spb/wp18_uqo

1% ============================================================================2\section{The model}3\label{sec:model}4% ============================================================================56The analysis uses two economies that share notation and mechanisms. The \emph{benchmark} (Section~\ref{subsec:benchmark}) compresses the life cycle into three stages and a generation into two periods; it delivers closed forms and the propositions of Section~\ref{sec:equilibrium}. The \emph{quantitative} economy (Section~\ref{subsec:quant}) has twelve five-year periods, housing on an intensive margin, persistent income risk, long-term nominal annuity mortgages, two bequest dates, durable housing stocks and an explicit lender; it is solved numerically in Sections~\ref{sec:computation}--\ref{sec:sensitivity}. Every reduction made in the benchmark is a candidate explanation for a discrepancy between the propositions and the simulations, and Section~\ref{sec:equilibrium} names those reductions explicitly.78\subsection{Units and timing conventions}9\label{subsec:units}1011Time is discrete. All quantities labelled ``real'' are in units of the period consumption good. The consumer price index is $P_t$, gross inflation $1+\pi_t = P_t/P_{t-1}$, expected inflation over $(t-1,t)$ as of $t-1$ is $\pi^e_t$, and the surprise is defined multiplicatively, $1+\pi^u_t \equiv (1+\pi_t)/(1+\pi^e_t)$; in a stationary equilibrium $\pi_t = \pi^e_t = \bar\pi$. The real price of one unit of owner-occupied housing is $p_t$, so the house-price index $P_t p_t$ is distinct from the CPI. The real rent per unit of rental services is $\rho_t$. A mortgage originated at $t$ has nominal principal $B^{nom}_t$; its real value at a later date $t+k$ is $b^m_{t+k} \equiv B^{nom}_{t+k}/P_{t+k}$, so the nominal balance is what the contract fixes and the real balance is what matters for consumption. The real gross rate is $R \equiv 1+r$, exogenous (a small open economy), and the nominal rate on contracts written at $t$ satisfies the Fisher relation $1+i_t = R(1+\pi^e_{t+1})$ because the lender is competitive and risk neutral. A fixed-rate mortgage (FRM) keeps $i_t$ for the life of the loan; an adjustable-rate mortgage (ARM) resets every period; a term-$k$ contract resets every $k$ periods, which in the five-year calibration of the quantitative model represents the Canadian renewable term. Households belong to dynasties, infinite sequences of parent--child pairs; the labour-income process is identical across dynasties, so dynasties differ only through the transfers they receive. This ``equal-income'' design is what makes the owner--renter gaps of the paper attributable to transfers, leverage and constraints rather than to earnings.1213\subsection{The benchmark economy}14\label{subsec:benchmark}1516\paragraph{Demographics and timing.}17A household lives three stages, $j = 1,2,3$: a \emph{buyer} (young adult, who chooses tenure, buys or rents, and borrows), a \emph{payer} (prime age, who pays rent or mortgage interest and amortisation; under an ARM or term contract the rate is reset at the start of this stage) and a \emph{seller--donor} (who sells the house, repays the remaining balance, consumes and gives an inter-vivos transfer to the child). The child enters the model at stage~1 exactly when the parent enters stage~3, so a generation lasts two periods and the transfer arrives at the moment of the tenure decision, the ``gift at first purchase'' channel documented by \citet{engelhardt1998intergenerational} and \citet{blickle2019borrowing}. A unit mass of stage-1 households is born each period. Stage-3 households do not demand housing in the two markets (their retirement housing is outside the model and netted out of $y_3$); this is the one demographic simplification of the benchmark, and the quantitative model removes it. Within a period: the price level and the stage-3 income shock are realised; real prices $(p_t,\rho_t)$ are determined; stage-3 households sell, repay, consume and transfer; stage-1 households receive the transfer and choose tenure, housing and the mortgage while stage-2 households pay and consume; the owner-occupied and rental markets clear.1819\paragraph{Preferences and the housing need.}20A household born at $t$ with tenure $o \in \{O,R\}$ has lifetime utility21\begin{equation}22U^o = \ln c_1 + \chi\,\mathbf 1\{o = O\} + \beta\,\mathbb E[\ln c_2] + \beta^2\,\mathbb E[\ln c_3 + \nu\ln b'],23\qquad \beta\in(0,1),\ \nu>0,\ \chi\ge 0,24\label{eq:utilityA}25\end{equation}26where $b'$ is the transfer to the child. Each household needs exactly one unit of housing services at stages~1 and~2 (unit demand): an owner holds one unit of the owner-occupied stock, a renter rents one unit of the rental stock. Housing therefore enters utility only through the constant ownership premium $\chi$ \citep{henderson1983model}; $\chi = 0$ is allowed. The warm-glow term is the \citet{denardi2004wealth} specification with the luxury shift set to zero so that the transfer rule is linear. Incomes $y_1, y_2 > 0$ are deterministic and common; $y_3$ is drawn from a distribution $G$ with a continuous density $g > 0$ on $[\underline y_3, \bar y_3]$, independently across households and generations.2728\paragraph{Mortgage, deposits and the lender.}29A stage-1 owner buys one unit at $p_t$, pays the buying cost $\kappa_b p_t$ and maintenance $m p_t$, makes a down payment of at least $\theta p_t$ and borrows the nominal principal $B^{nom}_t \le P_t(1-\theta)p_t$. The contract pays interest $i_1 B^{nom}_t$ plus amortisation $a B^{nom}_t$ at stage~2, $a\in(0,1]$, and $(1+i_2)(1-a)B^{nom}_t$ at stage~3, with $i_1 = R(1+\pi^e_{t+1})-1$ fixed at origination for every contract and $i_2 = i_1$ (FRM) or $i_2 = R(1+\pi^e_{t+2|t+1})-1$ (ARM or term contract renewed at $t+1$). Real payments are $D_2 = (i_1+a)(1-\theta)p_t/(1+\pi_{t+1})$ and $D_3 = (1+i_2)(1-a)(1-\theta)p_t/[(1+\pi_{t+1})(1+\pi_{t+2})]$. In the stationary state write $D_2 = \delta_2 p$, $D_3 = \delta_3 p$ with30\begin{equation}31\delta_2 \equiv (1-\theta)\Big[R - \frac{1-a}{1+\bar\pi}\Big], \qquad32\delta_3 \equiv \frac{(1-a)(1-\theta)R}{1+\bar\pi}, \qquad33e_3 \equiv (1-\kappa_s) - \delta_3, \qquad d \equiv \theta + \kappa_b + m,34\label{eq:deltas}35\end{equation}36so that $d$ is the stage-1 cash outlay per unit of price and $e_3$ the net equity share at sale. The identity37\begin{equation}38\delta_2 + \frac{\delta_3}{R} = (1-\theta)R39\label{eq:fisher}40\end{equation}41is the present-value (Fisher) neutrality of the nominal contract: anticipated inflation front-loads real payments ($\partial\delta_2/\partial\bar\pi > 0$, $\partial\delta_3/\partial\bar\pi < 0$), the ``tilt'', without changing their present value. The house is net wealth at the end of life, $e_3 > 0$, if and only if $(1-\kappa_s)(1+\bar\pi) > (1-a)(1-\theta)R$, that is, if amortisation and the down payment are large enough; a two-period version in which the whole mortgage is a balloon fails this condition for generation-length periods, which is why the benchmark has three stages. The liquid asset is a one-period nominal deposit $n\ge 0$ paying $i_t$; its real gross return is $R/(1+\pi^u_{t+1})$, so a surprise erodes nominal savings exactly as it erodes nominal debt. The lender is a competitive risk-neutral intermediary that funds mortgages and takes deposits at the same nominal rate; its realised real profit in a surprise period accrues to rentiers outside the dynasties, and is the mirror image of the households' revaluation gains.4243\paragraph{Rental and owner-occupied supply.}44Landlords are competitive and risk neutral and the rental stock is segmented from the owner-occupied stock. Per cohort, owner-occupied supply is $S(p) = \bar H p^{\varepsilon}$, $\varepsilon\in[0,\infty)$, with $\varepsilon = 0$ a fixed stock, and rental supply is $S_r(\rho) = \bar H_r\rho^{\varepsilon_r}$, $\varepsilon_r\in(0,\infty]$. In the benchmark $S$ is a per-period supply curve (a reversible land-conversion technology); the quantitative model uses the durable stock-flow version whose impact elasticity is $\varepsilon\delta_H$. Nominal wages and rents are indexed to a surprise with elasticities $\eta_w,\eta_\rho\in[0,1]$ (full indexation in the baseline). The no-segmentation alternative, one stock converted costlessly by landlords, is the \citet{kaplan2020housing} and \citet{greenwald2025do} benchmark and is analysed in Proposition~\ref{prop:noseg}.4546\paragraph{Budgets.}47For an owner born at $t$, with $n_1$ and $n_2$ the deposits carried out of stages~1 and~2,48\begin{align}49c_1 + d\,p_t + n_1 &= b + y_1 + \big[(1-\theta)p_t - B_t\big], \label{eq:budgetO1}\\50c_2 + D_2 + m\,p_t + n_2 &= y_2 + n_1 \tfrac{R}{1+\pi^u_{t+1}}, \label{eq:budgetO2}\\51c_3 + b' &= W^O_3 \equiv y_3 + (1-\kappa_s)\,p_{t+2} - D_3 + n_2\tfrac{R}{1+\pi^u_{t+2}}, \label{eq:budgetO3}52\end{align}53where the bracket in \eqref{eq:budgetO1} is non-negative and zero when the loan-to-value constraint binds. For a renter, $c_1 + \rho_t + n_1 = b + y_1$, $c_2 + \rho_{t+1} + n_2 = y_2 + n_1R/(1+\pi^u_{t+1})$ and $c_3 + b' = W^R_3 \equiv y_3 + n_2R/(1+\pi^u_{t+2})$. Deposits cannot be negative, there is no refinancing, and no default: every stage-2 payment can be met out of income and every owner can repay the stage-3 balance out of income plus the sale proceeds (Assumption~\ref{ass:feas} below). Consumption of a hand-to-mouth household, which sets $n_1 = n_2 = 0$ and borrows the maximum, is written $c^O_1(b) = b + y_1 - dp$, $c^R_1(b) = b + y_1 - \rho$, $c^O_2 = y_2 - \delta_2 p - mp$, $c^R_2 = y_2 - \rho$, $W^O_3(y_3) = y_3 + e_3p$ and $W^R_3(y_3) = y_3$.5455\subsection{The quantitative economy}56\label{subsec:quant}5758The quantitative economy keeps the dynasty structure, the nominal contracts and the segmented markets of the benchmark and relaxes everything else. Figure~\ref{fig:timing} summarises its timing and Table~\ref{tab:notation} in Appendix~\ref{app:tables} collects the notation.5960\begin{figure}[htbp]61\centering62\includegraphics[width=\textwidth]{fig01_model_timing.pdf}63\caption{Timing of the overlapping-generations model. One period is five years; a household lives twelve periods (ages 25--84) and retires at 65; its child enters when the parent is 55--59, receives the inter-vivos gift $b^g$ at entry and the estate $b^e$ at the parent's death, when the child is itself 55--59. A mortgage originated at age 2 is amortised over $N = 5$ periods.}64\label{fig:timing}65\end{figure}6667\paragraph{Demographics and transfers.}68Ages are $j = 1,\dots,J = 12$ (periods of five years, 25--84), retirement at $j_R = 9$. Each household has one child who enters the model at the parent's age $j_g = 7$; the child receives an inter-vivos gift $b^g$ at its age~1, chosen by the parent at $j_g$, and the parent's estate $b^e$ at the parent's death, when the child is $j_e = 7$. Both are chosen with warm-glow utility; ``bequests off'' sets the warm-glow weight to zero for both. Population is constant with a unit mass per age, and there is no survival risk in the baseline.6970\paragraph{Preferences and income.}71Period utility over non-housing consumption and housing services is72\begin{equation}73u(c,s) = \frac{\big(c^{1-\alpha}s^{\alpha}\big)^{1-\gamma}}{1-\gamma}, \qquad74\phi(x) = \nu\,\frac{(x+\underline x)^{1-\gamma}}{1-\gamma},75\label{eq:uB}76\end{equation}77with $s = (1+\chi)h$ for an owner of size $h$ and $s$ the rented size for a renter, and $\phi$ the warm glow over a transfer $x\in\{b^g,b^e\}$ with luxury shift $\underline x > 0$ \citep{denardi2004wealth}; the benchmark's linear rule is the limit $\underline x\to 0$, $\gamma\to 1$. Real labour income at $j < j_R$ is $y_j = \bar y_j z$, with $\bar y_j$ a hump-shaped age profile and $z$ a five-state Rouwenhorst chain for the five-year aggregate of an annual AR(1); the pension is a flat replacement of the last working-age income. The child's initial state $z_1$ is drawn from the stationary distribution independently of the parent in the baseline (equal-income design); an intergenerational earnings correlation $\rho_g$ is a sensitivity switch. A small unanticipated transitory component of income, realised before decisions, spreads the atoms of the cash-on-hand distribution that would otherwise make aggregate demand discontinuous in prices (Section~\ref{sec:computation}).7879\paragraph{Housing, tenure and transactions.}80Owner-occupied houses come in four sizes $h\in\mathcal H$, the smallest being the indivisibility that creates an entry threshold; rented services $s\ge\underline s$ are continuous with $\underline s < \underline h$. Maintenance costs $mph$ per period; buying and selling cost $\kappa_b ph$ and $\kappa_s ph$; moving between owned sizes is a sale plus a purchase; renters move costlessly. Each period an owner chooses among \emph{stay} (possibly refinancing at cost $\kappa_f ph$), \emph{move} and \emph{sell-and-rent}; a renter chooses between \emph{rent} and \emph{buy}.8182\paragraph{Nominal annuity mortgages and origination constraints.}83A mortgage with real principal $B$, nominal contract rate $i_0$ and $N = 5$ remaining payments (25 years) has the constant nominal payment $\text{pay}^{nom} = B^{nom} i_0/[1-(1+i_0)^{-N}]$, and the real balance evolves as84\begin{equation}85b^{m\prime} = \frac{b^m(1+i_c) - \text{pay}_t}{1+\pi_{t+1}}, \qquad \text{pay}_t = \text{pay}^{nom}_t/P_t,86\label{eq:debtlaw}87\end{equation}88where $i_c = i_0$ for an FRM, $i_c$ is the current Fisher rate for an ARM (with the nominal payment recomputed on the remaining balance and term), and $i_c$ resets at every period boundary for the five-year term contract. The real payment of an FRM falls with realised inflation; that of an ARM is re-priced. At origination and refinancing the loan must satisfy the loan-to-value constraint $B\le(1-\theta)p_th$ and a nominal payment-to-income constraint at a qualifying rate $i_q = i_t+\Delta_q$ (the stress test): $\text{pay}^{nom}(i_q)/(P_ty) \le \psi$. Because the qualifying payment is nominal and income is current, expected inflation raises $i_t$ and tightens this constraint, which is how the benchmark's tilt reappears in the quantitative model. There is no unsecured borrowing, $a'\ge 0$, and no default; the solver enforces a consumption floor and reports the (zero) mass of forced sales.8990\paragraph{Rental sector, construction and the lender.}91Rental units form a separate stock owned by competitive risk-neutral landlords who buy at $p^r_t$, pay maintenance and an operating wedge $\phi p^r_t$, and charge the zero-profit rent92\begin{equation}93\rho_t = p^r_t(1+m+\phi) - \mathbb E_t p^r_{t+1}/R .94\label{eq:rent}95\end{equation}96A segmentation parameter $\sigma\in[0,1]$ gives the share of rental units that are structurally distinct; the remainder converts one-for-one from the owner-occupied stock at $p^r_t = p_t$, so that $\sigma = 1$ is the benchmark and $\sigma = 0$ the single-stock economy of Proposition~\ref{prop:noseg}. Rent rigidity $\eta_\rho$ applies to sitting tenants in a surprise period, the difference being a transfer from landlords. The two stocks are durable,97\begin{equation}98H_{t+1} = (1-\delta_H)H_t + \iota\,p_t^{\varepsilon}, \qquad H^r_{t+1} = (1-\delta_H)H^r_t + \iota_r\,(p^r_t)^{\varepsilon_r},99\label{eq:supply}100\end{equation}101so the stationary supply curves are $S(p) = \iota p^{\varepsilon}/\delta_H$ and $S_r(p^r) = \iota_r (p^r)^{\varepsilon_r}/\delta_H$ and the one-period elasticity is $\varepsilon\delta_H$. Construction profits go to a class outside the dynasties. The lender holds mortgage balances against household deposits and rentier funding at the world real rate; its real net worth obeys $A^L_{t+1} = (1+i_t)(\text{Mortgages}_t - \text{Deposits}_t)/(1+\pi_{t+1}) + \text{rentier flow}$, so a surprise revalues $-(\text{Mortgages}_t-\text{Deposits}_t)(1+i_t)\pi^u_{t+1}/[(1+\pi^e_{t+1})(1+\pi^u_{t+1})]$ for the lender, the mirror image of the households' aggregate real gain. A transfer tax $\tau_b$ on gifts and estates, rebated lump-sum to entering households, is the policy experiment of Section~\ref{sec:supply}.102103\paragraph{The household problem.}104The individual state at age $j$ is $x = (j,z,o,h,b^m,n_m,a)$ with $o$ tenure, $h$ house size (zero for renters), $b^m$ the real mortgage balance, $n_m$ the remaining payments and $a\ge 0$ the real deposit; along a transition the origination date, hence the FRM rate, is identified by $n_m$ and calendar time. Writing $R_a\equiv(1+i_{t-1})/(1+\pi_t)$ for the realised real deposit return and $\ell$ for the discrete branch,105\begin{equation}106V_j(x) = \max_{\ell\in\mathcal L(x)} V^{\ell}_j(x), \qquad107V^{\ell}_j(x) = \max_{c,a',(h',B',s)}\Big\{u(c,s) + \mathbf 1_{j=j_g}\phi(b^g) + \beta\,\mathbb E\big[V_{j+1}(x')\mid z\big]\Big\},108\label{eq:bellman}109\end{equation}110subject to the branch budget. A renter who rents has $c + \rho s + a' = y_j + aR_a - \mathbf 1_{j=j_g}b^g$; a renter who buys has $c + (1+\kappa_b+m)ph' - B' + a' = y_j + aR_a - \mathbf 1_{j=j_g}b^g$ with the origination constraints; an owner who stays has $c + \text{pay}(b^m,n_m,i_c) + mph + a' = y_j + aR_a - \mathbf 1_{j=j_g}b^g$; an owner who moves sells at $(1-\kappa_s)ph$, repays $b^m(1+i_c)$ and proceeds as a buyer; an owner who sells and rents has $c + \rho s + a' = y_j + aR_a + (1-\kappa_s)ph - b^m(1+i_c) - \mathbf 1_{j=j_g}b^g$. At $J$ the house is sold, the balance repaid and $V_{J+1}\equiv\phi(b^e)$ with $b^e$ the terminal wealth. The estate received at $j_e$ is anticipated as a lottery whose points are equilibrium objects (Section~\ref{sec:computation}). The solver works in real variables; inflation enters only through $R_a$ and the real FRM payment, through the nominal rate in the qualifying payment and in ARM repricing, and through the indexation parameters. Net worth is $a + \mathbf 1_O[(1-\kappa_s)ph - b^m]$.111