2.1. Returns

We now focus our attention on returns because they represent the link between finance and the housing market. In general equilibrium, housing returns and bond returns need indeed to be equalized. Proposition 3 shows how interest rates are structurally linked to house prices and rent-price ratios.

Proposition 3. In the neighborhood of the deterministic steady state, housing prices, rents, and interest rates are linked by the following relationship:

(2) \begin{equation} R_{t}=\mathbb{E}_{t}\frac{p_{t+1}}{p_{t}}\left [ \left ( 1-\delta _{x}\right ) +l_{t+1}\right ] \varepsilon _{t+1}, \end{equation}

where

(3) \begin{equation} \varepsilon _{t+1}\,:\!=\,\frac{\beta _{1t+1}u_{c_{1t+1}}^{\prime }}{\mathbb{E}_{t}\beta _{1t+1}u_{c_{1t+1}}^{\prime }}. \end{equation}

Proof. See Online Appendix A.

Without uncertainty, $\varepsilon _{t+1}=1$ , and equation (2) is a standard arbitrage equation. It equates the gross interest rate on bonds to the return of a unit of non-depreciated housing.Footnote ²² With stochastic shocks, the gap between the expected and the realized discounted utility of the most patient agent (i.e. the dominant consumer) influences the relationship. Indeed, risk and volatility can affect the nature of the relationship between interest rates, house prices, and the rent-price ratios.

Equation (2) has similar counterparts in the literature and, notably, is reminiscent of the definition of housing returns in Favilukis et al. (Reference Favilukis, Ludvigson and Van Nieuwerburgh2017), once accounting for the housing premium between bond rates and housing returns. However, as Favilukis et al. (Reference Favilukis, Ludvigson and Van Nieuwerburgh2017) do not explicitly model rental markets, they replace the term $l_{t+1}$ by a marginal utility ratio—which is the result of a weighted average of the population’s marginal utilities and accounts for a weighted average of discount factors. The wealth distribution of the population is key in their article as it determines risk sharing and risk premia. In contrast, in our work, both $l_{t+1}$ and $\varepsilon _{t+1}$ are functions of the marginal utilities of the dominant consumer only. The latter also pins down both interest rates and rents. This is consistent with the main mechanism in Kaplan et al. (Reference Kaplan, Mitman and Violante2020), where non-constrained homeowners play a key role in housing market dynamics. Notice in particular that the arbitrage equation itself is pinned down by the dominant consumer. Indeed, the landlord is the only agent who is an active player on the (i) credit market, (ii) the real estate market and the (iii) rental market.Footnote ²³ Therefore, the equilibrium is imposed by the portfolio arbitrage of landlords.

3.1. Households

The three different types of agents are indexed by $i=1,2,3$ and have different time preference parameters, where $\beta _{i}$ satisfies $1\gt \beta _{1}\gt \beta _{2}\gt \beta _{3}\gt 0$ . Agent 1 (characterized by $\beta _{1})$ is the most patient ones and is thus landlords. Consistently with our results in Section 2, the discount factors of agents 2 and 3 are chosen such that they are owner-occupiers and renters, respectively (see Online Appendix A for a characterization of thresholds). As in Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017), $\tilde{\beta }_{t}$ is a random variable that captures shocks to intertemporal preferences and satisfies:Footnote ²⁴

(4) \begin{equation} \log \tilde{\beta }_{t}=\rho _{\beta }\log \tilde{\beta }_{t-1}+\epsilon _{\beta t}. \end{equation}

In each period, agent $i$ discounts date $t$ with $\beta _{i}^{t} \tilde{\beta }_{t}$ . Notice also that the way we incorporate $ \tilde{\beta }_{t}$ into agents’ preferences ensures that the status of agents does not change in response to the shock. We allow the population of each type to be of different size, so as to match empirical evidence. The size of the population of agent 1 is used as a normalizer and is set to 1. We then denote by $\omega _{2}\gt 0$ and $\omega _{3}\Omega _{t}\gt 0$ the relative shares of agents of type 2 and 3 with respect to the one of type 1. To account for changes in the homeownership rate, we introduce $\Omega _{t}$ , which is a random variable that captures shocks to the proportion of renters in the population.Footnote ²⁵ It satisfies:

(5) \begin{equation} \log \Omega _{t}=\rho _{\Omega }\log \Omega _{t-1}-\epsilon _{{\Omega }t}. \end{equation}

and is consistent with the stationarity of the model. The homeownership rate at time $t$ is thus defined according to agents’ population shares as $(1+\omega _{2})/(1+\omega _{2}+\omega _{3}\Omega _{t})$ , and the share of landlords among homeowners is $1/\left (1+\omega _{2}\right )$ .

Let us first consider the individual problem for agent 1, who are the landlords and the only ones willing to save in equilibrium (see Propositions 1 and 2). They are thus the sole owners of both the capital stocks and land. Following Iacoviello (Reference Iacoviello2005), the expected utility of an agent of type 1 is:

(6) \begin{equation} \mathbb{E}_{0}\sum _{t=0}^{\infty }\beta _{1}^{t}\tilde{\beta }_{t}\left [ \ln c_{1t}+j_{t}\ln \left ( h_{t}x_{1t-1}\right ) -\chi _{1}\frac{n_{1t}^{\eta }}{\eta }\right ], \end{equation}

where $c_{1t}$ is the consumption of non-durables, $h_{t}x_{1t-1}$ is the consumption of housing services, and $n_{1t}$ is hours worked. An agent of type 1 owns $x_{1t-1}$ units of housing but rents a share $(1-h_{t})$ of this housing stock to other agents, thus effectively consuming $h_{t}x_{1t-1}$ units of housing services. The parameter $\eta \gt 1$ determines the elasticity of labor supply, while the parameter $\chi _{1}\gt 0$ captures the preference for leisure. As in Iacoviello (Reference Iacoviello2005), we allow for intratemporal shocks to housing preference, $j_{t}$ with:

(7) \begin{equation} \log j_{t}=\rho _{j}\log j_{t-1}+(1-\rho _{j})\log j_{ss}+\epsilon _{jt}, \end{equation}

where $j_{ss}\gt 0$ is the steady-state value of $j_{t}$ and where $\epsilon _{jt}$ is a shock correlated to $\epsilon _{{\Omega }t}$ . The increase in the demand for housing is thus correlated to the increase of the share of homeowners.

The budget constraint of agents of type 1 is as follows:

(8) \begin{equation} \begin{array}{c} p_{t}\left [ \left ( 1-\delta _{x}\right ) +l_{t}\left ( 1-h_{t}\right ) \right ] x_{1t-1}+r_{t}^{c}k_{t-1}^{c}+r_{t}^{h}k_{t-1}^{h}+p_{t}^{q}\left ( 1+l_{t}^{q}\right ) q_{t-1} \\ +d_{1t}+w_{t}n_{1t}=c_{1t}+R_{t-1}^{d}d_{1t-1}+i_{t}^{c}+i_{t}^{h}+p_{t}x_{1t}+p_{t}^{q}q_{t}\end{array} \end{equation}

The left-hand side of the equation corresponds to the resources of an agent of type 1. They are the sum of the value of the previous-period housing stock, $p_{t}\left ( 1-\delta _{x}\right ) x_{1t-1},$ where $p_{t}$ is the housing price and $\delta _{x}\in \left [ 0,1\right ]$ the depreciation rate of housing; the rent received on the $\left ( 1-h_{t}\right )$ share of rented units, where $l_{t}$ is the rent-price ratio; the returns on investments in the two sectors, $r_{t}^{s}k_{t-1}^{s},$ where $r_{t}^{s}$ is the rental price of capital of sector $s$ and $k_{t-1}^{s}$ is the capital invested in sector $s$ ; the value of the land owned by agent 1, $p_{t}^{q}q_{t-1},$ where $p_{t}^{q}$ is the price and $q_{t-1}$ the quantity; the rent received from the land, $p_{t}^{q}\left ( 1+l_{t}^{q}\right ) q_{t-1}$ , where $l_{t}^{q}$ is the rent-price ratio for the land; new debt/credit, $d_{1t}$ (as discussed above, agent 1 is always a lender in equilibrium, which implies $d_{1t}\lt 0);$ labor income, $w_{t}n_{1t}$ , where $w_{t}$ is the wage. The right-hand side of (8) represents expenditures. They include consumption of non-durables, the payment of the debt interests at rate $R_{t-1}^{d}$ , capital investment in both sectors, $i_{t}^{s}$ , the purchases of housing and land.Footnote ²⁶

As commonly assumed, land does not depreciate and its quantity is constant and normalized to $1$ . It effectively acts as an adjustment cost which can increase the volatility of house prices [see, for instance, Iacoviello and Neri (Reference Iacoviello and Neri2010)]. Capital in each sector depreciates at rate $\delta _{s}\in \left [ 0,1\right ]$ . Investments in those sectors are affected by efficiency shocks as in Christiano et al. (Reference Christiano, Motto and Rostagno2014), which are denoted $\Upsilon _{t}^{s}$ and satisfy:

(9) \begin{equation} \log \Upsilon _{t}^{s}=\rho _{\Upsilon ^{s}}\log \Upsilon _{t-1}^{s}+\epsilon _{\Upsilon ^{s}t}. \end{equation}

Moreover, we introduce adjustment costs [as e.g. in Justiniano et al. (Reference Justiniano, Primiceri and Tambalotti2011), Christiano et al. (Reference Christiano, Trabandt and Walentin2011) or Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017)], so that capital accumulation in sector $s$ follows:

(10) \begin{equation} k_{t}^{s}=\Upsilon _{t}^{s}i_{t}^{s}\left [ 1-\frac{\phi _{s}}{2}\left ( \frac{i_{t}^{s}}{i_{t-1}^{s}}-1\right ) ^{2}\right ] +\left ( 1-\delta _{s}\right ) k_{t-1}^{s}, \end{equation}

where $\phi _{s}\geq 0$ gives the intensity of the inefficiency. As a result of these adjustment costs, the relative price of capital fluctuates.

The problem of agent 1 is to maximize (6) subject to (8) and (10) given $\left ( x_{1t-1},k_{t-1}^{c},k_{t-1}^{h},q_{t-1},d_{1t-1}\right )$ . The full list of first-order conditions can be found in Online Appendix B.1.

Let us now consider the individual problem of agents 2. Following the previous theoretical results, $\beta _{2}$ is such that, in equilibrium, these agents are owner-inhabitants who borrow up to the limit set by the collateral constraint, and who do not hold any assets except housing (which serves as a collateral for loans). For simplicity, variables referring to agents 2 and 3 are defined as those for agent 1 but are indexed by 2 and 3 respectively. The expected utility of an agent of type 2 is:

(11) \begin{equation} \mathbb{E}_{0}\sum _{t=0}^{\infty }\beta _{2}^{t}\tilde{\beta }_{t}\left [ \ln c_{2t}+j_{t}\ln x_{2t-1}-\chi _{2}\frac{n_{2t}^{\eta }}{\eta }\right ], \end{equation}

with the budget constraint given by:

(12) \begin{equation} p_{t}\!\left ( 1-\delta _{x}\right ) x_{2t-1}+d_{2t}+w_{t}n_{2t}=c_{2t}+R_{t-1}^{d}d_{2t-1} +p_{t}x_{2t} \end{equation}

As in (1), the collateral constraint writes:

(13) \begin{equation} d_{2t}\leq m\!\left ( 1-\delta _{x}\right ) x_{2t}\mathbb{E}_{t}p_{t+1}, \end{equation}

where $m\gt 0$ is given.Footnote ²⁷ Following Proposition 2, the constraint in (13) is binding. The problem of agent 2 consists in maximizing (11) subject to (12) and (13) for $\left ( x_{2t-1},d_{2t-1}\right )$ given. The full list of first-order conditions can be found in Online Appendix B.2.

Let us finally consider agent 3. Following our theoretical results, we assume that $\beta _{3}$ is such that these agents are renters in equilibrium and therefore hold no assets. The expected utility of an agent of type 3 writes:

(14) \begin{equation} \mathbb{E}_{0}\sum _{t=0}^{\infty }\beta _{3}^{t}\tilde{\beta }_{t}\left [ \ln c_{3t}+j\ln \left ( \phi z_{3t}\right ) -\chi _{3}\frac{n_{3t}^{\eta }}{\eta }\right ], \end{equation}

where $z_{3t}$ is the amount of housing rented by this agent and $\phi$ is kept for consistency with problem (1). We also assume that preference shocks for housing only affect homeowners, so that for agent 3, $j=j_{ss}$ . $\epsilon _{jt}$ is thus a preference shock for owned housing units as commonly done in the literature.

The budget constraint applying to agent 3 is:

(15) \begin{equation} w_{t}n_{3t}=c_{3t}+p_{t}l_{t}z_{3t}. \end{equation}

Note that, since agent 3 holds no assets, their problem is static and thus does not depend on $\beta _3$ . The latter only determines the fact that they are renters in equilibrium (see Proposition 2). First-order conditions are presented in Online Appendix B.3.

3.2. Firms

Firms produce in a perfectly competitive environment. As mentioned above, there are two sectors that produce non-durable goods and housing, respectively. The production function of the firms of the consumption sector is given by: $Y_{ct}=A_{ct}K_{ct}^{\gamma _{c}}L_{ct}^{\alpha _{c}}$ (with $\gamma _{c}+\alpha _{c}=1$ ), where $K_{ct}$ and $L_{ct}$ are respectively the capital and labor used by the firms and $A_{ct}$ is an exogenous productivity factor specific to the consumption sector which satisfies:

(16) \begin{equation} \log A_{ct}=\rho _{A_{c}}\log A_{ct-1}+\epsilon _{Act}. \end{equation}

The production function of the firms of the housing sector is given by: $Y_{ht}=A_{ht}K_{ht}^{\gamma _{h}}L_{ht}^{\alpha _{h}}Q_{t}^{1-\alpha _{h}-\gamma _{h}}$ where $K_{ht}$ , $L_{ht}$ , and $Q_{t}$ are respectively capital, labor, and land used by the firm [see e.g. Iacoviello and Neri (Reference Iacoviello and Neri2010)] and $A_{ht}$ is a productivity factor which satisfies:

(17) \begin{equation} \log A_{ht}=\rho _{A_{h}}\log A_{ht-1}+\epsilon _{Aht}. \end{equation}

where we allow $\epsilon _{Aht}$ to be correlated with $\epsilon _{Act}$ . First-order conditions of the firms’ problem are derived in Online Appendix B.4.

3.3. Equilibrium

There are 8 markets in the economy, which clear in equilibrium. Let us start with the asset markets. Equilibrium in the capital markets of both sectors implies: $K_{ct}=k_{t-1}^{c}$ and $K_{ht}=k_{t-1}^{h}$ . The equilibrium in the market for land is $Q_{t}=q_{t-1}=1$ , and the one for debt is $d_{1t}+\omega _{2}d_{2t}=0$ . Moreover, the rental market for housing satisfies: $\omega _{3}\Omega _{t}z_{3t}=\left ( 1-h_{t}\right ) x_{1t-1}$ .

As we assume that labor is perfectly mobile across sectors, the equilibrium condition for the labor market is:

(18) \begin{equation} L_{ht}+L_{ct}=n_{1t}+\omega _{2}n_{2t}+\omega _{3}\Omega _{t}n_{3t}. \end{equation}

We now turn to the equilibrium for the two goods markets. The equilibrium condition for non-durable goods is:

(19) \begin{eqnarray} Y_{ct} &=&c_{1t}+\omega _{2}c_{2t}+\omega _{3}\Omega _{t}c_{3t}+i_{t}^{c}+i_{t}^{h}, \end{eqnarray}

while the equilibrium for the housing good is:

(20) \begin{equation} Y_{ht}=x_{1t}+\omega _{2}x_{2t}-\left ( 1-\delta _{x}\right ) \left [ x_{1t-1}+\omega _{2}x_{2t-1}\right ]. \end{equation}

In the next section, we discuss our procedure to calibrate and estimate the model.

4.1. Data

We estimate our model using series for the following variables which are crucial for housing markets: house prices, the rent-price ratio, consumption, non-residential (i.e. capital) investment, residential investment (which corresponds to the output in the housing sector), private debt, interest rates, and homeownership rates. GDP is not included as it is the sum of residential investment (multiplied by house prices) and output in the rest of the economy.

We estimate our model from 1965 to 2019 with quarterly logged HP-filtered data (the smoothing coefficient is 1,600Footnote ²⁸ ). Following Pfeifer (Reference Pfeifer2020), we use a one-sided HP filter.Footnote ²⁹ For robustness, we repeat the exercise for the period 1965–2006. The latter period has the advantage of excluding the years of the Great Recession and thus significant non-linearities in housing markets such as the zero lower bound for interest rates, as well as significant variations in house prices and debt.

The choice of our data generalizes Iacoviello and Neri (Reference Iacoviello and Neri2010) to account for rental markets and is detailed in Online Appendix C, which also details the construction of the series we use. Figure 1 shows the series (in levels) used for the estimation.

Figure 1. Data.

Notes: Data used for estimation. Period: 1965Q1–2019Q4. The plotted rent-price ratio and real interest rate are annualized. The construction of the series is detailed in Online Appendix C.

4.2. Parameters calibration

We calibrate $\beta _{1}$ so that the annual real interest rate is 3% [as in Iacoviello and Neri (Reference Iacoviello and Neri2010) or Kaplan et al. (Reference Kaplan, Mitman and Violante2020)]. $\beta _{2}$ is set to 0.98 in line with Hendricks (Reference Hendricks2007), and $\beta _{3}$ can assume any value as it does not play any role in the equilibrium equations. The same applies to $\phi$ . We only assume that $\beta _{3}$ and $\phi$ are low enough such that agent 3 is a renter in equilibrium (see Proposition 2). The steady-state value of the preference for housing, $j_{ss}$ , is set to have a ratio of residential real estate over quarterly GDP of about 4 (its average between 1965 and 2016).

The parameters driving the preference for leisure $\chi _{1}$ , $\chi _{2}$ and $\chi _{3}$ are set such that $n_{1}=n_{2}=n_{3}=1/3$ . The underlying assumption is that agents of different types spend the same amount of time working at the steady state.

The relative shares $\omega _{2}$ and $\omega _{3}$ are calibrated to have a homeownership rate of 66.6% and so that landlords represent 10% of the homeowners, as in Sommer et al. (Reference Sommer, Sullivan and Verbrugge2013). This approach has the advantage of setting these parameters on clear empirical evidence without the need to use wealth holdings as a proxy, as in previous works in this literature. Indeed, wealth holdings data entail several measurement problems to calculate shares (see discussion in Justiniano et al. (Reference Justiniano, Primiceri and Tambalotti2015) among others).

The quantity of aggregate land is normalized to 1. The parameter $\delta _{x}$ is such that the annual rent-price ratio is 4.9% (its average between 1965 and 2016). The corresponding annual depreciation rate for housing is about 1.8%, which is close to micro-estimates (e.g. Nakajima and Telyukova (Reference Nakajima and Telyukova2020) use a value of 1.7%). We set the loan-to-value ratio $m$ to 0.85 as in Iacoviello and Neri (Reference Iacoviello and Neri2010), and in line with Calza et al. (Reference Calza, Monacelli and Stracca2013).Footnote ³⁰

We set $\gamma _{c}$ to 0.3 in line with the literature [e.g. Iacoviello (Reference Iacoviello2005)], and thus, $\alpha _{c}$ is set to 0.7. Following Iacoviello and Neri (Reference Iacoviello and Neri2010), the land share and the capital share $\gamma _{h}$ in the construction sector are both set to 0.1. The share of labor $\alpha _{h}$ is thus 0.8, in line with Iacoviello (Reference Iacoviello2005). Capital depreciation is assumed to be the same in both sectors and $\delta _{h}=\delta _{c}=0.0139$ in line with Davis and Heathcote (Reference Davis and Heathcote2005). Table 1 summarizes the calibration.

Table 1. Calibrated parameters

4.3. Parameters estimation

We estimate the persistence and standard deviations of the shocks that hit our economy, as well as capital adjustment costs.

Our list of shocks includes the technology shocks in the consumption and housing sectors, respectively [see equations (16) and (17)], the preference shock for housing [see equation (7)], the discount factor shock [see equation (4)], the homeownership shock [see equation (5)], and efficiency shocks in both sectors [see equation (9)].

Table 2 displays the priors and posteriors of the parameters we estimate. Our priors for the parameters driving the adjustment costs for capital in the two sectors ( $\phi _{c}$ and $\phi _{h}$ ) are based on Iacoviello and Neri (Reference Iacoviello and Neri2010).

Table 2. Estimated parameters

As Iacoviello and Neri (Reference Iacoviello and Neri2010), the prior means and standard deviations of the persistence parameters are set to 0.8 and 0.1, except for those related to the marginal efficiency of investment, which are based on the priors used in Christiano et al. (Reference Christiano, Motto and Rostagno2014), and for those related to the shock affecting the proportion of renters. For the latter, we chose a moderate value of 0.5 for the prior mean, with a relatively large standard deviation of 0.2.

Finally, we allow for a degree of correlation between the two productivity shocks and between the preference shock for owned housing units and the shock to the proportion of renters. We use diffuse priors for these correlations, with a prior mean of 0.

The posterior distributions are estimated using the Metropolis-Hastings algorithm.Footnote ³¹ Table 2 shows key statistics of these posterior distributions. In line with Iacoviello and Neri (Reference Iacoviello and Neri2010), we find that technological shocks in the consumption and housing sectors are quite persistent. We also find that productivity innovations in the housing and consumption sector are positively correlated. As expected, housing preferences and the homeownership rate shocks, respectively, are also positively correlated.

We find a relatively high volatility for the shock on the marginal efficiency of investment in the housing sector, together with a relatively strong persistence. This is explained by the high volatility of residential investment observed in the data, to which the shock contributes significantly (see the analysis of moments and the one for the variance decomposition below).

Finally, we find that both intertemporal and intratemporal shocks to preferences are not very persistent. This comes from the fact that the real interest rate displays significant short-run volatility (see Figure 1). Similar considerations apply for the rent-price ratio, to which the intratemporal shock contributes significantly (see Figure 1 and Table 4).

4.4. Business cycle properties: model versus data

We now turn to the business cycle properties of our model when the parameters are set at their estimated posterior means.

Table 3 shows standard statistics of the main aggregates and prices of our model. The first column shows the statistics of our data for the interval 1965–2019. The second and third columns show the simulated moments by using first and second-order approximations of the model.Footnote ³² In the Online Appendix we also provide two robustness checks. First, we repeat the same exercise with shorter series (1965–2006) so as to exclude the Great Recession. Indeed, its aftermath featured unconventional monetary policies and non-linear trends in housing markets. Secondly, we repeat the same exercise by using alternative data for housing prices (using the FHFA house price index instead of the U.S. Census Bureau house price index on single-family homes).

Table 3. Baseline model, moments for period 1965–2019

Notes: Data include series for the period 1965-2019. “1st order”: predictions from the model simulated with a first-order approximation. “2nd order”: predictions from the model simulated with a second-order approximation with pruning. Results come from a simulation of the model of a 30000-period length. The first 500 periods have been truncated. One-sided HP filter, lambda = 1600. Aggregates are expressed in per capita units.

We see that the relative standard deviations of our key variables of interest are very close to those in the data. The model reproduces the greater volatility of rent-price ratios with respect to house prices [see discussion in Dong et al. (Reference Dong, Liu, Wang and Zha2022)]. The model can also reproduce the fact that the volatility of residential investment is much greater than the one for non-residential investment. Moreover, by including renters, the share of collateral-constrained agents decreases and our model provides a relatively good match of debt volatility (for a discussion, see Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017), who match debt volatility by introducing occasionally binding collateral constraints).

Analogously, the correlations predicted by the model are in line with the data and the literature. Our model does a particularly good job in reproducing the correlations between the variables representing agents’s arbitrage when investing in housing, which is represented by equation (2). This confirms the importance of the main mechanisms at play in our model. While the correlation between house prices and rent-price ratios is in line with data in all samples, the one between house prices and interest rates improves significantly when we use shorter series excluding the Great Recession (see our robustness exercise in the Online Appendix). This should not surprise the reader as we do not model (unconventional) monetary policies and have thus a disadvantage in reproducing the series including the financial crisis. Our results are also consistent with Favilukis et al. (Reference Favilukis, Ludvigson and Van Nieuwerburgh2017) who underline the role of large risk premia for returns throughout that period.

Our model slightly underestimates the correlation between house prices and consumption, because of the absence of nominal rigidities, but still delivers positive correlations.Footnote ³³ However, for the same reason [see the discussion in Iacoviello and Neri (Reference Iacoviello and Neri2010)] the correlations of residential investment with output and house prices are smaller than those in the data. This is not new in the literature and is in line with Favilukis et al. (Reference Favilukis, Ludvigson and Van Nieuwerburgh2017), who also have difficulties in reproducing the correlation between output and house prices.Footnote ³⁴

Table 4 presents a brief inspection of the variance decomposition of our variables of interest. Technological shocks in both sectors are the key drivers of the dynamics of output and consumption. Technological shocks in the housing sector also have a strong impact on the rent-price ratio and house prices. Those in the consumption sector have also a large impact on non-residential investment and house prices. Discount rate shocks are necessary in our model for the dynamics of the interest rates and explain more than 3/4 of their volatility.

Table 4. Variance decomposition simulating one shock at a time (in percent)

Notes: Results come from a simulation of the model of a 30000-period length. The first 500 periods have been truncated. Second-order approximation with pruning. HP filter, lambda = 1600. Aggregates are expressed in per capita units.

Finally, while investment shocks in the consumption sector explain a large share of the dynamics of capital investment and output, those in the housing sector explain large shares of the fluctuations of house prices, residential investment,Footnote ³⁵ rent prices, private debt, and interest rates. Indeed, shocks affecting house prices are transmitted to rent-price ratios and interest rates because of the strong linkage in equation (2) and debt because of the collateral constraint.

5.1. Responses to a technological shock in the consumption-good sector

Following a positive technological shock in the consumption-good sector [an increase $\epsilon _{Act}$ in equation (16)], output and aggregate consumption increase, as expected (see Figure 2). Moreover, the model generates a positive response of capital investment in both sectors (see Figure 3). Absent adjustment costs, there would be a full shift of resources from one sector to the other [see Davis and Van Nieuwerburgh (Reference Davis and Van Nieuwerburgh2015)]. The presence of adjustment costs for sector-specific capital as well as the fixed supply of land (which effectively acts as an adjustment cost) prevents this type of effect, which would otherwise generate too much volatility of investment. Adjustment costs also play a role in the dynamics of the interest rate on loans because capital and financial returns are linked, and the former are affected by the fluctuations in the price of capital.

At the time of the shock (relative), house prices increase because the productivity in the housing sector falls with respect to the one in the consumption sector (see Figure 2).

As house prices increase, the value of the collateral of agents 2 rises, which relaxes their debt limit and enables them to borrow more (Figure 2). Agents of type 2 use these additional funds to increase non-durable consumption and lower their worked hours. This behavior is reinforced by an income effect associated to the increase in wages (Figure 3), which further dampens worked hours.

The interest rate dynamics are the result of the initial increase in the marginal productivity of capital in the consumption sector, the upward pressure due to the increase of credit demand from agents 2 and then their subsequent deleveraging. Since after the shock house prices stabilize, the rent-price ratio follows closely the dynamics of the interest rate because of the strong link in equation (2). The associated increase in rents explains why the demand of agent 3 for housing services decrease, even if their wage increases (see Figure 3).

5.2. Demand shocks for owned housing units

We now shift our attention to the effects of an intratemporal preference shock for owned housing [an increase in $\epsilon _{jt}$ in equation (7)]. According to Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017), these shocks explain the “lion’s share in the movement of house prices” and caused more than 70% of the decrease in consumption during the Great Recession.

The increase in $j_{t}$ triggers a jump in housing demand of both agents 1 and 2. As expected, the demand shock pushes up house prices (see Figure 4), which is magnified by the fact that land is in fixed supply. The increase in demand also leads firms to produce more housing and allows our model to reproduce the positive correlation between house prices and residential investment observed in the data [see Iacoviello and Neri (Reference Iacoviello and Neri2010)]. Collateral effects associated with the behaviors of agents of type 2 amplify the upward pressure on both house prices and residential investment and thus on total output.

Figure 4. Responses to a preference shock for owned housing.

Notes: Simulated impulse response functions after a calibrated positive preference shock for owned housing. Residential investment refers to output in the housing sector while investment refers to capital accumulation in both sectors. Aggregates are expressed in per capita units.

As the calibrated shock has little persistence, the effect on house prices is short-lived. For agent 1, the direct effect of making housing consumption more attractive (relative to the consumption good) is dampened by an indirect one. Indeed, the short rise in house prices makes it worthwhile to sell housing in period 1 and generates a wealth effect. Agent 1 tradeoffs these two effects by increasing the share of his/her housing stock it consumes and by selling some of the housing stock it owns (which will only affect its housing consumption in the next period). The second effect together with the increase in resources associated with the wealth effect pushes up both non-durable consumption and saving, entailing a short-lived and small-sized decrease in interest rates.

Since the impact of the shock on interest rates is short-lived and interest rates come back quickly towards the steady state, the dynamics of the rent-price ratio essentially tracks the one of house prices [see equation (2)]. Therefore, in response to the demand shock for housing, house prices and rent-price ratios (together with nominal rents) increase. This is because the increased preference for housing services makes landlords less willing to rent to tenants. This pushes up rents and rent-price ratios. Notice also that if the preference shock also hit rented units, this effect would be magnified.

These underlying mechanisms highlight how the shock emphasized by Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017) cannot reproduce the joint dynamics of housing prices and rents during the Great Recession if we account for rental markets. Indeed, during the financial crisis, house prices have plummeted but rent-price ratios have increased.

6.1. Responses to a pandemic-induced tilt in intertemporal preferences

We now calibrate the intertemporal preference shock of equation (4) so as to track the evidence in the aftermath of the Covid outbreak. Starting in Spring 2020, the Fed injected liquidity so as to cut interest rates. As a result, the (annualized) interest rate on 3-month T-bills went from about 1.50% in February 2020 to close to 0% in April 2020 (see Figure 5). The announcement of Jerome Powell on June 10th 2020 for the Fed to keep the funds rate between 0% and 0.25% throughout 2022 contributed to drive down long-term interest rates as well (e.g. the interest rate on 10-year treasuries went down by 0.84 percentage points during the same period).

We therefore pin down the persistence of the shock during the pandemic so that it mostly dies out after 3 years, which is globally in line with Powell’s commitment. More precisely, we set $\rho _{\beta }$ so that after 12 quarters the shock is only 5% of its initial value. As a result, $\rho _{\beta }=0.05^{1/12}\simeq 0.78$ . We then calibrate the standard deviation of $\epsilon _{\beta t}$ so that the initial decline in the quarterly interest rate is about 0.375 percentage points, consistent with the data.Footnote ⁴²

Figure 6 shows the response of the economy to the calibrated intertemporal preference shock, which is in line with the stylized facts highlighted in Figure 5.Footnote ⁴³ The shock directly affects intertemporal decisions and thus hits the Euler equation of agents of type 1 (i.e. the dominant consumers). They choose to postpone current consumption of non-durable goods and housing services, while they supply more savings, labor and invest in housing (to rent). This is consistent with the strong jump in personal savings starting from March 2020. As agents of type 1 pin down the interest rate, this change in intertemporal behavior pushes the interest rate down. This also mimics the Fed’s policy, that accommodated the surge of savings with large liquidity injections to stimulate the economy.

Figure 6. Responses to a shock to intertemporal preferences.

Notes: Simulated impulse response functions after a calibrated shock to intertemporal preferences. Debt is expressed in per capita units.

In equilibrium, these additional savings are associated with an increase of loans supplied to agents of type 2 and with an increase in capital in both sectors. As their intertemporal preferences are also modified, agents of type 2 also reduce their current consumption and accumulate more housing (and thus debt). Contrarily to the two previous ones, agents of type 3 solve a static problem and are not directly affected by the shock, although their behaviors are impacted by the change in equilibrium prices.

The shock we consider in this section thus tracks both the large rise in aggregate savings and the decrease in the interest rate. The implied increase in the demand for housing, from both agents 1 and 2, pushes house prices up. Importantly, this does not translate into an increase in rents. In contrast, the rent-price ratio decreases. This is due to the fact that agents of type 1 are supplying more housing on the rental market. It explains why agents of type 3 are consuming more housing services and less non-durables.Footnote ⁴⁴ Notice finally that the effect of the pandemic-induced shock is amplified by housing markets, who drive up both residential investment (i.e. housing output) and aggregate GDP.

The relationship between interest rates, housing prices, and the rent-to-price ratio is clearly characterized by equation (2). It shows that the interest rate is strongly linked with the product of housing prices and the rent-price ratio. Following a shock that reduces the interest rate, a rise in housing price is associated with a decline in the rent-price ratio [see the results of the SVAR of Dias and Duarte (Reference Dias and Duarte2019)]. The return on housing investments is in this case positively correlated with the interest rate.

6.2. Housing-induced welfare effects

The exercise in the previous section has highlighted that the policy-induced change in intertemporal preferences has had an expansionary impact on the housing market. We now shift our attention to its heterogeneous welfare implications, depending on agents’ statuses in the housing market.Footnote ⁴⁵

Welfare is simply defined here as an agents’ instantaneous utility.Footnote ⁴⁶ We measure welfare changes as follows. Let us denote by $u_i(\cdot )$ the instantaneous utility of agent $i$ , which is given by

(21) \begin{equation} u_{i}\left (c,hs,n\right )=\ln c+j_{ss}\ln hs-\chi _{i}\frac{n^{\eta }}{\eta } \end{equation}

where $c$ is consumption, $hs$ are housing services and $n$ are hours worked. Let us denote by $u_{i,ss}$ the value that this function takes at the steady state with

(22) \begin{equation} u_{i,ss} = u_{i}\left (c_{i,ss},hs_{i,ss},n_{i,ss}\right ) \end{equation}

We aim at grasping the change in welfare in response to the intertemporal preference shock. To this purpose, we proxy by $\Delta _{it}$ the instantaneous welfare change relative to the steady state where:

(23) \begin{equation} u_{i}\left (c_{it},hs_{it},n_{it}\right )=u_{i}\left (c_{i,ss}\left (1+\Delta _{it}\right ),hs_{i,ss}\left (1+\Delta _{it}\right ),n_{i,ss}\right ) \end{equation}

Hence, $\Delta _{it}$ is a consumption-equivalent variation measure. A positive $\Delta _{it}$ indicates that instantaneous utility in $t$ is higher than in the steady state (and vice versa), and the welfare change is expressed in percentage of the steady-state consumptions of non-housing goods and housing.Footnote ⁴⁷

Figure 7 plots the response of agents’ changes in (housing and goods) consumption-equivalent losses or gains in response to the shock. Notice in particular that the shock has a large negative impact on the welfare of landlords. This is because of a negative direct effect on non-durable consumption and labor supply, that is associated with the tilt in preferences (due to an increase of landlords’ labor supply and a decrease in consumption). But there is also a significant indirect effect of the shock, that is transmitted to their welfare via their revenues. Indeed, because of the fall in interest rates and the associated one in rents, landlords experience a significant decrease in financial revenues (see Figure 8). In contrast, the welfare of renters is barely affected. Indeed, the fall in rent-price ratios partly compensates for the decrease in wages and thus the fall in renters’ incomes. Finally, the fall in interest rates drives down credit costs, benefiting borrowers.

Figure 7. Welfare effect of the intertemporal preferences shock.

Notes: Simulated impulse response functions after a calibrated intertemporal preference shock. IRFs represent percent deviations from the steady state of instantaneous utilities of agents of type 1, 2, and 3, respectively.

Figure 8. Asset income and interests on debt.

Notes: Simulated impulse response functions after a calibrated intertemporal preference shock. IRFs represent percent deviations from steady state of asset income for agent 1 (first panel) and for agent 2 (second panel). The assets income of agent 1 is: $p_{t}\left [ l_{t}-\delta _{x}\right ] h_{1t} x_{1t-1}+r_{t}^{c}k_{t-1}^{c}+r_{t}^{h}k_{t-1}^{h}+p_{t}^{q}l_{t}^{q}q_{t-1} -\left ( R_{t-1}^{d}-1\right ) d_{1t-1}$ , consistently with Propositions 1 and 2. The assets income of agent 2 is: $-\left ( R_{t-1}^{d}-1\right ) d_{2t-1}$ .

Contrarily to the common view, our analysis suggests thus that policies entailing lower interest rates have particularly weighted on landlords although lower interest rates have contributed to drive up the value of housing (and the wealth of homeowners). Our framework shows how low rates can be beneficial to lower-income households, who rent and borrow.

There is thus an important channel linking agents’ welfare with the role they play on housing markets because of heterogeneous financial revenues. In our model, this is due to changes in rents and interest rates. As remarked by Pereira Da Silva,Footnote ⁴⁸ because “households at the bottom of the distribution are renters, the middle class owns mainly real estate and bank deposits, and those at the top own more sophisticated (and higher-yielding) financial assets,” monetary expansion seems to increase wealth inequality.

The debate is far from being closed and recent evidence points instead in favor of a decrease of inequality when the monetary policy is expansionary.Footnote ⁴⁹ While the evaluation of the distributional effects of current policies is beyond the scope of our work, our results contribute to the debate by pointing out the importance of the social stratification induced by the housing market. Our parsimonious framework helps clarifying the heterogeneous impacts of low-interest rate policies on households depending on the role they play on housing markets.