Impact of weak substitution between owning and renting a dwelling on housing market

Abstract

According to economic theory, an economically rational market agent searching for permanent housing in a particular stage in his/her life cycle should base his/her tenure considerations also by comparing rent to the user costs of homeownership, and among flats with otherwise identical housing services and security will select the cheaper of two alternative tenures. Economic theory perceives rental and owner-occupied housing as ‘communicating vessels’—a change in conditions in one necessarily entails changes in the other. This is called the ‘substitution effect’ and represents an important balancing mechanism in the housing market. Our hypothesis is that if the substitution between rental and owner-occupied housing in particular culture/society is weak, then the demand for owner-occupied housing becomes more income-elastic than vice versa. Consequently, in the case of a weak substitution effect, changes in house prices will closely mimic changes in household incomes, while in the case of a strong substitution effect this relationship will be much weaker. We confirmed our hypothesis by employing both a theoretical model and an empirical analysis of price data. If we assume poor responsiveness of housing supply, the main implication of our findings is that in societies with weak substitution effect between owning and renting we can expect higher house-price volatility and thus a higher chance of price bubbles appearing.

Appendices

Appendix 1: The simulation model details

1.1 Modelling household income structure

The household income structure is described by the density distribution function h(x) expressing the number of households with an annual income of x. The integral form of this function is H(x). The value of H(x) is the number of households with an annual income below or equal to x.

$$H(x) = \int\limits_{ - \infty }^{x} {h(x)dx}$$

The density function h(x) can be modelled by a surrogate function h_s(x) constructed as a linear combination of n Gaussians:

$$h_{s} (x) = \sum\limits_{i = 1}^{n} {a_{i} \frac{{e^{{ - \frac{{(x - \mu_{i} )^{2} }}{{2\sigma_{i}^{2} }}}} }}{{\sqrt {2\pi } }}} \quad {\text{then}}\quad H_{s} (x) = a_{0} + \sum\limits_{i = 1}^{n} {\frac{{a_{i} }}{{\sqrt {2\pi } }}\int\limits_{ - \infty }^{x} {e^{{ - \frac{{(x - \mu_{i} )^{2} }}{{2\sigma_{i}^{2} }}}} } dx} = a_{0} + \sum\limits_{i = 1}^{n} {a_{i} \varPhi \left( {\frac{{x - \mu_{i} }}{{\sigma_{i} }}} \right)dx}$$

where µ_i and σ_i are the mean value and the variance of the i-th gaussian, a_i is the amplitude (weight) of the i-th gaussian, a₀ is the integration constant assuring the zero value of H_s(x)= 0 at x = 0, Φ(x) is the cumulative distribution function of the standard normal probability distribution (with a zero mean value and unitary variance).

For practical reasons we set H_s(x)= 0 for all x < 0 since there are no households with negative income. Then the integral in the term for H_s(x) goes from 0 to x. The value of the integration constant a₀ is set to ensure H_s(0)= 0. The model parameters (n, µ_i, σ_i and a_i) are selected according to statistical data that minimise the difference between the model and data using the method of least squares. An example of a model for particular data set is shown in Fig. 5.

Fig. 5

Modelling household income. Note: The example compares recorded statistical data (shown as dots in the chart) and the model (the red line). The difference between the model and the data is shown as well (yellow line). The maximal difference is 3.5%, which is much better than the accuracy of the statistical data (10%). The model consists of 3 gaussians only (n = 3). Its parameters are shown in the table (right)

The percentage H_N(x₁,x₂) of households with an annual income within a certain interval (x₁,x₂) is then calculated as:

$$H_{N} (x_{1} ,x_{2} ) = H_{s} (x_{2} ) - H_{s} (x_{1} )$$

1.2 Modelling economic cycles

Housing affordability changes with economic cycles. The economic cycle is simplified here as the periodic alternation of periods of growth and recession. The duration of a single full economic cycle consisting of a single growth and a single recession phase is T. The relative duration of the growth part of the cycle is x_g. Its absolute duration is then t_g= x_gT. The growth rates during growth phases and decline rates during correction phases of the cycle are set by the constants r_g and r_r. Temporal scaling of the economy is then expressed using a multiplicative scaling function of time E(t) with a value of 1 for starting time t = 0.

$$E(t) = E_{n} \left( {\frac{t}{T}} \right)\quad {\text{where}}\quad E_{n} \left( x \right) = e^{{\left( {\left[ x \right] + \chi_{{ < 0,x_{g} )}} \left( {\left\{ x \right\}} \right)} \right)\ln (r_{g} ) + \left( {\left[ x \right] + \chi_{{ < x_{g} ,1)}} \left( {\left\{ x \right\}} \right)} \right)\ln (r_{r} )}}$$

where [x] is the floor function that gives the greatest integer value smaller or equal to x, {x} is the fraction function that gives the fractional part of x, so {x} = x − [x], Χ_<a,b)(t) is the indicator function that gives a value of 1 for t within a given interval < a,b) and a value of 0 elsewhere.

1.3 Modelling the price of owner-occupied housing and the costs of homeownership

All households demand one type of dwelling that may be owned or rented. To model house price P(t) we define a relative price change as:

$$P_{r}^{\prime } (t) = \frac{dP(t)}{dt}\frac{1}{P(t)}$$

and the relative change in demand for an owner-occupied dwelling as:

$$D_{Or}^{\prime \prime } (t) = \frac{{d^{2} D_{O} (t)}}{{dt^{2} }}\frac{1}{{D_{O} (t)}}$$

where D_O(t) is the number of households in owned flats. The change in house price P_r(t) of a dwelling is described by the following differential equation:

$$P_{r}^{\prime } (t) = D_{Or}^{\prime \prime } (t - t_{dD} )\Theta (t - t_{{d\Theta }} )\quad {\text{with initial condition}}\;P\left( t \right) = P_{0} (constant)\;{\text{for}}\;t \le 0$$

where Θ(t) is the supply factor that reflects changes in the number of flats available for sale on the market in time t, t_dD and t_dΘ are the time delay constants (reaction times).

The expression for $P_{r}^{\prime } \left( t \right)$ means that the house price follows changes in the number of households demanding owner-occupied property per time unit with a certain time delay. Where the derivative dD_O/dt is constant, this means that a constant number of households per time unit demand homeownership, which implies a stable market situation resulting in a constant house price. A change in dD_O/dt (which means the second derivative d²D_O/dt²≠ 0) induces a house price change.

A house price change is not determined solely by changes in demand D_O(t) but is also partly determined by changes in supply, which are represented in the previous equation by Θ(t). The supply factor Θ(t) representing the scale of new housing output produced by commercial development is determined by a ratio of price P(t) to reproduction costs B(t)—the so-called Tobin’s q. For the purpose of this study we use the following definition of Θ(t):

$$\frac{{d\Theta (t)}}{dt} = \left( {\frac{{P(t - t_{ds} )}}{{B_{{}} (t - t_{ds} )}} - 1} \right)c_{\Theta } \quad {\text{with initial condition}}\quad\Theta \left( t \right) = 1\;{\text{for}}\;t \le 0$$

This expression means that if P(t)> B(t), new flats are beginning to be developed, and after a certain reproduction time t_ds the increasing supply of flats will diminish the effect of demand on house price. Changes in Θ(t) follow the ratio of P/B with a delay of t_ds (the new supply responds to the market situation with a delay, owing to how long the residential construction process is). The scaling constant c_Θ means sensitivity to Θ(t) change, i.e. the greater the difference between P(t) and B(t), the greater the response from supply and thus the change in Θ(t). To avoid model instability, we limit the value of Θ(t) to an interval of< 0.5, 1> . So whenever Θ(t) exceeds one of the interval limits it is set as equal to that limit. If it is equal to 1, house price changes are determined solely by changes in demand (B(t)> P(t)). The reproduction costs function B(t) develops with economic cycles:

$$B(t) = B_{0} E(t)\quad {\text{where}}\;B_{0} \;{\text{are the initial reproduction costs in time}}\;t = 0.$$

In the discrete case with a time granularity of t_s, the equation for P(t) in the n-th time step can be rewritten as:

$$\frac{{\Delta P(nt_{s} )}}{{t_{s} }}\frac{1}{{P(nt_{s} )}} = \frac{{\Delta^{2} D_{O} \left( {(n - n_{dD} )t_{s} } \right)}}{{t_{s}^{2} }}\frac{1}{{D_{O} \left( {(n - n_{dD} )t_{s} } \right)}}\Theta \left( {(n - n_{{d\Theta }} )t_{s} } \right)$$

where our operators of difference are Δ and Δ² defined as:

$$\Delta P(t) = P(t) - P(t - t_{s} )\quad {\text{and}}\quad \Delta^{2} Q(t) = Q(t + t_{s} ) - 2Q(t) + Q(t - t_{s} )$$

The constants n_dD, n_dΘ and n_ds signify the delay expressed in time steps: n_dD= t_dD/t_s, n_dΘ= t_dΘ/t_s and n_ds= t_ds/t_s.

In the discrete case the equation for Θ(t) rewrites as:

$$\frac{{\Delta\Theta (nt_{s} )}}{{t_{s} }} = \left( {\frac{{P((n - n_{ds} )t_{s} )}}{{B((n - n_{ds} )t_{s} )}} - 1} \right)c_{\Theta }$$

The annual user costs of homeownership U(t) are defined as:

$U(t) = P(t)\left( {i + \delta } \right)$ where i is the interest rate and δ is the deprecation rate (constants).

The total housing costs of homeownership C_O(t) are equal to U(t) increased by the mortgage principal repayment costs coefficient c_m:

$C_{O} (t) = c_{m} U(t) = c_{m} P(t)\left( {i + \delta } \right)$ where c_m is the mortgage principal repayment costs coefficient.

1.4 Modelling rents and the costs of renting

The change in the annual rent price R(t) is modelled the same way as the change in house price P(t). We define the relative annual rent change $R_{r}^{\prime }$ as:

$$R_{r}^{\prime } (t) = \frac{dR(t)}{dt}\frac{1}{R(t)}$$

and the relative change in the demand for rental housing as:

$$D_{Rr}^{\prime \prime } (t) = \frac{{d^{2} D_{R} (t)}}{{dt^{2} }}\frac{1}{{D_{R} (t)}}$$

where D_R(t) is the number of households in rented dwellings. The change in annual rent can be expressed by the following differential equation:

$$R_{r}^{\prime } (t) = D_{Rr}^{\prime \prime } (t - t_{dN} )\alpha (t - t_{d\alpha } )\quad {\text{with initial condition}}\;R\left( t \right) = R_{0} \;(constant)\;{\text{for}}\;t \le 0$$

where α(t) is the supply factor reflecting the changing number of rental flats available on the market in time t, t_dN and t_dα are the time delay constants (response times).

The changes in the supply factor α(t) representing the scale of new supply of secondary market dwellings intended for rent is defined as follows:

$$\frac{d\alpha (t)}{dt} = \left( {I_{M} - \frac{{R(t) + G(t,t_{dG} )}}{{1 + i_{{}} + r(t)}}} \right)c_{\alpha } \quad {\text{with the initial condition}}\;\alpha \left( t \right) = 1\;{\text{for}}\;t \le 0$$

where I_M is the minimum total annual revenue from residential investment asked by investors [this can be scaled by the E(t) function], G(t,t_dG) is the average annual capital gain computed from changes in property price P(t) for time period t_dG: $G(t,t_{dG} ) = \frac{{P(t) - P(t - t_{dG} )}}{{t_{dG} }}$, i is the interest rate (constant), r(t) is the risk premium determined by the relative amount of low-income households (the first decile of income distribution) to the total number of households living in rental housing (its initial value is 0 and maximum value is 0.03), c_α is the scaling constant signifying a sensitivity to change of α(t); i.e. the greater the difference between I_M and the recent discounted total revenue from residential investment (rent and capital gain), the greater the supply response and thus the higher the change in α(t).

The expressions for R(t) and α(t) can be rewritten for the discrete case with a time granularity of t_s:

$$\frac{{\Delta R(nt_{s} )}}{{t_{s} }} = R(nt_{s} )\frac{{\Delta^{2} D_{R} \left( {(n - n_{dN} )t_{s} } \right)}}{{t_{s}^{2} }}\frac{1}{{D_{R} \left( {(n - n_{dN} )t_{s} } \right)}}\alpha \left( {(n - n_{d\alpha } )t_{s} } \right)$$

$$\frac{{\Delta \alpha (nt_{s} )}}{{t_{s} }} = \left( {I_{M} - \frac{{R(nt_{s} ) + G(nt_{s} ,n_{dG} t_{s} )}}{{1 + i_{{}} + r_{t} }}} \right)c_{\alpha }$$

$$G(nt_{s} ,n_{dG} t_{s} ) = \frac{{P(nt_{s} ) - P\left( {(n - n_{dG} )t_{s} } \right)}}{{n_{dG} t_{s} }}$$

1.5 Modelling the number of households

The total number of households D(t) in time t is structured into three categories:

$$D(t) = D_{P} (t) + D_{R} (t) + D_{O} (t)$$

• (a) Households sharing a flat with parentsD_P(t) This category is continuously increased by the constant number of new households D_Pn(t) per time unit (new families). The demographic changes/trends are not taken into account. The income structure of new households is described by distribution function H (see its definition above) and this function also does not change in time. D_P(t) is continuously reduced by households who move to rental or owner-occupied housing.

• (b) Households living in rental flatsD_R(t) These households are recruited mostly from the category of households living with parents D_P(t), especially new households D_Pn(t), when they assess the market situation and make a tenure choice.

• (c) Households living in owner-occupied flatsD_O(t) These households are recruited from both previous categories (D_P(t) and D_R(t)) when they assess the current market situation and make a tenure choice. We do not assume homeowners would move back to any of the other two tenure categories.

These categories are sorted in ascending order: Parents (lowest), Rental and Owner-occupation (highest). This study expects unidirectional movements of households across categories; it does not expect a flow in the reverse direction. Households in the two ‘lower’ tenure categories continuously assess the market situation and make possible tenure choices about moving to a ‘higher’ tenure category.

1.6 Modelling tenure choice and thresholds

Tenure choice is primarily determined by the affordability of alternative housing tenures, i.e. by the annual costs of rental R(t) and owner-occupied C_O(t) housing and annual household income. Each household that does not live in owner-occupied housing instantly will at a given moment compare its current relative housing expenses (evaluated as a percentage of household income) with the threshold value T_h(t). Then, if they are lower than threshold value, the tenure choice is made with a probability of ξ.

The threshold value T_h(t) changes with the periods of the economic cycle and in this way serves as a substitute for the trend in the income/employment rate during the economic cycle: during the economic growth phase it is increasing (this serves as a proxy for any increase in the income/employment rate) at a constant annual speed of T_g [%/year], while during the economic recession phase it is decreasing at a constant annual speed of T_r [%/year]. In the discrete case with a time granularity t_s, we calculate the value of T_h(nt_s) in the n-th step using its previous value T_h((n-1)t_s) while adhering to the following recursive rule:

$$T_{h} (nt_{s} ) = \left\langle {\begin{array}{*{20}c} {T_{h} ((n - 1)t_{s} ) + T_{gr} \frac{1}{{t_{s} }}} & {during\;growth} \\ {T_{h} ((n - 1)t_{s} ) - T_{rec} \frac{1}{{t_{s} }}} & {during\;recession} \\ \end{array} } \right.$$

Certain randomness can be observed in the tenure choice process: some households are either not informed about the current market situation or they hesitate even when their income is already sufficient to make a change in housing tenure. In this study, this randomness is modelled by introducing a single probability ξ constant, meaning that just ξ percent of all households who can make the decision really do so at the given time step. Others can make a decision during the next step if their income level still allows for it.

Since the threshold is set as a percentage, it has to be multiplied by rent R(t) or total costs of homeownership C_O(t). Therefore:

The threshold income for moving to a rental dwelling is: $Y_{R} (t) = T_{h} (t)R(t)$

The threshold income for moving to an owner-occupied dwelling is: $Y_{O} (t) = T_{h} (t)C_{O} (t)$

However, sufficient affordability is only the first necessary condition for a change in housing tenure. The main goal of our study is to test the impact on house price in two alternative situations:

• (a) Substitution between tenures exists In this case, households who make a tenure choice and have an income higher than the threshold income for moving to an owner-occupied dwelling additionally compare annual rent R(t) and the annual user costs of homeownership U(t) and select the less expensive tenure option.

• (b) Substitution between tenures does not exist In this case, households who make an actual tenure choice and have an income higher than the threshold income for moving to an owner-occupied dwelling always choose the owner-occupied dwelling.

1.7 The simulation method

The discrete model is used for a numeric simulation performed in temporal steps t_s. After each step (iteration) a new market situation is calculated and evaluated. The market situation is described as a set of measures affecting the possible tenure choice of households listed in Table 3.

Table 3 Variables affecting tenure choice

The n-th iteration is performed in several steps:

• (i) The new households are added to the D_P(nt_s) housing category (living with parents).

• (ii) A new set of market measures is calculated, including U(nt_s), R(nt_s) and household income thresholds Y_R(nt_s) and Y_O(nt_s).

• (iii) The number of households in the D_R(nt_s) category (living in rental flats) with income above the income threshold for owner-occupied housing Y_O(nt_s) and eligible to move (percentage ξ) is determined and they make a tenure choice. The decision is performed for them according to the selected alternative (substitution/no substitution) and they move to the chosen housing category D_R(nt_s) or D_O(nt_s); for the ‘no substitution’ alternative they move to owner-occupied housing D_O(nt_s).

• (iv) The number of households in the D_p(nt_s) category (living with parents) with income above both income thresholds Y_R(nt_s) and Y_O(nt_s) and eligible to move (percentage ξ) is determined. The decision is performed for them according to the selected alternative (substitution/no substitution) and they move to the chosen housing category D_R(nt_s) or D_O(nt_s).

• (v) Other households in the D_p(nt_s) category with income above the income threshold for rental housing Y_R(nt_s) but below the income threshold for owner-occupied housing Y_O(nt_s) and eligible to move (percentage ξ) are determined and they move to rental housing D_R(nt_s).

Appendix 2

See Table 4.

Table 4 Regression models explaining the changes in nominal house prices (24 countries)