Durable goods, Investment Shocks and the Comovement
Problem
Been-Lon Chen†
Academia Sinica
Shian-Yu Liao‡
National Taiwan University
January 2014
Abstract
Recent research based on sticky price models suggests that investment shocks are the most important
drivers of business cycle fluctuations. Yet, an investment shock generates the comovement problem
because there is an intertemporal substitution effect away from consumption and toward investment.
This paper resolves the comovement problem by extending the standard neoclassical sticky-price
model to a two-sector model with durable consumer services. When investment goods are used as
capital investment and durable consumer services, a positive investment shock also generates an
intratemporal substitution effect away from durable consumer services toward consumption. As the
intratemporal effect dominates the intertemporal effect, consumption increases and thus the
comovement problem is resolved.
Keywords: Investment shocks; Durables; Sticky prices; Comovement
JEL classification: E21; E22; E32
_____________________
†
Correspondence author and address: Institute of Economics, Academia Sinica, 128 Academia Road
Section 2, Taipei 11529, Taiwan. Phone (886-2)27822791 ext. 309; fax: (886-2)27853946; e-mail:
bchen@econ.sinica.edu.tw.
‡
Department of Economics, National Taiwan University, 21 Hsu-Chow Road, Taipei 10055, Taiwan.
e-mail: d95323013@ntu.edu.tw.
1.
Introduction
Recent research based on dynamic stochastic general equilibrium models suggests that
investment shocks are the most important drivers of business cycle fluctuations in the post-World
War II US economy. Justiniano et al. (2010) studied a sticky price-wage, one-sector model with a
variety of real and nominal frictions, along the line of Christiano et al. (2005), and several shocks, as
in Smets and Wouters (2007), including a shock to total factor productivity (or neutral technology
shock), as in the RBC literature; a shock to the marginal productivity of investment (or, for simplicity,
investment shock), as in Greenwood et al. (1988); and a shock to desired wage markups (or,
equivalently, to labor supply), as in Hall (1997). It found that over 50% of the fluctuations in output
and hours, and over 80% of the fluctuations in investment are driven by investment shocks. These
shocks may manifest as shocks either to the marginal efficiency of investment, as in Greenwood et al.
(1988), or to the investment-specific technology as in Greenwood et al. (1997). Previously, using a
structural vector auto regression methodology in a one-sector model, Fisher (2006) also found that
investment-specific shocks are the dominant source of business cycles in the US.
Despite their quantitative importance in explaining business cycle fluctuations, one difficulty
remains in these models: consumption typically falls (or does not rise immediately) after a positive
investment shock in the model. Such a comovement problem remains in two-sector models.1 The
comovement problem emerges because a positive investment shock generates a substitution effect
away from current consumption and toward current investment and thus future consumption which
dominates the income effect. As a result, the model does not produce comovement among
macroeconomic aggregates in response to an investment shock, unlike observed business cycles in
which output, consumption, investment, hours worked, and the real wage all rise and fall together.
This lack of comovement is clearly problematic in viewing investment shocks as an important source
of business cycles. In this paper, we resolve the comovement problem by extending the standard
neoclassical sticky-price model to a model with durable consumer services.
In a paper that empirically estimated the link between interest rates and durable consumer
services, Mankiw (1985) has stated the importance of knowing fluctuations in consumer durables for
understanding economic fluctuations. Baxter (1996) set up a two-sector model with durable
consumer services and two types of capital and investigated the effects upon business cycles of a
single shock to a Solow residual that is shared by both sectors and two less-than-perfectly-correlated
sector shocks to Solow residuals. More recently, Barsky et al. (2003, 2007) studied a two-sector model
with durable consumer services and with sticky nondurable prices and envisaged fluctuations in
1
Guerrieri et al. (2010) studied a model with the machinery and non-machinery sectors and found that the
multi-factor productivity shock generated a positive comovement between consumption and investment, but
the investment-specific shock did not.
1
nondurables and durable consumer services in response to monetary policy shocks. This brings about
a surge of research interests in studying the comovement between nondurables and durable
consumer services in response to monetary policy shocks; see Erceg and Levin (2006), Monacelli
(2009), Carlstrom and Fuerst (2010), Huang et al. (2013) and Chen and Liao (2014). Except for
Mankiw (1985) and Baxter (1996), all these papers of consumer durables envisage sticky-price models
with more sticky nondurable prices.2 Our model may be thought of as an extension of the line of
research led by Barsky et al. (2003, 2007) and studies the effects of a positive investment shock on
the comovement between consumption and investment.
Specifically, our model includes one sector that produces consumption goods and the other
sector that produces investment goods. Consumption goods are nondurable and are used only for
consumption. Investment goods are durable and used as investment that accumulates the stock of
capital and investment that accumulates the stock of durable consumer services. Durable prices are
more flexible than nondurable prices. Capital investment is subject to a positive shock. The
household obtains utilities from an index of consumption which is a composite of nondurable
consumption and durable consumer services. Without durable consumer services, our model is a
standard two-sector neoclassical sticky-price model. Like existing papers, in such a model, a positive
investment shock results in the comovement problem as it gives a higher relative price of investment
goods and generates an intertemporal substitution effect away from consumption and toward
investment and thus future consumption. In contrast, with durable consumer services, a higher
relative price of investment goods also generates an intratemporal substitution effect away from
durable consumer services and toward nondurable consumption. Since the intratemporal substitution
effect dominates the intertemporal substitution effect, nondurable consumption rises. As a result, the
comovement problem is resolved.
Our study complements three recent studies which resolved the comovement problem using
alternative mechanisms. First, Kahn and Tsoukalas (2011) solved the comovement problem by
studying a one-sector sticky price-wage model of variable capacity utilization with the cost of higher
capital utilizations in terms of a higher depreciation rate of capital, as in Greenwood et al. (1988).
Since a higher utilization rate increases the marginal productivity of labor, this generates a
substitution effect, away from leisure and toward consumption. Next, Eusepi and Preston (2009) also
studied a model of variable capacity utilizations in terms of a higher depreciation rate of capital. They
found that the heterogeneity in labor supply and consumption of employed and unemployed workers
can generate comovement in response to investment shocks since individual consumption is affected
by the number of hours worked with employed consuming more on average than unemployed and
2
Bils and Klenow (2004) have documented that the price of durable goods changes more frequently than that
of the overall consumption goods.
2
changes in the employment rate then affects aggregate consumption. Finally, Furlanetto and Seneca
(2010) envisaged a model of variable capacity utilizations in terms of the maintenance cost of capital,
as in Christiano et al. (2005), and they reconciled the comovement problem when variable capacity
utilizations are combined with nominal rigidities (in prices and wages) and nonseparable preferences
with a zero wealth effect in labor supply, as in Greenwood et al. (1988). Unlike these papers, the
mechanism to resolve the comovement problem in our paper does not rely on variable capacity
utilizations. In particular, the innovation in our paper is to consider durable consumer services so
that a positive investment shock generates an intratemporal substitution effect away from durable
consumer services toward consumption.
This paper is organized thus. In Section 2, we set up basic two-sector sticky-price models. In
Section 3, we calibrate the model and envisage the impulse responses of a positive investment shock.
Finally, concluding remarks are offered in Section 4.
2.
Two-sector sticky-price models with and without consumer durable goods
Two two-sector neoclassical sticky-price models with and without consumer durable goods are
analyzed. We start with the benchmark model with durable consumer goods.
The economy includes two sectors, the consumption and investment sectors. The consumption
sector produces nondurable goods only for consumption, while the investment sector produces
durable goods for capital investment which accumulates the stock of capital and consumer
investment which accumulates the stock of durable consumer services. 3 Each sector has a
continuum of firms which produce and sell final goods at competitive prices and has a continuum of
producers which produce and sell intermediates at monopolistic prices. The economy also consists
of a continuum of households with a unit mass. Households supply labor elastically, consume
nondurable goods and durable consumer services and offer capital.
2.1
Final Goods Producers
In each sector, there is a perfectively competitive final goods producer which produces Yj, j=C, I.
The final goods producer in sector j produces Yj by combining a continuum of intermediate goods
Yjt(z), z[0, 1], according to the following technology.
1
Y jt  [  (Y jt ( z ))
0
(  j 1)/  j
 j /(  j 1)
dz ]
, j=C, I,
(1a)
where εj>1 is the elasticity of substitution between intermediates. Consumption goods YC are only
used for nondurable consumption C. Investment goods YI are used either for capital investment IK
3
As in Baxter (1996), durable consumer goods are regarded as a type of investment goods in the business cycle
literature. See also Cooley and Prescott (1995), Christiano et al. (2005) and Del Negro et al. (2007).
3
which forms capital K or for consumer investment ID which forms durable consumption services D.
The laws of motions for capital and durable consumption services will be specified in the
household’s problem below.
Maximization of profits in sector j gives the demand for the intermediate z in sector j.
Y jt ( z )  (
Pjt ( z )   j
Pjt
)
Y jt , z[0, 1], j=C, I,
(1b)
where Pjt(z) is the price of the intermediate z in sector j.
Denote PCt as the price of consumption goods and PIt the price of consumption goods. A zero
profit of final goods in sector j gives the following price of final goods j
1
1 j
Pjt  [  ( Pjt ( z ))
0
2.2
1/( 1 j )
dz ]
.
Intermediate Goods Producers
In each sector, there is a continuum of intermediate producers of a unit mass indexed by z[0, 1]. A
producer rents capital and hires labor to produce intermediate goods according to the following
technology.

1 j
Y jt ( z )  A j K jt ( z ) j L jt ( z )
, j=C, I,
(2a)
where Kjt(z) is capital and Ljt(z) is labor used by an intermediate producer z in sector j. Parameter
αj(0, 1) is the capital share and Aj>0 is a productivity coefficient in sector j.
An intermediate firm z in sector j sells intermediates to final goods producers in sector j at a
monopolistic price. Following Rotemberg (1982), in setting its price Pjt(z), an intermediate producer
z faces the following adjustment cost.
( Pjt ( z )) 
j
2
P (z)
( Pjtjt1 ( z )  1) 2 PjtY jt ,
(2b)
where ϑj measures the degree of nominal rigidities in sector j.4
In each period, the representative intermediate firm z has to decide how much labor to hire,
how much capital to rent and what prices to set. Thus, managers of the firm maximize the value to
its owners which is the present discounted value of all current and future expected cash flows.

E0  0 t [ Pjt ( z )Y jt ( z )  Wt L jt ( z )  Qt K jt ( z )  ( Pjt ( z ))] , j=C, I,
t
(3)
t 0
where Wt is the nominal wage and Qt is the nominal capital rentals. In the objective, Et is conditional
expectations at any given period t, and β(0, 1) and Λt are, respectively, the discount factor and the
period-t marginal utility of nominal income of the representative household who owns the firm.
4
An alternative method of price adjustments is random price durations based on Calvo (1983). According to
Galí (2008, p.41), these two methods generate the same inflation dynamics.
4
Given the technology (2a), managers choose {Ljt(z), Kjt(z), Pjt(z)} t0 to maximize the cash
flows in (3) subject to the demand for intermediates in (1b). Let λjt(z) denote the current-valued
Lagrange multiplier of the demand for intermediate z in (1b), j=C, I. Moreover, denote MPjtL(z) and
MPjtK(z) as the period-t marginal product of labor and capital for z in sector j, respectively. The
first-order conditions for Ljt(z), Kjt(z) and Pjt(z) are
wt   jt ( z )
qt   jt ( z )
[1(1 (z)) ](
jt
j
Pjt (z)  j
Pjt
Pjt ( z )
PC t
Pjt ( z )
PCt
MPjtL ( z ),
(4a)
MPjtK ( z ),
(4b)
P (z) 

) j ( jt (z) 1) Pjt1jt(z) jt (z)[ jtYjt ( jtPjt ) j ] Yjt  Et [tt1 j ( jt1(z) 1)
P
Y (z)
 jt1(z) Pj,t1
Ct1 Pjt (z) j,t1
Y ]  0, (4c)
where πjt(z)≡Pjt(z)/Pjt-1(z) is the gross inflation of intermediate z in sector j. In (4a) and (4b),
wt≡Wt/PCt and qt≡Qt/PCt are, respectively, the real wage and the real capital rental.
By imposing the symmetry conditions of Kjt(z)=Kjt, Ljt(z)=Ljt, Pjt(z)=Pjt, Yjt(z)=Yjt and
λjt(z)=λjt for all z, (4a)-(4c) give, respectively,
wt   jt
Pjt
PCt
MPjtL , j=C, I,
(5a)
qt   jt
Pjt
PCt
MPjtK , j=C, I,
(5b)
1
 jt

j
 j  j , t
, j=C, I,
where  j ,t  1   j ( j ,t  1) j ,t  Et [ tt1  j ( j ,t 1  1)
( j ,t  1 ) 2 Y j ,t  1
 C ,t 1 Y j ,t
(5c)
], j=C, I.
The multiplier λjt stands for the real marginal cost of intermediates in sector j in period t, and its
inverse is the markup over the marginal cost. In determining the demand for labor and capital, the
real marginal cost of intermediates is needed in order to fill the wedge between the real wage and the
real marginal product of labor in (5a) and the wedge between the real rental and the real marginal
product of capital in (5b). Data indicates that durable prices are more flexible than nondurable prices
(cf. Bils and Klenow, 2004). To simplify the analysis, we focus on the case wherein durable prices are
flexible and nondurable prices are sticky. A flexible durable price gives ϑI=0, and thus ΩIt=1 for all t.
In this case, the markup 1/λIt=εI/(εI-1) is constant for all t. Moreover, in a steady state, πjt=πjt+1=1 for
both j=C, I, so Ωjt=Ωjt+1=1. Then, the markup 1/λj=εj/(εj-1) is constant in a steady state for both
j=C, I.
2.3 Households
In each period t, the representative household consumes an index of consumption Xt. Following
Baxter (1996), Barsky et al. (2007) and Monacelli (2009), the index of consumption is defined as
5
1
 1
1
 1

X t  [(1   )  (Ct )     ( Dt )  ] 1 ,
(6a)
where Ct is nondurable consumption and Dt is the stock of consumer durables which offers services
to the household. The parameter μ>0 is the share of durable consumer services, and η≥0 is the
elasticity of substitution between nondurable consumption and durable consumer services.
The household maximizes the following expected lifetime utility


E0   (  )t log X t  
 t 0
( Lt )1
1
 ,
(6b)
where Lt is the hours of work. Parameter ν>0 is the coefficient associated with the disutility of work,
and ϕ>0 is the inverse of the Frisch labor supply elasticity. The instantaneous utility in (6b) is
separable in consumption and leisure with the logarithmic form of consumption. The utility function
is consistent with the balanced growth path (e.g., King and Rebelo, 1999).
In each period t, the representative household faces the following budget constraint.
Ct  pt I Dt  pt I Kt  bt  Rt 1 btCt1  wt Lt  qt K t 
Tt
PC t

Ft
PC t
,
(7)
where Kt is the household’s capital holding in the beginning of period t and pt≡PIt/PCt is the relative
price of investment goods. πCt≡PCt/PCt-1 and πIt≡PIt/PIt-1 are the gross inflation of nondurable goods
and the gross inflation of investment goods, respectively. Bt-1 is the household’s nominal bond
holdings in the end of period t-1, with bt-1≡Bt-1/PCt-1 being its real value. Rt-1 is the gross nominal
interest rate on a bond holding between t-1 and t. Tt is nominal lump-sum transfers in t, and Ft is
nominal profits/dividends remitted from firms.
Thus, in each period t, with real income flows from returns to bond, wage, returns to capital,
lump-sum transfers and dividends, the household chooses consumption, consumer investment,
capital investment and bond holdings. While capital investment accumulates the stock of capital,
consumer investment forms the stock of durable consumer services. The stock of capital and the
stock of durable consumer services evolve according to
K t 1  (1   ) K t  t I Kt ,
(8a)
Dt  (1   ) Dt 1  I Dt ,
(8b)
where δ is the depreciation rate.
In (8a), ξt is a source of exogenous variations in the efficiency with which capital investment in
period t can be transformed into the stock of capital in period t+1, and thus into next period’s capital
input.5 Following Greenwood et al. (1988), we assume that the investment-specific technology
5
According to Justiniano et al. (2011), this variation might stem from technological factors specific to the
production of investment goods, as in Greenwood et al. (1997), and from disturbances to the process by which
these investment goods are turned into productive capital. Like these authors, we ignore that distinction and
6
change ξt follows the first-order stochastic process as follows.
log t   log t 1  et ,
(9)
where the innovation et is assumed to be independent and identically distributed normally with mean
0 and variance σ2.
With the laws of motion in (8a) and (8b), the representative household’s problem is to maximize
the lifetime utility (6b) subject to the budget constraint (7). We denote UCt, UDt and ULt, respectively,
as the marginal utility of nondurable consumption, durable consumer services and labor in t. The
first-order conditions for Ct, Lt, bt, Kt+1 and Dt are
U Ct  t ,
(10a)
U Lt
U Ct
(10b)
 wt ,


U Ct   Et U Ct 1  CtRt1 ,

(10d)
U Ct pt  U Dt   Et [U Ct 1 pt 1 (1   )],
(10e)
U Ct
pt
t

(10c)
  Et U Ct 1 qt 1  ptt11 (1   )  ,


along with limt→∞(β)tΛtbt=0, limt→∞(β)tΛtKt+1/ξt=0 and limt→∞(β)tΛtDt=0 which ensure that there is
no “Ponzi scheme.” Note that Λt is the current-valued Lagrange multipliers of the constraint (7) and
is thus the marginal utility of nominal income of the representative household.
In these conditions, (10a) and (10b) are standard, and (10c) is the consumption-Euler equation
which equates the marginal utility of nondurable consumption in this period to the discounted
expected marginal utility of shifting one unit of nondurable consumption to the next period.
Condition (10d) determines the demand for capital in the next period. The marginal cost of
increasing one unit of capital investment to the next period is the investment-shock-adjusted relative
price of investment goods in terms of the marginal utility of consumption in this period. The
marginal
benefit
is
the
discounted
expected
sum
of
the
capital
rental
and
the
investment-shock-adjusted relative price of investment goods of the undepreciated part of capital in
terms of the marginal utility of consumption in the next period.
Finally, (10e) determines the demand for durable consumer services in this period. The
condition equalizes the real marginal cost of investment in durable consumer services in this period
and the real marginal benefit from a marginal unit of durable consumer investment. The real
marginal cost in this period is the relative price of investment goods in terms of the marginal utility
of nondurable consumption in this period. The real marginal benefit is the sum of the immediate
marginal utility of durable consumer services in this period and the discounted expected relative
maintain an agnostic stance on the ultimate source of these disturbances.
7
price of investment goods of the undepreciated part of durable consumer services in terms of the
marginal utility of nondurable consumption in the next period. Through (10e), variations in the
relative price of investment goods are expected to affect the demand for nondurable consumption as
analyzed in the next section.
2.4 Equilibrium
In equilibrium, the consumption goods and the investment goods markets are clear.
YCt  2C ( Ct  1) 2 YCt  Ct ,
(11a)
YIt  2I ( It  1) 2 YIt  I Kt  I Dt .
(11b)
We abstract from redistribution via the fiscal policy, and hence Tt=0. Moreover, the capital, the
labor and the bond markets are all clear.
K t  KCt  K It ,
(12a)
LCt  LIt  Lt ,
(12b)
bt  0.
(12c)
The model is closed by the following monetary policy rule.
Rt
R
 ( t )  , χ>1,
(13)
where πt≡(πCt)1-μ(πIt)μ is a composite inflation index with the weight for investment goods inflation
being the share of durable consumer services in consumption μ, and both R and π are steady-state
values. We assume that χ is sufficiently large in order to ensure equilibrium determinacy.
2.4 The model without durable consumer goods
A special case is a two-sector model without durable consumer services. Even with two sectors,
because of no durable consumer services, as will be seen below, a positive investment-specific
technology shock leads to an increase in investment goods and a decrease in consumption goods,
thus the comovement problem, as in Fisher (2006) and Justiniano et al. (2010). In the model without
durable consumer goods, the utility function in (6b) is reduced to logCt-ν(Lt)1+ϕ/(1+ϕ). Moreover,
(10e) is not an equilibrium condition. As equilibrium conditions (7) and (11a) involve durable
consumer goods, they are also modified.
3.
Comparisons of the both models
In this section, we study the effects of investment-specific technology shocks on the impulse
responses of aggregate macro and other relevant variables.
8
3.1 Calibration
The time frequency is quarters. The steady-state real rate of return R is pinned down by
households’ discount factor β. We choose the real rate of return per annum of 4%. This implies a
quarterly discount factor of β=0.99.
For production, the total factor productivity in the consumption- and investment-goods sectors
are normalized to unity, so AC=AI=1. We follow Acemoglu and Guerrieri (2008) and set the capital
shares of the investment-goods and consumption-goods sectors equal to αI=0.27 and αC=0.47,
respectively, to match with the average capital share in the respective sector in 1987-2005.6 Following
Monacelli (2009), the elasticity of substitution between intermediate varieties in the final goods
production is εC=εI=6, which implies a steady-state markup rate of 20%. We follow Hansen (1985)
and set the depreciation rate of capital equal to δ=0.025 which implies a 10% annual depreciation
rate.7
For the preference, with hours of work L=1/3, we use the consumption-leisure tradeoff
condition in (10b) to calibrate the parameter of leisure in preference and obtain ν=5.573. The
elasticity of substitution between nondurable consumption and durable consumer services is set to
be η=1, implying a Cobb-Douglas form for the consumption index. We choose the share of durable
consumer services in the consumption index at μ=0.2 in order to match with 20% share of the
spending of consumer durables in total private spending in the US. We set the inverse of the Frisch
labor supply elasticity at ϕ=1, which is within the range of values used in existing literature.8
For the degree of price-stickiness, we set ϑI=0 so durable prices are flexible, as shown in Bils
and Klenow (2004), among others. We target the stickiness of nondurable prices at four quarters and
pin down ϑC=58.25.9 As for the monetary policy rule, we set the coefficient of the inflation in the
policy rule at χ=1.5, which is a standard value in the literature regarding Taylor rules.
Finally, the autocorrelation of the investment shock and the standard deviation of error to the
investment shock is set to be ρ=0.72 and σ=0.0603, respectively, which are within the range of the
6
Baxter (1996) estimated αI=0.32 and αC=0.52 over the period 1948-1985. Our results are the same if we set
αI=0.32 and αC=0.52.
7 This value of the depreciation rate gives a quarterly capital-to-output ratio of 9.5 which is in the range of the
data. Alternatively, if we use a lower value of δ=0.02 or a higher value of δ=0.03, the quarterly
capital-to-output ratio would be 11.2 and 8.2, respectively. Our results are robust under these different values.
8
The results are robust under ϕ[0.01, 10].
9 To obtain the value of ϑ , we use the log-linearization of the optimal pricing condition in (5c) to yield the
C
slope of the Phillips curve equal to (εC -1)/ϑC. Then, we equate the slope to the slope of the New Keynesian
Phillips curve in the standard Calvo-Yun model, which is (1-θC)(1-βθC)/θC, where θC is the probability of not
resetting prices for nondurables. Thus, we obtain ϑC=(εC -1)θC/[(1-θC)(1-βθC)]. See Appendix A. By setting the
stickiness of nondurable prices at four quarters so 1-θC=0.25, with εC=6 and β=0.99, we obtain ϑC=58.25.
9
estimated values in literature, such as Smets and Wouters (2007), Justiniano et al. (2010) and Khan
and Tsoukalas (2011).
Parameter values in the benchmark model are summarized in Table 1. In the Table, we also list
parameter values used in the model without consumer durable services wherein μ=0.
[Insert Table 1 here]
3.2 Effects of a positive investment shock
We carry out a positive investment shock in the same way as Justiniano et al. (2010). Specifically,
the innovation of the investment shock et is increased by one standard deviation in the stochastic
process with the initial value of the investment shock ξt normalized at unity. The results of impulse
responses are illustrated in Figure 1. An increase in et raises the marginal efficiency of investment. As
a result of a positive shock to the marginal efficiency of investment, there is an increase in the
demand for investment goods on impact (cf. Panel C). Since durable prices are more flexible than
nondurable prices, the relative price of investment goods increases (cf. in Panel G). The nominal
interest rate and inflation also rise (cf. Panels H and I).
[Insert Figure 1 here]
3.2.1 The Model without Consumer Durables
To see the impulse responses of other variable, we begin by envisaging the results in a
neoclassical sticky-price model without durable consumer services. The results are delineated by
dashed red lines in Figure 1.
A higher relative price of investment goods generates an intertemporal substitution effect away
from consumption and toward investment and thus future consumption. The higher return on
currently available resources would, at the same time, operate to persuade individuals to postpone
leisure. Consequently, there is an expansion in hours worked and output. Moreover, given the fixed
stock of capital, an increase in working hours lowers the marginal product of labor on impact.
Therefore, all real variables increase except for consumption and the labor productivity, measured as
average output per hour worked (cf. dashed red lines in Panels B and F). As a result, the model fails
to generate the comovement between investment and consumption in response to an investment
shock.
3.2.2 The Model with Consumer Durables
Next, we come to an otherwise the same model except for durable consumer services. Durable
consumer goods are a substitute for nondurable consumption. Now, there is another substitution
effect. A higher relative price of investment goods also generates an intratemporal substitution effect
away from durable consumer services and toward nondurable consumption. As the intratemporal
10
substitution effect dominates the intertemporal substitution effect, nondurable consumption goes up
(cf. solid blue lines in Panel B in Figure 1). As shown in Panels A–E, nondurable consumption
co-moves with output, investment, hours and real wage. Moreover, the labor productivity also
increases because output rises more than hours (cf. Panel F). In particular, it should be noted that
with durable consumer services, our model amplifies the impulse responses of real variables to a
positive investment shock.
To evaluate the extent to which our model reproduces the key quantitative features of business
cycles, Table 2 provides statistics summarizing contemporaneous correlations of key macro variables
with output. Column 1 in the table reports the US quarterly data over the period
1947:Q1–2013:Q3.10 It is clear to see that all key real variables in the US data are procyclical in the
postwar era. Columns 2 and 3 of this table compare simulated correlations of key macro variables
with output in the models with and without durable consumer services to those in the data. Column
3 is the simulated correlations for the model without durable consumer services. The results in
column 3 indicate a weak correlation of the labor productivity with output and, in particular, the
correlation between consumption and output is negative rather than positive. In contrast, as seen in
column 2, by adding durable consumer services into an otherwise standard neoclassical sticky-price
model, the simulated correlation between consumption and output can match well with the data and
all other variables move with output procyclically.
[Insert Table 2 here]
3.4 Why does the model with durable goods resolve the comovement problem?
This subsection explains the underlying reasons why, in an otherwise standard two-sector
neoclassical sticky-price model with more flexible durable prices, adding durable consumer services
can resolve the comovement problem. At the center of the analysis is the fact that in a two-sector
model with durable consumer services, the demand for durable consumer services changes the
household’s expenditure behavior.
First, we analyze the model without durable consumer services. By (10b) and (5a), we can obtain
U Lt
U Ct
  Lt Ct   jt
Pjt
PCt
MPjtL , j=C, I.
(14)
With standard preferences and technology, the marginal rate of substitution (-ULt/UCt) depends
positively on consumption and working hours, while the marginal product of labor MPjtL is
10
The source of data is the Federal Reserve Economic Data, Federal Reserve Bank of St. Louis. We use the
seasonally adjusted data, deflated by the GDP deflator, taken logarithms and de-trended with the
Hodrick-Prescott filter with a smoothing parameter of 1,600. Following Del Negro et al. (2007), consumption
corresponds to personal consumption expenditures on nondurables and services, while investment is the sum
of gross private domestic investment and personal consumption expenditures on durables. Following Hansen
(1985), the labor productivity is measured by real output per hour.
11
decreasing in working hours. For ease of exposition, we focus on a one-sector model with perfectly
competitive markets, so λjt=1 and Pjt/PCt=1 for j=C, I. As a result, any shock that increases working
hours and thus, decreases the marginal product of labor, on impact must also generate a fall in
consumption in order for (14) to hold at the new equilibrium. This is exactly what happens in
response to an investment shock in the model without durable consumer services, as was described
above. Even though we have extended a one-sector model with nondurable consumption and
durable investment to a two-sector model with nondurable consumption goods and durable
investment goods, the mechanisms are still the same and thus consumption falls in response to a
positive investment-specific technology shock.
By contrast, in our model with consumer durables, there is an additional optimization condition
which is a household’s shadow value of durable consumer services in period t given by (10e). In
optimum, this shadow value is equal to the real marginal cost of investment in durable consumer
services. If we denote Vt≡UCtpt, the household’s shadow value of durable consumer services in (10e)
can be rewritten as

Vt  U Dt   (1   ) Et (Vt 1 )  Et    (1   ) U Dt  ,
(15)
 0
in which the second equality follows from the law of iterated expectations.
The condition above indicates that a household’s shadow value of durable consumer services in
this period is equal to the expected discounted sum of the marginal utility of undepreciated durable
consumer services from period t onward. The household’s shadow value of durable consumer
services is quasi-constant, since variations in consumer durable flows in this period have little effects
on the stock of durable consumer services upon which the marginal utility of durable consumer
services depends. Further, the shadow value of durable consumer services depends for an important
part on the marginal utility of durable consumer services in the distant future, which is even less
sensitive to temporary shocks. With a quasi-constant shadow value of durable consumer services for
households on the right-hand side of (15), the relative price of investment goods pt and the marginal
utility of nondurable consumption UCt in this period on the left-hand side of (15) move in opposite
directions. As a positive investment shock increases the relative price of investment goods, a rise in
the relative price of investment goods is accompanied by a decrease in the marginal utility of
nondurable consumption which is associated with an increase in expenditure on nondurable
consumption. As a result, nondurable consumption rises.
4.
Concluding remarks
Recent research based on sticky price models suggests that investment shocks are the most
12
important drivers of business cycle fluctuations in the post-World War II US economy. Despite their
quantitative importance in explaining business cycle fluctuations, the comovement problem emerges
because a positive investment shock generates an intertemporal substitution effect away from
consumption and toward investment. Thus, investment increases but consumption decreases. This
lack of comovement is problematic in viewing investment shocks as an important source of
business cycles.
In this paper, we resolve the comovement problem by extending the standard neoclassical
sticky-price model to a two-sector model with durable consumer services. With durable consumer
services, a positive investment shock also generates an intratemporal substitution effect away from
durable consumer services toward consumption. As the intratemporal substitution effect dominates
the intertemporal substitution effect, consumption increases. As a result, consumption comoves with
investment, along with output, hours worked, the real wage and the labor productivity. We also find
that the model without durable consumer services gives a weak correlation of the labor productivity
with output and even a negative rather than positive correlation between consumption and output. In
contrast, the model with durable consumer services generates the simulated correlation between
consumption and output which can match well with data, and all other variables move with output
procyclically.
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Appendix A.
Let superscript “~” denote a percentage deviation of a variable from its steady-state level.
Log-linearization of the optimal pricing condition in (5c) gives a New Keynesian Phillips curve,
 j ,t 
 j 1
j
 j ,t   Et ( j ,t 1 ),
(A.1)
where the slope is (εj -1)/ϑj, j= C, I. The slope of the New Keynesian Phillips curve in the Calvo-Yun
model is (1-θj)(1-βθj)/θj, where θj is the probability of not resetting prices. (See for example, Eq. (3)
in Galí and Gertler, 1999.) By equating these two slopes, we can obtain the coefficient of the cost of
price adjustment ϑj.
15
Table 1: Parameter Setting
Description
Parameter
TFP in consumption and investment sector
Elasticity of sub. between nondurables and durables
Elasticity of sub. between intermediates
Share of durable services
Inverse elasticity of labor supply
Discount factor
Hours of work
Autocorrelation of the investment shock
Standard deviation of error to the investment shock
Coefficient of inflation rates in Taylor rule
Capital share in investment sector
Capital share in consumption sector
Depreciation rate
Parameter of labor in utility
Coefficient of price adjustment in investment sector
Coefficient of price adjustment in consumption sector
A C, A I
η
εC, εI
μ
ϕ
β
L
ρ
σ
χ
αI
αC
δ
ν
ϑI
ϑC
(frequency: quarterly)
Model with
Model without
consumer
consumer
durables
durables
1
1
1
1
6
6
0.2
0
1
1
0.99
0.99
1/3
1/3
0.72
0.72
0.0603
0.0603
1.5
1.5
0.27
0.27
0.47
0.47
0.025
0.025
5.573
5.794
0
0
58.25
58.25
Table 2: Contemporaneous Correlations with Output
Model
Data
with consumer
without consumer
1947-2013
durables
durables
Output
1.00
1.00
1.00
Consumption
0.76
0.75
-0.01
Investment
0.84
0.95
0.85
Hours
0.85
0.92
0.86
Wages
0.89
0.98
0.98
Labor productivity
0.41
0.95
0.12
16
A. Output
B. Consumption
.14
C. Investment
.03
.35
.30
.12
.02
.25
.10
.20
.01
.08
.15
.06
.00
.10
.04
.05
-.01
.02
.00
-.02
.00
5
10
15
-.05
5
D. Hours
10
15
5
E. Real wage
.08
10
15
F. Labor productivity
.10
.07
.07
.06
.08
.06
.05
.05
.04
.06
.04
.03
.03
.02
.04
.02
.01
.01
.00
.02
.00
-.01
.00
-.01
5
10
G. Relative price of investment goods
.05
-.02
5
15
10
15
H. Nominal interest rate
10
15
I. Inflation
.020
.014
.012
.016
.04
5
.010
.012
.008
.03
.008
.006
.02
.004
.004
.002
.01
.000
.00
.000
-.004
5
10
15
-.002
5
10
15
5
10
15
The model with consumer durables (our model)
The model without consumer durables
Figure 1. Impulse responses to a positive investment shock
Note: The horizontal axis is divided in quarters; the vertical axis represents percentage deviations from a steady state.
Investment is the sum of durable consumer investment and capital investment.
17