Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Tale of Two Monopolies

Yi-Chun Chen, Zhengqing Gui

TL;DR

This paper develops a margin-based approach to optimal selling mechanisms for multiproduct monopolies by perturbing prices over bundles and using the marginal-revenue (MR) calculus. Grounded in the taxation principle, it recasts incentive-compatibility constraints into self-selection pricing and derives first- and second-order conditions (FOCs and SOC) that pin down the structure of revenue-maximizing mechanisms. For symmetric two-dimensional (and, more generally, convex type/allocation spaces) under mild regularity, the framework yields complete characterizations across independence, substitutability, and complementarity, linking MR to the geometry of demand sets via a threshold function $\zeta$ and a one-dimensional indirect utility $\bar u$. The results explain when pure bundling, separate selling, or probabilistic/mixed bundling is optimal, and uncover general features such as pooling bundles, active bundles, and the possible existence of an exclusion set. Overall, the MR approach provides a unified, tractable route to tractable optimal mechanisms in multidimensional screening with broad applicability and clear comparative statics with respect to substitution/complementarity.

Abstract

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.

A Tale of Two Monopolies

TL;DR

This paper develops a margin-based approach to optimal selling mechanisms for multiproduct monopolies by perturbing prices over bundles and using the marginal-revenue (MR) calculus. Grounded in the taxation principle, it recasts incentive-compatibility constraints into self-selection pricing and derives first- and second-order conditions (FOCs and SOC) that pin down the structure of revenue-maximizing mechanisms. For symmetric two-dimensional (and, more generally, convex type/allocation spaces) under mild regularity, the framework yields complete characterizations across independence, substitutability, and complementarity, linking MR to the geometry of demand sets via a threshold function and a one-dimensional indirect utility . The results explain when pure bundling, separate selling, or probabilistic/mixed bundling is optimal, and uncover general features such as pooling bundles, active bundles, and the possible existence of an exclusion set. Overall, the MR approach provides a unified, tractable route to tractable optimal mechanisms in multidimensional screening with broad applicability and clear comparative statics with respect to substitution/complementarity.

Abstract

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.

Paper Structure

This paper contains 49 sections, 38 theorems, 118 equations, 14 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Main Theorem
  4. Demand Set and Marginal Revenue.
  5. Statement of the Main Theorem.
  6. When is MR well-defined?
  7. A Sufficient Condition for MR=0.
  8. Discussion of the Literature
  9. Two Additive Products
  10. Optimal Mechanisms
  11. Nondecreasing or Convex $\zeta$
  12. Concave $\zeta$
  13. Sufficiency
  14. Optimality of Bundling Strategies
  15. Summary
  16. ...and 34 more sections

Key Result

Lemma 1

For any feasible mechanism, if $Q \subseteq \mathcal{Q}$ is Borel measurable with $\mathbf{0} \notin Q$, then Moreover, if $\mu (\mathcal{D}(Q)\cap \mathcal{D}(Q^{c}))=0$, then $\mathop{\mathrm{MR}}\nolimits (Q)$ exists and satisfies

Figures (14)

  • Figure 1: Two products with uniformly distributed types.
  • Figure 2: Substitutes or complements with uniformly distributed types.
  • Figure 3: Uniform distribution on $[\underline{x},\underline{x}+1]^2$ with $\underline{x}>0$.
  • Figure 4: Using $\zeta$ to verify \ref{['Eqn_MR']} and \ref{['Eqn_SOC']}.
  • Figure 5: Nondecreasing or convex $\zeta$.
  • ...and 9 more figures

Theorems & Definitions (64)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • proof
  • Example 1: Uniform Distribution Revisited
  • Lemma 3
  • Definition 1
  • Lemma 4
  • Proposition 1
  • Corollary 1
  • ...and 54 more