3.1.1 Solution of household's problem

The individual maximizes utility (1) subject to the budget constraint of the young (2) and of the old (3). Combining equations (2) and (3), we construct the intertemporal budget constraint of the household:

(4)$$c_t^y + c_t^y \displaystyle{{\,p_{t + 1}} \over {\,p_t}} = w_{t + 1}^o + \lpar {w_t^y -T_t} \rpar \displaystyle{{\,p_{t + 1}} \over {\,p_t}}.$$

Maximizing lifetime utility (1) subject to the intertemporal budget constraint (4) gives the following first-order condition:

(5)$$c_{t + 1}^o = \beta c_t^y \displaystyle{{\,p_{t + 1}} \over {\,p_t}}.$$

Expression (5) implies that an individual's consumption may be increasing or decreasing over time depending on whether the asset price in period t + 1 is higher or lower than the asset price for period t and also the discount factor β. For example, if p _t+1/p _t >1/β then young will save more in period t anticipating higher returns on their savings relatively to their rate of impatience. This implies that consumption of young $c_t^y$ will decrease and consumption of old $c_{t + 1}^o$ will increase.

Combining the first-order condition (5) with the intertemporal budget constraint (4) yields consumption of the young in period t:

(6)$$c_t^y = \displaystyle{{w_{t + 1}^o} \over {1 + \beta}} \displaystyle{{\,p_t} \over {\,p_{t + 1}}} + \displaystyle{{w_t^y -T_t} \over {1 + \beta}}. $$

Equation (6) shows that the income and asset prices determine the fraction of income the individual consumes in the first period. The expression highlights the intuition described above namely that higher p _t+1, relatively to p _t, decreases consumption of the young $c_t^y$. Moreover, if $w_t^y$ increases, then $c_t^y$ of the young increases as well because young have more resources. If $w_{t + 1}^o$ increases, then consumption of the young $c_t^y$ increases as well expecting higher income in the future. If tax T increases, then consumption $c_t^y$ decreases as income after taxes decreases. Substituting the previous equation (6) into the budget constraint of young (2) helps to find the fraction of income saved s _t:

(7)$$s_t^d = \displaystyle{{w_t^y -T} \over {\,p_t}}\displaystyle{\beta \over {1 + \beta}} -\displaystyle{{w_{t + 1}^o} \over {\,p_{t + 1}}}\displaystyle{1 \over {1 + \beta}}. $$

Equation (7) gives us the flow of savings $s_t^d$. The equation shows that if p _t+1 increases relatively to p _t, then $s_t^d$ increases as individuals will save more today and consume more tomorrow. Same logic applies with $w_t^y$. However, if $w_{t + 1}^o$ increases, then $s_t^d$ will decrease because to cover future consumption old need less savings. In sum, higher wage $w_t^y$ increases both consumption of the young and savings of the young. This reflects the consumption smoothing motive in the model. If social security benefits $w_{t + 1}^o$ increase, then consumption of young $c_t^y$ increase but the amount of savings $s_t^d$ decreases. So the total average savings of each individual depends on incomes and on return of capital that savers expect in the next period. We can therefore rewrite the previous equation of savings $s_t^d$ times prices at p _t at time t of each individual as follows:

(8)$$p_ts_t^d = \lpar {w_t^y -T_t} \rpar \displaystyle{\beta \over {1 + \beta}} -\displaystyle{{w_{t + 1}^o} \over {1 + \beta}} \displaystyle{{\,p_t} \over {\,p_{t + 1}}}.$$

3.1.2 Government

This model assumes that the government always runs a balanced budget. Its expenses $w_t^o$ are financed with taxes T. We assume that all taxes are distributed as social security benefits in the same time period. The government's budget constraint for the period t is written as follows:

(9)$$T_tN_t^y = w_t^o N_t^o. $$

The government levies taxes T on young population $N_t^y$ and pay social security benefits $w_t^o$ to old population $N_t^o$. The taxes that finance the government's costs affect the household's budget constraint (2). However, the amount of income tax T is given in the model, so the government sets only the amount of social security benefits:

(10)$$w_t^o = T_t\displaystyle{{N_t^y} \over {N_t^o}}. $$

If the population is aging, which means $N_t^y /N_t^o$ decreases, then for the same amount of T _t social security benefits will decrease. The intuition is that if there are not enough young people contributing to the system, the benefits necessarily will decrease.

3.1.3 Goods market

As described before, there is no firm in the model, hence no endogenous production (we assume that output is exogenous). The good markets equilibrium condition for the period t is the following:

(11)$$c_t^y N_y^y + c_t^o N_t^o = w_t^y N_t^y. $$

Equation (11) reflects a closed economy without any imports or exports. If we had assumed an open economy, then total consumption by young and old $c_t^y N_t^y + c_t^o N_t^o$ would not necessarily equal the total output $w_t^y N_t^y$. The demand consists of consumption $c_t^y$ of young with population $N_t^y$ and consumption $c_t^o$ of old with population $N_t^o$. The good market supply is the total amount of exogenous labor income $w_t^y$ provided to the young generation.

3.1.4 Asset market

Derivation of asset market clearing: Given the goods market equilibrium and the government budget constraint, we can derive the equilibrium of the asset market. We start from the household's budget constraints for period t of young and old.

(12)$$w_t^y -T_t = c_t^y + p_ts_t^d, $$

(13)$$w_t^o + p_ts_{t-1}^s = c_t^o. $$

The reason that in equation (13) s _t−1 is indexed by s and not by d is to represent that the asset is being sold by the old and demanded by the young.

We multiply equations (12) and(13) with population measures to get the aggregate quantities.

(14)$$N_t^y w_t^y -N_t^y T_t = N_t^y c_t^y + N_t^y p_ts_t^d, $$

(15)$$N_t^o w_t^o + N_t^o p_ts_{t-1}^s = N_t^o c_t^o. $$

We add equations (14) and (15) and derive the following:

(16)$$N_t^y w_t^y -N_t^y T_t + N_t^o w_t^o + N_t^o p_ts_{t-1}^s = N_t^y c_t^y + N_t^y p_ts_t^d + N_t^o c_t^o. $$

Equation (16) gives intertemporal resource constraint. Next we use the government budget constraint which is described in equation (9) and derive:

(17)$$N_t^y w_t^y + N_t^o p_ts_{t-1}^s = N_t^y c_t^y + N_t^y p_ts_t^d + N_t^o c_t^o. $$

Finally, we use equilibrium in the goods market (11) and derive:

(18)$$N_t^o s_{t-1}^s = N_t^y s_t^d. $$

Equation (18) is the asset market clearing which we describe below.

Asset market clearing: Equilibrium in the asset market implies that the demand for the asset by the young (saving of the young) equals the supply of the asset by the old. The demand for the asset by the young is given by equation (7). As we mentioned before, this is the flow of savings in the economy. Furthermore, the price of the asset p _t will enter asset market clearing through $s_t^d$.

In every period t old individuals bring to the asset market their present holdings of assets $s_{t-1}^s$ to exchange it at the current price p _t. The supply of asset by the old is given and it is the total stock of assets in the economy. We denote $s_{t-1}^s$ as h _t since we think of the asset as being houses. With the above assumptions, the asset market clearing condition for the period t can be written as:

(19)$$N_t^y s_t^d = N_t^o h_t.$$

The left-hand side of equation (19) is the aggregate demand for the asset and the right-hand side is the aggregate supply of the asset.

3.1.5 Steady-state equilibrium

We solve for an equilibrium where prices are constant for every period. This allows us to abstract from time subscripts. This also implies that p _t = p _t+1 = p*. As a result we rewrite equation (7) as:

(20)$$s^d = \displaystyle{{w^y-T} \over {\,p^{^\ast}}}\displaystyle{\beta \over {1 + \beta}} -\displaystyle{{w^o} \over {\,p^{^\ast}}}\displaystyle{1 \over {1 + \beta}}. $$

Now we use the equation for asset market clearing (19) to derive the price level:

(21)$$p^{^\ast} = \displaystyle{{N^y} \over {N^oh}}\left[ {\displaystyle{{w^y-T} \over {\,p^{^\ast}}}\displaystyle{\beta \over {1 + \beta}} -\displaystyle{{w^o} \over {\,p^{^\ast}}}\displaystyle{1 \over {1 + \beta}}} \right].$$

Equation (21) describes how the equilibrium price level of houses is determined. It shows that if the number of young people increases relative to the number of old people, then house prices increase as well. In contrast, if the population ages which means we have more old relative to young, then house price will decrease. This is in line with the notion that aging population could lead in the future to a decline in house prices, all else equal. Factors other than the population structure that might influence house prices are the available stock of houses h, the level of net income w ^y − T _t, and the social security benefits w ^o. As we can see from equation (21) if the available stock of houses increases, then house prices decrease. Moreover, if the net income increases, then demand for houses increases and house prices increase. Finally, if social security benefits increase, the old individuals have more income so their willingness to sell the asset decreases. As a result, house prices decrease.

4.3.1 The effect of population growth on house prices

Coefficient β gives the change (in percentage points) of house price growth if population growth increases by one percentage point. In all our specifications, we find a sizable and statistically significant positive effect of population growth on house price growth. In Table 2, which does not include any country or time fixed effects, the coefficient varies in the range of 1.47–3.21 percentage points. The inclusion of other controls does not substantially affect the coefficient. However, when we include immigration flows, the population growth coefficient increases dramatically which is perhaps explained by the sharp decrease in our available observations.

In Table 3, which controls for country fixed effects, the coefficient increases to the range of 1.74–3.17. So this implies that population growth has a stronger effect on house prices when we consider time variation within countries rather than looking across countries. In this case (country fixed effects), the coefficient decreases as we are adding more control variables with the exception of immigration flows which (as explained) substantially decreases the number of observations.

In Table 4, which controls for both country and time fixed effects, the coefficient is in the range of 2.14–2.58. This implies that when we control for both time and country fixed effects, the coefficient is less volatile across specifications with different controls.

Overall, we find a positive and statistically significant effect of population growth on house prices growth. What is worth nothing is that the effect is very robust to the inclusion of country fixed effects, time fixed effects, and other variables representing economic activity such as GDP, interest rate, and immigrations flows. This is exactly what our theoretical model predicts. A higher fraction of young people in the economy increases the demand for houses and increases house prices.

4.3.2 The effect of other economic and demographic characteristics on house prices

We discuss here the effect of our control variables on house prices. There is no clear effect of GDP on house prices from our regressions. Most specifications indicate a negative relationship, while others suggest a positive relationship (e.g., Table 4, columns 2 and 4). Moreover, very often the relationship is not statistically significant and it seems to be affected by the inclusion of interest rates.

The relationship between house prices and interest rate is more robust. Higher interest rate is associated with a smaller growth in house prices. Intuitively, when the cost to purchase a house is low, the demand is high which accelerates house price growth. Finally, we get a negative coefficient between house price growth and immigration flows. Nonetheless, these estimates are based on a small number of observations and the coefficients are not statistically significant.

4.4.1 Clustering standard errors by country

It is possible that our standard errors are contaminated by correlation between the observations. To check this, we use the same regression as in section 4.3, but now we cluster standard errors at the country level. Our assumption implies that observations are correlated within a country but independent across countries (clusters). Table 5 replicates the specification in Table 4 (with country and time fixed effects) but cluster standard errors at the country level. So naturally, the coefficients are identical but the standard deviation changes. As we can see from Table 5, when we cluster standard errors at the country level, the standard errors increase slightly but coefficients are statistically significant at 1%. Once more, the only exception is the specification which controls for immigration flows which feature a small number of observations.

Table 5.

House prices and demographics (cluster S.E. by country)

Note: The standard errors are given in parentheses. One, two, and three stars denote significance at the 10%, 5%, and 1% levels, respectively.

4.4.2 Fixed effects vs. random effects: Hausman test

To evaluate if a fixed effect or a random effect model is more appropriate for our exercise, we run a Hausman test (Table 6). The test evaluates whether errors are correlated with the regressors or not. The null hypothesis in the test is that they are correlated.

Table 6.

Hausman test

Note: The table reports the Hausman test for analyzing the difference between fixed effect and random effect coefficient. One, two, and three stars denote significance at the 10%, 5%, and 1% levels, respectively.

Since the probability is higher than 5%, which implies that the difference is not significant, we conclude that the fixed effect model does not produce significantly different results from the random effect model.