One thing is to desire that at some point in the future computers will help us to record and properly allocate labour time. In the USSR the researches concerning these possibilities existed and, as Cockshott wrote, Market Socialists like Lange envisaged that possibility. Even Kantorovich so willing to these arrangements wrote: «The computation of optimal solution has its difficulties as well. In spite of the presence of efficient algorithms and codes practical linear programmes are not too simple since they are very large. The difficulties grow significantly when the linear model is modified by any of its generalization.» (pp. 21 in Leonid Kantorovich, "Mathematics in Economics: Achievements, Difficulties, Perspectives", The American Economic Review, Vol. 79, Nº6.) C&C have proposed to use the method of Gaussian elimination to deal with the empty boxes in a Leontief matrix, but what is more interesting from an institutional point of view is their thoughts concerning a feasible mechanism to feed the allocation machinery with human preferences as they are, using personal computers. Kind regards Alejandro Agafonow ____________________________________________________________________________________ Sé un Mejor Amante del Cine ¿Quieres saber cómo? ¡Deja que otras personas te ayuden! http://advision.webevents.yahoo.com/reto/entretenimiento.html ________________________________ Sé un Mejor Amante del Cine ¿Quieres saber cómo? ¡Deja que otras personas te ayuden! <http://us.rd.yahoo.com/mail/es/tagline/beabetter/*http:/advision.webevents.yahoo.com/reto/entretenimiento.html> .
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