Equilibrium and comparative statics

As in standard DMP models, the equilibrium is made of three equations with three un-knowns: r_c, θ_c and u. The first two fundamental equations are equation (3.8) which describes the supply side of the housing market based on the location of the good, and equation (3.11) that describes the determination of rents in each municipality. For sake of clarity, these equations are reported below.

r_c= (ρ+δ) κ

q(θ_c) and r_c= (1−β)u₀−βκ



θ_c+^X

i6=c

θ_i





These two equations are represented in the (θc;rc) diagram in Figure 3.4. The relationship that describes the supply side of the housing market shows that r_c is strictly increasing inθ_c. Based on the properties of the matching function, a rise in θ_c increases the average duration of a vacancy in the municipality and therefore pushes rents up. In contrast, the rent bargaining shows that a rise inθ_cis associated with lower rents. Indeed, renters’ power when negotiating is greater when the number of vacant dwellings is high. Therefore, more vacancies imply lower rents. This is shown by a negatively sloped line, therent bargaining line in the diagram. The equilibrium value θ_c^∗ is determined by the intersection between these two curves. This equilibrium value does not depend on u and is unique.

Figure 3.4: Equilibrium rent and perceived market tightness in c

The last fundamental relation to determine the equilibrium values of u and v is the traditional Beveridge curve, that describes the relationship between the relative number of vacancies and the number of flat-hunters in the long-run. This relationship is easily obtained by computing the long-run share of working population that will not find a suitable match in the housing market because of market frictions. This share is given

by¹¹:

u= δ

δ+^PC_i=1p(θ_i) (3.12)

This relationship shows that, when the market tightness increases, it becomes easier for renters to find a housing good in the city. As a result, the long-term number of homeless people in the city decreases. I represent in Figure 3.5 this Beveridge curve. Based on the properties of m(·), this curve is downward sloping and convex to the origin in the (u; v_c) diagram.

I also represent the two first fundamental equations in this figure. By equalizing equations (3.8) and (3.11), we know that the equilibrium market tightness θ_c^∗ (which is unique) is such that

ρ+δ+β(^P_iθ_i)q(θ_c)

q(θ_c) κ−(1−β)u₀ = 0

Given the optimal value of θ_c^∗, the combinations of (v_c; u) are such that:

θ^∗_c = s_cv_c

u ⇒ v_c

u = θ^∗_c s_c

The value θ_c^∗ is independent on u. Therefore, the ratio ^v_u^c is constant and equal to θ^∗_c/s_c. In the (u; vc) plan, this relationship is represented by an increasing line with a slope equal to θ^∗_c/s_c. N being constant, the intersection of the Beveridge curve and the housing

Figure 3.5: Beveridge curve and equilibrium uand v inc

creation linedetermines the proportion of working population who does not find a housing good in the metropolitan area at the equilibrium (u^∗), as well as the number of vacant dwellings in the municipality (v_c^∗).

Effect of geographic distance to the CBD

Let us now compare the equilibrium that prevails in two different municipalities c and c⁰, with d_c⁰ > d_c. For the moment, I still do not consider the effect of commuting costs

11This steady state equilibrium level ofuis obtained by equalizing the inflow of new workers active in the housing marketδ(1−u)N to the outflow of renters who manage to find an appropriate match each period of timePC

i=1p(θi)uN

but focus instead on ex ante distance related frictions. Rents being equalized across jurisdictions (as shown by equation (3.11)), the equilibrium value θ_c^∗ is the same across municipalities. Therefore, the graphical analysis provided in Figure 3.4 holds for all municipalities. In contrast, the equilibrium number of vacancies v^∗ differs across space.

c⁰ being more distant from the CBD than c, the visibility and attractiveness of dwellings incare greater than in c⁰, such thats_c> s_c⁰. Thus, the slope of the housing creation line changes from one jurisdiction to the other. Indeed, θ^∗_c =θ_c^∗0 =θ^∗ implies that

v_c u < v_c⁰

u

i.e. the ratio of vacant dwellings to non-residents increases with the distance to the CBD.

This is shown in Figure 3.6 with the new line HCc⁰. Similarly, the position and the slope of the Beveridge curve change. Everything else constant, an increase in v_c⁰ has a smaller effect on θ_c⁰ than a similar increase in v_c onθ_c, as shown in the figure.

Figure 3.6: Beveridge curve and equilibrium u and v inc and c⁰

The equilibrium number of vacancies is greater in c⁰: because of a lower visibility or attractiveness of vacancies in distant locations, a higher number of vacancies is required to ensure that the equilibrium described in Figure 3.4 is satisfied. The equilibrium value u^∗ is the same in the two scenarios as it is defined at the level of the metropolitan area.

Proposition 2 In the absence of commuting costs, if d_c⁰ > d_c, the equilibrium number of vacant dwellings in each jurisdiction is such that v_c⁰ > v_c. Hence, as the geographical distance from the central business district increases, the number of vacant dwellings rises.

Effect of commuting costs to the CBD

In this subsection, I introduce commuting costs. Eventually, geographic distance to the CBD might have two different effects: either it reduces the ex ante attractiveness or visibility of distant dwellings (as captured by the parameter s_c), or it affects renters’ ex post utility, as they have to pay the commuting cost to go to work each unit of time.

While the former can be considered as a sunk cost faced whilst searching for a dwelling, the latter can be considered as a variable cost incurred each period.

To evaluate the effect of this second type of costs, I introduce τ_c the commuting cost that a worker has to bear when living in jurisdiction cand working in the CBD. This cost is increasing in the distance from the CBD such that τ_c = τ(d_c) with τ⁰(d_c) > 0. This

commuting cost will affect the expected lifetime utility of residents such that equation (3.3) becomes

ρWc=w−rc−τc+δ(U −Wc)

Hence, the rent differential between two jurisdictions will reflect both the differences in utility and in commuting costs across locations.

The fundamental equations of the model are marginally affected by the inclusion of commuting costs in the model. Equations (3.8) and (3.12) are not altered by this change. The only equation to be affected is the equilibrium rent that results from the Nash bargaining between renters and landlords. The bargained rent becomes

r_c= (1−β) (u₀−τ_c)−βκ

C

X

i=1

θ_i

On the (θ_c; r_c) diagram displayed in Figure 3.4, this changes the intercept of the rent bargaining line which falls as the distance to the CBD rises. As a result, the equilibrium combination (θ_c^∗;r_c^∗) will not be equalized across space. In distant municipalities, the equilibrium that prevails is associated with a lower rent (as renters have to be compensated for these commuting costs) and a lower market tightness. This result is shown in Figure 3.7.

Figure 3.7: Equilibrium rent and market tightness with commuting costs

The perceived market tightness now differs across space. More specifically, we find that θ_c^∗ > θ^∗_c0 when d_c⁰ > d_c. As a result, the equilibrium number of vacant dwellings in two jurisdictions is such that:

s_cv_c > s_c⁰v_c⁰

when c⁰ is more distant from the CBD than c. As s_c⁰ < s_c, this condition might be guaranteed even when v_c⁰ ≤v_c. Therefore, commuting costs increasing with the distance to the CBD have a mitigating effect on the main result described in Proposition 2. This yields Proposition 3.

Proposition 3 Commuting costs increasing with the distance to the CBD affect the num-ber of vacancies in suburban jurisdictions. When geographical distance mainly affects renters’ ex ante decision through s_c, the number of vacant dwellings is greater in distant locations. If geographical distance disproportionately increases commuting costs to the CBD, the number of vacancies falls as distance to the CBD rises.

The intuition of Proposition 3 relies on the differences between the two distance related costs. While the first occurs during the housing search, it can be considered as a sunk cost.

Therefore, it will only affect theex ante probability of match between renters and buyers.

By contrast, commuting costs can be considered as variable costs that are incurred each period of time. Therefore, such costs translate – at least partially – into housing price differentials. Renters being compensated for this loss, this pushes the attractiveness of distant locations up. As a result, the equilibrium vacancy rate tends to fall in the presence of large commuting costs.