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Econ-265 · economics · Economics

Optimal inflation target

LLM status Open unverified

Statement

Original subject group: Inflation and monetary policy

Audit classification: Background; proposed extension. Strategy comparisons do not publish an unresolved optimal numerical target; the robust-target optimization is editorial. 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.
Fix a fully specified stochastic monetary economy with household types $i$, social weights $\omega_i$, sticky prices, downward nominal wage constraints, and an effective lower bound $i_t\ge\underline i$ on nominal policy rates. Preferences $u_i(c,\ell)$, shock laws, fiscal financing, and the admissible class $\mathcal R(\bar\pi)$ of policy rules around a constant inflation target $\bar\pi$ are given. Restrict targets to a compact interval $[\pi_L,\pi_U]$ and require a declared equilibrium selection and integrable discounted utility for every evaluated rule. Here $\beta\in(0,1)$, and nonnegative welfare weights sum to one; assume a maximizing target exists or report only the supremum. Define \[ W(\bar\pi)=\sup_{r\in\mathcal R(\bar\pi)} E_r\sum_{t\ge0}\beta^t\sum_i\omega_i u_i(c_{it},\ell_{it}),\qquad \bar\pi^*\in\arg\max_{\bar\pi\in[\pi_L,\pi_U]}W(\bar\pi). \] Determine an empirically credible set for $\bar\pi^*$ when wage rigidity, neutral-rate dynamics, and household inflation exposure are only partially identified. Quantify the tradeoff between lower-bound stabilization benefits, price dispersion, and heterogeneous purchasing-power costs within the same welfare criterion. Report whether the ranking of targets is robust to admissible parameter uncertainty and policy-rule restrictions. An optimum calculated in one calibrated model is a conditional answer, not a universal resolution. This specification asks for a defensible robust target in a declared environment; social weights and institutional policy choices cannot be inferred solely from inflation data.

Inflation targets must constrain implemented rules; otherwise identical unrestricted rule classes make $W(\bar\pi)$ independent of the label. For example restrict to a fixed family $i_t=\max\{\underline i,\bar r+\bar\pi+\phi_\pi(\pi_t-\bar\pi)+\phi_xx_t\}$, with $(\phi_\pi,\phi_x)$ in a stated compact admissible set and rates and inflation in the same units. For parameter $\theta$ in the population-compatible set $\Theta(P)$, let $W_\theta(\bar\pi)$ be the optimized welfare. The conditional optimum set is $\bigcup_{\theta\in\Theta(P)}\arg\max_{\bar\pi}W_\theta(\bar\pi)$; a robust commitment instead maximizes $\inf_{\theta\in\Theta(P)}W_\theta(\bar\pi)$, or minimizes specified welfare regret. These are different criteria. Compact targets do not guarantee attainment without nonempty rule feasibility and appropriate continuity or upper semicontinuity.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Heterogeneous inflation exposure. Retain the same target/rule family but add household-specific nominal debt, wage indexing, and consumption baskets. Compare welfare-optimal targets under fixed social weights with the representative-household benchmark.
  2. Minimax regret. Choose a target minimizing $\sup_{\theta\in\Theta(P)}[\max_{p}W_\theta(p)-W_\theta(\bar\pi)]$. This differs from worst-case welfare and allows comparisons when model utility levels have different arbitrary normalizations.

References and source audit

  • Hess Chung; Callum Jones; Antoine Lepetit; Fernando M. Martin. Implications of Inflation Dynamics for Monetary Policy Strategies. 2025. Locator: Abstract, printed p.1; Introduction, printed pp.2--3; strategy comparisons rather than choice of target level. Primary text checked. https://www.federalreserve.gov/econres/feds/files/2025072pap.pdf.
Strategy comparisons do not publish an unresolved optimal numerical target; the robust-target optimization is editorial.
  1. Hess Chung; Callum Jones; Antoine Lepetit; Fernando M. Martin. 2025. Implications of Inflation Dynamics for Monetary Policy Strategies. Abstract, printed p.1; Introduction, printed pp.2--3; strategy comparisons rather than choice of target level. https://www.federalreserve.gov/econres/feds/files/2025072pap.pdf. DOI: https://doi.org/10.17016/FEDS.2025.072.

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-265-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-265

Inflation targets under partially identified lower-tail risks

Problem ID: E52-261. Status: Unresolved. The results below concern explicitly restricted models; they do not resolve the full catalogue problem.

Question and restricted welfare model

The catalogue asks how to choose an inflation target when low-inflation constraints, inflation costs, and empirical uncertainty are treated consistently. The policy-strategy analysis [FEDS] provides context, rather than a numerical solution to this target-choice problem. We solve a reduced stationary model, distinguish the identified set of conditional optima from a robust optimum, and compute a separate minimax-regret benchmark.

Targets satisfy $\pi\in[0,P]$, $P>0$. In one interpretation a natural real rate $r$ and nominal lower bound $\underline i$ produce policy $i=\max\{\underline i,r+\pi\}$ and an output shortfall $x=-\sigma(\underline i-r-\pi)_+$. A quadratic shortfall loss is consequently proportional to a squared positive part. A similar reduced term can represent costly downward nominal wage adjustment. We collect the relevant nonnegative shortfall threshold in a random variable $Z\in[0,Z_{\max}]$ and consider loss \[ L_{a,e,b,F}(\pi)=a\pi^2+e\pi+b{\mathbb{E}}_F[(Z-\pi)_+^2], \qquad a\ge a_{\min}>0,\quad b\ge0,\quad e\in{\mathbb{R}}. \tag{1} \] For a repeated independent stationary environment with a fixed discount factor, total expected loss is (1) times a positive constant, so its minimizer is unchanged. The coefficients encode fixed normative weights as well as technology. Their uncertainty does not mean that welfare weights can be inferred from outcome data alone. This is not a complete New Keynesian equilibrium.

The conditional optimum and sensitivity

Theorem (Unique optimum and a distributional sensitivity bound): Loss (1) has a unique minimizer $p(a,e,b,F)$. Its derivative is \[ L'(\pi)=2a\pi+e-2b{\mathbb{E}}_F[(Z-\pi)_+]. \tag{2} \] The minimizer is zero if $L'(0)\ge0$, is $P$ if $L'(P)\le0$, and otherwise is the unique root of (2). It increases weakly with $b$ and with a first-order stochastic increase in $Z$, and decreases weakly with $a$ and $e$. For fixed $a,e,b$, two threshold laws satisfy \[ |p(a,e,b,F)-p(a,e,b,G)|\le\frac ba W_1(F,G). \tag{3} \]
Proof: The function $(z-\pi)_+^2$ has continuous derivative $-2(z-\pi)_+$, including at $z=\pi$. Bounded thresholds justify differentiation under expectation. Its derivative is nondecreasing, so (1) is strongly convex with parameter $2a$. Continuity on the compact interval gives existence, strict convexity uniqueness, and the endpoint and root conditions.

Increasing $b$ or stochastically increasing $Z$ weakly lowers (2) at every $\pi\ge0$; increasing $a$ or $e$ weakly raises it. A strictly increasing derivative with the stated endpoint convention yields the comparative statics.

For (3), the function $z\mapsto(z-\pi)_+$ is $1$-Lipschitz. Any coupling of $Z\sim F$ and $Z'\sim G$ therefore bounds the expectation difference by ${\mathbb{E}}|Z-Z'|$, and minimization over couplings gives derivative difference at most $2bW_1(F,G)$. Let the two constrained optima be $p,q$ and their derivatives be $f,g$. First-order optimality gives $f(p)(q-p)\ge0$ and $g(q)(p-q)\ge0$. Adding and using strong monotonicity of $f$ yields \[ 2a(p-q)^2\le (f(p)-f(q))(p-q) \le (g(q)-f(q))(p-q)\le2bW_1(F,G)|p-q|. \] Division proves the claim, including constrained endpoint solutions. ∎

The bound remains valid for distributions with atoms at the lower-bound threshold. Smooth densities are unnecessary for this quadratic shortfall loss.

Partial identification and robust choice are different objects

Assume the admissible parameter set is the rectangular product \[ a\in[a_-,a_+],\quad e\in[e_-,e_+],\quad b\in[b_-,b_+], \qquad F_-\preceq_{\mathrm{st}}F\preceq_{\mathrm{st}}F_+, \] where $a_->0$, $0\le b_-\le b_+$, both endpoint laws have the same bounded support, and every law between them in stochastic order is admitted. All combinations are allowed. Rectangularity is an assumption; empirical restrictions coupling parameters can invalidate the corner calculation.
Theorem (The identified interval and the worst-loss policy): The set of optimal targets across admissible structures is exactly the closed interval \[ [p(a_+,e_+,b_-,F_-),\ p(a_-,e_-,b_+,F_+)]. \tag{4} \] The minimax-loss policy instead is \[ \arg\min_{\pi\in[0,P]}\sup_{a,e,b,F}L_{a,e,b,F}(\pi) =p(a_+,e_+,b_+,F_+). \tag{5} \] It is unique. Formula (5) need not coincide with either endpoint in (4).
Proof: The comparative statics bound every conditional optimum by the two corner values in (4), and both corners are admitted. To fill the interval, move continuously from the first corner to the second: linearly decrease $a,e$, increase $b$, and mix the laws as $F_t=(1-t)F_-+tF_+$. Such mixtures stay between the endpoint laws in stochastic order. The losses vary uniformly continuously on $[0,P]$ along this path because support and parameters are bounded. Their unique minimizers vary continuously: any subsequential limit of minimizers along converging parameters minimizes the limiting loss by uniform convergence, and uniqueness identifies the limit. The intermediate value theorem proves (4).

For every nonnegative target, the terms $\pi^2$, $\pi$, and $(Z-\pi)_+^2$ have the signs and monotonicity needed to maximize loss by choosing $a_+,e_+,b_+,F_+$. Those choices are jointly feasible by rectangularity. The supremum loss is therefore exactly their strongly convex loss, proving (5). For example, increasing $a$ moves the optimum downward while increasing $b$ moves it upward; their simultaneous worst-loss corner is different from either corner optimizing the conditional target bound. No equality with an endpoint is implied. ∎

The identified set describes what the unknown true structure would recommend under the fixed welfare criterion. Minimax loss chooses a policy for a particular attitude toward uncertainty. Neither is a confidence interval until the admissible set itself is supported by a statistical coverage argument.

A fully solved minimax-regret comparison

Proposition (Two quadratic conditional models): Suppose two admissible conditional losses are $L_i(\pi)=A_i(\pi-p_i)^2+C_i$, with $A_i>0$ and $0\le p_1\le p_2\le P$. The unique minimax-regret target is \[ p_R=\frac{\sqrt{A_1}p_1+\sqrt{A_2}p_2}{\sqrt{A_1}+\sqrt{A_2}}, \] with worst regret \[ \frac{A_1A_2(p_2-p_1)^2}{(\sqrt{A_1}+\sqrt{A_2})^2}. \]
Proof: Subtracting each model's own minimum removes $C_i$, so regret is $A_i(\pi-p_i)^2$. A point outside $[p_1,p_2]$ can be moved to the nearest endpoint to weakly reduce both regrets and strictly reduce their maximum unless the endpoints coincide with that point. Within the interval the first regret is increasing and the second decreasing. Their maximum is minimized where their square roots are equal, namely $\sqrt{A_1}(\pi-p_1)=\sqrt{A_2}(p_2-\pi)$. Solving gives the target and substitution the value. If $p_1=p_2$, the common point is the unique zero-regret optimum. ∎
This is a genuine subcase of (1): if $Z\ge P$ almost surely, the shortfall term expands as a quadratic, with $A=a+b$ and an appropriate center whenever that center lies in $[0,P]$. In a general ambiguous model, minimax regret subtracts model-specific optima and cannot be replaced by maximizing the uncentered loss. The broader catalogue problem still requires equilibrium-consistent inflation and wage dynamics, identification of lower-tail risks, and defensible normative weights. No numerical target is inferred from this restricted calculation.

Completion Estimate

40%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt characterizes conditional inflation optima and their distributional sensitivity, distinguishes rectangular identified sets from worst-case choices and solves a quadratic minimax-regret comparison. An empirically credible target set still requires equilibrium-consistent wage and inflation dynamics, identified lower-tail risks and heterogeneous welfare costs under actual policy-rule restrictions.

References

  • [FEDS] H. Chung, C. Jones, A. Lepetit and F. M. Martin (2025), "Implications of Inflation Dynamics for Monetary Policy Strategies," Finance and Economics Discussion Series 2025-072, Board of Governors of the Federal Reserve System. doi:10.17016/FEDS.2025.072; official paper.

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