3.1 Households

Agents. Throughout the paper, we use the term “agent” to refer to the main economic decision unit in our model. This decision unit can be thought of as representing a generation, where age-specific variations in its size (e.g., getting married and/or having kids) are not explicitly modeled and taken as exogenous. As described below, we introduce equivalence scales in order to capture variations in the size of the decision unit, beyond co-residence.

Demographics. The household model is cast at an annual frequency. We denote by $j = \{1,\ldots,J=65\}$ the age of an agent in our model. Agents enter the economy at model age $j = 1$ , which corresponds to the biological age of 17 years, and survive with certainty in each model period until they die with certainty at the end of age $J$ . Each agent is part of a dynasty where successive generations of parents and children are born 30 years apart. Agents live through three life-cycle stages. During the first 30 years of their lives (first stage), each period, an agent has the option to co-reside with her/his parents, in which case multiple members of the family live in the same dwelling. Through this mechanism, the number and the composition of households in the model economy is determined endogenously. If an agent chooses to own or rent a housing unit instead, s/he forms her own household. At model age $31$ (second stage), the agent becomes part of the parent generation and is now linked to her/his own children, which are born into the economy at age 31. Since the measure of newborns is the same as the measure of parents, the population size is constant and normalized to unity. At model age $61$ , the agent becomes part of the grandparent generation (third stage). Overlaying these life-cycle stages are two life-cycle phases, a working phase spanning the model ages $j \in \{1,\ldots,j_r=48\}$ and a retirement phase for all model ages $j \in \{j_r+1=49,\ldots,J=65\}$ . Fig. 5 gives a stylized representation of the time line of the overlapping generations model by showing the life cycles of three overlapping generations, labeled as “parents,” “children,” and “grandchildren.”

Figure 5. Life cycle of parents, children and grandchildren.

Notes: Stylized representation of three overlapping generations (parents, children, and grandchildren).

Preferences and co-residence. Agents maximize expected lifetime utility with time discount factor $\beta$ and period utility $U \big (c,\hat{s},x;j,\unicode{x1D7D9}_{h \gt 0})$ , where $c$ denotes consumption of nonhousing goods, $\hat{s}$ is consumption of housing services, and $x \in \{0,1\}$ is an indicator variable for co-residence. We assume that housing services are provided by the parents and shared by all co-residing members of the family. Co-residence affects agents in two ways. First, due to the sharing of resources, per-capita consumption of housing services $\hat{s}$ differs from overall housing services purchased by the household, $s$ . Specifically, we will use household equivalence scales that depend on co-residence choices.Footnote ⁶ Second, for children we will embed an age-dependent utility cost from co-residing with one’s parents, capturing the desire for independence. Following Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021), we further include an additional utility gain for retired homeowners, reflecting the notion that retirees may enjoy own housing more than rented housing. When parameterizing our utility function, this utility shifter will only apply to retired agents ( $j\gt j_r$ ) who own $h \gt 0$ housing units.Footnote ⁷

Labor and pension income. Gross labor income during working age consists of an age-dependent, deterministic component $\varepsilon (j)$ , and a residual random component $\eta (j) \in E(j) = \left [\underline{\eta }_j,\ldots,\overline{\eta }_j\right ]^\prime$ ,

(1) \begin{equation} \ln y(\eta,j) = \varepsilon (j) + \eta (j). \end{equation}

The residual income component $\eta (j)$ evolves according to a Markov chain with age-dependent transition matrix $\pi \left (\eta (j+1) \mid \eta (j); j\right )$ and age $j$ specific stationary invariant distribution $\Pi (\eta ;j)$ . Retired agents receive nonstochastic and constant pension benefits. We assume that retirement benefits are a function of the last realization of the income shock prior to retirement, $\eta (j_r)$ , partially reflecting the fact that higher earnings lead to higher pension income.Footnote ⁸

3.2 Assets

Housing. Housing units $h$ can be owned by private agents and rental firms. Houses are restricted to the ordered set of discrete sizes $\mathcal{H} \in \{\underline{h},\ldots,\bar{h}\}$ , and they are traded at unit price $p$ at the end of the period. One unit of housing provides one unit of shelter. A homeowning agent ( $h \gt 0$ ) obtains $s = h$ units of housing services. Renting agents ( $h = 0$ ) buy housing services $s$ at price $\rho$ on the rental market. In every period, the housing stock depreciates at rate $\delta$ . As in İmrohoroğlu et al. (Reference Imrohoroğlu, Matoba and Tüzel2018) and outlined in Piazzesi and Schneider (Reference Piazzesi and Schneider2016), we assume that homeowners have to pay this fraction of their home value which make it conceptually similar to maintenance costs. When an agent buys or sells housing units, s/he incurs transaction costs, which are fractions $t^b$ (buyer) and $t^s$ (seller) of the house price:

(2) \begin{equation} \gamma (h,h^{\prime }) = \begin{cases} t^s p h + t^b p h^{\prime }, & \text{if}\; h \ne h^\prime, \\ 0 & \text{else}. \end{cases} \end{equation}

Financial assets. Agents can save in a risk-free bond at interest rate $r$ , and they can borrow against their houses at mortgage rate $r+\iota ^m$ , $\iota ^m\gt 0$ . The mortgage premium $\iota ^m$ is exogenous, potentially reflecting monitoring and administrative costs of mortgage lenders. Let $a$ denote an agent’s net financial assets. Newborn agents hold zero assets. Mortgage borrowing is subject to a downpayment constraint,

(3) \begin{equation} a' \geq -p (1-\theta _j) h', \end{equation}

where $\theta _j$ is an age-specific downpayment parameter, and $a'$ and $h'$ denote the agent’s choice for next period’s net financial assets and housing assets.

3.3 Recursive formulation of the decision problems

Our model features intergenerational links between parents and their children, which implies that both groups have to take into account the respective state variables of the other age group. The state vector of an agent is $(a,h,\eta,\tilde{a},\tilde{h},\tilde{\eta },j)$ , where the first three components $(a,h,\eta )$ summarize her/his own net financial assets, housing assets, and residual income. If $j \leq 30$ , the next three components $(\tilde{a},\tilde{h},\tilde{\eta })$ reflect the state variables of the parents; if $30 \lt j \leq 60$ , they denote the state variables of the children; if $j \geq 61$ , for brevity of notation, we simply set $(\tilde{a},\tilde{h},\tilde{\eta }) = (\emptyset,\emptyset,\emptyset )$ . Note that due to the fixed age difference between subsequent generations, $j$ determines the age of the other group $\tilde{j}$ as well: $\tilde{j} = j + 30$ if $j \leq 30$ , and $\tilde{j} = j - 30\;$ if $j \gt 30$ .

Let $V(a,h,\eta,\tilde{a},\tilde{h},\tilde{\eta },j)$ denote an agent’s value function. Agents choose consumption of nonhousing goods $c$ , housing services $s$ (unless they are co-residing in which case this will be chosen by their parents), financial assets $a'$ , and housing assets $h'$ for next period. During the first 30 years, they also choose whether to co-reside with their parents $x$ . We denote the policy functions by $C(\!\cdot\!)$ for consumption, by $S(\!\cdot\!)$ for housing services, by $A(\!\cdot\!)$ and $H(\!\cdot\!)$ for financial and housing assets, and by $X(\!\cdot\!)$ for co-residence. Thus, for example, $X(a,h,\eta,\tilde{a},\tilde{h},\tilde{\eta },j)=1$ means that an agent of age $j$ with own state variables $(a,h,\eta )$ chooses to co-reside with its parents of age $j+30$ with state variables $(\tilde{a},\tilde{h},\tilde{\eta })$ , and $X(\tilde{a},\tilde{h},\tilde{\eta },a,h,\eta,\tilde{j})=1$ means that an agent of age $\tilde{j}+30$ with own state variables $(a,h,\eta )$ has co-residing children of age $\tilde{j}$ and state variables $(\tilde{a},\tilde{h},\tilde{\eta })$ .

Housing decision. The discrete choice problem between owning and not owning can be concisely described as

(4) \begin{equation} V(a,\eta,\tilde{a},\tilde{h},\tilde{\eta },j) \:=\: \max _{h \in \{0,\mathcal{H}\}} \Big \{ V(a,h,\eta,\tilde{a},\tilde{h},\tilde{\eta },j) \Big \}. \end{equation}

Note that this formulation accommodates both the possibility of not becoming a homeowner at all ( $h = 0$ ) and choosing from the menu of the various discrete house sizes conditional on becoming a homeowner ( $h \gt 0$ ).

Homeowners. Let $V(a,h\gt 0,\eta,\tilde{a},\tilde{h},\tilde{\eta },j)$ denote the value function of an agent who currently owns a house of some size $h \in \mathcal{H}$ . This agent solves the recursive problem

(5) \begin{equation} V(a,h\gt 0,\eta,\tilde{a},\tilde{h},\tilde{\eta },j) \;=\; \max _{c,a',h'} U \big (c,\hat{s},0;j,1) + \beta \mathbb{E}_{\eta ^\prime,\tilde{\eta }^\prime \mid \eta, \tilde{\eta }, j} \left [V(a^{\prime },\eta ^{\prime },\tilde{a}^{\prime },\tilde{h}^{\prime },\tilde{\eta }^{\prime },j+1)\right ] \end{equation}

subject to

(6a) \begin{align} c + a' + p h' + \delta p h + \gamma (h,h') \;&=\; y(\eta,j) - T_j(\bar{y}) + p h + \big [ 1 + r \unicode{x1D7D9}_{a \gt 0} + (r+\iota ^m) \unicode{x1D7D9}_{a \lt 0} \big ] a, \end{align}

(6b) \begin{align} \bar{y} \;&=\; y(\eta,j) + r \max (a,0), \end{align}

(6c) \begin{align} a' \;&\geq \; -p (1-\theta _j) h', \text{and} h' \in \mathcal{H} \cup \{0\}, \end{align}

(6d) \begin{align} \hat{s} \;&=\; h/n(X(\tilde{a},\tilde{h},\tilde{\eta },a,h,\eta,\tilde{j}),j) \end{align}

(6e) \begin{align} \tilde{a}^{\prime } \;&=\; A(\tilde{a},\tilde{h},\tilde{\eta },a,h,\eta,\tilde{j}), \end{align}

(6f) \begin{align} \tilde{h}^{\prime } \;&=\; H(\tilde{a},\tilde{h},\tilde{\eta },a,h,\eta,\tilde{j}). \end{align}

Equation (6a) is the budget constraint, which states that expenditures on consumption, financial and housing assets, maintenance costs, and transaction costs for buying/selling must be equal to the sum of labor (or pension) income net of taxes, and financial and housing assets. Taxes are a function $T_j(\bar{y})$ of the tax base $\bar{y}$ , which is defined in equation (6b) as the sum of labor and asset income. For simplicity, we assume that during retirement income is not taxed. Equation (6c) is the borrowing constraint. Equation (6d) maps overall housing services into per-capita consumption of housing services by using equivalence weights, denoted by $n(x,j)$ , for households that have co-residents ( $n(x=1,j)$ ) and those that do not ( $n(x=0,j)$ ). Equations (6e) and (6f) define the laws of motion for financial and housing assets by the parents/children, respectively, as described by their policy functions.

None-homeowners. Next, denote by $V(a,h=0,\eta,\tilde{a},\tilde{h},\tilde{\eta },j)$ the value function of an agent who currently does not own a house. This agent can choose housing services $s$ on the rental market. If the agent is of age $j \leq 30$ , s/he also has the alternative option of co-residing with her/his parents. The recursive problem reads as follows:

(7) \begin{equation} V(a,h=0,\eta,\tilde{a},\tilde{h},\tilde{\eta },j) \;=\; \max _{c,s,x,a',h'} U \big (c,\hat{s},x;j,0) + \beta \mathbb{E}_{\eta ^\prime,\tilde{\eta }^\prime \mid \eta, \tilde{\eta }, j} \left [V(a^{\prime },\eta ^{\prime },\tilde{a}^{\prime },\tilde{h}^{\prime },\tilde{\eta }^{\prime },j+1) \right ] \end{equation}

subject to

(8a) \begin{align} c + a' + p h' + \gamma (0,h') \;&=\; y(\eta,j) - T_j(\bar{y}) - \unicode{x1D7D9}_{x = 0} \rho s + \big [ 1 + r \unicode{x1D7D9}_{a \gt 0} + (r+\iota ^m) \unicode{x1D7D9}_{a \lt 0} \big ] a, \end{align}

(8b) \begin{align} \bar{y} \;&=\; y(\eta,j) + r \max (a,0), \end{align}

(8c) \begin{align} a' \;&\geq \; -p (1-\theta _j) h', \text{and} h' \in \mathcal{H} \cup \{0\}, \end{align}

(8d) \begin{align} \hat{s} \;&=\; \begin{cases} S(\tilde{a},\tilde{h},\tilde{\eta },a,0,\eta,\tilde{j})/n(x=1,j) & \:\text{if}\: x = 1, \\ s/n(X(\tilde{a},\tilde{h},\tilde{\eta },a,h,\eta,\tilde{j}),j) & \text{ otherwise}, \end{cases} \end{align}

(8e) \begin{align} x \in \{0,1\} & \:\text{ if }\: j \leq 30, \;\text{and } x = 0 \text{ otherwise}, \end{align}

(8f) \begin{align} \tilde{a}^{\prime } \;&=\; A(\tilde{a},\tilde{h},\tilde{\eta },a,h,\eta,\tilde{j}), \end{align}

(8g) \begin{align} \tilde{h}^{\prime } \;&=\; H(\tilde{a},\tilde{h},\tilde{\eta },a,h,\eta,\tilde{j}). \end{align}

The distinction between renting and co-residing is reflected in equation (8d), where $S(\!\cdot\!)$ denotes the policy function for housing services. Note that $S(\tilde{a},\tilde{h},\tilde{\eta },a,0,\eta,\tilde{j})$ describes the housing services that are chosen by the parents. Thus, if $j \leq 30$ and $x=1$ , then $S(\tilde{a},\tilde{h},\tilde{\eta },a,0,\eta,\tilde{j})/n(x=1,j)$ is the housing service flow utility that a child of age $j$ experiences from co-residing with its parents. The second line in equation (8d) reflects the case where the agent is a renter, in which case the housing service flow depends on whether there are co-residents, $n(x=1,j)$ , or not, $n(x=0,j)$ . If the agent chooses to rent ( $x = 0$ ), s/he can choose $s$ freely, but has to pay expenditures on rent as reflected in the budget constraint (8a). Equation (8e) imposes an age restriction on the option of co-residing, and equations (8f) and (8g) mirror those from an owner’s problem above.

3.4 Rental firms and construction sector

We follow Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021) in their modeling of the real-estate market and assume that rental firms need to pay monitoring costs $\mu$ per unit of rented housing. These costs reflect the information asymmetry between a real-estate firm and its renters, and they create a wedge between prices and rents that imply a motive for homeownership. The market of real-estate firms is competitive. As a result, the zero-profit condition implies that house prices and rental rates are related as

(9) \begin{equation} (r + \delta ) p = \rho - \mu. \end{equation}

There is a construction sector producing housing units. Production entails costs which are convex and increasing in housing units, $K(I)$ with $K^\prime (I) \gt 0$ , and $K^{\prime \prime }(I) \gt 0$ . The maximization problem of the construction sector implies

(10) \begin{equation} p = K^\prime (I). \end{equation}

Denote by $\bar{H}$ the total housing stock. In steady state,

(11) \begin{equation} \delta \bar{H} = I, \end{equation}

that is, investment in new housing equals depreciation.

3.5 Government

The government levies taxes on income for working-age agents, and it pays out pension benefits to retirees. The tax base $\bar{y}$ is composed of labor and asset income and is taxed according to the nonlinear function

(12) \begin{equation} T_j(\bar{y}) \:=\: \begin{cases} \bar{y} - (1-\tau ) (\bar{y})^{1-\lambda } & \text{if }\: j \leq 48, \\ 0 & \text{else}, \end{cases} \end{equation}

where $\tau \in [0,1)$ and $\lambda \in [0,1]$ are two parameters characterizing the level and progressivity of the tax-transfer system (see Benabou (Reference Benabou2002) and Heathcote et al. (Reference Heathcote, Storesletten and Violante2017)), with higher values of $\tau$ reflecting higher average tax levels and higher values of $\lambda$ reflecting a higher degree of progressivity of the tax code. We assume a balanced budget and assume that any excess tax revenue is spent on public goods $G$ , which do not affect agents’ decisions.

3.6 Equilibrium

A stationary equilibrium in this economy is a set of value functions $\{ V \}$ , a set of policy functions $\{ C, S, A, H, X \}$ , a stationary distribution $\Phi$ of agents over states $(a,h,\eta,\tilde{a},\tilde{h},\tilde{\eta },j)$ , a house price $p$ , a rental rate $\rho$ , construction $I$ , and a housing stock $\bar{H}$ , such that:Footnote ⁹

1. 1. Value and policy functions solve the problems specified in equations (5)–(8g).

2. 2. Real-estate firms maximize profits which implies (9).

3. 3. Construction firms maximize profits which implies (10).

4. 4. All housing units are occupied

   \begin{equation*} \bar {H} \:=\: \int S(a,h,\eta,\tilde {a},\tilde {h},\tilde {\eta },j)\: d \Phi (a,h,\eta,\tilde {a},\tilde {h},\tilde {\eta },j). \end{equation*}

5. 5. The housing stock is stationary and satisfies $\delta \bar{H} = I$ .

6. 6. The government budget constraint clears:

   \begin{align*} G = \int T_j(\bar{y}(a,\eta,j)) \; d \Phi (a,h,\eta,\tilde{a},\tilde{h},\tilde{\eta },j). \end{align*}

7. 7. $\Phi (\!\cdot\!)$ is a stationary distribution induced by the exogenous processes and the policy functions.

We note that the last condition of this definition also entails a constraint on the distribution of newborns. In a stationary equilibrium, the distribution of newborns over parents’ states has to be identical to the actual distribution of parents at age 31. Since this distribution of parents is itself a function of the initial distribution, this implies a fixed-point equilibrium constraint.

4.1 Calibration

Our calibration strategy proceeds in two steps. First, we set some parameter values outside of the model using external estimates from our data or other studies. Second, we calibrate the remaining subset of parameters internally in order to match selected data targets.

4.1.1 Equivalence scales

We apply household equivalence weights to capture variations in household size over the life cycle. In our model, there are two components affecting the size of the household. The first component is age dependent and deterministic, reflecting variations in family size over the life cycle that we do not explicitly consider in our model (singles vs. married/cohabiting couples, children, etc.). The second component is the endogenous co-residence decision, which can result in multiple generations living in the same household. Equivalence weights are real number representations of these two components. To derive these equivalence weights, we proceed as follows. First, we compute the average number of adults and children for each age group.Footnote ¹⁰ Second, we weight the average number of adults and kids in accordance with the modified OECD equivalence weights (Hagenaars et al. (Reference Hagenaars, de Vos and Zaidi1994)). Table 1 reports our parameter values.

Table 1. Equivalence weights

4.1.2 Preferences

The utility function for an agent is specified as

\begin{equation*} U \big (c,\hat {s},x;j,\unicode {x1D7D9}_{h \gt 0}) \;=\; \frac {n(x=0,j)}{1-\sigma } \left [ \left (\frac {c}{n(x=0,j)}\right )^{\zeta } \left ( \left (1+ \xi \cdot \unicode {x1D7D9}_{h \gt 0} \unicode {x1D7D9}_{j\gt j_r}\right ) \cdot \hat {s} \right )^{1-\zeta } \right ]^{1-\sigma } - \unicode {x1D7D9}_{x = 1} \alpha (j), \end{equation*}

where $\sigma$ is the coefficient of relative risk aversion,Footnote ¹¹ and $\zeta$ is the expenditure share for nonhousing consumption goods. Per-capita consumption of nonhousing goods is obtained by dividing by the equivalence scale $n(x=0,j)$ . For housing services, per-capita consumption $\hat{s}$ depends on whether the agent is co-residing with her/his parents or children, $\hat{s} = s/n(x=1,j)$ , or not, $\hat{s} = s/n(x=0,j)$ . The shift parameter $\xi \geq 0$ reflects additional utility benefits for retired homeowners ( $h \gt 0 \wedge j \gt j_r$ ). We set this parameter to match the homeownership rate among retired agents, reflecting the idea that retirees may enjoy own housing more than rented housing. Finally, we include an age-dependent utility cost from co-residing with one’s parents, capturing the desire for independence. We model the disutility as a simple linear function of age, $\alpha (j) \:=\: \alpha _0 \cdot j$ , where $\alpha _0$ is a parameter.

4.1.3 Income process and pensions

For our model, we cannot take household labor income for the estimation of the exogenous income process because household sizes are partly an endogenous outcome. Therefore, we proceed in two steps. First, we estimate gross labor income profiles at the individual person level for non-retired male individuals. Second, we multiply this value with the average number of working individuals in an age group. We pool all waves in our sample and posit the following specification,

(13) \begin{equation} \ln y_{i,t} \;=\; \sum _{j = 1}^{j_r} \nu _j d_{j} + \sum _{t = 1993}^{2015} \iota _t D_t + \epsilon _{i,t}, \end{equation}

where $D_t$ are dummy variables for all years in our sample, and $d_j$ is a set of age dummies translated into model age. We estimate (13) via ordinary least squares (OLS) and use the estimated coefficients $\{\nu _j\}_{j = 1}^{j_r}$ to construct the age-dependent deterministic labor income component in our model. The estimation of the stochastic component follows a strategy similar to De Nardi et al. (Reference De Nardi, Fella and Pardo2020) and Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021). We use the residuals from the previous regression to construct age-dependent discrete Markov processes for residual income dynamics. To this end, we define five bins $\eta _{j,pc} \in \{\eta _{j,1},\ldots,\eta _{j,5}\}$ representing the quintiles of the residual income distribution at age $j$ (with $\eta _{j,1} \lt \eta _{j,2} \lt \ldots \lt \eta _{j,5}$ ). For each $\eta _{j,pc}$ , we assign its mean value. The elements of the Markov transition matrices are calculated as the proportions of individuals that are in bin $pc$ at age $j$ and move to bin $pc^\prime$ at age $j+1$ . In order to make the transition matrices uniformly stationary, we translate the transition matrices into doubly stochastic matrices following a Sinkhorn–Knopp algorithm (Sinkhorn (Reference Sinkhorn1964)) as in Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021).Footnote ¹² Finally, retirement income is calculated as follows. We assume that pension benefits are a function of the last realization of $\eta$ prior to retirement. Following Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021), we set pension income at $42\%$ of average earnings in the respective quintile across all ages, and we apply caps at 7200 euros and 38,400 euros. The upper cap is based on a maximum contribution threshold for the public retirement system, while the lower cap is based on old-age security.

4.1.4 Externally calibrated parameters

Table 2 summarizes the externally calibrated parameters. The coefficient of relative risk aversion $\sigma$ is set to a standard value of 2. The share of expenditures on nonhousing consumption $\zeta$ is set to 0.717 as in Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021). We also adopt the housing transaction costs from their study and set $t^b = 0.108$ and $t^s = 0.029$ . The depreciation rate is set to 0.01, in accordance with a housing lifespan of 100 years. Downpayment requirements are based on estimates from Chiuri and Jappelli (Reference Chiuri and Jappelli2003) and set to 20% of the house value during working age ( $j \leq j_r$ ). We disallow mortgages during retirement by setting the downpayment requirement for $j \gt j_r$ to $1$ . The real interest rate and the real mortgage rate are based on estimates for the yield on 10-year government bonds and 10-year fixed rate mortgage rates giving $r=0.0255$ and $\iota ^m=0.0176$ . As for the set of house sizes, we specify an equidistant 10-node grid between 80k euros and 400k euros. The minimum house size corresponds to a value just below the 10th percentile of the housing wealth distribution in the SOEP sample as estimated by Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021). The maximum house size is chosen large enough such that the fraction of agents who would buy an even larger house is negligible (we verify this in our numerical solution). Turning to the construction technology, we follow Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021) and assume $K(I) = \kappa _0 I^{1+\varphi }/(1+\varphi )$ , where $\kappa _0$ is a construction cost parameter. Caldera and Johansson (Reference Caldera and Johansson2013) estimate the long-run price elasticity of new housing supply in Germany at 0.428, which leads to $\varphi = 2.34$ . The parameters of the income tax function are again based on Kaas et al. (Reference Kaas, Kocharkov, Preugschat and Siassi2021)’s estimates. Using SOEP data, these authors estimate $\tau$ and $\lambda$ for different age groups by relating taxable income with net income. We compute an average value of their estimates for working-age periods and set $\tau = 44.3\%$ and $\lambda = 37.5\%$ accordingly. The value for $\lambda$ indicates that the German tax-transfer system exhibits a strong degree of tax progressivity.

Table 2. Externally calibrated parameters

4.1.5 Internally calibrated parameters

The remaining parameters, summarized in Table 3, are calibrated jointly within the model to match the following data moments, where we report in parenthesis the most closely associated parameter and its calibrated value: (1) average wealth across households is 193.7k euros ( $\beta =0.966$ ); (2) the overall homeownership rate is 42.8% ( $\mu =1.012\%$ ); (3) the homeownership rate among retired households is 58% ( $\xi =1.555$ ); (4) the overall co-residence rate is 20.5% ( $\alpha _0=0.00093$ ); and (5) the construction cost parameter $\kappa _0$ is set using conditions (10) and (11), where we normalize the price per housing unit to $p = 1$ ( $\kappa _0=0.1218$ ).

Table 3. Internally calibrated parameters

4.2 Model fit

We now assess the fit of the benchmark model, with a focus on the empirical facts documented in Section 2. Fig. 6 compares the life-cycle patterns of owning, renting, and co-residence between the data and our benchmark model. The top panel in this figure is identical to the graph that we presented in Section 2, and it shows the split of living arrangements in the data. The bottom panel depicts the corresponding life-cycle patterns in our model. There are three important features worth noting. First, the model generally does a good job of capturing the evolution of owning, renting, and co-residing by age. While young agents tend to either live with their parents or be renters, the share of owner-occupiers gradually increases during working age and then reaches a value of about 60% upon retirement. Second, despite our simple and parsimonious specification for the disutility from living with one’s parents, the model generates an age profile of co-residence that comes very close to the empirical profile (recall that we only target the overall average). In the youngest age group, the model predicts a co-residence rate of about 65%, slightly exceeding the empirical value of 59%. It then predicts a steep decline for agents in their 20s and 30s, consistent with the data, and a very low co-residence share for people in their 40s. Third, the division between owners and renters is captured reasonably well. While there are some quantitative differences in the second half of the life cycle, the fit is quite good in the first half, which is more relevant for the focus of this paper.

Figure 6. Split of living arrangements (Data vs. Model).

Source: Own calculations based on HFCS (second wave, 2014) and SOEP.

In Fig. 7, we plot the age profiles of net wealth and housing wealth from the model in comparison to the empirical profiles. The model captures rather well the hump-shaped pattern of wealth (left panel) which peaks during the pre-retirement years. Remarkably, the model comes very close to matching the empirical profile of housing wealth (right panel).Footnote ¹³ This profile increases steeply between the ages of 25 and 40, when many people choose to become homeowners, and then flattens out in the following years. Succinctly, the model does a good job of matching the age patterns of wealth accumulation in the data, considering that none of these moments were explicitly targeted in the calibration.

Figure 7. Net wealth and housing wealth over the life cycle (Data vs. Model).

Source: Own calculations based on HFCS (second wave, 2014) and SOEP.

Figure 8. Average income by living arrangement (Data vs. Model).

Notes: The top panel is based on calculations from the HFCS (second wave, 2014) and SOEP, and the bottom panel shows results from the model. The unit of measurement in the data is an individual, while in the model it is an agent (with age-specific variation in size as explained in the text). To ease comparability, we rescale income values from the data so that they match average income by age following the same logic as explained in the calibration of the income process.

Fig. 8 shows average income levels by living arrangement and age. Again, the top panel repeats the graph presented in Section 2; the bottom panel displays the model equivalent.Footnote ¹⁴ An important observation is that the choice of living arrangement by agents in the model is strongly linked to their current income. Agents forming their own households tend to earn more than those living with their parents, and the difference between these groups increases considerably with age. In the data, while income levels among co-residents are generally also lower than among renters/owners, the disparity is much smaller. One potential explanation is that economic reasons may not be the only determinant of co-residence choices, and that non-economic reasons may play an increasingly important role later in life. Regarding the split between renting and owning, the model generally predicts that, conditional on age, higher incomes are more likely to lead to homeownership. This fact is consistent with the data as well, albeit the differences are even more pronounced there.

4.3 Co-residence: Inspecting the mechanism

Before turning to our main counterfactual experiments, our goal is to shed more light on the key mechanisms related to co-residence in our model. Specifically, we conduct two sets of exercises. First, we investigate the dynamics surrounding the co-residence choice at a disaggregate level by focusing in on the period when children decide to move out of the parental household. Second, we assess the impact of co-residence on the aggregate level, with a particular focus on house prices.

In our first set of experiments, we consider the population of children in our model who co-reside with their parents at a given time $t$ (i.e., $x_t = 1$ ). Then we split this sample according to their co-residence choice in the subsequent period, that is, we distinguish between children who choose to move out ( $x_{t+1} = 0$ ) and those who resume in co-residence ( $x_{t+1} = 1$ ). Table 4 documents statistics related to the growth rate of childrens’ income from $t$ to $t+1$ (Panel A). As can be seen, the decision to move out is strongly correlated with income growth: On average, movers earn 84% more income than in the previous period, while for stayers income only grows by 6%. We also report the population shares of children who experience positive income growth, negative income growth, and those where income stays the same. Roughly 60% of children who decide to move out have seen their income increase, while among the population of stayers only 24% have experienced positive income growth. Succinctly, these findings suggest that the decision to move out of the parental household is often triggered by higher income realizations.

Next, we look at the implications for house size dynamics on the part of the parents. Panel B in Table 4 reports statistics related to the growth rate of housing services consumed by the parents from $t$ to $t+1$ . We find that parent households where the children have moved out consume less housing services in the next period, in other words, there is some downsizing. Notably, this reaction is strongly concentrated among renters: On average, their chosen house size declines by almost 13% after the children have left. Among owners, the reaction is much more muted. This finding is intuitive, because for owner-occupiers downsizing comes at additional transaction costs.

We conclude this section by studying the aggregate implications of co-residence for house prices. To this end, we conduct the following exercise. First, starting from our benchmark economy, we shut down co-residence altogether by increasing the taste for independence parameter $\alpha _0$ to a prohibitively high level. Second, we take the opposite approach and set $\alpha _0$ to zero, thereby making co-residence more attractive. Our findings are as follows. If we shut down co-residence, house prices increase by 9.0%. If we set $\alpha _0 = 0$ , the co-residence rate increases from 20.8% to 35.8%, and house prices decline by 22.0%. The upshot is that, through the lens of our model, co-residence depresses house prices.