A Tale of Two Monopolies

We apply marginal analysis à la Bulow and Roberts (1989) to characterize revenue-maximizing selling mechanisms for a multiproduct monopoly. We derive marginal revenue from price perturbations over arbitrary sets of bundles and show that optimal mechanisms admit no revenue-increasing perturbation for bundles with positive demand, nor revenue-decreasing perturbations for zero-demand bundles. For any symmetric two-dimensional type distribution under mild regularity, this analysis fully characterizes the optimal mechanism across independence, substitutability, and complementarity. For general type distributions and allocation spaces, our approach identifies bundles that must carry positive demand and provides conditions under which pure bundling or separate selling is suboptimal.

💡 Research Summary

The paper revisits the classic problem of revenue maximization for a monopolist selling multiple products, extending the marginal‑revenue framework of Bulow and Roberts (1989) to a multidimensional setting. The authors introduce a marginal‑revenue (MR) functional defined with respect to infinitesimal price perturbations on arbitrary bundles. Their main result, Theorem 1, states two complementary optimality conditions: (i) for any bundle that attracts positive demand, a marginal increase in its price cannot raise expected revenue (First‑Order Condition, FOC); (ii) for any bundle with zero demand, a marginal price decrease cannot raise revenue (Second‑Order Condition, SOC). These conditions are purely primal, avoiding the dual transport‑measure machinery employed in the Daskalakis‑Dekel‑Tzamos (DDT) literature.

The authors apply the framework to the leading case of two additive items with a symmetric two‑dimensional type distribution (SRS). They define a threshold function ζ(x₁) that marks the price level where MR for the bundle (u′(x₁), 1) is zero. The shape of ζ determines the structure of the optimal mechanism: if ζ is non‑decreasing, the optimal menu is deterministic; if ζ is non‑increasing and concave, the optimal menu may be infinite (probabilistic bundling); if ζ is non‑monotone but concave (e.g., truncated normal), at most two cut‑offs appear, leading to a mixed deterministic‑probabilistic mechanism. For the uniform distribution on