When Firm-Level Demand with Price Competition and Inverse Demand with Quantity Competition Coincide

The first part of the paper reviews an analytical method for inverting demand in price competition to find the corresponding inverse demand under quantity competition. The method yields a simple function relating demand elasticities with inverse demand elasticities. The second part of the paper uses this relationship to determine the conditions under which demand and inverse demand are the same. It is found that demand and inverse demand coincide, with both price and cross-price elasticities equal, when the firm's price elasticity of demand is proportional to the number of competitors the firm faces in the product group. The constant of proportionality is found to equal the product group elasticity of demand. The coincidence of demand and inverse demand is found under conditions of perfectly isolated duopoly all the way continuously through to the case of perfectly homogeneous competition among a very large number of firms in which any degree of insulation that may have existed elsewhere along the way has been removed