The Marxian transformation problem revisited Peter Fleissner,

1. The First Constitutive Congress of theWorld Advanced Research Project (WARP) Analytical Principles for the Transition Model to 21st Century Participatory DemocracyCTS) Universidad Autonoma Metropolitana, Mexico City The Marxian transformation problem revisited Peter Fleissner, transform!at, http://transform.or.at, TU-Vienna Vienna, Austria

2. Summary The 20th century was the battlefield of the “right” and “left” schools of economics with respect to the determination of the value/price of commodities. Marxian scholars insisted, that human labour being the essence of value to have a good argument for demonstrating exploitation, mainstream economists in the West focused on marginal utility theory to keeping up their basic axiom of methodological individualism. In the beginning of the 21st century this paper demonstrates how the Marxian labour theory of value and neo-classical economic analysis can be used simultaneously to see the transformation problem under a new, and joint perspective.

3. Introduction My understanding of the transformation problem: Marx posed it on the very abstract level of labour values only, neglecting possible changes on the level of corresponding use values He explained the modification of labour values of commodities as an effect of capitalist competition He assumed movements of capital will come to an end if profit rates are all equal In mathematically terms, he described the first step of an iterative process – like A. Kliman’s Temporal Single System Interpretation (TSSI). The repeated application of TSSI leads to the Simultaneous Single System Interpretation (SSSI) (i.e. input prices equal output prices) => SSSI and TSSI should not only be seen in opposition, but are at the same moment are very close to each other!

4. Transformation seen as “Gedankenexperiment” Two different “ideal types” (Max Weber) of economies: Petty commodity production: small competitive producers have ownership of the means of production, no wage labor, capital accumulation neglected. Examples: farmers, artisans, lawyers, physicians etc. Prices finally proportional to labor values. Capitalistproduction: wage labor, capitalistsmovetowardthesectorswithhighest rate ofprofit. Rates ofprofitsbecomeequalizedandarefinallyequaltotheaverage rate. (sinceFarjoun/Machoverthisis not realistic) Prices finally proportional tocapitaladvanced

5. Marx’ transformation problem: on a high level of abstraction If we want to apply it to actually existing economies empirically, we have also to add some non-essentials. I include marginalist theory of supply and demand By this we should come closer to empirical reality. Nevertheless, even with this extension we are far away from the surface we actually can see. In this paper we still do not include financial markets, we neglect the function of credits, we do not cover monopolistic power etc.

6. Economic Reality – A Complex Construction Contemporary Capitalism market prices (observed) 7 6 5 4 3 2 1 commodification of information goods/services Information Society: information as commodity, communication as commercial service Public sector taxes, subventions transfers, social insurance Globalized economy International financial capital markets for money, credit, stocks, derivatives Capitalism with perfect competition and fixed capital prices of production labor market Commodity production of self employed exchange values prices ~ labor values commodity/service markets Physical basis use values collective production/appropriation

7. New feature Changes in the demand of physical goods are possible and are computed. They are caused by price variations during transformation. Before transformation, the system of relative prices is proportional to labour values. After the transformation the relative prices represent a competitive capitalist price system where profit rates are equalized and consumers utility is maximized

8. Input-output scheme used (2 sectors) „=“ „+“ „+“

9. Marx’ version of the transformation problem* Marx started with prices proportional to labor values (we denote them by p0on unit level) and multiplied capital advanced by the average rate of profit π increased by 1. Prices of production p1: p1= pMarx= p0( A + C )( 1 + π ) , with ( 1 + π ) = p0 x / p0( A + C ) x , where p0 = f l ( E – A )-1, l …(row) vector of direct unit labor input f …proportionality factor (money value per labor value) f = pobs x / l ( E – A )-1x,pobsobserved unit prices. A…matrix of technical coefficients, C…consumption coeff. *without fixed capital

10. Bortkiewicz’ version of the transformation problem If we apply Marx’ method iteratively, p1= pMarx= p0( A + C )( 1 + π0 ) , by writing p i+1= p i ( A + C )( 1 + πi ) with ( 1 + πi ) = pi x / pi ( A + C ) x , pi converges to Bortkiewicz’ solution p¥ which is equal to the left eigenvector of the matrix (A+C) p¥ =p¥( A + C )( 1 + π¥ ) 1/( 1 + π¥ ) is equal to the largest eigenvalue of (A+C)

11. Bortkiewicz’ version of the transformation problem p i+1= p i ( A + C )( 1 + πi ), iterative solution:

12. Bortkiewicz’ version of the transformation problem keeps the value of total turnover invariant. Proof: By substituting 1+πi by 1 + πi = p ix / p i ( A + C ) x and right multiply the following equation pi+1 = p i ( A + C ) p ix / p i ( A + C ) x , by x, we arrive at pi+1x = p i ( A + C ) x [ p ix / p i ( A + C ) x ] and pi+1x = p ix. q.e.d.

13. A “more concrete” transformation problem After repetition of the basics demand functions are implemented. For reasons of simplicity change in demand only for consumers goods. The consumer demand functions may have the following simple form: {Cij } = { wj xj bij / pi } = diag-1(p) B diag(w) diag(x) where the bij’s are constants, wj..wage sums.  If one believes in utility functions one could derive the demand functions for each sector of production from logarithmic utility functions Nj. Maximize Nj w.r.t. a budget constraint Nj = b1j log( C1j )+ b2j log( C2j )+ lamdaj ( wj – p1 C1j – p2 C2j ), j = 1,2

14. A “more concrete” transformation problem Because by any transformation of prices final demand y will be affected, Leontief inverse is applied to determine x*, the output needed to produce y (inv is the given and constant physical column vector of capital investment goods) yi = Ci1 + Ci2 + invi , i = 1,2; x = ( E – A )-1y To perform the transformation we look for new relative prices p* and modified values of output x* that fulfil the following conditions: The first two equations for the vector variables x* and p* are described by x = ( E – A )-1 [ diag-1(p*) B diag(w) diag(x*) 1 + inv ], where B is a matrix of constants that determine consumer demand. inv is the column vector of capital investment goods.

15. A “more concrete” transformation problem The third equation equalizes the two industrial rates of profit. Capital advanced (including wages) per sector, K, are described a row vector K = p* (A diag(x*) + { Cij }) = p* [A diag(x*) + diag-1(p*) B diag(w) diag(x*)] K = p*A diag(x*) + 1’ B diag(w) diag(x*) By division of the elements of the row vector of the value of output 1’ diag(p*) diag(x*) by the respective elements of capital advanced, K, we get the industrial rates of profit, πi , or the growth of capital advanced, gj = (πi + 1). With these definitions we get the third equation from equal rates of profits by right-multiplication of the vectors K and the turnover 1’ diag(p*) diag(x*) by diag(x*)-1 as g1 = g2

16. A “more concrete” transformation problem or explicitly p1 / [ p1 a11 + p2 a21 + w1 (b11 + b21)] = p2 / [ p1 a12 + p2 a22 + w2 (b12 + b22)] The fourth and last equation assures the equality of the total value of output before and after the transformation p x = p*x*. We applied the software Maxima (see http://maxima.sourceforge.net/) to find the solution of the resulting polynomial of 4th order. The program came up with the following four solutions: Solution 1: p1 = 10.494, p2 = 0.941, x1 = 9.928, x2 = 101.780 Solution 2: p1 = 3.641, p2 = 0.308, x1 = 21.835, x2 = 390.685 Solution 3: p1 = 26.977, p2 = - 0.848, x1 = 8.750, x2 = 42.538 Solution 4: p1 = 0, p2 = - 1, x1 = 16, x2 = - 200. Only the first two solutions are feasible

17. A “more concrete” transformation problem * *For both industries we used the same utility function: N = Nj = 0.41667*log( C1j )+ 0.58333*log( C2j ), j = 1,2

18. Two iterative solutions are possible It is also possible to arrive at these two solutions by iteration, starting from labour values. As shown above for the solution of the classical transformation problem we can establish similar iteration processes. In the case of the concrete transformation we have to define two different ways how the iterations are defined. The first solution, close to the one by von Bortkiewicz can be found by firstly determining the overall rate of profit expressed at labour values. Secondly, multiplying cost prices at labour values by the factor (1 + rate of profit at labour values) results in a first approximation of prices of production p1.

19. Two iterative solutions possible With these prices consumption levels c1 are computed, which will change overall final demand y1. Right-multiplying the Leontief inverse (E – A)-1 by y1 results in changed output x1. To keep the total turnover at the same level we have to standardize p1 x1 =p0 x0. Applying the same steps again on the basis of p1and x1, and so on, we finally reach equilibrium with equal rates of profit in both sectors, and supply equals demand for consumer markets. The iteration process illustrates that starting in an economy with small commodity producers a new rule of price formation is established, leading step by step to prices of production in a capitalist economy.

20. Dxi+1 = f( xi ) - xi Dxi+1 = - ( f( xi ) - xi ) x2 x0 x1 x1 x2 x0 x1 x1 A more concrete transformation problem Utility function: Nj = d1j log( C1j )+ d2j log( C2j )+ lamdaj ( wj – p1 C1j – p2 C2j ), j = 1,2 Demand function : Cij = wj xj bij / pi = diag-1(p) B diag(w) diag(x) With 2 sectors: quadratic equation: p1 / [ p1 a11 + p2 a21 + w1 (b11 + b21)] = p2 / [ p1 a12 + p2 a22 + w2 (b12 + b22)] direct or iterative solution for prices and volumes/amounts

21. Results There is no longer only one solution, but two. As result of an iterative process, solution (4) is asymptotically stable, solution (5) is unstable. After the “concrete” transformation, there are higher utility levels for all groups of workers both solutions, (4) and (5), represent a kind of economic equilibrium (stationary points), because there is no need for capital to move to the other sector, as rates of profit are already equal.

22. Results Although all the unit wages (2 resp. 0,16 units) and also the wage per worker in the resp. industry (0,286 resp. 0,228 units) remain invariant over the “concrete” transformation, one can see from solution (5) that much more labour is needed for the economy than in all other cases. Open question: Can the unstable solution be transformed in a stable one?

23. Thanks for your attention Contact: fleissner@arrakis.es http://transform.or.at