Derivation of the organizing equation

This appendix derives equation 2 and states the assumptions behind it. The framework starts with optimizing households and adds a reduced-form external cost of financial vulnerability of the kind studied in the normative macroprudential literature (Lorenzoni 2008; Bianchi 2011; Farhi and Werning 2016; Korinek and Simsek 2016; Dávila and Korinek 2018). The purpose is not to calculate welfare, but to show how the objects discussed in the main text enter a first-order welfare expression.

The notation and sign convention are the same as in the main text. A higher $\tau$ permits more borrowing, so a tightening has $\mathrm{d}\tau<0$. Statutory capacity is $\kappa_i$, effective capacity is $\tilde{\kappa}_i$, and implementation pass-through $\epsilon_i$ is defined in equation 1. We use $\eta_i$ for the Lagrange multiplier on the borrowing constraint in utility units and $\lambda_i$ for the same object in money units.

B.1Environment

Households and markets. There is a unit continuum of households indexed by $i$. Household $i$ chooses tenure (own or rent), a consumption plan, housing $h_i$ (quantity, quality, and location), mortgage debt $b_i$ ($b_i=0$ for renters), liquid saving, and other financing such as unsecured credit or family transfers. We collect these choices in the vector $a_i$, and preferences are $U_i(a_i;Z)$, where $Z$ is defined below.

Households take as given a vector of market outcomes $x=(x_1,\dots,x_J)$, which includes the house price $P$, the rent $R$, mortgage spreads and fees $s$, and lender underwriting standards beyond the statutory rule. The budget constraint is written as $B_i(a_i;x)\le 0$, where $B_i$ is the present value of expenditure less the present value of resources. For a buyer, $B_i$ includes the purchase price $Ph_i$, less the mortgage $b_i$, plus the present value of debt service. For a renter, it includes the present value of rent. Payments at later dates are valued at the household’s own discount factor, so all objects below are measured in units of the numéraire at the date of the policy change.

B.1.1Borrowing constraints

Household $i$ faces the constraint $$\begin{equation} \label{eq:constraint} b_i \le \tilde{\kappa}_i(\tau,x), \end{equation}$$(B.1) where effective capacity $\tilde{\kappa}_i$ is statutory capacity $\kappa_i$ adjusted for how the rule is implemented. The instruments in the main text differ in which variables enter $\kappa_i$:

• LTV limit: $\kappa_i=\tau Ph_i$, where $\tau$ is the maximum loan-to-value ratio. Capacity rises with the house price, $\partial\kappa_i/\partial P=\tau h_i>0$, which is the collateral constraint of Kiyotaki and Moore (1997).

• LTI limit: $\kappa_i=\tau y_i$, where $y_i$ is the income the lender uses. Capacity does not depend on the house price. A DTI limit has the same form applied to total debt, so mortgage capacity is $\tau y_i$ less the household’s other debt.

• DSTI limit: required payments may not exceed a share $\tau$ of income. With payments equal to $\alpha(r+s,T)\,b_i$, where $\alpha$ is the annuity factor for the mortgage rate $r+s$ and maturity $T$, capacity is $\kappa_i=\tau y_i/\alpha(r+s,T)$. Because $\alpha$ increases with the rate, $\partial\kappa_i/\partial r<0$ and $\partial\kappa_i/\partial s<0$.

• Amortization requirement: the requirement restricts the repayment path rather than the loan at origination. It gives one constraint per period, and $\lambda_i$ below is the present value of the money-metric multipliers on these constraints. A household can therefore be constrained by an amortization requirement while having unused capacity at origination, as in the Swedish evidence (Bäckman et al. 2026).

Implementation enters through effective capacity. With lender exemptions $e_i$, for instance allocated under a quota (Kinghan and McCann 2019), effective capacity is $\tilde{\kappa}_i=\kappa_i+e_i$. If lenders grant more exemptions when the rule is tightened, $\partial e_i/\partial\tau<0$, and equation 1 gives $$\epsilon_i=1+\frac{\partial e_i/\partial\tau}{\partial\kappa_i/\partial\tau}<1 .$$ Lender underwriting standards enter $\tilde{\kappa}_i$ in the same way, which is why we include them in $x$.

B.1.2The household problem

Conditional on owning, household $i$ solves $$\begin{equation} \label{eq:hhproblem} V_i^{o}(\tau,x,Z)=\max_{a}\;U_i(a;Z) \quad\text{s.t.}\quad B_i(a;x)\le 0\;\;[\mu_i],\qquad b(a)\le\tilde{\kappa}_i(\tau,x)\;\;[\eta_i], \end{equation}$$(B.2) where $\mu_i>0$ is the marginal utility of the numéraire and $\eta_i\ge 0$ is the utility value of one more unit of borrowing capacity. The value of renting, $V_i^{\rho}(x,Z)$, is defined in the same way without the mortgage. The household’s value is $V_i=\max\{V_i^{o},V_i^{\rho}\}$, and $S_i\equiv V_i^{o}-V_i^{\rho}$ is its ownership surplus.

B.1.3Aggregates and the external cost

The vector $Z$ collects aggregate outcomes associated with financial vulnerability, such as total household credit $\int b_i\,\mathrm{d}i$, the distribution of leverage and debt service, the composition of new lending, and house-price growth. These outcomes affect households, for instance through the probability and depth of a financial crisis, but each household takes them as given. The external cost $\Psi(Z)$ is the money-metric value of these effects summed over households. Only its gradient enters the derivation, so we define it directly: $$\begin{equation} \label{eq:psi} \nabla\Psi(Z)\equiv-\int\frac{1}{\mu_i}\,\frac{\partial V_i}{\partial Z}\,\mathrm{d}i . \end{equation}$$(B.3) As an example, suppose a crisis occurs with probability $\pi(Z)$ and that $$U_i(a;Z)=\big(1-\pi(Z)\big)\,U_i^{NC}(a)+\pi(Z)\,U_i^{C}(a),$$ where $U_i^{NC}$ and $U_i^{C}$ are utility in the no-crisis and crisis states. Then $\partial V_i/\partial Z=-\nabla\pi\,(U_i^{NC}-U_i^{C})$ at the household’s optimum, and $$\nabla\Psi(Z)=\nabla\pi(Z)\;\mathcal{C},\qquad \mathcal{C}\equiv\int\frac{U_i^{NC}-U_i^{C}}{\mu_i}\,\mathrm{d}i\ \ge 0 .$$ The external cost thus equals the change in crisis probability times the money-metric loss in a crisis, summed over households. Households take their own losses in a crisis into account when they choose how much to borrow, given $\pi$. What they do not take into account is the effect of their borrowing on $\pi$. The evidence that credit-financed booms are followed by deeper recessions (Jordà et al. 2013; Cerra and Saxena 2008) bears on $\mathcal{C}$, and the evidence that credit growth predicts crises (Schularick and Taylor 2012; Greenwood et al. 2022) bears on $\nabla\pi$. Since $\nabla\pi$ is evaluated at $Z$, the marginal effect of credit on crisis probability can depend on the other elements of $Z$. The state dependence documented by Greenwood et al. (2022) and Krishnamurthy and Muir (2025) is consistent with positive cross-partials $\partial^2\pi/\partial Z_k\,\partial Z_l$ between credit growth and house-price growth or credit spreads, which is why the crisis map in Section 6.1 sorts tightening episodes on both dimensions.

The market outcomes $x$ and the aggregates $Z$ are determined in equilibrium by housing and rental market clearing, lender behavior, and expectations. We do not solve for this equilibrium. We only require that the equilibrium path $x(\tau)$, $Z(\tau)$ is differentiable, and we treat $\dot{x}\equiv\mathrm{d}x/\mathrm{d}\tau$ and $\dot{Z}\equiv\mathrm{d}Z/\mathrm{d}\tau$ as the responses estimated in the empirical literature. These are present values of the response paths, and $\lambda_i$ is the present value of the multipliers on the borrowing constraint at each date and in each state.

B.1.4Assumptions

Assumption B.1 is the source of the policy rationale. Because no household or lender takes into account the effect of its own borrowing on $Z$, private optimization does not lead to the social optimum, and the external cost remains in the welfare expression. We abstract from compliance and screening costs and from lender profits other than those captured by spreads.

B.2Derivation

Step 1: Envelope conditions. Fix tenure at the household’s optimum. The Lagrangian of problem B.2 is $$\mathcal{L}_i=U_i(a;Z)-\mu_i\,B_i(a;x)+\eta_i\big[\tilde{\kappa}_i(\tau,x)-b(a)\big].$$ By Assumption B.2, the derivatives of the value function with respect to $\tau$, $x$, and $Z$ equal the partial derivatives of $\mathcal{L}_i$ at the optimum. Changes in the household’s own choices have no first-order effect on its value: $$\begin{align} \frac{\partial V_i}{\partial\tau}&=\eta_i\,\frac{\partial\tilde{\kappa}_i}{\partial\tau}=\eta_i\,\epsilon_i\,\frac{\partial\kappa_i}{\partial\tau},\label{eq:envtau}\\ \frac{\partial V_i}{\partial x_j}&=-\mu_i\,\frac{\partial B_i}{\partial x_j}+\eta_i\,\frac{\partial\tilde{\kappa}_i}{\partial x_j} =\mu_i\,m_{ij}+\eta_i\,\frac{\partial\tilde{\kappa}_i}{\partial x_j},\label{eq:envx} \end{align}$$(B.4)
(B.5) where the second equality in B.4 uses equation 1, and $$\begin{equation} \label{eq:netposition} m_{ij}\equiv-\frac{\partial B_i}{\partial x_j} \end{equation}$$(B.6) is household $i$’s net position in market $j$. For the house price, $m_{ij}$ is net housing sold, negative for buyers and positive for sellers. For the rent, it is net rental supply, negative for renters and positive for landlords. For the mortgage spread, it is negative and proportional to $b_i$ for borrowers and positive for the owners of lenders. The derivative with respect to $Z$ is $\partial V_i/\partial Z=\partial U_i/\partial Z$, which enters equation B.3.

Step 2: Money metric. We measure welfare in units of the numéraire by weighting each household by $\omega_i\equiv1/\mu_i$, evaluated before the policy change, so that one unit of the numéraire has the same value whoever receives it. Welfare is $$\begin{equation} \label{eq:W} W(\tau)=\int\omega_i\,V_i\big(\tau,x(\tau),Z(\tau)\big)\,\mathrm{d}i , \end{equation}$$(B.7) and the money-metric shadow value of borrowing capacity is $$\begin{equation} \label{eq:weights} \lambda_i\equiv\frac{\eta_i}{\mu_i}\ge0 . \end{equation}$$(B.8)

Step 3: Aggregate. Differentiating equation B.7 along the equilibrium path and substituting equations B.3–B.8 gives the organizing equation.

Multiplying equation B.9 by $\mathrm{d}\tau$ and using $\mathrm{d}x_j=\dot{x}_j\,\mathrm{d}\tau$ and $\mathrm{d}Z=\dot{Z}\,\mathrm{d}\tau$ gives equation 2 in the main text. Grouping the same terms by whether a household is constrained ($\lambda_i>0$) or not, instead of by term, gives a decomposition into an effect on constrained households, an effect on unconstrained households, and crisis prevention. The two decompositions add up the same terms in a different order. Ours follows the objects that the different empirical literatures estimate.

B.3Properties used in the main text

Household adjustments. No behavioral response appears in equation B.9. When a household responds to a tightening by drawing down liquidity, buying a cheaper home, delaying a purchase, or relying on family support, the change in its welfare is fully captured by the shadow value $\lambda_i$ times the change in capacity, because the household was already optimizing. The adjustments documented in Section 4 should therefore not be added to term (I) as separate costs. They remain informative, however, as they show that the constraint binds, they help discipline estimates of $\lambda_i$, they change the composition of debt that enters $Z$, and they generate the market responses $\dot{x}$.

Transfers and constraint revaluation in term (II). Each market response enters term (II) through two channels. The first, $\int m_{ij}\,\mathrm{d}i$, is a transfer between households. Let $n_{ijt}$ be household $i$’s net sales in market $j$ at date $t$ and $Q_{it}$ its discount factor for date $t$, with $Q_{i0}=1$, so that $m_{ij}=\sum_tQ_{it}\,n_{ijt}$. Market clearing requires $\int n_{ijt}\,\mathrm{d}i=0$ at each date, so that $$\begin{equation} \label{eq:transfer} \int m_{ij}\,\mathrm{d}i=\sum_{t\ge1}\mathop{\mathrm{Cov}}\big(Q_{it},\,n_{ijt}\big). \end{equation}$$(B.10) For a price paid once at the date of the policy change, such as the purchase price of a home, the transfer channel sums to zero and affects only who gains and who loses. For payments spread over time, such as rents and mortgage spreads, the sum is zero only if households on the two sides of the market value future payments equally. Constrained households value future payments less: in a two-period model in which household $i$ can borrow at rate $r$, $Q_{i1}=(1-\lambda_i)/(1+r)$. The covariance in equation B.10 is the distributive pecuniary externality of Dávila and Korinek (2018), and it is small when the exposed households on the two sides of a market have similar shadow values. The main text treats the first channel as a transfer, which is exact for house prices and an approximation for rents and spreads.

The second channel, $\int\lambda_i\,\partial\tilde{\kappa}_i/\partial x_j\,\mathrm{d}i$, does not net out. Under an LTV limit, $\partial\tilde{\kappa}_i/\partial P=\tau h_i$, so a lower house price reduces capacity for every constrained household. The two channels can work in opposite directions for the same household. For a first-time buyer with $m_{iP}=-h_i$, the net exposure to the house price is $-h_i(1-\lambda_i\tau)<0$, so a lower price is still a gain, although a smaller one than the transfer channel alone suggests. For an owner who sells a home of size $h_i^{0}$ to buy a home of size $h_i$, the net exposure is $-(h_i-h_i^{0})+\lambda_i\tau h_i$, which is positive when the household trades up only modestly and is constrained, so a lower price is a loss (Stein 1995). Under LTI and DTI limits, $\partial\tilde{\kappa}_i/\partial P=0$ and only the transfer channel remains.

Households displaced from ownership. Under free tenure choice and a smooth distribution of ownership surplus, a small tightening moves only households that are close to indifferent between owning and renting ($S_i\approx0$). Their welfare loss is second order, because the envelope theorem applies to the chosen tenure, while their effect on credit is first order, because each of them removes an entire loan. If instead households with $S_i>0$ are kept out of ownership by a down-payment requirement, a rejected application, or an exhausted exemption quota, their loss is first order and is not captured by $\lambda_i$. Let $n_R(\tau)$ be the mass of such households and $\bar s_R$ the average money-metric surplus $S_i/\mu_i$ of those who are newly kept out by the tightening. The tightening then adds $$\begin{equation} \label{eq:rationing} -\,\bar s_R\,\frac{\mathrm{d}n_R}{\mathrm{d}\tau}\,\mathrm{d}\tau<0 \end{equation}$$(B.11) to the direct effect. Missing mass in bunching designs and entry responses, such as those in Eerola et al. (2023), measure $\mathrm{d}n_R$ only if displaced households are rationed rather than close to indifferent. They are also silent on $\bar s_R$.

B.3.1Internalities

Suppose households overvalue borrowing, for instance because they expect house prices to grow faster than they will or neglect repayment risk. Let households choose as if they maximized $\hat U_i=U_i+\mu_i\delta_ib_i$, where $\delta_i$ is the money-metric overvaluation of a marginal unit of debt, while welfare is evaluated with $U_i$. The envelope theorem applies to the problem households actually solve, so terms (I)–(III) are unchanged, with $\lambda_i$ and $m_{ij}$ evaluated at the households’ choices. Since $U_i=\hat U_i-\mu_i\delta_ib_i$, welfare picks up one additional term: $$\begin{equation} \label{eq:termIV} \frac{\mathrm{d}W}{\mathrm{d}\tau}=\big[\text{right-hand side of (B.9)}\big] \;\underbrace{-\;\int \delta_i\,\frac{\mathrm{d}b_i}{\mathrm{d}\tau}\,\mathrm{d}i}_{\text{(IV) Internality correction}} , \end{equation}$$(B.12) where $\mathrm{d}b_i/\mathrm{d}\tau$ is the total change in household $i$’s borrowing, including the response to market outcomes. A tightening reduces borrowing, so with $\delta_i>0$ term (IV) times $\mathrm{d}\tau$ equals $-\int\delta_i\,\mathrm{d}b_i\,\mathrm{d}i>0$, a benefit. The behavioral response that drops out of the welfare expression under Assumption B.3 now matters, as in the theory of corrective taxation. If overvaluation is concentrated among constrained borrowers, terms (I) and (IV) partly offset for the same households, and the net effect on these households can be positive if $\delta_i$ is large. Data linking house-price expectations to leverage choices (Bailey et al. 2019) offer one way to measure $\delta_i$.

B.4Limits

The organizing equation has four limits. First, it is a first-order expression. Observed changes in LTV and income limits are often several percentage points, and the second-order terms are then not negligible. Larger reforms require a structural model. Second, the envelope argument does not hold when households are rationed out of ownership or borrow more than is good for them. We state both cases as separate terms, equations B.11 and B.12, but for down-payment constraints rationing may be common. Third, the equation adds money-metric gains and losses without distributional weights, and the transfer channel nets out exactly only for one-time prices. The main text therefore reports incidence by group, so that other weights can be applied. Fourth, the present-value convention compresses timing. Announcement effects, transition paths, and changes in beliefs are part of $\dot{x}$ and $\dot{Z}$ but are not modeled separately.