In this talk, I will present the proportional response dynamics and show its convergence to the market equilibrium in two families of economies.

In the proportional response dynamics, each user follows the intuitive behavior of responding his goods to the others proportional to the utility he received from each of them. Compared to the standard tatonnement and its variants, the proportional response dynamics is much simpler and ensures the clearance of the goods in the process.

I will show that, in spite of its simplicity, the dynamics converges to the market equilibrium in the Fisher market with constant elasticity of substitution (CES) utilities and in a general exchange economy that models peer-to-peer file downloading systems. In addition, when the utility functions are strictly concave, the dynamics converges rapidly, with a running time comparable to that of the existing optimization based methods. The proof uses Eisenberg-Gale characterization of the market equilibrium and makes connection to the classic matrix scaling process.