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\section{Introduction}
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Among Canadian families under 35, the median net worth of those who own their home is about ten times that of those who do not, and the gap is of the same order among families aged 55--64 \citep{statcan2024survey}. Differences in labour income between the two groups are far smaller than this. Housing tenure is, in other words, a sorting device for wealth: it converts a modest difference in initial resources, often a parental transfer at the time of first purchase, into a large and durable difference in balance sheets, through the down payment that gates entry, the leverage that scales the return on the house, and the bequest that carries the result into the next generation. This paper asks under what conditions inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner--renter wealth inequality among households with the same labour-income process, and what an unanticipated inflation changes in that mechanism.

The question is theoretical and the answer is given in two steps. The first is a benchmark economy in which a household lives three stages, buys or rents once, repays a nominal mortgage, sells, and transfers part of the proceeds to a child who enters the model at that moment. Because the benchmark has closed forms, it delivers nine propositions: an ownership-entry threshold that equals the down-payment bound plus the stage-1 consumption of the marginal buyer (Proposition~\ref{prop:threshold}); the signs of its comparative statics, all of them liquidity effects of present-value-neutral parameters (Propositions~\ref{prop:cs}--\ref{prop:tilt}); a leverage multiplier net of transaction costs whose break-even price growth is independent of the loan-to-value ratio (Proposition~\ref{prop:leverage}); a closed-form intergenerational tenure persistence, the second eigenvalue of the dynasty chain, which is positive only if the house is net wealth at the end of life and which is a half-life rather than a trap (Proposition~\ref{prop:persist}); existence of a stationary equilibrium and a dominance condition for its uniqueness, with the corollary that a fixed stock converts every credit or transfer shift into price and composition (Proposition~\ref{prop:exist}, Corollary~\ref{cor:fixed}); price neutrality without segmentation under unit demand (Proposition~\ref{prop:noseg}); the full accounting of a price-level surprise under fixed-rate, renewable-term and adjustable-rate contracts, with the exact condition under which an indebted owner gains (Proposition~\ref{prop:accounting}); and the general-equilibrium impact of a surprise, whose sign is a statement about the net nominal position of the parents whose children are at the tenure margin (Proposition~\ref{prop:impact}, Corollary~\ref{cor:creditor}).

The second step is a quantitative overlapping-generations economy that relaxes the benchmark along the dimensions that matter for the question: twelve five-year periods with housing at every age and on an intensive margin, persistent income risk, nominal constant-payment mortgages amortised over 25 years under three contract types with a loan-to-value and a nominal payment-to-income constraint, an inter-vivos gift at the child's entry and an estate at the parent's death, segmented rental and construction sectors with durable stocks, and a lender with a balance sheet. It is solved in stationary general equilibrium and along perfect-foresight transitions after an unanticipated inflation, with the two housing markets clearing to a stated tolerance, and every result at fixed prices is labelled as such. The calibration is illustrative: no moment is fitted, verified Canadian comparison points are reported next to the model's moments, and no verified estimate of the Canadian housing-supply elasticity was found, so that parameter is swept rather than set.

Three findings are the paper's main results and all three are stated with their equilibrium status. First, transfers generate the equal-income gaps the question asks about and are the whole of their intergenerational persistence in this economy. In the baseline, owners hold \GapAgeFour{} years of average income more than renters at 40--44, children of owners are \qOOageThree{} rather than \qROageThree{} likely to own at 35--39, and the rank--rank slope of net worth between parents and children is \RankRankNW{}; switching the bequest motive off lowers the house price from 1 to \BeqOffP{} and sets the tenure persistence and the rank--rank slope to zero. The persistence is a half-life of \HalfLifeGen{} generations, not a trap: the search for multiple stationary equilibria, including under near-fixed supply, finds a monotone excess-demand function with a single root. Second, prices absorb most of what credit constraints would otherwise do to ownership. Lowering the minimum down payment from 20\% to 5\% raises ownership among households under 35 from \OwnershipUnderThirtyFive{} to \ThetaFivePEOwnUthirtyfive{} at fixed prices, but general equilibrium removes \AbsorbThetaFivePct\% of the aggregate ownership effect through the price and the rent; the long-run supply elasticity matters little for this because the demand shift is small and, in the transitions, because the impact elasticity of a durable stock is the product of the long-run elasticity and the depreciation rate. Third, and contrary to the benchmark's Proposition~\ref{prop:impact}, a 10\% unanticipated rise in the price level \emph{lowers} the real house price on impact, by \FrmLevelPzeroAbs\% under fixed-rate contracts and \FrmPersPzeroAbs\% when the surprise persists, because households in this economy are net nominal creditors: their deposits exceed their mortgages, so the surprise lowers aggregate household wealth and with it housing demand. The direct redistribution is as the accounting predicts, owners with a mortgage gain \FrmLevelCEVOM\% of lifetime consumption and renters lose \FrmLevelCEVRAbs\%, and the contract ordering, fixed-rate at least as good as the renewable term and both better than the adjustable rate, holds, with the renewable term penalised precisely when the surprise is persistent and the real rate responds. General equilibrium absorbs \FrmLevelAbsorbPct\% of the partial-equilibrium ownership response to the surprise, and the perfect-foresight transition differs from the static-expectations impact of the benchmark by a tenth of a percentage point.

The contributions are stated relative to the closest prior work and without inflation. The entry threshold with a liquidity component is an incremental extension of \citet{ortalomagne2006housing} and \citet{kaplan2020housing} to transfers, a rental sector and nominal debt; its general-equilibrium offset is a numerical statement, and the price feedback is shown to be governed by segmentation and the intensive margin, as \citet{greenwald2025do} would predict. The persistence result rationalises in closed form what \citet{charles2002transition} and \citet{daysal2023intergenerational} document empirically; it is new as theory and is deliberately not a multiplicity result. The inflation results are incremental on the direct revaluation of \citet{doepke2006inflation} and \citet{meh2011inflation} and on the contract comparison of \citet{garriga2017mortgages}, and new in two respects: the sign of the general-equilibrium price response is tied to a density-weighted net nominal position, and the Canadian five-year term is placed between the long fixed-rate and the adjustable contract and shown to lose its advantage when a persistent surprise is met by a real-rate response. The supply-elasticity and contract effects on mobility are comparative statics of a mobility measure, incremental on \citet{kiyotaki2011winners} and \citet{chambers2009accounting}, and small in this calibration.

Three results in this paper are negative relative to the hypotheses with which it started, and they are reported as results. Surprise inflation lowers rather than raises the real house price when households are net nominal creditors; price neutrality without segmentation does not carry over to an economy with an intensive housing margin; and the entry threshold is not monotone in the down payment at every productivity state once households can buy later. In each case the responsible benchmark assumption is identified and the proposition is restated under it, rather than the theory being adjusted to fit the simulations or the simulations being discarded.

The paper proceeds as follows. Section~\ref{sec:literature} reviews the literature by mechanism. Section~\ref{sec:model} sets out the benchmark and the quantitative economy. Section~\ref{sec:equilibrium} states the propositions and the remarks that scope them. Section~\ref{sec:computation} describes the algorithm, the calibration and the verification. Section~\ref{sec:baseline} reports the baseline, the equal-income paths and the down-payment sweep; Section~\ref{sec:inflation} the inflation and real-rate transitions; Section~\ref{sec:supply} supply, bequests, policy and welfare; Section~\ref{sec:sensitivity} sensitivity, limitations and conclusions. Appendix~\ref{app:proofs} contains the proofs, Appendix~\ref{app:computation} the numerical details and Appendix~\ref{app:tables} the supplementary exhibits.