The Model

In our model, a market is defined by which state (j ∈ J) it is in, whether it is in urban or rural areas (u), and which year it is in (t). Therefore, for each year we have 2J segregated markets.6 _{There is no interaction across markets. The supply side of the housing markets is}

fairly standard. For each market, the housing supply is a deterministic function of the housing price. The aggregate housing demand in each market can be computed by combining the housing demand of different age groups and the age structure of the population. While Mankiw and Weil (1989) focus on the changes in the age structure by keeping the housing demand for each age group fixed, our framework captures the possibility that longer life expectancy can lead to an increase in the housing demand for senior households. In other words, when people live longer, they keep their houses longer.

To capture the impact of life expectancy, or equivalently survival probabilities, on the hous- ing demand of individual households, we consider a model where the average housing demand for households from age group a, which we denote Fa, depends not only on the household

incomewaand the housing priceP, but also on the survival probabilitySa:

Fa = Fa(wa,Sa,P). (4.1)

Then, the aggregate demand of all households is described as follows:

D=X

a

FaNa, (4.2)

whereNais the total number of households in age groupa.

Demand

Consider the housing demand for households in state jand areau(urban area: u= 1; rural area:

u = 0) at timet. For this market, we assume that households from the same age groupahave identical income wa,j,u,t and survival probability Sa,t.7 Households choose whether to buy a

house. Houses are identical, and each household can only buy one house.8 For a representative household of agea, the probability of owning a housing unit,λa,j,u,t, is given by:

6_{This setting allows us to estimate the model through cross-state and cross-time variation.} 7_{The assumption that the survival probabilityS}

a,t does not vary by state and area is made because of data

availability issues.

8_{This assumption is well justified by the data. According to the American Community Survey, the fraction of}

λa,j,u,t =

exp(β0+β1log(wa,j,u,t)+β2Sa,t−β3log(Pj,u,t)+γa+Et+ηa,j,u,t)

1+exp(β0+β1log(wa,j,u,t)+β2Sa,t−β3log(Pj,u,t)+γa+Et +ηa,j,u,t)

. (4.3)

In this equation,Pj,u,t is the housing price in state j, areauat timet. γais the age fixed effect. Et is the time fixed effect, which may capture the macroeconomic environment that affects the

housing demand, such as mortgage interests and access to mortgages.ηa,j,u,t is an idiosyncratic

shock.

Equation 4.3 shows the extensive margin of the housing demand, which is consistent with a standard logit model (see, e.g. Gyourko and Linneman, 1996).9 As households are ex ante identical,λa,j,u,talso corresponds to the homeownership rate of age groupa.

Then, the aggregate housing demand in state jand areauat timetis given by:

Dj,u,t = X a Fa,j,u,tNa,j,u,t =X a λa,j,u,tNa,j,u,t, (4.4)

whereNa,j,u,t is the number of households of age groupaliving in state jand areauat timet.

Our specification of the demand side can be considered as a reduced-form approximation of fully specified life-cycle models used in the literature (see e.g. Chambers, Garriga, and Schlagenhauf, 2009). We choose this parsimonious approach mainly for two reasons. First, the economy in our model consists of many segregated markets which allows us to explore both the cross-sectional variation and cross-time variation to estimate the model. Hence, a relatively simpler demand side helps keep the computation and estimation of our model more computa- tionally tractable. Second, this parsimonious specification allows for a more transparent and direct investigation of the impact of key demographic variables on housing demand, and con- sequently equilibrium housing prices. For example, to examine the importance of survival probabilitySa,t on the housing demand profile, we can estimate Equation 4.3 using cross-state

and cross-time variations inλa,j,u,t and examine the economic and statistical significance of the

relevant coefficientsβ2.

9_{We abstract away from the intensive margin of the housing demand by assuming that all owner-occupied}

units are identical. This assumption is made due to a lack of appropriate measures of housing quantity and quality in the Census and the American Community Survey.

Supply

For state jand areauat timet, housing supply is determined through the following fixed effect model.

log(Pj,u,t)= αu,0+αu,1log(Dj,u,t)+θj,u+j,u,t, (4.5)

whereαu,1is the price elasticity of supply that may differ between rural and urban areas. θj,uis

an area fixed effect which may capture permanent housing market conditions, such as regula- tion or construction costs, for state jand areau. j,u,t is an idiosyncratic shock.