Az egyensúlyi ráták unicitása és a bérráta pozitivitása a Neumann-modell általánosításaiban (Uniqueness of equilibrium rates and the positive value of the wage rate in generalizations from the Neumann model)

Cikkünk arról a paradox jelenségről szól, hogy a fogyasztást explicit módon megjelenítő Neumann-modell egyensúlyi megoldásaiban a munkabért meghatározó létszükségleti termékek ára esetenként nulla lehet, és emiatt a reálbér egyensúlyi értéke is nulla lesz. Ez a jelenség mindig bekövetkezik az olyan dekomponálható gazdaságok esetén, amelyekben eltérő növekedési és profitrátájú, alternatív egyensúlyi megoldások léteznek. A jelenség sokkal áttekinthetőbb formában tárgyalható a modell Leontief-eljárásra épülő egyszerűbb változatában is, amit ki is használunk. Megmutatjuk, hogy a legnagyobbnál alacsonyabb szintű növekedési tényezőjű megoldások közgazdasági szempontból értelmetlenek, és így érdektelenek. Ezzel voltaképpen egyrészt azt mutatjuk meg, hogy Neumann kiváló intuíciója jól működött, amikor ragaszkodott modellje egyértelmű megoldásához, másrészt pedig azt is, hogy ehhez nincs szükség a gazdaság dekomponálhatóságának feltételezésére. A vizsgált téma szorosan kapcsolódik az általános profitráta meghatározásának - Sraffa által modern formába öntött - Ricardo-féle elemzéséhez, illetve a neoklasszikus növekedéselmélet nevezetes bér-profit, illetve felhalmozás-fogyasztás átváltási határgörbéihez, ami jelzi a téma elméleti és elmélettörténeti érdekességét is. / === / In the Marx-Neumann version of the Neumann model introduced by Morishima, the use of commodities is split between production and consumption, and wages are determined as the cost of necessary consumption. In such a version it may occur that the equilibrium prices of all goods necessary for consumption are zero, so that the equilibrium wage rate becomes zero too. In fact such a paradoxical case will always arise when the economy is decomposable and the equilibrium not unique in terms of growth and interest rate. It can be shown that a zero equilibrium wage rate will appear in all equilibrium solutions where growth and interest rate are less than maximal. This is another proof of Neumann's genius and intuition, for he arrived at the uniqueness of equilibrium via an assumption that implied that the economy was indecomposable, a condition relaxed later by Kemeny, Morgenstern and Thompson. This situation occurs also in similar models based on Leontief technology and such versions of the Marx-Neumann model make the roots of the problem more apparent. Analysis of them also yields an interesting corollary to Ricardo's corn rate of profit: the real cause of the awkwardness is bad specification of the model: luxury commodities are introduced without there being a final demand for them, and production of them becomes a waste of resources. Bad model specification shows up as a consumption coefficient incompatible with the given technology in the more general model with joint production and technological choice. For the paradoxical situation implies the level of consumption could be raised and/or the intensity of labour diminished without lowering the equilibrium rate of the growth and interest. This entails wasteful use of resources and indicates again that the equilibrium conditions are improperly specified. It is shown that the conditions for equilibrium can and should be redefined for the Marx-Neumann model without assuming an indecomposable economy, in a way that ensures the existence of an equilibrium unique in terms of the growth and interest rate coupled with a positive value for the wage rate, so confirming Neumann's intuition. The proposed solution relates closely to findings of Bromek in a paper correcting Morishima's generalization of wage/profit and consumption/investment frontiers.

On axiomatizations of the Shapley value for assignment games

We consider the problem of axiomatizing the Shapley value on the class of assignment games. We first show that several axiomatizations of the Shapley value on the class of all TU-games do not characterize this solution on the class of assignment games by providing alternative solutions that satisfy these axioms. However, when considering an assignment game as a communication graph game where the game is simply the assignment game and the graph is a corresponding bipartite graph buyers are connected with sellers only, we show that Myerson's component efficiency and fairness axioms do characterize the Shapley value on the class of assignment games. Moreover, these two axioms have a natural interpretation for assignment games. Component efficiency yields submarket efficiency stating that the sum of the payoffs of all players in a submarket equals the worth of that submarket, where a submarket is a set of buyers and sellers such that all buyers in this set have zero valuation for the goods offered by the sellers outside the set, and all buyers outside the set have zero valuations for the goods offered by sellers inside the set. Fairness of the graph game solution boils down to valuation fairness stating that only changing the valuation of one particular buyer for the good offered by a particular seller changes the payoffs of this buyer and seller by the same amount.