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Econ-511 · economics · Economics

Equilibrium effects of nudges at scale

LLM status Partially solved unverified

Statement

Original identifier: B6 (atlas item 66). Classification: Equilibrium policy evaluation research program. Original tractability label: Medium (editorial, not a complexity result).

Definitions and assumptions

Let markets $m$ be clusters with no cross-market interference, and let $s\in[0,1]$ denote the fraction of eligible consumers assigned a specified nudge. Consumers respond to assignment, prices $p$, product quality $q$, and social-state variables $n$. Firms choose prices, quality, and product offerings under a specified game. Let $\mathcal E(s)$ be its equilibrium set and let a declared selection rule $\sigma$ choose $e_\sigma(s)\in\mathcal E(s)$. A welfare functional $W^w$ includes consumer utility under evaluator $w$, producer surplus, implementation costs, and externalities. Prices are transfers where appropriate, not social resource costs counted twice.

Formal research question

Identify or bound the market-level effects \[ \Delta_Y^\sigma(s)=\mathbb E[\overline Y_m(s,e_\sigma(s))-\overline Y_m(0,e_\sigma(0))],\qquad \Delta_W^{w,\sigma}(s)=\mathbb E[W_m^w(s,e_\sigma(s))-W_m^w(0,e_\sigma(0))]. \] Here $\overline Y_m$ is the average outcome in a baseline-defined market population. Determine restrictions under which an individual-level low-saturation treatment effect predicts $\Delta_Y^\sigma(1)$, and characterize deviations induced by supplier responses or peer spillovers. When selection is unidentified, report the corresponding range over admissible $\sigma$ rather than one arbitrary equilibrium.

Interpretation and scope

A saturation experiment randomizes coverage at the market level and assignment within markets. It can separate direct and indirect effects under a specified interference model, provided prices and supplier behavior have time to adjust. Randomizing only individuals usually measures a policy embedded in a particular market equilibrium; it does not identify the full-coverage counterfactual. Scaling also changes the composition of implementers and recipients, which is distinct from equilibrium feedback. A successful extension would combine saturation variation, firm outcomes, and validated demand and supply restrictions to predict unobserved coverage levels. Identification at observed saturation values does not automatically justify extrapolation to one hundred percent coverage. Welfare must include inconvenience, privacy, or moral-pressure costs if the evaluator treats these as relevant; a behavioral outcome moving in the desired direction is insufficient. Long-run entry requires an additional horizon and dynamic model.

Source audit and status

[S1] supplies the nudge concept; [S2] measures welfare for energy social comparisons; [S3] compares academic and agency trials at scale. The last study does not establish general-equilibrium invariance and its set of interventions excludes some default changes. All citations are verified but source roles are narrowed. The displayed market-level target is an editorial extension, not a theorem or published conjecture attributed to these authors, and its openness depends on the market and response model.

References

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  1. Richard H. Thaler and Cass R. Sunstein (2008). Nudge: Improving Decisions About Health, Wealth, and Happiness. Yale University Press. https://yalebooks.co.uk/book/9780300146813/nudge/

    Source role: Conceptual policy motivation. Locator: Publisher description and original 2008 ebook metadata.

  2. Hunt Allcott and Judd B. Kessler (2019). The Welfare Effects of Nudges: A Case Study of Energy Use Social Comparisons. American Economic Journal: Applied Economics 11(1), 236--276. https://www.aeaweb.org/articles?id=10.1257/app.20170328

    Source role: Welfare measurement for a particular nudge. Locator: Abstract and article metadata.

  3. Stefano DellaVigna and Elizabeth Linos (2022). RCTs to Scale: Comprehensive Evidence from Two Nudge Units. Econometrica 90(1), 81--116. https://doi.org/10.3982/ECTA18709

    Source role: Comparison of academic trials with agency trials; not a general-equilibrium theorem. 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-511-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-511

Nudges at scale: equilibrium attenuation, amplification, and welfare

Scope

An individual experiment at one market saturation identifies a policy embedded in that equilibrium. It does not automatically identify full-coverage effects. This attempt solves two explicit market benchmarks, gives sharp extrapolation bounds under a stated smoothness restriction, and derives finite-sample simultaneous inference for randomized saturations. The general market game, endogenous entry and evaluator-dependent welfare in the catalogue remain unresolved.

First consider a unit mass of consumers with quantity demands \[ q_i=a-bp+\tau z_i+\gamma\overline q,\qquad b>0, \] where $z_i\in\{0,1\}$ is assignment and $s=E[z_i]$. Competitive supply is $p=c+\kappa\overline q$, $\kappa\ge0$. All compared quantities are assumed nonnegative and interior; these linear equations are not extrapolated beyond that domain. Market aggregation and substitution give \[ \overline q(s)=\frac{a-bc+\tau s}{1-\gamma+b\kappa}. \tag{1} \] Assume $D=1-\gamma+b\kappa>0$. The algebraic equilibrium is then unique. If an adaptive best-response process is intended, its convergence requires its own stability condition; uniqueness alone does not prove dynamical stability.

An exact comparison between direct and rollout effects

Holding the market state fixed, the individual assignment contrast equals $\tau$. In the continuum model one consumer has no effect on the aggregate, so this is also the direct contrast within a fixed-saturation equilibrium. The full market change is instead \[ \overline q(1)-\overline q(0)=\tau/D. \tag{2} \] Thus supply feedback attenuates the effect when $D>1$, while positive peer feedback can amplify it when $0<D<1$. With $b\kappa=1,\gamma=0$, rollout equals half the direct response. With $b\kappa=0.2,\gamma=0.8$, it equals $2.5$ times the direct response. These are feasible constructed coefficients, subject to choosing $a,c,\tau$ so quantities remain interior.

For a treated consumer the total change from the zero-coverage state includes both its own $\tau$ and the common price/peer term. Untreated consumers also change quantities through that common term. Therefore a difference between treated and untreated consumers can remove precisely the equilibrium effect relevant to the market mean. Randomizing coverage across independently operating markets supplies the variation needed to observe (1) as a function of $s$; randomizing only individuals at one $s$ does not.

More quantity need not mean more welfare

For a separate homogeneous benchmark, let the true marginal willingness to pay be $a-bQ$ and real marginal supply cost $c+\kappa Q$, with $b>0,\kappa\ge0$. Consumers perceive an intercept $a-u+\tau s$, where $u\ge0$ is a baseline underestimation and the nudge corrects perceived value by $\tau s$. Then \[ Q(s)=\frac{a-c-u+\tau s}{b+\kappa},\qquad Q^*=\frac{a-c}{b+\kappa}. \] Require interior quantities over the examined range. The evaluator uses true utility, includes producer surplus, and subtracts real implementation cost $ks$, $k\ge0$: \[ W(s)=(a-c)Q(s)-\tfrac12(b+\kappa)Q(s)^2-ks. \] Payments cancel between consumers and producers. Completing the square yields \[ W(s)-W(0)=\frac{u\tau s-\tfrac12\tau^2s^2}{b+\kappa}-ks. \tag{3} \] This is an exact welfare comparison, not an assertion that the chosen evaluator is uniquely correct. With $u=0$, any nonzero demand-raising nudge reduces surplus before costs because the benchmark started at the efficient quantity. With $u>0$, initial correction may help but excessive coverage can overcorrect. If $\tau>0$, the welfare-maximizing coverage on $[0,1]$ is \[ s^*=\operatorname{clip}_{[0,1]} \left(\frac{u}{\tau}-\frac{k(b+\kappa)}{\tau^2}\right). \] Different privacy, inconvenience or moral-pressure costs alter (3) and the optimum. The first quantity model's peer term cannot be inserted into this welfare formula without defining whether peer effects enter true utility or merely perceived utility; these are intentionally separate benchmarks.

Sharp bounds when full saturation was not observed

Let $f(s)$ be the population market-level outcome under the declared equilibrium selection rule. Suppose experiments identify $f(s_j)=y_j$ at finitely many saturation levels and economic reasoning supplies a global Lipschitz constant $L$ on $[0,1]$. Assume the observations satisfy $|y_i-y_j|\le L|s_i-s_j|$. Then \[ f(1)\in\left[ \max_j\{y_j-L(1-s_j)\},\quad \min_j\{y_j+L(1-s_j)\}\right]. \tag{4} \]
Proposition: The bounds in (4) are sharp over all real-valued $L$-Lipschitz functions matching the experimental means.
Proof: Necessity follows from the Lipschitz inequalities between $1$ and every observed level. For any value $v$ inside (4), augment the finite data by $(1,v)$. All pairwise inequalities still hold. The function $F(s)=\min_i\{y_i+L|s-s_i|\}$, with the augmented point included, is $L$-Lipschitz and agrees with every augmented value. Each cone has the same Lipschitz constant, and the pairwise inequalities force equality at the data points. This constructs an admissible extension attaining $v$. ∎
Known outcome support can be enforced by clipping the extension to that support, preserving both the Lipschitz constant and values already in the support. An untested assumed $L$ is not identified by observing a few small saturations. With no continuity restriction, $f(1)$ can take any supported value independently of the observed interior levels. Multiple equilibria may also make a selected $f$ discontinuous; a Lipschitz assumption on primitives does not automatically bound a selection rule that jumps.

Finite-sample inference at the market level

Suppose each tested saturation has $n_j$ independent markets whose bounded outcomes belong to $[A,B]$, and the assignment design identifies its target mean. With $J$ saturation levels, \[ r_j=(B-A)\sqrt{\log(2J/\alpha)/(2n_j)} \] gives simultaneous intervals $[\widehat y_j-r_j,\widehat y_j+r_j]$ with coverage at least $1-\alpha$, by Hoeffding and a union bound. Empty cells use the full outcome range. Intersect the set of all Lipschitz functions with these interval constraints and project to $f(1)-f(0)$. On the common event, the true function belongs to the feasible set, hence the projection covers the rollout contrast. This projection also preserves the coupling between baseline and full saturation. Separate coordinate confidence limits can be combined conservatively, but should not be called a sharp joint confidence region.

The procedure remains valid near zero direct effects because it uses no division by an estimated response. It can be wide if all observed saturations are low. Larger cluster counts improve sampling precision but cannot remove the distance-to-full-coverage term in (4) without stronger restrictions or high-saturation data.

Unresolved empirical and dynamic content

The results show precisely how a supply slope, peer effect or welfare evaluator changes an extrapolation. A completed catalogue application needs markets, an operational nudge, equilibrium adjustment time, supplier quality and entry data, and validation away from fitted saturations. Individual take-up effects, market quantity effects and social welfare are separate outcomes. None of the formulas claims a universal attenuation factor for nudges.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves supply-peer attenuation and welfare benchmarks, derives sharp Lipschitz rollout bounds, and gives simultaneous market-saturation inference. Full nudge-scale effects still require validated demand and supplier quality, dynamic entry, adjustment horizons, and equilibrium-selection sensitivity beyond a fixed smooth outcome function.

References

  • [ak] H. Allcott and J. B. Kessler (2019), The Welfare Effects of Nudges: A Case Study of Energy Use Social Comparisons, AEJ: Applied Economics 11(1), 236--276. https://www.aeaweb.org/articles?id=10.1257/app.20170328.
  • [dl] S. DellaVigna and E. Linos (2022), RCTs to Scale: Comprehensive Evidence from Two Nudge Units, Econometrica 90, 81--116. https://doi.org/10.3982/ECTA18709. The cited scale comparison does not supply the equilibrium coefficients assumed above.

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