In the previous sections, we have described the main factors that are a hindrance to rental market development as well as the aggregate effects of rental market underdevelopment. In this section, we propose a theoretical macroeconomic framework that can be used to assess the effects of changes in the organization of the housing rental market. The result from the previous section will be crucial to the development and implementation of the macro model. To be more precise, we use the micro results to feed the macro model and assess the effects of the following reforms in housing markets on the macroeconomy:
- (i)
Decreasing the impact of “bad tenant” risk on the level of rents,
- (ii)
Removing fiscal incentives to own,
- (iii)
Increasing the professionalism of landlords, for example by means of regulations on house inspections and maintenance, which would lower the psychological disadvantages of renting.
The choice of these reforms is made in light of the factors that appear to be most relevant in the survey. Therefore, this part of the paper comes as a natural consequence of the results found in Sect. 2.
The proposed DSGE model is based on the framework of Iacoviello (2005), in the sense that it includes housing and a collateral constraint for borrowers, whereas the description of the rental market is closely related to the recent works by Ortega et al. (2011) and Rubio (2014b). We use the latter two models, in which the rental market is well characterized for our purpose. However, we had to adapt them to study specifically the Polish market. In particular, we use a two-sector housing model with a rental market, as in Ortega et al. (2011). However, since that model is designed for Spain, it is a small open economy inside a monetary union. We, instead, propose a monetary policy framework in which the central bank is able to set interest rates, as in Rubio (2014b). The main structure of the model is as follows:
- 1.
There are two types of consumers, savers and borrowers, with different discount factors.
- 2.
Savers are the landlords and provide rental services to borrowers.
- 3.
Borrowers face collateral constraints when applying for a mortgage.
- 4.
There are two production sectors, construction and consumption goods.
- 5.
Housing can be purchased or rented.
- 6.
There are fiscal incentives to purchase or rent a dwelling.
A more elaborated description, with optimization problems, is presented below.
Savers
Savers maximize their utility from consumption \(C_{s,t}\), housing services \(H_{s,t}\), and working hours \(N_{s,t}\):
$$\begin{aligned} \max E_{0}\sum _{t=0}^{\infty }\beta _{s}^{t}\left( \log C_{s,t}+j\log H_{s,t}-\frac{\left( N_{s,t}\right) ^{1+\eta }}{1+\eta }\right) , \end{aligned}$$
(2)
where \(\beta _{s}\in \left( 0,1\right) \) is the discount factor and \(E_{0}\) the expectation operator. \(1/\eta >0\) is the labor-supply elasticity and \(j>0\) constitutes the relative weight of housing in the utility function. \(N_{s,t}\) is a composite of labor supplied to the consumption \(N_{cs,t}\) and housing sector \(N_{hs,t}\),
$$\begin{aligned} N_{s,t} =\left[ \omega _{l}^{1/\varepsilon _{l}}\left( N_{cs,t}\right) ^{\left( 1+\varepsilon _{l}\right) /\varepsilon _{l}} +\left( 1-\omega _{l}\right) ^{1/\varepsilon _{l}}\left( N_{hs,t}\right) ^{\left( 1+\varepsilon _{l}\right) /\varepsilon _{l}}\right] ^{\varepsilon _{l}/\left( 1+\varepsilon _{l}\right) }, \end{aligned}$$
(3)
where \(\omega _{l}\) is a weight parameter and \(\varepsilon _{l}\) the elasticity of substitution between both labor types.
The budget constraint is:
$$\begin{aligned}&C_{s,t}+b_{s,t}+q_{h,t}\left[ \left( 1-\tau _{h}\right) \left( H_{s,t}-\left( 1-\delta _{h}\right) H_{s,t-1}\right) +\left( H_{z,t}-\left( 1-\delta _{z}\right) H_{z,t-1}\right) \right] \nonumber \\&\quad \le \frac{R_{t-1}b_{s,t-1}}{\pi _{t}} +w_{cs,t}N_{cs,t}+w_{hs,t}N_{hs,t}+q_{z,t}H_{z,t}+S_{t}-T_{t}, \end{aligned}$$
(4)
where \(q_{h,t}\) is the real housing price and \(w_{cs,t}\) (\(w_{hs,t}\)) denotes real wages in the consumption (housing) sector. Savers can purchase or sell houses, either to live in \(H_{s,t}\) or to rent it \(H_{z,t}\) at price \(q_{z,t}\). \(\delta _{h}\) and \(\delta _{z}\) are the depreciation rates for owner-occupied and rented dwellings, respectively. They might differ if tenants utilize the dwelling more intensively than owners, which is discussed in Sect. 2. We call this phenomenon “bad tenant” risk. We also allow for the existence of tax incentives to own, in particular a subsidy \(\tau _{h}\). Next, the level of savings is given by \(b_{s,t}\) and the risk-free interest rate by \(R_{t-1}\). \(\pi _{t}\) is the inflation rate at period t. Finally, \(S_{t}\) are the profits of firms and \(T_{t}\) a lump-sum government transfer.
The first-order conditions for this optimization problem are as follows:
$$\begin{aligned}&\frac{1}{C_{s,t}}=\beta _{s}E_{t}\left( \frac{R_{t}}{C_{s,t+1}\pi _{t+1}} \right) , \end{aligned}$$
(5)
$$\begin{aligned}&\frac{j}{H_{s,t}}=\left( 1-\tau _{h}\right) \left[ \frac{q_{h,t}}{C_{s,t}} -\beta _{s}\left( 1-\delta _{h}\right) E_{t}\left( \frac{q_{h,t+1}}{C_{s,t+1} }\right) \right] , \end{aligned}$$
(6)
$$\begin{aligned}&\frac{q_{h,t}}{C_{s,t}}=\frac{q_{z,t}}{C_{s,t}}+\beta _{s}\left( 1-\delta _{z}\right) E_{t}\frac{q_{h,t+1}}{C_{s,t+1}}, \end{aligned}$$
(7)
$$\begin{aligned}&\frac{w_{cs,t}}{C_{s,t}}=\left( N_{s,t}\right) ^{\eta }\omega _{l}^{1/\varepsilon _{l}}\left( \frac{N_{cs,t}}{N_{s,t}}\right) ^{1/\varepsilon _{l}}, \end{aligned}$$
(8)
$$\begin{aligned}&\frac{w_{hs,t}}{C_{s,t}}=\left( N_{s,t}\right) ^{\eta }\left( 1-\omega _{l}\right) ^{1/\varepsilon _{l}}\left( \frac{N_{hs,t}}{N_{s,t}}\right) ^{1/\varepsilon _{l}}. \end{aligned}$$
(9)
Equation (6) is the standard Euler equation for consumption. Equations (7) and (8) represent the intertemporal conditions for housing purchased to own and let, respectively. In these equations, the benefits of purchasing a housing unit equate the alternative costs of forgone consumption. Finally, Eqs. (8) and (9) describe the labor-supply conditions for the consumption goods and the housing sectors.
Borrowers
Borrowers solve a similar optimization problem as savers:
$$\begin{aligned} \max E_{0}\sum _{t=0}^{\infty }\beta _{b}^{t}\left( \log C_{b,t}+j\log \widetilde{H}_{b,t}-\frac{\left( N_{b,t}\right) ^{1+\eta }}{1+\eta }\right) , \end{aligned}$$
(10)
where \(\beta _{b}<\beta _{s}\) is the discount factor for borrowers, and
$$\begin{aligned} N_{b,t}=\left[ \omega _{l}^{1/\varepsilon _{l}}\left( N_{cb,t}\right) ^{\left( 1+\varepsilon _{l}\right) /\varepsilon _{l}}+\left( 1-\omega _{l}\right) ^{1/\varepsilon _{l}}\left( N_{hb,t}\right) ^{\left( 1+\varepsilon _{l}\right) /\varepsilon _{l}}\right] ^{\varepsilon _{l}/\left( 1+\varepsilon _{l}\right) }. \end{aligned}$$
(11)
The key difference in the optimization problems of savers and borrowers is that \(\tilde{H}_{b,t}\) is a composite of owned housing purchased with a mortgage \(H_{b,t}\) and rental services \(H_{z,t}\):
$$\begin{aligned} \tilde{H}_{b,t}=\left[ \omega _{h}^{1/\varepsilon _{h}}\left( H_{b,t}\right) ^{\left( \varepsilon _{h}-1\right) /\varepsilon _{h}}+\left( 1-\omega _{h}\right) ^{1/\varepsilon _{h}}\left( H_{z,t}\right) ^{\left( \varepsilon _{h}-1\right) /\varepsilon _{h}}\right] ^{\varepsilon _{h}/\left( \varepsilon _{h}-1\right) }. \end{aligned}$$
(12)
The parameter \(\omega _{h}\) is very important in our analysis, as it approximates the preference for owning a house (purchased on credit) versus the rental housing. In turn, \(\varepsilon _{h}\) describes the elasticity of substitution between preferences for owner-occupied and rental housing. In this way, borrowers derive utility from the two types of housing. It should be emphasized that this does not literally mean that each borrower lives simultaneously in their own house and in a rented house. Instead, the interpretation is that there exists a large representative borrower-type household with a continuum of members, some of whom live in owner-occupied houses, the rest of whom live in rented houses. This composite index in the equation thus represents the aggregate preferences of all household members with respect to each kind of housing service.
For borrowers, we also allow for tax incentives to rent, in particular a subsidy to rent \(\tau _{z}\) is considered. Thus, the budget constraint and the collateral constraint for the borrowers are as follows:
$$\begin{aligned}&C_{b,t}+\frac{R_{t-1}b_{b,t-1}}{\pi _{t}}+q_{h,t}\left( 1-\tau _{h}\right) \left( H_{b,t}-\left( 1-\delta _{h}\right) H_{b,t-1}\right) +q_{z,t}\left( 1-\tau _{z}\right) H_{z,t} \nonumber \\&\quad \le b_{b,t}+w_{cb,t}N_{cb,t}+w_{hb,t}N_{hb,t},\end{aligned}$$
(13)
$$\begin{aligned}&b_{b,t}\le E_{t}\left( \frac{1}{R_{t}}k_{t}q_{h,t+1}H_{b,t}\pi _{t+1}\right) , \end{aligned}$$
(14)
where \(b_{b,t}\) represents the level of debt and \(k_{t}\) is a maximum loan-to-value ratio (LTV) that follows an autoregressive process \(\log k_{t}=(1-\rho _{k})\log (\overline{k})+\rho _{k}\log k_{t-1}+\zeta _{t}\) with normally distributed shocks, where \(\overline{k}\) is a steady-state value of the LTV. A shock to the LTV represents a credit constraint loosening or tightening.
The first-order conditions of this maximization problem are:
$$\begin{aligned}&\frac{1}{C_{b,t}}=\beta _{b}E_{t}\left( \frac{R_{t}}{C_{b,t+1}\pi _{t+1}}\right) +\lambda _{t}R_{t},\end{aligned}$$
(15)
$$\begin{aligned}&\frac{j}{\tilde{H}_{b,t}}\left( \frac{\omega _{h}\tilde{H}_{b,t}}{H_{b,t}}\right) ^{1/\varepsilon _{h}}=\left( 1-\tau _{h}\right) \left( \frac{q_{h,t}}{C_{b,t}}-\beta _{b}\left( 1-\delta _{h}\right) E_{t}\frac{q_{h,t+1}}{C_{b,t+1}}\right) \nonumber \\&\qquad \qquad \qquad \qquad \quad -\lambda _{t}k_{t}E_{t}q_{h,t+1}\pi _{t+1}, \end{aligned}$$
(16)
$$\begin{aligned}&\frac{j}{\tilde{H}_{b,t}}\left( \frac{\left( 1-\omega _{h}\right) \tilde{H}_{b,t}}{H_{z,t}}\right) ^{1/\varepsilon _{h}}=\left( 1-\tau _{z}\right) \frac{q_{z,t}}{C_{b,t}}, \end{aligned}$$
(17)
$$\begin{aligned}&\frac{w_{cb,t}}{C_{b,t}}= \left( N_{b,t}\right) ^{\eta }\omega _{l}^{1/\varepsilon _{l}}\left( \frac{N_{cb,t}}{N_{b,t}}\right) ^{1/\varepsilon _{l}},\end{aligned}$$
(18)
$$\begin{aligned}&\frac{w_{hb,t}}{C_{b,t}}=\left( N_{b,t}\right) ^{\eta }\left( 1-\omega _{l}\right) ^{1/\varepsilon _{l}} \left( \frac{N_{hb,t}}{N_{b,t}}\right) ^{1/\varepsilon _{l}}, \end{aligned}$$
(19)
where \(\lambda _{t}\) is the Lagrange multiplier of the collateral constraint. The above conditions can be interpreted analogously to those for savers. The most important difference is in the demand equation for owned and rented housing (17 and 18), which now equates the marginal utility from housing services (and the marginal value of housing as collateral in the case of Eq. 17) with the alternative cost of forgone consumption.
Firms
The intermediate consumption goods market is monopolistically competitive. The individual firm production function is:
$$\begin{aligned} Y_{t}\left( z\right) =A_{t}\left( N_{cs,t}\left( z\right) \right) ^{\gamma } \left( N_{cb,t}\left( z\right) \right) ^{\left( 1-\gamma \right) }, \end{aligned}$$
(20)
where the only factor of production is labor supplied by each agent, with \(\gamma \in \left[ 0,1\right] \) measuring the relative size of each group in terms of labor. \(A_{t}\) represents technology, which is an autoregressive process \(\log A_{t}=\rho _{A}\log A_{t-1}+u_{t}\) with normally distributed shocks. The symmetry across firms allows for avoiding the index z and rewriting the above equation in the form of the aggregate production function for consumption goods:
$$\begin{aligned} Y_{t}=A_{t}N_{cs,t}^{\gamma }N_{cb,t}^{\left( 1-\gamma \right) }. \end{aligned}$$
(21)
The intermediate housing investment goods market is subject to the same technology shock \(A_{t}\). Both types of households also supply labor to the construction sector, with the same relative size as in the consumption sector. The aggregate production function for housing investment is therefore:
$$\begin{aligned} IH_{t}=A_{t}N_{hs,t}^{\gamma }N_{hb,t}^{\left( 1-\gamma \right) }. \end{aligned}$$
(22)
Intermediate goods producers maximize profits:
$$\begin{aligned} \max _{N_{cs,t},N_{hs,t},N_{cb,t},N_{hb,t}}\frac{Y_{t}}{X_{t}} +q_{h,t}IH_{t}-w_{cs,t}N_{cs,t}-w_{hs,t}N_{hs,t}-w_{cb,t}N_{cb,t}-w_{hb,t}N_{hb,t}, \end{aligned}$$
(23)
where \(X_{t}\) is the markup that is equal to the inverse of real marginal costs.Footnote 2 The first-order conditions are the following:
$$\begin{aligned} w_{cs,t}= & {} \frac{1}{X_{t}}\gamma \frac{Y_{t}}{N_{cs,t}}, \end{aligned}$$
(24)
$$\begin{aligned} w_{cb,t}= & {} \frac{1}{X_{t}}\left( 1-\gamma \right) \frac{Y_{t}}{N_{cb,t}}, \end{aligned}$$
(25)
$$\begin{aligned} w_{hs,t}= & {} \gamma \frac{q_{h,t}IH_{t}}{N_{hs,t}},\end{aligned}$$
(26)
$$\begin{aligned} w_{hb,t}= & {} \left( 1-\gamma \right) \frac{q_{h,t}IH_{t}}{N_{hb,t}}. \end{aligned}$$
(27)
These first-order conditions represent the labor demanded for each type of consumer by each sector, respectively. The price setting problem for the intermediate goods producers is a standard Calvo–Yun case. They sell goods at price \(P_{t}\left( z\right) \). They can re-optimize the price with \(1-\theta \) probability in each period. The optimal reset price \(P_{t}^{OPT}\left( z\right) \) solves:
$$\begin{aligned} \sum _{k=0}^{\infty }\left( \theta \beta \right) ^{k}E_{t}\left\{ \Lambda _{t,k}\left[ \frac{P_{t}^{OPT}\left( z\right) }{P_{t+k}}-\frac{\varepsilon _{p}/\left( \varepsilon _{p}-1\right) }{X_{t+k}}\right] Y_{t+k}^{OPT}\left( z\right) \right\} =0, \end{aligned}$$
(28)
where \(\varepsilon _{p}\) represents the elasticity of substitution between intermediate goods.
The aggregate price level is therefore:
$$\begin{aligned} P_{t}=\left[ \theta P_{t-1}^{1-\varepsilon _{p}}+\left( 1-\theta \right) \left( P_{t}^{OPT}\right) ^{1-\varepsilon _{p}}\right] ^{1/\left( 1-\varepsilon _{p}\right) }. \end{aligned}$$
(29)
By combining (28) with (29) and log-linearizing, we can obtain the standard forward-looking Phillips curve.
Monetary authority and equilibrium conditions
To close the model, we assume that the central bank sets interest rates according to a Taylor rule that responds to inflation and output growth:Footnote 3
$$\begin{aligned} R_{t}=\left( R_{t-1}\right) ^{\rho }\left[ \pi _{t}^{\left( 1+\phi _{\pi }\right) }\left( \frac{Y_{t}}{Y_{t-1}}\right) ^{\phi _{y}}R\right] ^{\left( 1-\rho \right) }\varepsilon _{R,t}, \end{aligned}$$
(30)
where \(0\le \rho \le 1\) is the parameter associated with interest rate smoothing. \(\phi _{\pi }>0\) and \(\phi _{y}>0\) measure the interest rate response to inflation and output growth, respectively. R is the steady-state value of the interest rate. \(\varepsilon _{R,t}\) is a white noise shock with 0 mean and \(\sigma _{\varepsilon }^{2}\) variance.
The equilibrium conditions for the consumption goods and housing investment markets are:
$$\begin{aligned} Y_{t}= & {} C_{s,t}+C_{b,t},\end{aligned}$$
(31)
$$\begin{aligned} IH_{t}\equiv & {} \left( H_{s,t}-\left( 1-\delta _{h}\right) H_{s,t-1}\right) +\left( H_{b,t}-\left( 1-\delta _{h}\right) H_{b,t-1}\right) \nonumber \\&+\left( H_{z,t}-\left( 1-\delta _{z}\right) H_{z,t-1}\right) . \end{aligned}$$
(32)
Finally, the equilibrium government budget constraint is:
$$\begin{aligned} T_{t}=\tau _{z}q_{z,t}H_{z,t}+\tau _{h}q_{h,t}\left[ \left( H_{s,t}-\left( 1-\delta _{h}\right) H_{s,t-1}\right) +\left( H_{b,t}-\left( 1-\delta _{h}\right) H_{b,t-1}\right) \right] . \end{aligned}$$
(33)