spb/wp18_uqo

1% ============================================================================2\section{Numerical algorithm, calibration and verification}3\label{sec:computation}4% ============================================================================56\subsection{Solution method}7\label{subsec:algorithm}89The household problem \eqref{eq:bellman} is solved by backward induction over the twelve ages. The discrete branches other than \emph{stay} depend on the state only through cash on hand, so the values of renting, buying, refinancing and selling are computed on a cash-on-hand grid of $120$ points and the value of staying on the full state grid (five productivity states, four house sizes, $20$ mortgage balances, five remaining-payment counts, $50$ liquid-asset points). The continuous choice of next-period liquid assets is an exhaustive search on the grid refined by a golden-section step; loans are chosen among eleven fractions of the binding constraint, gifts among twelve fractions of cash, and the estate is anticipated as a five-point lottery whose points are the conditional means of the simulated distribution of estates by quintile. Endogenous-grid methods were not used because the combination of discrete tenure and size choices with the two origination constraints and a mortgage grid makes the discrete-choice upper envelope fragile, while the cost of the exhaustive search under just-in-time compilation is low (one evaluation of the household block and the distribution takes about one second).1011The distribution is a Monte-Carlo panel of \NDynasties{} dynasty slots, each holding a parent and a child whose life-cycle phases are equidistributed, simulated with common random numbers for \HouseholdPeriods{} household-periods after a burn-in of ten generations. A panel rather than a histogram is used because the dynastic link, a gift at the child's age~1 and an estate at age~7 both correlated with the parent's state, requires the joint parent--child distribution, which a histogram over individual states does not carry. Two approximations are deliberate: the estate is anticipated as the unconditional lottery, although the realised estate in the panel is exactly linked to the parent; and along a transition the lottery is held at its stationary value. A small unanticipated transitory income component, realised before decisions, is added because entrants without a gift and liquidity-constrained renters otherwise share the same cash at a given productivity state and switch tenure simultaneously, which produced a discontinuity in aggregate rental demand of about 1.5\% over a 0.05\% price interval in a first run; with the component, excess demand is continuous and the root is bracketed.1213Market clearing uses a nested regula-falsi (Illinois) search, the rental price inside the owner-occupied price, followed by a damped fixed point on the estate lottery and on the transfer-tax rebate. A Broyden update on the two prices was tried first and abandoned: the excess-demand functions are smooth but strongly non-linear (tenure shares move by 15--20 percentage points for a 5\% change in the relative price) and the Monte-Carlo noise of about $10^{-3}$ in differences of excess demand for a 0.5\% price step corrupted the secant updates. The supply scales $\bar H$ and $\bar H_r$ are set so that the baseline clears at $p^* = p^{r*} = 1$, the normalisation of \citet{kaplan2020housing}; every counterfactual keeps these scales. Transitions are solved by shooting in the sequence space on the log price paths over $T = 14$ periods (70 years) with the durable stocks \eqref{eq:supply}, a finite-difference Jacobian computed in parallel, Broyden updates with periodic refresh, and, for large shocks, a damped t\^atonnement pre-phase. The tolerances are $5\times10^{-4}$ on relative excess demand in the stationary equilibrium, which pins the equilibrium price to about $\pm0.05\%$ given the Monte-Carlo noise, and $1.5\times10^{-3}$ per period along a transition, where each cross-section has one sixth of the stationary sample. Every run records its residuals; runs that stall above tolerance are labelled with their residual wherever they appear.1415Welfare is measured by the consumption-equivalent variation $\lambda^{CE}$ such that scaling non-housing consumption by $1+\lambda^{CE}$ in every period of the baseline makes an age-1 household with state $(z_1,a_1)$ as well off as in the alternative,16\begin{equation}17V^{alt}_1(z_1,a_1) = (1+\lambda^{CE})^{(1-\alpha)(1-\gamma)}\,U^{base}_1(z_1,a_1) + \Phi^{base}_1(z_1,a_1),18\label{eq:cev}19\end{equation}20where $U$ is discounted flow utility and $\Phi$ the discounted warm glow, the latter obtained from cohort simulations under baseline policies and the former as $U = V-\Phi$, an identity at the baseline; the simulated $U$ is kept as a check and differs from $V-\Phi$ by at most \WelfareCheckMaxPct\% because simulated policies use nearest-grid loan and size choices, a discrepancy that would inflate the variation if used directly. For households alive at a shock the same measure is applied to their remaining lifetime. Comparisons across scenarios that change preference parameters are reported but are not welfare statements, since they compare value functions under different utilities.2122\subsection{Calibration}23\label{subsec:calibration}2425The calibration is illustrative. Table~\ref{tab:parameters} in Appendix~\ref{app:computation} lists every parameter, its unit and its status; no moment is fitted, and the verified Canadian comparison points of Table~\ref{tab:moments} are reported next to the model's moments without being targeted. The period is five years. The annual discount factor is $0.96$, the curvature $\gamma = 2$, the housing share $\alpha = 0.20$; the warm-glow weight $\nu = 2$ and luxury shift $\underline x = 1$ were chosen so that mean gifts are about two years of income. Productivity follows an annual AR(1) with persistence $0.95$ and innovation standard deviation $0.15$, converted to the five-year period, with a hump-shaped age profile and a 50\% replacement rate in retirement. The four house sizes are $\{1,1.5,2.25,3.4\}$ units and rented services are continuous above $0.3$ units. Maintenance is 1\% of value per year, buying costs 2.5\% of the price, selling 5\%, refinancing 1\%, depreciation of the stocks 1\% per year. Mortgages are amortised over $N = 5$ periods, the Canadian 25-year standard; the baseline minimum down payment is $\theta = 0.20$, the uninsured threshold; the payment-to-income cap is $\psi = 0.39$ at a qualifying rate two percentage points above the contract rate. The real rate and trend inflation are 2\% per year, the latter the Bank of Canada target, so the five-year nominal rate is $0.219$. The landlord wedge is 5\% of value per period, segmentation is full and both supply elasticities are set to one, a value with no verified Canadian source, which is why Section~\ref{sec:supply} sweeps it from $0.05$ to $2$.262728\subsection{Baseline equilibrium and verification}29\label{subsec:verification}3031The baseline stationary equilibrium clears at $p^* = \Pstar$ and $\tilde p_r^* = 1$ by normalisation, with an annual rent--price ratio of \RentPriceAnnual{}; relative excess demands are $\ExcessOwner$ in the owner-occupied market and $\ExcessRental$ in the rental market, the lottery converges in \LotteryRounds{} rounds and the whole equilibrium takes \BaselineSeconds{} seconds. Bellman residuals, measured as the relative difference between the interpolated value function and a re-optimisation on a 400-point asset grid at 400 random off-grid states, average $\BellmanOwnerMean$ for owners (99th percentile \BellmanOwnerPnn\%, maximum \BellmanOwnerMax\%) and $\BellmanRenterMean$ for renters; no household hits the forced-sale rule and \ClipSharePct\% of household-periods use a clipped interpolated policy. Table~\ref{tab:convergence} in Appendix~\ref{app:computation} collects these diagnostics with the grid, tolerance, sample-size and seed variants, and Figure~\ref{fig:diagnostics} plots them. The budget identities of every branch, the law of motion of nominal and real debt, the origination constraints, the terminal condition, the dynastic link, the lottery, the common random numbers, the monotonicity of the value functions and the single crossing of the tenure decision are unit-tested, and the zero-sum revaluation identity holds to machine precision in every transition.3233\input{tables/tab03_baseline_moments}3435Table~\ref{tab:moments} compares the model's moments with the verified comparison points. The ownership rate is \OwnershipAll{} overall, \OwnershipUnderThirtyFive{} under 35, \OwnThirtyFiveSixtyFour{} at 35--64 and \OwnSixtyFivePlus{} at 65 and over; \ShareMortgage{} of households hold a mortgage against 0.39 in the Survey of Financial Security; the mean net worth of owners is \NWRatioYoung{} times that of renters under 35, against a median ratio of about 10 in the survey. Two moments are flagged as not supported by the comparison points. The ownership profile peaks at \OwnPeak{} at 55--64 and falls to \OwnAgeTwelve{} at 80--84, because households with a warm glow over a liquid estate sell late in life to consume and to bequeath cash, whereas \citet{hou2010homeownership} report a plateau near three quarters after 65; this is a known consequence of the preference specification and is discussed in Section~\ref{sec:sensitivity}. And \ShareFirstBuyersGiftPct\% of first-time buyers receive a positive gift, well above the 28\% quoted by \citet{mirdamadi2023parents}, because every parent in the model has a warm-glow motive and the gift is the only transfer available at entry. These discrepancies do not affect the mechanisms the paper studies but they bound what the magnitudes can claim.3637\begin{figure}[htbp]38\centering39\includegraphics[width=\textwidth]{fig02_policy_functions.pdf}40\caption{Policy functions on the cash-on-hand grid in the baseline stationary equilibrium. Panel (a) plots the value of buying minus the value of renting at age~1 (25--29) by productivity state, clipped at $-0.6$; the single sign change along cash is the ownership-entry threshold. Panel (b) plots liquid saving at age~1 and panel (c) the house size bought at age~3 (0 = rent). Source: \texttt{results/data/baseline/policy\_age\{1,3\}.csv}.}41\label{fig:policy}42\end{figure}4344Figure~\ref{fig:policy} shows the policy functions at the entry ages. The buy-minus-rent value crosses zero exactly once along cash at every age from 1 to 3 and every productivity state, which is the single-crossing property that Proposition~\ref{prop:threshold} proves only on the hand-to-mouth region. At the crossing the buyer sets next-period liquid assets to zero for every state, so Assumption~\ref{ass:htm} holds for the marginal buyer; the marginal renter, however, saves, because in the quantitative economy renters accumulate towards a later purchase, which the benchmark rules out. This asymmetry is the origin of the scope restriction in Remark~\ref{rem:cs_scope}.45