Location, location, other locations: on spatially
dependent housing prices
Nicol´as Dur´an
∗Supervisor: dr. Pim Heijnen
August 22, 2015
We seek to explain the spatial dependence in housing prices which is well doc-umented in the spatial econometrics literature, yet insufficiently supported theoretically. We model a regional housing market with locations that dif-fer in their amenities, and households that enjoy those within a random utility framework. Assuming that a shock hits one of the regions and that house-holds bare financial costs if selling their house at a loss, we compute the equi-librium for the shocked region in four cases: (1) without financial costs nor shock; (2) without financial costs, but after the shock hits; (3) with financial costs that dominate the shock; and finally, (4) with the shock dominating the financial costs. In every case the market equilibrates with inter-regional de-pendent prices; however, in the last case studied we also account for intra-regional spatial spillovers between equilibrium prices. We fit our conclusions to recent studies on the effect of earthquakes on housing prices in the northern provinces of the Netherlands.
1
Introduction
“The three most important things to look for when buying a house are: location, location, and location.”1
When searching for a house to purchase, households put special emphasis in its location. The former expression has typically been used to stress its relevance. However, the value of a house depends on many other features as well. Empirical studies using hedonic methods to estimate housing prices fit them to amenities and local public goods (e.g. schools and air quality in the district2, or local crime rates3), as well as to characteristics of the house (e.g. size, years since
construction, number of bedrooms) 4. Moreover, other variables that are increasingly being
employed in explaining price formation in the housing market are prices at which other houses have been sold, as well as amenities offered in other neighborhoods or even cities, sometimes distant ones (Bailey et al., 2014; Baltagi and Li, 2014; Baltagi and Bresson, 2011; Fingleton, 2008).
These variables are typically addressed with spatial hedonic models which incorporate ei-ther spatial lags of the dependent variable (oei-ther house prices, nearby or in oei-ther locations), a spatially autocorrelated error term, spatial lags of explanatory variables (characteristics of other houses and amenities offered in other locations), or any combination of the three. The spatial econometrics literature has been fitting spatial hedonic models to explain housing prices for over two decades since the early models by Can (1992) and Dubin (1992). Yet, although these improvements are important to better understand price formation in the housing market, there is a vale of criticism over this literature since very little to no theory on spatial dependency or spillovers in real state markets has been developed (P´aez, 2009; Tsutsumi and Seya, 2009; Corrado and Fingleton, 2012; Gibbons and Overman, 2012). Further theoretical development is needed, particularly on questions such as which the mechanism behind these spatial interactions is. This paper faces said theoretical challenge and fills the gap in the literature by modeling a
1Shapiro (2006) attributes the expression to The Van Nuys News (CA, 1956), although it is believed to have
first appeared even sooner, in an advertisement in the Chicago Tribune as early as 1926.
2See Brasington and Hite (2005). 3See Bishop and Murphy (2011).
regional housing market which equilibrates for prices that depend on both, other prices, and amenities in other locations.
We model a housing market that features an outside and a central region with the latter of-fering differentiated benefits, and with households indexed by the utility they derive from those. We also divide the central region in two locations which receive a shock that reduces their amenities in different magnitudes. Post-shock, indifferent households may see their locational preferences change, which could drive each location’s market to new equilibrium conditions. Moreover, we assume that households finance their houses through mortgages. After the shock hits, because of the reduction in benefits, they might prefer to sell the house. However, since prices have dropped, selling the house might not suffice to cover what is still due of the mort-gage imposing financial costs on households and reducing their mobility (Ferreira et al., 2010). By distinguishing between a case in which the shock has a larger effect on households than the financial costs, and vice-versa, we find that the housing market equilibrates for spatially dependent prices.
We account for two types of spatial dependencies in our model: on the one hand, there is a positive and direct effect form improvements in amenities offered in the outside region. This spatial dependency is sourced in households having the outside region’s cost of living as a benchmark, which they compare to that in the central region in order to decide where to demand housing. Moreover, it is present in every equilibrium we compute.
Finally, we make use of recent empirical studies on the effect of earthquakes on housing prices in the northern provinces of the Netherlands to test how our theoretical framework per-forms in predicting these findings. We argue that our model predicts well the trends observed in housing prices in this region in the context of earthquakes taking place.
The paper is organized as follows: in the next section we develop a literature review on the empirical studies that use spatial hedonic models for housing prices. Section 3 presents the pre- and post-shock models without financial costs. Section 4 develops an equilibrium in the housing market for the two cases that compare the financial costs and the shock’s effect. Section 5 present the empirical studies on earthquakes and housing prices in Groningen and explain their results through insights drawn from our theoretical approach, and Section 6 concludes.
2
Literature Review
The purpose of this paper is to develop theory able to explain spatial interaction in the housing market. Most of the literature concerning spatial hedonic models usually deals with speci-fication issues5, such as which type of spatial interaction should be modeled for a better fit, rather than its theoretical background6. However, very little concern is placed on the
mecha-nism behind the observed spatial interaction between house prices. That is, why do housing prices depend on each other and, moreover, why do we observe that quality improvements on amenities in one location affect prices in houses located elsewhere?
Addressing this issue, and on the specification of spatial hedonic models and their relation with households’ willingness to pay for amenities, Small and Steimetz (2012) show that, in a competitive housing market, quality improvements on amenities are welfare enhancing only if they somehow enter the utility function of households located elsewhere. This, in turn, implies positive externalities on other locations from improvements on local amenities, which appear as spatial multipliers in the empirical literature. However, the authors claim that a spatial lag on the dependent variable in hedonic spatial models can be fitted to both, a mechanism in which externalities are solely pecuniary and hence not welfare enhancing, as well as to improvements which are technological, and therefore do impact welfare. Therefore, they advocate for the
5See Can (1992) for a seminal study on this.
inclusion of a spatial lag in the spatial hedonic model if and only if the improvement of local amenities enters the utility function of non-local households, so as to measure the real impact of a quality enhancement on some local amenity. However, despite theoretically approaching the relevance of introducing spatially lagged variables and its interpretation, the authors do not provide any mechanism behind the spatial multiplier effect on these models.
The study by Small and Steimetz directly undermines other studies whose main purpose is to determine the benefits behind a quality improvement in some local amenity. Only if the marginal benefits induced by the latter enter non-local households’ utility function, a spatial lag and hence a spatial multiplier effect can be modeled. Otherwise, there would be significant overestimation in welfare enhancement from improving amenities. Kim et al. (2003), analyzing the benefits of a reduction in air pollution in Seoul through a spatial hedonic model on real state values, estimate a spatial multiplier and indirect effect of 2.22 from a 1% improvement of the former on the latter, ceteris paribus, deduced from a spatial direct effect of 0.55. However, the authors do not argue for a technological externality from the air quality improvement and hence, provided the improvement was solely pecuniary and according to Small and Steimetz, they are overestimating those benefits by 122%. Similarly, Anselin and Lozano-Gracia (2008) show the magnitude in the errors incurred if spatial correlation is not taken under consideration when measuring the willingness to pay for improvements in air quality. The authors would be overestimating their results by almost 50% were there no technological externalities from local air quality improvements. Several other more recent papers are subject to this problem and not only regarding air quality, but also the benefits of reducing airport noise (Cohen and Coughlin, 2008); distance to swine production establishments (Kim and Goldsmith, 2009); improving access to potable water (Anselin et al., 2008); shortening the distance to high speed rail stations (Chen and Haynes, 2014); and whether road or railway noise is more disliked (Andersson et al., 2010). Therefore, improving our understanding of these externalities would enhance the fit of these estimates.
interpreted as the marginal willingness to pay for an improvement on them. Obtaining signif-icant coefficients for the spatially lagged variables (dependent, explanatory, or autocorrelated error term) when fitting spatial hedonic models imply that the marginal willingness to pay for a quality improvement in some amenity internalizes positive externalities form improvements on the same amenity, but in other locations. However, little explanation on the mechanism behind this externalities is usually offered.
A first example on this literature is found in Dubin (1992) who motivated her research ana-lyzing why neighborhood attributes would not explain house prices as much as it was thought they would. According to the author, the multicentric feature of cities and measurement er-rors on neighborhood characteristics’ quality might be behind the lack of results. Typically, accessibility and other important neighborhood attributes are measured by the distance to focal points of which there are many in a city. Moreover, quality measures are always problematic and neighborhood boundaries imprecise. Therefore, rather than modeling these features, she proposes to model the correlation in the error terms resulting after fitting the house price to its own characteristics, attending to the possible biased results by explicitly taking into account the spatial relationship using kriging7. Nevertheless, no emphasis is put on the fact that there
could be non-zero elements in the off diagonal terms of the regional matrix used to model the errors’ spatial autocorrelation. Her results on the effect of neighborhood attributes on Balti-more’s housing prices are sizable, as opposed to what previous studies showed, giving some of the earliest proof of spatial interaction among housing prices through the use of spatial hedonic technics.
Recent literature on spatial hedonic models for housing prices is not extensive, as opposed to its non-spatial version8. Many examples on the former seek to determine unbiased price elas-ticities of demand for quality improvements in house or neighborhood specific characteristics. Brasington and Hite (2005) estimate a spatial hedonic model for housing prices, with spatial lags on the endogenous and explanatory variables, by which the authors find a price elasticity of demand for environmental quality of -0.12. Moreover, they find significant spatial effects between prices, as well as characteristics, of houses sold nearby. However, the motivation
be-7See Oliver and Webster (1990) for an explanation on the kirging method for geographical prediction.
8See Farber (1998) for an early literature survey on hedonic methods employed in the study of the effect of
hind the use of spatial statistics methods is to account for the known spatial correlation that is typically found in the housing market and hence obtain unbiased estimates, regardless of the mechanism that determines these spatial correlations. Some previous examples use simpler models (e.g. Gawande and Jenkins-Smith (2001), Leggett and Bockstael (2000)), that are spe-cial cases of that used in Brasington and Hite (2005) and do not account for the theory behind their spatial estimates either.
All these studies that employ spatial hedonic methods would be benefited from filling the gap in the literature on the theory behind the spatial spillovers modeled by them.
3
Frictionless regional housing market
The model we propose features a housing market with two main regions, outside and central, in which a mass 1 of households own their houses. The central region is divided into two locations named 1 and 2 for lack of better names, which are in principle equal to each other, except for their inelastic housing supply, which may differ and are denoted by si, with i = 1, 2. Total
housing supply in the central region cannot host the entire mass of households in this market, namely s1+ s2 = sR< 1.
Flow costs for living in either central location, and outside are denoted by pi with i =
1, 2, and ¯p, respectively. Households are indexed by c, a random variable with cumulative distribution function F (·), with F (0) > 0 and support in R, which represents the benefits enjoyed by amenities offered in the central region.
In what follows we determine the frictionless equilibrium for this housing market, with regional spatial dependency in prices. We do so for two cases: first, a pre-shock equilibrium in which we get prices for the model as laid out so far. Second, a post-shock equilibrium, in which we introduce a shock that reduces benefits at both central locations, but does so asymmetrically.
3.1
Frictionless pre-shock equilibrium
region is relatively less expensive and hence ¯p < pR9.
A household c prefers living in the central region if its net utility is at least as large as the cost of living outside:
c − pR≥ −¯p.
Then, the indifferent household between the central and outside region can be denoted as c∗ = p∗R− ¯p. It follows that every household c ≥ c∗prefers living in the central region. This condition determines the demand for housing in the latter, since 1 − F (c∗) households satisfy the latter condition.
Equilibrium is reached if demand equals supply (sR) at the equilibrium price p∗R, such that
1 − F (p∗R− ¯p) = sR.
Therefore, the pre-shock, frictionless, equilibrium price in the central region equals p∗R= F−1(1 − sR) + ¯p,
with 1 − F (c∗) households living within either one of the two central locations at a price given by p∗R> ¯p.
Equilibrium prices in the central region depend on the outside price. The latter is a bench-mark that gives a price floor above which the housing bench-market in the central region equilibrates. This suggests the presence of direct spatial effects consistent with a sales comparison approach, which posits a direct causal mechanism through comparing prices of similar houses to deter-mine the value of a real state in a transaction10.
In the next subsection, an asymmetrical shock is introduced that drives the market into a new equilibrium with different conditions than those just given.
3.2
Frictionless post-shock equilibrium
Suppose that the central region is hit by a shock which affects the amenities enjoyed at each of its locations in such a way that, without any loss of generality, at location i = 1, 2, benefits
9It is also possible to think of a contrary case in which the central region is cheaper, while the outside is in
general more expensive. A clear example of this could be to consider the northern provinces of the Netherlands as the central region, while the more expensive south is the outside.
10See (Kim et al., 2003) for an example of a hedonic spatial model for the housing market under the sales
enjoyed are reduced by µi, with µ1 > µ2. Moreover, suppose that there are two types of
households: “new” and “old”. The latter are households who already own a house. The former are newcomers to homeownership and replace “old” households at a death rate equal to λ. It follows to analyze how this asymmetry affects housing demands in the central region for each household type: central households (already living in the central region); outsiders (households living in the outside region); and newcomers.
3.2.1 Indifferent households
A central household indifferent between staying and moving to the outside region satisfies the following condition:
c − µi − p∗R = pi− p∗R− ¯p,
where the left hand side is the total utility derived from living in central location i = 1, 2, and the right hand side is the total utility derived from moving to the outside region. Notice that, since prices are in fact flow costs, namely mortgage payments, and households moving out still have to make payments worth p∗Runtil their debt is cancelled, then profits collected from selling the house are the difference between the price they get for their house and what is still due. Then, denote c∗i as the indifferent household living in i = 1, 2 and define it as
c∗i = µi+ pi − ¯p.
Similarly, denote by c∗λi the newcomer who is indifferent between the outside and central region i, with i = 1, 2, and define it as
c∗λi= µi+ pi− ¯p. (1)
The indifferent outsider is the same as in (1) and therefore, the threshold for them equals that for newcomers. These thresholds determine every household who is willing to purchase a house at any of the two locations in the central region. Those with c larger or equal to the latter are willing to do so.
j, with i, j = 1, 2 and i 6= j, as follows:
c + µj + pj − pi ≥ c + µi (2)
Similarly, newcomers and outsiders decide to move to central location i rather than j, with i, j = 1, 2 and i 6= j provided
c + µi + pi− ¯p ≥ c + µj + pj − ¯p.
Hence, both sets of households demand housing at either location according to whether µ1 +
p1 ≥ µ2+ p2 or the opposite holds.
Notice that if conditions are such that a given type of household demands housing at location 1, for example, then every household of the same type will do so, since the condition does not involve c. Therefore, it is only possible for an equilibrium to hold provided households are indifferent between one or the other central location (Glaeser et al., 2008), which occurs only if µ1 + p1 = µ2 + p2. Intuitively, if any other case than equality holds, then one of
both central housing market collapses because there would be no demand for houses there. Moreover, with inelastic supplies, the central location receiving the entire market demand would see prices rise up to a point at which the latter equality is reached (Glaeser and Nathanson, 2014). Therefore, without frictions, the reduction in the cost of living at either central location must fully compensate for the fall in benefits equating both locations’ full cost of living, namely µi+ pifor i = 1, 2.
3.2.2 Demands and equilibrium
Equilibrium conditions are obtained by equating demands at each location to their respective supplies. We have shown that, post-shock, both flow costs need to stabilize at a point in which µ1+ p1 = µ2+ p2, so that an equilibrium can exist. Given this condition, notice that
c∗i = c∗λi= ¯c, for i = 1, 2, (3) and hence, equating demand and supply we get the following
which holds for equilibrium prices at i = 1, 2
p∗i = ¯p + F−1(1 − sR) − µi = p∗R− µi.
Without financial costs, prices at the central region are lowered by as much as benefits were reduced after the negative shock hit. This accounts for a direct effect on prices through a reduc-tion in the marginal willingness to pay for the amenity that received the shock. Moreover, there is still spatial dependency since both prices fully internalize those in the outside region as it was the case before any shock hit.
It remains to determine how newcomers and outsiders distribute between the two central locations. To do so, let us compute equilibrium conditions for each central location as follows
s1 = (1 − λ)s1(1 − F (c∗1)) + λδ(1 − F (c ∗ λ1)) + (1 − λ)δ [F (c ∗ ) − F (c∗λ1)] s2 = (1 − λ)s2(1 − F (c∗2)) + λ(1 − δ)(1 − F (c ∗ λ2)) + (1 − λ)(1 − δ) [F (c ∗ ) − F (c∗λ2)] (5)
Equations in (5) state which households demand housing at which location, with δ the fraction of newcomers and outsiders moving into 1. Incentives to move to the central region for one or the other group are exactly the same since the distribution of household is the same for both groups. Hence, the way in which they sort between locations should be identical as well. Moreover, notice that the last term in both equations of (5) equals zero, implying that no outsider moves to the central region. This is a consequence of prices dropping just as much as benefits, which keeps indifferent households unchanged.
Substituting (4) into the equation for s1 in (5), and given that (3) holds, we get
δ∗ 1 − δ∗ =
s1
s2
. In the following proposition we summarize this equilibrium:
Proposition 1. In a frictionless housing market with three locations as described in this section, after a shock hits the central region such that amenities are reduced in each locationi = 1, 2 byµi, withµ1 > µ2, andµ1+ p1 = µ2+ p2 holds, then new equilibrium prices are such that
p∗i = p∗R− µi. Moreover, households do not change regions nor locations; however, newcomers
We have established that if a shock were to hit the central region in such a way so as to reduce the quality in one or more of its amenities, housing prices would be reduced in the same magnitude as the reduction in benefits at each central location. The effect is direct and the mechanism works through a drop in the marginal willingness to pay for the amenity that suffered the shock. Prices at each location, however, would still depend on the benchmark in the outside, hence spatial dependency persists as a sales comparison mechanism, with the price in the outside as benchmark for costs of living.
We presented a model that, so far, has established a link between prices in an outside and cheaper region as a benchmark to those in the central region, addressing the sales comparison approach. However, we are yet to determine spatial spillovers from one central location to the other from the shock received at both. In order to do so, in the next section we study a more realistic model, one that features frictions given by households unwilling to sell their houses at a lower price than purchased.
4
Regional housing market with financial costs
Households take mortgage loans to purchase their houses. They commit to make mortgage payments for extended periods of time, which correspond to the price at which the house was bought. If at any moment the household is financially distressed and needs to sell the house to cancel the mortgage, it might find the market depressed and see that it is worth much less than when purchased. Selling the house would not cover for the entire debt assuring that, if getting a new mortgage, it will not be for a house as valuable, since its finances would be deteriorated. Therefore, households bare a cost from selling at a loss with financial consequences that exceed the straightforward difference in prices (Chan, 2001).
We introduce said costs to the previous model allowing for a financial friction if households sell at a loss. The latter is denoted by a parameter α > 1 which augments the loss from the difference in prices, namely pi− p∗R. We have shown in the previous section that in a frictionless
the other central location, or stay and enjoy reduced benefits.
Assuming that shocks occur among households with mortgage payments due, the financial cost from selling at a loss characterized by α is defined such that, payoffs earned in housing transactions in i = 1, 2 are denoted by
x (pi) =
( α (pi− p∗R) if pi < p∗R
pi − p∗R if pi ≥ p∗R.
(6) Without financial frictions, central households do not move since the indifferent ones re-main unchanged after the shock hit. Introducing the friction may have modified indifferent households’ preferences. In the next subsection, we study those.
We structure the analysis as follows: first, we determine each indifferent household within the new market conditions. Second, we show that if the negative shock on benefits from both central locations is dominated by the financial costs, then no spatial spillover arise between the latter two. Finally, we show that in this context, only if the financial costs are dominated by the shock an equilibrium with dependency among prices in the central region is possible. Moreover, in this context the dependence between prices arise in the form of spatial spillovers. We also provide conditions for both equilibriums to hold.
4.1
Indifferent households with financial costs
Let us first introduce the new indifferent households in this context. First, we compute the central household who is indifferent between staying and moving to the outside region. Then, we do so for newcomers and outsiders indifferent between staying out and moving into the central region. Finally, we compare these to the current indifferent household (¯c = c∗), and between them as well. The first set of comparisons allows us to determine both, whether there will be any outsider demanding housing in the central region, and whether any central household prefers to move out. By comparing them we can determine the size of these “crossed” demands, were there any.
Central households form location i = 1, 2 prefer staying in the central region provided the following condition is satisfied:
and therefore, the indifferent central household is such that c∗i = µi+ p∗R− ¯p + x (pi) .
For both, newcomers and outsiders, the threshold above which they prefer the central region over the outside is still the same as denoted in (1). In the case of outsiders, they have not received any negative impact on benefits and therefore face no financial costs. For newcomers, in turn, they had no houses to suffer any kind of cost in the first place. Hence, (6) does not apply for the computation of neither set of payoffs. We now compare these thresholds to c∗ and see for which case, or under which conditions do outsiders demand housing in the central region, and/or central households do so in the outside.
There is a distinction to be made with respect to whether financial costs are stronger or weaker than the negative impact from the shock. Intuitively, on the one hand, if they are stronger households might find it profitable not to sell their house and enjoy the reduced benefits. On the other hand, if they are weaker, they might find it more profitable to sell the house and bare the financial cost, but enjoy a larger set of benefits elsewhere. We address each case at a time, beginning with the former.
4.2
Equilibrium with dominating financial costs
Assume that the costs from the financial loss are larger than those of the reduction in amenities from the shock, or formally that
µi < α(pR− pi), for i = 1, 2. (7)
Let us first compute c∗i − c∗
for i = 1, 2 and get
c∗i − c∗ = µi+ α(pi− p∗R) < 0, (8)
Similarly, we compute c∗λi− c∗ and see whether there are outsiders demanding housing in
the central region:
c∗λi− c∗ = µi+ pi− p∗R. (9)
Without any frictions, c∗λi = c∗ since p∗R = µi + pi, as discussed in the previous section.
However, provided central households are reluctant to sell at a loss, the latter is no longer straightforward. Only outsiders with c such that c∗ > c > c∗λi demand housing in the central region. Hence, there will only be outsiders willing to move to the central region provided p∗R > µi+ pi. This condition is quite straightforward since it basically states that the absolute
cost of living in the central region (µi+ pi) needs to drop from its previous level (p∗R) for some
outsiders to find it profitable to move.
Comparing c∗i and c∗λiby computing c∗i − c∗λi, we get
c∗i − c∗λi= (1 − α)(p∗R− pi) < 0, (10)
since α > 1 and thus, c∗i < cλialways holds. From (9) and (10) possibilities are twofold: either,
c∗λ ≥ c∗ > c∗
i holds and therefore, central households stay in the central region due to financial
costs dominating without outsiders moving in; or c∗ ≥ c∗ λ > c
∗
i holds, with some outsiders
demanding housing in the central region with central households staying in the latter.
The following proposition summarizes preferences for the case in which financial costs dominate:
Proposition 2. Provided financial costs dominate at both locations i = 1, 2, namely µi <
α(p∗R − pi) holds, then either c∗ ≥ c∗λ > c∗i or c∗λ ≥ c∗ > c∗i hold, meaning that central
households stay in the central region, while outsiders either demand housing in the central region, or stay outside, respectively.
No central household moves post-shock if financial costs dominate the reduction in benefits. Since the indifference threshold below which central households move to the outside lays to the left of the previous one, then every household already living in the central region prefer staying there. Do central households stay in their location? Or are there incentives for them to move from one to the other? To determine this, we analyze the following condition for i, j = 1, 2 and i 6= j:
which implies that households at central location i 6= j are indifferent between moving or staying in i. Let us propose the conditions under which central households, newcomers, and outsiders, choose among central locations:
Proposition 3. If central households are indifferent between central locations and therefore condition (11) holds with equality, it follows thatµ1+ p1 6= µ2+ p2, and therefore, newcomers
and outsiders, if any, either demand housing in location 1, or do so in location 2, exclusively. Conversely, if newcomers are indifferent between central locations becauseµ1+ p1 = µ2+ p2
holds, it follows that central households living in 1 prefer moving to 2, while those living in 2, prefer staying in 2.
Proof. See Appendix.
Intuitively, Proposition 3 suggests that households in location 1 are better off moving to location 2 since the impact from the shock they suffered is both, larger than that in 2 and dominates the financial costs of moving, and hence it would gain them µ1− µ2in utility, as they
would also gain from paying p2 < p∗R. However, it is easy to see that only for the case in which
absolute costs of living at each location equalize, the market can achieve an equilibrium, namely the second part from Proposition 3 holds. Notice that, whichever the case of those described in Proposition 2, and if the first part of Proposition 3 holds and hence, µi + pi > µj + pj for
i, j = 1, 2 and i 6= j, we get the following equilibrium equation for central location i:
si = si(1 − λ)(1 − F (ci)), (12)
which cannot be satisfied since 0 < λ < 1 and 0 < F (·) < 1. Intuitively, there is always excess supply in i that reduce prices until µi+ pi = µj+ pj, and hence, only the case for indifference
between locations for newcomers and outsiders can hold in an equilibrium11. This implies that central households living in 1 demand housing in 2, while newcomers and outsiders, if any, demand housing in the central region sorting randomly between locations.
Given these conditions, in what follows we propose equilibrium prices in both central loca-tions:
11See Glaeser and Gottlieb (2009) for an analysis on consumers’ indifference as a prerequisite for regional
Proposition 4. Given financial costs dominating the shock’s reduction in benefits at both lo-cations and that µ1 + p1 = µ2 + p2 holds, then equilibrium prices in locations i = 1, 2 are
such thatp∗R ≥ µi+ pi. Moreover, if outsiders do not demand housing in the central region and
thereforec∗λ > c∗, then prices are formed as follows:
p∗i = ¯p − µi+ F−1(1 − sR) = p∗R− µi.
Given the same conditions but with outsiders demanding housing in the central region, then p∗i = ¯p − µi+ c∗λ, with c∗λ = F−1 1 −sR λ + 1 − λ λ δs2 − λ(1 − δ)s1 δ(1 − λ)sR . Proof. See Appendix.
Proposition 4 insights are twofold: on the one hand, for a dominating financial cost but a shock’s effect large enough so as for µi+ pi ≥ p∗Rto hold, which is what was assumed with the
condition c∗λ ≥ c∗
, we get prices in both central locations as if no financial friction existed. As should be easy to see, this in turn implies that the condition assumed regarding the indifferent households can only hold with equality, namely that c∗λ = c∗. Central households are not moving from the central region and with a price that exactly compensates for the amenities lost, then indifferent households remain unchanged, as does the marginal willingness to pay for each type of household. Hence, conditions are the same as in the case of a shock without financial costs.
On the other hand, again for a dominating financial cost, but rather a shock’s effect that is not as strong so as for p∗R ≥ µi + pi to hold instead, which is the condition for c∗ ≥ c∗λ, then
prices in equilibrium are as proposed with c∗λ < F−1(1 − sR). The shock is not as strong, and
conditions. This in turn weakens demand, which reduces prices and sends central homeowners even deeper “underwater” with their mortgages. Equilibrium is reached for a price that makes the outsider indifferent between enjoying the benefits from the central region, although reduced by the shock, or stay outside.
Regarding spatial interaction, prices in both central locations fully internalize the benchmark price from the outside as it was the case above. However, there is still no spatial lag for the price in the other central location. Prices only incorporate the marginal effect from a reduction in benefits elicited by the shock in their own location, as well as the marginal willingness to pay for a house of outsiders and newcomers, namely c∗λ− µi.
We now turn to the analysis of the housing market conditioned on the amenities reduction dominating the financial costs.
4.3
Equilibrium with dominating shock
Assume that the reduction in benefits due to the shock dominates the financial costs, and hence the following holds:
µi > α(pR− pi), for i = 1, 2. (13)
We can immediately conclude that an equilibrium in which p∗R ≥ pi + µi for i = 1, 2, is
impossible to reach, since from (13) we get that µi+ pi >
µi
α + pi > p
∗
R. (14)
If the reduction in benefits dominates the financial costs, then prices do not drop as much as benefits do. In a frictionless market, the reduction in benefits is entirely reflected by a drop in prices equating p∗R to pi + µi. Moreover, if financial costs dominate the shock, the drop in
prices overcompensate for the benefits reduction. However, in the context of a µi larger than
the former, prices do not fully compensate for the reduced utility.
Proposition 5. Provided the reduction in benefits dominates the financial costs in both locations i = 1, 2, namely µi > α(p∗R− pi) holds, then it follows that c∗λ > c
∗ i > c
∗ hold, meaning that
Proof. The first inequality was proven in (10).
The second is reached straightforward by noting that c∗i = c∗+ µi+ α(pi − p∗R) > c
∗for i = 1, 2,
where the inequality holds given that (13) also holds.
From Proposition 5 we get that, first, outsiders do not demand any housing in the central region but rather prefer to stay living in the outside. Second, some central households demand housing in the outside while the rest stay in the central region. Proposition 3 states preferences regarding which central location is chosen by either central households staying in the central region or newcomers. Moreover, since the same reasoning applies, then it has been shown already in (12) that an equilibrium cannot hold unless µ1+p1 = µ2+p2holds as well. Therefore,
we propose the following equilibrium prices for this particular case:
Proposition 6. Provided the shock dominates the financial costs and hence, formally, µi >
α(p∗R − pi) for i = 1, 2, holds, and moreover, the second case in Proposition 3 holds, and
µ1+ p1 = µ2+ p2, then prices at each central location are such that
p∗i = ¯p + α − 1 α F −1 (1 − sR) + µj − c∗j − µi+ c∗j (15) withi = 1, 2, j 6= i, and c∗2 = F−1 δ (1 − λF (c ∗) − (1 − λ)s R) − s1 δ(1 − λ)(1 − sR) , c∗1 = F−1 s1 s2 1 + λ 1 − λ F (c∗) − sR 1 − sR − 1 − s2 δ(1 − λ)(1 − sR) . Proof. See Appendix for proof and conditions for the equilibrium to hold.
If the shock dominates, prices in equilibrium incorporate the latter’s marginal effect on their own location as it was the case before, namely µifor i = 1, 2. However, notice that, since c∗jis in
fraction of α that is purely financial cost rather than the nominal loss from the price difference, namely (α − 1)/α. Therefore, the price in i incorporates the entire marginal effect from the shock, that is µi+ µj, yet only internalizes the marginal willingness to pay at the other central
location after considering the financial constraint for moving.
This is an indirect effect from the shock in one central location to prices in the other, namely a spatial spillover between central regions. A reduction in amenities across both locations not only impacts housing prices directly, but also does so through a spatially lagged indirect effect which is mediated by the financial constraint. Moreover, spatial dependence with the outside price remains. This implies that a quality improvement in some outside amenity still affects prices in the central region, although in a one to one basis, and regardless of the financial friction.
5
Application to the case of earthquakes in Groningen
The northern provinces of the Netherlands have experienced earthquakes for the past thirty years, which are induced by the extraction of natural gas that takes place mainly in the province of Groningen (van Eck et al., 2006). It is now well understood that this extraction leads to soil subsidence inducing earthquakes some decades after it began (Ellsworth, 2013). Although not too strong (between magnitudes of 2 and 4 in the Richter scale), these earthquakes affect households in many ways (van der Voort and Vanclay, 2015), one of which is reducing housing prices.
which households internalize the increased probability of the house collapsing, inducing major injuries to its inhabitants, which reduces the willingness to pay for a house in Groningen.
The authors focus on the last mechanism, by arguing that the first two monetary effects would not plausibly affect housing prices since, regarding the first one, damages are sunk costs which is rational to repair amid the sale of the house. With respect to the second mechanism, they argue that plausible future damages should not reduce housing prices since they are fully compensated by the firm extracting the natural gas. Hence, the fall in prices should be driven by non-monetary mechanisms, such as a reduction in the amenities enjoyed in the location, due to an increased likelihood of some disastrous event taking place.
The context of this study fits well to that of a shock in our model, as do the conclusion that prices of houses in Groningen fall, due to a reduction in the amenities that the province provides. In every equilibrium we computed, prices post-shock were reduced with respect to the previous equilibrium price. Households after the shock feel insecure because of an increased risk of major injuries which reduces marginal willingness to pay for a house located in Groningen. However, in order to fully relate this case to our model’s predictions, we are left to see whether prices in close by locations have also dropped, and how do financial costs affect the mechanism. We turn the attention to an empirical assessment done by Francke and Lee (2014) who find no significant differences between prices of houses sold in different areas in the northern region of the Netherlands. The authors define three areas: a risky one which consists of municipalities in the north of Groningen; and two reference areas that differ from each other on how distant they are to the risky one. They consist mostly of municipalities in the neighboring provinces of Friesland and Drenthe that are near the border with Groningen.
prices fully incorporate prices in the outside region (e.g. Amsterdam12). Prices in Amsterdam,
although not estimated with the same hedonic method as in Francke and Lee (2014), increased by more than 300% between 1985 and 2004, whereas those in Groningen did so by 250% in the same period (Kagie and Wezel, 2007). Although not adapted perfectly, this example is opposite to ours because of the price relation between the central and outside regions, since Amsterdam is more expensive and provide better amenities. However, the example still holds with northern provinces’ prices not growing as much as those in Amsterdam because, in part, of the earthquakes.
From the brake of the financial crisis until the first quarter of 2013, prices in the risky area, as measured by the authors, fell by 14% while those in the reference areas did so by 18% (Francke and Lee, 2013). This is a period in which financial costs dominate the shock’s impact, as in section 4.2 above. According to our model, in this situation equilibrium is reached for prices that fall and more than compensate the reduction in benefits elicited by the shock, namely p∗R ≥ µi + pi, with equality reached when no outsider demands housing in the central
region (Proposition 4). Moreover, it is the period of the downturn in the housing market amid the international financial crisis which our model predicts well as a scenario in which prices drop heavily and a negative spiral of prices and deteriorated credit conditions are reflected in the demand for houses in the central region heavily falling. Indeed, in the period post-crisis the drop in home sales (40%) is even sharper than that on prices, as reported by the authors.
In August 2012 a particularly large earthquake took place. The authors focus on the after-maths of this earthquake and the increase in the frequency of smaller ones to see whether there is a significant difference between the evolution of housing prices in the risky and the reference areas. They find no statistically significant difference between prices. However, other market indicators such as number of sales, sales velocity, and spread between asking and selling price are shown to be much more deteriorated in the risky area. This is a period related to that in which the shock dominates financial costs in our model. Moreover, by 2012 mortgage lending
12It is arguable whether or not Amsterdam’s metropolitan area and surroundings are in fact good benchmarks. It
for new home purchases in the Netherlands had increase by almost 70% since 2008 already, whereas Groningen was among the top 10 provinces on taking mortgage loans that year (NHG, 2012). Furthermore, non performing mortgages were less than 1.5% of total household lending by Dutch banks (OECD, 2014) in 2012. Therefore, we can argue that financing conditions were not binding at the time and that hence, costs from selling at a loss can be assumed to have been lower than the shock’s effect in the northern provinces.
Notice that, from Proposition 5, prices in this equilibrium would incorporate the marginal willingness to pay for a house in the other location that has not been affected neither by the shock nor by the financial friction. For a reasonably low financial cost parameter α, as was argued above, then most of the marginal willingness to pay in the other location is being in-ternalized by prices which, in turn, would imply similar prices, as was argued by the authors. However, because of the importance that prices in other locations hold for housing price forma-tion, it is probably the case that the estimated results in Francke and Lee (2014) be biased.
In a follow up study, we approach the question of whether there is spatial dependency among house prices in the northern region of the Netherlands, directly testing the predictions of this theoretical approach. Besides comparing prices between one and the other region as in Francke and Lee (2014), we directly estimate the role that earthquakes in Groningen have on the price formation in other areas besides that which is directly affected.
6
Conclusions
We develop a model for a regional housing market with an outside and a central region. The latter is divided into two locations which, as opposed to the outside region, offer amenities to households within a random utility framework. Moreover, households die at a rate λ and those in the central region are replaced by newcomers to homeownership that demand housing according to which location has a lower full cost of living, namely µi+ pi with i = 1, 2.
The model is made more realistic by introducing financial costs for households who want to sell their houses at a loss. We find that, when financial costs dominate the reduction in benefits from the shock, prices in both central locations show regional spatial dependency by internal-izing the price in the outside as a benchmark. In this situation, every central household prefer staying in the central region, and every household in location 1 prefers moving to location 2, while those in the latter prefer staying. Prices fall at least to those when no financial friction is present, implying that outsiders start demanding housing in the central region as prices fall below those in the equilibrium without frictions. Moreover, equilibrium prices internalize out-siders’ marginal willingness to pay net of the shock’s effect at the corresponding location.
Appendix
Proof of Proposition 3
Proof. For the first part of the proof, take condition (11), and notice that it has to hold for both central locations and therefore the following two conditions must hold for central households to be indifferent between central locations:
µ1+ p1− p∗R = µ2+ α(p2− p∗R) > 0,
µ2+ p2− p∗R = µ1+ α(p1− p∗R) > 0.
Subtract the second equation to the first and get the following condition 1 + α
2 =
µ1− µ2
p2− p1
> 1,
since α > 1 and implying that µ1+ p1 6= µ2+ p2, and therefore newcomers are not indifferent
between central locations.
Similarly, for the second part notice that the following must hold for the case proposed: µ1+ p1− p∗R> µ2+ α(p2− p∗R),
µ2+ p2− p∗R< µ1+ α(p1− p∗R)
and repeating the same steps as in the first part and get 1 + α
2 >
µ1− µ2
p2− p1
= 1
where the last equality holds provided newcomers are indifferent and therefore, µ1+p1 = µ2+p2
suggesting that the proposed behavior of central households is the only one for which the last inequality can hold. This concludes the proof.
Proof of Proposition 4
Proof. The first part of the proof is arrived at by noting that since outsiders do not demand housing in the central region, central households in 1 do so for location 2, and newcomers sort according to δ then equilibrium is reached in the market provided the following conditions hold:
s1 = λδ (1 − F (c∗λ))
Solve the system for c∗λ and get the expression proposed for p∗i with i = 1, 2.
The second part involves outsiders demanding housing in the central region and therefore the system solved in the first part has to be slightly modified:
s1 = δ [(1 − λ) (F (c∗) − F (c∗λ)) + λ (1 − F (c ∗ λ)] s2 = (1 − λ) [s1(1 − F (c∗)) + s2(1 − F (c∗))] + (1 − δ) [(1 − λ) (F (c∗) − F (c∗λ)) + λ (1 − F (c ∗ λ))]
Solving this system for c∗λ, getting proposed prices p∗i with i = 1, 2 from its expression, and notting that, since p∗R> p∗i + µimust hold, then it must follow that
sR > sR λ − 1 − λ λ δs2− λ(1 − δ)s1 δ(1 − λ)sR , which holds if and only if
δ > λs1
s2+ λs1− (1 − λ)
which in turn implies that s1should be larger than s2.
Proof of Proposition 6
Proof. The dominance of the shock rather than the financial restriction puts us in the condition proven in Proposition 5, which in turn determines that no outsider seeks housing in the central region while some central households do so in the outside.
Moreover, from the fact that the second case in Proposition 3 holds, we get that those who stay in the central region demand housing in 2, while newcomers are indifferent. Therefore, let us compute equilibrium conditions equating demand and supply for each central location:
s1 = δλ (1 − F (cλ)) s2 = (1 − λ) [s1(1 − F (c∗1)) + s2(1 − F (c∗2))] + (1 − δ)λ (1 − F (cλ) sR= 1 − F (c∗) − (1 − λ) [F (c∗1) − F (c ∗ )] − s2(1 − λ) [F (c∗2)) − F (c ∗ 1)] .
The third equilibrium equation states that total supply should be the same as before, except for those central households who left.
From solving the system for F (c∗1), and F (c∗2), getting prices from the expressions of c∗i for i = 1, 2, and noting that p∗1+ µ1 = p∗2+ µ2has to hold as well we get
with Σ = δ (1 − λF (c ∗) − (1 − λ)s R) − s1 δ(1 − λ)(1 − sR) , Ω = s1 s2 1 + λ 1 − λ F (c∗) − sR 1 − sR − 1 − s2 δ(1 − λ)(1 − sR) , and Ω > Σ ⇐⇒ δ > 1 − sR− s2 (1 − λ)(1 − sR(1 − s2)) + λF (c∗)(1 + s2) − sR− s2 ⇐⇒ s2 < sR (1 − λ)sR+ (1 − sR)λ .
Moreover, since the latter hold and µ1 > µ2 then it must be that F−1(Σ) > F−1(Ω), which
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