housing and the business cycle - duke...
TRANSCRIPT
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.
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Political Economy 99, 1166-1187.
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[3] Boldrin, M., L. Christiano, and J. Fisher, 2001, �Habit Persistence, Asset
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alternative theories of house prices.
36
[7] Domeij, D. and J. Heathcote, 2001, �Factor Taxation
with Heterogeneous Agents�, working paper, available at
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Time to Build�, Journal of Political Economy 109 (5), 1115-1131.
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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-
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38
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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.