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Econ-143 · economics · Economics

Data externalities

LLM status Open unverified

Statement

Original subject group: Digital, platform and data economics

Audit classification: Background; proposed extension. The official chapter and closing research topics were verified through primary indexing; this rights-design mechanism is a new specification. Its fully known finite benchmark is computable by enumeration.

Definitions and precise research target

Consider a finite population of $n$ consumers whose private characteristics are jointly distributed according to a known prior $P$. Consumer $i$ chooses a disclosure indicator $d_i\in\{0,1\}$; the vector $d$ determines signals available to a data intermediary. Correlation means that signals about one consumer may change posterior beliefs about another. A downstream firm uses those posteriors to choose products or prices. Let $v_i(d)$ be consumer $i$'s expected service benefit minus payments and privacy losses, and let $\pi(d)$ be firms' expected profit net of real data-processing costs. Define total welfare $W(d)=\sum_{i=1}^n v_i(d)+\pi(d)$. A rights mechanism $m$ specifies consent, compensation and allowable downstream uses; $\mathcal M$ is a stated feasible class with voluntary participation and truthful reporting. Under $m$, disclosure choices form equilibrium set $\mathcal E(m)$. Seek mechanisms maximizing a declared selection-weighted expectation of $W(d)$ over $d\in\mathcal E(m)$, subject to each nonconsenting person's specified privacy-risk bound. Compare individual consent, collective representation and limits on inference about nonparticipants. Quantify informational spillovers separately from transfers. The specification requires a clear privacy-loss metric and equilibrium rule; it is an original design proposal motivated by data externalities, without a verified canonical open formulation.

Definitions, assumptions and mathematical qualifications. The prior covers latent types and all signals jointly. A data-use rule maps revealed signals to downstream actions; a privacy-risk functional $\rho_i(d,a)$ must be supplied, for example expected monetary harm rather than unspecified privacy. A mechanism defines transfer funding, consent contingencies and agents information; truthful reporting and voluntary participation are constraints for each type or in expectation, as explicitly chosen. Fix $\kappa_m$, a probability distribution on a nonempty equilibrium set, and define $\mathcal W(m)=\int W(d)d\kappa_m(d)$. Constraint $\rho_i\leq\bar\rho_i$ applies to each nonconsenting individual. A finite mechanism menu has a maximizer only when the feasible subset is nonempty; otherwise use a supremum with existence conditions.

Variants and assumption changes

  1. Editorial, independent types. Set cross-person statistical dependence to zero while retaining any service or price spillovers; identify which externalities disappear and which remain.
  2. Editorial, correlated types. Permit inference about nonparticipants, compare individual and collective consent mechanisms using a supplied loss metric, and optimize robust welfare over admissible signal and harm distributions.

References and source audit

Checked source. The Economics of Privacy, Chapter 3 (2024), pp. 73--96; indexed closing passage p. 93. The indexed conclusion names data combination and regulatory evaluation as further work. Primary chapter metadata and indexed passage checked; full PDF returned HTTP 403. The rights-mechanism question is editorial. https://www.nber.org/system/files/chapters/c14782/c14782.pdf.
  1. Alessandro Bonatti. 2024. The Platform Dimension of Digital Privacy. The Economics of Privacy, Chapter 3 (2024), pp. 73--96; indexed closing passage p. 93. https://www.nber.org/system/files/chapters/c14782/c14782.pdf.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-143-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-143

A Gaussian privacy externality and an implementable disclosure frontier

Status: Unresolved. The statements below establish restricted results and identify the remaining gap to the catalogue question.

Abstract

For two correlated Gaussian types, we compute the exact disclosure interval allowed by a nonparticipant's information-loss cap. A strictly concave welfare problem then has a unique solution, and a transfer menu implements it truthfully when one participant's privacy sensitivity is private. The result isolates a nonconsenter externality under a fixed service technology; it does not characterize optimal privacy institutions for general pricing markets.

Short target and model

The question is how correlated information changes consent, compensation, and welfare. Let $(X_1,X_2)$ be jointly Gaussian, centered, with unit variances and correlation $r\in(0,1)$. Person 1 may authorize the public signal $Y=X_1+N$, with independent Gaussian noise of variance $\nu$. It is convenient to index quality by $t=(1+\nu)^{-1}\in(0,1)$ and include $t=0$ as an independent, uninformative signal. Person 2 does not consent and receives no transfer.

Information loss is mutual information in nats, $c_i(t)=I(X_i;Y)$. Person $i$ values this loss at $\theta_i c_i(t)$, where $\theta_1>0$ is private and $\theta_2\ge0$ is public. Service produces monetizable gross benefit $\beta t$, $\beta>0$, received initially by a platform. Transfers between the platform and person 1 cancel in welfare. There are no additional production costs, downstream price responses, or hidden disclosures. A regulator imposes $c_2(t)\le\rho$, $\rho\ge0$. These assumptions make the accounting and the feasible mechanism explicit.

The exact nonconsenter constraint

Proposition (Privacy frontier): For $0\le t<1$, \[ c_1(t)=-\tfrac12\log(1-t),\qquad c_2(t)=-\tfrac12\log(1-r^2t). \] The privacy cap is equivalent to $t\le T$, where \[ T=\min\{1,(1-e^{-2\rho})/r^2\}. \] The endpoint $t=1$ is excluded whenever $\theta_1>0$, since $c_1(t)$ diverges there. If $\rho=0$, only $t=0$ is feasible, despite person 1's ability to consent.
Proof: Gaussian conditioning gives ${\operatorname{Var}}(X_1\mid Y)=1-1/(1+\nu)=1-t$ and ${\operatorname{Var}}(X_2\mid Y)=1-r^2/(1+\nu)=1-r^2t$. The difference between unconditional and conditional Gaussian differential entropies is one half the logarithm of their variance ratio. This proves the formulas, with their continuous values at $t=0$. Exponentiating the cap inequality proves the interval. The final two assertions follow from these same formulas. ∎
Thus individual consent cannot eliminate the information revealed about a correlated nonparticipant. The conclusion depends on this particular information-loss metric; it is not a universal measure of privacy harm.

Welfare optimization and comparative statics

For a reported or actual type $\theta>0$ let \[ W_\theta(t)=\beta t-\theta c_1(t)-\theta_2c_2(t), \qquad 0\le t\le T,\quad t<1. \]
Theorem (Unique optimal disclosure): The welfare maximum exists and is unique. It is zero disclosure when $\beta\le(\theta+\theta_2r^2)/2$. Otherwise let $t_0\in(0,1)$ be the unique solution of \[ \beta={\theta\over2(1-t_0)}+ {\theta_2r^2\over2(1-r^2t_0)}. \] Then $t^*(\theta)=\min\{T,t_0\}$. Optimal quality is weakly decreasing in $\theta$ and $\theta_2$, and weakly increasing in $\beta$ and $\rho$. With the other primitives fixed, it is weakly decreasing in $r\in(0,1)$.
Proof: The second derivative is \[ W_\theta''(t)=-{\theta\over2(1-t)^2} -{\theta_2r^4\over2(1-r^2t)^2}<0. \] The first derivative at zero gives the no-disclosure condition. In the other case it is positive at zero and tends to minus infinity at one, so continuity and strict decrease give the unique root. A finite cap truncates the root. If $T=1$, divergence of $\theta c_1$ ensures attainment away from one; otherwise ordinary compactness does so. Increasing either harm weight decreases the derivative at each fixed quality; increasing $\beta$ increases it. These observations and the cap formula prove the first four monotonicities. Increasing $r$ increases $r^2/(1-r^2t)$ and weakly lowers the cap, proving the last one. ∎
The total welfare externality is included once. If a transfer $p$ is paid, person 1's utility is $p-\theta c_1(t)$, person 2's is $-\theta_2c_2(t)$, and platform profit is $\beta t-p$. Their sum is exactly $W_\theta(t)$.

A truthful, voluntary implementation

Define $A(t)=\beta t-\theta_2c_2(t)$ and offer all feasible qualities satisfying $A(t)\ge0$, with transfer $p(t)=A(t)$. Include $t=0$, whose transfer is zero. The platform commits to deliver the selected signal quality and transfer.
Theorem (Implementation for a privately known harm weight): For every $\theta>0$, person 1's optimal menu choice equals $t^*(\theta)$. The equivalent direct mechanism selecting $t^*(\widehat\theta)$ on report $\widehat\theta>0$, and paying $p(t^*(\widehat\theta))$, is dominant-strategy truthful and individually rational. Its transfers are nonnegative and never exceed gross platform benefit. The nonconsenter's cap holds for every report.
Proof: At any offered quality, the person's utility is $A(t)-\theta c_1(t)=W_\theta(t)$. Every maximizer of welfare has value at least $W_\theta(0)=0$, hence $A(t)\ge\theta c_1(t)\ge0$ and is included in the menu. The unique welfare maximizer therefore remains the optimal menu choice. Under the direct mechanism a false report merely chooses another point from this menu, which cannot improve actual utility. Choosing zero is available and yields zero, proving voluntary participation. Finally $0\le p(t)=\beta t-\theta_2c_2(t)\le\beta t$, and every menu quality obeys the cap by construction. ∎
The transfer rule internalizes the nonparticipant's harm in the participant's choice. It need not maximize the platform's profit among truthful mechanisms, and person 2 is not compensated. If person 2 has a right to refuse any uncompensated loss rather than a supplied cap, this is a different feasible mechanism problem; for zero permitted loss the proposition forces zero quality.

Limitations and next step

The proof assumes a known Gaussian prior, a verifiable signal technology, a known $\theta_2$, and a fixed benefit function. It does not solve learning of harm, multiple private types, collective consent, monopolistic downstream pricing, or robustness to misspecified correlation. A next extension would require a specified family of priors and a mechanism whose incentive constraints remain valid across that family. Mere substitution of a worst-case correlation into the welfare formula would not by itself prove robust truthfulness when preferences and information jointly change.

References

Completion Estimate

25%

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