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A Real Explanation for
Heterogeneous Investment Dynamics
Jonas D.M. Fisher∗
Federal Reserve Bank of Chicago
First Draft: July 23, 2001
This Draft: October 25, 2001
Abstract
Household investment, that is investment in consumer durables and
housing, leads non-residential fixed investment over the U.S. business cycle.
This observation represents a potent challenge to real business cycle (RBC)
theory. First of all the theory has been unable to account for it. In addition,
research suggests the observation is driven by monetary shocks, supporting
the view that these shocks play a leading role in the U.S. business cycle.
This paper shows that RBC theory is consistent with the investment dynamics after all. It does so by generalizing the standard home production
environment to take into account the fact that household capital is useful
in market production.
∗
Thanks to Lisa Barrow, Lawrence Christiano, Martin Eichenbaum, Eric French, Lars
Hansen and seminar participants at the 2001 NBER Summer Institute for useful input. I
would also like to thank Martin Eichenbaum and Randall Wright for encouraging me to write
this paper. The views expressed herein are those of the author and not of the Federal Reserve
Bank of Chicago or the Federal Reserve System.
1. Introduction
Since the work of Burns and Mitchell (1946) it has been recognized that business fixed investment and household investment in residential capital and consumer durables behave differently over the business cycle: household investment
leads business investment over the business cycle. This phenomenon has prevailed throughout the post-WWII period. As illustrated in Figure 1, around most
of the nine post-war recessions, household investment reaches its peak and then
its trough before business investment.
This evidence represents a potent challenge to real business cycle (RBC) theory. First of all the theory has been unable to account for the investment dynamics. The most success these models have achieved is to account for the fact
that the two kinds of investment co-move positively over the cycle. Indeed the
literature is replete with papers which have succeeded with respect to the comovement phenomenon but failed with respect to the dynamics.1 In addition,
both empirical and theoretical research suggests that the observation is driven by
monetary shocks.2 When combined with the failure of RBC theory this research
supports the view that monetary and not real shocks play the leading role in the
U.S. business cycle.
This paper shows that RBC theory is consistent with the investment dynamics
after all. It does so by generalizing the home production framework typically
used to analyze these dynamics. The generalization introduced is motivated by
the idea that the effectiveness of hours worked in the market is positively related
to the quantity and quality of household capital. For example, analogous to the
maintenance required to keep business capital in operating condition, the quality
of market labor hours is improved by activities at home such as rest, relaxation,
and personal care. Another example is the role of automobiles in commuting, a
crucial input into market production. These observations suggest that household
1
I believe a complete list of RBC studies which have examined the behavior of home and business investment in closed economies includes: Baxter (1996), Benhabib, Rogerson and Wright
(1991), Campbell and Ludvigson (1998), Chang (2000), Einarsson and Marquis (1997), Fisher
(1994, 1997), Gomme, Kydland and Rupert (2000), Greenwood and Hercowitz (1991), Greenwood, Rogerson and Wright (1995), Hornstein and Praschnik (1997), McGrattan, Rogerson and
Wright (1997), and Perli (1998).
2
Papers providing empirical support for the view that monetary factors can generate the
heterogeneous investment dynamics include Bernanke and Gertler (1995) and Fisher (1994).
Two recent papers in which the dynamics arise out of general equilbrium models of money are
Edge (2000) and Li and Chang (2001).
1
capital should be modeled as a complement to time in the supply of effective
hours used to produced market goods. Despite this seemingly obvious feature of
household capital, the standard home production model assumes that household
capital has no use in market production.3
The generalized home production framework can account for the investment
dynamics because it reverses a basic asymmetry in the treatment of capital goods
in the standard home production model. In the standard model business capital
is useful in the production of market consumption and business and household
investment goods. Crucially, it is useful for producing new household capital
goods which are an input into home produced consumption goods, while household
capital has only one purpose as an input into home produced consumption goods.
This asymmetry implies a strong incentive to build up business capital before
household capital in response to a positive innovation in market productivity.
If household capital has some use in market production then this asymmetry
is reversed since household capital is useful in producing both market and home
produced goods, while business capital can only be used to produce market goods.
I analyze the generalized home production formulation in the context of alternative assumptions on technologies for producing new investment goods. These
assumptions are motivated by the work of Topel and Rosen (1988) and the empirical studies summarized by Kydland and Prescott (1982) and Christiano and
Todd (1996). In particular the short run elasticity of household investment is
much greater than for business investment, in response to aggregate disturbances.
As well, there is plenty of evidence justifying this difference in elasticities. There
are significant gestation lags in new non-residential fixed investment projects as
well as a significant period of planning which takes time but requires relatively
few resources. These observations justify incorporating into the home production
framework differences in the way investment is undertaken across household and
business capital, with longer planning and gestation lags for the latter. Realistic
gestation and planning lags reinforce the incentives to build up household capital
in advance of business capital in the generalized home production framework.4
3
The standard calibration procedure for a single-capital-good RBC model involves aggregating household and business capital to arrive at a single capital stock measure. Implicit in this
practice is the assumption that household capital is useful in producing market goods and in particular is perfectly substitutable with business capital. The current analysis involves household
and business capital being less than perfect substitutes.
4
It should be noted that the notion of gestation lags being important for understanding
the heterogeneous investment dynamics has a long tradition in macroeconomics and is even
enshrined in macroeconomics textbooks. See for example, Hall and Taylor (1997).
2
I show that the general home production formulation can reconcile RBC models with the observed lead-lag pattern between household and business investment,
but gestation lags and planning differences across types of investment in the context of standard home production models cannot. Moreover, the general home
production framework can account for the fact that household investment is more
volatile than business investment. The standard home production models studied here, even with realistic gestation and planning lags in market investment,
consistently predict the opposite.
In a recent paper Gomme, Kydland and Rupert (2001) study gestation lags in
the context of standard home production models. They conclude that these lags
can move the standard home production framework into substantial conformity
with the data in terms of the observed investment dynamics. One contribution
of this paper is to show that the results underlying this conclusion are fragile
and that once new data is incorporated into the analysis, particularly new capital
stock data, they no longer hold. That is, gestation lags cannot account for the
data unless they are included in a model of home production of the general form
introduced in this paper.
The rest of the paper proceeds as follows. In the next section I describe the
model. After this I describe how I selected parameter values. This is followed by
a discussion of the findings. The final section concludes and suggests avenues for
future research.
2. The Model
The model consists of households, firms and a government acting in a competitive equilibrium. It departs from the standard home production model in two
ways. First, household capital is assumed to be complementary with labor and
business capital in market production. Second, there can be differences across
household capital and business capital in terms of the time required for planning
and completing new investment projects.
2.1. Households
The representative household has preferences over a consumption good purchased
from firms, cmt , a consumption good produced in the home, cht , hours devoted to
home production, nht and hours sold in the labor market, nmt . These preferences
3
are given by
Et
∞
X
j=t
β j−t [ln cmj + ψ ln chj + η ln (1 − nmj − nhj )] ,
(2.1)
where Et is the mathematical expectations operator conditional on time t information and η, ψ > 0. This particular specification of preferences is chosen because
it can reconcile the trend in the price of household investment goods in terms
of non-durable and service consumption and the stable nominal share of expenditures on household investment goods observed in U.S. data. Assuming less or
more substitution between home and market consumption would not be consistent with this evidence (see Fisher 1994, 1997 and Kydland 1995). Aside from its
use in pinning down preferences, the relative price trend plays no direct role in
the analysis. This helps maintain comparability with previous studies.
The key difference with standard home production models is in the nature of
labor hours supplied to the market. I assume that household’s are compensated
e
by firms for effective hours, nmt , where effective hours are derived from inputs of
market time and household capital. Specifically,
e
nmt ≤ hµ (zt nmt )1−µ ,
mt
(2.2)
e
cmt + xmt + xht ≤ (1 − τ k )rt kt + (1 − τ n )wt nmt + δ m τ k kt + ξ t .
(2.3)
where zt is the level of labor-augmenting technology, hmt is the stock of household
capital devoted to maintaining and enhancing the quality of market hours, and
0 ≤ µ < 1. As described in the introduction, the essential idea here is that
the effectiveness of hours worked in the market is enhanced by the quality and
quantity of household capital used in activities at home and in commuting to
work. In general it is reasonable to suppose that both household capital and
non-market time are inputs into producing effective market hours. I assume that
only the former is used to make the model easily comparable to standard home
production models.
At each date t the household faces the following budget constraint
Here xmt is the household’s investment in business capital, xht is investment in
household capital, kt is the household’s stock of business capital, rt is the rental
rate on that capital and wt is the wage for effective hours worked. Business
capital and effective market hours are taxed at the rates τ k and τ n , respectively.
Consistent with the U.S. tax code, there is a depreciation allowance for capital
4
taxation, δ m τ k kt , where 0 < δm < 1 is the rate of depreciation on business capital.
Notice that since household capital contributes to effective market hours it is
subject to the labor tax. Finally, ξ t is a lump-sum transfer from the government.
Throughout the analysis, I assume that household capital projects take one
period to complete. That is, the stock of household capital, ht accumulates according to
ht+1 = (1 − δ h )ht + xht .
(2.4)
In addition to being an input into effective market hours, household capital is an
input in the production of home goods. Specifically,
cht ≤ hφ (zt nht )1−φ ,
ht
(2.5)
hht + hmt ≤ ht .
(2.6)
where hht is the amount of household capital devoted to home production and
0 < φ < 1. As in Greenwood and Hercowitz (1991) the level of labor-augmenting
technology is identical in home and market production. However, as these authors
emphasize, given the specification of preferences in (2.1), variation in zt will have
no impact on equilibrium outcomes through home production. Its inclusion here
is merely to facilitate balanced growth. Finally, in any given period the uses
of household capital in enhancing effective market hours and in producing home
goods are subject to the availability of household capital in that period. Hence,
Business capital is accumulated according to the technology considered by
Kydland and Prescott (1982). Specifically, business capital projects require a
flow of investment lasting J periods until they are completed. Let sjt denote
the number of projects j periods from completion at time t and let ω j denote
flow of investment in a project j periods from completion. Then, total business
investment at date t is
xmt =
J
X
ω j sjt
(2.7)
j=1
and projects evolve according to
sjt+1 ≤ sj+1t ,
(2.8)
kt+1 = (1 − δ k )kt + s1t .
(2.9)
j = 1, 2, . . . , J − 1. Given this structure, business capital accumulates as follows
The problem of the representative household is: maximize (2.1) subject to
(2.2)-(2.6) by choice of cmt , cht , ht+1 , kt+1 , s1t+1 , s2t+1 , ..., sJ−1t+1 , sJt , hmt , hht ,
nmt and nht .
5
2.2. Firms
Firms produce a single output good, yt , which can be used as market consumption
as well as both kinds of investment. They choose business capital and effective
labor services to maximize profits,
e
yt − rt kt − wt nmt ,
where
α
e mt
yt ≤ kt n1−α .
e
Recall from (2.2) that effective market hours, nmt , are influenced by the level of
labor augmenting technical change, zt . As is standard, I assume this grows at the
deterministic gross rate γ ≥ 1 and is subject to exogenous disturbances:
zt = γ t exp(θt ), θt = ρθt−1 + εt , εt ∼ N(0, σ 2 ).
2.3. Government
Government is included in the model to make the analysis comparable to the
literature. Its only function in the model is to raise revenues and rebate these
revenues lump-sum to households:
e
ξ t = τ k rt kt + τ n wt nmt − δ k τ k kt .
3. Parameter Selection
In order to implement the model I must select values for the following parameters:
β, ψ, η, φ, µ, δ h , δ k , J, {ω i : i = 1, 2, ..., J} , α, γ, ρ, σ, τ k , τ n .
The strategy I use for doing this closely follows the one outlined in Greenwood,
Rogerson and Wright (1995) which is also implemented by Gomme, et. al. (2001).
The main difference with the calibration procedure used in these studies is the data
underlying the parameter choices. In particular, the new capital stock estimates
used here imply noticeable changes to long-run averages which are an important
input into the calibration procedure. As is outlined in the following section, these
changes have a significant impact on inference.
I analyze six versions of the model, which differ in terms of their assumptions
on the market investment technology and the use of household capital in determining effective market hours. Conditional on the parameters underlying these
6
differences, the remaining parameters are either identical or chosen in a consistent
manner across versions of the model. The parameter values corresponding to the
six versions of the model considered are summarized in Table 1. I now explain
how these parameter values were chosen.
Common across the different models are the time discount factor, the two tax
rates and the process governing labor-augmenting technology. The discount factor
β is set so that a period in the model is a quarter of a year and the real interest rate
is 6 percent at an annual rate along a non-stochastic balanced growth path. The
tax rates are the values used by Greenwood, et. al. (1995), τ n = 0.25 and τ k = 0.7.
The value for the labor tax is close to values used elsewhere in the RBC literature.
Greenwood, et. al. (1995) use the relatively large value of the capital tax rate
because without it the home production framework cannot simultaneously account
for observed capital output ratios and labor’s share of national income. In addition
they defend the value as being reasonable because it is the mean of reported values
in Feldstein, Dicks-Mireaux and Poterba (1983) and because it captures “all forms
of government regulation, interference, or any other institutional disincentive to
invest in business capital, not only direct taxation (p.165).” As in Greenwood,
et. al. (1995) I choose the growth rate of labor-augmenting technology, γ, using
the growth rate of per-capita GDP (excluding housing services) over the sample
period of this study, 1948-1999. The remaining parameters for the technology
process are chosen to be consistent with other papers in the literature: ρ = 0.95
and σ is set so that the innovation of exp((1−α)(1−µ)θt ) has a standard deviation
of 0.00763.5
I consider three versions of the market investment technology. The first corresponds to the technology which is the mainstay of the RBC literature, the oneperiod time-to-build case. This involves setting J = 1 and ω 1 = 1. The second is
the case considered by Kydland and Prescott (1982) and Gomme, et. al. (2000).
This is the four quarter time-to-build formulation and involves setting J = 4 and
ω 1 = ω 2 = ω 3 = ω 4 = 0.25. This captures the fact that business capital projects
generally take longer than household capital projects to complete. The third case
is motivated by Christiano and Todd (1996). They discuss evidence on both the
planning and building of new investment projects and argue that in the early phase
5
The Solow residual computed from the models considered in this paper will not in general
be identical to ones arising from a single-capital-good RBC model. In principle this suggests
an adjustment to the assumed technology process is in order. However, since capital does not
vary much over short horizons in the model, the differences with other models are very small so
that modifications to the technology process are unnecessary to retain consistency with previous
studies.
7
of many non-residential fixed investment projects relatively few resources are consumed. Instead, time is spent drawing up plans, gaining regulatory approval and
in other relatively low resource cost tasks. These “time-to-plan” considerations
are more relevant for business investment than household investment, and are at
least as empirically justified as Kydland and Prescott’s (1982) original time-tobuild formulation.6 Following Christiano and Todd (1996) I model time-to-plan
by keeping the four-period time-to-build formulation and assuming the resource
cost in the first period of a market investment are lower than in the last three
periods. Specifically I set J = 4, ω 4 = 0.1 and ω 1 = ω 2 = ω 3 = 0.3.
The preference parameters ψ and η, the home production parameters φ and
δ h , and the market production parameters α and δk are chosen using the exact
procedure described by Greenwood, et. al. (1995). Specifically, conditional on
the other parameter values, these parameters are chosen so that along the nonstochastic balanced growth path, the model matches six calibration targets: the
fractions of time devoted to market and non-market work, the ratios of market and
household capital to output, and the shares in real output of market investment
and household investment. The first two targets are taken from the same timeuse studies discussed by Greenwood, et. al. (1995): 1/3 of the time endowment
is devoted to market work and 1/4 to non-market work. The other four targets
are based on the most recently available data on capital stocks, investment and
output. Business capital is measured as private non-residential fixed capital and
household capital is measured as the chain-weighted sum of the stock of private
residential capital and consumer durables. The two investment series are measured
analogously and output is measured as GDP less consumption of housing services.
All variables are measured in chained 1996 dollars.
As describe in Katz and Herman (1997), the Bureau of Economic Analysis has
revised its methodology for computing capital stock and investment series. In addition to incorporating the new chain-weighting methodology, the biggest change
is in the way depreciation is calculated. Depreciation is now calculated geometrically instead of in the straight-line way used before, and data on secondary capital
markets is used to estimate appropriate rates of depreciation. An outcome of using
the revised methodology is that in many cases depreciation estimates have been
substantially reduced and consequently estimates of capital stocks are, overall,
much larger than previously estimated.7 For investment, another change is that
6
See Edge (2000) for a discussion of the empirical evidence showing significant differences in
gestation lags and planning periods for residential and non-residential structures.
7
For example, Katz and Herman (1997, p. 75, Table C) report that the total net stock of
8
software is now included in the measure of private non-residential fixed capital.
This also has the effect of increasing the market capital stock.
These changes in methodology have a substantial impact on estimates of
capital-output ratios. Using the sample period 1948-1999, I estimate the mean
market capital:output ratio to be 4.79 and the mean household capital:output
ratio to be 5.77. Greenwood, et. al. (1995) report mean values for these ratios
of 4.00 and 5.00, respectively.8 The investment shares are changed by less. These
are now 0.094 and 0.116 for business investment and household investment, respectively, compared to the values of 0.118 and 0.135 used by Greenwood, et. al.
(1995). As is described in the next section, these new estimates have a substantial
impact on inference in home production models.
It remains to describe how I select the share of household capital in producing
effective market hours, µ. I consider two cases for µ. One case corresponds to
the standard formulation of home production in which household capital is not
useful in market production at all. This involves setting µ = 0. I refer to versions
of the model with this parameter choice as pure home production models. The
other case incorporates the notion of household capital being useful in market
production and is referred to as general home production.
In the absence of independent evidence on µ, I select µ to demonstrate the
potential for general home production models to account for the observed investment dynamics. To this end, for each of the general home production models
studied, µ is chosen by minimizing a measure of the distance between the model
and empirical dynamic correlations between detrended log household and business investment. Let Ψ(µ) denote the mapping from µ to the model correlations
b
between xht+j and xmt , j = −2, −2, 0, 1, 2, and let Ψ denote the corresponding
empirical estimates. My estimator for µ is the solution to
³
´
³
´
0
b
b
min Ψ − Ψ(µ) V −1 Ψ − Ψ(µ) .
µ
b
Here V is a diagonal matrix with the sample variances of the Ψ along the diagonal. These variances are the basis for confidence intervals reported below for the
fixed reproducible tangible wealth in 1994 is 22.4 percent higher using the new methodology
compared to previous estimates. The largest changes are in the stock of structures. In 1994 the
stock of non residential structures is estimated to be 44.3 percent larger than before and the
stock of residential structures 26.6 percent larger.
8
Using the recent data, the market capital to market output ratio does not display a trend
over the sample period, while the household capital to market output ratio displays a slight
upward trend. In both cases the ratios are uniformly above the values reported by Greenwood,
et. al. (1995).
9
dynamic correlations between household and business investment. So, with this
choice of V , µ is effectively chosen so that Ψ(µ) lies as much as possible inside
these confidence intervals. For the versions of the model with general home production µ does not differ very much and is roughly equal to 0.20. The share of
household capital in the reduced form market production function is (1 − α)µ and
this share is about 0.14 in the versions of the model with general home production.
4. Findings
In this section I show that the general home production formulation can reconcile
RBC models with the observed lead-lag pattern between household and business
investment dynamics, but time-to-build and time-to-plan in the context of pure
home production cannot. Moreover, the general home production framework can
account for the fact that household investment is more volatile than business
investment. The pure home production models, even with realistic gestation lags
in business investment, consistently predict the opposite. The discussion below is
organized in terms of the different investment technologies considered.
4.1. One Quarter Time-to-Build
The pure home production model with one quarter time-to-build has the same
structure as ‘Model 1’ in Greenwood, et. al. (1995) and the identically labelled
model in Greenwood and Hercowitz (1991). The only difference with those models
is the parameterization. The general home production model with one quarter
time-to-build has not been studied before.
Table 2 shows standard business cycle statistics implied by these two models
as well as the other models considered below. Tables 3 and 4 show statistics which
summarize the investment dynamics which are the focus of this study. The layout
of the tables is similar, with the column headings indicating the source of the
reported statistics, either the U.S. data or a particular model. For the statistics
in the U.S. data columns, the numbers in parenthesis are standard errors. In the
column headings ‘General’ indicates models with general home production and
‘Pure’ indicates models with pure home production. Figures 2 to 4 present the
information contained in Table 3 graphically. Specifically, the vertical lines with
hash marks indicate plus and minus two standard deviation error bands about
the point estimates for the indicated correlations. The black dots in these figures
show the predictions of the various models considered for the same correlations.
10
The left hand columns of these figures show predictions of the pure home production models and the right hand columns show predictions of the general home
production models. The rows correspond to different assumptions on the business
investment technology. All statistics are based on data which has first been logged
and detrended with the Hodrick-Prescott filter.
The second column of Table 2 indicates the pure home production model
has implications for statistics typically studied in RBC studies that are little
distinguished from the identical but differently calibrated home production models
studied previously in the literature. The general home production model shares
these predictions (the first column of Table 1). In fact, all the models discussed
here have similar predictions for the statistics displayed in Table 2. Since these
predictions have been discussed at length elsewhere, for example in Greenwood,
et. al. (1995), I do not elaborate on them further and instead focus the rest of
the discussion on the behavior of household and business investment.
Table 3 and 4 confirm that the predictions for investment dynamics in the
pure model are strongly at variance with the data. Household investment comoves strongly negatively with business investment with the contemporaneous
correlation between these two variables equal to -0.60. In the data these variables
co-move positively with the contemporaneous correlation statistically significant
and equal to 0.39. In addition, household investment lags business investment over
the business cycle, contrary to the U.S. data.9 These dynamics are evident in the
relatively large positive correlations of household investment with lags in business
investment, the relatively large positive correlations of household investment with
lags in output and the relative large positive correlations of business investment
with future output. The final major failure of this model is respect to the relative
volatility of household and business investment. The second column of Table 4
shows that the pure home production model predicts business investment is much
more volatile than household investment: the ratio of the standard deviation of
household investment to the standard deviation of business investment is 0.58. In
the data this ratio is 1.43 and significantly greater than unity.
The pattern of co-movement and the lag-lead behavior of household and business investment in the pure home production model were first pointed out by
Greenwood and Hercowitz (1991). As they emphasized, the reason is due to a
basic asymmetry in pure home production models that household capital is only
9
Formally, for variable xt to lag (lead) variable yt , the peak correlation of xt+j with yt would
be for a positive (negative) value of j. This is the standard way of assessing lead-lag patterns
between two variables. See, for example, Christiano and Vigfusson (2001).
11
useful for producing home consumption goods, but business capital can produce
market consumption and both kinds of investment goods. This means business
capital is useful for producing household capital, an input into home production,
but household capital is not useful at all in market production. After a positive
technology shock which raises the productivity of the market sector there is a
strong incentive to move resources out of the home to build up business capital.
Only later is household capital built up.
Introducing general home production, where the basic asymmetry is reversed,
has a dramatic impact. Not only is there an effect encouraging positive comovement, there is also an effect which encourages agents to build up household
capital before business capital. The former arises because of the complementarity
between the two kinds of capital in market production. The latter arises because
household capital can not only help in the production of home goods, but in all
market goods as well. Of course the magnitude of these effects depends on the
share of household capital in market production, as well as other elements of the
calibration. Figure 5, which displays the responses of household (dashed lines)
and business investment (solid lines) to innovations in technology (the basic layout
of this figure is the same as in Figures 2-4), illustrates that the effects are quite
strong. With pure home production and one quarter time-to-build (first column
and row) the two investments move in opposite directions in the period of a shock,
with household investment dropping sharply. When general home production is
added to this model (second column, first row), household investment now responds positively in the period of the shock and business investment responds
with a lag.
These responses have a substantial impact on the statistics in Tables 3 and
4, moving the home production framework a long way toward the data. First,
household investment co-moves positively with business investment in the general model. The contemporaneous correlation between these variables is 0.28 and
within two standard deviations of the point estimate from the data. This prediction alone stands as a significant finding, given the extensive amount of research
that has been conducted to overturn the pure home production model’s prediction
of a negative correlation.10
Second, the lead correlations of household investment with business investment
10
If the share parameter µ is increased sufficiently then the two investments become negatively
correlated again. This occurs since household capital eventually becomes so important in market
production that after a positive technology shock agents find it optimal to substitute out of
business capital and into household capital.
12
are larger than the contemporaneous and lag correlations. That is, household
investment leads business investment over the business cycle. This is a significant
finding since no previous model has been shown to be consistent with the leadlag relationship between these variables found in the data. While qualitatively
successful, this model does over predict the one period lead of household over
business investment.
Third, the lead correlations of household investment with output are larger
than the corresponding lag correlations and the lag correlations for business investment with output are larger than the corresponding lead correlations. These
predictions also move the model closer to the data compared to the pure home
production model and represent a substantial improvement compared to that
model.
Finally, the first column of Table 4 indicates that in the general model household investment is more volatile than business investment and that the degree of
this excess volatility is within two standard deviations of the data. This pattern
of volatility has been almost as hard to achieve as the lead-lag pattern. For example, Gomme, et. al. (2001), while claiming success with respect to the lead-lag
pattern, make no such claims with respect to the relative volatility phenomenon.
In summary, even without any differences in the investment technology across
business and household investment, the general home production model represents
a dramatic improvement over the pure home production model. It overcomes the
latter model’s three main drawbacks in terms of its predictions for household
and business investment: their co-movement, lead-lag relationship and relative
volatility.
4.2. Four Quarter Time-to-Build
Gomme, et. al. (2001) argue that adding four quarter time-to-build to a model
with pure home production brings that framework into substantial conformity
with the data in terms of the co-movement and lead-lag properties of household
and business investment. The intuition for this potentially being the case is
straightforward. With time-to-build, only a fraction of the total resources for
business investment are needed in the period of a positive technology shock; the
effect of a positive technology shock on business investment is spread out over the
time it takes to complete a project. Compared to a model with only one quarter
time-to-build, this makes it more costly to quickly install new business capital and
lowers the opportunity cost of immediate investment in household capital. Since
13
household capital is not subject to the same gestation lags, there is an increased
incentive to accumulate it over business capital.
Ultimately the impact of this mechanism on the dynamics of household and
business investment depends on how it matches up with competing forces, such
as those induced by the fact that in pure home production models household
capital is not used to produce market goods. This in turn depends on the parameterization of the model. The pure home production model with four quarter
time-to-build described in this paper is the same as the model emphasized in
Gomme, et. al. (2001). However, these authors, while using virtually the same
procedure to calibrate their model, use the calibration targets in Greenwood, et.
al. (1995), which are based on obsolete data. Using this calibration they find that
the correlation of household investment at t − 1 with output at t is slightly larger
than the opposite cross-correlation, but that there is very little difference in the
corresponding business investment-output correlations. In addition, they find that
household and business investment are contemporaneously positively correlated.
These observations form the principal basis of their conclusion that time-to-build
“is an essential feature of reasonably calibrated household production models to
match the cyclical properties of business and household investment (p. 3).”11
This conclusion may end up being correct, but the fourth column in Table 3
and the first column, second rows of Figures 2 to 4 establish that the results presented by Gomme, et. al. (2001) supporting it are fragile. In particular, when the
new data is used to calibrate the pure home production model with four quarter
time-to-build, it fails on the two dimensions emphasized as positive contributions
by Gomme, et. al. (2001).12 Household investment is slightly negatively corre11
If one applied the criteria used by Gomme, et. al. (2001) to the evidence presented in the
previous subsection, this evidence would refute their conclusion. That is, a model without any
differences in investment technologies across household and business investment, the general
home production model with one period time-to-build, would satisfy their criteria. I think their
criteria fail to capture the essence of what the data say about the dynamics of business and
household investment. A better criteria for confirming the lead-lag pattern is the standard one
involving the dynamic correlations between two variables.
12
There is one other difference between the two calibrations. In a departure from usual
practice and the calibration procedure described by Greenwood, et. al. (1995), Gomme, et.
al. (2001) do not account for growth in their model. In the notation of this paper, they
assume γ = 1. I have reproduced the findings in their paper using their calibration. In this
case ρ(xht−1, yt ) − ρ(xht+1, yt ) = 0.10 and ρ(xmt−1, yt ) − ρ(xmt+1, yt ) = 0.01, where ρ(lt , qs )
denotes the correlation between variables lt and qs . When their calibration is adjusted to
take into account the same value for γ estimated here, ρ(xht−1, yt ) − ρ(xht+1, yt ) = 0.05 and
ρ(xmt−1, yt ) − ρ(xmt+1, yt ) = 0.04. Even using the Gomme, et. al. (2001) criteria for model
14
lated with business investment and still lags business investment (Panel A, Table
3 and Figure 2). In addition, the correlations of household investment with future
output continue to exceed the corresponding correlations with past output (Panel
B, Table 3 and Figure 3) and the reverse continues to be true for the correlations
of business investment with output (Panel C, Table 3 and Figure 4). The forces
emphasized by these authors are certainly at work in this model. This is seen
by the improvement in the investment correlations compared to the pure home
production model with only one quarter time-to-build. Nevertheless, these forces
are relatively weak when the model is calibrated using the new data and so the
investment correlations are still far from those estimated in the data.13
The model with four quarter time-to-build and general home production performs much better. Consider first the correlations between household and business
investment in Table 3 and in Figure 2. The incentives that come along with differences across investment in gestation lags reinforce the strong performance of the
general home production model without such differences and imply a correlation
pattern between household and business investment which is clearly consistent
with the observed lead-lag pattern in the data. In fact the predicted correlations
are remarkably close to those in the data, all falling within two standard deviations of the corresponding point estimates from the data. This strong performance
is roughly matched by the investment correlations with output shown in Table 2
and in Figures 3 and 4. The correlations of household investment with output
are clearly larger than the corresponding correlations with past output and the
reverse is true for the correlations between business investment and output, just
as in the data. Quantitatively, the model correlations are almost all within two
standard deviations of the point estimates.
The impulse response functions in the second row of Figure 5 are helpful in
gaining insight into the source of the substantially improved investment dynamics
in this model compared to the pure home production model. In the case of pure
home production, there seems to be little evidence of a consistent relationship
between the two forms of investment. Sometimes they are moving together, and
sometimes they move in opposite directions. To the eye, there is no apparent
evidence of a lead-lag relationship. The picture for the general home production
evaluation, this does not seem to be strong evidence in favor of their model.
13
The forces fighting against the time-to-build effects are stronger when the model is calibrated
using the new data. This can be seen by comparing the impulse response function for household
investment in the pure home production one quarter time-to-build model in Figure 5 with the
analogous response in Figure 1 of Gomme, et. al. (2001). The impact response is about -6
percent with the new data compared to about -1.5 percent with the obsolete data.
15
model is quite different. Immediately after the innovation to technology, household investment surges. At the same time there is a slow build up of business
investment. These response functions also highlight that in the pure home production case business investment seems more volatile than household investment,
whereas in the general home production case the opposite seems to be true. This
difference in relative volatilities is confirmed by the relevant entries in Table 4.
Overall the general home production model with four-quarter time-to-build
performs much better than the corresponding pure home production model, but
only moderately better than the general model with identical investment technologies for business and household investment. Accordingly it does not seem
justified to conclude that time-to-build is “essential” to account for the observed
heterogeneous investment dynamics.
4.3. Four Quarter Time-to-Build with a Planning Phase
Is the poor performance of the pure home production model with differences in
investment gestation lags due to an improper specification of time-to-build? One
way to assess this is to consider the addition of a planning phase to the four
quarter time-to-build formulation. On the one hand this is well motivated on
empirical grounds and seems like a reasonable alternative to the conventional
time-to-build specification. On the other hand, it seems possible that adding
another reason why business investment may lag household investment should
improve the performance of this model.
To some extent this latter possibility turns out to be true. Table 3 and the
last row of Figure 2 show that the maximal lead correlation exceeds the maximal lag correlation of household versus business investment. Unfortunately the
contemporaneous correlation is now much more negative than in the conventional
time-to-build case. The reason for this can be seen by studying the impulse response functions in the last row, left-hand column of Figure 5. Time-to-plan
certainly encourages a surge in household investment in advance of a build-up of
business investment. However, as soon as the resource flow into the newly initiated business investment projects rises, there is a swift cut-back in household
investment. Lowering the initial resource flow into business investment projects
exacerbates this problem and contributes to a much larger negative contemporaneous correlation between the two kinds of investment.
If one ignores the household-business investment correlations and focuses only
on the investment-output correlations, the pure home production model does look
16
better. However, it does not look as good when it is compared to the model with
the same market investment technology, but with general home production. The
empirically desirable lead-lag and co-movement properties of household and business investment present in the general home production model considered in the
previous subsection continue to be present when a planning phase is added to that
model. Finally, Table 4 indicates the greater volatility of household investment
compared to business investment continues to be the case with this version of the
general home production model, in stark contrast to the comparable pure home
production model.
5. Conclusion
This paper has demonstrated that RBC models can be reconciled with the comovement, lead-lag pattern and relative volatility of household and business investment. The feature of the models considered here which contributes most to
this finding is the fact that household capital is useful in market production, what
I have termed general home production. Differences in the time-to-build and plan
characteristics of household and business investment add somewhat to the success of the general home production framework, but do not seem to be crucial to
its success. When these model characteristics are incorporated into a standard
home production framework they do not account for the observed heterogeneous
investment dynamics. The results support the conclusion that future models of
home production which seek to address aggregate data should adopt the general
formulation developed in this paper.
The main drawback of the current research is that the use of household capital
in market production was not calibrated with independent evidence. It was estimated using some of the correlations of interest. This is not as much a drawback
as might at first seem, since only one new parameter has been added to previously studied models, yet the model does substantially better than these models
on multiple dimensions. Still, it would be very interesting to identify the magnitude of the complementarity of household capital with other inputs into market
production using micro data. This task is left to future research.
17
Data Appendix
The flow data sources are from the Haver Analytics database while the capital
stock series were downloaded directly from the web site www.stat-usa.gov. All
real series are in chained 1996 dollars. Except where noted, the original source
for the series is the Bureau of Economic Analysis. To form data counterparts of
model variables some series were added together or a series was subtracted from
another. In these cases, the appropriate chain-weighting procedure was used. This
requires price deflators and so for the series requiring price deflators I list both
real and nominal series. Otherwise I just list the real series.
The mnemonics for the series taken from the Haver Analytics database are
in parenthesis after a description of the series as follows: gross domestic product (GDPH), nonresidential fixed investment (FNH), residential fixed investment
(nominal — FR, real — FRH), consumption of non-durables (nominal — CN, real —
CNH), consumption of services (nominal CS, real CSH), consumer durable expenditures (nominal — CD, real — CDH), consumption of housing services (nominal
— CSR, real — CSRH), hours of private wage and salary workers on nonfarm payrolls (LHTPRIVA, original source is Bureau of Labor Statistics). The downloaded
BEA capital stock series are taken from data files 1KCU.TXT (nominal series)
and 15KRE.TXT (real series).
The data counterparts for the model variables are as follows: output is GDP
less consumption of housing services, consumption is the sum of nondurables and
services consumption less housing services, business investment is nonresidential
fixed investment, household investment is the sum of residential fixed investment
and durables consumption expenditures, total investment is the sum of household
and business investment, hours is private hours on nonfarm payrolls. The capital
stock series are derived analogously to the investment series.
18
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21
Table 1: Parameter Values
Parameter
One Quarter
Time-to-Build
General
Pure
Four Quarter
Time-to-Build
General
Pure
Preferences
0.991
0.476
0.652
Four Quarter
Time-to-Build with
Planning Phase
General
Pure
β
ψ
η
0.991
0.479
0.661
0.991
0.713
0.828
0.991
0.706
0.817
0.991
0.470
0.650
0.991
0.707
0.819
µ
φ
δh
0.201
0.172
0.015
Home Production
0
0.201
0
0.303
0.178
0.306
0.015
0.015
0.015
0.207
0.171
0.015
0
0.305
0.015
α
δk
J
ω1
ω2
ω3
ω4
0.302
0.014
1
1
0
0
0
Market Production
0.302
0.312
0.312
0.014
0.014
0.014
1
4
4
1
0.25
0.25
0
0.25
0.25
0
0.25
0.25
0
0.25
0.25
0.310
0.014
4
0.30
0.30
0.30
0.10
0.310
0.014
4
0.30
0.30
0.30
0.10
τn
τk
0.25
0.70
0.25
0.70
0.25
0.70
0.25
0.70
γ
ρ
σ
1.00528
0.95
0.014
Labor-augmenting Technology
1.00528
1.00528 1.00528
0.95
0.95
0.95
0.011
0.014
0.011
1.00528
0.95
0.014
1.00528
0.95
0.011
0.25
0.70
Government
0.25
0.70
Note: See the text for the meaning of the column headings.
Table 2: Standard Business Cycle Statistics in Various Models
Statistica
σy
σnm /σ y
σ c /σy
σ x /σy
ρ(nmt , yt )
ρ(ct , yt )
ρ(xt , yt )
U.S
Datab
1.91
(0.17)
1.04
(0.04)
0.46
(0.03)
2.59
(0.15)
0.89
(0.02)
0.79
(0.04)
0.76
(0.08)
One Quarter
Time-to-Buildc
General Pure
1.34 1.52
Four Quarter
Time-to-Build
General Pure
1.31 1.37
Four Quarter
Time-to-Build with
Planning Phase
General
Pure
1.31
1.37
0.42
0.43
0.43
0.42
0.42
0.42
0.38
0.37
0.39
0.39
0.39
0.39
3.39
3.44
3.39
3.38
3.39
3.38
0.99
0.99
0.99
0.99
0.99
0.99
0.97
0.97
0.96
0.97
0.96
0.97
0.99
0.99
0.99
0.98
0.99
0.99
Notes: a — σl denotes the standard deviation of variable l, ρ(lt , qs ) denotes the cross-correlation between
variable lt and variable qs , and x = xm + xh ; b — Point estimate with standard error in parenthesis based on
U.S. data over the sample period 1948:I-1999:IV. Standard errors computed using a GMM procedure. For
estimation of the relevant zero-frequency spectral density, a Bartlett window truncated at lag five was used.
For data sources, see the appendix; c — For parameter values corresponding to the different models indicated
in the column headings, see Table 1.
23
Table 3: Dynamics of Household and Business Investment in Various Models
Statistic
U.S
Data
One Quarter
Time-to-Build
General Pure
Panel A — Household
0.69
ρ(xht−2 , xmt )
(0.08)
0.59
ρ(xht−1 , xmt )
(0.07)
0.39
ρ(xht , xmt )
(0.07)
0.07
ρ(xht+1 , xmt )
(0.08)
-0.20
ρ(xht+2 , xmt )
(0.10)
Four Quarter
Time-to-Build
General Pure
Four Quarter
Time-to-Build with
Planning Phase
General
Pure
Investment Correlations with Business Investment
0.65 -0.08
0.78
0.10
0.81
0.87
-0.07
0.68
0.15
0.66
0.26
0.28
-0.60
0.33
-0.02
0.30
-0.35
0.15
0.53
0.04
0.27
0.03
-0.01
0.04
0.42
-0.09
0.35
-0.08
0.22
Panel B — Household Investment Correlations With Output
0.63
0.47
0.01
0.45
0.15
0.45
(0.09)
0.67
0.64
0.08
0.65
0.31
0.65
ρ(xht−1 , yt )
(0.09)
0.60
0.86
0.23
0.89
0.57
0.88
ρ(xht , yt )
(0.10)
0.36
0.25
0.80
0.34
0.38
0.33
ρ(xht+1 , yt )
(0.12)
0.08
0.11
0.59
0.01
0.32
0.00
ρ(xht+2 , yt )
(0.13)
ρ(xht−2 , yt )
ρ(xmt−2 , yt )
ρ(xmt−1 , yt )
ρ(xmt , yt )
ρ(xmt+1 , yt )
ρ(xmt+2 , yt )
0.20
Panel C — Business Investment Correlations With Output
0.24
0.32
0.43
0.28
0.48
0.28
(0.09)
0.51
0.51
0.57
0.47
0.63
0.47
(0.07)
0.75
0.72
0.63
0.73
0.80
0.71
(0.04)
0.81
0.99 -0.02
0.91
0.51
0.91
(0.05)
0.71
0.69 -0.08
0.86
0.25
0.88
(0.08)
Note: See the notes to Table 2.
24
0.12
0.25
0.48
-0.01
0.12
0.41
0.53
0.64
0.71
0.33
Table 4: Relative Volatility of Household and Business Investment in Various Models
Statistic
σxh /σ xm
U.S
Data
1.43
(0.12)
One Quarter
Time-to-Build
General Pure
1.24 0.58
Four Quarter
Time-to-Build
General Pure
1.53 0.60
Note: See the notes to Table 2.
25
Four Quarter
Time-to-Build with
Planning Phase
General
Pure
1.50
0.74
Figure 1: Household Investment Leads Business Investment Over the Business Cycle
30
36
20
24
10
20
15
10
5
12
0
0
0
-10
-20
-12
-30
-5
-10
-24
1947
1948
1949
-15
-20
1950 1951 1952 1953 1954
1955
40
100
75
40
20
50
20
10
25
1957 1958
60
30
1956
0
0
0
-20
-10
-25
-40
-20
-50
-60
-30
-75
1959 1960 1961 1962 1963
-80
1964 1966 1968 1970
50
150
25
100
0
50
-25
0
-50
-50
-75
-100
-100
1978
1979
1980
1974
1976
80
60
40
20
0
-20
-40
-60
-80
-100
-150
1977
1972
1981
1983
1985
1987
1989 1991
1993
Note: The plots cover consecutive sample periods from 1947:I to 1994:IV. The solid line is the level of
business investment and the dashed line is the level of household investment, both in billions of chained
1996 dollars. In each plot the series have been de-meaned using the mean of the series over the sample
period corresponding to the plot. The shaded regions correspond to business cycle recessions as dated by
the NBER.
26
Figure 2: Model and Estimated Correlations Between Household Investment at t+j and Business Investment
at t
Note: The vertical lines with hash marks indicate plus and minus two standard deviation error bounds
around the point estimates of the dynamic correlations between household investment at time t + j and
market investment at time t, j = −2, −1, ..., 2. The black dots show the corresponding correlation implied
by the indicated model.
27
Figure 3: Model and Estimated Correlations Between Household Investment at t + j and Output at t
Note: The vertical lines with hash marks indicate plus and minus two standard deviation error bounds
around the point estimates of the dynamic correlations between household investment at time t + j and
market output at time t, j = −2, −1, ..., 2. The black dots show the corresponding correlation implied by the
indicated model.
28
Figure 4: Model and Estimated Correlations Between Business Investment at t + j and Output at t
Note: The vertical lines with hash marks indicate plus and minus two standard deviation error bounds around
the point estimates of the dynamic correlations between market investment at time t + j and market output
at time t, j = −2, −1, ..., 2. The black dots show the corresponding correlation implied by the indicated
model.
29
Figure 5: Model Responses of Investment to Innovations in Technology
Note: The solid lines correspond to responses of business investment and the dashed lines to responses of
household investment.
30
@FedFRASER