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Competitive and Revenue-Optimal Pricing with Budgets

Simon Finster, Paul W. Goldberg, Edwin Lock

TL;DR

This paper studies budget-constrained buyers in markets with multiple divisible goods under a price-only mechanism. It introduces constrained social welfare and shows that competitive equilibria maximize it, while budgets can prevent classical utilitarian efficiency. When buyers have linear valuations, the authors prove a revenue-welfare coincidence: the competitive equilibrium prices are unique and revenue-maximising for a zero-cost seller, and these prices also maximize the quantity allocated to buyers. The results have practical implications for digital advertising platforms and financial asset markets, suggesting price-based mechanisms can align welfare objectives with seller revenue under budget constraints.

Abstract

In markets with budget-constrained buyers, competitive equilibria need not be efficient in the utilitarian sense, or maximise the seller's revenue. We consider a setting with multiple divisible goods. Competitive equilibrium outcomes, and only those, are constrained utilitarian efficient, a notion of utilitarian efficiency that respects buyers' demands and budgets. Our main contribution establishes that, when buyers have linear valuations, competitive equilibrium prices are unique and revenue-optimal for a zero-cost seller.

Competitive and Revenue-Optimal Pricing with Budgets

TL;DR

This paper studies budget-constrained buyers in markets with multiple divisible goods under a price-only mechanism. It introduces constrained social welfare and shows that competitive equilibria maximize it, while budgets can prevent classical utilitarian efficiency. When buyers have linear valuations, the authors prove a revenue-welfare coincidence: the competitive equilibrium prices are unique and revenue-maximising for a zero-cost seller, and these prices also maximize the quantity allocated to buyers. The results have practical implications for digital advertising platforms and financial asset markets, suggesting price-based mechanisms can align welfare objectives with seller revenue under budget constraints.

Abstract

In markets with budget-constrained buyers, competitive equilibria need not be efficient in the utilitarian sense, or maximise the seller's revenue. We consider a setting with multiple divisible goods. Competitive equilibrium outcomes, and only those, are constrained utilitarian efficient, a notion of utilitarian efficiency that respects buyers' demands and budgets. Our main contribution establishes that, when buyers have linear valuations, competitive equilibrium prices are unique and revenue-optimal for a zero-cost seller.
Paper Structure (11 sections, 11 theorems, 7 equations, 2 figures)

This paper contains 11 sections, 11 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Examples: Efficiency, Constrained Efficiency, and Revenue
  4. The Market
  5. Market Outcomes
  6. Constrained Efficiency
  7. The Revenue-Welfare Coincidence
  8. Elementwise-Minimal Feasible Prices
  9. Maximising Revenue and Welfare
  10. Arctic Auctions
  11. Conclusion

Key Result

Proposition 1

Let $v$ be differentiable and strongly concave with parameter $m$ for some $m>0$, and let $\beta = \infty$. Then there exists some supply $s \in \mathbb{R}$ so that revenue is not maximised at the market-clearing price.

Figures (2)

  • Figure 1: The feasible region in price space corresponding to \ref{['example:two']}.
  • Figure 2: The demand of buyers with quasi-linear utilities, linear valuations and budget constraints, which divides price space into convex regions. In each region, we specify the bundle demanded, which depends on prices $\bm{p}$. Left: The demand of a single buyer with linear values $\bm{v} = (5,3)$ and budget $\beta$ leads to three regions. Right: The aggregate demand of two buyers, one with values $\bm{v}^1 = (3,6)$ and budget $\beta^1$, and the other with values $\bm{v}^2 = (6,3)$ and budget $\beta^2$.

Theorems & Definitions (28)

  • Example 1
  • Example 2
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Example 3
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • ...and 18 more