Strategic advance sales, demand uncertainty and overcommitment

We study a game in which producers can sell in two periods: one before a random demand is fully revealed and one after. This type of game corresponds to models of strategic forward trading or of advance sales to intermediaries or consumers. Demand variations and committed advance sales results in the possibility that the net residual demand in the final stage may be so low that it is not profitable for producers make additional sales, or indeed, may even drive the final period price to zero, introducing some convexity into producers’ payoffs. If this possibility of ex post overcommitment occurs on the equilibrium path, it reduces the level of advance sales chosen by producers, muting the pro-competitive effects found under deterministic demand. We establish a condition that determines whether or not demand uncertainty is “minor”, in the sense that the equilibrium depends only on the expected value of the demand shock. In addition, we demonstrate that when the support of demand shocks is narrow enough compared to the marginal cost of production, there exists a unique symmetric subgame-perfect equilibrium in pure strategies. When the support of demand shocks is wider, we establish a regularity condition on the distribution of demand shocks and the model parameters that ensures the existence of a unique equilibrium in pure strategies. We illustrate through examples that commonly used uni-modal distributions satisfy this condition, while bi-model distributions may not.