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Screening in digital monopolies

Pietro Dall'Ara, Elia Sartori

TL;DR

This paper extends classic quality-based screening to a setting with nonseparable production costs that depend on the top quality produced, capturing features of digital goods where replication is costless. The monopolist chooses a top quality $q^{M}$ and then degrades to lower reliabilities, resulting in two intertwined inefficiencies: productive underinvestment in the highest quality and a distributional downgrading for some buyers, with $q^{M}<q^{oldsymbol{ ag{star}}}$ and a cap-driven allocation $m{q}^{M}( heta)=m{eta}( heta) ext{ capped at }q^{M}$. Competition intensifies productive inefficiency but can improve distributional efficiency, producing a contraction of the monopolist allocation; the results depend on the shape of production costs and utility curvature. The framework offers interpretable insights for digital markets, where development costs are upfront and distributional rents interact with the top-quality externality, and provides extensions to no-screening settings and multi-firm competition. Overall, the paper delivers a tractable, general mechanism for screening with top-quality coupling and delivers both qualitative and quantitative predictions about how costs and competition shape welfare in digital economies.

Abstract

A defining feature of digital goods is that replication and degradation are costless: once a high-quality good is produced, low-quality versions can be created and distributed at no additional cost. This paper studies quality-based screening in markets for digital goods, exploring how the insights of the canonical model of Mussa and Rosen (1978) change when production costs are nonseparable and, instead, depend only on the highest quality developed. The monopolist allocation exhibits two interdependent inefficiencies. First, a productive inefficiency arises: the monopolist underinvests in the highest quality relative to the efficiency benchmark. Second, due to a distributional inefficiency, certain buyers receive degraded versions of the produced good. Competition exacerbates productive inefficiency, but improves distributional efficiency relative to monopoly.

Screening in digital monopolies

TL;DR

This paper extends classic quality-based screening to a setting with nonseparable production costs that depend on the top quality produced, capturing features of digital goods where replication is costless. The monopolist chooses a top quality and then degrades to lower reliabilities, resulting in two intertwined inefficiencies: productive underinvestment in the highest quality and a distributional downgrading for some buyers, with and a cap-driven allocation . Competition intensifies productive inefficiency but can improve distributional efficiency, producing a contraction of the monopolist allocation; the results depend on the shape of production costs and utility curvature. The framework offers interpretable insights for digital markets, where development costs are upfront and distributional rents interact with the top-quality externality, and provides extensions to no-screening settings and multi-firm competition. Overall, the paper delivers a tractable, general mechanism for screening with top-quality coupling and delivers both qualitative and quantitative predictions about how costs and competition shape welfare in digital economies.

Abstract

A defining feature of digital goods is that replication and degradation are costless: once a high-quality good is produced, low-quality versions can be created and distributed at no additional cost. This paper studies quality-based screening in markets for digital goods, exploring how the insights of the canonical model of Mussa and Rosen (1978) change when production costs are nonseparable and, instead, depend only on the highest quality developed. The monopolist allocation exhibits two interdependent inefficiencies. First, a productive inefficiency arises: the monopolist underinvests in the highest quality relative to the efficiency benchmark. Second, due to a distributional inefficiency, certain buyers receive degraded versions of the produced good. Competition exacerbates productive inefficiency, but improves distributional efficiency relative to monopoly.
Paper Structure (40 sections, 28 theorems, 70 equations, 7 figures)

This paper contains 40 sections, 28 theorems, 70 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related literature
  3. Outline
  4. Model and results
  5. Model
  6. Efficiency
  7. The monopolist allocation
  8. Inefficiencies
  9. Discussion of the model primitives
  10. Comparative statics
  11. Interpretation
  12. Digital goods
  13. Single agent
  14. Extensions
  15. Non-discriminating monopolist
  16. ...and 25 more sections

Key Result

Proposition 1

Let $q^{\star}$ be the quality $q$ solving $g'(q)+\int_{\Theta}\theta\mathop{}\!\mathrm{d} F(\theta)=c'(q)$. The efficient allocation is given by $\bm{q}^{\star}(\theta)=q^{\star}$ for all $\theta$.

Figures (7)

  • Figure 1: Panel (a) illustrates the efficient quality $q^{\star}$; Panel (b) illustrates the efficient allocation. For these graphs and the following ones, $\theta$ is uniformly distributed, $c(q)=\frac{1}{2}q^{2}$, and $g(q)=\sqrt{{q}}$, unless specified otherwise.
  • Figure 2: The solution $\bm{q}$ to $\mathcal{P}(q)$ caps the virtual-surplus maximizer $\bm{\beta}$ at $q$. Panel (a) illustrates the solution for $q>\bm{\beta}(0)$; Panel (b) illustrates the solution for $q\le\bm{\beta}(0)$.
  • Figure 3: Panel (a) illustrates the monopolist quality $q^{M}$; Panel (b) illustrates the monopolist allocation.
  • Figure 4: With linear utility, the problem $\mathcal{P}(q)$ is solved by excluding types lower than the zero of the virtual value $\varphi$, and allocating $q$ to higher types.
  • Figure 5: Panel (a) compares the monopolist allocation with $\bm{q}^{MS}$ if the monopolist does not engage in full bunching, so $b(q^{M})>0$; Panel (b) compares the monopolist allocation with $\bm{q}^{MS}$ if the monopolist optimally offers a pooling contract, so $b(q^{M})=0$. The two figures have different allocations only because of different cost functions, see Proposition \ref{['prop:mon:comparative']}.
  • ...and 2 more figures

Theorems & Definitions (59)

  • Proposition 1
  • Remark 1
  • Lemma 1
  • Definition 2.1
  • Proposition 2
  • Remark 2.1
  • Remark 2
  • Remark 3
  • Proposition 3
  • Remark 4
  • ...and 49 more