house prices, sales, and time on the market in competitive...

Introduction Environment Equilibrium Quantitative Exercise Walrasian Economy Final Comments Appendix

House Prices, Sales, and Time on the Market inCompetitive Search

Antonia Dıaz Belen Jerez

Universidad Carlos III de Madrid

Universidad de Salamanca, January 2009

Dıaz, Jerez House Prices, Sales, and Time on the Market


Motivation

Features of US housing market:

Rios-Rull and Sanchez Marcos (2007):

High price volatility (1.3-2.3 higher than GDP volatility)Units sold co-move with prices (correlation 0.78), with evenlarger volatility (6.7 times GDP volatility)

Krainer (2001): Time on the market negatively correlatedwith prices and volume

“Hot” market: prices and volume rise, time on the market falls“Cold” market: prices and volume fall, time on the market rises



Introduction



This paper

Search equilibrium model that captures illiquidity of housingassets

We ask whether the model delivers observed features inrelation to the co-movement of house prices, sales and timeon the market



This paper

Key features of the model:

Houses: indivisible durable big-ticket items

Trading frictions: search and informational (US-NAR: 10 unitsvisited on average prior to purchase; time to buy 8 is weeks)

Competitive forces present: sellers listing lower prices sell theirproperty faster (Merlo and Ortalo-Magne 2004 for UK)



Households

Measure N of infinitely lived symmetric households

Value of services from housing units. Want at most 1 unit

Idiosyncratic flow value v > 0:

Matched: v = vMismatched: v = v < v

Matched households become mismatched with prob. α(e.g. changes in job or household size)

Quasi-linear preferences

Housing stock fixed (for now): H ∈ (N, 2N)Households hold 1 or 2 units

No rental market for houses



Market arrangementsDistribution of population

Households can be in 3 states each period t

non-traders: matched with 1 unitbuyers: mismatched with 1 unitsellers: matched with 2 units

nt + bt + st = N

nt + bt + 2st = H (or st = H −N)

Assume: Mismatched households must buy new unit beforeselling old one



Market arrangementsHousing Market with Search frictions

Buyers and sellers randomly matched in pairs

Matching function: M(bt, st) (CRS, incr., concave, C2)

Matching probabilities:

Sellers: ms(θt) ≡ M(bt,st)st

=M(θt, 1) where θt ≡ bt/stBuyers: mb(θt) ≡ M(bt,st)

bt=M(1, θ−1

t ) = ms(θt)/θtConditional on a match, buyers like seller’s unit with prob. q

Trading probabilities:

Buyers: πb(θt) ≡ qmb(θt)Sellers: πs(θt) ≡ qms(θt) = qmb(θt)θt

NOTE: Negative correlation between average time to sell 1/πs(θt)and volume πs(θt)st



Market arrangementsPrice Determination

Competitive search equilibrium (Montgomery 1991, Peters1991, Moen 1997, Shimer 1995)

With differentiable matching function and ex ante symmetricbuyers & sellers, outcome equivalent to Bargaining rule inHosios’s 1991:

Sellers get a share η(θt) of bilateral surplus, where η(θt) is theelasticity of mb(θt); i.e.,

η(θt) =−mb′(θt)θtmb(θt)

Also, outcome is constrained Pareto Optimal



Equilibrium

Evolution of the population

bt+1 = αnt + [1− πb(θt)]btst+1 = πb (θt) bt + [1− πs (θt)]st = H −Nnt+1 = (1− α)nt + πs (θt) st

Economy wide state variable: zt = (nt, bt), or zt = (nt, θt) whereθt = bt/(H −N)



Household’s problemBuyers (I)

Wb (z) = πb (θ) max {v − p(z) + βWs (z′) , v + βWb (z′)}

+[1− πb (θ)] [v + βWb (z′)]

given z′ = g (z) , z = (n, θ)



Household’s problemBuyers (II)

Equivalently,

Wb (z) = v + βWb (z′) + πb (θ) max{Sb(z, z′), 0}

where Sb (z, z′) = v − v − p(z) + β [Ws (z′)−Wb (z′)]

given z′ = g (z) , z = (n, θ)



Household’s problemSellers (I)

Ws (z) = v + πs (θ) max {p(z) + βWn (z′) , βWs (z′)}

+[1− πs (θ)]βWs (z′)

given z′ = g (z) , z = (n, θ)



Household’s problemSellers (II)

Equivalently,

Ws (z) = v + βWs (z′) + πs (θ) max{Ss(z, z′), 0}

where Ss(z, z′) = p(z) + β [Wn (z′)−Ws (z′)]

given z′ = g (z) , z = (n, θ)



Household’s problemNon-traders

Wn (z) = v + (1− α)βWn (z′) + αβWb (z′)

= v + βWn (z′)− αβ [Wn (z′)−Wb (z′)]

given z′ = g (z)



Equilibrium price

Bargaining rule in Hosios 1990:

Given z = (n, θ), share η (θ) = −mb′(θ)θmb(θ)

of bilateral surplus

allocated to sellers. Hence, p(z) solves

Sb

Ss=v − v − p+ β [Ws (z′)−Wb (z′)]

p+ β [Wn (z′)−Ws (z′)]=

1− η (θ)η (θ)

where z′ = g(z)

Bilateral surplus is

S = Sb + Ss = v − v + β[Wn

(z′

)−Wb

(z′

)]



Steady State

s = H −Nb = θs,

n = N − b− s = N − (1 + θ)sαn = q mb(θ)θ s,

(1− β)Wb = v + qmb(θ)[v − v − p+ β(Ws −Wb)],(1− β)Ws = v + qmb(θ)θ[p+ β(Wn −Ws)],(1− β)Wn = v − βα(Wn −Wb),

v − v − p+ β(Ws −Wb)p+ β(Wn −Ws)

=1− η(θ)η(θ)

,

η(θ) =−mb′(θ)θmb(θ)

,



Steady-State Comparative StaticsChanges in the vacancy rate

PropositionWhen the vacancy rate H−N

H falls (e.g. population N increases):

θ increases, so time to sell 1/πs falls & time to buy 1/πb incr.

Sales πs(H −N) increase for θ < θs, and fall for θ > θs,where 2 = η(θs)(2 + θs)Bilateral surplus increases

The price p increases for θ > θp, where θp is another threshold(otherwise the effect is ambiguous)

Also, θs > θp if β is not too low. So both sales and pricesincrease for θ ∈ (θp, θs) (e.g. in our quantitative exercise)



Steady-State Comparative StaticsChanges in the mismatch rate

PropositionWhen the rate α at which households become mismatchedincreases (e.g. greater geographical mobility, younger population):

θ increases, so time to sell 1/πs falls and time to buy 1/πbincreases

Sales πs(H −N) increase

Effect on bilateral surplus and price p ambiguous in general(but positive in our quantitative exercise)



Steady-State Comparative StaticsChanges in the severity of frictions q

PropositionWhen the probability q that a buyer likes units visited increases(e.g. improvement in information or intermediation technology):

θ falls, so 1/πs increases and 1/πb increases


Bilateral surplus falls

Price effect ambiguous in general (e.g. hump shaped inquantitative exercise)



Calibration

Model period is a week.Exponential matching function in Peters 1991:ms(θ) = 1− exp{−θ}

Table: Calibration

Param. Observation Value

N - 100H vac.rate 1985-2001=(H-N)/H= 1.5 % 101.52q time to buy = 1

qmb(θ)= 8 weeks 0.1972

α tenure = 1−αα + 1

qmb(θ)= 10 years 0.0019

β - 0.961/52

v - 1 = 1274.32(1− β)v - 0.1 = 127.43(1− β)



Steady State

Table: Steady State

θ 0.99Sellers s/N (%) 1.52Buyers b/N (%) 1.51Non-traders n/N (%) 96.96Sales πs(θ)(H −N)/H (%) 0.19Average time to sell 1/πs(θ) 8.06Price p 814.57Seller surplus Ss/S (%) 41.56Bilateral surplus S 11.91



Steady-State Comparative StaticsChanges in the population N (I)



Steady-State Comparative StaticsChanges in population size N (II)



Steady-State Comparative StaticsChanges in population size N (III)

Table: Increase in vacancy rate 1−N/H from 1.5 to 4 %

Vac. rate 1.5 % Vac. rate 4 % % Change

Price 814.57 52.23 −0.936Sales (% of stock) 0.186 0.176 −0.053Aver. time to sell 8.057 22.69 1.81Aver. time to buy 8 5.74 −0.28Seller’s share 41.56 12.12 −0.71



Steady-State Comparative StaticsChanges in mobility parameter α (I)



Steady-State Comparative StaticsChanges in mobility parameter α (II)



Steady-State Comparative StaticsChanges in mobility parameter α (III)

Table: Decrease in tenure from 10 to 7 years

Tenure 10 years Tenure 7 years % Change(α = 0.0019) % (α = 0.0028)

Price 814.57 6469.3 6.942Sales (% of stock) 0.186 0.265 0.425Aver. time to sell 8.057 5.652 −0.298Aver. time to buy 8 12.87 0.608Seller’s share 41.56 73.96 0.779



Summary

Sales and time on the market negatively correlated

Demand changes: Sales and price both increase

Very high elasticity of price to demand changes



Limiting Frictionless Economy

Consider limiting competitive situation where frictions vanish:

Buyers can perfectly identify the units they like q = 1Matching is efficient: M(bt, st) = min{bt, st} so

ms(θt) = min{θt, 1}mb(θt) = min{1, θ−1

t }

Competitive prices: agents on long side of the market get zerosurplus, so

Ss(θt) = 0, Sb(θt) = St if θt < 1Ss(θt) = St, S

b(θt) = 0 if θt > 1



Steady-State Comparative StaticsCase A: b > s or θ > 1

Steady state as before, except that now trading prob. and p are:

πb = θ−1; πs = 1

p = v − v + β(W s −W b) =1

1− β(v − v) +

β

1− βS

PropositionWhen the population N increases:

θ increases, so time to buy 1/πb increases and time to sell1/πs is constant

Sales πs(H −N) fall

Bilateral surplus S constant

Price p constant



Steady-State Comparative StaticsCase B: b < s or θ < 1

In this case,

πb = 1; πs = θ (1)

p = β(W s −Wn) = 0; S = v − v (2)

PropositionWhen the population N increases:

θ increases, so time to sell 1/πs falls, and time to buy 1/πb isconstant


Bilateral surplus constant

Price p constant



Steady-State Comparative StaticsCase A: θ > 1 so 1/πb(θ) = θ > 1 and 1/πs(θ) = 1

PropositionWhen α increases

θ increases, so time to buy 1/πb increases and time to sell1/πs is constant

Sales πs(H −N) constant

Bilateral surplus falls

Price p falls



Steady-State Comparative StaticsCase B: θ < 1 so πs = θ < 1 and 1/πb(θ) = 1

PropositionWhen α increases

θ increases, so time to sell 1/πs falls, and time to buy 1/πb isconstant


Bilateral surplus constant

Price p constant



Quantitative exercisesChanges in population size N (I)



Quantitative exercisesChanges in population size N (II)



Quantitative exercisesChanges in mobility parameter α (I)



Quantitative exercisesChanges in mobility parameter α (II)



Equilibrium with production

Variant of the model where: (i) non-traders can build new unitseach period (decreasing returns), (ii) old units depreciate

Sales and time on the market are negatively correlated

Demand changes: Sales and price positively correlated

Sales change more than the price

Qual. and quant. effect on prices, sales and time on themarket are not so different now in Walrasian model becausethere production instantaneously adjusts so θ = 1



Related literature

Walrasian models with frictions (adjustment costs, creditconstraints, incomplete markets...):

- Price volatility: Nakayima 2005, Ortalo-Magne and Rady 2003,Rıos-Rull and Sanchez-Marcos 2007...

Search (or island) models of housing

- Benchmark with Nash bargaining: Wheaton 1990- Similar qualitative results but different mechanism: Krainer

2001 (aggregate shocks to valuations), Yui-Zhang 2007(heterogeneous buyers and search intensity & free entry ofsellers)

- Different focus: Nieuwerburgh and Weill 2007 (price volatilityin island model w/ housing supply regulation), Hendel, Nevo,Ortalo-Magne 2007 (sale by owner vs. realtor in stock-flowmodel)



Future work

Lot’s of work to do!

Allow non-traders to buy housing units

Aggregate uncertainty (fluctuations in vacancy and mismatchrate)

Investigate production further (housing supply cannot adjustinstantaneously)

Rental market and population growth (e.g. add a flow ofindividuals with no houses each period)

Credit constraints

Policy analysis (e.g. taxes)



Appendix

Some preliminary results on transitional dynamics



A change in populationVacancy rate goes from 4% to 1.5%



A change in mobilityTenure decreases from 10 years to 7 years


house prices, sales, and time on the market in competitive...

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