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Econ-520 · economics · Economics

Privacy valuation

LLM status Open unverified

Statement

Original subject group: Digital, platform and data economics

Audit classification: Background; proposed extension. The publication exists and is relevant background. No exact published open theorem for this CV identification exercise was verified.

Definitions and precise research target

Consider adult users of an online service in a fixed country and enrollment period. A data-sharing contract $d\in\{0,1\}$ either limits collection to essential service data or permits a specified additional use; the contract's actual data flows are auditable. Let $u_i(c,d,\ell)$ be user $i$'s utility from baseline consumption $c>0$, sharing regime $d$, and future loss $\ell$, with utility strictly increasing in $c$. The additional use induces an uncertain loss distribution $F_i$ over a fixed five-year horizon, including discrimination and unwanted disclosure. Define compensating variation $CV_i$ by $E_{F_i}[u_i(c+CV_i,1,\ell)]=u_i(c,0,0)$, with expectation over the user's stated informed loss beliefs. Require finite expected utility and $c+CV_i>0$; report nonexistence if no finite compensation attains the comparison utility. The target is the population distribution of $CV_i$, alongside sensitivity to alternative credible loss distributions. Incentivized contract choices, comprehension checks and belief elicitation provide distinct observations; default effects cannot automatically be interpreted as valuations. Determine which additional information treatments, repeated choices or structural restrictions distinguish preferences from misunderstanding and friction. When distant harms cannot be learned from the observation window, bound valuations under explicit probability and loss limits. This is a proposed identification exercise rather than a verified universal unresolved privacy-valuation theorem.

Definitions, assumptions and mathematical qualifications. The contract defines actual additional data recipients, retention, permitted inference and the five-year loss variable, rather than a vague sharing label. Normalize $d=0$ loss to zero only if baseline risks are incorporated in baseline utility. Define $f_i(v)=\int u_i(c+v,1,\ell)dF_i(\ell)$ on $v>-c$. Continuity, strict increase and integrability imply at most one $CV_i$; existence further requires $u_i(c,0,0)$ to lie in the range of $f_i$. Compensation may be negative when extra services outweigh privacy losses. The population CV distribution is $G(v)=\Pr(CV_i\leq v)$, with nonexistence separately tabulated. Stated subjective loss beliefs and externally estimated loss frequencies are distinct estimands.

Variants and assumption changes

  1. Editorial, informed benchmark. Under quasi-linear utility $u_i(c,d,\ell)=c+b_i(d)-h_i(d)-\ell$, $CV_i=b_i(0)-b_i(1)+h_i(1)-h_i(0)+E_{F_i}\ell$; the issue is measuring these components.
  2. Editorial, uncertain harms. Let loss laws belong to a stated set with bounded probability and monetary severity; combine randomized information and comprehension measures to bound informed valuation without treating default behavior as direct willingness to pay.

References and source audit

Checked source. NBER Working Paper 30943, February 2023, indexed primary abstract. The publication supports privacy cost--benefit and externality background. Primary metadata and abstract indexed; direct PDF returned HTTP 403. An exact open-question passage was not verified. https://www.nber.org/papers/w30943.
  1. Avi Goldfarb; Verina F. Que. 2023. The Economics of Digital Privacy. NBER Working Paper 30943, February 2023, indexed primary abstract. https://www.nber.org/papers/w30943. DOI: https://doi.org/10.3386/w30943.

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-520-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-520

Privacy valuation: existence, risk premiums, and choice frictions

Problem ID: D12-516. Status: Unresolved. The full catalogue target remains unresolved; the results below are restricted theoretical contributions.

Contract and valuation target

Consider the catalogue's auditable binary data-sharing contract $d\in\{0,1\}$, baseline consumption $c>0$, and the informed subjective distribution $F$ of privacy-related losses over the stipulated five-year horizon. Discount losses into the same consumption units before aggregation. A compensating payment $v$ solves \[ E_F[u(c+v,1,\ell)]=u(c,0,0),\qquad v>-c. \tag{1} \] The target is a population distribution of such values, allowing negative compensation and recording nonexistence. It is not automatically the distribution of payments accepted in a default-dependent click interface. Goldfarb and Que's review distinguishes benefits and costs of data use; this attempt treats the contract, service benefits, beliefs and restrictions as explicit primitives. The risk and identification results below are analytical examples, not population valuations.

Existence before estimation

Define $H(v)=E_Fu(c+v,1,\ell)$ on $(-c,\infty)$. Assume $H$ is finite, continuous and strictly increasing; dominated convergence under appropriate local bounds is one sufficient continuity condition. Individual strict monotonicity alone does not ensure expected utility is finite or continuous.
Proposition: Equation (1) has at most one solution. It has a solution precisely when the baseline utility belongs to the image of $H$. In particular, if \[ \lim_{v\downarrow-c}H(v)<u(c,0,0)<\lim_{v\to\infty}H(v), \] it has a unique solution. If the baseline equals either limiting endpoint, a solution need not exist because the domain is open or unbounded.
Proof: Strict monotonicity gives uniqueness. Membership in the image is the exact existence criterion. Under the two strict inequalities, two finite domain points bracket the baseline value, and continuity plus the intermediate value theorem supplies a solution between them. A strictly increasing function cannot attain either of its endpoint limits at an interior point. ∎
This is a small but important bookkeeping requirement: coding everyone who never accepts as having an arbitrarily large finite value changes the target. With a finite experimental price grid, a high but finite value, a failed comprehension screen, a binding prohibition, and mathematical nonexistence can all produce the same responses.

An exact risk-sensitive benchmark

Take constant absolute risk aversion with known $\gamma>0$ and monetary service benefit $b$, intrinsic privacy cost $a$: \[ u(c,d,\ell)=-\exp[-\gamma(c+(b-a)d-\ell)]. \] The primitive utility is defined for all net resources in the exponent, while the payment convention still requires $c+v>0$. Assume losses are nonnegative. Let $M_F(\gamma)=E_Fe^{\gamma\ell}$.
Proposition: If $M_F(\gamma)<\infty$, the unique algebraic compensation is \[ v^*=a-b+\gamma^{-1}\log M_F(\gamma). \tag{2} \] It is an admissible valuation exactly when $v^*>-c$. If $M_F(\gamma)=\infty$, expected sharing utility is $-\infty$ for every finite payment and no finite compensation exists.
Proof: Expected utility under sharing is $-e^{-\gamma(c+v+b-a)}M_F(\gamma)$. Equating it to $-e^{-\gamma c}$ and taking logarithms gives (2). The payment-domain condition must still be checked. When the moment is infinite, multiplying by a strictly positive finite exponential leaves infinity, so equality with finite baseline utility is impossible. ∎
A heavy-tailed loss can have finite mean but no positive exponential moment, so finite expected monetary loss does not guarantee finite risk-sensitive compensation. Conversely $v^*$ may be negative when service benefits dominate privacy costs and risk. If it lies at or below $-c$, sharing is already preferred at every admissible compensation in this parameterization; the absence of an interior equality should not be interpreted as extreme aversion.

If $\ell$ is bounded, a Taylor expansion as $\gamma\downarrow0$ yields \[ v^*=a-b+E\ell+\frac\gamma2{\operatorname{Var}}(\ell)+O(\gamma^2). \] The boundedness assumption controls the remainder through finite higher moments. Thus observed willingness to accept depends on risk aversion and tail beliefs even if the contractual data use and expected loss are held fixed. There is no reason that an uninformed subjective law coincides with the informed law required by the catalogue.

Sharp bounds from limited loss information

Suppose $0\le\ell\le L$, $L>0$, and only $E\ell=\mu\in[0,L]$ is known. Jensen's inequality and the chord bound for the convex exponential give \[ e^{\gamma\mu}\le M_F(\gamma) \le 1+\frac\mu L(e^{\gamma L}-1). \] Consequently the algebraic compensation lies between \[ a-b+\mu \quad\hbox{and}\quad a-b+\gamma^{-1}\log\left[1+\frac\mu L(e^{\gamma L}-1)\right]. \tag{3} \] Both endpoints are sharp: the lower law is a point mass at $\mu$, and the upper law puts mass $\mu/L$ at $L$ and the remainder at zero. Convexly mixing these two laws preserves the mean and interpolates the exponential moment, so every value between the endpoints is attained. Intersecting with the admissible payment domain addresses finite existence separately.

For a check, set $a=b$, $\gamma=1$, $L=2$, $\mu=1$. Deterministic loss requires compensation $1$, while the equally likely losses $0$ and $2$ require $\log[(1+e^2)/2]\approx1.434$. The same expected monetary damage generates different privacy valuations solely through risk. This is an illustrative certainty-equivalent calculation and carries no empirical magnitude claim.

What binary choices identify

For a comprehending individual with stable informed valuation $v^*$, add an explicit monetary decision wedge $\delta_f$ under interface frame $f$ and suppose acceptance at offer $t$ follows \[ A(t,f)={\mathbf 1}\{t-v^*+\delta_f\ge0\},\qquad |\delta_f|\le\kappa. \] This is an additional behavioral model, not a consequence of expected utility. A finely measured switching threshold $\theta_f$ identifies $v^*-\delta_f$, yielding the sharp interval $[\theta_f-\kappa,\theta_f+\kappa]$ for $v^*$. With several frames, intersect these intervals. Sharpness follows by setting each wedge to $v^*-\theta_f$; every value in the intersection reproduces all switches. An empty intersection rejects the stable-value bounded-wedge model.

For a population with one measured threshold per person and a common bound $\kappa$, the valuation CDF obeys \[ F_\theta(v-\kappa)\le F_{v^*}(v)\le F_\theta(v+\kappa). \] The lower bound uses $v^*\le\theta+\kappa$ and the upper uses $v^*\ge\theta-\kappa$. Constant wedges at opposite extremes attain the bounds pointwise. Randomizing defaults identifies their effects on choices, but it does not identify the level of informed value unless some frame has a known wedge, a symmetry condition is justified, or additional measurement identifies the wedge.

If comprehension failures create arbitrary responses, the threshold model applies only after a justified comprehension measurement model. Conditioning on passing an endogenous test can itself change the population. Likewise learning about a rare future breach changes $F$ and hence (2), rather than merely removing a default wedge. These are distinguishable mechanisms requiring distinct interventions.

Remaining task

The analytic formulas solve a restricted CARA contract with specified beliefs. The catalogue's distributional target requires auditable contract versions, informed and internally coherent loss beliefs, consumption and risk-preference measurement, comprehension validation, and a selection model for participants. The existence calculation and bounds expose which missing inputs matter but do not supply them. No empirical CDF, privacy externality estimate, or social optimum is claimed.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt establishes compensation existence, sharp risk-sensitive loss bounds, and distributional valuation bounds under bounded interface wedges. Full informed privacy valuations still need auditable contract versions, credible long-run harm beliefs, risk and comprehension measurement, and participant-selection restrictions beyond the CARA benchmark.

References

  • [privacy] A. Goldfarb and V. F. Que (2023), The Economics of Digital Privacy, NBER Working Paper 30943. Primary bibliographic record and abstract: https://www.nber.org/papers/w30943. The reference supplies the economic context; the compensation equations and counterexamples here are self-contained.

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