# WP18 — Model specification (Phase 2)

Status: written incrementally on 2026-10-20 by the theory team. Part A is the tractable benchmark (three-period lifecycle, two-period generations) that delivers the analytical propositions proved in paper/appendix_proofs.tex; Part B is the full dynamic OLG model to be implemented in Phase 3. Notation is chosen to be consistent across the two parts and, where possible, with Kaplan–Mitman–Violante (2020) for the household block, Garriga–Kydland–Šustek (2017) for nominal mortgages and Sommer–Sullivan–Verbrugge (2013) for the rental block. Results of the simulations are never called theorems; every analytical claim below has a label (Proposition/Lemma) matching the appendix.

# 0. Conventions, units and notation

• Time is discrete, $t = 0, 1, 2, \dots$. One model period is a block of years (Part A: a "stage of life", roughly 10–15 years; Part B: 5 years, see §B.9).
• Numéraire. All quantities marked "real" are in units of the consumption good of the period. The consumer price index is $P_t$ (nominal price of one unit of consumption); gross CPI inflation is $1+\pi_t = P_t/P_{t-1}$.
• Expected and unexpected inflation. $\pi^e_{t}$ is the inflation rate over $(t-1,t)$ expected at $t-1$; the surprise is defined multiplicatively, $1+\pi^u_t \equiv (1+\pi_t)/(1+\pi^e_t)$. In the stationary equilibrium $\pi_t = \pi^e_t = \bar\pi$ and $\pi^u_t = 0$.
• House price vs CPI. $p_t$ is the real price of one unit of owner-occupied housing (units of consumption per unit of housing). The nominal house price is $P_t p_t$; the house-price index is therefore $P_t p_t$ and is distinct from the CPI $P_t$. "Real house-price growth" means $p_{t+1}/p_t - 1$.
• Rent. $\rho_t$ is the real rent per unit of rental housing services for one period, paid at the start of the period.
• Nominal debt. A mortgage originated at $t$ has nominal principal $B^{nom}t$ (dollars). Its real value is $B^{nom}t/P_t$. We write $B_t \equiv B^{nom}t/P_t$ for the real principal at origination and track the real balance $b^m{t+k} \equiv B^{nom}{t+k}/P{t+k}$ in later periods; the nominal balance is what the contract fixes, the real balance is what matters for consumption.
• Interest rates. $r$ is the real one-period rate (gross $R \equiv 1+r$). The nominal rate on contracts written at $t$ for the period $(t,t+1)$ satisfies the Fisher relation $1+i_t = R,(1+\pi^e_{t+1})$ (competitive risk-neutral lender, Assumption A2). An FRM keeps $i_t$ for the life of the loan; an ARM resets to the current $i$ every period; a term-$k$ contract (the Canadian five-year term) resets every $k$ periods.
• Households and dynasties. Each household belongs to a dynasty; a dynasty is an infinite sequence of parent–child pairs. Labour income processes are identical across dynasties (the "equal income" design); dynasties differ only through the transfers they receive.
• Indices: $j$ age, $z$ idiosyncratic income state, $o \in {O, R}$ tenure (owner, renter). Hats denote thresholds ($\hat b$), stars stationary-equilibrium values ($p^*$).

Table of symbols (both parts):

# Part A — Tractable benchmark: three-stage lifecycle, two-period generations

# A.1 Demographics and timing

A household lives three stages, $j = 1, 2, 3$: buyer (young adult; chooses tenure, buys or rents, borrows), payer (prime age; pays rent or mortgage interest and amortisation; under an ARM/term contract the mortgage rate is reset at the start of this stage), seller/donor (sells the house, repays the remaining balance, consumes and gives an inter-vivos transfer to the child). The child is born into the model (stage 1) at the date at which the parent enters stage 3, so a generation lasts two periods and the transfer is received exactly when the child makes its tenure decision — the empirical "gift at first purchase" channel (Engelhardt–Mayer 1998; Blickle–Brown 2019). Each period a unit mass of stage-1 households is born; the cross-section at any date has one unit mass of each stage.

Stage-3 households do not demand housing in the two housing markets of the model (their retirement housing is outside the model and its cost is netted out of $y_3$). This is the only demographic simplification of Part A; Part B has housing at all ages.

Within-period timing (date $t$). (1) The price level $P_t$ is realised (possibly with a surprise $\pi^u_t$) and the income shock $y_{3}$ of stage-3 households is realised. (2) Real prices $p_t$ and $\rho_t$ are determined. (3) Stage-3 households sell, repay, consume and transfer $b'$ to their stage-1 child. (4) Stage-1 households receive $b'$, choose tenure, housing and the mortgage; stage-2 households make payments and consume. (5) The owner-occupied and rental markets clear.

# A.2 Preferences and the housing need

Lifetime utility of a household born at $t$ with tenure $o \in {O,R}$: $$ U^o = \ln c_1 + \chi,\mathbf 1{o = O} + \beta,\mathbb E\big[\ln c_2\big] + \beta^2,\mathbb E\big[\ln c_3 + \nu \ln b'\big], \qquad \beta \in (0,1),\ \nu > 0,\ \chi \ge 0 . $$ Each household needs exactly one unit of housing services in 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 $\chi$ (an ownership premium, Henderson–Ioannides 1983; $\chi = 0$ is allowed). The warm-glow term $\nu \ln b'$ is the De Nardi (2004) specification with the luxury shift set to zero so that the transfer rule is linear (Lemma 1). Income $y_1, y_2$ are deterministic and common; $y_3$ is drawn from a continuous distribution $G$ with density $g > 0$ on $[\underline y_3, \bar y_3]$, i.i.d. across households and generations. The income process is the same for every dynasty.

# A.3 Markets, contracts and the lender

Mortgage. A stage-1 owner buys one unit at $p_t$, pays the buying cost $\kappa_b p_t$ and maintenance $m p_t$, makes the down payment $\theta p_t$ and borrows the nominal principal $B^{nom}_t = P_t (1-\theta) p_t$ (the LTV constraint binds under Assumption A4; in general $B^{nom}_t \le P_t(1-\theta)p_t$). The contract specifies the nominal payments

• stage 2 (date $t+1$): interest $i_{1} B^{nom}_t$ plus amortisation $a B^{nom}_t$, $a \in (0,1]$;
• stage 3 (date $t+2$): $(1+i_{2})(1-a) B^{nom}_t$,

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 / term contract renewed at $t+1$). Real payments are $$ D_2 = \frac{(i_1 + a)(1-\theta)p_t}{1+\pi_{t+1}}, \qquad D_3 = \frac{(1+i_2)(1-a)(1-\theta)p_t}{(1+\pi_{t+1})(1+\pi_{t+2})}. $$ In the stationary state ($\pi = \pi^e = \bar\pi$) write $D_2 = \delta_2 p$, $D_3 = \delta_3 p$ with $$ \delta_2 = (1-\theta)\Big[R - \frac{1-a}{1+\bar\pi}\Big], \qquad \delta_3 = \frac{(1-a)(1-\theta)R}{1+\bar\pi}, \qquad \delta_2 + \frac{\delta_3}{R} = (1-\theta)R . $$ The last identity is Fisher neutrality in present value: anticipated inflation front-loads real payments (the "tilt", $\partial\delta_2/\partial\bar\pi > 0$, $\partial\delta_3/\partial\bar\pi < 0$) without changing their present value.

Liquid asset. A one-period nominal deposit paying $i_t$; real balance $n \ge 0$ (no unsecured borrowing). Its real gross return is $R/(1+\pi^u_{t+1})$: a surprise erodes nominal savings exactly as it erodes nominal debt.

Lender. A competitive, risk-neutral intermediary funds mortgages and takes deposits at the same nominal rate, earning zero expected real profit; its equity is held by rentiers outside the dynasties (equivalently, a small open economy facing the world real rate $r$). Its realised real profit in a surprise period is the mirror image of the households' revaluation gains (Proposition 8). The alternative closure (mortgages funded by domestic deposits, endogenous $r$) is a Part B extension.

Rental sector. Landlords are competitive and risk neutral; the rental stock is segmented from the owner-occupied stock (different units; no costless conversion). Rental supply per cohort is $S_r(\rho) = \bar H_r \rho^{\varepsilon_r}$, $\varepsilon_r \in [0,\infty]$ ($\varepsilon_r = \infty$: rent fixed at $\bar\rho$). This reduced form stands for landlords who hold a stock $H^r$ priced at $p^r$ with the arbitrage rent $\rho = p^r(1+m+\phi) - \mathbb E p^{r\prime}/R$, so that $S_r$ is the supply of rental units expressed in rent space. Real wage and rent indexation to a surprise are $\eta_w, \eta_\rho \in [0,1]$ (baseline 1).

Owner-occupied supply. $S(p) = \bar H p^{\varepsilon}$ units per cohort, $\varepsilon \in [0,\infty)$; $\varepsilon = 0$ is a fixed stock. In Part A, $S$ is used as a per-period supply curve (the stock is re-supplied each period along $S$, equivalently a reversible land-conversion technology). The durable stock-flow version $H_{t} = (1-\delta)H_{t-1} + \iota p_t^{\varepsilon}$ with stationary curve $S(p) = \iota p^\varepsilon/\delta$ and impact elasticity $\varepsilon\delta$ is the Part B specification; the Part A propositions on impact effects are stated under the per-period curve and the Phase 3 code will report both.

No segmentation (benchmark). If instead owner and rental units are the same stock and landlords convert costlessly, the total demand for units is fixed by unit demand (two per cohort) and $p$ is pinned by $S(p) = 1$ (per cohort), the rent follows $\rho = p(1+m+\phi) - p'/R$, and tenure has no effect on the price (Proposition 7). This is the Kaplan–Mitman–Violante/Greenwald–Guren neutrality; price feedback in Part A requires segmentation.

# A.4 Household budgets

Owner (born at $t$; stationary prices unless noted): $$ \begin{aligned} \text{stage 1:}\quad & c_1 + (\theta + \kappa_b + m),p_t + n_1 = b + y_1 + \underbrace{\big[(1-\theta)p_t - B_t\big]}{\ge 0 \text{ if the LTV constraint is slack}}, \ \text{stage 2:}\quad & c_2 + D_2 + m,p_t + n_2 = y_2 + n_1 \tfrac{R}{1+\pi^u{t+1}}, \ \text{stage 3:}\quad & c_3 + b' = W_3^O \equiv y_3 + (1-\kappa_s),p_{t+2} - D_3 + n_2 \tfrac{R}{1+\pi^u_{t+2}} . \end{aligned} $$ (Maintenance in stage 2 is $m p_t$: it is a share of the purchase price, i.e. a physical cost; the distinction is immaterial in the stationary state.) We write $d \equiv \theta + \kappa_b + m$ for the stage-1 cash outlay per unit of price. Define the net equity share at sale $e_3 \equiv (1-\kappa_s) - \delta_3$ (stationary), so that $W_3^O = y_3 + e_3 p$ for an owner with $n_2 = 0$; $e_3 > 0$ requires $(1-\kappa_s)(1+\bar\pi) > (1-a)(1-\theta)R$, i.e. enough amortisation and down payment for the house to be net wealth at the end of life.

Renter (born at $t$): $$ c_1 + \rho_t + n_1 = b + y_1, \qquad c_2 + \rho_{t+1} + n_2 = y_2 + n_1\tfrac{R}{1+\pi^u_{t+1}}, \qquad c_3 + b' = W_3^R \equiv y_3 + n_2\tfrac{R}{1+\pi^u_{t+2}} . $$ Constraints: $c_j > 0$, $n_1, n_2 \ge 0$ for renters; $n_1 \ge -(1-\theta)p_t$ for owners (a negative $n_1$ is the mortgage; the payment schedule above applies to the mortgage part), $n_2 \ge 0$ (no refinancing in Part A; Part B allows it). No default: Assumption A3 guarantees $c_2, W_3 > 0$ on the support of $y_3$.

Stage-3 split (Lemma 1). Maximising $\ln c_3 + \nu \ln b'$ subject to $c_3 + b' = W_3$ gives $c_3 = W_3/(1+\nu)$, $b' = \nu W_3/(1+\nu)$: the transfer is a fixed share of end-of-life wealth, so the dynasty map is linear in $W_3$.

# A.5 The hand-to-mouth (HTM) regime and closed-form values

Assumption A4 (liquidity-constrained at the margin) requires that at the wealth level at which the tenure choice is marginal, both the would-be owner and the would-be renter would like to borrow against future income but cannot: $1/c_1 > \beta R,\mathbb E[1/c_2]$ and $1/c_2 > \beta R (1+\nu), \mathbb E[1/W_3 \mid y_2]$ at $n_1 = n_2 = 0$ (renter) and at $B = (1-\theta)p$, $n_2 = 0$ (owner). Under A4, the closed-form values are $$ V^O(b) = \ln(b + y_1 - d p) + \chi + \beta,\mathbb E\ln(y_2 - D_2 - m p) + \beta^2(1+\nu),\mathbb E\ln\big(y_3 + (1-\kappa_s)p'' - D_3\big) + K, $$ $$ V^R(b) = \ln(b + y_1 - \rho) + \beta,\mathbb E\ln(y_2 - \rho') + \beta^2(1+\nu),\mathbb E\ln y_3 + K, $$ with $K = \beta^2[\nu\ln\nu - (1+\nu)\ln(1+\nu)]$ common. Define the ownership premium in utility units $$ \Delta \equiv \chi + \beta,\mathbb E\big[\ln(y_2 - D_2 - mp) - \ln(y_2 - \rho')\big] + \beta^2(1+\nu),\mathbb E\big[\ln(y_3 + (1-\kappa_s)p'' - D_3) - \ln y_3\big], \qquad \Lambda \equiv e^{-\Delta}. $$ $\Delta$ collects everything that happens after the purchase: the stage-2 cash-flow comparison (mortgage payment plus maintenance versus rent) and the stage-3 housing equity. The tenure decision compares the stage-1 cash sacrifice $d p - \rho$ with $\Delta$.

# A.6 Ownership-entry threshold (Proposition 1) and comparative statics (Proposition 2)

Under A1–A5 (A5: $\Delta > 0$ and $dp > \rho$, i.e. owning costs more cash up front and pays off later), the household owns iff $b \ge \hat b$ with $$ \boxed{;\hat b(p,\rho) = \frac{d,p - \Lambda,\rho}{1-\Lambda} - y_1;}, \qquad \hat b > b_{\min} \equiv d p - y_1, $$ and the stage-1 consumption of the marginal owner is $c_1^O(\hat b) = \Lambda(dp-\rho)/(1-\Lambda)$. If $dp \le \rho$ (owning is cheaper even in cash), $\hat b = b_{\min}$: only feasibility (the down payment plus costs) binds. The threshold is the sum of a feasibility component (down payment, costs) and a preference component (the sacrifice of stage-1 consumption is worth it only if the household is rich enough that the marginal utility of stage-1 consumption is low).

Comparative statics (Proposition 2; all under strict A4 and $r \ge 0$): $\partial\hat b/\partial p > 0$, $\partial\hat b/\partial\rho < 0$, $\partial\hat b/\partial\theta > 0$, $\partial\hat b/\partial\kappa_b > 0$, $\partial\hat b/\partial\kappa_s > 0$, $\partial\hat b/\partial m > 0$, $\partial\hat b/\partial\chi < 0$, $\partial\hat b/\partial y_1 = -1$ (direct), and $\partial \hat b/\partial\bar\pi > 0$ (anticipated inflation raises the threshold through the tilt of real mortgage payments, Proposition 3). The sign of $\partial\hat b/\partial\theta$ is not a present-value effect — a higher down payment is PV-neutral under Fisher pricing — it is a liquidity effect, which is why A4 is needed; for an unconstrained household $\theta$ and $\bar\pi$ are neutral (Proposition 3).

Interpretation of $\partial\hat b/\partial p > 0$: a higher price raises the cash outlay $dp$ and the stage-2 payments, and raises the equity carried into stage 3 ($e_3 p$); for a liquidity-constrained household the first effect dominates whenever $r \ge 0$ (the proof bounds the stage-3 term by the Euler inequalities). The opposite sign would require a household that values forced saving more than current consumption, which contradicts A4.

# A.7 Leverage amplification with costs (Proposition 4)

For an owner, end-of-life wealth is $W_3^O = y_3 + (1-\kappa_s)p'' - D_3$. Its response to the resale price is $(1-\kappa_s)$ per unit of price, i.e. $(1-\kappa_s)/d$ per unit of cash invested at purchase: the leverage multiplier net of costs, decreasing in $\theta$, $\kappa_b$, $\kappa_s$, $m$. Compared with a renter who follows the same consumption path and invests the cash-flow differences at $R$, the owner ends with more wealth iff $$ (1-\kappa_s)(1+g) ;\ge; R^2(1+\kappa_b+m) + mR - \frac{\rho}{p},(R+R^2), \qquad g \equiv p''/p - 1, $$ i.e. iff real price growth exceeds a break-even rate $g^$ that is increasing in $\kappa_b, \kappa_s, m$ and decreasing in the rent–price ratio, and independent of $\theta$: leverage scales gains and losses but does not change the break-even. A negative price shock $x$ (resale at $(1-x)p$) wipes out the owner's housing equity iff $x \ge x^ \equiv 1 - \delta_3/(1-\kappa_s)$; the renter is unaffected. In the stationary state ($g = 0$) with $\rho/p$ at the segmented-market level, the owner–renter end-of-life wealth gap among HTM households with the same income is exactly $e_3 p$ and the transfer gap is $\nu e_3 p/(1+\nu)$.

# A.8 Dynasties: transitions, persistence and the intergenerational tenure correlation (Proposition 5)

With HTM households, $b' = \nu W_3/(1+\nu)$ with $W_3^R = y_3$ and $W_3^O = y_3 + e_3 p$ (stationary). The child owns iff $b' \ge \hat b$. Hence, with $\hat y_R \equiv (1+\nu)\hat b/\nu$ and $\hat y_O \equiv \hat y_R - e_3 p$, $$ q_{RO} = \Pr(\text{child owns}\mid\text{parent rents}) = 1 - G(\hat y_R), \qquad q_{OO} = \Pr(\text{child owns}\mid\text{parent owns}) = 1 - G(\hat y_O). $$ The tenure of a dynasty is a two-state Markov chain; its second eigenvalue $$ \lambda \equiv q_{OO} - q_{RO} = G(\hat y_R) - G(\hat y_R - e_3 p) $$ is the parent–child correlation of the ownership indicator and the per-generation persistence of any tenure gap (half-life $\ln 2/(-\ln\lambda)$). $\lambda > 0$ iff $e_3 > 0$ (the house is net wealth at the end of life) and the density $g$ is positive between $\hat y_O$ and $\hat y_R$; $\lambda$ is increasing in $e_3 p$ for fixed $\hat b$. Given prices, the unique stationary owner share is $\mu^* = q_{RO}/(1-\lambda)$ and the dynasty distribution converges to it geometrically at rate $\lambda$. In the no-risk limit ($G$ degenerate) $\lambda \in {0, 1}$: tenure classes become absorbing when $b'_R < \hat b \le b'_O$, and any $\mu$ consistent with that inequality is stationary — a continuum of history-dependent steady states that disappears as soon as income risk mixes dynasties across the threshold (Remark in the appendix; see also failed_hypotheses.md).

# A.9 Stationary equilibrium (Proposition 6), fixed supply (Corollary) and neutrality without segmentation (Proposition 7)

Definition (stationary equilibrium, Part A). Given $(\bar\pi, r, G)$ and parameters, a stationary equilibrium is $(p^, \rho^, \mu^, \hat b^)$ such that (i) $\hat b^* = \hat b(p^, \rho^)$ from A.6 with $p'' = p^$, $\rho' = \rho^$; (ii) $\mu^* = q_{RO}/(1-\lambda)$ evaluated at $(p^,\rho^)$; (iii) owner-occupied market clearing $S(p^) = \mu^$ (per cohort; two cohorts hold units, two cohorts are supplied); (iv) rental market clearing $S_r(\rho^) = 1 - \mu^$; (v) the lender earns zero real profit; (vi) the distribution of transfers is the stationary distribution of the dynasty chain: a mass $\mu^$ of stage-1 households draw $b' = \nu(y_3 + e_3 p^)/(1+\nu)$ and a mass $1-\mu^*$ draw $b' = \nu y_3/(1+\nu)$, $y_3 \sim G$.

Existence follows from the intermediate value theorem applied to $f(\mu) = \mu^*(S^{-1}(\mu), S_r^{-1}(1-\mu)) - \mu$ on $[0,1]$ (continuity under A3). Uniqueness holds when the entry-threshold channel dominates the ladder channel: a higher $p$ raises $\hat b$ (fewer renter children qualify) but also raises the equity $e_3 p$ bequeathed by owners (more owner children qualify); condition (C): $\tfrac{1+\nu}{\nu},\tfrac{d\hat b}{dp} \ge e_3$ along the clearing curve makes $f$ decreasing. If (C) fails, multiple stationary equilibria are possible (a Stein-type property-ladder multiplicity); we do not establish them and leave the search to Phase 3.

Fixed supply (Corollary). With $\varepsilon = 0$ the owner share is $\mu^* = \bar H$ in every stationary equilibrium: any change in $\theta$, transfers, $y$ or $\chi$ is absorbed by $p^$ and the composition of owners, not by the ownership rate; $\lambda$ can still change through $e_3 p^$.

No segmentation (Proposition 7). With one stock and costless conversion, $p^*$ solves $S(p) = 1$ (per cohort) and is independent of $\theta$, $G$, the transfer distribution and of inflation surprises; tenure, $\hat b$ and $\lambda$ are determined given that price, and there is no price feedback from credit, inheritance or inflation. Price feedback in this model is therefore a statement about segmentation and supply, not about credit per se.

# A.10 Surprise inflation: accounting (Proposition 8) and general equilibrium impact (Proposition 9)

Experiments. The economy is in the stationary equilibrium up to $t$. At $t+1$: (E1) level surprise: $1+\pi_{t+1} = (1+\bar\pi)(1+\pi^u)$, expectations for $t+2$ onward unchanged at $\bar\pi$; (E2) persistent surprise: $\pi_{t+1} = \pi_{t+2} = (1+\bar\pi)(1+\pi^u)-1$, learned at $t+1$ (so ARMs reset to the higher nominal rate at $t+1$, FRMs do not). Optional monetary response: the real rate for $(t+1,t+2)$ becomes $r' = r + \phi_r\pi^u$.

Direct revaluation (partial equilibrium, prices fixed). Real payment changes relative to the no-surprise path, for the cohort aged 2 at $t+1$ (bought at $t$):

• $D_2$: multiplied by $1/(1+\pi^u)$ for every contract (the stage-2 nominal payment was fixed before the shock);
• $D_3$: (E1) multiplied by $1/(1+\pi^u)$ for every contract; (E2) multiplied by $1/(1+\pi^u)^2$ under an FRM and by $1/(1+\pi^u)$ under an ARM (the reset recovers one factor for the lender); with a real-rate response, the ARM payment is further multiplied by $R'/R$.

For the cohort aged 3 at $t+1$ (bought at $t-1$): $D_3$ multiplied by $1/(1+\pi^u)$ for every contract. Nominal savers lose $n R,\pi^u/(1+\pi^u)$. Real wages change by the factor $(1+\pi^u)^{-(1-\eta_w)}$, real rents paid by sitting tenants by $(1+\pi^u)^{-(1-\eta_\rho)}$. Summing over households, the real revaluation of nominal claims is zero-sum: the lender's realised real loss equals the households' net real gain on nominal positions, and landlords bear the rent-rigidity transfer. An owner with a mortgage loses from the surprise iff the real-income loss from imperfect wage indexation plus the loss on nominal savings exceeds the debt erosion: with full ARM repricing and a real-rate response $\phi_r > 0$ in (E2), the stage-3 payment can rise in real terms, so the net sign depends on $(1-a)R\phi_r$ versus the erosion of the stage-2 payment (explicit condition in the appendix).

General-equilibrium impact at $t+1$ (Proposition 9; under per-period supply and static expectations of post-$t+1$ prices, Assumption A6). The stage-3 owner-parents at $t+1$ have $W_3 = y_3 + (1-\kappa_s)p_{t+1} - D_3/(1+\pi^u)$, so the mass of their children who qualify rises: $\partial\mu_{t+1}/\partial\pi^u|{p,\rho} = \mu^* g(\hat y_O) D_3/(1+\pi^u)^2 > 0$. With stage-2 owners not transacting, owner-market clearing is $S(p{t+1}) = (\mu^* + \mu_{t+1})/2$ and rental clearing $S_r(\rho_{t+1}) = 1 - (\mu^* + \mu_{t+1})/2$. Under condition (C):

1. $dp_{t+1}/d\pi^u \ge 0$ and $d\rho_{t+1}/d\pi^u \le 0$ (strict iff $g(\hat y_O) > 0$ and the respective elasticity is finite); the rent–price ratio falls.
2. $d\mu_{t+1}/d\pi^u \ge 0$, strict iff $\varepsilon > 0$; with a fixed owner stock the ownership rate does not move and the entire effect is a composition shift: owner-dynasty children replace renter-dynasty children at the margin.
3. $d\hat b_{t+1}/d\pi^u \ge 0$: the renter-to-owner transition probability $q_{RO}$ falls on impact — surprise inflation lowers upward tenure mobility of dynasties without nominal debt, and raises $\lambda$.
4. Winners: stage-3 FRM/ARM owner-sellers (debt erosion plus a higher sale price), stage-2 owners (erosion of $D_2$, and of $D_3$ under an FRM in E2). Losers: the lender (full nominal erosion), renter-dynasty children at the margin (priced out), nominal savers, and — a case that shows inflation does not enrich every owner — inframarginal owner-dynasty children who buy at $t+1$: their cash position changes by $\tfrac{\nu}{1+\nu}\big[D_3\tfrac{\pi^u}{1+\pi^u} + (1-\kappa_s)dp_{t+1}\big] - d,dp_{t+1}$, negative when supply is inelastic and the transfer share $\nu/(1+\nu)$ is small. Renter-dynasty children who remain renters gain from the lower rent if $\varepsilon_r < \infty$.
5. Under no segmentation (Proposition 7) items 1–3 vanish and only the direct revaluation remains.

The full perfect-foresight transition (durable stock, forward-looking prices, two affected parent cohorts, propagation through the dynasty composition) is computed numerically in Phase 3 and is not covered by Proposition 9; the toy script src/toy_model.py simulates it under the Part A structure as a check on the sign results.

# A.11 What the propositions do and do not deliver

Delivered analytically: threshold with feasibility and preference components; signs of its comparative statics incl. LTV, costs, rent and anticipated inflation; leverage multiplier and break-even growth with costs; closed-form tenure persistence and half-life; existence of a stationary GE and a uniqueness condition; price-neutrality without segmentation; the complete accounting of a surprise under FRM/ARM, level/persistent, with wage and rent indexation and lender wealth; signed impact effects of a surprise on price, rent, ownership, threshold and mobility under segmentation. Not delivered (honest gaps): single crossing of the tenure decision outside the HTM regime; multiplicity of stationary equilibria (possible, not established); the full transition path; anything about the intensive margin of housing (unit demand).

# Part B — Full dynamic model: lifecycle OLG with nominal mortgages, bequests, rental and construction sectors

Part B generalises Part A along the dimensions that Part A freezes: many ages, housing at all ages and on an intensive margin, persistent income risk, long-term amortising nominal mortgages with refinancing and three contract types, a payment-to-income constraint, bequests at two dates, a durable housing stock, and a lender with an explicit balance sheet. The equilibrium concept is a stationary recursive competitive equilibrium with constant inflation $\bar\pi$, and a perfect-foresight transition after an unanticipated inflation shock. Everything is written so that src/model.py, household_solver.py, equilibrium.py, distributions.py and simulation.py can be coded from this section alone.

# B.1 Demographics and the dynasty link

• Ages $j = 1, \dots, J$ (baseline $J = 12$ periods of 5 years, model age 1 = 25–29, death after age $J$ = 80–84; retirement at $j_R = 9$, 65 years). No survival risk in the baseline (a survival profile ${s_j}$ is a switch; accidental bequests then go to the child).
• Each household has one child who enters the model (age 1) when the parent is aged $j_g = 1 + \Delta_g$ with $\Delta_g = 6$ (30-year generation gap). The child receives an inter-vivos transfer $b^g$ at its age 1, chosen by the parent at age $j_g$, and the parent's estate $b^e$ at the parent's death, i.e. at the child's age $j_e = J - j_g + 2 = 7$. Both transfers are chosen with warm-glow utility (B.2). Baseline experiments switch on both; "bequests off" sets $\nu = 0$ for both.
• Population is constant: a unit mass per age. A dynasty is indexed by its phase; the cross-section at any date contains all phases, so the stationary distribution is over $(j, \text{state})$ with equal mass at each $j$.

# B.2 Preferences

Period utility over non-housing consumption $c$ and housing services $s$: $$ u(c, s) = \frac{\big(c^{1-\alpha},s^{\alpha}\big)^{1-\gamma}}{1-\gamma}, \qquad \alpha \in (0,1),\ \gamma > 1, $$ with $s = h$ for an owner of a house of size $h$ scaled by the ownership premium $s = (1+\chi) h$ if desired ($\chi \ge 0$; $\chi = 0$ baseline, so that any owner premium comes from the rent–user-cost wedge), and $s$ = rented size for a renter. Warm glow for a transfer $x \in {b^g, b^e}$ (De Nardi 2004): $$ \phi(x) = \nu,\frac{(x + \underline x)^{1-\gamma}}{1-\gamma}, \qquad \nu \ge 0,\ \underline x > 0, $$ so that transfers are a luxury ($\underline x > 0$) — Part A's linear rule is the limit $\underline x \to 0$, $\gamma \to 1$. Discount factor $\beta$. Preferences are identical across dynasties.

# B.3 Endowments and income

Real labour income at age $j < j_R$: $y_{j} = \bar y_j,z$, with $\bar y_j$ a deterministic age profile (hump-shaped) and $z$ a first-order Markov chain on ${z_1,\dots,z_{n_z}}$ with transition $\Pi_z$ (Rouwenhorst discretisation of $\log z' = \rho_z \log z + \sigma_z\epsilon$). Pension at $j \ge j_R$: $y_j = \varrho,\bar y_{j_R-1},\bar z$ (flat, with $\bar z$ the last working-age state in the baseline). The child's initial state $z_1$ is drawn from the stationary distribution of $\Pi_z$ independently of the parent in the baseline (equal-income design: dynasties differ only through transfers); an intergenerational earnings correlation $\rho_g \in [0,1)$ is a robustness switch (experiment 8). Nominal income is $P_t y_j$; a surprise with wage indexation $\eta_w$ changes real income by $(1+\pi^u)^{-(1-\eta_w)}$ in the surprise period.

# B.4 Housing, tenure and transaction technology

• Owner-occupied houses come in sizes $h \in \mathcal H = {h_1 < \dots < h_{n_h}}$ with $h_1 = \underline h > 0$ the minimum owner-occupied size (the indivisibility that creates an entry threshold). Rented sizes $s \in \mathcal S = [\underline s, \bar s]$ (continuous or a fine grid), $\underline s < \underline h$.
• Maintenance $m,p_t h$ per period keeps the unit intact (owners); landlords pay the same $m$ plus an operating wedge $\phi$ (B.6).
• Buying costs $\kappa_b p_t h$, selling costs $\kappa_s p_t h$ (shares of value). Moving between owned sizes = sell + buy. Renters move costlessly.
• Discrete tenure/adjustment choice each period: for an owner with $(h, b^m)$: stay (keep $h$, keep mortgage, possibly refinance), move (sell, buy $h'$, new mortgage), sell and rent; for a renter: rent ($s$ chosen) or buy ($h$, mortgage). All households must consume housing every period ($s \ge \underline s$).

# B.5 Mortgage contracts (nominal), amortisation, refinancing, constraints

A mortgage originated at $t$ with real principal $B$ ($B^{nom} = P_t B$), nominal contract rate $i_0$ and amortisation length $N$ periods (baseline $N = 5$, 25 years) has the constant nominal payment $$ \text{pay}^{nom} = B^{nom},\frac{i_0}{1 - (1+i_0)^{-N}} . $$ Let $n \in {N, N-1, \dots, 1}$ be the remaining number of payments. The real balance $b^m \equiv B^{nom}{\text{outstanding}}/P_t$ evolves as $$ b^{m\prime} = \frac{b^m(1+i{c}) - \text{pay}t}{1+\pi{t+1}}, \qquad \text{pay}_t = \frac{\text{pay}^{nom}_t}{P_t}, $$ where $i_c$ is the rate currently applying to the contract: $i_c = i_0$ for an FRM; $i_c = i_t$ (the current nominal rate, Fisher) for an ARM, with the nominal payment recomputed each period on the remaining balance and term; for a term-$k$ contract $i_c$ resets every $k$ periods (the Canadian five-year term with $k = 1$ in a 5-year-period calibration is an ARM at the period frequency with an annual-within-period fixed rate; the US 30-year FRM is $k = N$). The real payment of an FRM falls with realised inflation (nominal payment fixed), that of an ARM is re-priced each period: this is the only difference between contracts and it is exactly what Part A's $\omega$ summarises.

Origination constraints (at purchase or refinancing; $y$ = current real income):

• LTV: $B \le (1-\theta),p_t h$;
• payment-to-income (DSTI) at the qualifying rate $i_q = i_t + \Delta_q$ (stress test, $\Delta_q \ge 0$): $\text{pay}^{nom}(i_q)/(P_t y) \le \psi$. Because the qualifying payment is nominal and income is current, expected inflation raises $i_t$ and tightens this constraint: the tilt effect of Part A reappears here as a constraint on leverage.
• No unsecured borrowing: liquid asset $a' \ge 0$.

Refinancing. An owner may at any period replace the outstanding balance by a new loan at the current rate and remaining term reset to $N$, paying $\kappa_f p_t h$, subject to the origination constraints (cash-out allowed up to LTV). Prepayment without a new loan is the case $B' < b^m$.

Default. None in the baseline: the solver enforces $c \ge \underline c > 0$ and reports any state where the payment cannot be met (a "forced sale" rule — sell, repay, rent — is applied in those states and their mass is reported as a diagnostic; if it exceeds 0.1 % of households the calibration is rejected).

# B.6 Rental sector, landlords and segmentation

Rental units are a separate stock $H^r$ (units) owned by competitive risk-neutral landlords who buy rental units at price $p^r_t$, pay maintenance $m p^r_t$ and an operating wedge $\phi p^r_t$ (management, vacancy, tax treatment — the Henderson–Ioannides source of an owner premium), and rent at $\rho_t$: $$ \rho_t = p^r_t,(1 + m + \phi) - \frac{\mathbb E_t,p^r_{t+1}}{R} \qquad (\text{zero expected profit}). $$ Segmentation parameter $\sigma \in [0,1]$: a share $\sigma$ of rental units is structurally distinct (supplied by its own construction curve), a share $1-\sigma$ can be converted one-for-one and costlessly from the owner-occupied stock so that $p^r_t = p_t$ on that margin. Baseline $\sigma = 1$ (full segmentation, as in Part A); $\sigma = 0$ is the no-segmentation benchmark (Proposition 7). Implementation: $p^r_t = \sigma,\tilde p^r_t + (1-\sigma),p_t$ with $\tilde p^r_t$ clearing the structurally rental stock. Rent rigidity $\eta_\rho$: in a surprise period sitting tenants pay $\rho_t (1+\pi^u)^{-(1-\eta_\rho)}$; new tenants pay $\rho_t$; the difference is a transfer from landlords.

# B.7 Construction and the housing stocks

Owner-occupied stock $H_t$ and rental stock $H^r_t$ (units) evolve as $$ H_{t+1} = (1-\delta_H)H_t + I_t, \qquad I_t = \iota,p_t^{\varepsilon}, \qquad H^r_{t+1} = (1-\delta_H)H^r_t + I^r_t, \qquad I^r_t = \iota_r,(p^r_t)^{\varepsilon_r}, $$ with $\delta_H$ physical depreciation of the stock (units retired), $\varepsilon$, $\varepsilon_r$ long-run supply elasticities, $\iota,\iota_r$ scales. Stationary: $H = \iota p^\varepsilon/\delta_H \equiv S(p)$ and the impact (one-period) elasticity is $\varepsilon\delta_H$. Construction profits are rebated lump-sum to a construction-owner class outside the dynasties (equivalently, the construction sector uses land owned by rentiers) so that household budgets are unaffected; this is recorded in the aggregate resource constraint. Near-fixed supply (experiment 7) is $\varepsilon \to 0$.

# B.8 The lender and the government

The lender is a competitive intermediary with nominal assets (mortgage balances) and nominal liabilities (household deposits $a$ and rentier funding at the world real rate $r$). The nominal rate is $1+i_t = R(1+\pi^e_{t+1})$ (plus a constant spread $\varsigma \ge 0$ on mortgages, baseline 0). Its real net worth $A^L_t$ obeys $$ A^L_{t+1} = \frac{(1+i_t),\big(\text{Mortgages}t - \text{Deposits}t\big)}{1+\pi{t+1}} + \text{Rentier flow}, $$ and a surprise $\pi^u{t+1}$ generates the real revaluation $-(\text{Mortgages}t - \text{Deposits}t)(1+i_t)\pi^u{t+1}/((1+\pi^e{t+1})(1+\pi^u_{t+1}))$ for the lender, the mirror image of the households' aggregate real gain on nominal positions (Proposition 8 holds state by state). Baseline closure: rentiers absorb the lender's gains and losses (exogenous $r$); extension: a closed economy where deposits must fund mortgages and $r$ clears the credit market (recorded as experiment 8c).

Government: none in the baseline. Experiment 6 adds a transfer tax $\tau_b$ on $b^g$ and $b^e$ with lump-sum rebate to age-1 households.

# B.9 State variables, timing and the household problem

Individual state at age $j$: $x = (j, z, o, h, b^m, n_m, \iota_c, a, \beta^{rec})$ where $o \in {R, O}$; $h \in \mathcal H$ (0 for renters); $b^m \ge 0$ real mortgage balance; $n_m$ remaining payments; $\iota_c$ contract-rate state (FRM only; in a stationary equilibrium a single value, during a transition "pre-shock rate" vs "post-shock rate": two values); $a \ge 0$ liquid real asset; $\beta^{rec} \in {0, 1}$ whether the estate has already been received (only needed at $j_e$). The aggregate state in the stationary equilibrium is the distribution $\Phi$ over $x$.

Timing within period $t$ for a household of age $j$:

1. Income shock $z$ realised; aggregate surprise $\pi^u_t$ (if any) realised; real prices $(p_t, p^r_t, \rho_t)$ and $i_t$ announced.
2. Transfers: an age-$j_g$ household chooses $b^g$ (paid now to its age-1 child); an age-$J$ household's estate $b^e$ (chosen at $J$ out of terminal wealth after selling the house, net of $\kappa_s$ and the mortgage balance) is paid to its age-$j_e$ child. Age-1 households start with $a_1 = b^g$; age-$j_e$ households add $b^e$ to $a$.
3. Mortgage payment due: pay$t$ (FRM: fixed nominal; ARM: repriced at $i_t$). Real wages and rents indexed per $\eta_w, \eta\rho$ in a surprise period.
4. Discrete choice (stay / move / sell-and-rent / rent / buy / refinance), then $c$, $a'$.
5. Markets clear; stocks evolve.

Bellman equations (stationary prices; $V_j(x)$): $$ V_j(x) = \max_{\text{discrete } \ell \in \mathcal L(x)} V^{\ell}j(x), \qquad V^{\ell}j(x) = \max{c, a', (h', B', s)} \Big{ u(c, s) + \mathbf 1{j=j_g}\phi(b^g) + \beta,\mathbb E\big[V_{j+1}(x') \mid z\big] \Big}, $$ subject to the budget of branch $\ell$ (all real):

• Renter stays renter ($\ell = RR$): $c + \rho s + a' = y_j + a,R_a - \mathbf 1_{j=j_g} b^g$, with $R_a \equiv (1+i_{t-1})/(1+\pi_t)$ the realised real deposit return; $x' = (j+1, z', R, 0, 0, 0, \cdot, a')$.
• Renter buys ($\ell = RO$): $c + (1 + \kappa_b + m),p,h' - B' + a' = y_j + a R_a - \mathbf 1_{j=j_g} b^g$, with $B' \le (1-\theta)p h'$, DSTI, $a' \ge 0$; $s = (1+\chi)h'$; $x' = (j+1, z', O, h', b^{m\prime}(B'), N, i_t, a')$.
• Owner stays ($\ell = OO$): $c + \text{pay}(b^m, n_m, i_c) + m p h + a' = y_j + a R_a - \mathbf 1_{j=j_g} b^g$; $s = (1+\chi)h$; $x' = (j+1, z', O, h, b^{m\prime}, n_m - 1, i_c, a')$. Refinancing ($\ell = OF$): replace pay by the new schedule with $B'$, pay $\kappa_f p h$, constraints at origination; cash-out $B' - b^m(1+i_c)$ enters the budget.
• Owner moves ($\ell = OM$): sell at $(1-\kappa_s)p h$, repay $b^m(1+i_c)$, then as $RO$ with $h'$.
• Owner sells and rents ($\ell = OR$): $c + \rho s + a' = y_j + a R_a + (1-\kappa_s)p h - b^m(1+i_c) - \mathbf 1_{j=j_g} b^g$.
• Terminal condition at $j = J$: the house is sold, the balance repaid, and $V_{J+1} \equiv \phi(b^e)$ with $b^e = a' $ (terminal wealth not consumed); with survival risk, the same applies at death at any age.
• Feasibility: $c \ge \underline c$, $s \ge \underline s$, $a' \ge 0$; if no feasible branch exists the forced-sale rule applies.

The transfer $b^g$ at age $j_g$ is a continuous choice with first-order condition $u_c = \phi'(b^g)$ (interior) and corner $b^g = 0$; the estate $b^e$ at $J$ is a continuous choice between $c_J$ and $b^e$.

Nominal vs real in the solver. The solver works in real variables; $B^{nom}$ never appears explicitly, only $b^m$ and the real payment. Inflation enters through (i) $R_a$ and the real payment of FRMs (both divided by $1+\pi_t$), (ii) the nominal rate in the DSTI constraint and in ARM repricing, (iii) wage and rent indexation parameters. Reporting must always give both $B^{nom}/P$ (real debt) and the house-price index $P p$ separately from the CPI.

# B.10 Equilibrium definitions

Stationary recursive competitive equilibrium (SRCE) with constant inflation $\bar\pi$ and exogenous $r$: value functions ${V_j}$, policies, prices $(p^, p^{r}, \rho^)$, nominal rate $i^ = R(1+\bar\pi) - 1$, stocks $(H^, H^{r})$, lender net worth $A^{L*}$ and a distribution $\Phi^*$ such that

1. policies solve the household problems given prices;
2. owner-occupied market clears: $\int h,\mathbf 1{o = O},d\Phi^* = H^* = \iota (p^*)^{\varepsilon}/\delta_H$;
3. rental market clears: $\int s,\mathbf 1{o = R},d\Phi^* = H^{r*}$, with $\rho^* = p^{r*}(1+m+\phi - 1/R)$ and $H^{r*} = \iota_r (p^{r*})^{\varepsilon_r}/\delta_H$ on the segmented margin ($p^{r*} = p^*$ on the converted margin);
4. dynasty consistency: the distribution of $a_1 = b^g$ across age-1 households equals the distribution of $b^g$ chosen by age-$j_g$ households, and likewise for $b^e$ at $j_e$ (the child's $z_1$ is drawn independently in the baseline);
5. $\Phi^*$ is invariant under the policies, the Markov chain and the dynasty link;
6. the lender earns zero expected real profit; construction profits and the lender's flows go to the outside classes;
7. aggregate resource constraint: $C + m p^* H^* + (m+\phi)p^{r*}H^{r*} + \text{transaction costs} + p^* I + p^{r*} I^r = Y + r,\text{(net foreign position)}$, reported as a residual.

Transition after an unanticipated inflation shock (MIT shock). Start from the SRCE. At $t = 0$ the inflation path ${\pi_t}{t \ge 0}$ is revealed (E1: $\pi_0 = (1+\bar\pi)(1+\pi^u)-1$, then $\bar\pi$; E2: elevated for $k$ periods), with $\pi^e_t = \pi_t$ for $t \ge 1$ (perfect foresight after the surprise), $\pi^e_0 = \bar\pi$ (the surprise). A transition equilibrium is a path ${p_t, p^r_t, \rho_t, i_t, H_t, H^r_t, \Phi_t}{t=0}^{T}$ with $T$ large such that households optimise with perfect foresight, markets clear every period, and the economy is within tolerance of the SRCE at $T$. Pre-shock FRM cohorts carry $\iota_c$ = old rate; new originations use the new $i_t$. Partial-equilibrium versions (prices fixed at $p^,\rho^$) are reported separately and are never labelled GE.

# B.11 Aggregation and accounting identities to be tested

• Budget identity per household and period (sum of uses = sum of sources), state by state, in the simulation (tests/).
• Debt law of motion: $b^{m\prime}(1+\pi') = b^m(1+i_c) - \text{pay}$; after $N$ payments $b^m = 0$ to machine precision.
• Zero-sum revaluation: $\sum_{\text{households}} \Delta(\text{real nominal position}) + \Delta A^L = 0$ in the surprise period.
• Market clearing residuals $|{\int h,d\Phi - H}|/H$, $|{\int s,d\Phi - H^r}|/H^r$ below tolerance; Bellman residuals on off-grid points; transfer consistency (item 4).
• Wealth decomposition: net worth $= a + \mathbf 1_O,[(1-\kappa_s)p h - b^m]$; owner–renter gap by age, by transfer quantile, holding $z$ history fixed (equal-income comparison, experiment 2).

# B.12 Recommended numerical choices (for Phase 3)

# B.13 Mapping between Part A and Part B