Optimal Price of Entry into a Competition

A continuum of contestants are choosing whether to enter a competition. Each contestant has a type, and of those who enter, the ones with highest types receive prizes. A profit-maximising firm controls entry, and charges a price for it. I analyse how the mass and value of prizes determine equilibrium price and intensity of entry. An increase in the value of each prize leads the firm to increase the price while keeping intensity of entry fixed. Conversely, when the mass of prizes increases, the firm initially keeps the price constant while entry increases; and later – raises the price.