We analyze the two-bidder discriminatory auction with downward sloping marginal valuations and a continuous, variable award. We allow for a common component in marginal valuations and affiliation. We focus on problems that admit solutions with strictly downward sloping bidding schedules. Using the method of characteristics, we reduce the first order conditions to a pair of ordinary differential equations where we fix the equilibrium quantities that are allocated to the two bidders. These ordinary differential equations are extensions of the Milgrom-Weber equation for the first price unit auction. A new ordinary differential equation that characterizes the relation between signals yielding the fixed quantity allocation is obtained. The equilibrium solution of the discriminatory auction is given by the solution of an initial value problem for these two coupled ODEs that generalize the Milgrom-Weber equation. We show the conditions for existence of solutions to this systems of ODEs. Some examples are analytically and numerically solved using our approach.
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