Computing Equilibrium Prices in Exchange Economies with Tax Distortions

We consider the computation of equilibrium prices in market settings where purchases of goods are subject to taxation. While this scenario is the standard one in applied computational work, so far it has not been an object of study in theoretical computer science. Taxes introduce significant distortions: equilibria are no longer Pareto optimal, sufficient conditions for uniqueness do not continue to guarantee it, existence itself must be revisited. We analyze the effects of these distortions on scenarios which, in the absence of taxes, admit polynomial time algorithms, and prove a number of results: For Fisher's model with homothetic preferences: if consumers are subject to a uniform tax regime, the model loses the {\em representative consumer} and becomes prone to multiple disconnected equilibria; however we show that the distortion has a structure which leads to two (as opposed to one) representative consumers. We take advantage of this property to develop polynomial time algorithms, in spite of the presence of multiple disconnected equilibria. To obtain the result above, we develop a technique to estimate the sensitivity of Fisher's equilibrium prices with respect to the consumers' income, which might find wider applications in the field. For the exchange model with Cobb-Douglas utility functions: if consumers are subject to a differentiated tax regime, we show that the model is equivalent to a Cobb-Douglas exchange economy with an extra good. This implies that the polynomial time computability is preserved. For the exchange model with utility functions satisfying weak gross substitutability: if consumers are subject to a differentiated tax regime, the techniques based on convex programming seem to break down. However we are again able to preserve the polynomial time computability, by means of suitable modifications of certain combinatorial algorithms.