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Econ-399 · economics · Economics

Heterogeneous monetary transmission

LLM status Open unverified

Statement

Original subject group: Inflation and monetary policy

Audit classification: Explicit published open question. The Fisher-channel magnitude is explicitly debated; the conditional causal response surface 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 households with predetermined type $h_i=(a_i^{L},a_i^{I},d_i,n_i)$, where liquid assets, illiquid assets, debt, and nominal asset exposure are measured before a monetary intervention. Let $m_t$ be an externally identified unexpected increase in the policy rate, and $c_{i,t+H}(m)$ household real consumption under shock size $m$, including equilibrium income and price adjustments. Define the conditional response \[ \tau_H(h)=\left.\frac{\partial E[c_{i,t+H}(m)\mid h_i=h]}{\partial m}\right|_{m=0}. \] Identify $\tau_H(h)$ and its aggregation $\int\tau_H(h)\,dF(h)$, where $F$ is the population distribution of predetermined types. Provide instruments that isolate monetary surprises from central-bank information about fundamentals, and address nonrandom refinancing, unobserved insurance, and changes in household composition. Separate direct interest-rate and nominal-revaluation channels from induced labor-income and asset-price responses through a stated structural model or independently credible channel interventions. Test whether identified heterogeneous effects aggregate to observed consumption responses across episodes. The target includes uncertainty about the quantitative Fisher redistribution channel; it does not assume that all transmission mechanisms are unknown or that conditioning on wealth alone identifies a causal channel.

Measure the shock $m$ in a fixed unit, such as one percentage point of the annualized nominal policy rate, and define consumption in common real currency units. A derivative requires differentiability near zero and two-sided treatment support; without these, target finite shock contrasts instead. For continuous household type $h$, conditional expectations are defined only $F$-almost everywhere. To interchange the derivative and population integral, require an integrable dominating bound on individual derivatives and hold the predetermined type distribution fixed. For a household with nominal net position $n_i$, a price-level surprise changes its real value locally by $-n_i\,d\log P$; a consumption channel also needs its marginal propensity to consume and general-equilibrium income response. This balance-sheet identity alone is not a causal estimate.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Finite shock asymmetry. Compare $E[c_H(a)-c_H(0)\mid h]$ and $E[c_H(-a)-c_H(0)\mid h]$ on overlapping states and treatment supports. Do not infer large-shock symmetry from a local derivative.
  2. Refinancing constraints. Separate fixed-rate borrowers, adjustable-rate borrowers, and refinancing-eligible borrowers using characteristics measured before the announcement. Compare direct debt-service responses with labor-income responses under a disclosed channel-intervention model.

References and source audit

The Fisher-channel magnitude is explicitly debated; the conditional causal response surface is editorial.
  1. Janet L. Yellen. 2016. Macroeconomic Research After the Crisis. Section Heterogeneity. https://www.federalreserve.gov/newsevents/speech/yellen20161014a.htm.
  2. Adrien Auclert; Matthew Rognlie; Ludwig Straub. 2025. Fiscal and Monetary Policy with Heterogeneous Agents. Nominal assets, printed p.16; final sentence on the Fisher-channel empirical magnitude. https://web.stanford.edu/~aauclert/annual_review_hank.pdf. DOI: https://doi.org/10.1146/annurev-economics-091624-044646.

Integrated refinement: ECON-025

Mortgage structure in heterogeneous monetary transmission

This refinement adopts the additional source conventions (audited:conventions), subject to its local restrictions. Mortgage-contract composition. Let the pre-shock state be $s_0=(x_0,\chi)$, with $x_0=(a,h,b,z)$ and mortgage contract $\chi$, and use the initial law $\mu_0$. Conditional expected consumption $c_t(s_0;\epsilon)$ integrates subsequent refinancing, default and state transitions. Define \[ \operatorname{IRF}_C(t;\mu_0) =\left.\frac{\partial}{\partial\epsilon} \int c_t(s_0;\epsilon)\,d\mu_0(s_0)\right|_{\epsilon=0}. \] Compare this response after replacing the supported conditional contract law by $H'(d\chi\mid x_0)$ while holding the $x_0$ marginal fixed and recomputing equilibrium. Specify maturity, payment resets, prepayment, collateral and refinancing eligibility; include lenders and securities holders as counterparts. Hold shock timing, normalization and the subsequent policy rule fixed. Use finite impulses when differentiation fails. Reweighting existing contracts is distinct from changing future menus or ownership selection, and does not freeze the endogenous date-$t$ state distribution.

Supplied source audit for ECON-025

Local [S1], [S2], etc. in this audit and the references immediately below belong to ECON-025, independently of the original entry's reference numbering. [S1], PDF p. 9, supplies the HANK/housing/mortgage topic. This reweighting estimand is editorial, not an identified formula supplied by the lecture.

References for integrated refinement ECON-025

[S1] Benjamin Moll (n.d.). Lecture 11: Good Research Topics and Open Questions. Distributional Macroeconomics lecture slides; publicly archived at a 2019 URL.

Verified locator / source role: PDF p. 9: housing, mortgages, and monetary transmission.

https://benjaminmoll.com/wp-content/uploads/2019/07/Lecture11_2149.pdf

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.

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 2

Econ-399-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-399

Heterogeneous monetary transmission: the Fisher channel as a covariance, the missing intercept, and refinancing asymmetry

Abstract

We prove the following. (i) To first order, the household response decomposes into earnings, Fisher (nominal revaluation), interest-exposure and substitution channels. When net nominal positions sum to zero, the aggregate Fisher channel equals $-\mathrm{Cov}(\mathrm{MPC}_i,n_i)\cdot d\log P/dm$ times population size. So its magnitude is identified by the cross-sectional covariance of MPCs with nominal positions and the price-level response. (ii) From $(h_i,c_i,m)$ data alone, the type-conditional total response $\tau_H(h)$ is identified under shock exogeneity, but channel shares are not. We construct observationally equivalent channel splits. (iii) Panel regressions with time fixed effects identify only differences $\tau_H(h)-\tau_H(h_0)$. The aggregate $\int\tau_HdF$ (the missing intercept) requires time-series identification of the aggregate response, so cross-sectional heterogeneity cannot be aggregated without it. (iv) Under fixed-rate mortgages with refinancing thresholds, the refinancing channel responds to cuts but not to hikes. The cut response is proportional to the mass of borrowers whose rate gap exceeds the threshold, which yields finite-shock asymmetry.

1. Decomposition

For household $i$ with marginal propensity to consume $\mathrm{MPC}_i$, earnings $Y_i$, net nominal position $n_i$, unhedged interest exposure $U_i$ (maturing assets minus liabilities) and elasticity of intertemporal substitution $\sigma$, a first-order approximation in the spirit of [auclert] gives \[ dc_i=\mathrm{MPC}_i\left(dY_i-n_i\,d\log P+U_i\,dr\right)-\sigma\,\chi_i\,dr, \] where $\chi_i$ is the Hicksian scaling of the substitution effect.
Proposition: If $\sum_in_i=0$ (a closed economy, with nominal claims in zero net supply among households and counterparties), then the aggregate Fisher channel is $-\sum_i\mathrm{MPC}_in_i\,d\log P=-N\,\mathrm{Cov}(\mathrm{MPC},n)\,d\log P$.
Proof: $\sum_i\mathrm{MPC}_in_i=N[\mathrm{Cov}(\mathrm{MPC},n)+\overline{\mathrm{MPC}}\,\bar n]$, and $\bar n=0$. ∎
So the Fisher channel's quantitative magnitude, which the source flags as debated [ars], is identified by $\mathrm{Cov}(\mathrm{MPC},n)$ and $d\log P/dm$. If the government or foreigners hold nominal claims, $\bar n\ne0$ and an additional term appears.

2. Channel shares are not identified from responses alone

Proposition: Given the joint law of $(h_i,c_{i,t+H}(m))$ with $m$ exogenous, the total $\tau_H(h)$ is identified. But for any $h$-measurable reassignment $(\Delta Y,\Delta P,\Delta r)\mapsto(\Delta Y',\Delta P',\Delta r')$ with $\mathrm{MPC}(\Delta Y'-n\Delta P'+U\Delta r')-\sigma\chi\Delta r'=\mathrm{MPC}(\Delta Y-n\Delta P+U\Delta r)-\sigma\chi\Delta r$, the data are unchanged. Separating channels therefore requires measured income responses $dY_i$, measured price responses, and MPCs from independent variation (for example lottery or rebate shocks).
Proof: Only the total enters $c_i$. Each channel's size is a product of a household characteristic and an aggregate response, and the aggregate responses are not separately pinned down by consumption data. ∎

3. The missing intercept

Proposition: Consider $c_{it}=\alpha_t+\sum_k\beta_kh_{ik}m_t+\varepsilon_{it}$ with time fixed effects $\alpha_t$. Then $\beta$ identifies $\tau_H(h)-\tau_H(0)$ for linear $\tau_H$. The level $\tau_H(0)$, and hence $\int\tau_HdF$, is absorbed by $\alpha_t$. It is identified only from the time-series regression of $\bar c_t$ on $m_t$, which uses as many observations as there are shock episodes.
Proof: Any component of the response common to all $i$ at date $t$ is collinear with $\alpha_t$. ∎
Hence "test whether heterogeneous effects aggregate to observed responses" is a test across episodes, with power limited by the number of identified shocks, not by the cross-sectional sample size. General-equilibrium income effects that are common across households live entirely in the intercept.

4. Mortgages and asymmetry

Proposition: Suppose borrowers hold fixed-rate mortgages with rates $r_i$, and refinance when the market rate $r^m$ satisfies $r_i-r^m\ge\theta$ (cost threshold). Borrowers already in the money at $r^m_0$ have refinanced. For a change $\Delta$ in $r^m$, the mass newly refinancing is $\Phi(\Delta)=\Pr(\theta-|\Delta|\le r_i-r^m_0<\theta)$ if $\Delta<0$, and $0$ if $\Delta>0$ (no refinancing into higher rates). The refinancing component of the consumption response is $\mathrm{MPC}\times$(payment saving)$\times\Phi$, so $\mathbb E[c(-a)-c(0)]\ne-\mathbb E[c(a)-c(0)]$ in general. The asymmetry is larger when the rate-gap distribution has more mass just below $\theta$.
Proof: Refinancing is an option, exercised only when it is in the money. ∎
Reweighting the contract distribution $H'(d\chi\mid x_0)$ in the integrated refinement changes $\Phi$ through the gap distribution. This is a well-defined counterfactual only with equilibrium recomputation, as the refinement requires.

5. Remaining

Identification of the full surface $\tau_H(h)$ needs shocks free of central-bank information effects, and enough shock episodes for the intercept. Unobserved insurance (family transfers) attenuates $\mathrm{MPC}_i$. Household composition changes require fixed baseline units. These remain empirical tasks [yellen,moll].

Completion Estimate

48%

This is a post hoc, heuristic estimate for the full catalogue target. Fisher redistribution accounting, heterogeneous-response intercept identification and a refinancing-threshold example separate several monetary transmission mechanisms. Independently identified channel interventions, unobserved insurance and endogenous mortgage selection remain missing, and contract-composition equilibrium feedback is described without being solved.

References

  • [yellen] J. L. Yellen, Macroeconomic research after the crisis (2016), section "Heterogeneity".
  • [ars] A. Auclert, M. Rognlie, L. Straub, Fiscal and monetary policy with heterogeneous agents (2025), p. 16.
  • [auclert] A. Auclert, Monetary policy and the redistribution channel, AER 109 (2019), 2333--2367.
  • [moll] B. Moll, Lecture 11: Good research topics and open questions, PDF p. 9.

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Econ-399-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-399

Heterogeneous monetary transmission: a research attempt

Target and outcome

Status: unresolved. This attempt derives a conditional instrument formula, a covariance formula for monetary redistribution, and an equilibrium correction for changes in mortgage composition in a linear benchmark. These results distinguish assumptions that identify a total response from assumptions that identify its channels. They do not measure the Fisher channel or resolve its empirical magnitude.

The primary review by Auclert, Rognlie and Straub [hank], printed p.16, explicitly discusses disagreement about that magnitude. Its role here is motivation; the finite-type moment model and feedback calculation below are additional specializations. Consumption is measured in real currency, and $m$ is measured in a fixed policy-rate unit.

Conditional responses and instrument validity

Use episodes $t$ and predetermined household types $h=1,\ldots,J$, holding population shares $f_h$ fixed. At a declared horizon suppose the structural conditional mean has the local linear form \[ C_{ht}=\alpha_h+\tau_h m_t+g_h q_t+u_{ht}, \] where $q_t$ is central-bank information about fundamentals, not the policy shock itself. Suppose a centered instrument $z_t$ satisfies \[ {\mathbb{E}}[z_tq_t]=0,\qquad {\mathbb{E}}[z_tu_{ht}]=0,\qquad \pi={\mathbb{E}}[z_tm_t]\ne0. \] The first exclusion is substantive: a high-frequency announcement surprise need not satisfy it simply because its time window is narrow.
Proposition (Conditional and aggregate responses): Under these assumptions, \[ \tau_h=\frac{{\mathbb{E}}[z_tC_{ht}]}{\pi},\qquad \bar\tau=\sum_hf_h\tau_h =\frac{{\mathbb{E}}[z_t\sum_hf_h C_{ht}]}{\pi}. \]
Proof: Multiply the structural equation by $z_t$ and take expectations. Centering removes the intercept, and the exclusions remove both nuisance terms. Divide by the nonzero first stage. The aggregation identity follows by linearity over the finite, fixed population weights and the common denominator. ∎

This proof requires the same episode law across types. If missingness or sample construction produces different first stages $\pi_h$, a pooled moment ratio weights $\tau_h$ by $f_h\pi_h/\sum_j f_j\pi_j$, not by $f_h$. For example, equal population weights, responses $(0,2)$ and first stages $(1,3)$ produce pooled IV $1.5$ while the population response is $1$. Type-specific estimation and valid population reweighting are necessary.

One can avoid division by a weak first stage when constructing uncertainty sets. For each candidate $\tau$, test the moment ${\mathbb{E}}[z(C_h-\tau m)]=0$ and retain candidates not rejected by a size-controlled test under the specified episode dependence. Projection then covers the true coefficient whenever the moment test is valid there. Such a set may be unbounded when relevance is weak; clipping it would sacrifice coverage. This is a construction principle, not a claim that a particular standard-error formula is uniformly valid for every monetary time series.

The Fisher channel is a covariance

Let $n_h$ be predetermined nominal net assets in baseline real currency and let $a_h$ be the marginal propensity to consume out of an unanticipated real wealth transfer at the specified horizon. If the shock causes a local log-price-level change $\pi_Pm$, the direct balance-sheet effect is \[ dW_h=-n_h\pi_Pm,\qquad \tau_h^{F}=-a_hn_h\pi_P. \] With $\sum_h f_hn_h=0$ across the complete set of counterparties, \[ \bar\tau^F=-\pi_P{\mathbb{E}}[an] =-\pi_P{\operatorname{Cov}}(a,n). \] The aggregate asset change itself is zero, but the induced consumption change need not be. If borrowers have higher MPCs, the covariance can be negative. A price-level increase then raises consumption through this direct redistribution calculation; a contraction that lowers the price level reverses the sign. No sign for $\pi_P$ is assumed automatically.

Counterparties matter. Household nominal claims need not sum to zero when government or foreign liabilities are omitted. Then the correct formula includes ${\mathbb{E}}[a]{\mathbb{E}}[n]$, and fiscal responses or external wealth transfers require their own equations. Similarly, using debt balances without nominal assets or using average MPCs discards the covariance required by the identity. This partial derivative holds other channels fixed; observational IV for the total shock does not separately identify it.

Mortgage composition changes the equilibrium response

Let $H$ be the distribution of pre-shock mortgage contracts conditional on baseline characteristics. In a scalar linear equilibrium benchmark let direct consumption response and income feedback satisfy \[ \Delta C=A(H)m+B(H)\Delta Y,\qquad \Delta Y=\kappa\Delta C+\nu m, \] where $A(H)$ incorporates direct contract resets and net asset effects, $B(H)$ is the aggregate response to income, and $\kappa,\nu$ are fixed structural parameters. Assume $1-\kappa B(H)>0$, and that these equations are the locally selected equilibrium under the stated policy rule.
Proposition (Reweighting with feedback): The unique response is \[ \tau_C(H)=\frac{A(H)+B(H)\nu}{1-\kappa B(H)}. \] Its differential under a small admissible change in composition is \[ d\tau_C= \frac{dA}{1-\kappa B} +\frac{\nu+\kappa A}{(1-\kappa B)^2}\,dB. \]
Proof: Substitute the income equation into consumption, rearrange and divide by the positive denominator. Differentiate the resulting quotient, collecting the $dB$ terms. The denominator's change is part of the equilibrium adjustment, not a statistical reweighting artifact. ∎

For example, let $\kappa=0.5$, $\nu=0$, and initially $(A,B)=(1,0.2)$, giving response $10/9$. Changing contracts to $(1,0.6)$ raises it to $10/7$ even though the aggregate direct coefficient $A$ is unchanged. Reweighting only direct debt-service responses would miss this change. The example illustrates feedback algebra; no magnitude is calibrated.

For a finite change from $H$ to $H'$, recompute the full quotient at both distributions. Holding the old endogenous income response fixed defines a different partial-equilibrium estimand. If new contracts lie outside the support of observed contracts, neither quotient is identified by sample weights alone. Nonlinear refinancing, defaults and eligibility cutoffs can also make the derivative fail; finite policy contrasts are then the proper target.

Design implications and remaining obstacles

A credible implementation first fixes the shock unit, horizon and predetermined type distribution, then establishes instrument exclusions and relevance within the relevant episode sample. Separate contract-price or reset interventions may identify pieces of $A$; income variation and a structural equilibrium model are needed for $B,\kappa,\nu$. Observed consumption aggregation provides an overidentifying consistency check only if the same population and episodes are used throughout.

Two-sided randomized shock support would identify finite symmetric contrasts under consistency. If a conditional mean response $g_h(m)$ has third derivative bounded by $M_h$ on $[-a,a]$, the central contrast $[g_h(a)-g_h(-a)]/(2a)$ differs from $g_h'(0)$ by at most $M_ha^2/6$, by Taylor's theorem with integral remainder. This explicit approximation error must be added to sampling uncertainty; smaller shocks can weaken statistical signal.

The full target remains unresolved because instrument validity, joint balance sheets and MPCs, endogenous contract selection and out-of-support policy changes are substantive empirical restrictions. The formulas supply transparent tests of proposed answers and an exact solution of the stated linear benchmark, not a universal response surface.

Completion Estimate

58%

This is a post hoc, heuristic estimate for the full catalogue target. Conditional policy-shock IV identifies heterogeneous total responses, while Fisher covariance accounting and a solved mortgage-composition equilibrium quantify separate benchmark channels. Credible information exclusion, latent insurance and refinancing selection, empirical Fisher magnitudes and general contract-counterfactual identification remain unresolved.

References

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