UQO Working Papers · WP18 · v1.0
Housing, Leverage, and Intergenerational Wealth Inequality: A Dynamic General Equilibrium Theory
- Year
- 2026
- First release
- October 2026
- Latest revision
- October 2026
- Publication status
- Working paper
Transfers, not income, make housing tenure a sorting device for wealth
Among households with identical income processes, gifts and estates generate a 0.95 income-year wealth gap at mid-life and a rank–rank slope of 0.34 that vanishes without bequests; an unanticipated 10% inflation lowers rather than raises the real house price.
Abstract
This paper asks under what conditions inherited wealth, mortgage leverage, nominal debt, housing supply and down-payment constraints generate persistent owner–renter wealth gaps among households with the same labour-income process, and what an unanticipated inflation changes. A three-stage benchmark with two-period generations yields a closed-form ownership-entry threshold that adds a liquidity component to the down-payment bound, a leverage multiplier whose break-even price growth is independent of the loan-to-value ratio, a closed-form intergenerational tenure persistence that is positive only if the house is net wealth at the end of life, existence and a uniqueness condition for the stationary equilibrium, and the accounting of a price-level surprise under three mortgage contracts.
A twelve-period overlapping-generations model with nominal annuity mortgages, gifts and estates, and segmented rental and construction sectors is solved in stationary general equilibrium and along perfect-foresight transitions under an illustrative calibration. Transfers generate a 0.95 income-year wealth gap at mid-life and a rank–rank wealth slope of 0.34 that vanishes without transfers; price adjustment removes 64–66% of the partial-equilibrium ownership response to the down-payment constraint; no multiple equilibria arise under near-fixed supply.
Because households are net nominal creditors, a 10% surprise moves the real house price by −0.45%, a fall where the benchmark predicts a rise; the contract ordering — fixed-rate above renewable term above adjustable — is confirmed. Three results that contradicted the benchmark are reported as results and the responsible assumptions identified.
Key findings
0.95 yr
of income: owner–renter wealth gap at 40–44, equal incomes
0.34
rank–rank wealth slope with transfers, −0.01 without
0.54 vs 0.35
child owns at 35–39 given parent owner versus renter
64–66%
of the partial-equilibrium down-payment effect absorbed by prices
−0.45%
real house price after a 10% surprise, fixed-rate contract
9
closed-form propositions from the three-stage benchmark
Scientific contributions
- Ownership entry threshold with a liquidity component
- The transfer required to own exceeds the down-payment bound by the stage-one consumption of the marginal buyer; in general equilibrium prices absorb 64% (θ = 0.05) to 66% (θ = 0.40) of what the constraint would do to ownership at fixed prices.
- Leverage scales exposure, not the break-even
- The leverage multiplier raises the sensitivity of terminal wealth to the resale price while the break-even price growth is independent of the loan-to-value ratio.
- Intergenerational persistence in closed form
- Parent–child tenure persistence is the second eigenvalue of the dynasty chain, positive only if the house is net wealth at the end of life — a half-life rather than a trap. Quantitatively, transfers are the whole of the model's persistence: the rank–rank slope of 0.34 falls to −0.01 without bequests.
- Existence, uniqueness and no multiplicity
- The stationary equilibrium exists and is unique when the entry-threshold channel dominates the ladder channel; three price scans, including a supply elasticity of 0.05, find monotone excess demand and a single root.
- Surprise inflation reverses sign in general equilibrium
- With deposits exceeding mortgages, a 10% price-level surprise lowers the real house price by 0.45% (fixed-rate), 0.47% (renewable term) and 0.79% (adjustable) where the benchmark predicted a rise; mortgaged owners gain +1.14%, +0.46% and −1.21% of lifetime consumption, and the Canadian five-year term loses its advantage when the real rate responds.
- Negative results reported as results
- Three simulation outcomes contradicted the benchmark — the inflation sign, price neutrality without segmentation, and monotonicity of the entry threshold in the down payment — and each proposition is restated under the identified premise rather than the simulations discarded.
Methodology
Methodology (5)
Closed-form benchmark theory
Nine propositions on entry, leverage, persistence, existence and uniqueness, and surprise-inflation accounting, with proofs in a standalone appendix and a general optimiser used to verify the hand-to-mouth assumption.
Household problem by backward value-function iteration
Twelve ages with rent, buy, refinance, sell and stay branches on a cash-on-hand grid; loans on eleven fractions of the binding constraint, gifts on twelve fractions of cash, a five-point estate lottery; numba-accelerated.
Stationary general equilibrium
Nested Illinois regula falsi (rental market inside the owner market) with a damped fixed point on the estate lottery and the transfer-tax rebate, tolerance 5e-4; supply scales normalised so that house and rental prices equal one.
Perfect-foresight transitions
Sequence-space shooting over 14 periods with durable stocks, a parallel finite-difference Jacobian, Broyden updates and a damped tâtonnement pre-phase; every run reported with its residual (12 of 16 within tolerance).
Welfare
Consumption-equivalent variation net of warm glow for entrants by transfer quintile and for households alive at a shock by age and tenure.
Research data
- Benchmark environment
- Three stages of life with two-period generations, unit housing demand, a nominal mortgage, an inter-vivos transfer at the child's purchase and a bequest; closed forms for the entry threshold, the leverage multiplier, the dynasty chain and the revaluation of a price-level surprise under fixed-rate, renewable-term and adjustable contracts.
- Quantitative OLG economy
- Twelve five-year periods, four house sizes and continuous rented services, Rouwenhorst income risk, 25-year constant-payment nominal mortgages under three contracts with loan-to-value and payment-to-income constraints, a gift at the child's entry and an estate at the parent's death, segmented rental and construction stocks and a lender balance sheet; simulated on a 96,000-dynasty Monte-Carlo panel.
- Illustrative calibration
- No microdata and no fitted moment; verified Canadian comparison points (Survey of Financial Security 2023 owner–renter net worth and mortgage share, Hou 2010 ownership profile, CIBC gift statistics) are reported next to model moments without being targeted; the housing-supply elasticity, lacking a verified Canadian estimate, is swept over 0.05–2.
Reproducibility
- make all: tests → simulation (45 stationary equilibria, 5 partial-equilibrium runs, 3 price scans, 16 transitions, welfare; about 60 minutes on 18 cores) → 14 figures and 9 tables → LaTeX
- 69 pytest tests on identities, debt dynamics, feasibility, market clearing and residuals (about 20 s), 44 of them on the closed-form benchmark
- Every number in the manuscript is one of about 2,500 macros in numbers.tex read from 358 committed CSV/JSON result files
- Research folder with 72 verified references, the model specification, propositions, failed hypotheses and a results-audit integrity gate; proofs compile as a standalone theory appendix
Research artifacts
Version history
| Version | Date | Description | File |
|---|---|---|---|
| v1.0current | October 2026 | Current manuscript as published on www.spboucher.ai (revised edition after the 2026-10-17 scientific review for papers 1–17). | PDF · 65 p. |