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Econ-547 · economics · Economics

State-dependent fiscal multipliers

LLM status Partially solved unverified

Statement

Original subject group: Fiscal policy and sovereign debt

Audit classification: Explicit published open question. Size/sign multiplier asymmetries are expressly debated; this causal surface also adds distinct state variation. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
Let $G_t$ be real government purchases and $Y_t$ real output. Define predetermined state $s_t$ using a predeclared recession indicator, inflation, debt, openness, and monetary-policy constraints. An instrument $Z_t$ identifies an unanticipated purchase innovation, excluding responses to anticipated future output and other simultaneous fiscal measures. For intervention magnitude $a$, let $Y_{t+h}(a)$ and $G_{t+h}(a)$ be potential outcomes under a stated financing and monetary-policy rule. Define the cumulative state-dependent multiplier \[ m_H(s,a)= \frac{\sum_{h=0}^H E[Y_{t+h}(a)-Y_{t+h}(0)\mid s_t=s]} {\sum_{h=0}^H E[G_{t+h}(a)-G_{t+h}(0)\mid s_t=s]}, \] when its denominator is nonzero. Identify the response surface for a fixed horizon and observed range of shock sizes, with confidence sets robust to weak instruments and overlapping outcomes. Determine which state differences remain after holding instrument composition, financing, and policy reactions comparable. Distinguish marginal from large-intervention multipliers and positive from negative spending shocks; linear local projections cannot automatically establish those equalities. Report support limitations when large shocks occur only in exceptional crises. The goal is a transportable conditional estimate under explicit identification assumptions, not one invariant fiscal multiplier applicable to every economy and spending program.

Purchases and output must be real flows in the same currency and period units; scaling a policy-rate shock or a tax change is not the same intervention. Specify the full intervention path $\delta G_{t:t+H}(a)$, fiscal financing, and monetary reaction, so that equal $a$ means comparable spending across states. In a derivative multiplier replace numerator and denominator by their derivatives at $a=0$, with a nonzero denominator. A just-identified instrumental-variable ratio generally identifies an instrument-specific local response and does not identify the full nonlinear surface $m_H(s,a)$ without richer support and exclusion assumptions. Predetermined recession status must be based on information available before the shock, not output subsequently changed by it. With continuously indexed states, identification is defined only on their population support.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Monetary constraint. Compare the same purchase and financing path under a predetermined effective-lower-bound regime and an unconstrained policy regime. Specify selection into the regime and sufficient instrument support before interpreting a multiplier difference causally.
  2. Spending composition. Split purchases into public investment and government consumption with separately identified innovations. Hold total spending and financing constant and estimate composition effects, rather than pooling instruments that change different programs.

References and source audit

  • Adrien Auclert; Matthew Rognlie; Ludwig Straub. Fiscal and Monetary Policy with Heterogeneous Agents. 2025. Locator: Section5, Nonlinearities, printed pp.19--20; size/sign asymmetries rather than all state-dependence questions. Primary text checked. https://web.stanford.edu/~aauclert/annual_review_hank.pdf.
Size/sign multiplier asymmetries are expressly debated; this causal surface also adds distinct state variation.
  1. Adrien Auclert; Matthew Rognlie; Ludwig Straub. 2025. Fiscal and Monetary Policy with Heterogeneous Agents. Section5, Nonlinearities, printed pp.19--20; size/sign asymmetries rather than all state-dependence questions. https://web.stanford.edu/~aauclert/annual_review_hank.pdf. DOI: https://doi.org/10.1146/annurev-economics-091624-044646.

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-547-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-547

Fiscal multiplier ratios, nonlinear dose responses, and weak relevance

Status: Partial; the full catalogue target remains unresolved.

Abstract

A bounded-moment inversion gives a finite-sample confidence set for an instrument-specific fiscal multiplier without division by an estimated first stage. A quadratic intervention model distinguishes marginal, positive-shock, and negative-shock multipliers and states exactly which randomized dose support identifies its parameters. A composition counterexample shows that observed state differences can arise from different spending mixes even when component multipliers do not vary by state.

Define the entire intervention

Fix a horizon $H$, a predetermined state $s$, the real output and purchase numeraire, and the financing and monetary reaction rule. Write \[ Y^\Sigma(a)=\sum_{h=0}^H\{Y_{t+h}(a)-Y_{t+h}(0)\},\qquad G^\Sigma(a)=\sum_{h=0}^H\{G_{t+h}(a)-G_{t+h}(0)\}. \] The catalogue multiplier is the ratio of their conditional expectations when the denominator is nonzero. Its state is fixed using pre-shock information; classifying a recession using output affected by the shock changes the conditioning event.

A single instrumental-variable ratio addresses a narrower object. In one state let observed cumulative variables be $Y,G$, and let instrument $Z$ satisfy the declared homogeneous moment model \[ {\mathbb{E}}[Z(Y-mG)\mid s]=0. \tag{1} \] Set $a_s={\mathbb{E}}[ZY\mid s]$ and $b_s={\mathbb{E}}[ZG\mid s]$. If $b_s\ne0$, the unique moment coefficient is $m_s=a_s/b_s$. The exclusion in (1) is not implied by calling $Z$ a fiscal news shock. It also does not identify a nonlinear response surface without additional restrictions.

A confidence set that does not divide by relevance

For a finite-sample benchmark observe $n$ independent episodes in the same state, with $|Z_i|\le1$ and $|Y_i|,|G_i|\le M$. Episodes are separated sufficiently to make the assumed independence meaningful; overlapping horizon sums from consecutive dates are not automatically independent.

Let $\widehat a=n^{-1}\sum_iZ_iY_i$, $\widehat b=n^{-1}\sum_iZ_iG_i$, and $t_n=\sqrt{2\log(2/\alpha)/n}$. Define \[ \mathcal C_n=\{m\in{\mathbb{R}}: |\widehat a-m\widehat b|\le M(1+|m|)t_n\}. \tag{2} \]

Theorem: For every true finite $m$ satisfying (1), the confidence set (2) covers it with probability at least $1-\alpha$, uniformly over the bounded independent law class. No lower bound on $|b_s|$ is required.
Proof: At the true coefficient, $X_i=Z_i(Y_i-mG_i)$ has mean zero and absolute bound $M(1+|m|)$. The bounded-mean exponential inequality gives ${\mathbb{P}}(|n^{-1}\sum_iX_i|>M(1+|m|)t_n)\le\alpha$. On the complementary event the defining inequality in (2) holds at that coefficient. Since the bound has the same probability for every admissible law, the coverage is uniform. Inverting one valid test at each candidate does not require a union bound over all real candidates to cover the true one. ∎
When the first stage is weak, (2) may be unbounded or contain two rays. That is compatible with valid uncertainty. Imposing an arbitrary finite plotting range and then describing the clipped interval as the full confidence set would hide the weak-identification problem.

A second construction starts from simultaneous intervals for $a_s,b_s$ and projects all ratios $a/b$ with $b\ne0$. If the first-stage interval is bounded away from zero, the ratio extrema occur among its four corner ratios. If it crosses zero, division of interval endpoints is generally incorrect: the set can be unbounded or disconnected. The projection is an uncertainty construction for the moment ratio, not proof that it equals a large spending intervention multiplier.

What two randomized nonzero doses identify

Consider the restricted state-specific response model \[ {\mathbb{E}}[G^\Sigma(a)\mid s]=a,\qquad {\mathbb{E}}[Y^\Sigma(a)\mid s]=\beta_s a+\gamma_s a^2 \tag{3} \] for doses in a known interval containing zero. Purchase flows and output are in comparable units, and all compared doses use the same financing and monetary rule. Negative doses represent purchase reductions under that rule.

For $a\ne0$ the average multiplier is $m_s(a)=\beta_s+\gamma_s a$. The derivative multiplier at zero is $\beta_s$, whereas the derivative ratio at nonzero $a$ is $\beta_s+2\gamma_sa$. Positive and negative shocks of equal magnitude differ by $2\gamma_sa$ in their average multipliers.

Proposition: If randomized doses $0,a_1,a_2$ have positive assignment probabilities, with $a_1a_2\ne0$ and $a_1\ne a_2$, the model identifies \[ \gamma_s= \frac{\mu_s(a_2)/a_2-\mu_s(a_1)/a_1}{a_2-a_1}, \qquad \beta_s=\mu_s(a_1)/a_1-\gamma_sa_1, \tag{4} \] where $\mu_s(a)={\mathbb{E}}[Y^\Sigma(a)\mid s]$.
Proof: Randomization identifies each mean response relative to dose zero. Dividing (3) at the two nonzero doses yields two linear equations $\mu_s(a_j)/a_j=\beta_s+\gamma_sa_j$. Their determinant is $a_2-a_1\ne0$. Subtraction and substitution give (4). ∎
With only dose zero and dose one, a response of $1.2$ is compatible with $(\beta,\gamma)=(1.2,0)$ or $(0.8,0.4)$. They agree on all tested outcomes but predict multipliers $1.2$ and $0.4$ respectively for dose minus one. One can construct identical outcome noise at the observed doses, making the full observed laws coincide. The missing negative-shock conclusion is therefore an extrapolation, not a small-sample issue.

Even two nonzero doses identify (4) only under the quadratic restriction. Adding a function proportional to $a(a-a_1)(a-a_2)$ leaves the tested means unchanged while altering other doses. A sufficiently small coefficient preserves any strict local outcome restrictions. Finite support does not identify an unrestricted smooth response surface everywhere.

Spending composition can mimic state dependence

Suppose investment purchases have constant multiplier two and current consumption purchases have constant multiplier one half. In each state an instrument induces a positive investment purchase response $g_I$ and consumption response $g_C$. With additive output effects its instrument ratio is \[ m^{IV}=\frac{2g_I+\tfrac12g_C}{g_I+g_C}. \] If the investment share is $0.8$ in recessions and $0.2$ in normal periods, the ratios are $1.7$ and $0.8$. Neither component multiplier varies by state. The difference comes entirely from instrument composition.

The same logic applies to financing, anticipation, and monetary accommodation. Comparing ratios across states requires either the same intervention mix or an explicit decomposition identifying how each component changes. If an instrument induces components with opposite signs, the weights need not be convex and the ratio can lie outside the component-multiplier range.

Remaining empirical and dynamic work

The moment confidence set, quadratic identification, and composition example are complete within their declared classes. They do not validate a fiscal instrument, identify the full response path, establish independence of historical episodes, or transport crisis-dose responses to ordinary times. Continuous states require support and smoothness assumptions, while overlapping outcomes need dependence-aware inference.

The broad catalogue problem remains unresolved. In particular, a narrow confidence interval for one well-identified local ratio is not evidence that all shock sizes, signs, states, financing arrangements, and purchase categories share its value. These are elementary restricted results rather than claims of a new general fiscal multiplier theory.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt gives weak-first-stage-safe inference for a bounded independent-episode moment ratio, identifies a quadratic randomized-dose model, and demonstrates composition-induced state differences. The full fiscal response surface still requires credible instruments, comparable financing and policy reactions, supported shock signs and sizes, and inference allowing overlapping outcomes and continuous states.

References

  • [ars] A. Auclert, M. Rognlie, and L. Straub (2025). Fiscal and Monetary Policy with Heterogeneous Agents, section 5. https://web.stanford.edu/~aauclert/annual_review_hank.pdf. The primary review discusses nonlinear size and sign questions; equations (1)--(4) are the explicit models used here.

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