Market Equilibrium with Transaction Costs

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Identical products being sold at different prices in different locations is a common phenomenon. To model such scenarios, we supplement the classical Fisher market model by introducing transaction costs. For every buyer i and good j, there is a transaction cost of c ij ; if the price of good j is p j , then the cost to the buyer i per unit of j is p j  + c ij . The same good can thus be sold at different (effective) prices to different buyers. We provide a combinatorial algorithm that computes ε-approximate equilibrium prices and allocations in $O\left(\frac{1}{\epsilon}(n+\log{m})mn\log(B/\epsilon)\right)$ operations - where m is the number goods, n is the number of buyers and B is the sum of the budgets of all the buyers.