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Econ-545 · economics · Economics

Vertical integration and self-preferencing

LLM status Partially solved unverified

Statement

Original identifier: I7 (atlas item 77). Classification: Vertical conduct and welfare research program. Original tractability label: Medium (editorial, not a complexity result).

Definitions and assumptions

Let an upstream firm supply an input to downstream firms, one of which may be integrated with it. A regime $v$ specifies ownership, wholesale contracting, access obligations, ranking rules, and permissible discrimination. Demand depends on consumer prices, quality, and accessibility. Supply behavior follows a declared bargaining or contracting game, including whether contracts are public, enforceable, and contingent. Let $e_\sigma(v)$ denote an equilibrium under a declared selection rule. Define foreclosure as a specified reduction in rivals' access, raising of their relevant costs, or exclusion relative to the baseline; do not identify it merely with a rival's lower profit.

Formal research question

Identify or bound total welfare effects $\Delta W^\sigma(v,v_0)$ and consumer surplus effects, and separate mechanisms where the counterfactuals are well defined. For example, define $v_E$ as allowing internal elimination of successive price distortions while requiring the baseline access terms, and compare \[ W(e_\sigma(v_E))-W(e_\sigma(v_0)),\qquad W(e_\sigma(v))-W(e_\sigma(v_E)). \] Determine conditions under which both comparisons are identified and their signs robust across admissible conduct and bargaining assumptions. Evaluate specific equal-access or ranking restrictions using causal policy variation or a validated structural model.

Interpretation and scope

The same integration can remove successive markups, coordinate investment, and change the integrated firm's incentive to disadvantage downstream rivals. The magnitude of each channel depends on contract possibilities and bargaining power; double marginalization is not guaranteed under every nonintegrated contract. A mechanism decomposition requires feasible counterfactual regimes and is generally path-dependent when channels interact. The displayed intermediate regime is therefore an explicit modeling input, not a uniquely natural attribution. Search self-preferencing is a particular conduct change whose welfare depends on match quality, ranking incentives, and outside options, and cannot be inferred solely from vertical ownership. Data on wholesale transfers may identify neither marginal costs nor bargaining threats without additional restrictions. Equal-access mandates can alter investment or participation, so the policy outcome should include these responses over a stated horizon. A successful extension would predict both rival access and final-consumer outcomes after the same intervention, with uncertainty over alternative bargaining models and selection rules.

Source audit and status

[S1] develops a foreclosure mechanism; paper authors Hart and Tirole are distinguished from discussion contributors in the Brookings record. [S2] surveys foreclosure theories, and [S3] quantifies integration in television markets. All original citations are verified, but television evidence is not direct proof about platform ranking. Existing theory already characterizes efficiency and foreclosure in particular models. The explicit regime comparison is an editorial extension directed at identification and application-specific welfare assessment.

References

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  1. Oliver Hart and Jean Tirole (1990). Vertical Integration and Market Foreclosure. Brookings Papers on Economic Activity: Microeconomics, 205--286 (article and discussion). https://www.brookings.edu/articles/vertical-intergration-and-market-foreclosure/

    Source role: Formal foreclosure mechanism; discussion authors are not paper coauthors. Locator: Abstract and article metadata.

  2. Patrick Rey and Jean Tirole (2007). A Primer on Foreclosure. Handbook of Industrial Organization, Volume 3, Chapter 33, 2145--2220, Elsevier. https://www.sciencedirect.com/science/chapter/handbook/pii/S1573448X06030330

    Source role: Review of foreclosure theories. Locator: Abstract and article metadata.

  3. Gregory S. Crawford, Robin S. Lee, Michael D. Whinston, and Ali Yurukoglu (2018). The Welfare Effects of Vertical Integration in Multichannel Television Markets. Econometrica 86(3), 891--954. https://doi.org/10.3982/ECTA14031

    Source role: Structural quantification of vertical efficiencies and foreclosure. Locator: Abstract and article metadata.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-545-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-545

Vertical efficiency and foreclosure under explicit contracts

Status: Partial; the full catalogue target remains unresolved.

Abstract

Successive linear markups create an exactly calculable integration gain, while efficient two-part contracting removes that gain before integration. In a separate competitive-input benchmark, excluding a downstream rival reduces welfare. These solved examples establish contract-dependent opposite signs. An exact path-decomposition identity explains why assigning interactions to efficiency or foreclosure requires a stated intermediate regime.

A contract-sensitive target

The catalogue asks for consumer and total welfare effects of a specified ownership and conduct regime. Ownership alone is insufficient: wholesale contracts, access obligations, ranking, entry, and investment responses determine the equilibrium. Here all demand and cost primitives are known and there is one static market. The models are intentionally restricted, with no empirical inference from observed margins.

In the first benchmark inverse final demand is $P(Q)=a-Q$ for $Q\in[0,a]$, upstream marginal cost is $c$ with $0\le c<a$, and downstream transformation is costless. Set $D=a-c>0$. Transfers between the two firms cancel in total welfare, which is \[ W(Q)=\int_0^Q(a-x-c)\,dx=DQ-Q^2/2. \tag{1} \] Consumer surplus at the market price is $Q^2/2$. The demand curve is a declared money-metric valuation schedule, so adding consumer surplus and joint profit reproduces (1).

Successive linear markups

There is one upstream monopolist and one downstream monopolist. The upstream firm first posts a nonnegative linear wholesale price $w$; the retailer then chooses quantity. For $w<a$, the retailer maximizes $(a-Q-w)Q$, giving $Q(w)=(a-w)/2$. The upstream firm maximizes $(w-c)(a-w)/2$, so its optimum is $w^*=(a+c)/2$. Prices above $a$ yield zero trade and cannot beat this positive-profit outcome.

After vertical integration the single firm maximizes $(a-Q-c)Q$, giving $Q_I=D/2$. The separated outcome is $Q_S=D/4$.

Proposition: In this linear-contract game, integration increases total welfare by $5D^2/32$ and consumer surplus by $3D^2/32$.
Proof: At separation, (1) gives $W_S=D^2/4-D^2/32=7D^2/32$. At integration, $W_I=D^2/2-D^2/8=3D^2/8$. Subtracting yields $5D^2/32$. Consumer surplus rises from $D^2/32$ to $D^2/8$, giving $3D^2/32$. The strict concavity of both firms' one-variable objectives establishes the stated optima. ∎
With $a=10,c=2$, quantities rise from two to four. Welfare rises from fourteen to twenty-four, and consumer surplus from two to eight. These figures check the formulas with $D=8$. The efficiency gain is specific to the restriction to successive linear prices, not a general property of separate ownership.

Two-part tariffs change the benchmark

Allow an enforceable public two-part contract consisting of per-unit price $w$ and fixed fee $F$, with a retailer outside option of zero and acceptance at indifference. Set $w=c$. The retailer then chooses $Q=D/2$ and earns operating profit $D^2/4$ before the fee. Set $F=D^2/4$ to extract that profit.

This contract reproduces the integrated quantity and joint profit under separate ownership. The retailer participates and chooses the efficient-for-joint-profit quantity, while the upstream firm obtains the same total profit as an integrated monopolist. No feasible contract can generate greater joint profit than the maximum of $(a-Q-c)Q$, so the proposed contract is upstream-optimal subject to participation. Integration then has no additional output or consumer-surplus benefit in this benchmark.

A fixed fee constrained by limited liability, private demand information, hidden wholesale terms, or noncontractible investment may fail to implement this outcome. Those are substantive contract restrictions that must be measured or modeled. Observing a high linear wholesale price does not tell whether unobserved rebates or fixed payments neutralize its marginal incentive.

A separate foreclosure loss

Consider instead a competitive upstream input supplied to two identical downstream firms at marginal cost $c$. They choose quantities simultaneously under the same inverse demand. Their first-order equations are $D-2q_i-q_j=0$, giving $q_1=q_2=D/3$ and total $Q_C=2D/3$. The input market is competitive, so there is no upstream markup to eliminate.

Suppose a specified integration-and-access regime gives one downstream firm exclusive control of an essential input and bars the rival from buying it from any source. This is a genuine reduction in access, imposed as a regime primitive. The remaining firm chooses monopoly quantity $Q_F=D/2$.

Proposition: Relative to competitive input access, this exclusion lowers total welfare by $5D^2/72$ and consumer surplus by $7D^2/72$.
Proof: Equation (1) gives $W_C=2D^2/3-2D^2/9=4D^2/9$ and $W_F=3D^2/8$. Their difference is $(32-27)D^2/72$. Consumer surplus is $2D^2/9$ before exclusion and $D^2/8$ afterward, differing by $(16-9)D^2/72$. ∎
This second economy does not combine with the first by mechanically adding their welfare changes: their baseline contract and competition structures differ. Their purpose is a rigorous sign counterexample. A rival's lower profit alone would not establish exclusion; the proof specifies an essential-input access change. Efficient expansion by an integrated firm could also lower rival profits with positive consumer effects.

Path dependence of mechanism accounting

For one fully specified economy suppose the four combinations of efficiency permission $e\in\{0,1\}$ and discriminatory access $f\in\{0,1\}$ are feasible and have selected welfare $W_{ef}$. The total change is $W_{11}-W_{00}$. Assigning efficiency first gives efficiency contribution $W_{10}-W_{00}$ and access contribution $W_{11}-W_{10}$. Reversing the order gives $W_{01}-W_{00}$ and $W_{11}-W_{01}$.

The change in the attributed efficiency contribution across these orders is \[ I=W_{11}-W_{10}-W_{01}+W_{00}. \] This is immediate by subtraction. If $I\ne0$, an order-free separation into those two marginal contributions is impossible without an additional attribution rule. Averaging the two orders assigns half the interaction to each channel, but that is a convention rather than an identified causal pathway.

If one intermediate regime is legally or technologically infeasible, its welfare cannot be estimated by pretending it is an available intervention. The catalogue's $v_E$ therefore needs a real contract and access interpretation.

A ranking example and the unresolved application

For two available offers, let their net real match values be $v_O,v_R$. A ranking policy changes the probability that the consumer receives the owned offer from $p_0$ to $p_1$, holding prices, quality, participation, and costs fixed. Expected welfare changes by $(p_1-p_0)(v_O-v_R)$. The sign depends on match values, not on ownership. Allowing price or quality responses adds terms and requires another equilibrium calculation; the simple formula is only a frozen-market benchmark.

The completed proofs show why vertical welfare claims need contract data and feasible counterfactuals. They do not identify actual access changes, bargaining threats, investment, or match values in a platform. No current legal conclusion or empirical merger recommendation is made. The broad agenda remains unresolved, with the familiar markup calculations used as transparent benchmarks rather than claimed novel theory.

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt computes contract-dependent integration gains and foreclosure losses, shows efficient contracting can remove successive markups, and makes mechanism path dependence explicit. Full vertical and ranking policy evaluation still requires identified access, bargaining, investment, and match-quality responses in one validated economy, with robustness across admissible conduct and equilibrium assumptions.

References

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