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1% ============================================================================2\section{Baseline simulations: inheritance, leverage and inequality}3\label{sec:baseline}4% ============================================================================56This section reports the baseline stationary equilibrium and the experiments that isolate the inheritance mechanism: life-cycle profiles, the entry threshold, equal-income wealth paths with different initial transfers, the dependence of inequality and mobility on transfers, the down-payment sweep in general and partial equilibrium, and the accounting of negative price shocks under leverage. All magnitudes are statements about this parameterisation.78\subsection{Life-cycle profiles and the aggregate nominal position}9\label{subsec:profiles}1011\begin{figure}[htbp]12\centering13\includegraphics[width=\textwidth]{fig04_lifecycle_profiles.pdf}14\caption{Baseline life-cycle profiles in the stationary general equilibrium: tenure and mortgage status (a), mean net worth of owners and renters (b) and balance sheets (c), by five-year age group. Units: average five-year labour income. Source: \texttt{results/data/baseline/profiles\_by\_age.csv}.}15\label{fig:profiles}16\end{figure}1718Figure~\ref{fig:profiles} shows the life-cycle profiles. Ownership rises from \OwnAgeOne{} at 25--29 to \OwnPeak{} at 55--64 and falls to \OwnAgeTwelve{} at 80--84; a share \ShareOwnersMortgage{} of owners carry a mortgage, with a mean loan-to-value ratio at origination of \MeanLTVNew{} and a median house value of \HouseValueIncome{} years of income. The net-worth gap between owners and renters, who have the same income process by construction, is \GapAgeTwo{} years of income at 30--34, \GapAgeFour{} at 40--44, \GapAgeSeven{} at 55--59 and \GapAgeTen{} at 70--74. The gap is housing equity carried forward: forced amortisation and the leverage multiplier of Proposition~\ref{prop:leverage} accumulate it for owners while renters, whose median net worth under 35 is zero, save little before purchase. The balance-sheet panel contains the fact that governs the inflation results of Section~\ref{sec:inflation}: aggregate real mortgage debt is \TotalDebt{} per unit of age mass against real deposits of \TotalDeposits{}, so households are net nominal creditors by a ratio of \DebtDepositRatio{} to one. Mortgage debt is concentrated among owners aged 30--50; deposits are held by the young saving for the down payment and, mostly, by owners aged 55--75 who have repaid their loans and are accumulating a liquid estate. The old-age sale of houses is the moment where the model departs from the Canadian profile, as noted in Section~\ref{sec:computation}, and it also means that the estate is liquid: \ShareEstatePositivePct\% of children receive a positive estate at 55--59, of mean size \MeanEstate{}, and \ShareGiftPositivePct\% receive a positive gift at entry, of mean size \MeanGift{} (\MeanGiftAnnual{} years of average income, median \MedianGift).1920\subsection{The ownership-entry threshold}21\label{subsec:threshold_num}2223\begin{figure}[htbp]24\centering25\includegraphics[width=\textwidth]{fig03_ownership_thresholds.pdf}26\caption{Ownership-entry thresholds (units: average five-year income). Panel (a): the smallest initial liquid wealth at which a renter buys, by age and productivity state, in the baseline. Panel (b): the age-1 threshold at the median productivity state as a function of the minimum down payment, in general equilibrium (prices re-solved) and at fixed prices. The grid sensitivity of this statistic is about $\pm0.07$. Source: \texttt{results/data/baseline/thresholds.csv}, \texttt{results/data/theta\_sweep/*.json}.}27\label{fig:thresholds}28\end{figure}2930The quantitative counterpart of $\hat b$ is the smallest liquid wealth at which the age-$j$ renter buys, $\hat a_j(z)$. Figure~\ref{fig:thresholds}(a) plots it. At age~1 it is \ThresholdAgeOneLowZ, \ThresholdAgeOneZtwo, \ThresholdAgeOneMidZ, \ThresholdAgeOneZfour{} and \ThresholdAgeOneHighZ{} years of income for the five productivity states from the lowest to the highest; it falls with age, and at ages~2 and~3 it is negative for the two highest states (\ThresholdAgeTwoHighZ{} and \ThresholdAgeThreeHighZ{} at the top), meaning that those households buy out of income alone. The dependence on productivity is the feasibility component of Proposition~\ref{prop:threshold} operating through the payment-to-income cap: at the lowest state the cap, not the loan-to-value constraint, binds, and the threshold is the cash needed to buy the smallest house with the largest loan the cap allows. The threshold is the least robust statistic of the paper, moving by \DevCoarseThr{} between the coarse and baseline grids and \DevFineThr{} between the baseline and fine grids (Table~\ref{tab:convergence}), which is why statements about it are confined to differences larger than that.3132Panel (b) is the check of Proposition~\ref{prop:cs} and the first of the three discrepancies. At fixed prices the age-1 threshold at the median state is \ThetaFivePEThr{} with a 5\% minimum down payment, \ThresholdAgeOneMidZ{} with 20\% and \ThetaFortyPEThr{} with 40\%: it rises with $\theta$ between 20\% and 40\%, as the proposition predicts, but it is not lower at 5\% than at 20\%. The difference is within the grid sensitivity, and its economic source is the one named in Remark~\ref{rem:cs_scope}: with a small down payment the option of renting now and buying later is more valuable, which raises the renter's continuation value and the cash at which buying now is preferred. For the highest productivity state the threshold falls monotonically with $\theta$ at fixed prices (\ThetaFivePEThrHighZ, \ThresholdAgeOneHighZ, \ThetaFortyPEThrHighZ); for the lowest it is unchanged (\ThetaFivePEThrLowZ, \ThresholdAgeOneLowZ, \ThetaFortyPEThrLowZ) because the loan-to-value constraint does not bind there. Proposition~\ref{prop:cs} is therefore supported where its binding-constraint and once-and-for-all premises hold and contradicted, within noise, where they do not. In general equilibrium the price and rent responses add a second layer, and the threshold is \ThetaFiveThr, \ThetaTenThr, \ThresholdAgeOneMidZ, \ThetaThirtyThr{} and \ThetaFortyThr{} for the five values of $\theta$.3334\subsection{Equal-income wealth paths}35\label{subsec:equalincome}3637\begin{figure}[htbp]38\centering39\includegraphics[width=\textwidth]{fig05_equal_income_paths.pdf}40\caption{Net worth along a deterministic median-income path under baseline policies, for entrants with initial transfers at the 10th to 90th percentiles of the gift distribution (a) and for two entrants whose transfers straddle the age-1 threshold, $a_1 = \EqIncAoneMinus$ and $\EqIncAonePlus$ (b). Units: average five-year income; the estate at 55--59 is set at the median lottery point. Source: \texttt{results/data/baseline/equal\_income\_paths.csv}.}41\label{fig:equal}42\end{figure}4344Figure~\ref{fig:equal} isolates the mechanism of Propositions~\ref{prop:leverage} and~\ref{prop:persist} by holding the income path fixed at the median and varying only the transfer at entry. Entrants with gifts at or below the median ($a_1\le\EqIncAoneMedian$) rent until 45--49 and then buy the smallest house with the maximum loan; the entrant just above the age-1 threshold ($a_1 = \EqIncAonePlus$) buys at 25--29 and the one just below ($a_1 = \EqIncAoneMinus$) at 40--44. Net worth of the two straddling households is \EqIncNWPlusAgeFour{} against \EqIncNWMinusAgeFour{} at 40--44 and \EqIncNWPlusAgeSeven{} against \EqIncNWMinusAgeSeven{} at 55--59: a \EqIncEndowDiffPct\% difference in endowment becomes a \EqIncGapPctAgeFour\% difference in wealth at 40--44 and a \EqIncGapPctAgeSeven\% difference at 55--59, before the paths converge after retirement when both households sell. The threshold converts small endowment differences into two-decade wealth gaps through earlier leverage and forced amortisation; the sign is that of Proposition~\ref{prop:leverage}'s multiplier and the magnitude is illustrative.4546\subsection{Transfers, inequality and mobility}47\label{subsec:transfers}4849\begin{figure}[htbp]50\centering51\includegraphics[width=\textwidth]{fig06_inheritance_quantile_trajectories.pdf}52\caption{Ownership rate (a) and median net worth (b) by age and by quintile of the inter-vivos transfer received at entry, baseline cross-section. The bottom two quintiles receive no transfer. Source: \texttt{results/data/baseline/by\_gift\_quintile.csv}.}53\label{fig:quintiles}54\end{figure}5556Figure~\ref{fig:quintiles} shows the same mechanism in the cross-section. Ownership at 25--29 is \OwnAgeOneQOne, \OwnAgeOneQTwo, \OwnAgeOneQThree, \OwnAgeOneQFour{} and \OwnAgeOneQFive{} from the bottom to the top quintile of the transfer received; the bottom two quintiles receive nothing. The gap between the second and the fifth quintile narrows from \GapQtwoQfiveAgeOnePP{} percentage points at 25--29 to \GapQtwoQfiveAgeSevenPP{} at 55--59 as the untransferred households buy later, but median net worth at 55--59 is still \MedNWAgeSevenQTwo{} in the second quintile against \MedNWAgeSevenQFive{} in the fifth. Measured at the age at which the gift is made, 55--59, the probability that the child owns is \qOOageSeven{} if the parent owns and \qROageSeven{} if the parent rents, so the persistence statistic of Proposition~\ref{prop:persist} is $\lambda_7 = \LambdaAgeSeven$, a half-life of \HalfLifeGen{} generations; it is weak at that age because \ShareParentOwnerPct\% of parents own by then. At 35--39 the same probabilities are \qOOageThree{} and \qROageThree{}, $\lambda_3 = \LambdaAgeThree$. The rank--rank slope of net worth between parents and children at 55--59 is \RankRankNW{} and the Gini of net worth is \GiniNW{} over all ages and \GiniAgeSeven{} at 55--59.5758\begin{figure}[htbp]59\centering60\includegraphics[width=\textwidth]{fig07_inequality_mobility.pdf}61\caption{Inequality and mobility across stationary equilibria (general equilibrium unless stated): Gini of net worth by age (a) and intergenerational tenure and wealth persistence (b) for the baseline, without bequests, with a 50\% rebated transfer tax, under near-fixed supply, and with minimum down payments of 5\% and 40\%. Source: \texttt{results/data/\{baseline,bequests,sensitivity\}/*.json}.}62\label{fig:inequality}63\end{figure}6465\input{tables/tab07_mobility_inequality}6667Table~\ref{tab:mobility} and Figure~\ref{fig:inequality} vary the transfers. Without bequests ($\nu = 0$) the price falls to \BeqOffP{}, the rent rises by \BeqOffRhoDevPct\%, ownership falls to \BeqOffOwn{} (\BeqOffOwnUthirtyfive{} under 35, against \OwnershipUnderThirtyFive), and the transition probabilities of the child become equal, $q_{RO} = \BeqOffqRO$ and $q_{OO} = \BeqOffqOO$, with a rank--rank slope of \BeqOffRR: intergenerational persistence disappears entirely, as Proposition~\ref{prop:persist} requires, and here it is the whole of it because incomes are independent across generations by design. The Gini rises to \BeqOffGini, since transfers lift the bottom of the young distribution more than the top, and the equal-income gap at 40--44 falls from \GapAgeFour{} to \BeqOffGapFour. At fixed prices the same change lowers ownership by \BeqOffPEDropPP{} percentage points; the general-equilibrium price fall offsets all but \BeqOffGEDropPP{} of them. Halving the bequest motive gives intermediate values (price \BeqHalfP, $q_{OO} = \BeqHalfqOO$, rank--rank \BeqHalfRR). The comparison with an intergenerational income correlation of 0.4 is the caveat of \citet{fagereng2021why} in model form: $q_{RO}$ falls to \RhoGqRO{} and $\lambda_3$ rises to \RhoGLamThree{} with the rank--rank slope at \RhoGRR, so most of the tenure persistence beyond the baseline comes from income transmission, not from housing, and the model's transfer channel should be read as one mechanism among several.6869\subsection{The down-payment constraint in general and partial equilibrium}70\label{subsec:thetasweep}7172\input{tables/tab05_scenarios}7374\begin{figure}[htbp]75\centering76\includegraphics[width=\textwidth]{fig14_down_payment_sweep.pdf}77\caption{Down-payment sweep: ownership rate (a), equilibrium prices (b) and intergenerational persistence (c) as functions of the minimum down payment $\theta$, in general equilibrium (prices, lottery and rebate re-solved) and with prices fixed at the baseline. Source: \texttt{results/data/theta\_sweep/*.json}.}78\label{fig:theta}79\end{figure}8081Table~\ref{tab:scenarios} and Figure~\ref{fig:theta} report the down-payment sweep. In general equilibrium a minimum down payment of 5, 10, 20, 30 and 40\% gives prices of \ThetaFiveP, \ThetaTenP, \PriceBaseline, \ThetaThirtyP{} and \ThetaFortyP{} and ownership rates of \ThetaFiveOwn, \ThetaTenOwn, \OwnBaseline, \ThetaThirtyOwn{} and \ThetaFortyOwn, with under-35 ownership from \ThetaFiveOwnUthirtyfive{} to \ThetaFortyOwnUthirtyfive. At fixed prices the 5\% constraint raises ownership to \ThetaFivePEOwn{} (\ThetaFivePEOwnUthirtyfive{} under 35) and the 40\% constraint lowers it to \ThetaFortyPEOwn{} (\ThetaFortyPEOwnUthirtyfive): the price and rent responses absorb \AbsorbThetaFivePct\% and \AbsorbThetaFortyPct\% of the partial-equilibrium ownership effect, the direction of Corollary~\ref{cor:fixed} with a unit supply elasticity rather than a fixed stock. Mobility moves with the constraint, $q_{RO}$ at 35--39 falling from \ThetaFiveqRO{} to \ThetaFortyqRO{} and $\lambda_7$ rising from \ThetaFiveLamSeven{} to \ThetaFortyLamSeven, while the rank--rank slope is flat at \ThetaFiveRR--\ThetaFortyRR: a tighter constraint lowers upward tenure mobility without changing the rank persistence of wealth, because the latter is dominated by the size of transfers rather than by who crosses the threshold. The payment-to-income cap has a smaller effect in the same direction (ownership \PsiThirtyOwn{} and \PsiFiftyOwn{} for $\psi = 0.30$ and $0.50$).8283\subsection{Leverage and negative price shocks: accounting}84\label{subsec:pricedrop}8586Proposition~\ref{prop:leverage} is an accounting statement and is checked as one, on the baseline cross-section with prices fixed after the drop. Existing owners carry a mean loan-to-value ratio of \MeanLTVOwners{} and no household is underwater after a 10, 20 or 30\% fall in the house price, because the first annuity payment, a fifth of the loan, is made within the purchase period and the balance carried across five-year periods never exceeds the underwater threshold $x^*$; the mean loss relative to net worth is \LossNWAllTen, \LossNWAllTwenty{} and \LossNWAllThirty\% for the three drops. New borrowers evaluated at origination, with a mean loan-to-value of \MeanLTVNew, are \UnderwaterNewTen, \UnderwaterNewTwenty{} and \UnderwaterNewThirty\% underwater, consistent with $x^* = 1-0.8/0.95\approx 0.16$ at the 80\% cap with a 5\% selling cost, and lose \LossNWNewTen, \LossNWNewTwenty{} and \LossNWNewThirty\% of net worth. The five-year period therefore understates peak exposure, a limitation noted in Section~\ref{sec:sensitivity}; the general-equilibrium counterpart of a price fall, a persistent rise in the real rate, is in Section~\ref{subsec:rateshock}.87