T\^atonnement Beyond Constant Elasticity of Substitution

In this paper, we bring consumer theory to bear in the analysis of Fisher markets whose buyers have arbitrary continuous, concave, homogeneous (CCH) utility functions representing locally non-satiated preferences. The main tools we use are the dual concepts of expenditure minimization and indirect utility maximization. First, we use expenditure functions to construct a new convex program whose dual, like the dual of the Eisenberg-Gale program, characterizes the equilibrium prices of CCH Fisher markets. We then prove that the subdifferential of the dual of our convex program is equal to the negative excess demand in the associated market, which makes generalized gradient descent equivalent to computing equilibrium prices via t\^atonnement. Finally, we use our novel characterization of equilibrium prices via expenditure functions to show that a discrete t\^atonnement process converges at a rate of $O\left(\frac{1}{t}\right)$ in Fisher markets with continuous, strictly concave, homogeneous (CSCH) utility functions -- a class of utility functions beyond the class of CES utility functions, the largest class for which convergence results were previously known. CSCH Fisher markets include nested and mixed CES Fisher markets, thus providing a meaningful expansion of the space of Fisher markets that is solvable via t\^atonnement.