Economics Letters 26 (1988) l-5 North-Holland zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA THE EXCESS UTILITY FUNCTIONS AND THE WELFARE ADJUSTMENT PROCESS Pablo SERRA Uniuersidad de Chile, Casilla 2777, Santiago, Chile Received 24 September 1987 This paper introduces the excess utility functions which are shown to have all the properties the excess demand functions have. Then, these functions are used to simplify the proofs of existing results about the stability of the welfare adjustment process. 1. Introduction In the second section of this brief paper I discuss the welfare maximization problem (WMP) in the particular case in which the welfare function is linear in the individual utility functions. A condition is stated for the weights of the welfare function so that the solution of the WMP will be a competitive equilibrium. In section 3 I introduce the excess utility functions. These functions have, at least, all the properties the excess demand functions have and allow us to characterize the competitive equilibrium in a similar way to the excess demand functions. The similarities between both types of functions are used to study the stability of the welfare adjustment process. The results are the same as those already established by Mantel (1971); the originality of my approach lies in the treatment. Finally, the last section presents a proof of the existence of a solution to the general equilibrium problem. This proof was originally done by Negishi (1960). Now I present a simpler version that uses the excess utility function concept. 2. Tbe welfare maximization problem Assume an exchange economy with zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFE n consumers and m commodities, where 0’ = (wl,, . . . , L$,) represents the initial endowment of the ith consumer, and x1 = (xi,. . . , xk) his consumption bundle. Assume also that the consumer’s preferences can be represented by utility functions u’, i = 1,. . . , n which are strictly concave and locally non-saturating. Finally, assume that the social welfare can be represented by a linear function in the individual utility functions. Let a, denote the weight given to the ith consumer, then the mathematical formulation of the WMP is max 5 a,u’(x’), i=l n n s.t. c xi< c ai, i=l zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA r=l xi > 0, zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA i=l ‘...> n. (3) 0165-1765/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)