The equivalence of price and quantity competition with delegation

In a two-stage differentiated-products oligopoly model, profit-maximizing owners first choose incentive schemes in order to influence their managers' behavior. In the second stage, the managers compete either both in prices, both in quantities, or one in price and the other in quantity. If the owners have sufficient power to manipulate their managers' incentives, the equilibrium outcome is the same regardless of how the firms compete in the second stage. If demand is linear and marginal cost is constant, basing the manager's objective function on a linear combination of the firm's profit and its rival's profit is sufficient for the equivalence result.