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Econ-481 · economics · Economics

Macroprudential leakage

LLM status Open unverified

Statement

Original subject group: Banking, credit and financial stability

Audit classification: Published research agenda. The IMF source discusses leakage and further evidence. The policy-normalized leakage ratio is editorial; credit categories must form a partition and ratios may be undefined under zero contraction.

Definitions and precise research target

Fix a jurisdiction adopting a bank-focused macroprudential tightening $u$, measured on a declared instrument scale. Let $B_h(u)$ be regulated-bank risky credit at horizon $h\geq0$, $N_h(u)$ domestic nonbank credit, $F_h(u)$ cross-border credit and $S_h(u)$ credit redirected to unregulated sectors. Classifications must be mutually exclusive, or double-counted claims must be reconciled. For the intervention $do(u)$, define the leakage ratio whenever its denominator is positive: \[\Lambda_h=\frac{E[N_h(u)+F_h(u)+S_h(u)-N_h(0)-F_h(0)-S_h(0)]}{E[B_h(0)-B_h(u)]}.\] The research task is to identify this ratio and the accompanying change in system-wide default or crisis risk. Specify the eligible borrowers, instrument, adoption date and untreated counterfactual; determine conditions under which policy endogeneity and anticipation can be controlled. Include lender and borrower adjustment, not only static accounting substitution. Estimate heterogeneity by instrument, financial development and stress state. If unregulated credit is only partially observed, derive bounds using transparent coverage assumptions. Credit displacement is not necessarily risky leakage: changes in borrower quality, collateral and maturity must be measured before interpreting $\Lambda_h$ as a financial-stability effect.

Additional formal definitions and scope (editorial). Partition credit by lender legal status, lender residency and borrower sector using a mutually exclusive reporting hierarchy: for example domestic regulated banks, domestic nonbanks and foreign lenders, with any further unregulated category excluding the first three. Claims are measured on the same consolidation, currency, maturity and borrower-population basis. $u$ is a finite policy change, so $\Lambda_h$ is a ratio of average changes rather than a marginal elasticity. Require $E[B_h(0)-B_h(u)]>0$; negative numerators describe crowding out rather than positive leakage. Specify crisis risk as $\Pr(C_h(u)=1)$, where the indicator uses a declared systemic-service or output-loss threshold. If the denominator is weakly identified or close to zero, bound numerator and denominator jointly rather than reporting an unstable point ratio. A marginal version uses derivatives at $u=0$, requiring a positive bank-credit contraction derivative.

For the identification part, declare an admissible class $\Theta$ of structural models, including preferences, technologies, shock laws, measurement/sampling rules and any equilibrium selector. If $O$ denotes the complete observable history and $T(\theta)$ the target defined above, the identified set is $\mathcal I_T=\{T(\theta):\theta\in\Theta,\ \mathcal L_\theta(O)=\mathcal L_{\rm obs}(O)\}$; point identification means this set is a singleton. A $do(a)$ comparison replaces stated policy/technology equations while fixing the declared exogenous law and re-solving the remaining equations. These definitions do not assert that an identification theorem holds without further restrictions.

Variants and assumption changes

  1. Complete credit coverage (editorial). Observe disjoint bank, nonbank and foreign credit categories and identify finite-change leakage jointly with risk composition.
  2. Partial coverage (editorial). Impose transparent reporting-coverage bounds and derive joint bounds for bank-credit contraction, displacement and crisis-risk changes; allow an undefined ratio when contraction is zero.

References and source audit

  1. Primary IMF DP 2023/002 cover confirms all seven authors; official landing-page author field omits Gelos and Martinez Peria. DOI verified at official metadata. Locator: Executive summary, printed pp.v--vi; Introduction, pp.viii--ix; Chapters 3--4 and 6. Support: Macroprudential costs, resilience and leakage agenda. PDF accessible at legacy IMF ashx URL; some repeated fetches throttled. URL: https://www.imf.org/-/media/Files/Publications/DP/2023/English/MPEEOQEA.ashx.
  1. Nina Biljanovska; Sophia Chen; Gaston Gelos; Deniz Igan; Maria Soledad Martinez Peria; Erlend Nier; Fabián Valencia. 2023. Macroprudential Policy Effects: Evidence and Open Questions. Executive summary, printed pp.v--vi; Introduction, pp.viii--ix; Chapters 3--4 and 6. https://www.imf.org/-/media/Files/Publications/DP/2023/English/MPEEOQEA.ashx. DOI: https://doi.org/10.5089/9798400226304.087.

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-481-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-481

Macroprudential leakage Equilibrium substitution, reporting bounds and risk composition

Problem ID: G28-477. Status: Unresolved; restricted results and explicit gaps.

Scope and accounting

The target is finite-change credit displacement following a bank-focused policy, together with system-wide risk. The IMF review [imf] poses the leakage and circumvention agenda. Its indexed official summary and bibliographic record were checked in this attempt; direct full-text retrieval failed, so no detailed empirical result from its chapters is claimed. We derive a competitive substitution benchmark, sharp partial reporting bounds and a joint optimization formulation for weak denominators. None supplies causal policy assignment from observational credit totals.

At a fixed horizon let $B(u)$ be regulated-bank credit and $U(u)$ the sum of disjoint domestic nonbank, foreign and remaining unregulated credit. All claims use the same borrower population, maturity, currency and consolidation basis. The bank contraction is $D=B(0)-B(u)$ and displacement is $N=U(u)-U(0)$, giving $\Lambda=N/D$ only when $D>0$. The expected-outcome version replaces these levels by causal regime means. Firm balance sheets financed by a bank loan and then lending onward must be consolidated before $B+U$ is called credit to final borrowers.

An equilibrium with borrower and lender adjustment

Let $m\geq1$ be the number of unregulated lending technologies. A representative borrower obtains benefit $v(Q)=aQ-bQ^2/2$, $b>0$, from total credit $Q=q_B+\sum_{j=1}^mq_j$. Competitive lending technologies have real costs \[ C_B(q_B)=c_Bq_B+\gamma_Bq_B^2/2, \quad C_j(q_j)=c_jq_j+\gamma_jq_j^2/2, \] with $\gamma_B,\gamma_j>0$. A policy adds a per-unit bank wedge $u$; its receipts may be rebated lump sum without affecting the credit demand in this quasilinear model. Lenders and borrowers are price takers, so a common loan price equals marginal borrower benefit and marginal costs, including the bank wedge. Consider a policy interval on which every lender quantity is strictly positive.
Theorem (Exact finite leakage in the interior quadratic economy): The competitive equilibrium is unique. Put $S=\sum_j1/\gamma_j$ and $T=1/\gamma_B+S$. On the stated interior interval, for every finite tightening $\Delta u>0$, \[ D=\frac{1+bS}{\gamma_B(1+bT)}\Delta u>0, \quad N=\frac{bS}{\gamma_B(1+bT)}\Delta u, \quad \Lambda=\frac{bS}{1+bS}\in(0,1). \tag{1} \] Thus total credit falls by $D-N>0$ despite positive leakage.
Proof: Equilibrium quantities uniquely maximize $v(q_B+\sum_jq_j)-C_B(q_B)-uq_B-\sum_jC_j(q_j)$ on the nonnegative orthant. The objective is strictly concave since its Hessian is $-b\mathbf1\mathbf1^T-{\operatorname{diag}}(\gamma_B,\gamma_1,\ldots,\gamma_m)$, negative definite. It tends to minus infinity as quantities grow, so a unique maximum exists; its marginal conditions implement competitive supply and demand with market clearing. In the interior, loan price $r$ satisfies $q_B=(r-c_B-u)/\gamma_B$, $q_j=(r-c_j)/\gamma_j$ and $r=a-bQ$. Differentiating gives \[ r'=\frac{b/\gamma_B}{1+bT},\quad q_B'=\frac{r'-1}{\gamma_B} =-\frac{1+bS}{\gamma_B(1+bT)},\quad U'=Sr'=\frac{bS}{\gamma_B(1+bT)}. \] Coefficients are constant while the active set is unchanged, so integrating over $\Delta u$ proves the finite-change formulas. Subtracting gives $D-N=\Delta u/[\gamma_B(1+bT)]>0$. ∎
The leakage ratio rises with the responsiveness of unregulated lenders and with the steepness of inverse borrower demand. It cannot exceed one in this maintained equilibrium. A value above one would therefore reject this benchmark or its measurement and causal assumptions, rather than be impossible in every credit market. New borrower entry, changing quality, collateral feedback, foreign demand shifts and policy anticipation can all change the benchmark.

For twice continuously differentiable costs and benefit with $C_B^{\prime\prime}>0$, $C_j^{\prime\prime}>0$ and $v^{\prime\prime}<0$ throughout an interior equilibrium path, the same implicit differentiation gives the local ratio $(-v''(Q))S/[1+(-v''(Q))S]$, where now $S=\sum_j1/C_j''(q_j)$. Over a finite interior policy path the finite ratio is its average weighted by positive bank contraction $-dq_B$. This follows by writing $dU=\Lambda_{\rm loc}(-dq_B)$ and integrating numerator and denominator. Thus the finite ratio lies between the smallest and largest local ratios on that path, without assuming their constancy.

Displacement need not mean increased or decreased risk

Proposition (A risk-composition threshold): Suppose each unit of bank credit has expected loss $\theta_B\geq0$, and all displaced credit has constant expected loss $\theta_U\geq0$, with no other portfolio changes. The change in expected aggregate credit loss is \[ \Delta L=D(\theta_U\Lambda-\theta_B). \tag{2} \] Consequently even $0<\Lambda<1$ and falling total credit are compatible with either an increase or a decrease in expected losses.
Proof: Bank exposure falls by $D$, changing expected loss by $-\theta_BD$; unregulated exposure rises by $N=\Lambda D$, changing it by $\theta_UN$. Adding proves (2). If $\theta_B>0$, choose $\theta_U<\theta_B/\Lambda$ or $\theta_U>\theta_B/\Lambda$ to obtain the two strict signs; with economically bounded loss fractions, the latter case is feasible whenever $\theta_B<\Lambda$. ∎
Expected losses still do not identify a crisis probability. For example, let two equal unit exposures have Bernoulli default indicators, each with probability $p\in(0,1/2]$, and define a crisis as both defaulting. They can default together, giving crisis probability $p$, or on disjoint events, giving crisis probability zero; marginal expected losses equal $2p$ in both. More generally, for $n$ observed marginal default probabilities $p_i$ and a declared loss threshold $K$, a sharp crisis bound in the unrestricted dependence class is the linear program over probabilities $\pi_s$ on $s\in\{0,1\}^n$: \[ \sum_s\pi_s=1,\quad \pi_s\geq0,\quad \sum_s s_i\pi_s=p_i, \qquad \min\ \text{or}\ \max\quad \sum_{s:\,\sum_i\ell_i s_i\geq K}\pi_s. \tag{3} \] Every feasible vector is a joint default law, and every joint law yields one, proving sharpness directly. Exponential size is disclosed; no polynomial-time claim is made. The same exposures must not appear twice in the threshold loss.

Sharp leakage bounds under partial reporting

Suppose causal means of reported unregulated credit are $R_a\geq0$ in regimes $a=0,1$. Coverage is a restriction on mean credit value: $R_a=\kappa_aU_a$, with $0<\underline\kappa_a\leq\kappa_a\leq\overline\kappa_a\leq1$. This is not automatically implied by a percentage of reporting firms. Assume the two coverages can vary independently and impose no extra joint restriction. Bank contraction $D=d>0$ is identified.
Theorem (Sharp independent-coverage interval): Under those assumptions the leakage identified set is exactly \[ \left[ \frac{R_1/\overline\kappa_1-R_0/\underline\kappa_0}{d}, \frac{R_1/\underline\kappa_1-R_0/\overline\kappa_0}{d} \right]. \tag{4} \] Negative endpoints are permitted. If a common coverage factor is imposed instead, with $\kappa\in[\underline\kappa,\overline\kappa]$, the sharp set is the image $\{(R_1-R_0)/(\kappa d):\kappa\in[\underline\kappa,\overline\kappa]\}$, which can be strictly narrower.
Proof: Each level lies in the interval $[R_a/\overline\kappa_a,R_a/\underline\kappa_a]$. Subtracting the independently feasible level intervals and dividing by positive $d$ gives (4). To verify sharpness, every level in an interval is obtained by $\kappa_a=R_a/U_a$ when $R_a>0$; if $R_a=0$, positive coverage forces $U_a=0$. Independence permits all pairs, and their differences fill the interval by continuity. One may realize the means with deterministic credit totals and a reported/unreported partition of those totals. Under common coverage, $U_1-U_0=(R_1-R_0)/\kappa$, and every allowed $\kappa$ is similarly realized, proving the second claim. ∎
An increase in the reporting population after regulation can otherwise look like displacement. Stable coverage is an economic restriction to check, not an implication of using the same database name.

Joint bounds with a weak contraction denominator

Let $x$ collect causal level means and any additional coverage or balance variables. Suppose the exact admissible set is a nonempty compact polytope $\mathcal P=\{x:Ax\leq b\}$, and write $N(x)=n_0+n^Tx$, $D(x)=d_0+d^Tx$. Linear accounting restrictions and level bounds fit this representation. It is exact only if every point is causally and observationally compatible.
Theorem (A linear-fractional reduction): If $D(x)\geq\delta>0$ throughout $\mathcal P$, put $M=\max_{x\in\mathcal P}D(x)$. The sharp lower and upper leakage bounds are attained and equal the minimum and maximum of \[ n^Tz+n_0t\quad\text{subject to}\quad Az\leq bt,\quad d^Tz+d_0t=1,\quad t\geq1/M. \tag{5} \] If admissible points with $D=0$ exist, the ratio is undefined there and must be reported separately. On the positive-denominator subset, if $x_k\to x_0$, $D(x_k)\downarrow0$ and $N(x_0)>0$, the ratio has no finite upper bound; if $N(x_0)<0$, it has no finite lower bound.
Proof: For $D(x)>0$, put $t=1/D(x)$ and $z=tx$. The constraints in (5) follow and its objective is $N(x)/D(x)$. Conversely any feasible $(z,t)$ has $t\geq1/M>0$ and gives $x=z/t\in\mathcal P$ and $D(x)=1/t$, recovering the original ratio. This is a bijection. With $D\geq\delta$ on compact $\mathcal P$, $D$ has a finite maximum and $t$ lies in a compact positive interval; the transformed set is compact, so the objective's extrema are attained. Compatibility of every point makes them sharp. The undefined case follows from the target's definition. Finally continuity makes $N(x_k)\to N(x_0)$, while the positive denominator tends to zero, proving the two divergence claims. ∎
If numerator and denominator both tend to zero, no general divergence claim follows; their joint restrictions decide the possible slopes. Independent coordinate intervals can miss those restrictions. For sample uncertainty, a simultaneous confidence region for causal levels can be projected through (5), including an undefined flag when needed. Its coverage is inherited on the event that the whole true level region is included, but a valid causal and sampling procedure must first be supplied.

Credible policy assignment, anticipation adjustment, complete borrower coverage and changing default dependence remain the central empirical gaps. The results give falsifiable equilibrium restrictions and explicit measurement bounds, while keeping credit quantities distinct from the financial-stability outcome the policy is intended to improve.

Completion Estimate

45%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives exact competitive leakage, sharp reporting-coverage intervals, crisis-dependence bounds, and joint optimization for positive contraction denominators. Causal policy assignment, anticipation, borrower risk adjustment, and dynamic lender responses remain necessary to identify the full leakage and financial-stability target.

References

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