February 15, 2007 N. L. Stokey

Very Preliminary

HOUSING AND NONDURABLE CONSUMPTION

Housing services are different from other consumption goods. First, moving typi-

cally entails substantial adjustment costs, so individuals adjust their consumption of

housing services infrequently. In addition, individuals who own their house or apart-

ment must hold a minimal level of equity in it, placing an additional constraint on the

set of portfolios available to owners. This paper studies the effects of these two factors

on the individuals’s demand for housing services and nondurables consumption, as

well as her asset holdings.

The model studied here applies to both renters and owners. For renters the ad-

justment cost for housing reflects search and moving costs, while for owners it also

includes brokers’ fees for buying and selling. On the portfolio side, a renter’s asset

holdings are constrained if short sales are prohibited or subject to a margin require-

ment. An owner faces a similar but tighter constraint on her portfolio, because of the

equity she must hold in her house. Thus, while owners face larger adjustment costs

and tighter portfolio constraints, both types of consumers face decision problems of

the same form.

Transaction costs and portfolio constraints affect choices in several ways. First con-

sider the mix of consumption goods, the ratio of nondurables to housing. Suppose that

preferences are homothetic, the relative price of housing is constant, and nondurable

consumption and the portfolio are costlessly adjustable. Then absent transaction

costs for housing, the individual would choose a constant mix of consumption goods.

With adjustment costs for housing, however, the ratio of nondurable consumption to

housing will vary over time. Moreover, the extent of the variation–the way the ratio

changes with changes in permanent income or wealth–depends on the elasticity of

1

substitution.

In particular, if the elasticity of substitution between nondurables and housing

is less than unity, as suggested by the data, then consumption of nondurables will

respond weakly to changes in permanent income or wealth during intervals when

housing is fixed. That is, consumption of nondurables will be unresponsive in the

short run, even if long run income elasticities are unity for both goods. Thus, the

basic idea here is similar in flavor to the one in Chetty and Szeidel (2004), where a

model with ‘fixed’ and ‘flexible’ components of consumption is studied. The model

here offers an explanation for the source of the ‘fixed’ component.

During intervals when the individual’s housing consumption is constant, her port-

folio and her consumption of nondurables adapt to the fixed level of housing. Em-

pirically, then, the presence of adjustment costs has several implications. First, while

an Euler equation for nondurables holds at the individual level during these intervals,

unless preferences are additively separable between durables and housing, it does not

have the usual form.

Second, because the income elasticity of demand for nondurables is affected by

the level of housing services, individuals who have moved recently–and hence are

unlikely to move again soon–should behave differently from those who expect to

move in the near future. Third, the consumption mix of owners should display wider

swings than the ratio for renters, since renters presumably face smaller adjustment

costs for housing.

Adjustment costs for housing also affect portfolio behavior. As noted already,

owners will differ from renters because they face tighter constraints. In addition

the implicit risk aversion of an individual consumer–either an owner or a renter–

changes depending on whether or not her wealth is near a threshold that will trigger

an adjustment in housing.

Overall, the goal here is to look at the effect of the transaction cost, the equity

2

constraint, and various preference parameters on:

(a) the consumption mix, the ratio of nondurables to housing services;

(b) the ratio of housing to wealth, which influences the expenditure rate; and

(c) the portfolio share of risky assets.

First a model without transaction costs is studied. In this setting owners differ from

renters only in terms of the portfolio constraints they face. Hence their behavior

differs if and only if the equity constraint binds for owners. Since that constraint

limits risky assets holdings, it binds for individuals who are sufficiently risk-tolerant.

Because they face a higher implicit ‘price’ for housing, these individuals reduce their

consumption of housing services.

In the model with transaction costs, housing consumption follows an (S, s) type

policy. Here we are interested in how various parameters affect the width of the (S, s)

bands and the behavior of (a)-(c) within that band.

1. Related literature

There is a sizable literature, going back two decades, studying whether including

durable goods can improve the fit of asset pricing models. Early attempts assumed

that consumption of durables is flexible in the sense that there are no adjustment

costs. In the absence of adjustment costs, homotheticity implies that the economy

can be described, as usual, by a representative agent. That is, except for scale the

behavior of all individuals and of the aggregate are identical. Under this assumption

an Euler equation of the usual type holds for nondurables, and that equation can be

tested on aggregate data. In this group are the papers by Dunn and Singleton (1986)

and Eichenbaum and Hansen (1990). They found that including durables does little

to improve the fit of the model.

The theoretical paper by Grossman and Laroque (1990) provided a framework for

studying the behavior of an individual who consumes only one good, housing services,

3

and faces adjustment costs for changing her level of consumption. They showed that

the adjustment cost affects the consumer’s portfolio behavior in a systematic way.

Specifically, consumers who have recently adjusted their housing stock, and hence

anticipate a long interval of time before another adjustment, are more risk averse

than those who anticipate making an adjustment in the near future. Their model

does not include nondurable consumption, however, so it provided no predictions

about the behavior of standard Euler equations.

Subsequent empirical work confirmed the importance of adjustment costs for a

variety of durables. Eberly (1994) looked at household-level data on automobile

purchases, while Caballero (1993), Carroll and Dunn (1997) and Attanasio (2000)

looked at aggregate durable purchases. All found evidence of adjustment costs.

More recently, several papers have revisited questions about fluctuations in con-

sumption, portfolio behavior, and asset pricing using models that incorporate adjust-

ment costs.

Chetty and Szeidl (2004) look at a model with committed and flexible consumption,

interpreted as housing and food. Preferences are additively separable across the two

goods and over time. Two variants are considered, one with fixed time intervals

between adjustments of the committed good, and one with stochastic (Calvo-type)

intervals. Because preferences are additively separable across goods, consumption of

the flexible good depends on the level of commitment good only through the budget

constraint. Hence food consumption, at the individual level and in aggregate, satisfies

the usual type of Euler equation. But since F = C − X, aggregate consumptionbehaves like a consumer with a habit, X, which here represents aggregate consumption

of housing and depends on a weighted average of net wealth at the most recent past

adjustment date, summed over all cohorts of adjusters. Chetty and Szeidl’s empirical

work focuses on portfolio behavior, and confirms that households who have adjusted

their committed consumption in the recent past–i.e., have moved recently and hence

4

are unlikely to move again in the near future–display more risk averse portfolio

behavior than those with longer tenures at their current residence.

Fukushima (2005) explores the asset pricing implications of this setup, using addi-

tively separable preferences and a stochastic (Calvo-type) adjustment rule. He uses a

GE endowment economy of the type in Lucas (1978), where the aggregate endowment

can be converted into either consumption good.

Kullmann and Siegal (2004) look at portfolios of homeowners, and find evidence of

state-dependent risk aversion. Specifically, as the model of Grossman and Laroque

predicts, owners who have recently adjusted their housing stock are more risk averse

than those who anticipate making an adjustment in the near future.

Siegel (2004) explores a number of empirical questions using a model that incorpo-

rates adjustment costs and postulates CES preferences over durables and nondurables

within the period. He looks at two data sets. The first is a quarterly data set covering

1952-2002, in which durables consist of automobiles and furniture. The other is an

annual data set covering 1929-2002 in which the durable is housing.

The first data set displays a time series pattern that is consistent with the presence

of adjustment costs. Specifically, Siegal looks at a cointegration residual that mea-

sures deviations in the ratio of durables to nondurables. He finds that it varies in a

systematic way over the business cycle, peaking during contractions and bottoming

out during booms. This is just what a model with adjustment costs would predict. At

the beginning of a boom, all households increase their consumption of nondurables,

while only a fraction incur the adjustment cost and increase their stock of durables.

Hence the residual falls. As the boom continues, the residual climbs back toward its

mean value. At the onset of a recession the argument is reversed. Siegal also looks

at the relationship between the residual and asset returns, and the evidence here is

consistent.

The results for the data set on housing are harder to interpret. Siegal notes that

5

the data used here include the Great Depression and World War II, both unusual

events that may overwhelm the rest of the series. Another issue with housing is

that increases in aggregate consumption are limited by the fact that the aggregate

stock cannot increase quickly. Although occupancy rates may rise during booms,

adjustment along this margin is limited. The fact that times-to-build are relatively

long for housing further complicates the aggregate adjustment process.

Lustig and Van Nieuwerburgh (2004) consider the role of housing as collateral in a

setting with no adjustment costs.

Piazzesi, Schneider, and Tuzel (2007) look at the asset pricing implications of a

model with housing and nondurables, but no adjustment costs. [To be completed.]

2. Preliminaries

There are two consumption goods, housing services H and a single composite

nondurable C. The price of the nondurable is normalized to one. The purchase

price of housing PH is taken to be constant. Since the relative price of housing varies

considerably across cities and regions, however, we will be interested in how differences

in PH affect behavior. The flow of housing services H is equal to the value of the

house, which reflects both quantity and quality, including features like location, lot

size, and other attributes.

Preferences over the two goods have the CES form

U(C,H) =

£ωC1−ζ + (1− ω)H1−ζ¤1/(1−ζ) , ζ 6= 1,CωH1−ω, ζ = 1,

where ω ∈ [0, 1) is the relative weight on nondurables, and 1/ζ is the elasticity ofsubstitution.

The consumer’s intertemporal utility function has the form

E0

"Z ∞0

e−ρt{U [C(t), H(t)]}1−θ

1− θ dt#, (1)

6

where θ is the coefficient of relative risk aversion and ρ is the rate of time preference.

The consumer’s only income is the return on her portfolio. She holds two assets,

a safe one with a constant rate of return r, and a risky one with mean return µ > r and

variance σ2 > 0. The return r on the safe asset is also the interest rate on mortgages.

Define the ratio

γ ≡ (µ− r) /σ2, (2)

so γ > 0 is the inverse ‘price’ of risk.

The rental cost of housing has three components: interest at the rate r, depre-

ciation at the rate δ, and maintenance at the rate m. Hence for a renter the cost of

housing is ph = rPH + δ +m. Since the price of housing is constant, owners do not

face any additional risk, and since the mortgage rate is r, for an owner wealth held in

the safe asset can be interpreted as equity in her house. Hence ph is also the ‘direct’

cost of housing for an owner. The difference between owning and renting arises from

an additional portfolio constraint on owners. This constraint will be discussed below.

Let Q denote the consumer’s total wealth, H the value of her house, and A her

holdings of the risky asset. Then Q − A denotes wealth in the safe asset. GivenC,H,A,Q, the change in the consumer’s total wealth over a short interval of time dt

is

dQ = {µA− r [H − (Q− A)]− (δ +m)H − C} dt+ σAdz (3)= {rQ+ (µ− r)A− phH − C} dt+ σAdz,

where z is a Wiener process. For an owner, the term in braces in the first line consists

of the return on her risky assets, her mortgage payment, depreciation and maintenance

on her house, and nondurable consumption. If Q − A ≥ H the consumer owns herhouse outright, and if the inequality is strict she has additional wealth invested at

the risk-free rate. The second line rewrites this expression in a way that reflects the

budget constraint of a renter: her returns on safe and risky assets, less expenditures

7

on the two consumptions goods.

The consumer also faces constraints on her portfolio. Assume the consumer

cannot short the risky asset, requiring A ≥ 0. Under the parameter restrictions wewill use here, this constraint does not bind. Let ∈ (0, 1] be the (exogenously given)minimal equity an owner must hold in her house. The consumer may be permitted to

buy the risky asset on margin, but assume that she cannot use the minimal equity in

her house as collateral. This leads to the constraint A ≤ ass (Q− H) , where ass ≥ 1reflects the size of the margin requirement. If ass = 1, the consumer cannot buy risky

assets on margin. Hence the portfolio constraint for the consumer is

A ∈ [0, ass (Q− H)] , (4)

For a renter there is no equity constraint, so = 0.

The following parameter restrictions will be used throughout.

Assumption 1:

0 ≤ ≤ 1, ass ≥ 1,δ,m ≥ 0,0 < r < µ, σ2 > 0,

0 ≤ ω ≤ 1, ζ > 0,θ > 0, θ 6= 1 ρ > 0,ρ+ (1− θ) δ > 0.

The case θ = 1, which represents lnU, can be treated along similar lines. The last

restriction will be used in section 4.

3. No transaction costs

A useful benchmark for comparisons is the model with no transaction cost. In

this case the consumer’s problem is to choose (C,H,A) to maximize (1) subject to

the budget constraint (3) and the portfolio constraint (4), given initial wealth Q0 > 0.

8

Since the objective function is homogeneous of degree (1− θ) in (C,H,A,Q) andthe constraints are homogeneous of degree one, the optimal ratios H/Q, A/Q, etc.

are constant over time. Hence the consumer’s problem can be written as

W (Q0) = maxc≥0,h∈[0,1/ ]a∈[0,ass(1− h)]

E0

"Z ∞0

e−ρt[u(c)hQ(t)]1−θ

1− θ dt#

(5)

s.t.dQ

Q= [r + a (µ− r)− (ph + c)h] dt+ aσdz,

where c ≡ C/H is the ratio of nondurable consumption to housing services, h ≡ H/Qis the ratio of housing to wealth, a ≡ A/Q is the portfolio share in the risky asset,and

u(c) ≡£ωc1−ζ + (1− ω)¤1/(1−ζ) , ζ 6= 1,cω, ζ = 1,

is the intensive form of the CES aggregator.

For any fixed (c, h, a), total wealth Q is a geometric Brownian motion with

constant drift and variance. Hence E0£Q(t)1−θ

¤grows at the constant rate

Γ(c, h, a; θ) ≡ (1− θ)·r + (µ− r) a− (ph + c) h− θ1

2(σa)2

¸. (6)

Consequently, if ρ > Γ the problem in (5) can be written as

maxc≥0,h∈[0,1/ ]a∈[0,ass(1− h)]

[u(c)h]1−θ

1− θQ1−θ0

ρ− Γ(c, h, a; θ) . (7)

The next assumption insures that Γ satisfies the required condition.

Assumption 2: If 0 < θ < 1,

ρ > (1− θ)× [r + (µ− r) ass − θa2ssσ2/2] , if θ < γ/ass,[r + (γ/θ) (µ− r) /2] , if θ ≥ γ/ass.

Under Assumptions 1 and 2, the maximum in (7) is attained and the maximizing

values for (c, h, a) are unique.

9

Lemma 1: If Assumptions 1 and 2 hold, then for any feasible c, h,

α(h; θ, ) = min {γ/θ, ass (1− h)} (8)

is the unique maximizing value for a in (7).

Proof: See the Appendix.

Unless the consumer is holding a house of maximal size, h = 1/ , and hence has

no discretionary wealth to allocate, the share of wealth in the risky asset, α(h; θ, ), is

strictly positive. Moreover, that share depends only on γ/θ, ass and h. For consumers

who are sufficiently risk averse (low 1/θ), the solution is interior, at a = γ/θ. For

these consumers the share of wealth in the risky asset is increasing in the inverse price

γ and in risk tolerance 1/θ, up to the point where the constraint ass (1− h) binds.For consumers who are sufficiently risk tolerant (high 1/θ), the constraint ass (1− h)

binds and the solution is at a corner. For an owner who is constrained, housing ser-

vices have an a extra cost at the margin. In addition to the direct cost ph, an increase

in consumption of housing services involves an incremental portfolio distortion. The

size of this distortion depends on and ass. For a renter, the constraint on risky asset

holdings may bind if she is sufficiently risk tolerant. But for a renter = 0, so her

level of housing consumption h does not affect the constraint.

To characterize the consumer’s choice of housing services and nondurables, we can

use substitute the portfolio a = α(h; θ, ) from (8) into (7). Then W (Q0) = Q1−θ0 w

∗,

where

w∗ ≡ maxc≥0,h∈[0,1/ ]

[u(c)h]1−θ

1− θ1

ρ− Γ(c, h, α(h; θ, ); θ) . (9)

Proposition 2 characterizes the solution for a renter.

Proposition 2: If Assumptions 1 and 2 hold, then for = 0, the unique solution

to the problem in (9) is

cR =

µωph1− ω

¶1/ζ, (10)

10

hR(θ) =1

cR + ph

1

θ

½ρ− (1− θ)

·r + σ2γaR(θ)− θ1

2σ2a2R(θ)

¸¾, (11)

where aR(θ) = min {γ/θ, ass} . Moreover,

h0R(θ) R 0 asr − ρσ2

+

µγ − 1

2θ2aR

¶aR R 0.

Proof: See the Appendix.

For a renter, the ratio cR of nondurable consumption to housing depends on the rental

price ph and the parameters ω and ζ, but not on θ. The ratio of expenditure to wealth,

(cR + ph)hR(θ), depends on θ but not on ph, ω or ζ.

More risk-averse renters, those with higher θ, hold portfolios with lower expected

returns. These consumers also have a lower elasticity of intertemporal substitution, so

they have a stronger incentive to smooth consumption over time. Both considerations

should lead them to choose a lower ratio hR, which implies a lower ratio of expenditure

to wealth, and this true for values of θ above a certain threshold. For θ below that

threshold, however, an increase in risk aversion raises the renter’s ratio of expenditure

to wealth. For example, suppose r = ρ. Then hR is increasing in θ if

θmin

½1,

θ

γass

¾< 2

If ass is large, so the short sale constraint does not bind, then hR is increasing in θ

for θ < 2. If ass = 1, so no short sales are allowed, the condition depends on γ. Asset

market data suggests γ ≈ 2, and in this case, too, hR is increasing for θ < 2.The next proposition characterizes the solution to (9) for owners. For owners the

solution depends on as well as θ.

Proposition 3: If Assumptions 1 and 2 hold, then for > 0, the solution to the

problem in (9) is as in (10)-(11) if ass [1− hR(θ)] ≥ γ/θ. Otherwise

aB(θ, ) = ass [1− hB(θ, )] , (12)

11

0 2 4 60.2

0.205

0.21

0.215

0.22

0.225

0.23

θ

nondurable consumption / durable value

µ = 0.07

σ = 0.1655

r = 0.01

δ = 0

m = 0.02

ρ = 0.04

ω = 0.75

ζ = 1.5

ε = 0.15a

ss = 1

0 2 4 60.08

0.1

0.12

0.14

0.16

0.18

0.2

θ

durable value / wealth

0 2 4 60.02

0.025

0.03

0.035

0.04

0.045

θ

expenditure flow / wealth

0 2 4 60

1

2

3

4

5

6

7

θ

safe assets / house value

0 2 4 6

0.4

0.5

0.6

0.7

0.8

0.9

1

θ

portfolio share in risky asset

0 2 4 60.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

θ

expected growth rate of wealth

u0(cB(θ, ))u(cB(θ, ))

=1

cB(θ, ) + ph + [γ − θaB(θ, )] σ2 ass .

For a constrained buyer

hB(θ, ) < hR(θ),

cB(θ, ) R cR as ζ R 1.

Proof: See the Appendix.

If ζ > 1, i.e. if the elasticity of substitution between housing and nondurables is less

than one, then a constrained owner chooses a higher ratio c of nondurables to housing

than a renter. (The reverse holds for ζ < 1, and for ζ = 1 the ratio is unchanged. But

the evidence is fairly strong that the elasticity is less than one between nondurables

and housing.)

In the absence of a transaction cost, the value for a consumer of type j = R,B,

with initial wealth Q0, is Q1−θ0 w

∗j , where w

∗ is the maximized value in (9).

4. With transaction costs

Suppose the consumer must pay a transaction cost of λH when she adjusts her

consumption of housing services, where λ > 0. Then she will adjust her housing

consumption only occasionally, by discrete amounts. Hence her budget constraint

has two parts. At dates when she adjusts her housing consumption, her wealth falls

by the amount of the transaction cost. At all other times the durable depreciates

deterministically and wealth grows stochastically.

With a positive transaction cost two state variables are needed. We will use total

wealth and the size of the house, (Q,H). Assumptions 1 - 2 insure that the Principle

12

of Optimality applies, so we can write the consumers’s problem in the recursive form

V (Q0, H0) = sup[C(t),A(t)],T 0,H0

E0

(Z T 00

e−ρt[U(C(t), H(t))]1−θ

1− θ dt+ e−ρT 0V (Q0,H 0)

),

(13)

s.t. dQ = [rQ+ (µ− r)A− phH − C] dt+ σAdz,dH = −δHdt,A ∈ [0, ass (Q− H)] , t ∈ [0, T 0),Q0 = Q(T 0)− λH(T 0),H 0 ≤ Q0,

where the stopping time T 0 is the time of the next adjustment, the random variable

H 0 is the size of the house purchased at that time, and the stochastic processes

{C(t), A(t), t ∈ [0, T 0)} describe her consumption and portfolio allocation before thetransaction. As before, the consumer is an owner if > 0 and a renter if = 0.

Since the transaction cost cannot raise the consumer’s utility, clearly V (Q0, H0)

is bounded above by Q1−θ0 w∗, where w∗ was defined in the previous section. In

addition, if Q0 > λH0, then under Assumption 1 it is always possible to choose a

feasible strategy for which expected utility is bounded below. One such strategy is as

follows. Set A ≡ 0, so the risky asset is not held; let C = cH, where c > 0 is small;set H = hQ when a transaction is made, where h > 0 is small; choose a long period

T between transactions; and make the first transaction at date 0. During intervals

when no transaction is made wealth grows at a constant rate

dQ

Q= [r − (ph + c)h] dt.

For c and h sufficiently small, this growth rate is positive. Hence Wn+1 > Wn, all n.

where Wn is wealth after the nth house sale. Over each interval when no transaction

occurs, utility is vW 1−θn , where v is a constant. Hence lifetime utility under this

13

strategy is∞Xn=0

e−ρnTvW 1−θn ≥ v (Q0 − λH0)1−θ1

1− eρT ,

where the right side is finite.

As before, the value function V is homogeneous of degree (1− θ) in the statevariables (here (Q,H)) and the policy functions for C,A, and H 0 are homogeneous of

degree one. Hence the problem can be written in terms of a single state variable, a

ratio. It is convenient to use q = Q/H = 1/h. Note that when a transaction is made

q0H 0 = Q0 = Q(T 0)− λH(T 0) = [q(T 0)− λ]H0e−δT 0,

so

H 0 =q(T 0)− λ

q0e−δT

0H0.

Also note that

{U [C(t),H(t)]}1−θ = £u(c(t))H0e−δt¤1−θ .Using these facts we can write the Bellman equation (13) in the intensive form

v(q0) = sup{c(t),a(t)},T 0,q0

E0

(Z T0

e−ηtu[c(t)]1−θ

1− θ dt+ e−ηT

µq(T )− λ

q0

¶1−θv (q0)

)(14)

s.t. dq = {[r + δ + (µ− r) a] q − (ph + c)} dt+ σaqdz,a ≤ ass

µ1−

q

¶, t ∈ [0, T 0),

q0 ≥ ,

where v(q) ≡ V (q, 1),η ≡ ρ+ (1− θ) δ,

and as before c = C/H and a = A/Q. Assumption 1 insures η > 0.

Two properties of the solution are immediate from (14). First, the optimal

stopping time has the form T = T (b) ∧ T (B), where T (β) denotes the first time the

14

stochastic process x reaches β, and 0 ≤ b < B < +∞ are optimally chosen thresholds.In addition, the optimal return point does not depend on the state q(T ) when the

adjustment is made. Let

M ≡ maxq0

v (q0)q01−θ

, (15)

denote the (normalized) value when a transaction is made, and let S denote the

optimal return point.

To characterize the optimal policy, note that for q ∈ (b, B) , v(q) satisfies theHJB equation

ηv(q) = maxc,a

½u(c)1−θ

1− θ + {[r + δ + (µ− r) a] q − (ph + c)} v0 +

1

2σ2 (aq)2 v00

¾. (16)

The FOCs for the optimal consumption mix and portfolio allocation are

u(c)−θu0(c)− v0(q) = 0, (17)(µ− r) v0 + σ2aqv00 = 0.

At the boundaries b and B, the usual value matching and smooth pasting condi-

tions must hold. In addition, the return point S solves the problem in (15). Hence

(b, B, S,M) must satisfy

v(b) = (b− λ)1−θM, v0(b) = (1− θ) (b− λ)−θM (18)v(B) = (B − λ)1−θM, v0(B) = (1− θ) (B − λ)−θM,v(S) = S1−θM, v0(S) = (1− θ)S−θM.

Note that M = v(S)/S1−θ is the value for an individual with net wealth Q = 1

when she purchases a house, and 1/S is the ratio of her house to her total wealth.

Hence as the transaction cost goes to zero,

limλ→0

M = νB and limλ→0

S = 1/hB.

15

where νB and hB are the solutions for a buyer in a world with no transaction costs.

The interval [b, B] shrinks to the single point S, and the functions c(q) and a(q)

approach constant functions, with levels cB and aB from the NTC problem.

Probabilities, expected duration

It is also interesting to look at the probabilities of adjustment at each threshold,

and at the expected time between adjustments. Let µ(z) and σ2(z) be the drift and

variance for assets, and define the normalized drift.

δ(z) ≡ 2µ(z)σ2(z)

.

Then define

s(z) ≡ exp½Z z

δ(ζ)dζ

¾, b ≤ z ≤ B,

and the integral of its inverse,

J(y, x) ≡Z yx

s−1(z)dz, b ≤ x ≤ y ≤ B.

Let x ∈ [b, B] be an initial condition, and let T = T (b) ∧ T (b) be the stopping timedefined as the first time the process reaches b or B. Then the probability that b is

reached before B, given the initial state x, is

θ(x) = Pr [T = T (b) | x] = J(B,x)J(B, b)

,

and the expected time until a threshold is reached is

τ (x) = E [T | x] = [1− θ(x)]Z Bb

2s(ζ)

σ2(ζ)J(B, ζ)dζ −

Z xb

2s(ζ)

σ2(ζ)J(x, ζ)dζ.

τ(b) =

·0×

Z Bb

2s(ζ)

σ2(ζ)J(B, ζ)dζ

¸− 0 = 0,

τ (B) =

·1×

Z Bb

2s(ζ)

σ2(ζ)J(B, ζ)dζ

¸−Z Bb

2s(ζ)

σ2(ζ)J(B, ζ)dζ = 0.

16

Calibration

Calibrating the preference parameters ζ, ω is challenging. In addition to data lim-

itations, two problems are inherent. First, housing is a product with many quality

dimensions, so controlling price differences for quality is difficult. In addition, the

substantial transaction costs associated with moving mean that empirical analyses

must rely on some sort of (untestable) assumptions about the adjustment process.

In the absence of transaction costs, the solution in (10) to the consumer’s (sta-

tic) problem of choosing the mix of housing and nondurables leads to the regression

equation

ln

µxh

1− xh

¶= a0 + (1− ε) ln ph,

where xh is the expenditure share for housing and ε = 1/ζ is the elasticity of substi-

tution. The Consumer Expenditure Survey reports an expenditure share of 32.9% for

housing for the U.S. as a whole. In principle a rough estimate of ε could be obtained

by exploiting cross-regional data, which shows considerable variation in the price of

housing relative to other goods. Data on the expenditure ratio are not available by

geographic region, however, so this avenue is a dead end.

Hanushek and Quigley (1980) look at data from the Housing Allowance Demand Ex-

periment, which involved a sample of low-income renters in Pittsburgh and Phoenix.

Households in each city were randomly assigned to treatment groups which received

rent subsidies that varied from 20% to 60% and a control group that received no sub-

sidy. The estimated price elasticities were -0.64 for Pittsburgh and -0.45 for Phoenix.

Taken together, these estimates suggest an elasticity of about -1/2, or ζ ≈ 2.Flavin and Nakagawa (2004) find a very low elasticity of substitution between

housing and nondurables. In Table 2 they report ζ = 1− (−6.5) = 7.5 or 7.7. Table 3gives about the same result for owners, a little lower for renters, ζ = 1−(−5.0) = 6.0.In either case the elasticity of substitution between housing and nondurables is very

17

low, on the order of ε = 1/7 = 0.15.

Regional price indices for 38 areas show considerable variation in the price of hous-

ing relative to nondurables. Normalized to have a mean of 1.00, they range from 1.34

in San Francisco to 0.85 in St. Louis and Dallas. If the elasticity of substitution is 1/2

the predicted range for the expenditure share for housing is from 37% to 31%, while

for an elasticity of 0.13 the predicted range is 39% to 29%. Since the expenditure

shares are not terribly sensitive to the elasticity parameter, we will use a wide range

of values, adjusting ω to keep the expenditure share for housing at the observed value

of 1/3.

REFERENCES

Campbell, John Y. and Joao R. Cocco. 2005. How do house prices affect consump-

tion? Evidence from micro data. NBER working paper 11534.

Chetty, Raj and Adam Szeidl. 2004. Consumption commitments: neoclassical

foundations for habit formation. Working paper.

Cocco, J., 2005. Portfolio choice in the presence of housing. Review of Financial

Studies 18, 535-567.

Dunn, K., Singleton, K., 1986. Modeling the term structure of interest rates under

non-separable utility and durability of goods. Journal of Financial Economics 17,

27-55.

Eichenbaum, Martin S. and Lars P. Hansen. 1990. Estimating models with in-

tertemporal substitution using aggregate time series data. Journal of Business and

Economic Statistics 8, 53-69.

Eichenbaum, Martin S., Lars P. Hansen, and Kenneth J. Singleton. 1988. A time

series analysis of representative agent models of consumption and leisure choice under

uncertainty, Quarterly Journal of Economics 103: 51-78.

Flavin, Marjorie and Shinobu Nakagawa. 2005. A model of housing in the presence

18

of adjustment costs: a structural interpretation of habit persistence. NBER working

paper 10458, April 2004.

Flavin, M., Yamashita, T., 2002. Owner-occupied housing and the composition of

the household portfolio. American Economic Review 92, 345-362.

Fukushima, Kenichi. 2005. Asset pricing implications of precommitted consump-

tion, working paper.

Lustig, H., Van Nieuwerburgh, S., 2006. Housing collateral, consumption insurance

and risk premia. Unpublished Working Paper, University of California, Los Angeles.

Grossman, Sanford and Guy Laroque. Asset pricing and optimal portfolio choice

in the presence of illiquid durable consumption good, Econometrica 58: 25-51.

Hanushek, Eric A. and John M. Quigley. 1980. What is the price elasticity of

housing demand? Review of Economics and Statistics 62: 449-454.

Ogaki, M., Reinhart, C.M., 1998. Measuring intertemporal substitution: the role

of durable goods. Journal of Political Economy 106, 1078-1098.

Piazzesi, Monika, Martin Schneider, and Selale Tuzel. 2006. Housing, consumption

and asset pricing, Journal of Financial Economics

Siegel, Stephen. 2004. Consumption-based asset pricing: durable goods, adjust-

ment costs, and aggregation, Working paper.

APPENDIX A: THE NTC MODEL

Proof of Lemma 1: If 0 < θ < 1, then [u(c)h]1−θ / (1− θ) > 0, and we needto restrict the parameters so that ρ > Γ for all feasible (c, h, a), so that expected

discounted utility does not diverge to +∞ along any feasible path. If θ > 1, then[u(c)h]1−θ / (1− θ) < 0, .and we need to insure that ρ > Γ for at least one feasible(c, h, a), so that expected discounted utility does not diverge to −∞ along everyfeasible path.

19

First we will show that if

ρ > Γ(c, h, a; θ),

all feasible (c, h, a) , if 0 < θ < 1,some feasible (c, h, a) , if θ > 1, (19)then the optimal portfolio satisfies (8). Then we will show that under Assumptions

1 and 2, (19) holds.

(i) Suppose (19) holds. Since a appears in (7) only as an argument of Γ, the optimal

portfolio solves

maxa∈[0,ass]

1

1− θ1

ρ− Γ(c, h, a; θ) .Hence the objective is to maximize Γ if 0 < θ < 1 and to minimize Γ if θ > 1. Note

that

Γa(c, h, a) = (1− θ)£(µ− r)− θaσ2¤

= (1− θ)σ2 (γ − θa) .

If 0 < θ < 1, then Γaa < 0, so Γ is concave. Since Γa(c, h, 0) > 0, there cannot be an

optimum at a = 0. If γ/θ > ass (1− h) , then

Γa(c, h, ass (1− h) ; θ) = (1− θ)σ2 [γ − θass (1− h)] > 0,

so the solution is at a corner, α(h; θ) = ass (1− h) . Otherwise the solution is interiorand satisfies Γa = 0. Hence the optimal portfolio is as in (8).

If θ > 1, the objective is to minimize Γ(c, h, a). In this case Γ is convex, and the

preceeding argument holds with a sign change. Hence (8) also holds for θ > 1.

(ii) Next we will show that Assumptions 1 and 2 insure (19) holds. Suppose

0 < θ < 1. Then

d

dhΓ(c, h, α(h; θ); θ) = Γh + Γaα

0(h; θ)

= − (1− θ) (ph + c) + (1− θ) σ2 [γ − θα(h; θ)]α0(h; θ)≤ − (1− θ) (ph + c)< 0, (20)

20

where the second line uses the fact that γ/θ ≥ α(h; θ) and α0(h; θ) ≤ 0. Hence forany ≥ 0,

ρ > Γ(0, 0, α(0; θ); θ) ≥ Γ(c, h, α(h; θ); θ) ≥ Γ(c, h, a; θ), all feasible (c, h, a) ,

where the first inequality uses Assumption 2, the second uses (20) and the fact that

∂Γ/∂c < 0, and the third uses the fact that α(h; θ, ) maximizes Γ(c, h, a; θ).

If θ > 1, then since r > 0, for a = 0 and all c, h sufficiently small,

ρ > 0 > (1− θ) [r − (ph + c)h] = Γ(c, h, 0). ¥

Proof of Proposition 2: The optimal portfolio aR is as in (8), with = 0.

For cR note that for any fixed expenditure flow E on housing and nondurables, the

optimal consumption mix (c, h) solves

maxc,h

u(c)h s.t. (ph + c) h = E.

Eliminating the constraint gives (10). Since aR does not involve h, maximizing (9)

with respect to h gives the FOC

0 =1− θhR

+Γh

ρ− Γ= (1− θ)

µ1

hR− ph + cR

ρ− Γ¶,

or

(ph + cR) hR = ρ− Γ [0, 0, αR(θ); θ] + (1− θ) (ph + cR)hRr +

1

θ

£ρ− r − σ2γaR(θ)

¤+ σ2γaR(θ) + (1− θ) 1

2σ2a2R(θ),

as in (11). Then

(ph + cR)h0R(θ) = −

1

θ2¡ρ− r − σ2γaR

¢− 12σ2α2R −

·µ1− θθ

¶γ − (1− θ)αR

¸σ2a0R

=1

θ2

·r − ρ+

µγ − θ21

2αR

¶σ2aR

¸,

21

where the last line uses the fact that either aR − γ/θ = 0 or a0R = 0. ¥

Proof of Proposition 3: The solution in (10)-(11) solves the problem with

> 0 if and only if aR, hR satisfies the tighter portfolio constraint, i.e. if and only if

then

aR(θ) =γ

θ≤ ass [1− hR(θ)] .

Otherwise the tighter portfolio constraint binds, and the owner solves

maxc,h

[u(c)h]1−θ

1− θ1

ρ− Γ[c, h, ass (1− h)] .

The FOCs for this problem are

0 =(1− θ)u0(c)

u(c)+

Γcρ− Γ ,

0 =1− θh

+Γh − Γa ass

ρ− Γ ,

or

u0(c)u(c)

=h

ρ− Γ ,1

h=

c + ph + [γ − θass (1− h)]σ2 assρ− Γ .

Combining these two gives gives

ω

ω + (1− ω) cζ−1 =c

ph + c+ [γ − θass (1− h)] σ2 ass .

The right side is increasing in c, taking values between 0 and 1. If ζ > 1 the left side

is decreasing in c, taking values between 1 and 0.

For a renter the right side of (x) is simply c/ (c + ph) . The additional term for the

constrained owner, which is positive, represents the higher implicit price of housing

that consumer faces.

22

The FOC for housing gives

(ph + cB) hB + (γ − θaB) σ2 ass= ρ− Γ (0, 0, aB; θ) + (1− θ) (ph + cB)hB,

or

hB =1

ph + cB

1

θ

£ρ− Γ (0, 0, aB; θ)− (γ − θaB) σ2 ass

¤,

while for a renter

hR =1

ph + cR

1

θ[ρ− Γ (0, 0, aR; θ)] .

23