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Econ-406 · economics · Economics

Welfare effects of nudges

LLM status Partially solved unverified

Statement

Original subject group: Behavioral and experimental economics

Audit classification: Background; proposed extension. Allcott et al. 2025 is genuine and its abstract presents welfare conditions as results. The general application-level identification specification is editorial; no remaining exact theorem was verified.

Definitions and precise research target

Model and research target. Consider consumers $i$ choosing quantities $q_i$ of a good. Let $v_i(q)$ be experienced monetary-equivalent benefit, $b_i(q,n)$ a choice distortion under nudge $n\in\{0,1\}$, and $p(n,t)$ the equilibrium price when a tax $t$ is fixed. A specified behavioral demand model is \[q_i(n,t)\in\arg\max_{q\geq0}\{v_i(q)+b_i(q,n)-(p(n,t)+t)q\}.\] Define welfare, including production costs $C$, aggregate external damage $H$, and real implementation cost $K$, by \[W(n,t)=\mathbb E[v_i(q_i(n,t))]-C(Q(n,t))-H(Q(n,t))-K(n),\qquad Q(n,t)=\mathbb E[q_i(n,t)].\] Transfers from prices and tax receipts cancel only under the chosen welfare weights and incidence assumptions. The target estimand is $\Delta W(t)=W(1,t)-W(0,t)$, or the corresponding difference with taxes reoptimized under each nudge. Determine which experiments and market data identify experienced benefits, heterogeneous distortions, price responses, and spillovers sufficiently to sign $\Delta W$ in a specified policy application. A mean change in purchases does not by itself identify welfare, and observed choice utility need not equal experienced benefit. Report sensitivity to benefit measurement, welfare weights, unobserved heterogeneity, and equilibrium assumptions.

Definitions and maintained assumptions (editorial unless a source definition is named). Normalize consumers to mass one on a probability space, with $q_i\in[0,\bar q]$, finite $\bar q$, integrable experienced benefits and continuous choice payoff. Fix a measurable tie-breaking rule for demand if maximizers are multiple. Supply has a specified competitive cost function $C$, market clearing and an equilibrium-selection rule determining $p(n,t)$; price cannot merely be named an equilibrium without that supply model. $v_i$ is the normative money-equivalent benefit and $b_i$ a decision distortion, not a welfare benefit unless the declared criterion includes it. $C(Q),H(Q),K(n)$ are total real costs per unit consumer mass, with $H$ external damage. Equal welfare weights and public revenue rebated as transfers justify dropping prices and taxes from total surplus; unequal weights require separately counting incidence. Fixed-tax $\Delta W(t)$ differs from $\sup_{t\in\mathcal T}W(1,t)-\sup_{t\in\mathcal T}W(0,t)$, where tax set $\mathcal T$ and fiscal constraints are declared. Observed purchase changes do not identify $v_i$, distortions or cost functions without restrictions or auxiliary measurements.

Variants and assumption changes

  1. Fixed tax/price experiment (editorial). Randomize labels holding product prices and taxes fixed, measure experienced benefits by a defended protocol, and bound welfare sensitivity to distortion heterogeneity and implementation costs.
  2. Equilibrium and tax optimization (editorial). Specify supply elasticity, pass-through and a compact admissible tax set. Identify nudge welfare after equilibrium response and compare separately with taxes optimized under each regime, preserving the same public budget and welfare weights.

References and source audit

  1. Publisher abstract states welfare results; full PDF inaccessible. No exact remaining identification question verified. https://www.aeaweb.org/articles?id=10.1257/aer.20231304
  1. Hunt Allcott; Daniel Cohen; William Morrison; Dmitry Taubinsky. 2025. When Do "Nudges" Increase Welfare?. Abstract; AER 115(5), pp. 1555–1596. Full paper was not accessible in this audit.. https://www.aeaweb.org/articles?id=10.1257/aer.20231304. DOI: https://doi.org/10.1257/aer.20231304.

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-406-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-406

Welfare effects of nudges

Target and substantive obstruction

The catalogue asks for $\Delta W(t)$ at a fixed tax or with taxes separately reoptimized. The observed purchase response alone cannot sign this quantity. This attempt supplies an exact observational equivalence, sharp welfare bounds under a marginal-benefit measurement restriction, and a finite-sample sign certificate. These are restricted identification results, not an empirical conclusion about any particular nudge.

Two economies with opposite welfare effects

Consider a unit mass of identical buyers, $q\in[0,1]$, a competitive technology $C(Q)=Q$, price one, zero tax, no external damage and no implementation cost. Observe quantities $q(0)=1$ and $q(1)=0$. In both regimes suppose total decision benefit is $2q$ before the nudge and zero afterwards. For any constant $a$ let experienced benefit and distortions be \[ v_a(q)=a q,\qquad b_a(q,0)=(2-a)q,\qquad b_a(q,1)=-a q. \] Decision utility net of price is respectively $q$ and $-q$, with the specified unique choices. All transactions, prices, tax receipts, production and choice responses are identical in all these economies. Experienced welfare, however, is $W_a(0)=a-1$ and $W_a(1)=0$, hence \[ \Delta W_a=1-a. \] With $a=1/2$ the nudge gains $1/2$; with $a=2$ it loses one. Both experienced benefits are increasing. A nonlinear supply extension does not remove this indeterminacy: adding any function to experienced benefit and subtracting it from distortion preserves the total decision payoff at every price.
Proposition: Without a restriction separating $v$ from $b$, even full experimental demand curves under both nudge regimes do not identify the sign of experienced-surplus welfare, including under a known supply system.
Proof: For arbitrary integrable $h_i(q)$ replace $v_i$ by $v_i+h_i$ and $b_i$ by $b_i-h_i$. Every consumer's optimization and thus every equilibrium selected by the same rule is unchanged. Welfare differences change by ${\mathbb{E}}[h_i(q_i(1))-h_i(q_i(0))]$. The displayed linear example changes its sign while preserving all observations and monotonic experienced benefits. ∎
This does not invalidate a defended normative measurement of experienced benefit. It identifies exactly which independent information the experiment must add. Preference reports that merely reproduce distorted choices are not such information by definition.

Sharp bounds with explicit benefit information

First suppose quantities $(q_{i0},q_{i1})$ and covariates are jointly available, for example in a stable-type crossover design with no carryover. Assume absolutely continuous $v_i$ and known integrable envelopes \[ \underline m_i(s)\le v_i'(s)\le\overline m_i(s),\qquad 0\le s\le1. \] No concavity restriction is imposed in the following sharpness assertion. Write $d_i=q_{i1}-q_{i0}$, $a_i=\min(q_{i0},q_{i1})$, $b_i=\max(q_{i0},q_{i1})$, and \[ L_i=\begin{cases}\int_{a_i}^{b_i}\underline m_i(s)ds,&d_i\ge0,\\ -\int_{a_i}^{b_i}\overline m_i(s)ds,&d_i<0,\end{cases} \quad U_i=\begin{cases}\int_{a_i}^{b_i}\overline m_i(s)ds,&d_i\ge0,\\ -\int_{a_i}^{b_i}\underline m_i(s)ds,&d_i<0.\end{cases} \] Let $D=C(Q_1)-C(Q_0)+H(Q_1)-H(Q_0)+K(1)-K(0)$.
Theorem: With known real-cost difference $D$ and unrestricted distortions, $\mathcal I_{\Delta W}=[{\mathbb{E}} L_i-D,{\mathbb{E}} U_i-D]$.
Proof: Integrate the derivative bounds in the correct orientation to obtain necessity. For the lower endpoint, choose the lower derivative on increasing-quantity segments and the upper derivative on decreasing segments, filling the remaining interval arbitrarily within the envelopes. Integrate from zero to construct $v_i$. Reverse the choice for the upper endpoint. Each construction can be paired with a continuous distortion $b_i(q,n)=- (q-q_{in})^2+(p_n+t)q-v_i(q)$, which makes the observed quantity the unique decision optimum at its observed price. Supply remains unchanged because the market-clearing quantities are unchanged. Convexly interpolate the chosen marginal-benefit functions and corresponding distortions to attain every intermediate welfare value. Measurable envelopes and quantity paths ensure measurable constructions and integrability. ∎
Sharpness here pertains to the observed regimes and supplied envelopes. If a whole demand curve, concavity, restrictions on distortion, or dynamic learning are additionally observed, those restrictions must also enter the feasible class; the simple endpoint construction may then cease to be sharp. If only a randomized parallel-arm experiment is observed, individual quantity pairs are generally unavailable. One must optimize these expressions over all couplings consistent with the marginals and envelope information. It is invalid to match each treated buyer with an arbitrary control and call the result sharp. An alternative is to bound each regime's experienced benefit level separately; that gives valid, potentially less informative bounds without a cross-world joint law.

A sign certificate with measurement error

A particularly transparent case has known common experienced benefit $v$ with $0\le v(q)\le B_v$, outcomes measured in independent randomized arms of sizes $n_0,n_1$. Let $\widehat V_n$ be each arm's mean measured benefit and assume its population measurement bias is bounded in absolute value by $\eta_n$. Measurements themselves lie in $[0,B_v]$. Suppose real-cost difference lies in a simultaneously valid interval $[D_L,D_U]$. A union bound and Hoeffding give with probability at least $1-\alpha$ for the two benefit means, \[ |\widehat V_n-{\mathbb{E}} v(q(n))|\le B_v\sqrt{\frac{\log(4/\alpha)}{2n_n}}+\eta_n=:r_n. \] Consequently a valid conditional-on-cost-bounds welfare interval is \[ [\widehat V_1-\widehat V_0-r_0-r_1-D_U, \widehat V_1-\widehat V_0+r_0+r_1-D_L]. \tag{1} \] If cost bounds have failure probability $\alpha_C$, unconditional coverage is at least $1-\alpha-\alpha_C$. A positive lower endpoint signs a welfare gain; a negative upper endpoint signs a loss. The shrinking sampling error cannot remove persistent normative measurement ambiguity $\eta_0+\eta_1$.

For illustration, paired reductions of $0.2$ with marginal experienced benefit in $[0.8,1.2]$, constant marginal production cost one, and nudge implementation cost $0.01$ give the sharp interval $[-0.05,0.03]$. If marginal benefit is instead in $[0.8,0.9]$, the interval is $[0.01,0.03]$. These are hypothetical arithmetic checks; they make the decisive measurement requirement explicit.

Equilibrium experiments and optimized taxes

The experiment must randomize a market-level saturation when it is meant to identify market equilibrium $Q_n,p_n$. Individual randomization at a fixed market price estimates a different partial-equilibrium contrast. With finitely many admissible taxes $t\in\mathcal T$, simultaneous valid bounds $L_n(t)\le W(n,t)\le U_n(t)$ imply \[ \max_tL_1(t)-\max_tU_0(t) \le\max_tW(1,t)-\max_tW(0,t) \le\max_tU_1(t)-\max_tL_0(t). \] The two regimes may select different taxes. Joint feasibility and fiscal rules must hold at every candidate. For a continuous compact tax interval and a proved welfare Lipschitz constant, a mesh adds its modulus error; compactness alone supplies neither that constant nor an identified welfare function.

The unresolved application-level work is benefit measurement, credible heterogeneity restrictions, supply and spillover identification, and welfare weights. The analysis assumes equal weights so that transfers cancel. Under unequal weights, taxes and price incidence are extra terms, not costs that can silently be omitted.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt establishes observational welfare equivalence, sharp benefit-envelope bounds, and a finite-sample sign certificate under explicit measurement assumptions. Application-level identification of experienced benefits, heterogeneous distortions, equilibrium spillovers, and unequal welfare weights remains unresolved.

References

  • [source] H. Allcott, D. Cohen, W. Morrison and D. Taubinsky (2025). When Do "Nudges" Increase Welfare? American Economic Review 115(5), 1555--1596. https://www.aeaweb.org/articles?id=10.1257/aer.20231304. Publisher abstract checked for context; the equivalence, envelopes and finite-sample calculation here are proved independently, not presented as unresolved theorems of the source.

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