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1% ============================================================================2\section{Surprise inflation, mortgage contracts and a real-rate shock}3\label{sec:inflation}4% ============================================================================56\subsection{Design}7\label{subsec:infl_design}89The economy starts in the baseline stationary equilibrium and at $t = 0$ the price level turns out 10\% above expectation over the shock period (a level surprise, about two percentage points per year for five years), or 10\% above in each of two consecutive periods with the second known at $t = 0$ (a persistent surprise); inflation then returns to target and expectations are correct thereafter. The transition is solved with perfect foresight of the post-shock price path over 70 years, durable stocks, and the three contracts: a fixed-rate mortgage whose nominal payment is fixed for the remaining term, the five-year term whose rate resets at the next period boundary, and a stylised adjustable-rate mortgage repriced within the period so that its balance is not eroded. Each transition is reported next to its partial-equilibrium counterpart, the same shock with prices fixed at the stationary values, and is labelled with its maximum market-clearing residual: the fixed-rate level surprise stalled at $\FrmLevelMaxRes$ after a Jacobian refresh and is labelled as not converged, while the term and adjustable level surprises and the three persistent surprises converged to $\TermLevelMaxRes$--$\FrmPersMaxRes$. For a level surprise the fixed-rate and five-year-term economies are identical at the five-year frequency, since both erode the balance once and both face the same post-shock rate, and every impact statistic of the two runs agrees to two decimals ($p_0 = \FrmLevelPzero$ against $\TermLevelPzero\%$); the pair therefore bounds the solver's price noise at about 0.02 percentage points. The price paths are back within 0.1\% of the stationary state after \FrmLevelPriceBackYears{} years and the residual at the terminal date is $\FrmLevelResT$. Zero-sum revaluation holds to machine precision.1011\input{tables/tab06_inflation}1213\subsection{Impact: the real house price falls}14\label{subsec:infl_impact}1516\begin{figure}[htbp]17\centering18\includegraphics[width=\textwidth]{fig08_inflation_transition.pdf}19\caption{Perfect-foresight transitions after a 10\% unanticipated price-level surprise (general equilibrium unless labelled): real house price (a), real rent (b), ownership rate (c) and real mortgage debt (d), in deviation from the stationary state, for the three contracts and the two persistence profiles; the dotted line in (c) is the fixed-rate level surprise with prices fixed. The fixed-rate level run is reported with its residual. Source: \texttt{results/data/inflation/*\_path.csv}.}20\label{fig:irf}21\end{figure}2223Table~\ref{tab:inflation} and Figure~\ref{fig:irf} give the central negative result of the paper. A level surprise lowers the real house price on impact by \FrmLevelPzeroAbs\% under the fixed-rate and term contracts and by \ArmLevelPzeroAbs\% under the adjustable contract, and lowers the real rent by \FrmLevelRhoZeroAbs{} and \ArmLevelRhoZeroAbs\%; a persistent surprise lowers the price by \FrmPersPzeroAbs\% (fixed rate), \TermPersPzeroAbs\% (term) and \ArmPersPzeroAbs\% (adjustable). Proposition~\ref{prop:impact} predicts a rise. The reason is the one stated in Corollary~\ref{cor:creditor} and visible in the aggregate balance sheet of Section~\ref{sec:baseline}: households are net nominal creditors. The surprise erodes real mortgage debt by \FrmLevelErosion{} per unit of age mass under the fixed-rate contract (and not at all under the adjustable one) but erodes real deposits by \FrmLevelDepLoss, so the lender gains \FrmLevelLender{} (\ArmLevelLender{} under the adjustable contract) and aggregate household wealth falls; the demand for housing, on both the extensive and the intensive margin, falls with it. The direction is the corollary's; the magnitude of the rent response is not, because the corollary's unit demand makes the two markets mirror images of a fixed number of units, whereas here the wealth loss lowers the demand for rented services as well, so both prices fall.2425The ownership rate barely moves in general equilibrium, by \FrmLevelDOwnGE{} percentage points on impact for the fixed-rate level surprise against \FrmLevelDOwnPE{} with prices fixed, \ArmLevelDOwnGE{} against \ArmLevelDOwnPE{} for the adjustable contract, and \FrmPersDOwnGE{} against \FrmPersDOwnPE{} for the persistent fixed-rate surprise: the price adjustment absorbs \ArmLevelAbsorbPct--\FrmLevelAbsorbPct\% of the partial-equilibrium ownership response, the composition result of Corollary~\ref{cor:fixed} in a durable-stock economy whose impact supply elasticity is $\varepsilon\delta_H\approx 0.05$. The price paths return with \FrmLevelPriceSignChanges{} sign change for the level surprise and \FrmPersPriceSignChanges{} for the persistent one, all after the impact and all smaller than 0.1\%; the cohort-crowding oscillation found in a myopic simulation of the benchmark does not appear under perfect foresight with durable stocks.2627The persistent surprise differs from the level surprise through the nominal rate expected for the second period, which jumps from \FrmLevelNominalRateZero{} to \FrmPersNominalRateZero{} per period. Real debt falls on impact by \FrmPersDebtZeroPctAbs\% under the fixed-rate contract, whose nominal payments are fixed through both inflation periods, by \TermPersDebtZeroPctAbs\% under the term contract and by \ArmPersDebtZeroPctAbs\% under the adjustable contract, where the fall is lower origination rather than erosion: the qualifying payment at the higher nominal rate tightens the payment-to-income cap, ownership falls by \FrmPersDOwnPEAbs{} to \ArmPersDOwnPEAbs{} percentage points with prices fixed and by \FrmPersDOwnGEAbs{} to \ArmPersDOwnGEAbs{} in general equilibrium. This is Proposition~\ref{prop:tilt}'s tilt operating along the transition through the nominal constraint, and it is why the persistent surprise lowers the price four times more than the level surprise.2829\subsection{Redistribution and the contract ordering}30\label{subsec:infl_redistribution}3132\begin{figure}[htbp]33\centering34\includegraphics[width=\textwidth]{fig09_inflation_redistribution.pdf}35\caption{Redistribution in the shock period of the fixed-rate level surprise: revaluation of nominal positions by age and tenure (a), consumption at impact relative to the no-shock path (b), and consumption-equivalent welfare of households alive at the shock by age and tenure (c), general equilibrium. Source: \texttt{results/data/inflation/frm\_level\_impact.csv}, \texttt{frm\_level\_welfare.csv}.}36\label{fig:redistribution}37\end{figure}3839Figure~\ref{fig:redistribution} shows who gains and who loses within the fixed-rate level surprise. Owners aged 30--44 gain on their nominal position, up to \FrmLevelOwnerYoungGainMax{} of a five-year income, and raise impact consumption by \FrmLevelOwnerYoungConsMin{} to \FrmLevelOwnerYoungConsMax\%; owners aged 60--74, deposit-rich and debt-free, lose \FrmLevelOwnerOldLossMin{} to \FrmLevelOwnerOldLossMax{} and cut consumption by \FrmLevelOwnerOldConsMinAbs{} to \FrmLevelOwnerOldConsMaxAbs\%; renters lose \FrmLevelNetRAbs{} on average and cut consumption by \FrmLevelConsRAbs\%. In lifetime terms, owners with a mortgage gain \FrmLevelCEVOM\% of consumption (\FrmLevelBetterOMPct\% of them are better off), owners without a mortgage lose \FrmLevelCEVONMAbs\%, renters lose \FrmLevelCEVRAbs\% and all households alive at the shock lose \FrmLevelCEVAllAbs\% on average, against \FrmLevelCEVAllPEAbs\% with prices fixed: the price response barely changes the incidence because prices move little. This is the age and tenure pattern of \citet{doepke2006inflation} and \citet{meh2011inflation}, with the model-specific addition that the losers include the young renters saving for a down payment.4041The contract ordering of Proposition~\ref{prop:accounting} is confirmed. For a level surprise the fixed-rate and term contracts give identical gains to mortgaged owners (\FrmLevelCEVOM\%) while the adjustable contract turns the gain into a loss (\ArmLevelCEVOM\%, \ArmLevelBetterOMPct\% better off), since no balance is eroded and the deposit loss remains. For a persistent surprise, mortgaged owners gain \FrmPersCEVOM\% under the fixed-rate contract (\FrmPersBetterOMPct\% better off), \TermPersCEVOM\% under the five-year term (\TermPersBetterOMPct\%) and lose \ArmPersCEVOMAbs\% under the adjustable contract, the ordering fixed rate above term above adjustable, with debt-free owners losing about \FrmPersCEVONMAbs\% and renters about \FrmPersCEVRAbs\% in every case. The Canadian renewable term is thus intermediate, as the institutional reading of Section~\ref{sec:literature} suggested, and the gap between it and the long fixed-rate contract opens only when the surprise outlasts the term.4243\subsection{Wage indexation, rent rigidity and a real-rate response}44\label{subsec:infl_variants}4546Three variants test the conditions of Proposition~\ref{prop:accounting}. With unindexed nominal wages ($\eta_w = 0$; the run stalled at $\UnindexedMaxRes$ and is labelled) the one-period real wage loss of \UnindexedWageAll{} per household dwarfs the nominal revaluation, the price falls by \UnindexedPzeroAbs\% and the rent by \UnindexedRhoZeroAbs\%, consumption falls by \UnindexedConsAllAbs\% on impact, and every group loses, mortgaged owners by \UnindexedCEVOMAbs\% with \UnindexedBetterOMPct\% better off: condition \eqref{eq:gaincond} fails for every indebted owner once the real-income loss exceeds the debt erosion. With rigid rents for sitting tenants ($\eta_\rho = 0$), incumbents receive a one-period transfer of \RigidRentsRentGainR{} from landlords, the market rent for new tenants rises by \RigidRentsRhoZero\% on impact because incumbents stay put, ownership falls by \RigidRentsDOwnPEAbs{} percentage points with prices fixed but only \RigidRentsDOwnGEAbs{} in general equilibrium, and the renters' welfare loss improves only from \FrmLevelCEVR{} to \RigidRentsCEVR\%: the hedge of \citet{pallotti2024who} is present but, at the five-year frequency, small. With a real-rate response to the persistent surprise ($\phi_r = 0.5$, a rise of five percentage points per period in the two surprise periods), the rate response dominates: the price falls by \FrmTaylorPzeroAbs\% (fixed rate) and \TermTaylorPzeroAbs\% (term), mortgaged owners gain \FrmTaylorCEVOM\% under the fixed-rate contract but only \TermTaylorCEVOM\% under the five-year term (\FrmTaylorBetterOMPct{} against \TermTaylorBetterOMPct\% better off), because the renewal at the higher real rate claws back most of the term borrowers' gain, exactly as Proposition~\ref{prop:accounting} states for $\phi_r>0$; debt-free owners and renters are cushioned by the higher deposit return, and the average household gains \FrmTaylorCEVAll\% under the fixed-rate contract and loses \TermTaylorCEVAllAbs\% under the term. The distinction between the Canadian term and the long fixed-rate contract therefore matters precisely when the surprise is persistent and monetary policy reacts.4748\subsection{Segmentation, supply elasticity and expectations}49\label{subsec:infl_structure}5051Three further variants check the structural assumptions of Proposition~\ref{prop:impact}. Without segmentation ($\sigma = 0$, one stock) the price falls by \NoSegPzeroAbs\% against \FrmLevelPzeroAbs\% with segmented stocks and the rent by \NoSegRhoZeroAbs\%, with ownership moving by \NoSegDOwnGE{} percentage points in general equilibrium against \NoSegDOwnPE{} with prices fixed and identical welfare: the neutrality of Proposition~\ref{prop:noseg} does not carry over, for the reason given in Remark~\ref{rem:noseg_scope}, and segmentation itself adds only about a seventh to the price response. With near-fixed long-run supply ($\varepsilon = \varepsilon_r = 0.05$; the run stalled at $\FixedSupplyMaxRes$ and is labelled) the price falls by \FixedSupplyPzeroAbs\% against \FrmLevelPzeroAbs\%, and ownership moves by \FixedSupplyDOwnGE{} percentage points against \FixedSupplyDOwnPE{} at fixed prices: the long-run elasticity barely matters on impact because the stock is durable and the shock-period supply is essentially fixed in both cases, so the price absorbs the demand shift and the composition result of Corollary~\ref{cor:fixed} is what the numbers show. Under static expectations (Assumption~\ref{ass:static}, with stationary continuation values and prices clearing period by period) the price falls by \StaticPzeroAbs\% against \FrmLevelPzeroAbs\% under perfect foresight, ownership moves by \StaticDOwnGE{} against \FrmLevelDOwnGE{} percentage points, welfare agrees to two decimals and the gift-age transition probability $q_{RO}$ falls from \FrmLevelqROsevenSS{} to \StaticqROsevenZero{} in both: the expectation scheme is immaterial for the size of the effects in this calibration, and the sign of the price response is negative under both. Mobility on impact is as Proposition~\ref{prop:impact} predicts in direction and negligible in size, $q_{RO}$ at the gift age falling to \FrmLevelqROsevenZero{} and $\lambda_7$ rising from \FrmLevelLamSevenSS{} to \FrmLevelLamSevenZero; the age-1 threshold at the median state rises from \FrmLevelThrSS{} to \FrmLevelThrZero{}, by \FrmLevelThrChangePct\%, through the erosion of the savings of would-be buyers rather than through a higher price, the third channel named in Remark~\ref{rem:impact_scope}.5253\subsection{A real-rate shock and leverage}54\label{subsec:rateshock}5556The general-equilibrium counterpart of a house-price fall is a persistent rise in the real rate: two percentage points per year for twenty years, unanticipated at $t = 0$, from the stationary equilibrium of each of three down-payment regimes, with fixed-rate contracts keeping their origination rate. The runs for $\theta = 0.20$ and $0.40$ converged ($\RateTwentyMaxRes$ and $\RateFortyMaxRes$); the run for $\theta = 0.05$ diverged from a cold start and, after a warm start and a damped t\^atonnement pre-phase, stalled at $\RateFiveMaxRes$, so its numbers are indicative only. The real house price falls on impact by \RateTwentyPzeroAbs\% at $\theta = 0.20$ and \RateFortyPzeroAbs\% at $\theta = 0.40$ (\RateFivePzeroAbs\% in the unconverged run), overshoots by \RateTwentyPriceMaxPos\% when the rate returns to 2\% twenty years later, and the rent falls by \RateTwentyRhoZeroAbs\%. Ownership on impact moves by \RateTwentyDOwnGE{} and \RateFortyDOwnGE{} percentage points in general equilibrium against \RateTwentyDOwnPE{} and \RateFortyDOwnPE{} with prices fixed: the ranking of the partial-equilibrium responses across $\theta$ (\RateFiveDOwnPE, \RateTwentyDOwnPE, \RateFortyDOwnPE{} for 5, 20 and 40\%) is the leverage-exposure ordering of Proposition~\ref{prop:leverage}, and general equilibrium removes it. Owners lose \RateTwentyNWOAbs{} of a five-year income of net worth on impact and cut consumption by \RateTwentyConsOAbs\%; renters' net worth rises slightly and their consumption falls by \RateTwentyConsRAbs\%. Welfare is nevertheless positive on average, \RateTwentyCEVAll\% for households alive at the shock, because net nominal creditors gain from the higher real return, which dominates the housing-wealth loss: mortgaged owners gain \RateTwentyCEVOM\% (\RateTwentyBetterOMPct\% better off, shielded by the fixed rate while their deposits earn more), debt-free owners \RateTwentyCEVONM\% and renters \RateTwentyCEVR\%. The price fall lowers the entry threshold (from \RateFortyThrSS{} to \RateFortyThrZero{} at the median state for $\theta = 0.40$) and raises upward mobility ($q_{RO}$ at 35--39 from \RateTwentyqROthreeSS{} to \RateTwentyqROthreeZero{} at $\theta = 0.20$). Leverage thus amplifies the housing-equity loss, as the accounting says, but with fixed-rate contracts, net creditor positions and no default the channel operates through consumption and entry rather than through foreclosure.57