Housing and the Business Cycle - Duke Universitypublic.econ.duke.edu/Papers//Other/Heathcote/housing.pdf · 2001-11-20 · Housing and the Business Cycle Morris Davis ReturnBuy and

Housing and the Business Cycle

Morris DavisReturnBuy and

Federal Reserve Board

Jonathan Heathcote∗

Duke University andStern School, NYU

November 5, 2001

Abstract

In the United States, the percentage standard deviation of residentialinvestment is more than twice that of non-residential investment. GDP,consumption, and both types of investment all co-move positively. At theindustry level, output and hours worked in construction are more than threetimes as volatile as in services, and output and hours co-move positivelyacross sectors. We reproduce all these facts in a multi-sector growth modelwith the following characteristics: different Þnal goods are produced usingdifferent proportions of the same set of intermediate inputs, constructionis relatively labor intensive, residential investment is relatively constructionintensive, and housing depreciates much more slowly than business capital.Previous empirical work exploring the determinants of residential invest-ment is re-examined in light of the model�s equilibrium relationship betweenresidential investment, house prices, and the rental rate on capital.

Keywords: Residential investment; House prices; Multi-sector modelsJEL classiÞcation: E2; E3; R3

∗Corresponding author: New York University, Stern / Economics, 44 W. 4th Str. 7-180,

New York NY 10012. Email: [email protected]. We thank seminar participants at

Duke University, the Board of Governors of the Federal Reserve, NYU, North Carolina State

University, the Stockholm School of Economics, the Society for Economic Dynamics Meetings

in Costa Rica 2000, the Macro / Labor Conference at the University of Essex 2000, the Bank of

Canada Conference on Structural Macro-Models 2000, and the NBER / Cleveland Fed Economic

Fluctuations and Growth Workshop in Philadelphia 2000. Heathcote thanks the Economics

Program of the National Science Foundation for Þnancial support.

1. Introduction

The size of the housing stock is large: similar in value to private non-residential

structures and equipment combined, similar in value to annual GDP, and three

times as large as the total stock of all other consumer durables. Although housing

is typically considered part of the economy�s capital stock in one-sector models

(see, for example, Cooley and Prescott in Cooley 1995), there are good reasons

for distinguishing between housing on the one hand and non-residential struc-

tures and equipment on the other. A conceptual reason is that to the extent that

housing is used for production, it is for production at home that for the most part

is not marketed and not measured in national accounts. A practical reason for

making the distinction is to attempt to account for differences between the busi-

ness cycle dynamics of residential and non-residential investment. In particular,

residential investment is much more volatile than business Þxed investment, and

strongly leads the cycle whereas non-residential investment lags.

The primary goal of this paper is to examine the extent to which a neoclassical

multi-sector stochastic growth model can account for the dynamics of residential

investment. A model economy with explicit microfoundations is calibrated using

industry-level data. The production structure is such that data on the empirical

counterpart to each variable in the model is available.1 There are two Þnal goods

sectors. One produces the consumption / investment good, while the second pro-

duces the residential investment good. Final goods Þrms use three intermediate

inputs produced in the construction, manufacturing and services sectors. These

intermediate inputs are in turn produced using capital and labor rented from

a representative household. Productivity is stochastic as a result of exogenous

sector-speciÞc labor-augmenting technology shocks.

The representative household maximizes expected discounted utility over per-

capita consumption, housing services and leisure. Each period it decides how

much to work and consume, and how to divide savings between physical capital

and housing, both of which are perfectly divisible. There is a government which

levies stochastic taxes on capital and labor. In this framework, changes in the

1This is not the case in the home production literature in which inputs to and productivity

within the home sector are imperfectly observed.

1

tax rate on capital income will affect the relative returns to saving in the forms

of capital and housing, since the implicit rents from owner-occupied housing are

untaxed.

We use the model to ask three main questions. First, to what extent can a

simple multi-sector model driven primarily by technology shocks account for the

business cycle facts relating to residential investment, non-residential investment

and house prices? Second, given the methodology for identifying productivity

shocks, to what extent can the model account for United States economic history

over the post War period? Third, can we use structural equilibrium relation-

ships implied by the model to reinterpret previous empirical work addressing the

determinants of residential investment?

Relation to literature

The model is speciÞcally designed to explore the unusual business cycle dy-

namics of residential investment. In most previous multi-sector real business cycle

models it is not possible to focus squarely on residential investment or house prices

since housing is typically not distinguished from other consumer durables (see, for

example, Baxter 1996 or Hornstein and Praschnik 1997).2 Failure to model hous-

ing explicitly may be important because there are important respects in which

housing differs from cars or televisions. First, housing is a much better store of

value since houses depreciate at a rate of only 1.6 percent per year, compared

to 21.4 percent for other durables. Second, the production technology for pro-

ducing houses is more construction intensive and thus more labor intensive than

the technology for producing consumer durables. We shall Þnd that these two

features are crucial in accounting for the dynamics of residential investment.

Multi-sector models typically have trouble replicating the strong positive co-

movement between consumption of durable goods and residential investment on

the one hand with consumption of nondurables and business Þxed investment on

the other. Furthermore labor supply tends to co-move negatively across sectors

in models, while hours worked co-move positively in different sectors of the U.S.

2One exception is an exploratory paper by Storesletten 1993 who found that the process for

sector-speciÞc shocks can not account for the fact that residential investment leads the cycle.

In a recent paper, Edge 2000 considers the differential effects of monetary shocks on residential

and structures investment in a multi-sector model with sticky prices.

2

economy. The reasons for these failures are simple. First, the models incorporate

a strong incentive to switch production between sectors in response to sector

speciÞc productivity shocks. Second, even if shocks are perfectly correlated across

sectors, there is an incentive to increase output of new capital prior to expanding

anywhere else, since additional capital is a pre-requisite for efficiently expanding

output in other sectors.3 Previous authors have also had trouble accounting for

the fact that residential investment (or residential investment plus spending on

consumer durables) is more volatile than business investment.4 We shall also

be interested in assessing the extent to which our novel production structure and

calibration methodology can address both the co-movement and relative volatility

puzzles.

An alternative to our approach of introducing the housing stock directly in

the utility function would have been to assume that housing and non-market time

are combined according to a home production technology to give a non-marketed

consumption good. Greenwood, Rogerson and Wright (1995 p.161) show that

these alternative modelling approaches are closely related. In particular, given

3Various Þxes have been proposed to solve the co-movement problem. Fisher 1997 assumes a

non-linear function for transforming output into non-durable consumption goods, new consumer

durables, and new physical capital. Since in the limit different goods must be produced in Þxed

proportions it is easy to see how this approach can resolve the co-movement problem. Baxter

1996 estimates a high correlation between productivity growth across sectors and also introduces

sectoral adjustment costs for investment, which dampens the incentive to increase investment

in the sector producing capital while reducing investment elsewhere. Chang 2000 combines

adjustment costs with substitutability between time and durable goods; thus when households

work more in periods of high productivity they also demand more durables. Gomme, Kydland

and Rupert 2001 introduce time-to-build in the sector producing new market capital, which has

a similar effect to introducing adjustment costs in that it dampens the investment boom in the

capital-producing sector and allows investment to rise in all sectors simultaneously. Boldrin,

Christiano and Fisher 2001 Þnd that a combination of limited labor mobility across sectors and

a habit in consumption can generate co-movement in hours worked across sectors.4Fisher 1997 Þnds that none of his speciÞcations give household investment more volatile

than business investment. In Baxter�s 1996 model, consumption of durables (which includes

residential investment) is too smooth and is less volatile than business Þxed investment in either

sector. For all but one of the parameterizations they consider, Gomme, Kydland and Rupert

2001 Þnd market investment to be more volatile than home investment, contrary to the pattern

in the data.

3

(i) a Cobb Douglas technology for producing the home good from capital and

labor, and (ii) log-separable preferences over leisure, market consumption and

home consumption, the home production model has a reduced form in which

only market consumption, market hours and the stock of home capital enter the

utility function. The benchmark calibration adopted by Greenwood and Her-

cowitz (1991) satisÞes these functional-form restrictions, which suggests that our

model is closely related to theirs. The main difference is that contrary to Green-

wood and Hercowitz, we do not assume a single (market) production technology.

In the results section we systematically compare and contrast the two economies.

All the papers described above are based on representative agent economies.

We remain within the representative agent framework since our focus is on un-

derstanding business cycle dynamics. Models with incomplete markets environ-

ments typically focus on steady states (see for example Platania and Schlagenhauf

2000 or Fernandez-Villaverde and Krueger 2001).5 While frictions such as poorly

functioning rental and mortgage markets are likely important in accounting for

cross-sectional issues (such as life-cycle consumption / savings patterns or hetero-

geneity in asset holding portfolio choices) it is not obvious that they are important

for housing dynamics at the aggregate level.6 In any case it would appear to be

sensible to ask whether a representative agent model with complete asset markets

is broadly able to capture observed aggregate dynamics before turning to richer

environments.

The predominant theory of residential investment in the real estate literature

(see, for example, Topel and Rosen 1998 or Poterba 1984 and 1991) may be

summarized as follows. There is a price for residential housing that clears the

market given the existing stock of homes. Developers observe this price and build

5Diaz-Gimenez, Prescott, Fitzgerald and Alvarez 1992 and Ortalo-Magne and Rady 2001

both consider the effects aggregate shocks in model economies which incorporate indivisibilities

in housing, credit constraints, and life cycle dynamics. However the focus on Diaz-Gimenez et.

al. is on monetary policy rather than the business cycle dynamics, while Ortalo-Magne and

Rady assume a constant stock of housing, and therefore have nothing to say about residential

investment.6Krusell and Smith 1998 and Rios-Rull 1994 study the aggregate dynamics of economies in

which households face large amounts of uninsurable idiosyncratic risk. They Þnd that they are

virtually identical to those observed when markets are complete.

4

more houses the larger is the gap between this price and the cost of construction,

which is often proxied by the real interest rate. Changes in prices are primarily

driven by demand-shocks which affect future expected rental rates. Regressions

of residential investment on house prices and the real interest rate have typically

generated positive and strongly signiÞcant estimates for the coefficient on the

house price term. This has been presented as evidence in support of the demand-

shock driven view of residential investment.

The Þrst order conditions of the model developed imply a particular equi-

librium relationship between residential investment and new house prices. This

relationship indicates a negative relationship between residential investment and

house prices, holding other variables constant. However, the equation implied by

the model also includes the marginal product of capital and terms involving sec-

toral capital stocks and the existing stock of houses. Using simulated data from

the model we measure the size of omitted variable bias on the house price term

when estimating residential investment equations of the type used in previous

empirical work.

Findings

The calibrated model economy is found to account for the following facts:

(i) the percentage standard deviation of residential investment (at business cycle

frequencies) is almost three times that of non-residential investment, (ii) hours

worked and output are most volatile in the construction sector and least volatile

in the services sector, (iii) consumption, non-residential investment, residential

investment and GDP all co-move positively, (iv) hours worked and output in the

construction, manufacturing, and services sectors all co-move positively, (v) house

prices are procyclical and positively correlated with residential investment, (vi)

residential investment (weakly) leads the cycle, and (vi) the percentage standard

deviation of GDP is around two. Among the moments we compute, there are two

respects in which the model performs poorly. First the model cannot account for

the observed volatility in house prices. Second the model does not reproduce the

observation that non-residential investment lags the cycle.

Note that the model resolves both the co-movement and relative volatility puz-

zles. These successes are attributable to certain characteristics of the calibrated

5

production technologies.

First, while our Solow residual estimates suggest only moderate co-movement

in productivity shocks across intermediate goods sectors, co-movement in effective

productivity across Þnal goods sectors is ampliÞed by the fact that both Þnal

goods sectors use all three intermediate inputs, albeit in different proportions. A

second feature we emphasize is that construction and hence residential investment

are relatively labor intensive. Residential investment volatility rises when capital�s

share in construction is reduced because following an increase in productivity

less additional capital (which takes time to accumulate) is required to efficiently

increase the scale of production. A third important aspect of the calibration is

that the depreciation rate for housing is much slower than that for capital. This

increases the relative volatility of residential investment, since it increases the

incentive to concentrate production of new houses in periods of high productivity.

When we ask the model how well it can account for observed U.S. history, we

Þnd that it does an good job in accounting for the observed time paths for GDP,

consumption and both non-residential and residential investment.

Estimating the correct and mis-speciÞed equations for residential investment

on simulated model output, we Þnd that for the mis-speciÞed equation (loosely

based on previous empirical work) the point estimate for the house price coeffi-

cient is positive. Recall that this co-efficient is negative in the correctly speciÞed

equation, reßecting the fact that residential investment is driven by supply-side

productivity shocks. Thus we argue that care should be taken in drawing in-

ferences from previous estimation results as to the appropriate way to model

residential investment.

2. The Model

In each period t the economy experiences one event et ∈ E. We denote by etthe history of events up to and including date t. The probability at date 0 of any

particular history et is given by π(et). The population grows at a constant rate

η. In what follows all variables are in per-capita terms.

A representative household supplies homogenous labor and rents homogenous

capital to perfectly competitive intermediate-goods-producing Þrms. These Þrms

6

allocate capital and labor frictionlessly across three different technologies. Each

technology produces a different good which we identify as construction, manu-

factures and services, and which we index by the subscripts b, m and s respec-

tively. The quantities of each good produced after history et are denoted xi(et),

i ∈ {b,m, s} . Output of intermediate good i is a Cobb-Douglas function of thequantity of capital ki(et) and labor ni(et) allocated to technology i :

xi(et) = ki(e

t)θi³zi(e

t)ni(et)´1−θi

. (2.1)

Note that the three production technologies differ in two respects. First, the

shares of output claimed by capital and labor, determined by capital�s share θi,

differ across sectors. For example, our calibration will impose θb < θm, reßecting

the fact that construction is less capital intensive than manufacturing. Second,

each sector is subject to exogenous sector-speciÞc labor-augmenting productivity

shocks. We let zi(et) denote labor productivity in sector i after history et.

Let pi(et) denote the price of good i in units of the Þnal consumption good

delivered after history et. Let w(et) and r(et) be the wage and rental rate on

capital measured in the same units. The intermediate Þrms� static maximization

problem after history et is

max{ki(et),ni(et)}i∈{b,m,s}

Xi

npi(e

t)xi(et)o− r(et)k(et)−w(et)n(et)

subject to eq. 2.1 and to the constraints7

kb(et) + km(e

t) + ks(et) ≤ k(et),

nb(et) + nm(e

t) + ns(et) ≤ n(et),n

ki(et), ni(e

t)oi∈{b,m,s} ≥ 0.

The law of motion for intermediate Þrms� productivities has a deterministic

and a stochastic component. We assume a constant trend growth rate for each

7Note that while capital in sector i at date t is chosen at date t, aggregate capital in place

at date t is chosen at t− 1, in the standard way. Thus in equilibrium k(et) does not depend on

the shocks realized at date t.

7

technology, but permit the rate to vary across sectors. Let gzi denote the constant

gross trend growth rate of labor productivity associated with technology i.

The goods produced by intermediate goods Þrms are used as inputs by Þrms

producing two Þnal goods: a consumption / capital investment good and a resi-

dential investment good. Final goods Þrms are perfectly competitive and allocate

the intermediate goods freely across two Cobb-Douglas technologies. We use the

subscript c to index the consumption / capital investment good and h to index

residential investment (RESI). Let yj(et), j ∈ {c, h} denote the quantity of Þnalgood j produced after history et using quantities bj(et), mj(e

t) and sj(et) of the

three intermediate inputs. Thus

yj(et) = bj(e

t)Bjmj(et)Mjsj(e

t)Sj j ∈ {c, h} (2.2)

where Bj , Mj and Sj = 1−Bj −Mj denote the shares of construction, manufac-

tures and services respectively in sector j. The technology used to produce the

consumption good differs from that used to produce RESI with respect to the rel-

ative shares of the three intermediate inputs,. Thus, for example, our calibration

leads us to set Bh > Bc, reßecting the fact that the housing sector is relatively

construction-intensive.

We normalize the price of the consumption good after any history to 1, and let

pr(et) denote the price of RESI. The Þnal goods Þrms� static proÞt maximization

problem after history et is

max{bj(et),mj(et),sj(et)}j∈{c,h}

yc(et) + pr(e

t)yh(et)−

Xi∈{b,m,s}

npi(e

t)xi(et)o

subject to eq. 2.2 and to the constraints

bh(et) + bc(e

t) ≤ b(et),

mh(et) +mc(e

t) ≤ m(et),sh(e

t) + sc(et) ≤ s(et),n

bj(et),mj(e

t), sj(et)oj∈{c,h} ≥ 0.

8

There is a government which raises revenue by taxing labor income at rate

τn(et) and capital income (less a depreciation allowance) at rate τk(et). Tax rev-

enues are divided between non-valued government spending on the consumption

/ investment good denoted g(et), and lump-sum transfers to households denoted

ξ(et). Government consumption is assumed to be a Þxed fraction of output of the

consumption / investment sector. Tax rates, however, are stochastic, and follow

an autoregressive process jointly with sector-speciÞc productivity shocks:

bz(et+1) =³log ezb(et+1), log ezm(et+1), log ezm(et+1), eτk(et+1), eτn(et+1) ´0

= Bbz(et) + bε(et+1), ∀et+1 consistent with et.

Tildas are used to indicate that each element of bz(et) records the deviation fromtrend value at et, for example,

log ezb(et) = log zb(et)− t log gzb − log zb,0.The 5×5 matrix B captures the deterministic aspect of how shocks are trans-

mitted through time, and bε(et+1) is a 5 × 1 vector of shocks drawn indepen-dently through time from a multivariate normal distribution with mean zero and

variance-covariance matrix V.

The representative household derives utility each period from per-capita house-

hold consumption c(et), from per-capita housing owned h(et), and from leisure.

The size of the household grows at the population growth rate η. The amount of

per-household-member labor supplied plus leisure cannot exceed the period en-

dowment of time, which is normalized to 1. Period utility per household member

after history et is given by

U(c(et), h(et), (1− n(et)) =¡c(et)µch(et)µh(1− n(et))1−µc−µh¢1−σ

1− σwhere µc and µh determine the relative weights in utility on consumption, housing

and leisure. At date 0, the expected discounted sum of future period utilities for

the representative household is given by

∞Xt=0

Xet∈Et

π(et)βtηtU(c(et), h(et), (1− n(et)) (2.3)

9

where β < 1 is the discount factor.8 Note that the ßow of utility that households

receive from occupying housing they own will constitute an implicit rent that is

untaxed.

Households divide income between consumption, spending on new capital that

will be rented out next period and has price pnk(et), and spending on new housing

that will be occupied next period and has price pnh(et). Once rents have been paid

and utility from housing delivered, households sell old capital and housing to a

new type of Þrm, which we label the adjustment Þrm, at prices pok(et) and poh(e

t).9

The depreciation rate for capital is given by δk (houses depreciate at rate δh).

Thus the household budget constraint is10

c(et) + ηpnk(et)k(et+1) + ηpnh(e

t)h(et+1)− pok(et)k(et)− poh(et)h(et) (2.4)

=³1− τn(et)

´w(et)n(et) + r(et)k(et)− τk(et)

³r(et)− δk

´k(et) + ξ(et).

Adjustment Þrms combine �old� capital and housing with the capital invest-

ment good and the residential investment good (purchased from Þnal goods Þrms)

to produce new capital and new housing. Because the capital investment good

and the consumption good are perfect substitutes in production, they must have

the same price in equilibrium. Consumption is the numeraire good, and this price

is therefore unity.

The static maximization problem of an adjustment Þrm is

maxih(et),ik(et),k(et),h(et)

ηpnk(et)k(et+1) + ηpnh(e

t)h(et+1)

−pr(et)ih(et)− ik(et)− pok(et)k(et)− poh(et)h(et)subject to the production technologies

ηk(et+1) = (1− δk)k(et) + φkÃik(e

t)

k(et)

!k(et), (2.5)

8Note that the household weights per-household-member utility by the size of the household.9Note that this is slightly different than the standard formulation, in which households pur-

chase investment rather than new capital. However, equilibrium allocations here are identical

to those under the standard description; we adopt this particular decentralization because it is

convenient for pricing capital and housing at different moments within the period.10The population growth rate η multiplies variables dated t + 1 because all variables are in

per-capita terms.

10

ηh(et+1) = (1− δh)h(et) + φhÃih(e

t)

h(et)

!h(et). (2.6)

The production technology for new capital and new houses incorporates a

penalty for deviating from steady state investment rates. The functions φk and

φh are increasing, concave, and satisfy the following properties in steady state:

φk

Ãikk

!=ikkand φh

Ãihh

!=ihh, (2.7)

φ0k

Ãikk

!= φ0h

Ãihh

!= 1, (2.8)

ζk =−φ00k

³ikk

´ikk

φ0k³ikk

´ = ζh =−φ00h

³ihh

´ihh

φ0h³ihh

´ . (2.9)

The Þrst two properties imply zero steady state adjustment costs, while the

third implies that the elasticity of the price of new capital with respect to the

investment rate (ik/k) is equal across sectors.

The representative household chooses k(et+1), h(et+1), c(et) and n(et) for all

et and for all t ≥ 0 to maximize expected discounted utility (eq. 2.3) subject to asequence of budget constraints (eq. 2.4) and a set of inequality constraints c(et),

n(et), h(et), k(et) ≥ 0 and n(et) ≤ 1. The household takes as given a complete setof history dependent prices, tax rates and transfers pok(e

t), poh(et), pnk(e

t), pnh(et),

r(et), w(et), τk(et), τn(e

t), ξ(et), unconditional probabilities over histories given

by π(et), and the initial stocks of capital and housing.

2.1. DeÞnition of equilibrium

An equilibrium is a set of prices, taxes and transfers pi(et)i∈{b,m,s}, pok(et), poh(e

t),

pnk(et), pnh(e

t), pr(et), r(et), w(et), τk(e

t), τn(et), ξ(et) for all et and for all t ≥ 0

such that when households solve their problems and Þrms proÞt maximize taking

these prices as given all markets clear and the government�s budget constraint is

satisÞed.

Market clearing for Þnal goods implies that

c(et) + ik(et) + g(et) = yc(e

t),

11

ih(et) = yh(e

t).

Market clearing for intermediate goods implies that

bh(et) + bc(e

t) = xb(et), (2.10)

mh(et) +mc(e

t) = xm(et), (2.11)

sh(et) + sc(e

t) = xs(et). (2.12)

Market clearing for capital and labor implies that

kb(et) + km(e

t) + ks(et) = k(et),

nb(et) + nm(e

t) + ns(et) = n(et).

Since the government cannot issue debt, the government budget constraint is

satisÞed when

ξ(et) + g(et) = τn(et)w(et)n(et) + τk(e

t)³r(et)− δ

´k(et).

2.2. Equilibrium prices

The Þrst order conditions for the intermediate goods Þrms� problem are as follows

(suppressing history dependence).

With respect to capital by sector

r = piθik(θi−1)i (zini)

1−θi i ∈ {b,m, s} . (2.13)

With respect to labor by sector

w = zipi(1− θi)kθii (zini)−θi i ∈ {b,m, s} . (2.14)

The Þrst order conditions for the Þnal goods Þrms� problem are as follows.

With respect to construction goods, manufactures and services by sector

pb =Bcycbc

=Bhyhprbh

, (2.15)

pm =Mcycmc

=Mhyhprmh

, (2.16)

12

ps =Scycsc

=Shyhprsh

. (2.17)

The Þrst order conditions for the adjustment Þrms� problem are as follows.

With respect to non-residential and residential investment

pnk = 1/φ0k

µikk

¶, (2.18)

pnh(et) = pr(e

t)/φ0hµihh

¶. (2.19)

With respect to old capital and old housing

pokpnk= (1− δk) + φk

µikk

¶− φ0k

µikk

¶ikk, (2.20)

pohpnh= (1− δh) + φh

µihh

¶− φ0h

µihh

¶ihh. (2.21)

It is straightforward to show that all Þrms, including adjustment Þrms, make

zero proÞts in every state.

From eqs. 2.14 to 2.17 we derive the following expression for the price of

residential investment

log pr = {κ1 + (Bc −Bh)(1− θb) log zb + (Mc −Mh) (1− θm) log zm (2.22)

+(Sc − Sh) (1− θs) log zs}+ [(Bc −Bh)θb + (Mc −Mh)θm + (Sc − Sh) θs] (log km − lognm)

This expression indicates that a positive productivity shock in sector i tends

to reduce the relative price of residential investment if residential investment is

relatively intensive in input i. The size of the relative price change is increasing

in the difference in factor intensities across the two Þnal goods technologies, and

is increasing in the labor intensity of sector i.11

From eq. 2.19 it is clear that the price of new housing is closely related to

the price of residential investment. However, the two prices are not identical in

the presence of adjustment costs. For example, if the residential investment rate

is above average, an additional unit of new housing is more expensive than an

additional unit of residential investment.11To the extent that a productivity shock affects equilibrium sectoral capital-output ratios,

there is a second effect on the relative price of residential investment via the last term in eq.

2.22. This last effect disappears if θb = θm = θs.

13

2.3. Rental markets, mortgage markets, and house prices

In the description of the model economy above we abstract from many aspects

of housing that have attracted attention, such as the existence of rental markets,

the market for mortgages, and the deductability of mortgage interest payments.

Markets in our model are complete, however, so it is straightforward to imagine

rental or mortgage markets, and to price whatever is traded in these markets.

For example, one could imagine that each household rents out some or all of

the housing they own to its neighbor, thereby breaking the link between ownership

and occupation and establishing a rental market. Given equilibrium allocations

(which are independent of the size of this hypothetical rental market) the rental

rate for housing, denoted q, is such that households are indifferent to renting a

marginal unit of housing:

q =Uh(c, h, (1− n))Uc(c, h, (1− n)) .

If rental income is taxed at a positive rent, households will strictly prefer

owner-occupation to owner-renting since the implicit rents from owner-occupation

are untaxed. If rental income is not taxed, the size of the rental sector is indeter-

minate.

Suppose next that rather than buying housing out of income, households have

the option of borrowing on a mortgage market, where interest payments on these

loans are tax deductible. It is straightforward to see that if the rate at which

households can deduct mortgage interest payments against tax is exactly equal to

the tax rate on capital income, then households will be indifferent between paying

cash for housing versus taking out a mortgage. If the rate at which households

can deduct mortgage interest is less than this, households strictly prefer to pay

cash. The intuition is simply that the equilibrium net after tax rate of interest

on a mortgage loan in the economy is (r − δk)(1 − τ∗) where τ∗ is the fractionof interest payments that may be deducted against tax. The marginal beneÞt of

taking out a mortgage loan is the return on the extra dollar of savings that can

then be saved, with return (r − δk)(1− τk). Only if τ∗ = τk will households beindifferent between alternative ways of Þnancing house purchases.12

12Gervais 2001 conducts a richer analysis of the interaction between housing and the tax code.

14

What is the appropriate measure of the price of housing? Thus far we have

deÞned two house prices: pnh, the price of a unit of new housing to be delivered at

the start of the next period, and poh, the price of a unit of used housing delivered

at the end of the current period. For comparison to the data, we choose to deÞne

the price of housing as the price that would arise in a market for housing delivered

at the start of the current period (once shocks have been observed). Households

must be indifferent between buying a house on this market versus renting a house

for the current period, and buying a unit of used housing to be delivered at the

end of the period. Thus we deÞne

ph = q + poh

In the National Income and Product Accounts, private consumption includes

an imputed value for rents from owner-occupied housing. We choose to deÞne

private consumption and GDP consistently with the NIPA. Thus private con-

sumption expenditures is given by

PCE = c+ qh.

Analogously GDP is given by

GDP = yc + pryh + qh.

Note lastly that during a simulation of the economy, prices are changing,

both because sector speciÞc trends in productivity, and because of sector speciÞc

shocks around these trends. We deÞne real private consumption and real GDP

using balanced growth path prices, so that our measures of real quantities capture

trends in relative prices, but not short-run changes in relative prices.

2.4. Solution method

Our goal is to simulate a calibrated version of the model economy. The Þrst step

towards characterizing equilibrium dynamics is to solve for the model�s balanced

growth path. We have a multi-sector model in which the trend growth rate of

labor productivity varies across sectors. A balanced growth path exists since

15

preferences and all production functions have a Cobb-Douglas form. The gross

trend growth rates of different variables are described in table 1.

Several properties of these growth rates may be noted. The trend growth rates

of yc, phyh and pixi for i ∈ {b,m, s} are all equal to gk, the trend growth rateof the capital stock and consumption. This growth rate is a weighted product

of productivity growth in the three intermediate goods sectors. For example,

if capital�s share is the same across sectors then gk = gBczb gMczmg

Sczs . The trend

growth rates of intermediate goods prices exactly offset the effects of differences

in productivity growth across sectors such that (i) interest rates are trendless in

all sectors (see eq. 2.13) and (ii) wages in units of consumption grow at the same

rates across sectors (see eq. 2.14).

Given trend growth rates for variables, the next step is to use these growth

rates to take transformations of all the variables in the economy such that the

transformed variables exhibit no trends. We do this because for computational

purposes it is convenient to work with stationary variables. The new stationary

variables are deÞned as follows, where x denotes a generic old variable, gx is the

gross trend growth rate of the variable, and �xt is the stationary transformation:

�xt =xtgtx

The penultimate step in the solution method is to linearize a set of equations

in stationary variables that jointly characterize equilibrium around the balanced

growth path, which corresponds to a vector containing the mean values of the

transformed variables in the system. We solve the system of linear difference

equations using a Generalized Schur decomposition (see Klein 2000).

2.5. Data and calibration

The model period is one year.13 This is designed to approximately capture the

length of time between starting to plan new investment and the resulting increase

13For more detail on all data sources and calibration procedures, see the data appendix of

this paper which is available as Duke Economics Working Paper Number 00-09 and is also at

www.econ.duke.edu/~heathcote/research.htm

16

in the capital stock being in place.14 Edge (2000) reports that for non-residential

structures the average time to plan is around 6 months, while time to build from

commencement of construction to completion is around 14 months. For residential

investment the corresponding Þgures are 3 months and 7 months.15

Parameter values are reported in tables 2 and 3. The population growth rate,

η, is set to 1.8 percent per year, the average rate of growth of hours worked in

private sector between 1949 and 1998, which is the sample period used for cali-

bration purposes. Data from the National Income and Product Accounts Tables,

the Fixed Reproducible Tangible Wealth tables, the Gross Product by Industry ta-

bles, and the Benchmark Input-Output Accounts of the United States, 1992 (all

published by the Department of Commerce) are used to calibrate most remaining

model parameters.

The empirical analogue of the model capital stock is the stock of private Þxed

capital (excluding the stocks of residential capital and consumer durables) plus the

stock of government non-defense capital. The depreciation rate for capital, δk, is

set to 5.2 percent, which is the average annual depreciation rate for appropriately

measured capital between 1949 and 1998. The empirical analogue of the model

housing stock is the stock of residential Þxed private capital. The average annual

depreciation rate for housing, which identiÞes δh, is 1.6 percent.

14A yearly model is also convenient because data on inputs and output by intermediate indus-

try and tax rate estimates are only available on an annual basis. In the Þrst draft of this paper,

however, we set the period length to a quarter, and used an interpolation procedure to estimate

sector-speciÞc shocks (see the data appendix). Business cycle statistics are substantively the

same for both period lengths.15Gomme Kydland and Rupert 2001 argue that faster time to build for residential structures

can help account for the fact that non-residential investment lags the cycle, and that residential

and non-residential investment are positively correlated contemporaneously. In their calibration

they set the time to build for residential investment to 1 quarter, and the time for non-residential

investment to 4 quarters. This difference is probably too large, both because time to build for

residential investment is likely longer than a quarter, and also because private non-residential

structures only accounts for 28% of total non-residential investment over the 1949 to 1998 period;

the majority of non-residential investment is accounted for by investment in equipment and

software which can presumably be put in place more quickly. While there is probably still some

role for differential time to build, we abstract from it in this analysis to examine alternative

mechanisms for generating realistic dynamics for investment over the business cycle.

17

The coefficient of relative risk aversion, σ, is set equal to 2. All other preference

parameters are endogenous. The shares of consumption and housing in utility (µcand µh) are chosen so that in steady state, households spend 30 percent of their

time endowment working, and so that the value of the capital stock is equal to

annual GDP, which is the case, on average, for the sample period. The discount

factor, β, is set so that the annual after tax real interest rate in the model is 6

percent. The implied value for β in the benchmark model is 0.967.

Industry-speciÞc data (industries are deÞned according to the 1987 2-digit

SIC) are used to calibrate capital shares for the three intermediate sectors of the

model: construction (b), manufacturing (m), and services (s). For the model con-

struction sector, we use SIC construction industry data. For manufacturing, we

use all NIPA classiÞed �goods-producing� industries except construction: agri-

culture, forestry, and Þshing (AFF), mining, and manufacturing. For services we

use all �services-producing� industries except FIRE: transportation and public

utilities, wholesale trade, retail trade, and services.16

For each model sector i, the sectoral capital share in year t, θi,t, is deÞned as

θi,t =

PjCOMPj,tP

j{V Aj,t − IBTj,t − PROj,t} (2.23)

where the j subscript denotes speciÞc SIC industries included in sector i, and

COMPj,t, V Aj,t, IBTj,t and PROj,t denote, respectively, nominal compensation

of employees, nominal value added, nominal indirect business tax and non-tax

liabilities, and nominal proprietor�s income for industry j in year t. The av-

erage value of the construction sector capital share over the period is 0.13, for

manufacturing it is 0.31, and for services it is 0.24 (see table 3).

The logarithm of the (non-stationary) annual Solow residual in intermediate

sector i is given by

log (zi,t) =1

1− θi [log (xi,t)− θi log (ki,t)− (1− θi) log (ni,t)] . (2.24)

16The FIRE (Þnance, insurance, and real estate) industry is omitted when calculating the

capital share of the service sector of the model because much of FIRE value added is imputed

income from owner occupied housing.

18

where xi,t is real output of intermediate sector i in year t, ki,t is real sectoral

capital, and ni,t is sectoral hours worked.

The residual of a regression of log (zi,t) on a constant and a time trend over

the sample period deÞnes the logarithm of the detrended annual Solow residual

for industry i, denoted log (ezi,t). The annual growth rates of the non-stationarySolow residuals are −0.25 percent in construction, 2.77 percent in manufacturing,and 1.68 percent in services, identifying the quarterly growth rates gzb, gzm, and

gzs.

Government consumption is set equal to 18.1 percent of GDP, the period

average.17 The mean tax rates on capital and labor income, τk and τn, are set

so that along the balanced growth path the model matches two features of the

data over the period: the capital stock averages 1.52 times annual output, and

government transfers average 8.2 percent of GDP.18 To construct a series of shocks

to tax rates, we take the series estimated by McGrattan, Rogerson and Wright

(1997), who extend a methodology developed by Joines (1981). These rates are

reported from 1947 to 1992. We extend the series for the years 1993 to 1997 using

the method developed by Mendoza, Razin and Tesar (1994) and data from the

OECD Revenue Statistics and National Accounts.19

In the model, logged detrended sectoral productivities and detrended tax rates

are assumed to follow a joint autoregressive process. The estimates of the pa-

rameters deÞning this process are in table 4 at the end of the paper. A few

17Government consumption in the data is deÞned as NIPA government consumption expen-

ditures plus NIPA government defense investment expenditure.18Over the sample period, capital tax rates have trended downwards, while labor tax rates

have increased. When we solve and simulate the model, however, we assume that there are no

trends in tax rates. We do this because (i) this would considerably complicate computation

of the model�s balanced growth path, and (ii) in the long run tax rates are bounded; thus the

upward trend in labor tax rates is not sustainable.19The estimates for the last Þve years are thus constructed using a somewhat different method-

ology than the 1947 to 1992 Þgures, and generate different average tax rates. However, for the

period for which estimates based on the both methodologies are available, year to year changes

in the estimated rates track each other closely. We therefore identify a series for tax shocks by

(i) rescaling appropriately so that the two different sets of estimates for capital and labor tax

rates coincide in 1992, (ii) estimating linear trends in the two tax rate series by OLS, and (iii)

computing deviations in the series from this linear trend.

19

features of these estimates are worth mentioning. First, there is little evidence

that technology shocks spill-over across intermediate goods sectors. Second pro-

ductivity shocks appear to be rather more persistent that shocks to tax rates.

Third productivity shocks in the construction and manufacturing sectors appear

to be considerably more volatile than those in services. Productivity shocks are

weakly correlated across sectors, and in particular shocks to the construction sec-

tor are essentially uncorrelated with those in manufacturing. Capital tax rates

are considerably more volatile than labor tax rates.

An important aspect of the calibration procedure concerns the estimates for

{Bc,Mc, Sc} and {Bh,Mh, Sh}, the shares of construction, manufacturing andservices in production of the consumption-investment good (subscript c) and the

residential investment good (subscript h). These parameters determine the extent

to which residential investment is produced with a different mix of inputs than

other goods. At this point, we employ the �Use� table of 1992 Benchmark NIPA

Input-Output (IO) tables. The IO Use table contains two sub-tables. In the Þrst,

total spending on components of Þnal aggregate demand (personal consumption,

private Þxed investment, etc.) is decomposed into sales purchased from all in-

termediate industries. In the second, total sales for each private industry (and

for the government) are attributed to value-added by that industry, and sales

purchased from other industries. Thus, for example, Þnal sales of the construc-

tion industry include value added from construction and sales purchased from the

manufacturing and services sectors.

One possible approach would be to assume that the distribution of value-

added across intermediate sectors for each component of Þnal demand is equal

to the distribution of sales purchased from the different sectors (see table 5).

Rather than doing this, we use the second IO table to track down where value

was originally created in each intermediate industry�s sales. For example, some

portion of construction sales is attributed to purchases from manufacturing, which

in turn can implicitly be divided into manufacturing value added plus sales to

manufacturing from construction and services. Since this trail is never ending,

dividing the Þnal sales of a particular industry into fractions of value-added by

each intermediate industry requires an inÞnite recursion.

Once we have this breakdown, we use the Þrst IO Use table to compute,

20

for example, the fraction of value added in residential investment from the con-

struction industry (which will identify the parameter Bh). The results are given

in table 6. The shares of value added by construction, manufacturing and ser-

vices in the consumption-investment sector are respectively Bc = 0.0307, Mc =

0.2696, and Sc = 0.6998. For residential investment, the corresponding shares

are Bh = 0.4697, Mh = 0.2382, and Sh = 0.2921. Comparing tables 5 and 6 it

is clear that there are large differences between the distribution of value added

and the distribution of sales. For example, although we attribute residential in-

vestment entirely to sales from construction, these construction sales implicitly

contain large quantities of value originally created in the manufacturing and ser-

vice industries, such that only 47 percent of the value of residential investment is

ultimately attributable to the construction industry.20�21

The parameters determining the size of adjustment costs in residential and

non-residential investment are set such that adjustment costs are identical for the

two types of investment, and so that the average percentage standard deviation of

residential investment across simulations of the model is 5.11 times that of GDP,

the average empirical ratio over the sample period. One measure of the size of

adjustment costs is the elasticity of the price of new capital, pnk , with respect to

the investment rate, ik/k. In the model this elasticity (equal, by assumption for

both types of investment) is 0.137. This represents much smaller adjustment costs

than are often used.22

2.6. Questions

There are four sets of issues we use the model to address. First, we ask how

successful is the calibration procedure in terms of matching Þrst moments, such

20More details concerning the data and the matrix algebra used to construct table 6 are in

the data appendix.21Hornstein and Praschnik 1997 describe a model in which a non-durable intermediate input

is used in durable goods production, but they do not use Input-Output data in their calibration

procedure.22An alternative measure of the size of adjustment costs is the average value for£ik(e

t)− φk¡ik(e

t)/k(et)¢k(et)

¤/ik(e

t), which is the fraction of resources invested that do not

tranlate into additional new capital because of adjustment costs. Under the baseline calibration

this value is 0.037 percent.

21

as the average fraction of GDP accounted for by residential investment. Second,

we simulate the model and compare second moments of simulated model output

to the data. This is the standard exercise in the real business cycle tradition.

To gain some intuition about how the model works, we systematically compare

our model to Greenwood and Hercowitz (1991). Third, we feed in the actual

productivity and tax shocks suggested by our calibration procedure, and examine

the extent to which the model can account for the observed history of a set of

macroeconomic aggregates from 1948 to 1997. Lastly, we contrast our model with

the typical framework for thinking about residential investment, and argue that

previous empirical work does not necessarily support the view that residential

investment dynamics are driven by shocks on the demand side rather than the

supply side.

3. Results

3.1. Balanced growth path

Table 7 indicates that the model is very successful in terms of matching Þrst

moments. For example, the shares along the balanced growth path of the various

components of aggregate demand are virtually identical in the model and the

data. In particular, note that the model reproduces the observed shares of non-

residential and residential investment in GDP. This suggests that we are using the

correct depreciation rates for capital and housing, and appropriate growth rates

for productivity and population.23 In terms of the shares of gross private domes-

tic income accounted for by the three intermediate goods sectors, the calibration

delivers the correct average size of the construction industry, but delivers a man-

ufacturing share that is too small relative to the sample average in the data. The

reason is that intermediate goods shares in Þnal goods production were computed

23This is interesting in light of the fact that a depreciation rate of around 10 percent on an

annual basis is typically assumed for capital, while the annual rate in our calibration is only 5.2

percent. We attribute this difference to the facts that (1) we exclude consumer durables from

our measure of the capital stock, and (2) we explicitly account for both productivity growth and

population growth, both of which imply a relatively high investment rate along the balanced

growth path even with little depreciation.

22

using the 1992 Input - Output tables, and manufacturing�s share of the economy

has declined over the post War period. The average tax rates generated by the

calibration procedure are τk = 41.8 percent, and τn = 28.4 percent. These are

extremely close to standard estimates in the taxation literature (see, for example,

Domeij and Heathcote 2001).

3.2. Business cycle frequency ßuctuations

We simulate our model economy to determine whether it is capable of accounting

for some of the facts regarding the behavior of housing over the business cycle

in the United States. A large set of business cycle moments are presented in

table 8. We Þnd that our benchmark model can account for many of the features

of the data that we document in the introduction. In particular, we reproduce

almost exactly the volatilities of both non-residential investment and residential

investment relative to GDP, and residential investment is positively correlated

with consumption, non-residential investment and GDP.24 House prices are pro-

cyclical and positively correlated with residential investment. Labor supplies

and outputs are correlated across intermediate goods sectors, and output and

employment are most volatile in the construction industry and least volatile in

the services industry. Thus the model can account for both the relative volatility

and the co-movement puzzles.

There are, however, two respects in which the model performs poorly. First,

house prices (at the aggregate level) are slightly more volatile than GDP in the

U.S., while in the model they are only half as volatile. Second, a striking feature

of residential investment noted in the introduction is that it strongly leads the

cycle; the correlation between GDP and residential investment the previous year

is larger than the contemporaneous correlation between the two (see table 8). In

the model the strongest correlation is the contemporaneous one, and thus the

model fails to reproduce this feature of the data. The model can claim a more

limited success, however, in that the correlation between residential investment

24An implication of the Þrst Þnding is that our parameter values and simulation results would

have been virtually identical had we calibrated the adjustment cost parameter to match the

volatility of non-residential investment relative to GDP (recall that we instead chose to match

the observed relative volatility of residential investment).

23

at a lead of a year with GDP is larger than the correlation when residential

investment is lagged by a year.25

Comparison to Greenwood and Hercowitz (1991)

To understand which features of the model allow us to reproduce particular

features of the data, we consider several alternative parameterizations (see table

9). For each alternative parameterization we consider, we adjust a set of six

preference and Þscal parameters (β, µc, µh, g, τk and τn) so that the balanced

growth path ratios to GDP of government spending, transfers, capital and housing

are all unchanged, labor supply is 30 percent of the time endowment, and the after

tax interest rate is 6 percent. One way to think of this exercise is that for each

alternative parameterization we recalibrate the model so that it does a reasonable

job in terms of matching certain Þrst moments of the data, and then simulate to

assess how second moments vary across parameterizations.

Our Þrst alternative parameterization is essentially the benchmark model in

Greenwood and Hercowitz (1991). The model effectively has only one production

technology, capital and housing depreciate at the same rate, and there are no

adjustment costs or tax shocks. Thus this set up is a standard one-sector RBC

model, except that some fraction of the capital stock enters the utility function

rather than the production function. Utility is log separable in consumption,

housing and leisure, and thus the model can be reinterpreted as a reduced form

of an economy with home production (see the introduction). Simulation results

are in column GH of table 10. Since the different intermediate goods are pro-

duced using identical technologies and enter symmetrically in production of the

two Þnal goods, the model has nothing to say regarding the relative volatilities

and cross-sectoral correlations of construction, manufacturing and services. In

other respects the model performs poorly. For example, the model predicts very

little volatility in house prices, primarily because there is never a productivity dif-

ferential between production of consumption versus residential investment. Labor

supply and residential investment are much less volatile in the model than in the

data.25Our paper does about as well in terms of replicating observed lead-lag patterns as Gomme

et. al. 2001, who focus on differential time to build across sectors.

24

The main differences between the GH parameterization and the economy

labelled A in tables 9 and 10 is that the coefficient of relative risk aversion is

increased to 2, (the value in our baseline calibration), and productivity shocks

are stationary (but persistent) rather than unit root. These changes only worsen

the performance of the model. Non-residential investment becomes much more

volatile than residential investment, and the two types of investment co-move neg-

atively contemporaneously. Moreover, non-residential investment strongly leads

GDP, while residential investment strongly lags, exactly the opposite of what is

observed in the data. The explanation for this poor performance is that follow-

ing a good productivity shock, households want to immediately allocate a larger

fraction of output towards increasing the capital stock used in production, where

productivity is high, rather than towards increasing the stock of capital used in

utility production, where productivity is unchanged.

We now proceed to add increasing realism, layer by layer, to the straw-man

models described above. The Þrst thing we add are adjustment costs, where

adjustment costs are at the same level as in our benchmark parameterization. We

Þnd that adjustment costs have large effects on the behavior of the model (see

column B of table 10). In particular, the volatilities of both types of investment

fall signiÞcantly, and the two types of investment are now (perfectly) positively

correlated. The intuition is that with large adjustment costs it is optimal to

increase the business capital stock more slowly following a good shock, and this

means more new capital is available for adding to the housing stock. Introducing

adjustment costs also increases house price volatility, since the price of housing

now depends on the ratio of residential investment to the housing stock. There

are still many respects, however, in which the gap between the model and the

data remains large. One is the relative volatility puzzle; business investment is

more volatile than residential investment, contrary to the pattern in the data.

We next introduce sector speciÞc productivity shocks (see column C). The

shock process is parameterized following a procedure analogous to that described

in the calibration section, except that tax rates are not included in the system.

Since productivity shocks are estimated to be more volatile in construction and

manufacturing than services, this change has the effect of generating relatively

more volatile output (and employment) in these sectors. However, since all Þnal

25

goods are still produced using the same technology, this change has little effect

on the business cycle dynamics of any macro aggregates.

The next feature we add is sector speciÞc capital shares for intermediate goods

Þrms (column D). The fact that construction is relatively labor intensive means

that output of the construction sector becomes more volatile than in the previous

case, while the fact that manufacturing is relatively capital intensive reduces the

volatility of manufacturing output. The intuition is simply that following a good

productivity shock, it is easier to expand output rapidly the more important is

labor in production, since holding capital constant, the marginal product of labor

declines more slowly.

Next, in column E, we introduce difference depreciation rates for capital and

housing. The largest effect of this change is to increase the volatility of residential

investment, so that non-residential investment and residential investment are now

equally volatile. The reason reducing the depreciation rate for housing increases

the volatility of residential investment is that slower depreciation increases op-

portunities to concentrate residential investment in periods of high productivity;

conversely during a prolonged period of low productivity, it is possible to build

few or new no homes without bringing about a large fall in the stock.

Column F introduces different Þnal goods production technologies for the con-

sumption / business investment versus the residential investment sectors. This

is essentially the benchmark model, except that tax rates are still assumed con-

stant. This change has large effects. Because intermediate input intensities differ

across the two different Þnal goods, sector speciÞc productivity shocks change the

effective relative cost of building new houses versus other goods. This accounts

for the decline in the correlation between residential and non-residential invest-

ment. Allowing for two Þnal goods technologies more than doubles the percentage

standard deviation of residential investment, such that residential investment is

now much more volatile than non-residential investment, as in the data. The

explanation for this result hinges on the fact that construction is a much more

important input for residential investment than for the rest of the economy. Re-

call that productivity shocks in the construction industry have a larger variance

than those in the services industry, and that construction is also relatively labor

intensive. These characteristics of the construction technology tend to increase

26

the relative volatility of construction-intensive residential investment.

There is an additional effect in the opposite direction, however, in that output

and employment in the construction sector are now more volatile than in the

equivalent version of the model with a single Þnal goods sector (compare columns

E and F ). With a separate residential investment sector, a fall in the relative price

of construction inputs associated with an increase in construction productivity

translates into a fall in the price of residential investment (see eq. 2.22). Since

demand for residential investment is very price sensitive this in turn generates a

large boom in residential investment and in the demand for construction inputs.

Thus, part of the reason the construction industry is so volatile is that a large

fraction of the demand for its output is for residential investment.

Lastly consider the effects of introducing stochastic tax rates by comparing

column F with column G (our benchmark economy). The main effect of this

change is to increase the volatility of labor supply, which arises because tempo-

rary changes in the labor income tax rate change the relative returns to working

at different dates. The percentage standard deviation of hours in the services

sector (relative to GDP) is now 0.47, compared to 0.23 in the same economy

without tax shocks. More volatile labor supply translates into more volatile out-

put; the percentage standard deviation of GDP rises from 1.63 to 1.93. Thus the

model now comes to reproducing the observed volatility of output even though

all productivity shocks are sector-speciÞc. The volatilities of all other variables

also rise relative to the economies without tax shocks, but not proportionately;

the volatility of consumption rises slightly relative to GDP, while the volatilities

of both types of investment relative to GDP fall. The correlation between house

prices and residential investment increases from 0.18 to 0.36. All these changes

improve the overall success of the model in terms of replicating observed business

cycle dynamics.

Our initial expectation was that introducing stochastic capital income taxes

would increase the volatility of investment by increasing time variation in the

after-tax return to capital, and in the relative expected after-tax returns to saving

in the form of taxed capital versus untaxed labor. One reason investment volatility

does not rise much is that it is expectations over future capital tax rates that affect

27

investment decisions, and capital tax shocks are not very persistent.26

Co-movement and leading residential investment

What accounts for the ability of the benchmark model to reproduce the empir-

ical positive correlation between house prices and residential investment, and the

positive correlations between consumption and either type of investment? One

important component, as discussed above, is the presence of adjustment costs.

Without adjustment costs, increased relative productivity in construction would

tend to reduce house prices but increase production of new housing. At the same

time output of the relatively expensive consumption / investment good would

tend to fall. When adjustment costs are introduced, additional residential invest-

ment drives up house prices, which restores a positive residential investment /

house price correlation. Furthermore, a smaller increase in residential investment

leaves more resources for producing the consumption / investment good, helping

restore a positive correlation between different components of Þnal demand.

However, adjustment costs are small under the baseline calibration. Moreover,

the model continues to generate positive correlations between residential invest-

ment on the one hand and GDP, non-residential investment and house prices

on the other even when the elasticity of new house (capital) prices with respect

to the investment rate is reduced (for both types of investment) from the base-

line value of 0.137 to a value of 0.065.27 One reason for this is that while the

under-lying productivity shocks are only weakly positively correlated across inter-

mediate goods sectors, the correlation is effectively magniÞed at the Þnal goods

level, since both Þnal goods sectors use all three intermediate goods as inputs

(albeit in different proportions).28

26We also experimented with setting all shock innovations to either the capital tax rate or the

labor tax rate to zero. With no capital tax rate shocks, the main differences relative to economy

F are that GDP is slightly more volatile (1.80 versus 1.63) and the percentage standard deviation

of labor supply (relative to GDP) increases from 0.30 to 0.53. With no labor tax rate shocks,

the volatilities of GDP and labor supply are 1.79 and 0.42, and, relative to GDP, residential

investment is less volatile than in economy F (5.12 versus 5.43).27When the elasticity is 0.065, residential investment is (counter-factually) eight times as

volatile as GDP.28There is a sense in which the low depreciation rate for housing and the low capital share

in construction also contribute to resolving the co-movement puzzle. In particular, suppose

28

Part of the reason the residential investment weakly leads GDP in the model

is that output of the residential investment sector may be increased relatively

efficiently without waiting for additional capital to become available. This is

because the residential investment technology is construction and therefore ulti-

mately labor intensive. When all Þnal goods are produced according to the same

technology (and are therefore equally labor intensive) it is non-residential invest-

ment rather than residential investment that weakly leads the cycle (see column

E in table 10).

3.3. Accounting for U.S. history

The aim of this section of the paper is to compare the observed timepaths for

a set of macro variables over the post-War period to those predicted by our

model, given the estimated series of productivity and tax shocks. In particular,

we assume that the U.S. economy was on its balanced growth path until the

start of 1949, and that from 1949 until 1997 the shocks that generated deviations

from the balanced growth path were equal to the residuals generated from the

autoregressive estimation procedure described in the calibration section. This is a

more ambitious exercise than the simulation exercise conducted above in that we

are now assessing the performance of the model at all frequencies. In particular,

we shall address the extent to which the model can account for observed long run

trends, as well as for business cycle frequency ßuctuations. Moreover, we look at

sectoral output and house prices, in addition to the standard macro aggregates.

The series in Þgures 1 and 2 are all scaled so that they take the value one in 1949.

Since constant quality house price data is only available from 1963 we normalize

that we deviate slightly from the baseline calibration by setting δh = δk = 0.052, and assume

θb = θm = θs = 0.25. Suppose, in addition, that we reset the adjustment cost parameter so that,

as before, the model reproduces the observed relative volatility of residential investment (the

implied elasticity turns out to be 0.057). The values for corr(non−RESI, RESI) and corr(Ph,RESI) in this economy are counter-factually negative (−0.01 and −0.24 respectively), while thepercentage standard deviation of non-residential investment is larger than in the data (2.75 times

that of output). This suggests a connection between the co-movement and relative volatility

puzzles. In particular, the parameter values that are important for reproducing observed relative

volatiliy lead us to adopt, within the calibration procedure, investment adjustment costs large

enough to resolve the co-movement puzzle.

29

house prices to be one in 1963.

Across the sample period U.S. private consumption and GDP both increased

by approximately a factor of Þve. At the same time, non-residential investment

increased by a factor of seven whereas residential investment grew only half as

much. The model does an excellent job of matching trend growth in hours worked,

and a very good of capturing long run growth in consumption and output. Since

the focus of the paper is primarily on residential investment, it is reassuring that

the model correctly predicts residential investment to be the slowest growing

component of Þnal demand. The reason is simply that since residential investment

is construction intensive, and negative trend growth in construction productivity

means that the relative prices of residential investment and of housing tend to be

rising over time. This reduces growth in the demand for new housing; given Cobb

Douglas preferences expenditure shares on consumption, leisure and housing are

constant along the balanced growth path.

The biggest failures of the model at low frequencies are in replicating the

growth of non-residential investment and of manufacturing output. Both discrep-

ancies are readily accounted for. The model overpredicts manufacturing growth,

since manufacturing�s share of trend nominal output is constant in the model as

a consequence of assuming Cobb Douglas technologies and preferences. However,

manufacturing�s share of nominal output has fallen dramatically over the sample

period, as the service sector has grown. The fact that the model underpredicts

growth in non-residential investment is likely in part a consequence of assuming

a constant depreciation for capital; when we estimate depreciation rates using

NIPA nominal depreciation Þgures, we Þnd the rate of depreciation to be rising

over time. A second reason the model underpredicts non-residential investment

growth is that we assume a common production technology for consumption and

business investment, whereas in reality non-residential investment is more manu-

facturing intensive than consumption (see table 6). Thus the model generates too

little trend decline in the relative price of business capital (generated by manu-

facturing productivity growth) and too little real growth in business investment.

Consider next the ability of the model to account for U.S. macroeconomic

history at business cycle frequencies. To better assess the model�s cyclical per-

formance, Þgure 3 describes percentage deviations from a Hodrick Prescott trend

30

for various macro aggregates. Note Þrst that the model closely reproduces the

histories of deviations from trend in GDP and consumption. Slightly poorer per-

formance at the end points of the sample might reßect poor endpoint properties

of the Hodrick Prescott Þlter. The poor Þt in the Þrst few years of the sample

might alternatively be due to the U.S. economy being off its balanced growth

path prior 1949, contrary to the assumption made here. The model also does a

good job in accounting for historical output ßuctuations at the sectoral level (see

Þgure 2).

Deviations from trend in the data are much larger for non-residential invest-

ment than for GDP or consumption, and larger again for residential investment.

Comparing the predictions of the model with the data, the Þt for both types of

investment is generally good, suggesting that productivity and tax shocks can

largely account for observed investment dynamics. There a few caveats to this

conclusion, however. First, swings in non-residential investment are generally

somewhat smaller in the model than in the data. Second, non-residential invest-

ment in the data appears to slightly lag non-residential investment in the model

(which Þts with the fact that non-residential investment lags the cycle empiri-

cally) whereas the model delivers no strong lead-lag patterns. Third, residential

investment in the data appears to slightly lead residential investment in the model

during the early part of the sample (which again is consistent with residential in-

vestment leading the cycle empirically). Comparing the last two major recessions,

the model does a very good in accounting for the depth of the recession in the

early 1980�s, including a dramatic fall in residential investment which was 38 per-

cent below trend in 1982. However, the model underpredicts the depth of the

recession in the early 1990�s, a failing which is also clear from Þgure 1.29

One important respect in which the model performs poorly is in accounting

for house price dynamics. The model does at least correctly predict an upward

trend in the relative price of housing. This upward trend in the data looks more

dramatic if house prices are measured relative to the GDP deßator relative to the

CPI.30 The reason for the upward trend in the model is simply that productivity

29Hansen and Prescott 1993 also investigate the 1990-1991 recession in a multi-sector model.30We compare the price of housing relative to the consumption / investment good in the

model. In the data the price of output (GDP deßator) incorporates changes in the price of

31

in the construction industry has been declining over time relative to productivity

in other industries. In terms of cyclical volatility in house prices, the model does

not account for a large fraction of observed price dynamics, and in particular it

fails to capture the large boom in house prices that peaked in 1979. We return

to this issue in the conclusion.

3.4. Relation to empirical literature on residential investment

Our model is not the Þrst to address the determination of residential investment

and house prices. A simple version of what we label the traditional model may

be visualized as follows (see Kearl 1979, Topel and Rosen 1998, or Poterba 1984

and 1991). There is an upward sloping supply curve for residential investment,

since higher prices encourage developers to build more houses. The demand

curve for new houses is inÞnitely elastic, since houses are viewed as Þnancial

assets that must pay the market rate of return. Demand shocks affect future

expected rents (dividends on the housing asset) and shift the demand curve up

and down. Thus equilibrium price / investment pairs are traced out along the

residential investment supply curve. Topel and Rosen (p.737) conclude that �the

large price elasticity of supply of new houses estimated here must be an important

consideration for understanding the great variability in housing investment.�

Our model shares many features of the traditional view; for example, housing

may be viewed as an asset which in equilibrium offers the same expected return

as capital, provided implicit rents from owner-occupation are included in the

return calculation. However, there are some important differences between the

two frameworks, reßecting the fact that we are explicit about all the production

technologies in the economy and are therefore able to derive exact equilibrium ex-

pressions for all prices. One key difference is that the dynamics of both residential

investment and house prices in our model are primarily driven by supply-side pro-

ductivity shocks.31 In contrast, the typical assumption in the traditional model is

residential investment. At the same time the CPI incorporates a shelter component. When we

constructed a model equivalent of the CPI, we found slightly slower trend growth in the relative

price of housing.31Contrary to the traditional view, the demand for new housing is not perfectly elastic, since

additional housing implies less new capital, which drives up future interest rates, and increases

32

that the current price of housing is largely independent of construction costs, and

changes in house prices mostly reßect changes in the demand for housing relative

to other goods.

When residential investment is regressed on house prices and the real interest

rate, the coefficient on the house price term typically turns out to be positive

and strongly signiÞcant. This appears to conÞrm the traditional demand-shock

driven view of residential investment. At the same time, a positive house price

co-efficient is prima facie inconsistent with a productivity shock driven theory of

residential investment, according to which one might expect an increase in resi-

dential investment to be associated with higher productivity in the construction

sector and thus lower prices for construction output and housing. We shall argue,

however, that care should be taken in interpreting these regression results.

In the appendix we show that the Þrst order conditions of our model imply the

following equilibrium relationship between residential investment and new house

prices:

(1− ζh)log (yh) = log(r)− log(pnh)− ζhlog(h) + log³δ∗−ζhh λ

´. (3.1)

where λ is a term involving sector capital stocks, and δ∗h is the balanced growthpath value for ihh . This relationship indicates a negative relationship between res-

idential investment and house prices, holding other variables constant. However,

in addition to the rental rate on capital, the equation implied by the model also

includes terms involving sectoral capital stocks and the existing stock of houses.32

This relationship cannot be interpreted as a supply curve, since the variables in

the equation are all determined endogenously in equilibrium and depend on both

preference and technology parameters.

Equation 3.1 is not the residential investment equation that has typically

been estimated in the literature. In table 11 we consider various alternative

speciÞcations, in order to assess the potential importance of mis-speciÞcation. We

estimate each statistical model using simulated output from the model, where the

model simulation uses the identiÞed historical shocks (see the previous section).

the opportunity cost of saving in the form of housing.32 If there are no adjustment costs in the model (ζh = 0) then the housing stock term drops

out, but the term involving sectoral capital stocks remains.

33

In the Þrst column we estimate the correct model, using ordinary least squares,

as a check on our algebra. Given equation 3.1, the coefficients on the price of new

housing and the marginal product of capital should be minus one and one respec-

tively, and the regression should Þt perfectly. The reason the estimated coeffi-

cients do not exactly match the predicted coefficients is that given the numerical

solution method, simulation allocations are not exactly equilibrium allocations

away from the balanced growth path.

In alternatives (1) and (2) we re-estimate the residential investment equation

omitting the housing and sectoral capital stock terms. The estimated co-efficient

on the house price term now becomes positive (and large and signiÞcant if no

time trend is included). Thus omitting variables could lead one to interpret this

artiÞcial economy as being driven by shocks to the demand for housing, whereas

in fact (by construction) it is driven by supply-side productivity shocks.

The problem of omitted variables is not the only way in which previous es-

timation equations have been mis-speciÞed relative to the equilibrium relation

that obtains in our model. Two other problems are that in equation 3.1 it is the

pre-tax rental rate for capital that enters the equation, rather than the interest

rate on a risk-free bond (typically lagged) that has been used in empirical work.

In addition the house price term is the price of a new house to be delivered at the

start of the next period, rather than the current start of period market price of

housing. In columns (3) and (4) we explore the importance of mis-speciÞcation

in these dimensions. We compute risk free interest rates in the model using the

standard pricing formula for risk and tax free one period bonds. We Þnd that the

effect of using the risk free rate rather than the marginal product capital is to

further increase the estimated coefficient on the house price term, and to weaken

the coefficient on the interest rate term. However, exactly which measure of house

prices we use turns out not to be very important.

To summarize, previous empirical work estimating a residential investment

supply curve is only loosely grounded in theory, and the equations estimated

are mis-speciÞed relative to the structural equilibrium relationship implied by

the model developed here. Bias due to mis-speciÞcation is potentially large,

and thus previous Þndings do not necessarily support the view that ßuctuations

in residential investment are primarily driven by demand-side shocks to house

34

prices.

4. Conclusion

This paper suggests that many of the aggregate stylized facts relating to housing,

such as the high volatility of residential investment, are natural consequences of

the way houses and other goods are produced. Some key features of the calibrated

model economy�s production technologies are the importance of construction as

an input to residential investment, the importance of labor as an input in the

construction industry, and the fact that housing depreciates much more slowly

than capital.

An important part of the calibration procedure involves using Input Output

data to Þx the relative importance of different intermediate inputs in various

components of Þnal demand. Our focus was on residential investment, and we

therefore chose to emphasize the construction industry at the intermediate in-

dustry level, and to abstract from differences in the production technologies for

consumption versus non-residential investment at the Þnal goods level. However,

using the same calibration methodology it would be relatively straightforward to

consider a Þner disaggregation of the components of Þnal demand, or to increase

the number of intermediate sectors. For example, one could use the input mix

estimates in table 6 to introduce an explicit non-residential investment sector.

The paper leaves several open issues for exploration. First, what can account

for the strong lead of residential investment over the cycle? Both this paper and

Gomme, Kydland and Rupert (2001) make some progress on this dimension, but

neither succeed in reproducing the fact that the correlation between GDP at date

t and residential investment (or residential investment plus purchases of consumer

durables) at t− 1 is larger than the contemporaneous correlation.Second, the model does poorly in accounting for long swings in house prices

such as the large run up in house prices that occurred in the 1970�s. House prices

in the model are pinned down in the medium to long term by relative sectoral

productivities.33 This means that there is little prospect of being able to account

33This is a consequence of assuming that all markets are competitive and that new and (appro-

priately discounted) old houses are perfect substitutes. House prices would only reßect relative

35

for U.S. house price history by extending the model to capture demand side

factors that have received attention in the literature, such as the demographics

of the baby boom and bust, or changes through time in the effective size of the

tax advantage conferred by mortgage interest deductability. Introducing land as

a Þxed factor, however, could allow a larger role for demand side shocks.34 For

example, a surge in immigration would tend to drive up land and thus house

prices in both the short and long run, though it is not clear the effect would be

quantitatively important.

References

[1] Benhabib, J., R. Rogerson, and R. Wright, 1991, �Homework in Macro-

economics: Household Production and Aggregate Fluctuations�, Journal of

Political Economy 99, 1166-1187.

[2] Baxter, M., 1996, �Are Consumer Durables Important for Business Cycles?�,

Review of Economics and Statistics 78 (1), 147-155.

[3] Boldrin, M., L. Christiano, and J. Fisher, 2001, �Habit Persistence, Asset

Returns and the Business Cycle�, American Economic Review, forthcoming.

[4] Chang, Y., 2000, �Comovement, Excess Volatility and Home Production�,

Journal of Monetary Economics 46, 385-396.

[5] Cooley, T. and E. Prescott, 1995, �Economic Growth and Business Cycles�,

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[6] Diaz-Gimenez, J., E. Prescott, T. Fitzgerald, and F. Alvarez, �Banking in

Computable General Equilibrium Economies�, Research Department Staff

Report 153, Federal Reserve Bank of Minneapolis.

productivities even in the short run if there were no adjustment costs and all sectors were equally

capital intensive (see eqs. 2.19 and 2.22).34 See Poterba 1991 for some evidence on the importance of land prices and for a review of

alternative theories of house prices.

36

[7] Domeij, D. and J. Heathcote, 2001, �Factor Taxation

with Heterogeneous Agents�, working paper, available at

http://www.econ.duke.edu/~heathcote/

[8] Edge, R., 2000, �The Effect of Monetary Policy on Residential and Structures

Investment under Differential Project Planning and Completion Times�, In-

ternational Finance Discussion Paper Number 671, Board of Governors of

the Federal Reserve System.

[9] Fernandez-Villaverde, J. and D. Krueger, 2001, �Consumption and Saving

over the Life-Cycle: How Important are Consumer Durables?�, working pa-

per, Stanford University.

[10] Fisher, J., 1997, �Relative Prices, Complementarities and Comovement

among Components of Aggregate Expenditures�, Journal of Monetary Eco-

nomics 39 (3), 449-474.

[11] Gervais, M., 2001, �Housing Taxation and Capital Accumulation�, Journal

of Monetary Economics, forthcoming.

[12] Gomme, P., F. Kydland, and P. Rupert, 2001, �Home Production Meets

Time to Build�, Journal of Political Economy 109 (5), 1115-1131.

[13] Greenwood, J., R. Rogerson, and R. Wright, 1995, �Household Production in

Real Business Cycle Theory�, in T. Cooley ed., Frontiers of Business Cycle

Research, Princeton University Press, Princeton, 157-174.

[14] Greenwood, J. and Z. Hercowitz, 1991, �The Allocation of Capital and Time

over the Business Cycle�, Journal of Political Economy 99, 1188-1214.

[15] Hansen, G. and E. Prescott, 1993, �Did Technology Shocks Cause the 1990-

1991 Recession?�, American Economic Review 83 (2) 280-286.

[16] Hornstein, A. and J. Praschnik, 1997, �Intermediate Inputs and Sectoral

Comovement in the Business Cycle�, Journal of Monetary Economics 40,

573-595.

37

[17] Joines, D., 1981, �Estimates of Effective Marginal Tax Rates on Factor In-

comes, Journal of Business 54 (2), 191-226.

[18] Kearl, J., 1979, �Inßation, Mortgage, and Housing�, Journal of Political

Economy 87 (5), 1115-1138.

[19] Klein, P., 2000, �Using the Generalized Schur Form to Solve a Multi-variate

Linear Rational Expectations Model�, Journal of Economic Dynamics and

Control 24 (10), 1405-1423.

[20] Krusell, P. and A. Smith, 1998, �Income and Wealth Heterogeneity in the

Macroeconomy�, Journal of Political Economy 106, 867-896.

[21] McGrattan, E., R. Rogerson, and R. Wright, 1997, �An Equilibrium Model

of the Business Cycle with Household Production and Fiscal Policy�, Inter-

national Economic Review 38 (2), 267-290.

[22] Mendoza, E., A. Razin and L. Tesar, 1994, �Effective Tax Rates in Macro-

economics: Cross-Country Estimates of Tax Rates on Factor Incomes and

Consumption�, Journal of Monetary Economics 34 (3), 297-323.

[23] Ortalo-Magne, F. and S. Rady, 2001, �Housing Market Dynamics: On the

Contribution of Income Shocks and Credit Constraints�, CEPR Working

Paper 3015.

[24] Platania, J. and D. Schlagenhauf, 2000, �Housing and Asset Holding in a

Dynamic General Equilibrium Model�, working paper, Florida State Uni-

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[25] Poterba, J., 1984, �Tax Subsidies to Owner-occupied Housing: An Asset

Market Approach�, Quarterly Journal of Economics 99, 729-752.

[26] Poterba, J., 1991, �House Price Dynamics: The Role of Tax Policy and

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pleteness�, Journal of Monetary Economics 34, 462-496.

38

[28] Storesletten, K., 1993, �Residential Investment over the Business Cycle�,

working paper, Institute for International Economic Studies, Stockholm Uni-

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[29] Topel, R. and S. Rosen, 1988, �Housing Investment in the United States�,

Journal of Political Economy 96 (4), 718-740.

5. Appendix: residential investment and house prices

The total value of intermediate goods output must equal the total value of Þnal

goods output, since Þnal goods Þrms make zero proÞts:

pbxb + pmxm + psxs = yc + phyh. (5.1)

From the equilibrium expression for the price of manufactures (eq. 2.16)

yc =pmmc

Mc.

From the equilibrium expressions for the interest rate (eq. 2.13)

xi =rKipiθi

.

Thus eq. 5.1 may be rewritten as

r

µKbθb+Kmθm

+Ksθs

¶=pmmc

Mc+ phyh. (5.2)

Using the market clearing condition for manufactures (eq. 2.11) and reusing

eq. 2.13 and eq. 2.16 we can derive a convenient alternative expression for mc

mc = xm −mh (5.3)

=rKmpmθm

− Mhphyhpm

.

Substituting eq. 5.3 into eq. 5.2

r

µKbθb+Kmθm

+Ksθs

¶=

³rKmθm

−Mhphyh´

Mc+ phyh. (5.4)

39

Recall that the price of new housing is related to the price of residential

investment as follows

pnh =ph

φ0h(yh/h).

Thus eq. 5.4 may be rewritten as

yh =rλ

pnhφ0h(yh/h)

,

where

λ =Km

(Mh −Mc) θm− Mc

(Mh −Mc)

µKbθb+Kmθm

+Ksθs

¶.

In logarithms, the equilibrium relation between residential investment, house

prices and the real interest rate is

log(yh) = log(r) + log(λ)− log(pnh)− log(φ0h(yh/h)).To make further progress we need to specify an explicit functional form for the

adjustment cost function that satisÞes the properties deÞned in eqs. 2.7 through

2.9. One such function is

φh(x) =

µδ∗h −

δ∗hγ

¶+

Ã1

γδ∗(γ−1)h

!xγ.

where δ∗h is the steady state value forihh . Note that for this functional form, the

elasticity of the price of new capital or new housing with respect to the appropriate

investment rate is given by

ζk =−φ00k

³ikk

´ikk

φ0k³ikk

´ = ζh =−φ00h

³ihh

´ihh

φ0h³ihh

´ = 1− γ

The derivative of the adjustment cost function is

φ0(yh/h) =yγ−1h

(hδ∗h)γ−1

We use this expression to derive the following alternative equation relating resi-

dential investment to pnh, r, λ, and the housing stock h :

γlog (yh) = log(r)− log(pnh)− (1− γ)log(h) + log³δ∗(γ−1)h λ

´.

40

Table 1: Balanced Growth Path Growth Rates (gross growth rates of per-capita variables)

nb, nm, ns, n, r 1

kb, km, ks, k, c, ik, g, yc, w ( ) ( ) ( )[ ] scmcbcscmcbc SMBSzs

Mzm

Bzbk gggg θθθθθθ −−−−−−= 1

1111

bc, bh, xb bbzbkb ggg θθ −= 1

mc, mh, xm mmzmkm ggg θθ −= 1

sc, sh, xs sszsks ggg θθ −= 1

h, ih, yh hhh Ss

Mm

Bbh gggg =

phyh, pbxb, pmxm, psxs gk

Table 2: Tax Rates, Depreciation Rates, Adjustment Costs, Preference Parameters

Davis Heathcote Grenwood Hercowitz (GH)

Mean tax rate on capital income: kτ 0.4177 0.50

Mean tax rate on labor income: nτ 0.2843 0.25

Govt. cons. to GDP 0.181*1 0.0

Transfers to GDP 0.082*

Depreciation rate for capital: δk 0.052* 0.078

Depreciation rate for housing: δh 0.016* 0.078

Elasticity pnk of wrt ik/k: ξk 0.137 0.0

Elasticity of pnh wrt ih/h: ξh 0.137 0.0

Population growth rate: η 1.8* 0.0

Discount factor: β 0.9672 0.96

Risk aversion: σ 2.00* 1.00

Consumption�s share in utility: µc 0.3162 0.2600

Housing�s share in utility: µh 0.0381 0.0962

Leisure�s share in utility: 1-µc-µh 0.6457 0.6438

1 Starred parameter vales are chosen independently of the model.

Table 3: Production Technologies

Con. Man. Ser. GH

Input shares cons/inv production: Bc, M c, Sc 0.0307 0.2696 0.6997

Input shares housing production: Bh, M h, Sh 0.4697 0.2382 0.2921

Capital�s share by sector: θb, θm, θs 0.13 0.31 0.24 0.30

Trend productivity growth (%): gzb, gzm, gzs -0.25 2.77 1.68 1.00

Autocorrelation co-efficient: ρ 1.0

Std. dev. log productivity innovations: σ(ε)

0.022

Table 4: Estimation of Exogenous Shock Process

System estimated: 11 �� ++ ++= ttt zBAz ε

Standard deviation of innovations

σ(εb) 0.0409

σ(εm) 0.0351

σ(εs) 0.0176

σ(εk) 0.0236

where

������

�

�

������

�

�

=

nt

kt

st

mt

bt

t zzz

z

ττ

logloglog

� ,

������

�

�

������

�

�

=

nt

kt

st

mt

bt

t

εεεεε

ε and ),0(~ VNtε .2

σ(εn) 0.0062

Autoregressive coefficients in matrix B (standard errors in parentheses)

log zbt+1 log zm

t+1 log zst+1 τk

t+1 τnt+1

log zbt 0.636

(0.113) -0.036 (0.097)

0.047 (0.049)

0.046 (0.065)

-0.015 (0.017)

log zmt -0.121

(0.184) 0.714 (0.157)

0.081 (0.079)

0.036 (0.106)

-0.067 (0.028)

log zst 0.012

(0.153) -0.007 (0.131)

0.899 (0.066)

0.012 (0.088)

0.062 (0.023)

τkt 0.212

(0.279) -0.078 (0.239)

-0.257 (0.120)

0.525 (0.161)

0.104 (0.042)

τnt -1.094

(0.877) -0.900 (0.752)

0.264 (0.376)

-0.242 (0.506)

0.543 (0.132)

R2 0.58 0.69 0.91 0.36 0.82 Correlations of innovations

εb εm εs εk εn

εb 1.0 0.059 0.345 -0.055 -0.302

εm 1.0 0.579 0.048 -0.256

εs 1.0 0.040 -0.061

εk 1.0 0.513

2 All variables are linearly detrended prior to estimating this system.

Table 5: Decomposition of Final Demand into Final Sales From Industries (%)

(based on 1992 IO-Use Table)

PCE BFI + RESI RESI3 BFI G4

Construction 0.0 43.9 100.0 22.6 33.6

Manufacturing 23.3 41.3 0.0 56.9 44.2

Services 76.7 14.8 0.0 20.5 22.2

Table 6: Decomposition of Final Demand into Value Added by Industry (%)

PCE BFI + RESI RESI BFI PCE + BFI + GOVI5

Construction 1.4 21.3 47.0 11.6 3.1

Manufacturing 23.0 40.6 23.8 46.9 27.0

Services 75.7 38.1 29.2 41.5 70.0

Table 7: Properties of Steady State: Ratios to GDP %

Data (1949-1998)

Model

Capital stock (K) 152 152

Housing stock (Ph x H) 101 101

Private consumption (PCE) 63.6 63.8

Government consumption (G) 18.1 18.1

Non-residential inv (non-RESI) 13.4 13.6

Residential inv (Pr x RESI) 4.7 4.6

Construction (Pb x Yb) 5.3 5.2

Manufacturing (Pm x Ym) 33.4 26.8

Services (Ps x Ys) 61.3 68.0

Real after tax interest rate (%) 6.0

3 We attribute all $225.5bn of residential investment in 1992 to sales from the construction industry, since the first I/O use table does not have a �residential investment� column. We defend this choice in the data appendix. 4 G is government expenditure, which includes government consumption and government investment expenditures. 5GOVI is government investment. We assume that the value added composition of government investment by intermediate industry is the same as business fixed investment.

Table 8: Business Cycle Properties6

Data (1949-1998) Model

% Standard Deviations (rel. to GDP)

GDP 2.31 1.93

PCE 0.77 0.62

Labor (N) 1.03 0.54

Non-RESI 2.23 2.19

RESI 5.11 5.11

Construction output (Yb) 2.74 3.50

Manufacturing output (Ym) 1.84 1.44

Services output (Ys) 0.85 0.97

Construction hours (Nb) 2.33 1.92

Manufacturing hours (Nm) 1.52 0.52

Services hours (Ns) 0.64 0.47

House prices (Ph) (Data: 1963-1998) 1.29 0.54

Correlations

PCE, GDP 0.80 0.98

Ph, GDP (Data: 1963-1998) 0.53 0.88

PCE, non-RESI 0.63 0.96

PCE, RESI 0.67 0.56

non-RESI, RESI 0.24 0.45

Nb, Nm 0.77 0.75

Nb, Ns 0.89 0.64

Nm, Ns 0.78 0.99

Ph, RESI (Data: 1963-1998) 0.39 0.36

Lead lag patterns

i = 1 i = 0 i = -1 i = 1 i = 0 i = -1

corr(Non-RESI t-i, GDPt) 0.27 0.75 0.55 0.50 0.96 0.50

corr(RESI t-i, GDPt) 0.55 0.47 -0.17 0.38 0.68 0.25

6 Statistics are averages over 500 simulations, each of length 50 periods, the length of our data sample. Prior to computing statistics all variables are (i) transformed from the stationary representation used in the solution procedure back into non-stationary representation incorporating trend growth, (ii) logged, and (iii) Hodrick-Prescott filtered.

Table 9: Alternative Parameterizations

Model Selected parameter values

GH Greenwood and Hercowitz see tables 2 and 3

A One sector model, housing in utility (re-parameterized GH)

σ = 2, ρ = 0.85, σ(ε) = 0.022 δk = δh = 0.052

θb = θm = θs = 0.25 Bh, = B c, Mh, = M c, Sh, = S c

B A + adjustment costs ξk = ξh = 0.137

C B + sector-specific shocks see table 4

D C + sector-specific capital shares θb = 0.13, θm = 31, θs = 0.24

E D + different depreciation rates δh = 0.016

F E + two final goods technologies Bh, = 0.47, Mh, = 0.24, Sh, = 0.29

G (Benchmark) E + stochastic taxes Table 10: Alternative Parameterizations: Business Cycle Properties

Data GH A B C D E F G

GDP (% std dev) 2.31 1.39 1.93 1.72 1.60 1.56 1.60 1.63 1.93

Std. dev. relative to GDP

PCE 0.77 0.60 0.39 0.53 0.56 0.56 0.59 0.60 0.62

N 1.03 0.36 0.47 0.34 0.32 0.31 0.29 0.30 0.54

Non-RESI 2.23 2.73 4.01 2.41 2.36 2.35 2.39 2.31 2.19

RESI 5.11 2.08 2.92 1.97 1.88 1.87 2.40 5.43 5.11

Yb 2.74 1.25 1.14 1.13 1.90 2.27 2.20 3.87 3.50

Ym 1.84 1.25 1.14 1.13 1.80 1.68 1.62 1.59 1.44

Ys 0.85 1.25 1.14 1.13 1.04 1.08 1.04 0.97 0.97

Ph 1.29 0.11 0.07 0.35 0.34 0.34 0.38 0.55 0.54

Correlations

Non-RESI, RESI 0.24 0.88 -0.10 1.00 1.00 1.00 0.99 0.41 0.45

Ph, RESI 0.39 0.94 0.80 0.98 0.98 0.98 0.99 0.18 0.36

Lead-lag pattern: corr(xt-1, GDPt) – corr(x t+1, GDP t) x = RESI. 0.73 -0.10 -0.94 0.07 0.05 0.05 -0.00 0.11 0.13

x = Non-RESI -0.28 0.37 0.46 0.10 0.09 0.09 0.09 0.05 -0.01

Table 11: OLS Regressions on Simulated Data, Using Historical Productivity & Tax Shocks Dependent variable is γln(RESIt)7 (standard errors in parentheses)

Model (1) (2) (3) (4)

Constant - 0.78 (0.38)

0.41 (0.45)

-1.11 (0.29)

-1.21 (0.29)

Time trend - - 0.010 (0.00)

0.004 (0.00)

0.003 (0.00)

ln(Pnt) -0.96

(0.06) 3.31

(0.19) 0.46

(0.75) 2.07

(0.76) -

ln(MPKt) 0.98 (0.01)

1.01 (0.47)

2.00 (0.48)

- -

ln(λ t) 1.11 (0.02)

- - - -

ln(H t) -0.29 (0.02)

- - - -

ln(rfrt-1) - - - 0.25 (0.23)

0.24 (0.22)

ln(Pt) - - - - 2.29 (0.73)

R2 0.999 0.872 0.904 0.871 0.877

7 In the various regressions γ is the adjustment cost parameter (γ = 1 - ξh = 0.863), RESIt is residential investment at date t, Pn

t is the price at date t of a new unit of housing delivered at t+1, MPKt is the marginal product of capital at t, λ t

is the term involving sectoral capital stocks (see the appendix for details), Ht is the stock of housing at t, rfrt is the interest rate on a risk-free tax-free bond that is bought at date t and delivers one unit of consumption at t+1, and Pt is the price at date t of a unit of housing that may be enjoyed in date t (we think of this as the model counterpart to the empirical price of housing). The equation estimated in the Model column of the table is an equilibrium relationship. The model implied co-efficients on the house price term and the marginal product of capital are plus and minus one respectively.