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Econ-602 · economics · Economics

Adoption races and beliefs

LLM status Open unverified

Statement

Original subject group: Innovation and AI economics

Audit classification: Background; proposed extension. The cited 2025 study was revised in August 2026 and already reports adoption-belief effects. Its measured intentions and robotics outcomes are not the catalogue proposed AI discovery welfare effect.

Definitions and precise research target

Study firms in a specified industry that have not adopted a particular AI research tool at baseline. Firm $i$ holds belief $B_i\in[0,1]$ about the fraction of competitors adopting within a year, chooses research investment $I_i\geq0$, and later produces validated innovation value $Y_i$. Randomly provide accurate information $Z_i\in\{0,1\}$ about competitors' previously measured adoption plans, without changing tool prices or access. Let $B_i(z)$, $I_i(z)$ and $Y_i(z)$ denote potential beliefs and outcomes. Estimate $E[B_i(1)-B_i(0)]$, $E[I_i(1)-I_i(0)]$, and $E[Y_i(1)-Y_i(0)]$ for eligible firms. Interpreting the investment ratio \[ \beta=\frac{E[I_i(1)-I_i(0)]}{E[B_i(1)-B_i(0)]} \] as a belief response additionally requires a nonzero first stage and exclusion of other information channels; it is not automatically a structural derivative. Record actual competitor adoption to separate corrected expectations from subsequent strategic responses. To assess distortion, compare validated value net of real research costs under information provision and baseline, including spillovers within randomized market clusters. Does correcting adoption beliefs accelerate socially useful research or mainly induce duplication and defensive expenditure? This is a proposed extension of existing evidence on adoption expectations, with a defined information treatment and population.

Definitions, assumptions and mathematical qualifications. The signal treatment is a fixed truthful reporting rule about competitor adoption at a preregistered survey date. Potential outcomes are defined under that report and under no report, including each treatment version and access baseline. A market cluster rather than a single firm is the treatment unit if competitors react to others receiving information. The Wald ratio is defined only if $E[B_i(1)-B_i(0)]\ne0$; its interpretation as a causal belief response needs a structural belief-to-investment map, instrument exclusion and an appropriate monotonicity or homogeneity restriction. A continuous belief change does not automatically produce a LATE interpretation. Social value includes rivals gains, duplication, real investment costs and any information-production costs.

Variants and assumption changes

  1. Editorial, reduced form. Estimate assignment effects on reported beliefs, verified actual adoption and validated outcomes; reduced-form effects do not require treating beliefs as an instrumented endogenous dose.
  2. Editorial, welfare channel. Supply an identified belief-to-investment model and randomized market information saturation; assess net innovative value including duplication and strategic reactions.

References and source audit

Checked source. NBER Working Paper 34532, December 2025; revised August 2026, indexed abstract. The study reports experimental adoption-belief results, not the proposed innovation-welfare question. Primary metadata and abstract indexed; direct PDF returned HTTP 403. https://www.nber.org/papers/w34532.
  1. Zoë B. Cullen; Ester Faia; Elisa Guglielminetti; Ricardo Perez-Truglia; Concetta Rondinelli. 2025. The Innovation Race: Experimental Evidence on Advanced Technologies. NBER Working Paper 34532, December 2025; revised August 2026, indexed abstract. https://www.nber.org/papers/w34532. DOI: https://doi.org/10.3386/w34532.

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-602-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-602

Adoption races and beliefs: Network feedback separates belief effects from adoption multipliers

The intervention and estimands

Fix firms eligible before treatment and a consequential information intervention $Z$ about competitors' measured plans. The target includes effects on beliefs $B$, research investment $I$, and validated innovation value $Y$, as well as welfare after real costs and spillovers. A familiar ratio is \[ \frac{E[I_i\mid Z_i=1]-E[I_i\mid Z_i=0]} {E[B_i\mid Z_i=1]-E[B_i\mid Z_i=0]}. \tag{1} \] Its denominator must be nonzero, and its causal interpretation needs more than random assignment. Cullen and coauthors study information and advanced-technology adoption [cullen]; their study does not supply a validated-innovation welfare estimate for every AI research race. We analyze a declared linear network submodel in which (1) can differ from the direct investment response even with a clean own-belief exclusion restriction.

A unique investment equilibrium

Let $W$ be a known $n\times n$ nonnegative matrix, $n\ge2$, with zero diagonal and row sums one. Let $0\le\gamma<1$, $\theta\ge0$, and $\pi>0$. Information assignments form $z\in\{0,1\}^n$, and \[ B(z)=b+\pi z,\qquad I(z)=a+\theta B(z)+\gamma W I(z). \tag{2} \] Require $b$ and $b+\pi\mathbf1$ to lie in $[0,1]^n$, since beliefs are fractions, and require $a+\theta b\ge0$. Thus all displayed investments are nonnegative. One behavioral foundation is the best-response map from perceived objectives $(a_i+\theta B_i+\gamma\sum_jW_{ij}I_j)I_i-I_i^2/2$; zero diagonal ensures the peer term is fixed when firm $i$ optimizes. This is a restriction on behavior, not an assumption that perceived private returns equal validated social value.
Lemma: For every assignment the unique nonnegative equilibrium is \[ I(z)=R(a+\theta b+\theta\pi z),\qquad R=(I_n-\gamma W)^{-1}=\sum_{k=0}^{\infty}\gamma^kW^k. \tag{3} \]
Proof: The sup operator norm of $W$ is one. The geometric series therefore converges in that norm and multiplication by $I_n-\gamma W$ telescopes to the identity, proving invertibility and (3). Every term of the series is nonnegative, so its product with the nonnegative right side is nonnegative. The nonnegative best-response map, including projection at zero if needed, is a contraction with modulus $\gamma$ in the sup norm. Thus it has at most one fixed point, and (3) supplies it. ∎

Own randomization can include feedback through competitors

Suppose firm assignments are mutually independent Bernoulli variables with probabilities strictly between zero and one. Randomization is independent of the fixed primitives, and information has no direct effect on (2) except through the corresponding belief coordinate.
Theorem: Under individual assignment the Wald ratio (1) equals \[ \theta R_{ii} =\theta\left(1+\sum_{k\ge2}\gamma^k(W^k)_{ii}\right). \tag{4} \] If instead the entire isolated market receives a common assignment, the corresponding firm-level ratio is $\theta/(1-\gamma)$. For $\gamma>0$ and $\theta>0$, the individual ratio equals $\theta$ precisely when no positive-weight directed walk returns to firm $i$.
Proof: In (3), changing only $z_i$ shifts investment by $\theta\pi Re_i$. Independence ensures all other assignments have the same distribution conditional on either value of $Z_i$. Thus the numerator is $\theta\pi R_{ii}$ and the belief first stage is $\pi$, proving the first equality. The series representation and $W_{ii}=0$ give (4). With common assignment, $W\mathbf1=\mathbf1$ implies $R\mathbf1=\mathbf1/(1-\gamma)$, yielding the market ratio. Finally every summand in (4) is nonnegative, and $(W^k)_{ii}>0$ exactly when some length-$k$ positive-weight walk returns to $i$. With positive $\gamma,\theta$, equality holds exactly when all those return terms vanish. ∎
Thus exclusion of a direct information effect does not remove equilibrium feedback. The ratio can be meaningful as a response to this complete information regime while failing to equal the structural coefficient $\theta$. It is not automatically a continuous-treatment complier effect.

Recovering the structural response with network variation

Proposition: Under the individual randomization above, suppose $\theta>0$ and the full investment vector is observed. Define the population moment matrix \[ D=\frac1\pi\operatorname{Cov}(I,Z) \operatorname{Cov}(Z)^{-1}. \tag{5} \] Then $D=\theta R$ is invertible, $\theta=1/(D^{-1})_{ii}$ for every $i$, and for any $W_{ij}>0$, \[ \gamma=-\theta\,(D^{-1})_{ij}/W_{ij}. \tag{6} \] If $\theta=0$, these moments identify a zero direct response but do not identify $\gamma$.
Proof: Center (3) and multiply by the centered assignment vector to get $\operatorname{Cov}(I,Z)=\theta\pi R\operatorname{Cov}(Z)$. The assignment covariance is diagonal with strictly positive entries, so (5) equals $\theta R$. For positive $\theta$ its inverse is $\theta^{-1}(I_n-\gamma W)$. The diagonal and off-diagonal identities give (6); a positive entry of $W$ exists because its rows sum to one. For $\theta=0$, the entire moment matrix is zero for every allowed $\gamma$, proving the final claim. ∎
The scalar first stage $\pi$ is itself identified from beliefs by random assignment. Classical mean-zero reporting error independent of assignment preserves that first stage, but arbitrary systematic belief measurement errors need a separate model. If only own investment rather than the full cross-firm response is recorded, the matrix identification argument is unavailable.

Innovation value and real-cost welfare

Fix a social accounting convention $\mathcal W(I)=v^{\mathsf T}I-I^{\mathsf T}HI/2$, where $H$ is symmetric positive semidefinite and includes the declared real costs and net cross-firm effects. This quadratic is an additional restriction; beliefs are not themselves social benefits. For a common information treatment, let $I^0$ be baseline investment and $d=\theta\pi/(1-\gamma)$. Direct expansion gives \[ \Delta\mathcal W =d(v-HI^0)^{\mathsf T}\mathbf1 -\frac{d^2}{2}\mathbf1^{\mathsf T}H\mathbf1-c_{\mathrm{info}}, \tag{7} \] where $c_{\mathrm{info}}$ is the real information-production cost, counted once. For example, take two firms, each linked to the other, $\gamma=1/2$, $\theta=1$, $\pi=1/2$, and $a=b=0$. Investment rises from zero to one at each firm. If validated gross value is $Y_i=I_i$ and real research cost is $2I_i^2$, gross value rises by two but net welfare falls by two even with zero information cost. This uses $v=\mathbf1$ and $H=4I_n$ in (7), and all beliefs remain between zero and one. A positive adoption or innovation-value response therefore does not establish a positive net social gain.

Remaining gap

The exact network formulas require known links, stable linear responses, isolated markets, and the specified information exclusion. They neither establish those assumptions for AI research nor identify duplication, rival spillovers, or resource costs from adoption intentions alone. Market-level randomization and validated value and cost measurement can identify regime welfare without pretending that the individual Wald ratio is a universal structural parameter. Extending identification to unknown networks, heterogeneous nonlinear responses, and endogenous competitor entry remains unresolved here.

Completion Estimate

49%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt solves a linear network investment equilibrium, identifies feedback-distorted individual and market Wald ratios, and derives a welfare comparison separating investment from validated value. The full adoption-race target still needs measured belief changes and innovation outcomes, credible network and exclusion restrictions, saturation evidence, and real duplication and spillover costs.

References

  • [cullen] Z. B. Cullen, E. Faia, E. Guglielminetti, R. Perez-Truglia, and C. Rondinelli (2025), The Innovation Race: Experimental Evidence on Advanced Technologies, NBER Working Paper 34532, December 2025, subsequently revised. Primary abstract and record: https://www.nber.org/papers/w34532.

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