In empirical structural demand estimation, a common assumption is that firms set prices at a static equilibrium, a profit-maximising monopoly price or a Nash equilibrium in oligopoly, and that observed prices reflect that equilibrium exactly. This assumption, however, leaves little scope to explain variation in observed prices around the equilibrium, how firms might experiment or drift, how multiple equilibria may arise, or how firms could benefit from better pricing. We introduce a framework of supply-side models in which firms draw prices from a distribution that is an increasing function of profits: firms are more likely to choose prices that yield higher profits, but may sometimes choose less profitable prices. Concretely, each firm samples price \(p\) from a density \(f(p) \propto g\bigl(\pi(p)\bigr)\), where \(\pi(p)\) is the firm's profit as a function of price and \(g(\cdot)\) is a strictly increasing link function.
This formulation has several appealing properties. First, the mode of the price distribution coincides with the classical profit-maximising or Nash equilibrium price. Second, the curvature of the profit function governs price dispersion: sharply peaked profits concentrate prices near the optimum; flat profits yield higher variance. Third, certain demand and link function combinations yield tractable parametric distributions; under linear demand and an exponential link, the induced distribution is normal. Fourth, the framework accommodates multiple equilibria via multimodal price distributions and is well-defined for discrete price supports.
We extend the framework to oligopoly by modelling a sequential profit-weighted simulation process, in which firms repeatedly draw profit-weighted price responses conditional on rivals' prices. This defines a Markov chain over the price vector. We prove that, under mild regularity conditions, the chain is uniformly geometrically ergodic with a unique stationary distribution whose mode coincides with the Nash equilibrium prices. For estimation, we implement an Exchange MCMC algorithm for the monopoly case and exploit the stationary distribution of the ergodic Markov chain as a direct likelihood for the oligopoly case with discrete prices.
