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Econ-559 · economics · Economics

Common ownership and competition

LLM status Open unverified

Statement

Original identifier: I9 (atlas item 79). Classification: Ownership influence causal identification research program. Original tractability label: Hard (editorial, not a complexity result).

Definitions and assumptions

Let $b_{hj}\geq0$ be investor $h$'s cash-flow share in firm $j$, and $c_{hj}\geq0$ its effective influence weight in that firm's decision process. These are distinct inputs. An illustrative manager objective is \[ \sum_k\lambda_{jk}\pi_k,\qquad \lambda_{jk}=\frac{\sum_hc_{hj}b_{hk}}{\sum_hc_{hj}b_{hj}}, \] where the denominator is positive and $\lambda_{jj}=1$. This aggregation requires a specified governance mechanism; portfolio overlap alone does not imply it. Let $O$ describe holdings, influence, communications, and compensation. Product-market conduct, demand, costs, and ownership-induced governance changes form a declared equilibrium model with selection $\sigma$. If $Y$ is welfare, specify a finite real-valued evaluator with declared cardinal normalizations and weights on consumers, workers, and owners, accounting for monetary transfers consistently.

Formal research question

Identify or sharply bound the causal effects of a defined ownership or governance intervention $O_1$ versus $O_0$ on prices, output, quality, and welfare: \[ \Delta_Y^\sigma=Y(e_\sigma(O_1))-Y(e_\sigma(O_0)). \] Determine what observations and exclusions identify the influence weights or a less restrictive mechanism connecting investors to product decisions. Test whether direct governance measures predict conduct changes beyond market structure and cost changes. Compare the competition channel with governance-efficiency effects under explicitly defined counterfactual restrictions, and report ambiguity when the same price patterns are consistent with several mechanisms.

Interpretation and scope

Institutional investors can have shared cash-flow interests without coordinating or influencing pricing. Conversely, small holdings may carry substantial influence in a dispersed shareholder base. Modified concentration measures derived from ownership models therefore are model-implied proxies and not direct observations of managerial objectives. Holdings respond to firm prospects, and index changes can affect financing, liquidity, or attention as well as ownership overlap; an instrument requires exclusions covering these channels. Product-market concentration and common ownership often move together, demanding variation or restrictions that separate their roles. Direct voting, engagement, and compensation evidence can support a governance link but may also reflect anticipation of product-market changes. Holding demand and technology fixed is useful for a competition mechanism counterfactual, while total policy evaluation must allow the intervention's governance and financing effects. A successful extension would establish a credible chain from specified investor influence to strategic choices and customer outcomes, with uncertainty over the influence mapping, rather than interpret a single ownership--price regression as universal causal evidence.

Source audit and status

[S1] reports airline evidence; [S2] documents historical model-implied incentives; [S3] is Kennedy, O'Brien, Song, and Waehrer's 2017 working paper presenting economic foundations and empirical analysis. Its working-paper status is retained, and it is not confused with O'Brien--Waehrer's separately titled critique. All three original author-year entries are resolved. Competing interpretations make this an editorial causal and mechanism research program, not a theorem settled or certified open by one cited estimate.

References

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  1. José Azar, Martin C. Schmalz, and Isabel Tecu (2018). Anticompetitive Effects of Common Ownership. Journal of Finance 73(4), 1513--1565. https://doi.org/10.1111/jofi.12698

    Source role: Airline evidence supporting common-ownership hypothesis. Locator: Abstract and article metadata.

  2. Matthew Backus, Christopher Conlon, and Michael Sinkinson (2021). Common Ownership in America: 1980--2017. American Economic Journal: Microeconomics 13(3), 273--308. https://www.aeaweb.org/articles?id=10.1257/mic.20190389

    Source role: Historical patterns of model-implied ownership incentives. Locator: Abstract and article metadata.

  3. Pauline Kennedy, Daniel P. O'Brien, Minjae Song, and Keith Waehrer (2017). The Competitive Effects of Common Ownership: Economic Foundations and Empirical Evidence. Working paper, July 24, 2017; SSRN 3008331. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3008331

    Source role: Critical analysis; working-paper status retained. 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-559-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-559

Common ownership and competition

GPT 6 Astra Ultra October 9, 2026
Target and missing governance link.
The catalogue distinguishes investor holdings from managerial influence. Under its illustrative aggregation, \[ \lambda_{jk}=\frac{\sum_h c_{hj}b_{hk}}{\sum_h c_{hj}b_{hj}}, \] cash-flow shares $b$ do not determine influence weights $c$. Backus, Conlon, and Sinkinson [ownership] document historical model-implied ownership incentives; the aggregation still requires a governance assumption. This note first gives the exact set consistent with holdings and unrestricted admissible influence. It then propagates that set into a product-market comparison while allowing a separately bounded cost efficiency. The broader causal interpretation is left explicit.
An exact influence restriction.
Fix a manager $j$ and a rival $k$. Let $H_j$ be a finite nonempty set of investors eligible to influence $j$. Assume $b_{hj}>0$ for every $h\in H_j$. Influence vectors may be any $c_{hj}\geq0$, not all zero, and have no additional restrictions. Define $r_h=b_{hk}/b_{hj}$.
Theorem (Holdings identify an interval, not an objective): Under these assumptions the sharp set of feasible cross-profit weights is \[ \lambda_{jk}\in[\min_{h\in H_j}r_h,\ \max_{h\in H_j}r_h]. \tag{1} \] Every point is attained. If an investor with $b_{hj}=0<b_{hk}$ is additionally allowed arbitrarily large influence, while some own-share-positive investor retains positive influence, no finite upper bound follows from holdings alone.
Proof: Define $\omega_h=c_{hj}b_{hj}/\sum_g c_{gj}b_{gj}$. These weights are nonnegative and sum to one, and the target ratio equals $\sum_h\omega_h r_h$. This gives the interval outer bound. Conversely, every convex combination of the $r_h$ is generated by choosing $c_{hj}=\omega_h/b_{hj}$. In one dimension these convex combinations fill the displayed interval, including both endpoints. For the final claim, fix the positive denominator contribution of an own-share-positive investor and increase the zero-own-share investor's influence. The denominator stays fixed while the numerator grows without bound. ∎

The positivity requirement is consequential. Replacing it by a numerical regularization of the denominator would manufacture a governance bound. Direct evidence on voting or engagement could instead restrict $c$ and shrink (1), but that would be additional information.

A Cournot benchmark for the competition channel.
There are two firms producing a homogeneous good with inverse demand $P=A-Q$, $A>c\geq0$. Real marginal cost is $c$, and consumer gross benefit is $AQ-Q^2/2$. Managers choose nonnegative quantities to maximize $\pi_i+\lambda\pi_j$, with $\pi_i=(A-q_i-q_j-c)q_i$. Symmetric governance restricts both cross-profit weights to the same $\lambda\in[\underline\lambda,\overline\lambda]\subset[0,1)$. This interval may come from (1) and additional symmetry restrictions. All consumers and ultimate owners receive equal monetary weights, and there are no foreign-income omissions or public transfers.
Theorem (Unique equilibrium and welfare monotonicity): For each admissible $\lambda$, the unique equilibrium is \[ q_1=q_2=\frac{A-c}{3+\lambda},\qquad P=A-\frac{2(A-c)}{3+\lambda}. \] Total welfare is \[ W(\lambda,c)=\frac{2(A-c)^2(2+\lambda)}{(3+\lambda)^2}. \tag{2} \] At fixed cost, price strictly increases and welfare strictly decreases with $\lambda$.
Proof: Each manager's objective is strictly concave in its own quantity, with derivative $A-c-2q_i-(1+\lambda)q_j$. If one firm's equilibrium quantity were zero, the other's best response would be $(A-c)/2$; the zero firm's marginal gain would then be $(A-c)(1-\lambda)/2>0$, a contradiction. Thus both first-order equations hold. Subtracting them gives $(1-\lambda)(q_1-q_2)=0$, hence the displayed symmetric solution. It satisfies both global best responses, proving existence and uniqueness. The restriction $\lambda<1$ excludes the common-objective boundary where division of monopoly output can be nonunique.

Payments from consumers to owners cancel, leaving $W=(A-c)Q-Q^2/2$. Substitution gives (2). Price has derivative $2(A-c)/(3+\lambda)^2>0$, while \[ \frac{\partial W}{\partial\lambda} =-\frac{2(A-c)^2(1+\lambda)}{(3+\lambda)^3}<0. \] ∎

Allowing governance efficiencies.
A specified ownership intervention may also reduce real unit costs by $g\in[0,\bar g]$, where $\bar g\leq c_0$. Suppose its joint identified primitive set is the complete rectangle $[\underline\lambda,\overline\lambda]\times[0,\bar g]$; this rectangularity must be justified separately in an application. Baseline governance $0\leq\lambda_0<1$ and cost $c_0<A$ are known. Under intervention, cost is $c_0-g$, and all other primitives remain fixed.
Proposition (A sharp welfare prediction interval): The sharp intervention welfare-effect set is \[ \begin{split} \Delta W\in[&W(\overline\lambda,c_0)-W(\lambda_0,c_0),\\ &W(\underline\lambda,c_0-\bar g)-W(\lambda_0,c_0)]. \end{split} \tag{3} \] For a known postintervention weight $\lambda_1$, a nonnegative welfare effect occurs exactly when \[ g\geq (A-c_0)\left( \sqrt{\frac{(2+\lambda_0)/(3+\lambda_0)^2} {(2+\lambda_1)/(3+\lambda_1)^2}}-1\right). \tag{4} \] A negative right side means every nonnegative efficiency suffices.
Proof: Equation (2) decreases with $\lambda$ and increases with $g$ because $A-c_0+g>0$. The opposite corners therefore attain the extrema in (3). Continuity on the connected rectangle makes its image a complete interval, proving sharpness. For (4), compare the two expressions in (2), cancel the factor two, and take positive square roots. All quantities divided by are strictly positive, so the inequality is equivalent without reversing its direction. ∎
What a price regression does not resolve.
Even if demand intercept $A$ and symmetric output $q>0$ are observed, the equilibrium condition only gives $c=A-(3+\lambda)q$. Different admissible $\lambda$ can therefore be paired with different costs to generate the same price and quantity. Holdings alone narrow $\lambda$ through (1), and cost audits could narrow $c$; neither observation by itself identifies the full governance chain. An ownership instrument that also changes financing or productivity would not isolate the fixed-cost derivative in the theorem.

The preceding calculations attempt the original mechanism question by explicitly retaining both influence and efficiency. They do not infer that investors actually maximize the weighted objective from portfolio overlap. Empirical ownership changes are endogenous, quality can respond, and prices may be set through differentiated-product bargaining rather than quantities. The remaining target is to identify those governance and technology restrictions and validate the joint prediction set. Equations (1)--(4) solve a conditional model and its sharp bounds, not a universal causal effect of common ownership.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt sharply bounds governance weights from holdings and propagates them through a unique Cournot equilibrium with bounded cost efficiencies. Full common-ownership effects still require an identified investor-to-decision mechanism, credible ownership interventions and exclusions, direct governance validation, and quality and welfare responses beyond the symmetric quantity benchmark.

References

  • [ownership] M. Backus, C. Conlon, and M. Sinkinson (2021), "Common Ownership in America: 1980--2017," American Economic Journal: Microeconomics 13(3), 273--308. Official article and abstract.

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