3.1. Lenders

It is assumed that a set of households displays a high discount factor relative to other households. Their preferences take the standard following form:

(1) \begin{equation} \max E_0^P \sum _{t=0}^{\infty } \beta _P^t d_t [\,\ln(C_{t,P})+j \ln(H_{t,P}) + \psi \ln(1-N_{t,P})]. \end{equation}

Patient households thus value consumption $C_{t,P}$ , housing services provided by real estate assets $H_{t,P}$ , and leisure hours equal to $1-N_{t,P}$ , where $N_{t,P}$ are working hours. Patient households discount future periods with the discount factor $\beta _P$ , $j$ is the weight allocated to housing services in the utility function, and $\psi$ is the weight allocated to leisure. $d_t$ follows an autoregressive process in the form of:

(2) \begin{equation} \ln(d_t)= \rho _d \ln(d_{t-1}) + \varepsilon _{d,t}, \end{equation}

where $\rho _d\lt 1$ , and $\varepsilon _{d,t}$ is the discount factor shock which follows a normal distribution with mean zero and variance $\sigma _d$ . The interpretation of this shock is that time preferences are time-varying: patient households can suddenly display more or less preference for current consumption, housing services, and leisure. This shock is introduced to mimic a context of higher willingness to lend, independently of borrowers’ net worth. An exogenous decrease in the current period discount factor of lenders thus increases the credit supply, independently of borrowers’ ability to pay back the debt. Expectations are evaluated under the probability measure $P$ and can differ from rational expectations.

The intertemporal flow of funds constraint of patient households writes as follows:

(3) \begin{equation} C_{t,P}+q_t H_{t,P} + B_t= w_{t} N_{t,P} + R_{t-1} B_{t-1} + q_t H_{t-1,P}, \end{equation}

where $q_t$ is the price of houses, $B_t$ is the debt held by patient households, $R_t$ is the (gross) interest rate on debt, and $w_t$ is the wage. Housing assets are traded in each period. The intertemporal first-order conditions with respect to housing, debt, and hours worked are standard, except that the time preference shock is included:

(4) \begin{equation} d_t \frac{1}{C_{t,P}} q_t= \beta _P E_t^P\left [\frac{1}{C_{t+1,P}} q_{t+1} d_{t+1} \right ] + j d_t \frac{1}{H_{t,P}}. \end{equation}

(5) \begin{equation} \frac{d_t}{C_{t,P}}= \beta _P E_t^P\left [\frac{d_{t+1}}{C_{t+1,P}} R_t \right ]. \end{equation}

(6) \begin{equation} \frac{w_t}{C_{t,P}}= \frac{\psi }{1-N_{t,P}}. \end{equation}

3.2. Entrepreneurs

Entrepreneurs own the capital stock and maximize the intertemporal utility of consumption streams:

(7) \begin{equation} \max E_0^P \sum _{t=0}^{\infty } \beta _F^t [\,\ln(C_{t,F})], \end{equation}

subject to the following flow of funds constraint:

(8) \begin{equation} C_{t,F}+ q_t H_{t,F} + R_{t-1} B_{t-1,F} + w_t N_t + I_t = Y_t + B_{t,F} + q_t H_{t-1,F}, \end{equation}

where $\beta _F$ is the entrepreneurs’ discount factor, $C_{t,F}$ is consumption, $H_{t,F}$ represents real estate holdings, $B_{t,F}$ is debt, $N_t$ is labor demand, $I_t$ is investment, and $Y_t$ is output, which is produced according to the following production function:

(9) \begin{equation} Y_t= A_t K_{t-1}^{\alpha } H_{t-1}^v N_t^{1-\alpha -v}. \end{equation}

Total factor productivity $A_t$ follows a standard $AR(1)$ process in the log:

(10) \begin{equation} \ln(A_t)= \rho _a \ln(A_{t-1})+ \varepsilon _{a,t}, \end{equation}

where $\varepsilon _{a,t}$ follows a normal distribution with mean zero and variance $\sigma _a$ . Adjusting capital too fast is assumed to be costly, and the capital accumulation equation takes the standard following form under capital adjustment costs (Hayashi (Reference Hayashi1982)):

(11) \begin{equation} K_t=I_t+(1-\delta )K_{t-1} - \frac{\phi }{2}\left (\frac{I_t}{K_{t-1}}-\delta \right )^2 K_{t-1}, \end{equation}

where $K_t$ is the capital stock, $\delta$ is the capital depreciation rate, and $\phi$ is a parameter governing the size of the capital adjustment cost. Entrepreneurs can borrow a limited amount of debt and face a collateralized borrowing constraint in which housing assets play the role of pledgeable assets:

(12) \begin{equation} B_{t,F} \leq m E_t^P \left [q_{t+1} \frac{H_{t,F}}{R_t}\right ], \end{equation}

where the term $E_t^P\big[q_{t+1} \frac{H_{t,F}}{R_t}\big]$ represents expected future asset value and $m$ is the loan-to-value ratio. The lender can recover only some fraction of the pledgeable assets in case of default due to asymmetry of information between lenders and borrowers, implying that $m\lt 1$ . The first-order conditions for firms with respect to labor, debt, and real estate assets write:

(13) \begin{eqnarray} w_t= \frac{(1-\alpha -v)Y_t}{N_t}, \end{eqnarray}

(14) \begin{equation} \frac{1}{C_{t,F}}= \beta _F E_t^P \left [\frac{1}{C_{t+1,F}}\right ] R_t + \mu _{F,t}, \end{equation}

and

(15) \begin{equation} \frac{q_t}{C_{t,F}}= \beta _F E_t^P \left [\frac{1}{C_{t+1,F}} \left (q_{t+1} + \frac{v Y_{t+1}}{H_{t,F} } \right ) \right ] +\mu _{F,t} m E_t^P\left [\frac{q_{t+1}}{R_t}\right ], \end{equation}

where $\mu _{F,t}\geq 0$ is the Lagrange multiplier associated with the borrowing constraint. The complementary slackness condition writes

(16) \begin{equation} \mu _{F,t} \left [B_{t,F} - m E_t^P \left [q_{t+1} \frac{H_{t,F}}{R_t}\right ] \right ]=0. \end{equation}

The first-order condition with respect to labor is standard, except that the share of labor in the production function depends not only on the share of capital but also on the share of real estate assets in the production function. The Lagrange multiplier $\mu _{F,t}$ associated with the borrowing constraint appears in the previous two equations, which shows that financial frictions act as an intertemporal wedge in the first-order conditions by comparison to standard first-order conditions. Note that in the non-stochastic steady state, the Lagrange multiplier associated with the borrowing constraint of firms $\mu _F$ is equal to $\big(\frac{\beta _P-\beta _F}{\beta _F}\big) \frac{1}{C_F}$ . Therefore, the discount factor of lenders must be strictly higher than the discount factor of borrowers (i.e., $\beta _F\lt \beta _P$ ) to ensure that the Lagrange multiplier associated with the borrowing constraint is strictly positive. The combination of the first-order conditions with respect to capital and to investment yields

(17) \begin{align} &\frac{1}{C_{t,F} \left (1-\phi \left(\frac{I_t}{K_{t-1}}-\delta \right)\right )} \nonumber \\ &\quad =E_t^P\left [\!\beta _F \frac{1}{C_{t+1,F}} \!\left (\!\frac{\alpha Y_{t+1}}{K_t}+ \frac{1}{1-\phi \left(\!\frac{I_{t+1}}{K_t}-\delta \!\right)}\left(\!1-\delta - \frac{\phi }{2}\left (\!\frac{I_{t+1}}{K_t}-\delta \right )^2 + \phi \!\left(\! \frac{I_{t+1}}{K_t}-\delta \!\right) \frac{I_{t+1}}{K_t} \!\right) \!\right ) \!\right ]\!. \end{align}

3.3. Impatient households

The preferences of impatient households are similar to those of patient households except that their time preference rate $\beta _I$ differs ( $\beta _I\lt \beta _P$ ). This assumption, which is standard in a borrower–saver model, makes impatient households willing to borrow rather than lend. The maximization program of impatient households is thus the following:

(18) \begin{equation}\max E_0^P \sum _{t=0}^{\infty } \beta _I^t [\,\ln\,(C_{t,I})+j \ln(H_{t,I}) + \psi \,\ln\,(1-N_{t,I})]\end{equation}

s.t.

(19) \begin{equation} C_{t,I}+R_{t-1}B_{t-1,I}+q_t H_{t,I}= w_t N_{t,I} + q_t H_{t-1,I} + B_{t,I} \end{equation}

(20) \begin{equation} B_{t,I} \leq m E_t^P\left [q_{t+1} \frac{H_{t,I}}{R_t}\right ]. \end{equation}

All variables indexed by $I$ for impatient households are equivalent to similar variables indexed by $P$ for patient households. Impatient households face a borrowing constraint similar to that of entrepreneurs. The first-order conditions with respect to housing, labor supply, and debt write

(21) \begin{equation} \frac{q_t}{C_{t,I}} = \beta _I E_t^P \left [\frac{q_{t+1}}{C_{t+1,I}} \right ] + j\frac{1}{H_{t,I}} + \mu _{I,t} m E_t^P\left [\frac{q_{t+1}}{R_t}\right ], \end{equation}

(22) \begin{equation} \frac{w_t}{C_{t,I}}= \frac{\psi }{1-N_{t,I}}, \end{equation}

(23) \begin{equation} \frac{1}{C_{t,I}}=\beta _I E_t^P\left [\frac{1}{C_{t+1,I}} R_t\right ] + \mu _{I,t}, \end{equation}

where $\mu _{I,t}\geq 0$ is the Lagrange multiplier associated with the borrowing constraint of the impatient households. The complementary slackness condition writes

(24) \begin{equation} \mu _{I,t} \left [B_{t,I} - m E_t^P \left [q_{t+1} \frac{H_{t,I}}{R_t}\right ] \right ]=0. \end{equation}

3.4. Market clearing

Finally, the model is closed by adding market clearing conditions and standard transversality conditions. The market clearing condition on the goods market is

(25) \begin{equation} Y_t= I_t + C_{t,P}+ C_{t,F} + C_{t,I}. \end{equation}

Bonds are assumed to be in zero-net supply:

(26) \begin{equation} B_t= B_{t,F} + B_{t,I}. \end{equation}

The equilibrium condition on the labor market is

(27) \begin{equation} N_t=N_{t,P}+N_{t,I}. \end{equation}

Finally, the market clearing condition on the housing market is

(28) \begin{equation} H_{P,t}+H_{F,t}+H_{I,t}=1. \end{equation}

The model is solved under the assumption that the borrowing constraint of both entrepreneurs and impatient households is binding (and thus that the associated Lagrange multipliers $\mu _{F,t}$ and $\mu _{I,t}$ are strictly positive). When numerically solving the model, I verify that this assumption holds true in all simulations.

4.1. Perceived process for house price growth

Following a recent trend in the literature on learning regarding asset prices (Adam et al. (2012), Adam et al. (Reference Adam, Marcet and Nicolini2016), Adam et al. (2017), Winkler (Reference Winkler2020)), I now assume that agents in the economy do not understand the endogenous process through which house prices form. The actual equilibrium price results from the equalization of the demand for housing of the three sectors in the model to the exogenous supply of housing. Instead, in the learning model, agents observe house prices realizations and try to determine whether the actual evolution is permanent or temporary. They thus try to evaluate the persistence of the current variation in house prices based on their past experience. Therefore, instead of taking into account the housing market clearing condition, atomistic market participants believe that logged house prices follow an exogenous process which takes the following form:

(29) \begin{equation}\ln(q_{t})-\ln(q_{t-1})=\ln(\mu _{t})+\ln(\eta _{t}),\end{equation}

where $\eta _t$ is a temporary disturbance, and where the time-varying persistent component $\mu _t$ follows the process:

(30) \begin{equation}\ln(\mu _t)=\ln(\mu _{t-1})+ \ln(\nu _t),\end{equation}

where $\nu _t$ is an additional disturbance. Agents perceive the innovations $\eta _t$ and $\nu _t$ to be normally distributed according to the following joint distribution:

(31) \begin{equation}\begin{pmatrix} \ln(\eta _t)\\ \ln(\nu _t) \end{pmatrix} \sim N \begin{pmatrix} \begin{pmatrix} 0\\ 0 \end{pmatrix},\begin{pmatrix} \sigma _{\eta }^2 & 0\\ 0 & \sigma _{\nu }^2 \end{pmatrix} \end{pmatrix}.\end{equation}

Modeling perceived house price growth as an unobserved component model is grounded on several empirical elements. First, the implied learning mechanism not only gives rise to a neat optimal filter but is also very intuitive: when house prices grow, it is hard to disentangle whether the increase is persistent or only temporary. Observers thus try to evaluate the persistence of the increase based on their past experience. This is consistent with recent empirical evidence that shows that households extrapolate from recent changes in house prices to form expectations of future prices (Kuchler and Zafar (Reference Kuchler and Zafar2019)). Second, the perceived law of motion for house prices is consistent with the short-term empirical behavior of house prices. Indeed, US house prices present episodes of persistent increase followed by episodes of persistent decrease. Random walk specifications were thus a natural starting point in the literature for testing the behavior of house prices (Gau (Reference Gau1984), Case and Shiller (Reference Case and Shiller1989)) and are thus an intuitive device for modeling the unobserved component of perceived future house price growth. Third, Adam et al. (2012) and Caines (Reference Caines2020) show that this specification for perceived house price growth helps better understand the recent dynamics of house prices in G7 countries prior to the subprime financial crisis. Fourth, recent papers (Adam et al. (Reference Adam, Marcet and Nicolini2016), Adam et al. (2017), Winkler (Reference Winkler2020) and Adam et al. (Reference Adam, Pfauti and Reinelt2020)) emphasize that this specification is successful in explaining several features of survey data about future stock and housing returns.

4.2. Optimal Bayesian learning

Agents observe house price realizations $q_t$ without noise, but they are not able to separately observe the persistent component and the transitory component of what they believe to be the exogenous process driving house price dynamics. Therefore, they face an optimal filtering problem and come up with the best statistical estimate $\ln(\widehat{\mu _t})$ of the persistent component $\ln(\mu _t)$ in each period $t$ . Due to normality of residuals and the linearity of the process, Bayesian filtering amounts to standard Kalman filtering in the setup. Again following the related literature, the prior distribution of beliefs is assumed to be a normal distribution with mean parameter $\ln(\widehat{\mu _0})$ and dispersion parameter $\sigma _0$ . Because the deterministic steady state is the starting point in the simulations below, as is usual in DSGE models analysis, I set the prior mean and dispersion parameters at their steady state values. The prior mean belief about house price growth is thus set at $\ln(\widehat{\mu _0})=0$ , and prior uncertainty $\sigma _0^2$ is set at its Kalman filter steady state value $\sigma ^2$ :Footnote ²

(32) \begin{equation} \sigma ^2= \frac{-\sigma _{\nu }^2+\sqrt{(\sigma _{\nu }^2)^2+ 4 \sigma _{\nu }^2\sigma _{\eta }^2}}{2}. \end{equation}

Agents’ subjective probability measure $P$ is specified jointly by equations (29), (30), and (31), by prior beliefs and by knowledge of the productivity and lenders’ discount factor random processes. The posterior distribution of beliefs in time $t$ following some history up to period $t$ , $\omega _t$ , is $\ln(\mu _t)|\omega _t\sim N(\ln(\widehat{\mu _t}),\sigma ^2)$ , where $\ln(\widehat{\mu _t})$ is given by the following optimal updating rule:

(33) \begin{equation}\ln(\widehat{\mu _t})= \ln(\widehat{\mu _{t-1}})+g\left [\ln(q_t)-\ln(q_{t-1})-\ln(\widehat{\mu _{t-1}})\right ].\end{equation}

This unique recursive equation – in which $g$ is the Kalman filter gain, which optimal expression is $\tfrac{\sigma ^2}{\sigma ^2+\sigma _{\eta }^2}$ – fully characterizes agents’ beliefs about house price growth, which are summarized in each period $t$ by the state variable $\widehat{\mu _t}$ . The Kalman filter gain governs the size of the updating in the direction of the last forecast error.Footnote ³ Logically, the Kalman filter gain increases in the signal-to-noise ratio $\tfrac{\sigma _{\nu }^2}{\sigma _{\eta }^2}$ . A higher signal-to-noise ratio means that changes in house prices are driven to a higher extent by changes in the persistent component $\mu _t$ relative to changes in the transitory noise $\eta _t$ . Thus, the last forecast error is more informative for predicting future house prices. To solve the model under subjective expectations, I resort to lagged beliefs updating to avoid the simultaneous determination of beliefs and house prices. Indeed, according to equation (33), the mean belief about house price growth $\ln(\widehat{\mu _t})$ in period $t$ depends on current house prices $q_t$ . At the same time, house prices in period $t$ depend on the expectations of future house price growth and, thus, on the current mean belief $\ln(\widehat{\mu _t})$ . To avoid this issue, which is inherent in self-referential learning, lagged beliefs updating is assumed, that is, agents rely on lagged information when updating their beliefs. This assumption is common to all papers that model the same specification of asset prices and is also standard in the general self-referential learning literature. Adam et al. (2017) provide microfoundations for this updating rule with delayed information. Lagged beliefs updating consists in rewriting the beliefs updating equation rule (33) as:

(34) \begin{equation} \ln(\widehat{\mu _t})= \ln(\widehat{\mu _{t-1}})+g\left [\ln(q_{t-1})-\ln(q_{t-2})-\ln(\widehat{\mu _{t-2}})\right ]. \end{equation}

The slightly modified updating rule means that in period $t$ , agents update their mean belief in the direction of the forecast error of the previous period rather than of the current period. Consequently, the mean belief $\ln(\widehat{\mu _t})$ is now predetermined at time $t$ , and equilibrium house prices are determined by the housing market clearing condition. Lagged beliefs updating thus ensures that the equilibrium is unique. Given the perceived law of motion for house prices, agents believe that house prices in period $t$ are such that:

(35) \begin{equation} \ln(q_t)=\ln(q_{t-1})+ \ln(\widehat{\mu _{t}})+z_{1t}, \end{equation}

where $z_{1t}$ is seen by agents as an exogenous forecast error, normally distributed with mean 0 and variance $\sigma _z$ , whereas it is actually endogenous: it is equal to the difference between the expected growth rate of house prices and the actual growth rate formed endogenously on the housing market. Following Winkler (Reference Winkler2020), I treat the lagged forecast error $\ln(q_{t-1})-\ln(q_{t-2})-\ln(\widehat{\mu _{t-2}})$ as an exogenous disturbance $z_{2t}$ in the belief system of agents in period $t$ while ensuring that $z_{2t}$ is equal to the lagged forecast error. In the learning model, expected quarterly housing returns $E_t^P[\,\ln(q_{t+1})-\ln(q_t)]$ are given by $\ln(\widehat{\mu _t})$ . Thus, the perceived parameters of the unobserved component model $\sigma _\nu$ and $\sigma _{\eta }$ affect housing returns expectations through the Kalman filter gain $g$ . Therefore, these parameters affect the demand for housing and the related substitution effects.

4.3. Solving the model under imperfect market knowledge

The system of equations characterizing the subjective structural model includes the first-order conditions (4− 6), (13− 15), (17), and (21− 23), the flow of fund constraints (3), (8), and (19), the production function (9), the capital accumulation equation (11), the complementary slackness conditions (16− 24), market clearing conditions (26) and (27), random processes (2) and (10), and the beliefs’ updating equation (34). $P$ now represents the subjective probability measure. The market clearing condition on the housing market is not included in the perceived system of equations because agents do not understand how house prices form. Solving this subjective system of equations yields the subjective policy functions. However, subjective solution functions do not characterize the actual equilibrium house prices, which arise endogenously in the model through the market clearing condition.Footnote ⁴ Therefore, to solve the model under imperfect market knowledge, I rely on two steps, following the method proposed in Winkler (Reference Winkler2020). First, I numerically solve for the coefficients of the approximate subjective policy function of the above system of equations in the neighborhood of the deterministic steady state, relying on standard perturbation methods. Second, I solve for the approximate actual policy function, that is, the objective solution function, by deriving actual endogenous house prices from the subjective policy function, relying on chain rules derivation. I obtain the derivatives of the Taylor expansion of the actual policy function in the neighborhood of the deterministic steady state. This yields a numerical approximation for the objective policy function, which fully characterizes the numerical solution to the learning model.Footnote ⁵