Discussion Paper No. 1046

STICKY-WAGE MODELSAND KNOWLEDGE CAPITAL

Kevin X. D. Huang
Munechika Katayama
Mototsugu Shintani
Takayuki Tsuruga

January 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Sticky-Wage Models and Knowledge Capital
Kevin X. D. Huangy,

Munechika Katayamaz,

Mototsugu Shintanix,

and Takayuki Tsuruga{
This draft: January 2019

Abstract
We present a sticky-wage model with two types of labors: while worker’s labor contributes to current
production, researcher’s work helps develop new ideas to add to …rm’s knowledge capital that enhances
its productivity for many periods. The long-lived e¤ect of knowledge capital on productivity is analogous
to the long-lasting e¤ect of consumer durables on utility in the sticky-price model of Barsky, House andKimball (2007). Our sticky-wage model generates the near monetary neutrality result similar to the resultin their sticky-price model, if returns to researchers’labor are low in developing knowledge capital. We
show, however, that the relative role of the pricing of the two production inputs analogous to consumption
durables and nondurables in their sticky-price model is completely reversed in our sticky-wage model.
JEL Classi…cation: E22, E24, E31, E52
Keywords: Intangible capital, wage rigidity, monetary policy

We thank Miles Kimball and seminar participants at University of Tokyo for helpful discussions and comments. Katayama,
Shintani and Tsuruga gratefully acknowledge the …nancial support of Grant-in-aid for Scienti…c Research (15H05729, 15H05728,
18K01520) and Osaka University International Joint Research Promotion Program.
y Vanderbilt University; e-mail: kevin.huang@vanderbilt.edu.
z Waseda University; e-mail: mkatayama@waseda.jp.
x The University of Tokyo; e-mail: shintani@econ.rcast.u-tokyo.ac.jp.
{ Osaka University; e-mail: tsuruga@iser.osaka-u.ac.jp.

11Introduction

The importance of durable goods for economic ‡uctuations has long been studied in the business cycle models(e.g., Baxter 1996; Weder 1998). In a more recent paper, Barsky, House and Kimball (2007, hereafter BHK)
emphasize the dominant role of the pricing of durable goods in generating monetary non-neutrality in a New
Keynesian sticky-price model with durable and nondurable goods. They also provide an intriguing exampleof the near neutrality of money when prices of durables are ‡exible and those of nondurables are sticky.
This paper provides another example of perverse predictions of the monetary neutrality in a New Keynesian sticky-wage models with two types of labors. As emphasized by Galí (2011), the wage-setting blockin the New Keynesian model plays a central role in determining the response of the economy to monetaryshocks. As such, the sticky-wage approach has been considered as a workhorse in the New Keynesian literature along with the sticky-price approach.1 In sticky-wage models, wages are set by households in a way
symmetric to how prices are set by …rms in sticky-price models (e.g., Erceg, Henderson and Levin 2000).
Likewise, we turn BHK’s New Keynesian model on its head, replacing their two types of goods with our two
types of labors.
We examine the roles of the pricing of the two types of labors analogous to consumption durables andnondurables in BHK’s sticky-price model. In BHK’s sticky-price model, …rms set prices for two types ofgoods: once produced, one is immediately consumed, while the other adds to its stock that yields utility overtime and wears out only gradually. In our sticky-wage model, households set wages for two types of labors:while worker’s labor immediately contributes to current production, researcher’s labor develops new ideas to
add to …rm’s knowledge capital that enhances its productivity for many periods and becomes obsolete only
gradually.2 The long-lived e¤ect of knowledge capital on productivity in our sticky-wage model is therefore
analogous to the long-lasting e¤ect of consumer durables on utility in BHK’s sticky-price model.
Our …ndings are summarized as follows. First, we study the con…guration of nominal rigidities when
1 Huang and Liu (2002), Huang, Liu and Phaneuf (2004), Christiano, Eichenbaum and Evans (2005) and many others …ndthat wage rigidity is at least as important as price rigidity for explaining the empirical regularities in the U.S. economy.
2 This structure di¤ers from Carlstrom and Fuerst (2010) who simply added sticky wages in the BHK model with a single
type of workers’labor.

2

wages of durable input (i.e., researcher’s labor) are ‡exible and wages of nondurable input (workers’labor)
are sticky. We …nd that our sticky-wage model under this con…guration does not predict the near neutralityof money. In fact, while this con…guration of nominal rigidities corresponds to the case of the near neutralityof money in BHK’s sticky-price model, the non-neutrality in our sticky-wage model is as signi…cant as whenboth researchers’ and workers’ wages are sticky. Second, we consider the other con…guration of nominal
rigidities under which researchers’wages are sticky and workers’wages are ‡exible. We …nd that the stickywage model under this con…guration does predict the near neutrality of money as in the BHK’s sticky-pricemodel, if returns to researchers’labor in developing new ideas are reasonably low. This result means thatthe relative role of the pricing of the two production inputs analogous to the two consumption goods in
BHK’s sticky-price model is completely reversed in our sticky-wage model: to generate a prediction similar
to that of the standard New Keynesian model, the model needs stickiness in wages of workers rather than of
researchers. In the model, the pricing of workers’labor plays a dominant role in determining the responseof aggregate output to monetary shocks.

2

A sticky-wage model with knowledge capital

The model features a continuum of households indexed by i 2 [0; 1], each consisting of a worker and a
researcher, and a continuum of …rms indexed by j 2 [0; 1], each producing a di¤erentiated good. The labor
services of the workers are di¤erentiated and imperfectly substitutable, and so are the labor services of theresearchers. There is a government conducting monetary policy.
At any date t, the objective of household i 2 [0; 1] is to maximize
Et

1
X

s t

[U (Cs (i))

VW (NW;s (i))

VR (NR;s (i))] ;

(1)

s=t

where Et is the conditional expectations operator,
2 (0; 1) is a subjective discount factor, Cs (i) =hRi"P =("P 1)
1
Cs (i; j)("P 1)="P dj
is the household i’s total consumption with "P > 1, and NW;s (i) and0
NR;s (i) are worker’s and researcher’s labors, respectively. The functions U , VW , and VR are strictly increas-

ing and twice continuously di¤erentiable, with concave U and convex VW and VR . The household’s budget

3

constraint in period t is
Z 1

Pt (j) Ct (i; j) dj

WW;t (i)NW;t (i) + WR;t (i)NR;t (i)Et [Dt;t+1 Bt+1 (i)] + Bt (i) +t (i);(2)0

where Pt (j) is good j’s price, WW;t (i) and WR;t (i) are nominal wages of worker i and researcher i, respectively,
Dt;t+1 is the stochastic discount factor from date t + 1 to t, Bt+1 (i) is a random quantity representing
household i’s holdings of one-period state-contingent nominal bonds in period t, andt (i) is household i’s

claim to …rms’pro…ts.
Utility maximization gives rise to household i’s demand, Ct (i; j) = [Pt (j)=Pt ] "P Ct (i), where Pt =R1
R1
[ 0 Pt (j)1 "P dj]1=(1 "P ) . The total demand for …rm j’s good is given by Ct (j) = 0 Ct (i; j)di = [Pt (j)=Pt ] "P Ct ,where Ct =R1
0

Ct (i)di.

Firm j produces its output, using workers’labor inputs and its knowledge capital according toCt (j) = F (NW;t (j); Kt (j));

where NW;t (j) =hR

1
NW;t (i; j)("W
0

1)="W

di

i"W =("W(3)1)

with "W > 1, and Kt (j) is …rm j’s knowledge capital,

respectively. The function F is homogenous of degree one, strictly increasing, concave, and continuouslydi¤erentiable. Knowledge capital satis…es the following law of motion:

Kt (j) = (1

)Kt 1 (j) + Xt (j) ;(4)where Xt (j) denotes …rm j’s new ideas produced by R&D investment. Knowledge capital becomes obsolete
at a rate

2 (0; 1). We consider a small positive value of

to ensure the stationarity of Kt (j).3

New ideas are developed using researchers’labor NR;t (j), together with a stock of aggregate knowledge
capital Kt 1 , according toXt (j) = G(NR;t (j); Kt 1 );where NR;t (j) =hRi"R =("R 1)
1("R 1)="R
N
(i;
j)
di
with "R > 1.
R;t
0

(5)

The function G is homogenous of degreeone, strictly increasing, concave, and continuously di¤erentiable. Knowledge capital is accumulated by
3 See, for example, Comin and Gertler (2006) for an introduction of the possibility that knowledge becomes obsolete.

4

individual …rms and the economy-wide knowledge stock in turn helps each …rm develop new ideas to add on
its own knowledge capital (i.e., knowledge spillover). Therefore, Kt (j) can also be thought of as a measure ofintangible assets.4 The speci…cation here is in the spirit of the seminal works by Romer (1990) and Grossman
and Helpman (1991), among others.
Cost minimization gives rise to …rm j’s demand for worker i and researcher i,

NW;t (i; j) =WW;t (i)WW;t"W

NW;t (j) and NR;t (i; j) =WR;t (i)WR;t"R

NR;t (j);R1
R1
where WW;t = [ 0 WW;t (i)1 "W di]1=(1 "W ) and WR;t = [ 0 WR;t (i)1 "R di]1=(1 "R ) . Because households areindi¤erent about working at di¤erent …rms, wages WW;t (i) and WR;t (i) are independent of j.
At any date t, …rm j chooses fNW;s (j); NR;s (j); Ks (j)gs t to maximize
Et

1X

Dt;s [Ps (j) Cs (j)WW;s NW;s (j)WR;s NR;s (j)] ;

(6)

s=t

subject to (3)–(5). Here Dt;s =t, for all s > t, with Dt;tQs t

=1 Dt+

1;t+denotes the s-period stochastic discount factor from s to

1. This pro…t maximization problem takes into account the solution to theembodied cost minimization problem.
The …rst-order conditions for NW;t (j), NR;t (j), and Kt (j) are given by
WW;tPt= FN;t ;

(7)

WR;t
;
GN;t

(8)

Qt

=

Qt

= Pt FK;t + (1) Et Dt;t+1 Qt+1 ;(9)where FN;t and FK;t are the marginal product of inputs adjusted by price markups to simplify the notations. In particular, FN;t
"P = ("P1). In (8), GN;t1
P[@F (NW;t ; Kt ) =@NW;t ], FK;t1
P[@F (NW;t ; Kt ) =@Kt ] whereP=@G (NR;t ; Kt 1 ) =@NR;t . Here, due to symmetry across …rms, we drop theindex j from the …rm j’s variables. The Lagrange multiplier for (4) is Qt , which is interpreted as the nom4 Examples include patents, copyrights, trademarks and trade names, blueprints or building designs, engineering drawings,

and organizational expenses, as de…ned in the Compustat database. While not a pure public good, knowledge capital modeled
here is only partially excludable or non-rival and represents a cost independent from the level of output (see Romer 1986, 1990;
Arrow 2000). In a similar vein, it may also be reinterpreted as a form of organizational capital in the spirit of Beaudry andDevereux (1995), in the sense that its accumulation is an alternative rather than a complement to production.

5

inal marginal bene…t of increasing Kt . In equilibrium, the nominal marginal bene…t Qt is equalized to thenominal marginal cost of producing new ideas Xt , which is given by WR;t =GN;t .
Households are monopolistic competitors in the labor markets for workers and researchers. They set theirwages Wh;t (i) where h represents either W or R. The total demand for household i’s labor is given by

Nh;t (i) =Z 1

Nh;t (i; j)dj =0

where Nh;t =R1
0

Wh;t (i)Wh;t"h

Nh;t ;

Nh;t (j)dj for h = W; R. Taking the labor demand functions as given, households set Wh;t (i)in a staggered fashion with a probability of wage adjustment 1

h , respectively.

If worker (researcher) i

gets the chance to reset its wage in period t, then it will choose the wage to satisfy

Wh;t (i) =

Et

"h
"h

1

P1

"h
s t 0
Vh ([Wh;t (i)=Wh;s ] "h Nh;s )Wh;s
Nh;s
h)
P1
;
"h
s
t
0
Et s=t ( h ) U (Cs (i))Wh;s Nh;s =Ps

s=t (h = W; R:(10)

Analogous to BHK’s sticky-price model in which households must use cash to purchase goods, oursticky-wage model assumes that …rms, instead of households, must hold money since they face a cashin-advance (CIA) constraint for the payment of labors.5 Therefore, money demand is introduced here via
Mt = WW;t NW;t + WR;t NR;t .6 The money supply Mts grows at a rate e t , Mts = e t Mts 1 , wheret is awhite-noise process.
Finally, de…ne real GDP as Yt :Yt = P Ct + QXt ;(11)

where P and Q are the steady-state values of Pt and Qt , respectively. The latter is interpreted as the imputed
value of new ideas in the steady state, since Qt is equal to the nominal marginal cost of producing Xt . Notethat Yt includes R&D investment as it re‡ects recent changes in the de…nition of GDP. The nominal GDP
is Pt Ct + Qt Xt and the GDP de‡ator is the nominal GDP divided by real GDP.
5 To conserve space, we did not explicitly introduce the CIA constraint into the …rm problem. In an appendix, which isavailable upon request from the authors, we assume that …rms must make wage payment in advance by money. That modelwith …rms’money demand yields identical …rst-order conditions described in the main text in this paper.
6 See also Schmitt-Grohé and Uribe (2006, 2007) who assume the CIA constraint on the wage bills of …rms.

6

3

Results

In this section, we describe parametrization and present results. We then provide the intuition for our results.

3.1

Parametrization

In simulating our model, we closely follow the baseline parametrization in BHK unless otherwise noted. The
subjective discount factor

is set to ensure that the annual risk-free rate equals 2 percent. The utility1function is parametrized as U (Ct (i)) = Ct (i)=h = 1, for h = W , R.

= (1

1+

), Vh (Nh;t (i)) = Nh;t (i)h= (1 +h ), where

The parameter "P is set to 11 as in BHK, while "W and "R are set to 5,

consistent with the previous studies on sticky-wage models (e.g., Erceg, Henderson and Levin 2000; Huang
and Liu 2002). When we assume wage stickiness in simulations, the average duration between wage changes
is set to four quarters.
1

The production function of consumption goods is speci…ed as F (NW;t (j); Kt (j)) = [NW;t (j)]Here,[Kt (j)] .

is set to 0.5, due to the lack of a broad consensus on the knowledge input share, but our results

are robust to the choices of this parameter value. We choose the depreciation rate of knowledge capitalto ensure that the steady-state GDP share of R&D investment equals 0.03, consistent with the U.S. data.7
The production function of new ideas is speci…ed as G (NR;t (j); Kt 1 ) = [NR;t (j)] Kt 1 , whereandmeasure the degree of returns to researchers’labor and the degree of knowledge spillovers, respectively. For
the latter parameter , we set= 0:5, again due to a lack of broad consensus on knowledge spillovers in

the production of new idea. In parameterizing the former parameter , however, we allow for= 1 and

0.5 and contrast the impulse responses of output under both cases. In the next section, we will examine themodel under high returns ( = 1) and low returns ( = 0:5) to researchers’labor in knowledge production
and discuss the roles of nominal wage rigidities on real e¤ects of monetary shocks.
7 In our model, the steady-state GDP share of R&D investment is QX= (P C + QX) written as a function of

it is given by

[1

(1

)]

1

=

n

P +

[1

(1

)]

1

o

.

7

. In particular,

3.2

Main …ndings

We …rst investigate the real e¤ect of a monetary shock in the sticky-wage model with high returns toresearchers’ labor (= 1) in producing new idea. Figure 1 plots impulse responses of output to a one

percent increase in the money supply.8 In the left panel, we show the impulse responses under= 1 for three

con…gurations of nominal wage rigidities. The dotted line represents output responses when both workers’
and researchers’wages are sticky (All wages sticky). The dashed line corresponds to output responses when
workers’wages are sticky but researchers’wages are ‡exible (Sticky workers’wages). In both con…gurations,the monetary non-neutrality is signi…cant and output rises above the steady state by 0.47 and 0.48 percent on
impact of the monetary shock, respectively. The impacts of the monetary shock on output take more than ayear for them to go back to the steady state. The solid line points to output responses when researchers’wages
are sticky but workers’wages are ‡exible (Sticky researchers’wages). In this con…guration of nominal wage
rigidities, the impact converges to zero somewhat more quickly than those in the other two con…gurations.
Nevertheless, the output response on impact is very close to those in the other con…gurations.
The results suggest that the sticky-wage model with two types of labors does not generate the near
neutrality of money even when prices of durable inputs are ‡exible. This is a sharp contrast to BHK’s
sticky-price model. The model also predicts the substantial non-neutrality when prices of nondurable inputs
are ‡exible. In fact, when

= 1, monetary non-neutrality is substantial in any con…gurations of nominal

rigidities.
However, when returns to researchers’labor are reasonably low (i.e.,= 0:5), the real e¤ect of money canbe dramatically weak. In the right panel of Figure 1, we again show the impulse responses of output but inthe economy with

= 0:5. If researchers’wages are sticky and workers’wages are ‡exible, output increases

by a small amount (e.g., only 0.03 percent on impact), much smaller than the other two con…gurations of
nominal rigidities.
It should be emphasized that the near neutrality of money in our sticky-wage model is generated under
8 In all …gures of this paper, simulations are based on a period of 100th of a year, as in BHK. For convenience, quarters aremarked on the horizontal axes.

8the exactly opposite con…guration of nominal rigidities to that in BHK’s sticky-price model, regardless ofthe similarity between our sticky-wage and BHK’s sticky-price models. Monetary neutrality in BHK’s model
occurs with ‡exible durable prices even if nondurable prices are sticky, whereas the neutrality in our model
occurs with ‡exible nondurable input prices even if durable input prices are sticky.
We can examine whether the result in our sticky-wage model is numerically close to the monetary neutrality result in BHK’s sticky-price model. Figure 2 indicates that, in comparisons, the neutrality result ismore striking in our sticky-wage model than in BHK’s sticky-price model. For the sake of compatibility,
we set the wage rigidity such that impulse responses of output from the standard sticky-wage model match
those from the standard sticky-price model.9 Furthermore, physical capital is abstracted from BHK’s stickyprice model. The dashed lines display the impulse responses from the standard sticky-price and sticky-wagemodels (which by construction are identical across the two standard models). The solid line in the left panel
reproduces BHK’s simulation results under the assumption that prices of durables are ‡exible and prices
of nondurables are sticky. The solid line in the right panel presents the impulse response of output in our
sticky-wage model, but with sticky researchers’wages and ‡exible workers’wages. The …gure indicates that
output increases by about 0.03 percent in our model, but it rises by about 0.06 percent in BHK’s model.
The contrast between our sticky-wage model and BHK’s sticky-price model goes beyond the above two
instructive cases of nominal rigidities. To generate monetary non-neutrality, BHK’s sticky-price model
depends on stickiness in prices of durables rather than of nondurables. However, our sticky-wage model
depends on stickiness in wages of workers rather than of researchers. As the dashed and dotted lines in
the right panel of Figure 1 illustrate, once workers’wages become sticky, the monetary non-neutrality is as
signi…cant as when all wages are sticky. The similarity of output responses even suggests the irrelevanceof pricing of durable input in our sticky-wage model. As we will show below, the irrelevance of pricing of
durable input also holds broadly, unless returns to researchers’labor are high.10
9 The standard models are the sticky-price model with only nondurables and the sticky-wage model with only workers’labor.
We match the slopes of the New Keynesian Phillips curves in the two standard models to facilitate the comparison.
1 0 The irrelevance of pricing of durable input is fairly robust to changes in the model, including incorporation of sticky prices
into our model. The additional checks other than reported below are available upon request from the authors.

9

3.3

Analytical discussion

To understand the mechanism behind the monetary neutrality, we focus on dynamics of knowledge capital
Kt , workers’labor NW;t , and researchers’labor NR;t . This is because GDP is given by Yt = P Ct + QXt =P F (NW;t ; Kt )+QG (NR;t ; Kt 1 ). To help us understand the mechanism, Figure 3 plots the impulse responses
of a variety of variables to a monetary injection under the two con…gurations of nominal rigidities.
First, let us look at knowledge capital Kt . In a similar spirit as discussed in BHK, if the depreciation rate
is low, a ‡ow-stock ratio is so low that even large changes in the production of ideas have small e¤ects on
the total stock of knowledge capital. Therefore, as shown in the …rst row of Figure 3, a monetary injection
does not produce a sizable increment in the stock of knowledge capital, independent of con…gurations of
nominal rigidities. Consequently, it is helpful to treat knowledge capital as roughly constant: Kt ' K.
We then look at workers’ labor NW;t . In equilibrium, workers’ wage markups

W;tare equal to thegap between the marginal product of workers’ labor FN;t in (7) and its marginal rate of substitutions for0(NW;t ) =U 0 (Ct ). Therefore, we haveconsumption VW

W;t

V 0 (NW;t )= FN (NW;t ; Kt ) ;U 0 [F (NW;t ; Kt )](12)

where we used the market clearing condition for the composite good, Ct = F (NW;t ; Kt ).
Equation (12) suggests that NW;t e¤ectively has a one-to-one relationship to wage markups for workers
if Kt is treated as constant. Thus, NW;t does not respond to a monetary injection if workers’ wages are‡exible (i.e.,

W;t is constant for all t).

Similarly, consumption exhibits extremely small movement due toCt = F (NW;t ; Kt ) along with the muted responses of Kt and NW;t to the monetary shock.11 The responses
of NW;t and Ct are recon…rmed by the solid lines in the second and third rows of Figure 3.
Finally, look at researchers’ labor NR;t . Again, researchers’ wage markups

R;tare equalized to thegap between marginal product of researchers’ labor (Qt =Pt ) GN;t (in terms of consumption goods) and its
1 1 The validity of our analysis and basic conclusion does not depend on how we parametrize preferences and technology. In
0 > 0, V 00
fact, it is su¢ cient to assume U 0 > 0, U 00 0, VW
0, FN > 0, FK
0, FN K0, and FN N
0.
W

10

marginal rate of substitutions for consumption VR0 (NR;t ) =U 0 (Ct ). This relationship can be expressed as
R;t V

where

t

0

(NR;t ) =t GN (NR;t ; K) ;(13)

Qt Ut0 =Pt . In the equation, we also set Kt ' K.

Equation (13) implies that NR;t e¤ectively has a one-to-one relationship to wage markups for researchers
becauset is near constant under ‡exible workers’wages. To understand the near constancy offact that Dt;t+1 =t , use the[U 0 (Ct+1 ) =U 0 (Ct )] (Pt =Pt+1 ) and rewrite (9) ast =U

which indicates thatt0(Ct ) FK (NW;t ; K) +(1

) Et t+1 ;(14)

is stable over time since NW;t and Ct are both near constant under the ‡exible

workers’wages. Therefore, NR;t has a one-to-one relationship with

R;t when workers’wages are ‡exible. If

researchers’wages are sticky, a monetary injection moves researchers’wage markups countercyclically andthus NR;t increases. Because Kt 1 is slowly moving to the monetary shock and Xt = G (NR;t ; Kt 1 ), Xt isalso mainly driven by

R;t .

Returns to researchers’ labor critically in‡uences the sensitivity of Xt and Yt to a monetary shock. In
particular, when returns to researchers’ labor is high (i.e.,= 1), the incentive to employ researchers isextremely strong. In fact, an initial increase in Xt is 15.26 percent to a one percent increase in the money
supply. As a result, ‡uctuations in researchers’labor matter signi…cantly in business cycle frequencies and
GDP increases by 0.47 percent, despite a small steady-state GDP share of R&D investment. When returns
to researchers’ labor are reasonably low (i.e.,= 0:5), by contrast, the incentive to hire researchers islimited. An increase in Xt on impact of the monetary injection is now 0.96 percent, which is transmitted to
an increase in Yt by 0.03 percent. Consequently, together with essentially constant consumption, the output
response is near zero in response to an unexpected increase in the money supply.
The relative wage response is helpful to understand the sensitivity of researchers’labor. Based on (7),(8), and the de…nition0t = Qt Ut =Pt , the relative wage WR;t =WW;t equals the ratio of the marginal productof researchers’labor (Qt =Pt ) GN;t =0
t GN;t =Ut to that of workers’labor FN;t . Then, since responses of11t,Kt 1 , NW;t , and Ct are all muted when workers’wages are ‡exible, we sett ', Kt 1 ' K, NW;t ' NW ,and Ct ' C and obtain
WR;t1
GN (NR;t ; K)
= 0
:
WW;t
U (C) FN (NW ; K)

(15)

Therefore, NR;t moves together with the relative wages. When workers’wages are ‡exible and researchers’
wages are sticky, the adjustment of researchers’wages is slower than that of workers’wages, so a monetary
injection decreases the relative wage (as shown in the solid line in the bottom right corner of Figure 3).
Indeed, using our speci…cation of the function G (NR;t ; Kt 1 ), we can approximate the above equation by^R;t '
N

1
1

^ R;t
W

^ W;t ;
W

where a variable with a hat represents the log-deviation of that variable from its steady-state value. The
researchers’ labor is less sensitive to the relative wage as

decreases (i.e., as the marginal product of

researchers’ labor becomes more diminishing). This sensitivity of researchers’ labor to the relative wage
critically in‡uences the signi…cance of the monetary non-neutrality.
The mechanism of our monetary non-neutrality result under the opposite con…guration, that is to say, withsticky workers’wages and ‡exible researchers’wages, is more straightforward to explain. Under sticky workers’wages, the wage markups decrease in response to a monetary injection. Due to the one-to-one relationshipbetween NW;t andW;t , NW;t increases.

The increase in NW;t leads to an increase in Ct (= F (NW;t ; Kt )).

Consumption is dominant in the increase in GDP Yt = P Ct +QXt . See the dashed lines for consumption and
R&D in Figure 3.12 Given the dominance of Ct , the monetary non-neutrality in this opposite con…guration
hinges upon whether workers’ wages are ‡exible or sticky, but has little to do with rigidity in researchers’
wages and with the degree of returns to researchers’labor.
There is a …nal remark on our neutrality result. In BHK’s sticky-price model, the negative comovement
between labor inputs for producing nondurable and durable goods is critical in producing their neutrality
result. In our model, the neutrality result does not hinge upon any speci…c comovement patterns between
1 2 The e¤ect of X on Y is negligible as long as
is close to unity. In particular, under our assumption of the functionalttform of U and F , the …rst term of the right-hand side of (14) becomes Ct1 =K ' =K, implying the near constancy of t .
Together with the constancy of R;t in (13) in this con…guration, NR;t is near constant and thus Xt is small in response.

12

workers’labor and knowledge capital or researchers’labor. To see this more clearly, we invoke the functional
forms for U , VW , and F postulated in our numerical simulations. Under ‡exible workers’ wages in which
W;t is constant for all t, the log-linearized equation of (12) can be written as

[ +

W +

(1

^W;t =
)] N

(1

^ t:
)K

(16)

^W;t is negative whereas that on K
^ t can be negative, zero, or positive. By virtue
Here, the coe¢ cient on Nof (16), the correlation of workers’ labor with knowledge capital (thus with researchers’ labor as well) is
negative, zero, or positive, if

is greater than, equal to, or smaller than unity. However, as we will see in thenext section on robustness, the sign of the correlation does not exert any quantitatively signi…cant impact
on the real e¤ect of money.

4

Robustness checks

We here check the robustness of the results from our sticky-wage model with

= 0:5 to alternative valuesof parameters. The parameters we consider are the degrees of relative risk aversion, knowledge capital
contribution in production, and researchers’wage stickiness. Basically, comparisons in …gures of this section
are made for two con…gurations of nominal rigidities: sticky researchers’wages and ‡exible workers’wages(left panel) versus sticky workers’wages and ‡exible researchers’wages (right panel).13

Degree of relative risk aversion We …rst show that our results are robust to the degree of relativerisk aversion, which is set to unity in the baseline calibration. However, a lower value of(i.e., moreelastic intertemporal substitution in consumption) can weaken households’consumption smoothing motive.
This may result in greater ‡uctuations in aggregate demand following monetary shocks and in stronger reale¤ects of monetary shocks.
Figure 4 plots the impulse responses of output to a one percent increase in the money supply for a wide
range of values of
when

under the two con…gurations of nominal rigidities. As the right panel of the …gure shows,declines from 5 to 1, and then to 0:01, the monetary injection has a stronger real e¤ect. However,

1 3 We con…rm that output responses when all wages are sticky are similar to those when only workers’ wages are sticky.

Therefore, we will present the results for the two con…gurations of nominal rigidities.

13the three impulse response functions in the left panel shows that money remains nearly neutral for all thevalues of^W;t is almost invariant
when workers’wages are ‡exible. We can also recon…rm from (16) that N^ t ' 0) and thus money remains nearly
to the monetary shock because knowledge capital is slowly moving (Kneutral, given the relatively small response of Xt .

Degree of knowledge capital contribution in production We next show that our results are robust
to knowledge input share in production, which is set to 0.5 in the baseline calibration. The robustness

check is important because of a lack of consensus on the value of .
Figure 5 displays the impulse responses of output for a wide range of values ofunder the two alternative

con…gurations of nominal rigidities. As the left panel of the …gure shows, the impulse response of output
suggests that money remains nearly neutral with sticky researchers’ wage, even when the importance ofknowledge input in production increases from 0.05 to 0.70. On the other hand, when workers’ wages are
sticky, output responses substantially di¤er across di¤erent values of(shown in the right panel). Therefore,

we conclude that, at least for the near neutrality of money when researchers’wages are sticky, our model’s
predictions are robust to .14

Degree of researchers’ wage stickiness Recall that in the baseline calibration we set the degreeof wage stickiness to ensure that the average duration between wage changes is four quarters. Figure 6
plots output responses with varying average durations of researchers’ wages, from one to eight quarters,
under ‡exible (left panel) and sticky (right panel) workers’wages. When workers’wages are ‡exible, output
responses remain uniformly small so money remaining nearly neutral. When workers’wages are sticky (withan average duration of four quarters), signi…cant monetary non-neutrality is present. In both panels, increases
in researchers’wage stickiness only marginally strengthen the real e¤ects of monetary shocks, recon…rming theimportance of workers’wage stickiness (and the unimportance of researchers’wage stickiness) in determining
output responses.
1 4 We also con…rmed the robustness of the near neutrality of money to the degree of knowledge spillovers
2 [0; 1), since ourcalibration of = 0:5 does not necessarily rely on solid empirical …ndings. Changes in have only a tiny e¤ect on both Kt 1
and NR;t so that output responses remain essentially unaltered under the two con…gurations of nominal rigidities.

14

5

Concluding remarks

In an in‡uential paper, Barsky, House and Kimball (2007) show that the pricing of durable goods plays
a dominant role, whereas the pricing of nondurable goods is immaterial, in determining the real e¤ects ofmonetary shocks in a New Keynesian sticky-price model. Speci…cally, money can be neutral or have reale¤ects depending on whether prices of durable goods are sticky or ‡exible, but independent of rigidity in
nondurable goods’ prices. After concluding that “durables are the most important element in sticky-pricemodels,” they urge that “researchers must devote more e¤ort to empirical investigation of the pricing ofthese goods.”
The recent development in the New Keynesian literature has assigned a central role to the sticky-wage
approach, along with the sticky-price approach. Against the background, we turned BHK’s sticky-pricemodel on its head, replacing their two types of goods with our two types of labors. We have used a New
Keynesian sticky-wage model to study how the pricing of the two types of labors may a¤ect the real e¤ects
of monetary shocks.
We showed that the model’s prediction critically depends on stickiness in wages of workers (prices of
nondurable input) rather than wages of researchers (prices of durable inputs) in determining the real e¤ects
of monetary shocks.
Under low returns to researchers’ labor in producing new ideas, the relative role of the pricing of thetwo production inputs analogous to consumption durables and nondurables in BHK’s sticky-price model
is completely reversed in our sticky-wage model: whether money is neutral or not hinges upon rigidity orlack thereof in workers’wages rather than in researchers’wages. At the very general level, it is the pricing
of workers’ not researchers’ labor that plays a dominant role in shaping aggregate dynamics following amonetary shock. We demonstrated that this conclusion holds quite generally regardless of other details of
our model. Our results in this paper suggest that future research on the heterogeneity in wage stickinessacross labors should be a priority at least as high as that on the heterogeneity in price stickiness acrossgoods.

15

References
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Barsky, Robert B., Christopher L. House, and Miles S. Kimball. 2007. “Sticky-Price Models and Durable
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Baxter, Marianne. 1996. “Are Consumer Durables Important for Business Cycles?” The Review of Economics
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Galı́, Jordi. 2011. “The Return of the Wage Phillips Curve.” Journal of the European Economic Association
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Grossman, Gene M. and Elhanan Helpman. 1991. “Quality Ladders in the Theory of Growth.” The Review
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Huang, Kevin X.D., Zheng Liu, and Louis Phaneuf. 2004. “Why Does the Cyclical Behavior of Real Wages
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Romer, Paul M. 1986.
94 (5):1002–1037.

“Increasing Returns and Long-Run Growth.”

Journal of Political Economy———. 1990. “Endogenous Technological Change.” Journal of Political Economy 98 (5, Part 2):S71–S102.
Schmitt-Grohé, Stephanie and Martı́n Uribe. 2006. “Optimal Fiscal and Monetary Policy in a Medium-Scale
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Kenneth Rogoff. National Bureau of Economic Research, Inc, 383–462.
———. 2007. “Optimal Inflation Stabilization in a Medium-Scale Macroeconomic Model.” In Monetary
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16

High returns to researchers' labor (

0.6

= 1)

Low returns to researchers' labor (

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

= 0.5)All wages sticky
Sticky workers' wages
Sticky researchers' wages

0
0

1

2

3

4

6

8

12

0

1

2

3

4

6

8

12

Figure 1: Responses of GDP to the Monetary ShockNotes: The left panel shows responses of GDP to a one percent increase in the money supply when returns to researchers’
labor are high in knowledge production (φ = 1.0). The right panel presents those when returns to researchers’ labor are low(φ = 0.5). In each panel, dotted lines represent responses of GDP when both workers’ and researchers’ wages are sticky; dashedlines correspond those when workers’ wages are sticky, but researchers’ wages are flexible; solid lines are those when workers’
wages are flexible, but researchers’ wages are sticky. Vertical axis measures percentage deviations from the steady state. Timein quarters is on horizontal axis.

17

Sticky-price models

1.2

Sticky-wage models

1.2

Sticky nondurables prices
Standard sticky-price model

Sticky researchers' wages
Standard sticky-wage model

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0
0

1

2

3

4

6

8

12

0

1

2

3

4

6

8

12

Figure 2: The Near Neutrality of Money in Sticky-price and Sticky-wage Models
Notes: The solid line on the left panel corresponds to the near money neutrality result in BHK’s model with sticky nondurable
and flexible durable prices. The solid line on the right panel shows responses of GDP to a one percent increase in the money
supply in our model with sticky researchers’ wages and flexible workers’ wages. The dashed lines represent the responses of
GDP in the standard sticky-price model (left panel) and those in the standard sticky-wage model (right panel). Vertical axes
measure percentage deviations from the steady state. Time in quarters is on horizontal axes.

18

GDP

0.6

Knowledge

0.02

Sticky researchers' wages
Sticky workers' wages

0.4

0.01
0.2
0

0
0

1

2

3

4

6

8

12

0

Consumption

0.6

1

2

3

4

6

8

12

8

12

R&D

1

0.4
0.5
0.2
0

0
0

1

2

3

4

6

8

12

0

Workers' hours worked

1

1

3

4

6

Researchers' hours worked

2

0.5

2

1

0

0
0

1

2

3

4

6

8

12

0

1

2

Workers' wage

3

4

6

8

12

Researchers' wage

1

1

0.5

0.5

0

0
0

1

2

3

4

6

8

12

0

1

2

3

Consumption-good price

4

6

8

12

Relative wage
1

1

0.5

0

0

-1
0

1

2

3

4

6

8

12

0

Quarters

1

2

3

4

6

8

12

Quarters

Figure 3: Responses to a One Percent Increase in the Money Supply
Notes: Each panel shows responses of corresponding variables in our sticky-wage model to a one percent increase in the money
supply under two alternative configurations on wage stickiness. Solid lines represent the case when workers’ wages are flexible
and researchers’ wages are sticky. Dashed lines show responses with sticky workers’ wages and flexible researchers’ wages.
Vertical axes measure percentage deviations from the steady state. Time in quarters is on horizontal axes.

19

Sticky researchers' wage

0.6

Sticky workers' wage

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

= 0.01
=1
=5

0
0

1

2

3

4

6

8

12

0

1

2

3

4

6

8

12

Figure 4: Responses of GDP with Different Degrees of Relative Risk Aversion
Notes: The left panel shows responses of GDP to a one percent increase in the money supply when workers’ wages are flexible,
but researchers’ wages are sticky. The right panel displays those when workers’ wages are sticky but researchers’ wages are
flexible. Lines in each panel are for different values of σ. Vertical axes measure percentage deviations from the steady state.
Time in quarters is on horizontal axes.

20

Sticky researchers' wage

1

Sticky workers' wage

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

= 0.05
= 0.3
= 0.7

0
0

1

2

3

4

6

8

12

0

1

2

3

4

6

8

12

Figure 5: Responses of GDP with Different Degrees of Knowledge Capital Contribution
Notes: The left panel shows responses of GDP to a one percent increase in the money supply when workers’ wages are flexible,
but researchers’ wages are sticky. The right panel displays those when workers’ wages are sticky but researchers’ wages are
flexible. Lines in each panel are for different values of α. Vertical axes measure percentage deviations from the steady state.
Time in quarters is on horizontal axes.

21Flexible workers' wage /Sticky researchers' wage

0.6

Sticky workers' wage /Sticky researchers' wage

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

1 quarter
4 quarters
8 quarters

0
0

1

2

3

4

6

8

12

0

1

2

3

4

6

8

12

Figure 6: Responses of GDP with Different Degrees of Researchers’ Wage StickinessNotes: The left panel shows responses of GDP to a one percent increase in the money supply when workers’ wages are flexible,
but the stickiness of researchers’ wages varies across lines. The right panel displays those when the average duration for workers’
wage changes is four quarters. Solid, dashed, and dotted lines in each panel represent the average duration between researchers’
wage changes being one quarter, four quarters, and eight quarters, respectively. Vertical axes measure percentage deviations
from the steady state. Time in quarters is on horizontal axes.

22