Result: Existence and computation of short-run equilibria in economic geography

The new economic geography literature provides a general equilibrium framework that explains the emergence of economic agglomerations as a trade-off between increasing returns at the firm level and transportation costs related to the shipment of goods. The existence and uniqueness of short-run equilibria of this model has been shown for the case of two regions. The proposed approach employs the differential evolution algorithm to obtain estimates of the Lipschitz constant and the infinity norm of the function along the boundary and utilizes these values to investigate the existence of solutions of a function, and the computational burden of computing the topological degree of this function. This approach is employed to investigate the existence of short-run equilibria for more than two regions using fixed point and topological degree theory, as well as, the differential evolution algorithm. Irrespective of parameter settings the criteria from topological degree theory suggest that the model can have equilibria. The differential evolution algorithm identified such equilibria and for none of the parameter settings that were considered more than one equilibria were detected. The experimental results obtained also indicate that the computation of such equilibria has an exponential worst-case lower bound complexity, as the model yields a function that is neither contractive, nor nonexpanding. Finally, the computation of the topological degree to identify the number of equilibria also has a very high computational cost.