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 owneroccupied 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 owneroccupied housing in particular culture/society is weak, then the demand for owneroccupied housing becomes more incomeelastic 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 houseprice volatility and thus a higher chance of price bubbles appearing.
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Notes
 1.
Tobin’s q is in this case based on the assumption that people seeking housing are indifferent as to whether the housing is new or old; new housing development appears once the price of older flats comes close to the price of acquiring a new flat. The greater the disproportion between the reproduction costs and the market price of older flats, the longer the market supply of new flats will remain ‘deaf’ to the rise in demand.
 2.
If the substitution effect is constrained, rental housing is residualised and the tenants would consist mainly of lowincome, lowerclass households, or households renting housing as just a temporary solution to their needs. Rental housing becomes a residual and transitional form of housing. High turnover increases the costs incurred by landlords and the bias in the social structure of tenants increases landlords’ risks.
 3.
OECD Analytical House Price database.
 4.
Social and private renting are not distinguished in our empirical analysis due to following reasons: (1) social renting broadens tenure choice as private renting, despite the fact that its allocation and rent setting are not marketbased and (2) besides lacking official comparative statistics, there are diverse systems of social renting—some of them closer while others farer from market allocation—and it would be difficult to decide whether particular regime of social renting should or should not be included into analysis.
 5.
Based on data provided in Housing Statistics in the EU 2010.
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Acknowledgements
The research on this paper was sponsored by the Czech Science Foundation with grant number 1606335S. We would like to thank OECD (Analytical House Price Database Department), and especially Christophe André, for kind provision of house price and supplementary data for selected OECD countries.
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Appendices
Appendix 1: The simulation model details
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.
The density function h(x) can be modelled by a surrogate function h_{s}(x) constructed as a linear combination of n Gaussians:
where µ_{i} and σ_{i} are the mean value and the variance of the ith gaussian, a_{i} is the amplitude (weight) of the ith gaussian, a_{0} 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 a_{0} 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.
The percentage H_{N}(x_{1},x_{2}) of households with an annual income within a certain interval (x_{1},x_{2}) is then calculated as:
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_{g}T. 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.
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.
Modelling the price of owneroccupied 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:
and the relative change in demand for an owneroccupied dwelling as:
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:
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 owneroccupied 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 d^{2}D_{O}/dt^{2}≠ 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 socalled Tobin’s q. For the purpose of this study we use the following definition of Θ(t):
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:
In the discrete case with a time granularity of t_{s}, the equation for P(t) in the nth time step can be rewritten as:
where our operators of difference are Δ and Δ^{2} defined as:
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:
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.
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:
and the relative change in the demand for rental housing as:
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:
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:
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 lowincome 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}:
Modelling the number of households
The total number of households D(t) in time t is structured into three categories:

(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 owneroccupied 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 owneroccupied 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 Owneroccupation (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.
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 owneroccupied C_{O}(t) housing and annual household income. Each household that does not live in owneroccupied 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 nth step using its previous value T_{h}((n1)t_{s}) while adhering to the following recursive rule:
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 owneroccupied 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 owneroccupied 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 owneroccupied dwelling always choose the owneroccupied dwelling.
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.
The nth 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 owneroccupied 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 owneroccupied 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 owneroccupied 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.
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Lux, M., Sunega, P. & Jakubek, J. Impact of weak substitution between owning and renting a dwelling on housing market. J Hous and the Built Environ (2019). https://doi.org/10.1007/s10901019096613
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Keywords
 House prices
 Housing demand
 Homeownership