ON THE HIGHEST DEGREE OF SMOOTHNESS OF OUTCOME FUNCTIONS COMPATIBLE WITH QUANTITY CONSTRAINED EQUILIBRIUM NON-WALRASIAN PERFORMANCE (MECHANISM, MICROFOUNDATION, KEYNES)

In a class of exchange processes with an auctioneer, we ask: For outcome functions which embody quantity constrained behavior, what is the highest degree of smoothness that still allows the mechanism to realize a non-Walrasian allocation at an equilibrium? We show that the answer depends on whether the quantity constrained behavior is embodied in the price or the quantity outcome functions: (A) if quantity outcome functions, we can get non-Walrasian equilibrium allocations in the mechanism even if all quantity outcome functions have own-derivatives (but not cross-derivatives) everywhere; and (B) if price outcome functions, non-existence of even own-derivatives is necessary for the mechanism to achieve non-Walrasian performance. (In terms of the definition of (v) below, we are referring here to the derivatives of the perception function.); We adopt Hurwicz's resource allocation mechanism framework and study a mechanism for a two-person, two-good, pure-exchange economy with the following characteristics: (i) each consumer emits a quantity message concerning his excess demand for a non-numeraire good x; (ii) an auctioneer adjusts the price so as to clear the market; (iii) the budget equalities always hold; (iv) actual trades occur only at an equilibrium; (v) an outcome function is a composite of (1) a perception function and (2) two forecast functions. The forecast functions are functions of the other consumer's message and describe the upper and lower limits of a consumer's net trade, while the perception function transforms his attainable set from the whole price line into (a) a segment, or (b) a curve (some portion of which overlaps the price line). It turns out that the difference between (a) and (b) accounts for the difference between (A) and (B).; Our result (B), using price outcome functions, is analogous to Gale's result for Hahn's model (see Gale {lcub}1978{rcub}, Hahn {lcub}1978{rcub}). (A) and (B) answer, in the more general framework of resource allocation mechanisms, a version of Gale's question for both price and quantity outcome functions.