Complex market dynamics under Box–Cox monopoly

The monopoly (cartel) model is considered assuming that the monopolist tends to maximize expected profit with respect to output and price of some good. The demand function has a nonlinear form affiliated with the Box–Cox data transformation which depends on parameters determining the curvature of the demand. The supply reaction function under profit maximization is derived from the demand function. The situation where the demand is shifted and brings uncertainty to the market is considered. A simplified “price rule” for the search of the new optimal point gives the model features similar to the cobweb model. When the monopolist, under a time lag, tries to reach next period’s optimal point, the appearance of periodic and chaotic price oscillations is shown to depend on the curvature of the demand. The price dynamics are described by a one-dimensional map which is topologically conjugate in the specified parameter range to the well-known logistic map. Two-dimensional bifurcation diagrams in the parameter plane are obtained where regions of regular and chaotic behavior are indicated.