ISSN (Print) 0473-453X

Discussion Paper No. 1159

ISSN (Online) 2435-0982

STICKY WAGES IN A WORLD OF IDEAS
Kevin X. D. Huang
Munechika Katayama
Mototsugu Shintani
Takayuki Tsuruga

February 2022

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

Sticky Wages in a World of Ideas∗
Kevin X. D. Huang† Munechika Katayama‡ Mototsugu Shintani§ Takayuki Tsuruga¶
This draft: February 2, 2022

Abstract
The search for new ideas by profit-seeking firms and knowledge spillovers are well-known and fundamental sources of modern economic growth. This paper examines the implications of idea production
and knowledge capital for monetary business cycles. We construct a sticky-wage model where workers
produce goods based on firm-specific knowledge capital and researchers develop new ideas aided by the
economywide stock of knowledge. As a quantitatively small group in the economy, researchers are inconsequential for the real effects of monetary shocks when the returns to research are low. However, thisintuitive conclusion can be overturned when the returns to research are high. In this situation, monetaryshocks can have significant real effects as long as wages are sticky for researchers, even if wages are
perfectly flexible for workers, who are quantitatively dominant in the economy.
JEL Classification: E22, E24, E31, E52
Keywords: Ideas, nonrivalry, knowledge capital, sticky wages, monetary neutrality∗ We are grateful to Chad Jones and Miles Kimball for helpful discussions on the subject and to seminar participantsat the University of Tokyo for useful comments and suggestions. Katayama, Shintani, and Tsuruga acknowledge the financial
support from Grant-in-Aid for Scientific Research (17H02510, 18K01520, 18K01684, 20H01482, 20H05631, 20H05633, 21H00700,
and 21H04397), the Joint Usage and Research Centers at Osaka University and Kyoto University, and the Osaka University
International Joint Research Promotion Program.
† Vanderbilt University; e-mail: kevin.huang@vanderbilt.edu.
‡ Waseda University; e-mail: mkatayama@waseda.jp.
§ Correspondence to: Faculty of Economics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, JAPAN;
e-mail: shintani@e.u-tokyo.ac.jp.
¶ Osaka University; e-mail: tsuruga@iser.osaka-u.ac.jp.

11Introduction

There is long debate concerning the impact of the search for new ideas on economic growth and businesscycles. The economic growth literature has long recognized that how new ideas are produced lies at theheart of economic growth. For example, Jones (2019a) argues that the shape of the idea production function
is “an intriguing subject of study” in growth models including research and development (R&D) activity.
In contrast, the business cycle literature has revealed that R&D activity undertaken in the search for new
ideas is procyclical.1 Such procyclicality may influence fluctuations in output and employment, suggesting
the importance of idea production on output and employment in business cycles.
Unfortunately, the business cycle literature has paid little attention to how ideas are produced in considering output and employment fluctuations. For example, sticky wages are one of the standard explanations
for understanding the responses of output and employment to monetary shocks in New Keynesian models.2
However, idea production is not commonly included in standard sticky-wage models. One reason for thisis that researchers is quantitatively small in terms of economic activity. Table 1 shows the shares of R&D
expenditure to GDP in selected developed countries. As shown, the R&D expenditure share to GDP is onlyaround 1 to 3 percent in these countries. R&D by researchers is also small in terms of the total labor costof researchers relative to workers. In evidence, the US labor cost ratio of researchers to workers is only
about 0.08 according to the 2018 Occupational Employment Statistics (OES). Thus, while R&D activity ispotentially important for business cycles, it is often neglected in the standard business cycle literature.
To address this deficiency, this paper explores the importance of knowledge and ideas in a New Keynesian
sticky-wage model. We explicitly incorporate idea production and knowledge capital into a sticky-wage modelusing two types of labor, namely, workers and researchers. Workers produce traditional goods based on afirm’s knowledge capital. Researchers produce new ideas to add to the firm’s knowledge capital with theaid of the preexisting stock of economywide knowledge. These new ideas then enhance the economywide
knowledge stock, improve the productivity of all researchers, and incentivize researchers to search for other
1 Examples include Comin and Gertler (2006), Barlevy (2007), Ouyang (2011) and Mand (2019)
2 See Huang and Liu (2002), Huang et al. (2004), Christiano et al. (2005), and Galı́ (2011) among others.

2

Table 1: Share of R&D expenditure to GDP
Year
2000
2010
2019

Canada
1.86
1.83
1.59

France
2.09
2.18
2.20

Germany
2.41
2.73
3.19

Italy
1.00
1.22
1.47

Japan
2.91
3.14
3.20

UK
1.62
1.64
1.76

US
2.63
2.74
3.07

Source: OECD, available at https://data.oecd.org/rd/gross-domestic-spending-onr-d.htm. Numbers are percentages.

new ideas.
We investigate whether the idea production incorporated into a sticky-wage model matters for the realeffects of monetary shocks. Our calibrated parameters for knowledge capital replicate the R&D expenditure
share to GDP and labor cost ratio in the US economy. Using this calibration, we simulate the sticky-wagemodel to evaluate the real effects of monetary shocks.
We find that the shape of the idea production function can be an important factor in generating thereal effects of monetary shocks. When the returns to research are low in idea production, the quantitative
impact of researchers on the real effect of monetary shocks is small. To obtain the substantial real effect ofa monetary shock, the wages of workers rather than researchers must be sticky. However, when researcher
wages are sticky and worker wages are flexible, money is almost neutral. Indeed, the model dynamics are
effectively the same as the model without researchers. Thus, when the returns to research are low, ideas andknowledge do not have a substantial impact on the nonneutrality of money.
In contrast, when the returns to research are high, monetary shocks exert significant real effects, regardlessof the source of nominal wage rigidity. Remarkably, the real effect of money is substantial, even when worker
wages are flexible. In our model, the contribution of researchers to the economy is quantitatively small interms of the R&D expenditure share to GDP and the labor cost ratio of researchers to workers. Nevertheless,
we obtain the substantial real effect of money provided that researcher wages are sticky. The real effect iscomparable to that when there are nominal rigidities in the wages of workers, who are quantitatively dominant
in the economy.
Our findings shed new light on the conventional argument by Jones (2019a) from the viewpoint of3

monetary business cycle models. In the idea production function, two parameters determine how ideas areproduced: the standing-on-shoulders effect and the degree of returns to research. The growth literatureemphasizes the standing-on-shoulders effect as a crucial factor in understanding economic growth (Jones,
2005). By contrast, our sticky-wage model highlights the degree of returns to research, rather than thestanding-on-shoulders effect, as a crucial factor for the monetary transmission mechanism. We also find thatthe degree of returns to research is important for both real and nominal economic variables.
The rest of the paper is organized as follows. Section 2 describes our sticky-wage model incorporating
firm-specific knowledge capital generated by idea production of researchers. Section 3 presents our main
results. Section 4 conducts sensitivity analysis concerning some of the important parameters. Finally,
Section 5 concludes.

2

The model

The model features a continuum of households indexed by i ∈ [0, 1], each consisting of a worker and a
researcher, and a continuum of firms indexed by j ∈ [0, 1], each producing a differentiated good. The labor
services of workers are differentiated and imperfectly substitutable, as are those of researchers. A government
conducts monetary policy.

2.1

Households

At date t, the objective of household i ∈ [0, 1] is to maximize

Et

∞
X

β s−t [U (Cs (i)) − VW (NW,s (i)) − VR (NR,s (i))] ,

(1)

s=t

where Et denotes the conditional expectations operator, β ∈ (0, 1) is a subjective discount factor, Cs (i) =hR
iεP /(εP −1)
1(εP −1)/εP
C
(i,
j)
dj
is the household’s consumption basket, with εP > 1 and Cs (i, j) represents0
ing household i’s demand for goods j. Also, NW,s (i) and NR,s (i) are the household’s labor supply of a worker
and a researcher, respectively. The functions U , VW , and VR are strictly increasing and twice-continuously

4differentiable, with concave U and convex VW and VR . The household’s budget 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) + Πt (i),(2)

0where Pt (j) denotes good j’s price, WW,t (i) and WR,t (i) are the 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 the household i’s holdingsof one-period state-contingent nominal bonds, and Πt (i) is the household’s claim to firm profits.
Utility maximization gives rise to household i’s demand for goods j,Pt (j)Ct (i, j) =Pt

−εP
Ct (i),

R1
where Pt = [ 0 Pt (j)1−εP dj]1/(1−εP ) . The total demand for good j from all households in the economy isthen given by Ct (j) =R1
0

Ct (i, j)di = [Pt (j)/Pt ]−εP Ct , where Ct =R1
0

Ct (i)di.

Households are monopolistic competitors in labor markets, where they set wages for their worker andresearcher. The total demands for household i’s labor are given by
Z 1

Wh,t (i)Nh,t (i) =Nh,t (i, j)dj =
Wh,t
0

−εh
Nh,t ,(3)

where Nh,t (i, j) is the demand for household i’s type-h labor from firm j, as given below, and Nh,t =R10

Nh,t (j)dj, for h = W, R. Taking the corresponding labor demand schedules as given, households set

wages in a randomly staggered fashion à la Calvo (1983), with constant hazard rates θW and θR whenunable to adjust wages for their workers and researchers, respectively.3 If worker (researcher) i receives theopportunity to reset its wage in period t, then the wage will be chosen to satisfy

εh EtWh,t (i) =εh − 1

P∞

εh
s−t 0
Vh ([Wh,t (i)/Wh,s ]−εh Nh,s )Wh,sNh,ss=t (βθh )
P∞
,
εh
s−t
0
Et s=t (βθh ) U (Cs (i))Wh,s Nh,s /Ps

h = W, R.

(4)

Note that households are price takers in goods and bond markets. Irrespective of whether they can set wages
3 It is worth noting that we have fixed the measures of workers and researchers and do not allow ones to become the others as

responses to shocks to the economy. In this sense, as in typical business cycle studies, we abstract from modeling occupational
choice.

5

in labor markets, they can always choose their goods consumption and bond holdings at any given date.

2.2

Firms

A firm j hires two types of labor services from households indexed by i ∈ [0, 1]: namely, workers NW,t (i, j)
and researchers NR,t (i, j). The former contributes to the production of the differentiated output. The lattercreates and accumulates new ideas as firm-specific knowledge. These labor services employed by firm j are
aggregated by the following constant elasticity of substitution function. For h = W, R,
Z 1
Nh,t (j) =εh /(εh −1)
Nh,t (i, j)(εh −1)/εh di,0where εh > 1, and the wage index for labor is given by
Z 1
Wh,t =

Wh,t (i)1−εh di1/(1−εh ).

0Note that workers and researchers from household i are indifferent to working in different firms. Thus,
Wh,t (i) is independent of j.
Cost minimization gives rise to firm j’s demand for worker i:WW,t (i)NW,t (i, j) =
WW,t

−εW
NW,t (j),and for researcher iWR,t (i)NR,t (i, j) =
WR,t

−εR
NR,t (j).

The symmetry across firms implies that the total demand for household i’s labor Nh,t (i) is
Z 1Wh,t (i)Nh,t (i) =Nh,t (i, j)dj =
Wh,t
0

−εh
Nh,t ,

which is consistent with (3). While wage takers in the labor markets, firms are monopolistic competitors
in the goods market, where they set prices for their products at a markup µP = εP / (εP − 1) over their6marginal costs of production.
Workers hired by firm j, {NW,t (i, j)}i∈[0,1] , produce goods j, Yt (j), based on firm j’s knowledge capital
Kt (j) according to the following goods production function:4

Yt (j) = F (NW,t (j), Kt (j)).

(5)

The function F is strictly increasing in both NW,t (j) and Kt (j).
Knowledge can be forgotten or become obsolete over time but can also be enhanced by additional new
ideas. This notion is embedded in the following law of motion for firm j’s knowledge capital:

Kt (j) = (1 − δ)Kt−1 (j) + Xt (j) ,(6)where δ ∈ (0, 1) is the rate at which existing knowledge is forgotten or becomes obsolete over time.5
New ideas Xt (j) are produced by the labor inputs of researchers for firm j, {NR,t (i, j)}i∈[0,1] , with theaid of preexisting economywide knowledge Kt−1 which itself is derived from ideas developed in the past by
all researchers in the economy.6 Thus, the idea production function isXt (j) = G(NR,t (j), Kt−1 ),(7)

where the function G is strictly increasing in NR,t (j) but not necessarily increasing in Kt−1 . The dependenceof Xt (j) on Kt−1 reflects that knowledge becomes nonexcludable while it is nonrival. Individual firms andresearchers are so small that they take the time path of economywide knowledge as given when searchingfor new ideas. This modeling of ideas and knowledge capital follows the approach taken in the endogenous
growth literature emphasizing the nonrivalry of ideas.7 In (7), we assume a one-period delay until knowledge
4 We abstract away physical and human capital from the goods production function to make the key mechanism in our model
as transparent as possible. Including these forms of capital would not alter the main results of this paper or add new insights
in a substantive way.
5 See, for example, Comin and Gertler (2006) for an introduction of the possibility that knowledge can become obsolete overtime and Jones (2019a) for a reflection on this possibility.
6 Unlike workers, the labor inputs of researchers have a long-lasting impact on the economywide knowledge stock. This
durability in production technology resembles the role of durable consumption goods considered by Barsky et al. (2007).
7 See the original contribution of Romer (1990) and many subsequent works. Jones (2019a) provides a comprehensive reviewof this defining feature of an idea production function, which is crucial for understanding economic growth.

7becomes nonexcludable, but we can easily relax this assumption. For now, we leave the functional form ofG(NR,t (j), Kt−1 ) unspecified but will discuss two parameters in the idea production function, namely, thedegrees of the standing-on-shoulders effect and the returns to research.
We emphasize that knowledge capital in our model has three important differences from physical capital.
First, unlike physical capital, knowledge is nonrival and nonexcludable, as emphasized in Romer (1990)
and Jones (2002). Second, the idea production function G(NR,t (j), Kt−1 ) for producing knowledge capital
substantially differs from the good production function F (NW,t (j), Kt (j)) for producing physical capitalin their properties. Recent evidence provided by Bloom et al. (2020) suggests that idea production may
decrease with economywide knowledge. That is, ideas become harder to find as knowledge accumulates,i.e., ∂G(NR,t (j), Kt−1 )/∂Kt−1 < 0. Third, the depreciation rate of knowledge capital is often assumed to
be zero, in contrast to physical capital. In the growth literature, it is standard to assume that changes inknowledge capital are equal to idea production, ∆Kt (j) = G(NR,t (j), Kt−1 ). In our context, this equation
holds if δ = 0. Therefore, throughout the paper, we assume a positive value of δ to ensure the stationarityof knowledge capital but taking an extremely small value in line with the growth literature.
Following Schmitt-Grohé and Uribe (2006, 2007), we assume that firms face a cash-in-advance (CIA)constraint for their wage payments:
Z 1
Mt (j) =Z 1
WW,t (i) NW,t (i, j) di +0WR,t (i) NR,t (i, j) di,(8)0

where Mt (j) is firm j’s demand for money and the right-hand side of the equation is the wage bill paid byfirm j. The firm j’s profits Πt (j) is given by

Z 1
Πt (j) = Pt (j)Ct (i, j) di + Tt (j)


Z 1
Z 1
+ Mt−1 (j) −WW,t−1 (i) NW,t−1 (i, j) di −WR,t−1 (i) NR,t−1 (i, j) di − Mt (j) . (9)000In (9), the first two terms on the right-hand side of the equation represent cash received by firm j. The firm
sells its final goods to households at price Pt (j) and receives monetary transfers Tt (j) from the government.

8

The expression inside the brackets on the right-hand side of (9) represents the cash remaining at the beginningof period t. The firm must hold money to pay the wage bill at the end of period t.
Aggregating Mt (j) over j yields the total demand for money Mt in this economy:
Z 1
Mt

=

Mt (j) dj
Z 1
Z 1= WW,t
NW,t (j) dj + WR,t
NR,t (j) dj
000= WW,t NW,t + WR,t NR,t ,where Nh,t =

R1
0

(10)

Nh,t (j)dj for h = W, R.

At date t, firm j also chooses {NW,s (j), NR,s (j), Ks (j)}s≥t to maximize

Et

∞
X

Dt,s [Ps (j) Ys (j) − WW,s NW,s (j) − WR,s NR,s (j) + Ts (j)] ,

(11)

s=t

subject to (5)–(7).8 Here, Dt,s =Qs−tτ =1 Dt+τ −1,t+τ

denotes the s-period stochastic discount factor from s

to t, for all s > t, with Dt,t ≡ 1. This profit-maximization problem takes into account the solution of theembodied cost-minimization problem.
The first-order conditions for NW,t (j), NR,t (j), and Kt (j) are given by

WW,t
Pt

=

∗
FN,t
,

(12)

Qt

=

WR,t
,
GN,t

(13)

Qt

=

∗
Pt FK,t+ (1 − δ) Et Dt,t+1 Qt+1 ,(14)

where we impose symmetry across firms and drop the individual firm’s index j. Here Qt denotes the Lagrange
multiplier for (6) (or the shadow price of ideas). This measures the nominal marginal benefit to a firm of
increasing its knowledge capital and, in equilibrium, equals its nominal marginal cost of producing new ideas
given by WR,t /GN,t where GN,t ≡ ∂G (NR,t , Kt−1 ) /∂NR,t . To simplify the exposition, we also introduce∗∗
two auxiliary notations, FN,tand FK,t, to denote the marginal products of inputs adjusted for the price8 In (11), we substitute the CIA constraint (8) into the firm j’s profits because (8) always holds with equality.

9

−1
∗
∗
markup µP , that is, FN,t
≡ µ−1
P [∂F (NW,t , Kt ) /∂NW,t ] and FK,t ≡ µP [∂F (NW,t , Kt ) /∂Kt ].

2.3

The government

The government finances the transfers to firms by issuing money. The budget constraint is
Z 1

s
Tt (j) dj = Mts − Mt−1.

0

The money supply Mts grows at a rate eξt :s
Mts = eξt Mt−1,

where ξt is a white-noise process. Every period, the money market clears:

2.4

(15)

R1
0

Mt (j) dj = Mt = Mts .

GDP

In our model economy, nominal GDP is defined by Pt Ct + Qt Xt , where Qt is interpreted as the imputed
value of new ideas. We define real GDP as

YtGDP = Ct +

Qt
Xt .
Pt

(16)

With the inclusion of the imputed value of new ideas in GDP, we may interpret idea production by researchersin our model as the official measure of R&D in the recent redefinition of GDP where we treat R&D not as
an intermediate expense but as a final investment.9 Overall, the narrowly measured R&D included in official
GDP data is only a small part of real-world innovation processes.10 In the next section, we investigate theimpact of narrowly measured R&D investment on monetary nonneutrality.
9 We could also define real GDP as output at constant prices, as in Barsky et al. (2007). That is, Y GDP = P C + QX ,tttwhere P and Q are the steady-state values of Pt and Qt , respectively. While the definition slightly differs from (16), our results
are robust to the alternative definition.
10 See, for example, Isaacson (2014), Wolfe (2014) and Jones (2019b).

10

3Main results

We illustrate the main results of this paper using the model’s impulse responses to a monetary shock. To
generate the impulse response functions, we specify the utility and production functions and then assign
values to the model’s parameters. After providing the main results, we inspect the mechanism behind them.
We then derive the implications of idea production for monetary business cycle models.

3.1

Functional forms and parameter values

We postulate the following functional forms
1−σ

U (Ct (i)) =Ct (i),
1−σ(17)with σ > 0 for the period utility of consumption and1+ψVh (Nh,t (i)) =h
Nh,t (i),1 + ψh(18)

with ψh > 0 and h = W, R for the period disutility of labor. In this case, the first-order condition (4) canbe rewritten asεh Et
Wh,t (i) =εh − 1P∞σs−t
M RSh,s|t (i) Ps Nh,s|t (i)/ {[Cs (i)] Ps }s=t (βθh )P∞,σEt s=t (βθh )s−t Nh,s|t (i)/ {[Cs (i)] Ps }(19)

where Nh,s|t (i) = [Wh,t (i)/Wh,s ]−εh Nh,s and M RSh,s|t (i) isσ

M RSh,s|t (i) = [Cs (i)]Nh,s|t (i)

ψh

,

(20)

α

(21)

for h = W, R.
The goods production function is specified as

1−α

F (NW,t (j), Kt (j)) = NW,t (j)11

Kt (j) ,where α > 0. We assume the idea production function is given byλG (NR,t (j), Kt−1 ) = NR,t (j)φ Kt−1,

(22)where φ represents the degree of returns to research and λ captures the degree of the standing-on-shoulders
effect. We impose φ > 0 on the idea production function. However, as noted earlier, the sign of λ can be
negative. If λ < 0, ideas become harder to find as knowledge accumulates (i.e., the so-called fishing-out
effect).
We now assign values to the model’s parameters. The subjective discount factor β is chosen to beconsistent with an annualized steady-state real interest rate of 2 percent. We set the degree of relative risk
aversion in consumption to unity, along with the inverse of the Frisch elasticities of labor supply (σ = ψW =
ψR = 1). We set the elasticity of substitution between differentiated goods, εP , to 11, and the elasticity
of substitution between differentiated labor, εh , to 5, for h = W, R. When we consider wage stickiness forworkers (researchers), we set the hazard rates θW (θR ) to 0.75. Thus, the average duration of newly set
wages for individual workers (researchers) is four quarters.11 When we assume that the wages of workers(researchers) are flexible, we set θW (θR ) to 0, so that the average duration of newly set wages for individual
workers (researchers) is one quarter. These parameter values appear reasonable choices considering the
sticky-wage literature (e.g., Huang and Liu, 2002). As we demonstrate in later sensitivity analysis, thebasic conclusion of this paper is robust to alternative values of these parameters within empirically plausible
ranges.
The idea production function (22) has crucial implications for predictions of our model. To be specific,
combine (22) with (6). Given the symmetry of Kt (j) = Kt and NR,t (j) = NR,t , we haveφλ∆Kt + δKt−1 = NR,t
Kt−1
.

(23)

11 We treat the two hazard rates θW and θR symmetrically even if the labor types differ. The micro evidence on nominal wage
adjustment suggests that the degrees of stickiness in wages do not differ substantially regardless of labor type. For example,
Barattieri et al. (2014) show that differences in the frequency of wage changes are quantitatively small across occupations for the
US. Le Bihan et al. (2012) and Sigurdsson and Sigurdardottir (2016) report similar results for France and Iceland, respectively.
Barattieri et al. (2014) also finds that the frequency of wage changes ranges between 16.3 and 21.6 percent. The reported
values suggest that our parameterization of θh = 0.75 is not inconsistent with the data because the observed frequencies imply
a hazard rate that ranges from 0.78 to 0.83.

12In Romer’s original contribution (i.e., Romer, 1990), the degrees of returns to research and the standing-onshoulders effect were both set to unity. That is, φ = λ = 1. More recent studies argue that a λ less than 1 isimportant for reconciling idea-based growth theory with the data.12 A recent paper by Bloom et al. (2020)
even suggests λ < 0.
We parameterize φ and λ as follows. Rewrite (23) to obtain the steady-state knowledge capital K as afunction of the steady-state labor of researchers NR :

K=

 η
1
NRη ,
δ

where η = φ/(1 − λ). Bloom et al. (2020) suggest that a plausible estimate of η for the aggregate economy is0.32. They continue to assume that φ = 1 as in Romer (1990). Thus, the estimate of η can be translated into
λ = −2.13. If the returns to research are decreasing (φ < 1), the value of λ depends on how we calibrate φ.
To explore the impact of idea production on business cycles, we consider two values for φ, namely, φ = 0.5(low returns to research) and φ = 1 (high returns to research). According to these parameter values, λ isset to −0.56 and −2.13, respectively.
The other remaining parameters for technology are the returns to knowledge α in (21) and the knowledge
capital depreciation rate δ in (6). To calibrate these parameters, we target the steady-state share of R&D
expenditure to GDP and the steady-state ratio of labor costs of researchers to workers. The R&D expenditure
share is given by−1(Q/P )Xαδ [1 − β (1 − δ)]=−1 ,C + (Q/P )XµP + αδ [1 − β (1 − δ)](24)

and the labor cost ratio can be calculated as

WR NR
α
φδ
=
,
WW NW
1 − α 1 − β (1 − δ)(25)

where a variable without a time subscript denotes its steady-state value. To derive these expressions, we
assume that ψW = ψR .
12 See Jones (1995, 1999, 2002, 2005) and Fernald and Jones (2014) for details.

13

Regarding the share of R&D expenditure to GDP, we set the share at 3 percent based on the US data
in 2019 shown in Table 1. We set the labor cost ratio at 0.08 using data from the 2018 OES compiledby the Bureau of Labor Statistics. The OES reports the employment and annual mean wages of various
occupations. Our definition of “researchers” includes some research-related managers, computer and mathematical occupations, engineers, scientists and researchers, and doctors.13 All remaining occupations are
considered “workers.” Using this classification, we obtain the mean annual wage and the number of employed. As a result, the calibrated parameter values are (α, δ) = (0.79, 2.26 × 10−4 ) when φ = 0.5 and(α, δ) = (0.58, 3.15 × 10−4 ) when φ = 1.14 Interestingly, the parameter values of δ are positive but close tozero, consistent with our assumption (e.g., 3.15 × 10−4 ).
We emphasize that both the share of R&D expenditure to GDP and the labor cost ratio are too low to
generate significant aggregate effects on output. Below we show that the seemingly unimportant researchers
can have a substantial impact on real effect of monetary shocks, depending on the shape of the idea production
function.

3.2

Responses of real GDP

We investigate the real effect of a monetary shock in the sticky-wage model using the two levels of returns
to research (φ = 0.5 and 1) in idea production. Figure 1 plots the impulse responses of real GDP to a
one-percent increase in the money supply under different configurations of nominal wage rigidities. The
dotted line represents the responses of real GDP when the wages of both workers and researchers are sticky(“W-sticky/R-sticky,” θW = θR = 0.75). The dashed line corresponds to the responses when worker wages
are sticky, but researcher wages are flexible (“W-sticky/R-flexible,” θW = 0.75, θR = 0). However, we havea particular interest in the configuration where researcher wages are sticky, but worker wages are flexible(“W-flexible/R-sticky,” θW = 0, θR = 0.75).
13 The Appendix provides the details of all occupations included in our definition of “researchers.” Admittedly, classifying
occupations in the OES into our definition of “researchers” is somewhat arbitrary. For example, doctors may produce knowledge,
but their knowledge may not necessarily contribute to the production of consumption goods. However, we confirm that excluding
doctors from researchers leads to only a minor impact on our main results.
14 Jones (1995) also includes physical capital in a goods production function. We can do the same here without altering ourmain conclusion in any substantial way. We can also insert in (21) a term for physical capital that is kept constant at thebusiness cycle frequency (an assumption often made in the monetary business cycle literature) without changing any of ourresults.

14

Low returns to research ( = 0.5)

0.5High returns to research ( = 1)

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

W-sticky/R-sticky
W-sticky/R-flexible
W-flexible/R-sticky

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 1: Responses of real GDP to a monetary shock
Notes: The left panel depicts the responses of real GDP to a one-percent increase in the money supply when the returns toresearch are low in idea production (φ = 0.5). The right panel presents those when the returns to research are high (φ = 1). In
each panel, dotted lines represent the responses of real GDP when the wages of both workers and researchers are sticky; dashedlines are when worker wages are sticky, but researcher wages are flexible; solid lines are when worker wages are flexible, but
researcher wages are sticky. Vertical axes measure percentage deviations from the steady state. Time is given in quarters on
the horizontal axes.

The left panel of Figure 1 depicts the impulse responses under φ = 0.5. In the cases of W-sticky/R-sticky(the dotted line) and W-sticky/R-flexible (the dashed line), we confirm that the real effects of a monetaryshock are substantial because of sticky wages of workers. Indeed, the initial responses of real GDP are 0.19
and 0.18 percent, respectively. The effects of the monetary shock on real GDP take more than a year toreturn to the steady state of zero. Comparing the responses in these configurations, we also find that they
are similar. When researcher wages are sticky, but worker wages are flexible (i.e., W-flexible/R-sticky, being
the solid line), the real effect of a monetary shock on real GDP is small. The response is 0.02 percent on
impact and extremely insignificant compared to those in the other two configurations.
The reason for these results is straightforward. In this model, researchers are a quantitatively small groupin the economy. In our model, steady-state R&D expenditure accounts for only 3 percent of all expenditures
and the steady-state labor cost ratio of researchers to workers is only 0.08. Therefore, the presence of15

researchers in the model does not strongly affect the responses of real GDP. As a result, the responses of real
GDP under W-sticky/R-sticky and W-sticky/R-flexible will be similar. The extremely small responses ofreal GDP under W-flexible/R-sticky are also easy to understand because nominal rigidities are present onlyin the wages of researchers, who are quantitatively small in the economy. Thus, this observation suggests
that, not surprisingly, nominal rigidities in the wages of workers, who are quantitatively dominant in theeconomy, are necessary for the strong real effects of a monetary shock.
However, when the returns to research are high (i.e., φ = 1), this result can be overturned. In the rightpanel of Figure 1, we again plot the impulse responses of output, but in the economy with φ = 1. The solidline in the right panel shows that the real effect can be strong under W-flexible/R-sticky. More specifically,
even when nominal rigidities are present only in the wages of researchers, who are quantitatively small inthe economy, real GDP increases by a large amount (e.g., 0.26 percent on impact). The positive responsesof real GDP are visible and comparable to those in the other configurations.15 Although the real effectconverges to zero more quickly than in the other two configurations, the initial response of 0.26 percent
under W-flexible/R-sticky is not substantially lower than the 0.36 percent under W-sticky/R-sticky andW-sticky/R-flexible.
A key message is that the shape of the idea production function critically affects the real effect of amonetary shock. When the returns to research are low, researchers are insignificant in generating realeffects. However, when the returns to research are high, we find that researchers play a nonnegligible role ingenerating the real effects of a monetary shock. Importantly, even if wages are fully flexible in most labormarkets, stickiness only in the wages of researchers can generate strong real effects of monetary shocks toan extent comparable to the case where nominal wages are sticky in all labor markets.

3.3

Inspecting the mechanism

In this section, we inspect the transmission mechanism of monetary shocks. Using the goods and idea
production functions, real GDP is given by Ct + (Qt /Pt )Xt = F (NW,t , Kt ) + (Qt /Pt )G (NR,t , Kt−1 ). Thus,
15 We note that the calibrated values of returns to knowledge in (21) differ between the two panels. We calibrate α at 0.79 in
the left panel and 0.58 in the right panel. However, even if we use the same α for the right panel, the responses of real GDP in
the right panel do not substantially differ.

16

GDP

0.6

W-flexible/R-sticky
W-sticky/R-flexible

0.4

Knowledge

0.02
0.015
0.01

0.2

0.005

0

0
1

2

3

4

6

8

12

1

Consumption

0.6

2

3

4

6

8

12

8

12

R&D

10

0.4
5
0.2
0

0
1

2

3

4

6

8

12

1

Worker labor

1

2

3

6

Researcher labor

10

0.5

4

5

0

0
1

2

3

4

6

8

12

1

2

3

Worker wage

4

6

8

12

Researcher wage

1

1

0.5

0.5

0

0
1

2

3

4

6

8

12

1

Real shadow price

2

3

4

6

8

12

Relative wage

1

1
0.5

0.5

0

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 2: Responses to a monetary shock
Notes: Each panel depicts the responses of the corresponding variables in our sticky-wage model to a one-percent increase in themoney supply given the two alternative configurations for wage stickiness. Solid lines represent the case under W-flexible/Rsticky. Dashed lines are for W-sticky/R-flexible. In the figure, the returns to research are high (φ = 1). Vertical axes measure
percentage deviations from the steady state. Time is given in quarters on the horizontal axes.

17the real effect of a monetary shock on real GDP is determined by the responses of four variables: the labor
services of workers, NW,t , knowledge capital, Kt , the real shadow price of ideas, Qt /Pt , and the labor servicesof researchers, NR,t . Figure 2 plots the impulse responses of these variables to a monetary injection as wellas other variables of interest under our baseline parameterization. In the figure, the returns to research
are parameterized to be high (i.e., φ = 1). The figure compares the responses under the most importantconfiguration of W-flexible/R-sticky to those under W-sticky/R-flexible. As shown in Figure 2, NW,t , Kt ,and Qt /Pt are almost unchanged under W-flexible/R-sticky. In what follows, we explain why this is thecase. Then, we explore how increases in NR,t lead to the real effect of monetary shocks.
We first consider knowledge capital Kt . If the depreciation rate δ is low, the flow–stock ratio is so lowthat even large changes in idea production have small effects on the stock of knowledge capital. Therefore,
as shown in the first row of Figure 2, a monetary injection does not produce a sizable increment in the stockof knowledge capital, independent of the configurations of nominal rigidities. Consequently, it is helpful totreat knowledge capital as roughly constant: Kt ' K.
Second, we consider the labor services of workers NW,t . In equilibrium, wage markups for workers µW,t∗equal the gap between the marginal product of workers FN,tin (12) and the marginal rate of substitution of0(NW,t ) /U 0 (Ct ). This relationship can be expressed as
their labor for consumption VWµW,t = FN∗ (NW,t , Kt )

0
(NW,t )VW
0
U [F (NW,t , Kt )]


,

(26)

where we used Ct = F (NW,t , Kt ). Equation (26) suggests that NW,t effectively has a one-to-one relationshipto wage markups for workers µW,t given the near constancy of Kt . Thus, NW,t does not respond to amonetary injection if worker wages are flexible (i.e., µW,t is constant for all t). Similarly, consumption
exhibits extremely small movement due to Ct = F (NW,t , Kt ) along with the muted responses of Kt and
NW,t to the monetary shock.16 Therefore, as the solid lines in the second and third rows of Figure 2 show,
the responses of NW,t and Ct are almost zero under W-flexible/R-sticky.17
16 The validity of our analysis and the basic conclusion does not depend on how we parameterize preferences and technology.
0 > 0, V 00 ≥ 0, F ∗ > 0, F ∗ ≥ 0, F ∗∗In fact, it is sufficient to assume U 0 > 0, U 00 ≤ 0, VWWNKN K ≥ 0, and FN N ≤ 0.
17 Under W-sticky/R-flexible, µW,t decreases in response to a positive monetary shock. As a result, the labor services of18

Finally, consider the real shadow price of ideas Qt /Pt . To this end, we rewrite (14) as

Qt
Pt

=
'

U 0 (Ct+1 ) Qt+1
U 0 (Ct ) Pt+1


Qt+1
∗
FK (NW,t , Kt ) + β (1 − δ) Et
,
Pt+1
∗
FK
(NW,t , Kt ) + β (1 − δ) Et





(27)

where the approximate equality in the second line results from the fact that Ct = F (NW,t , Kt ) is almost
constant under W-flexible/R-sticky. Equation (27) implies that Qt /Pt is stable over time when worker wages
are flexible because NW,t and Kt are all near constant in response to a monetary shock. Indeed, Qt /Pt shownin the solid line of the bottom row of Figure 2 indicates that the responses of the shadow price of ideas areagain almost zero under W-flexible/R-sticky.
All the above observations imply that the real effect of a monetary shock under W-flexible/R-sticky
depends almost entirely on how NR,t responds. Note that wage markups for researchers µR,t equal the gap
between their marginal revenue product (Qt /Pt )GN,t and the marginal rate of substitution of their labor for
consumption VR0 (NR,t ) /U 0 (Ct ). This relationship can be expressed asQtµR,t =GN (NR,t , Kt−1 )Pt

VR0 (NR,t )U 0 [F (NW,t , Kt )]


,

(28)

where we replace Ct in U 0 (Ct ) by F (NW,t , Kt ). In the right-hand side of (28), Kt , NW,t , and Qt /Pt are allnear constant. The stability of these variables implies that NR,t has a one-to-one relationship with µR,t .
The transmission mechanism of a monetary injection under W-flexible/R-sticky is as follows. When Mt
increases, it relaxes the CIA constraint of firms and increases the nominal demand for labor inputs. Under
W-flexible/R-sticky, a monetary injection lowers the wage markups only for researchers. In response to thedecline in the wage markups for researchers, NR,t increases.18 The increase in NR,t leads to an increase inidea production Xt . An increase in Xt results in an increase in real GDP given by Ct + (Qt /Pt )Xt while Ct
and Qt /Pt remain unchanged.
workers NW,t and thus the consumption Ct increase (see the dashed lines in Figure 2). Given a large expenditure share of
consumption in GDP, the sticky wages of workers play a dominant role in generating the significant real effects of a monetaryshock.
18 We can confirm this inverse relationship from the concavity of G with respect to N
R,t and the convexity of VR in (28).

19Low returns to research ( = 0.5)
R&D

10High returns to research ( = 1)
R&D

10

8

8

6

6

4

4

2

2

0

0
1

2

3

4

6

8

12

1

Researcher labor

10

2

3

8

6

6

4

4

2

2

0

6

8

12

Researcher labor

10

8

4

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 3: Responses of R&D and researcher labor services to a monetary shock
Notes: Top panels depict the responses of R&D to a one-percent increase in the money supply, and the bottom panels thoseof researcher labor services. Left panels exhibit the case when the returns to research are low (φ = 0.5), while the right panels
correspond to those when the returns to research are high (φ = 1). For the configuration of nominal wage rigidities, worker
wages are flexible, but researcher wages are sticky (W-flexible/R-sticky). Vertical axes measure percentage deviations from thesteady state. Time is given in quarters on the horizontal axes.

3.4

Importance of the returns to research

In the previous subsection, we showed that the output responses to a monetary shock critically dependon responses of idea production. This subsection discusses the importance of returns to research in idea
production for the real effects of monetary shocks.
Figure 3 compares the responses of idea production (R&D) and those of researcher labor services to amonetary shock with different degrees of returns to research (φ = 0.5 on the left panels and φ = 1 on the20right panels). As the upper panels of the figure indicate, the responses of Xt under φ = 1 are substantiallylarger than those under φ = 0.5. In particular, a one-percent increase in the money supply leads to anincrease of Xt by 8.75 percent at impact under φ = 1, but the increase is only 0.76 percent under φ = 0.5.
Turning to the lower panels, the initial response of NR,t is 8.75 percent under φ = 1 while it is only 1.53
percent under φ = 0.5. Again, there is a substantial difference between the cases with high and low returns
to research.
When returns to research are high, there is a strong incentive to hire researchers in response to a monetary
injection. This incentive is reinforced by the strong demand for ideas because knowledge is a durable inputthat lasts many periods. As the returns to research are not decreasing in NR,t , the large increase in researcher
labor services NR,t is transmitted to idea production Xt . Thus, we obtain a substantial real effect of money
even though researchers are quantitatively small in the economy. By contrast, when the returns to research
are low, the incentive to hire researchers is weak. Given that the productivity of research decreases with
NR,t , the increase in NR,t is further weakly transmitted to Xt . Consequently, the output response to amonetary shock is near zero.
Different degrees of returns to research affect (i) the sensitivity of Xt to NR,t and (ii) the sensitivity ofφ
λNR,t to µR,t . Using our specifications for the idea production function G(NR,t , Kt−1 ) = NR,t
Kt−1
and the1+ψR
period disutility function V (NR,t ) = NR,t/(1 + ψR ), we can approximate (22) and (28) as

bt
dX
' φ,
bR,t
dN

(29)

bR,t
dN
−1
'
,
db
µR,t
1 + ψR − φ(30)where hatted variables represent the log deviation of the corresponding variables from the steady state.
In deriving these equations, we assume that Kt , NW,t , and Qt /Pt are approximately constant due to Wflexible/R-sticky. It is important to note that, as the returns to research φ increase from 0.5 to 1, thebt to NbR,t and N
bR,t to µ
sensitivity of both X
bR,t increase.
We emphasize that the main driver of idea production in this monetary business cycle model is the labor21

services of researchers rather than knowledge capital. Because Kt is near constant in response to a monetaryshock, the role of knowledge capital in idea production in business cycle frequency is limited. Variations inidea production mostly stem from changes in the labor services of researchers.
Therefore, our findings shed new light on the discussion regarding the shape of the idea production function. Jones (2005, 2019a) emphasize that the standing-on-shoulders effect determines the long-run trend
in economic growth. We find instead that the returns to research determine the real effect of monetaryshocks in business cycle frequency. In particular, the parameter φ capturing the returns to research inφλG(NR,t , Kt−1 ) = NR,tKt−1strongly influences idea production through changes in labor services of re-searchers. In contrast to φ, the parameter λ capturing the standing-on-shoulders effect plays only a minor
role in the real effect of monetary shocks because of the near constant knowledge capital. Nevertheless, theshape of the idea production function matters in generating the real effects of monetary shocks. In thissense, our study complements Jones (2005, 2019a).

3.5

Responses of prices and wages

Our monetary business cycle model allows us to further explore the implications of the shape of the idea
production function for nominal variables. Figure 4 compares the responses of nominal prices and wages
to a monetary shock between high and low returns to research. In the figure, the configuration of nominal
rigidities remains W-flexible/R-sticky.
The case with high returns to research delivers interesting implications for wages and prices. Under Wflexible/R-sticky, all prices other than researcher wages (Pt , WW,t , Qt ) are flexible. Interestingly, Figure 4
indicates that the adjustment of these prices to the new steady state slows as φ increases from φ = 0.5 to
φ = 1. When the returns to research are low (φ = 0.5), the consumption good price, the shadow price ofideas, and worker wages adjust immediately to a value close to one percent in response to a one-percent
increase in the money supply. By contrast, when the returns to research are high, the initial responses are
only about 0.35 percent. In other words, higher returns to research generate a slower adjustment of prices,even if they are completely flexible.

22

Low returns to research ( = 0.5)
Consumption-good price1

High returns to research ( = 1)
Consumption-good price

1

0.5

0.5

0

0
1

2

3

4

6

8

12

1

Shadow price of ideas

1

2

3

6

8

12

Shadow price of ideas

1

0.5

4

0.5

0

0
1

2

3

4

6

8

12

1

Worker wage

1

2

3

4

8

12

Worker wage

1

0.5

6

0.5

0

0
1

2

3

4

6

8

12

1

Researcher wage

1

2

3

6

8

12

Researcher wage

1

0.5

4

0.5

0

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 4: Responses of prices and wages to a monetary shock
Notes: Each panel depicts the responses of the corresponding variables in our sticky-wage model to a one-percent increase in
the money supply. Left panels show the case when the returns to research are low (φ = 0.5). Right panels correspond to thecase when the returns to research are high (φ = 1). For the configuration of nominal wage rigidities, worker wages are flexible,
but researcher wages are sticky (W-flexible/R-sticky). Vertical axes measure percentage deviations from the steady state. Timeis given in quarters on the horizontal axes.

This slower adjustment results from complementary nature across prices and wages in our model. Whenφ = 0.5, the responses of Pt , WW,t , and Qt are close to each other, and they substantially differ from thoseof WR,t . However, when φ = 1, these flexible prices move in tandem with sticky researcher wages.
We can easily observe this synchronization with sticky researcher wages from (12) and (13). These23

equations imply
WR,t
Qt GN (NR,t , Kt−1 )=.
WW,tPt FN∗ (NW,t , Kt )(31)

As discussed in Section 3.3, Qt /Pt and FN∗ (NW,t , Kt ) are stable over time under W-flexible/R-sticky. Thus,φ−1 λthe response of the relative wage critically depends on GN (NR,t , Kt−1 ) = φNR,tKt−1 , which is almost

unchanged when φ = 1. Thus, the relative wage is nearly constant.19 The nearly constant relative wage
implies that flexible WW,t synchronizes with sticky WR,t . It immediately follows from (12) and the nearconstancy of Qt /Pt that WW,t ' Pt ' Qt .
The above discussion reinforces the importance of the returns to research in idea production. The resultsin Section 3.4 highlight the role of φ in terms of real variables. Our results in this subsection also suggestthat the returns to research matter for nominal variables. The shape of the idea production function can be
an important factor in generating the slow adjustment of nominal variables as well as the real effects of amonetary shock.

4

Sensitivity analysis

In this section, we check whether the results for φ = 0.5 and φ = 1 are robust to alternative parameter values.
We consider the following parameters for the sensitivity analysis: the degree of relative risk aversion (σ); theinverse of the Frisch elasticities of labor supply (ψR and ψW ); the degree of stickiness in researcher wages(θR ); the depreciation rate of knowledge capital (δ); and the returns to knowledge in the goods production
function (α). The comparisons depicted in the figures in this section are for two values of the returns toresearch: low returns to research (left panel) versus high returns to research (right panel). In both panels,
worker wages are fully flexible and researcher wages are sticky. Once again, researchers are quantitatively
small in the economy. Nevertheless, if the returns to research are high, sticky researcher wages can generate
substantial real effects of monetary shocks. We also make some remarks on the robustness to alternative
functional forms of the idea and goods production functions.
19 We can reconfirm the responses of relative wages from the bottom row of Figure 2.

24

Low returns to research ( = 0.5)

0.5High returns to research ( = 1)

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

= 0.01
=1
=5

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 5: Responses of real GDP for different degrees of relative risk aversion
Notes: The left panel depicts the responses of real GDP to a one-percent increase in the money supply when the returns toresearch are low in idea production (φ = 0.5). The right panel is when the returns to research are high (φ = 1). For theconfiguration of nominal wage rigidity, worker wages are flexible, but researcher wages are sticky (W-flexible/R-sticky). In eachpanel, dotted lines represent the responses of real GDP when σ = 0.01; dashed lines correspond to when σ = 1; solid lines arewhen σ = 5. Vertical axes measure percentage deviations from the steady state. Time is given in quarters on the horizontal
axes.

4.1

Degree of relative risk aversion

We first show that our results are robust to the degree of relative risk aversion σ. In the baseline parameterization, we set σ to unity. However, a higher value of σ (i.e., more inelastic intertemporal substitution
in consumption) strengthens the consumption smoothing motive of households. This may result in smallerfluctuations in aggregate demand following monetary shocks and weaker real effects of monetary shocks,
even under φ = 1.
Figure 5 displays the impulse responses of real GDP to a one-percent increase in the money supply forthree different values of σ, with dotted lines for σ = 0.01, solid lines for σ = 1, and dashed lines for σ = 5. As
both panels of Figure 5 show, the responses of real GDP to monetary shocks are not essentially affected by
the value of σ. The intuition is simple. When worker wages are flexible, monetary shocks do not substantiallychange Ct . If consumption is constant, the parameter for the curvature of the period utility does not matter

25

Low returns to research ( = 0.5)

0.5High returns to research ( = 1)

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

= 0.01
=1
=5

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 6: Responses of real GDP with different values of ψNotes: The left panel depicts the responses of real GDP to a one-percent increase in the money supply when the returns toresearch are low in idea production (φ = 0.5). The right panel is when the returns to research are high (φ = 1). For theconfiguration of nominal wage rigidity, worker wages are flexible, but researcher wages are sticky (W-flexible/R-sticky). We
assume ψW = ψR = ψ. In each panel, dotted lines represent the responses of real GDP when ψ = 0.01; dashed lines correspondto when ψ = 1; solid lines are when ψ = 5. Vertical axes measure percentage deviations from the steady state. Time is given
in quarters on the horizontal axes.

for the model’s dynamics.

4.2

The inverse of the Frisch elasticity of labor supply

We next consider the robustness of our results to the inverse of the Frisch elasticities of labor supply (ψWand ψR ), which are set to unity in the baseline parameterization. A large ψW and ψR may also strengthen
the motive for smoothing labor supply. Moreover, unlike the case of σ, (30) indicates that the value of ψRdirectly affects the sensitivity of NR,t to their wages. Thus, the sensitivity analysis to the inverse of the
Frisch elasticity of labor supply is more important than that to σ. Below, we compute the responses of real
GDP to a monetary shock, maintaining the assumption of ψW = ψR in the robustness analysis.
Figure 6 displays the impulse responses of real GDP for three values of ψ (= ψW = ψR ), with dottedlines for ψ = 0.01, solid lines for ψ = 1, and dashed lines for ψ = 5. The left panel of the figure confirms

26that when the returns to research are low, the real effects of a monetary shock are weak for all three valuesof ψ. When φ is low, the increase in NR,t is weakly transmitted to Xt via the idea production function. The
right panel of the same figure corresponds to the case of φ = 1. When the returns to research are high, theincrease in real GDP is only slightly lower under ψ = 5 (the dashed line) than ψ = 1 (the solid line). In our
simulations, the impact of increasing ψ from 1 to 5 on the response of real GDP is small. On the contrary,
if we reduce ψ from 1 to 0.01, the real effect of a monetary shock becomes substantially larger.20 Therefore,
the real effect of a monetary shock under W-flexible/R-sticky remains robust to alternative values of ψ.

4.3

Degree of stickiness in researcher wages

In the third sensitivity analysis, we directly vary the degree of stickiness in researcher wages to observe howchanges in θR affect our main conclusion. We find that it has little effect. To illustrate, we consider threevalues for θR , θR = 0.5 (dotted lines), θR = 0.75 (solid lines), and θR = 0.875 (dashed lines), corresponding
to a two, four, and eight quarter average duration of newly set wages of researchers, respectively.
Figure 7 illustrates the results of the sensitivity analysis. When the returns to research are low (theleft panel), the real effects of a monetary shock remain weak, even under the higher degree of stickiness inresearcher wages. Qualitatively, increasing the degree of stickiness in researcher wages strengthens the realeffects of the monetary shock. However, the quantitative differences are small when the returns to research
are low and large when the returns to research are high. That is, an increase in the degree of wage stickiness
amplifies the real effect when the returns to research are high. Even when θR = 0.5, the initial responseof real GDP is 0.15 percent, which is much larger than any initial response of real GDP in the right panel.
Thus, the shape of the idea production function is much more important than the degree of nominal wage
stickiness in generating the real effect of monetary shocks.
20 This asymmetry is because of the effect of ψ
R appearing in the denominator on the right-hand side of (30).

That is, thesensitivity of the labor services of researchers to wage markups more strongly increases as ψR becomes smaller. In particular,bR,t /db
when the decline in ψR is from 5 to 1, the increase in dNµR,t is from 0.2 (= 1/5) to 1 (= 1/1). Alternatively, when thebdecline is from 1 to 0.01, it increases dNR,t /dbµR,t from 1 (=1/1) to 100 (=1/0.01).

27

Low returns to research ( = 0.5)

0.5High returns to research ( = 1)

0.5

R

0.4

R

0.4

R

0.3

0.3

0.2

0.2

0.1

0.1

0

= 0.5
= 0.75= 0.875

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 7: Role of stickiness in researcher wages
Notes: The left panel depicts the responses of real GDP to a one-percent increase in the money supply when the returns toresearch are low in idea production (φ = 0.5). The right panel is when the returns to research are high (φ = 1). For theconfiguration of nominal wage rigidity, worker wages are flexible, but researcher wages are sticky (W-flexible/R-sticky). In eachpanel, dotted lines represent the responses of real GDP when θR = 0.5; dashed lines correspond to when θR = 0.75; solid lines
are when θR = 0.875. Vertical axes measure percentage deviations from the steady state. Time is given in quarters on thehorizontal axes.

4.4

Depreciation rate of knowledge capitalIn the calibration, we have chosen the depreciation rate of knowledge capital δ, together with the returns toknowledge in the goods production function α, to match the data on the R&D expenditure share to GDP
and the labor cost ratio. However, the evidence on δ is not necessarily ample and the value of δ may vary
according to the interpretation of knowledge capital. We thus check the robustness of our results to different
values of δ. In this sensitivity analysis to δ, we keep α unchanged at 0.58, which is the baseline parametervalue when the returns to research are high (φ = 1).
We compare two cases: one to reduce the value δ to one-third of the baseline parameter value (i.e.,δ = 1.05 × 10−4 ), and the other to triple the value of δ (= 9.45 × 10−4 ). We confirm that the R&D
expenditure share to GDP and the labor cost ratio with these values of δ remain within reasonable ranges.
Table 2 shows the R&D expenditure shares to GDP and the labor cost ratios implied from δ where α is28

Table 2: Steady-state R&D expenditure shares and labor cost ratios for different values of δHigh returns to research (φ = 1)
δR&D exp. share
1.05 × 10−4
0.01
3.15 × 10−4
0.03
9.45 × 10−4
0.08
Low returns to research (φ = 0.5)
δR&D exp. share
1.05 × 10−4
0.01
3.15 × 10−4
0.03
9.45 × 10−4
0.08

Labor cost ratio
0.03
0.08
0.22
Labor cost ratio
0.01
0.04
0.11

Notes: The table reports steady-state R&D expenditure shares (denoted by“R&D exp. Share”) and labor cost ratios with different values of δ. We fix α
at α = 0.58, which is the baseline parameter value under φ = 1. Labor cost
ratios depend on the choice of φ. The steady-state R&D expenditure shareis given by (24) and the labor cost ratio is given by (25).

fixed at α = 0.58. The top panel of Table 2 shows the case when the returns to research are high (φ = 1).
As shown in the second column of the table, if we reduce δ from 3.15 × 10−4 to 1.05 × 10−4 , the R&D
expenditure share reduces to 1 percent. If we triple δ (= 9.45 × 10−4 ), the R&D expenditure share amounts
to 8 percent. We observe similar patterns regarding the labor cost ratio. As shown in the third column of
the table, the resulting labor cost ratio ranges from 0.03 to 0.22. The bottom panel of Table 2 presents thecase when the returns to research are low (φ = 0.5). While the steady-state R&D expenditure shares do notdepend on the value of φ, the labor cost ratios decrease by half.
Figure 8 plots the impulse responses. The dotted line represents the case of δ = 1.05 × 10−4 and thedashed line corresponds to the case of δ = 9.45 × 10−4 . The solid line uses the baseline parameterization
under φ = 1 (i.e., δ = 3.15 × 10−4 ) for comparisons. Overall, impulse responses are amplified through theincreased δ, especially when the returns to research are high. As shown in the right panel of the figure,
output increases at impact by 0.20 percent for δ = 1.05 × 10−4 and by 0.33 percent for δ = 9.45 × 10−4 .
The stronger real effect of money under a larger δ results from the increased steady-state expenditure
share of R&D to GDP (i.e., (Q/P )X/[C + (Q/P )X]). As the steady-state expenditure share of R&D to
GDP increases, the effect of Xt on total expenditure Ct + (Qt /Pt )Xt is amplified. For this reason, theoutput response is somewhat large when δ = 9.45 × 10−4 even under φ = 0.5. Nevertheless, the real effects

29

Low returns to research ( = 0.5)

0.5High returns to research ( = 1)0.5

= 1.05 10 -4= 3.15 10 -4

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

= 9.45 10 -4

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 8: Responses of real GDP with different values of δNotes: The left panel depicts the responses of real GDP to a one-percent increase in the money supply when the returns toresearch are low in idea production (φ = 0.5). The right panel is when the returns to research are high (φ = 1). For theconfiguration of nominal wage rigidity, worker wages are flexible, but researcher wages are sticky (W-flexible/R-sticky). In eachpanel, dotted lines represent the responses of real GDP when δ = 1.05 × 10−4 ; solid lines are when δ = 3.15 × 10−4 ; dashedlines correspond to when δ = 9.45 × 10−4 . Vertical axes measure percentage deviations from the steady state. Time is given inquarters on the horizontal axes.

of monetary shocks substantially differ between φ = 0.5 and φ = 1 for each value of δ. Therefore, our key
message that the shape of the idea production function matters for the real effect of monetary shocks remains
unchanged.

4.5

Returns to knowledge in the goods production function

In the calibration, we chose the returns to knowledge in the goods production function α under the assumptionof constant returns to scale. However, the evidence on knowledge as a factor of production is limited, andthe value of α may differ, depending on the interpretation of knowledge capital. Thus, it would be sensible
to check the robustness of our results, using a wide range of the value of α.21
To implement sensitivity analysis to the returns to knowledge in the goods production function, we
21 Jones (2019a) also considers constant and increasing returns to scale in the goods production function.

30

Table 3: Steady-state R&D expenditure shares and labor cost ratios for different values of αKHigh returns to research (φ = 1)
αK
R&D exp. share
0.28
0.02
0.58
0.03
0.88
0.05
1.18
0.06
Low returns to research (φ = 0.5)
αK
R&D exp. share
0.28
0.02
0.58
0.03
0.88
0.05
1.18
0.06

Labor cost ratio
0.04
0.08
0.12
0.16
Labor cost ratio
0.02
0.04
0.06
0.08

Notes: The table reports the steady-state expenditure shares of R&D to
GDP (denoted by “R&D exp. share”) and labor cost ratios with different
values of αK . We fix δ at δ = 3.15 × 10−4 , which is the baseline parametervalue. Labor cost ratios depend on the choice of φ. The R&D expenditure
share is given by αK δ[1 − β(1 − δ)]−1 /{µP + αK δ[1 − β(1 − δ)]−1 } and thelabor cost ratio is given by (αK /αN )(φδ)[1 − β(1 − δ)]−1 .

replace the goods production function (21) by

F (NW,t (j), Kt (j)) = NW,t (j)αN Kt (j)αK ,(32)

where αN > 0 and αK > 0 but not necessarily αN + αK = 1. In our sensitivity analysis, we consider
αK = {0.28, 0.88, 1.18} in comparison to the baseline parameter value α = 0.58. We fix αN at 0.42, which isthe baseline parameter value of the returns to workers under φ = 1 (i.e., 1−α = 0.42), so that we can consider
the decreasing returns to scale (αN +αK = 0.7), as well as the increasing returns to scale (αN +αK = 1.3 and1.6). In this robustness analysis, we fix the remaining parameters at the baseline parameter value includingδ (= 3.15 × 10−4 ).
We confirm that the R&D expenditure share to GDP and the labor cost ratio resulting from the newly
parameterized αK do not deviate substantially from the data. Table 3 shows how the R&D expenditure
share and the labor cost ratio vary as we change αK . The top panel of Table 3 shows the case when the
returns to research are high (φ = 1). Recall that our calibration achieves a steady-state R&D share to GDP
of 3 percent and the labor cost ratio of 0.08 when αK = α = 0.58 and αN = 1 − α = 0.42. The second31Low returns to research ( = 0.5)

0.5High returns to research ( = 1)

0.5

K

0.4

K

0.4

K
K

0.3

0.3

0.2

0.2

0.1

0.1

0

= 0.28= 0.58= 0.88= 1.18

0
1

2

3

4

6

8

12

1

2

3

4

6

8

12

Figure 9: Responses of real GDP for different values of αNotes: The left panel depicts the responses of real GDP to a one-percent increase in the money supply when the returns toresearch are low in idea production (φ = 0.5). The right panel is when the returns to research are high (φ = 1). For theconfiguration of nominal wage rigidity, worker wages are flexible, but researcher wages are sticky (W-flexible/R-sticky). In eachpanel, the dashed line is the case of αK = 0.28, the bold solid line is the case of αK = 0.58, the dashed line is the case ofαK = 0.88 and the thin solid line is the case of αK = 1.18. Vertical axes measure percentage deviations from the steady state.
Time is given in quarters on the horizontal axes.

column indicates that the R&D expenditure share to GDP increases from 2 to 6 percent when we increase
αK from 0.28 to 1.18. The third column represents the labor cost ratios that range from 0.04 to 0.16. The
bottom panel of Table 3 displays the steady-state R&D expenditure share and the labor cost ratio when the
returns to research are low (φ = 0.5).
Figure 9 depicts the impulse response functions for four values of αK . The results are robust to thevalue of returns to knowledge in goods production. When φ = 0.5, the real effect of a monetary shock underW-flexible/R-sticky is weak, regardless of αK . In contrast, when φ = 1, the output responses are large. The
size of the output responses also increases with αK . In particular, following a one-percent increase in themoney supply, output increases by 0.22 percent when αK = 0.28 (shown by the dotted line in the right panelof Figure 9) whereas output increases by 0.31 percent when αK = 1.18 (shown by the thin solid line in thesame panel).

32

As in Section 4.4, the stronger real effect of money under a larger αK results from the increased steadystate expenditure share of R&D to GDP (i.e., (Q/P )X/[C +(Q/P )X]). The effect of Xt on total expenditure
Ct + (Qt /Pt )Xt becomes larger when the steady-state expenditure share of R&D to GDP increases. The realeffects of monetary shocks substantially differ between φ = 0.5 and φ = 1 for each value of αK . Therefore,
our key message that the shape of the idea production function matters for the real effects of monetary shock
remains unchanged.

4.6

Functional forms of production functions

Before closing this section, we discuss the robustness of our results to alternative functional forms of theidea and goods production functions.
In the idea production function, we assume that knowledge becomes nonexcludable with only a oneperiod delay. This assumption may be too strong. We thus replace the assumption with a new assumptionthat the transformation of newly created ideas into nonexcludable knowledge requires multiple periods. In
particular, we can consider the following generalized idea production function with partial transformation:λG(NR,t (j), K̄t−1 ) = NR,t (j)φ K̄t−1,(33)

instead of (22). Here K̄t−1 is a function of lagged nonexcludable knowledgeK̄t−1 =TY

(Kt−τ )ντ ,

(34)

τ =1

where

PT

τ =1 ντ = 1 and T can be infinity. In (33), the knowledge becomes nonexcludable more slowly becauseTK̄t−1 is the weighted geometric mean of lagged nonexcludable knowledge of {Kt−τ }τ =1 , as indicated by (34).
This specification reduces to the previous idea production function when ν1 = 1 and T = 1.
Our results concerning the real effects of money do not change, irrespective of how we specify K̄t in themodel.22 The reason is straightforward, as suggested in Section 3.3. Under a sufficiently low δ, a monetaryshock does not generate a sizable change in the stock of knowledge Kt . Because K̄t is the geometric mean
22 The simulation results are available upon request.

33

of lagged Kt , K̄t is near constant. As long as K̄t is near constant, the model’s dynamics can be explainedby the mechanism discussed in Section 3.3.23
Turning to the goods production function, the generalization (e.g., a constant elasticity of substitution
production function to allow for the complementarity between workers and knowledge) does not affect our
main results under W-flexible/R-sticky. To see this, recall that real GDP can be expressed as F (NW,t , Kt ) +(Qt /Pt )G(NR,t , Kt−1 ). As long as Kt is nearly constant and worker wages are flexible, neither NW,t nor
Qt /Pt fluctuates in response to a monetary shock. As suggested by (26) and (27), this result holds nomatter how we generalize the goods production function F (NW,t , Kt ). Output responses are quantitatively
similar under any shape of the goods production function when the configuration of nominal wage rigidities
is W-flexible/R-sticky.

5

Concluding remarks

In the economic growth literature, it is well-known that the production of new ideas and knowledge spillovers
can be fundamental sources of endogenous growth. In contrast, in the business cycle literature, while theprocyclicality of R&D activity suggests their importance for business cycles, there has been little attention to
the role of idea production in understanding output and employment fluctuations. One may conjecture thatthe impact of R&D activity may be negligible in business cycles because both the shares of R&D expenditure
to GDP and the labor cost ratio of researchers to workers are small.
We show that this conjecture is not always correct in the transmission mechanism for monetary shocks.
Incorporating idea production and knowledge capital in a New Keynesian sticky-wage model provides new
insights into monetary business cycles. Ideas and knowledge can play a crucial role in generating the realeffects of monetary shock, depending on the degree of the returns to research in idea production. If the returns
to research are high in idea production, nominal rigidities only in researcher wages generate a substantial
real effect of money. Even though researchers are quantitatively small in the economy, the real effect of
ν1
23 We can also consider the idea production function (33), where K̄1−ν1 . In this specification, only
t−1 (j) = Kt−1 [Kt−1 (j)]some knowledge capital becomes nonexcludable while the remainder remains firm specific. For the same reason discussed, thesimulation results based on this idea production function are similar to the results described in the main text.

34

money is comparable to when there are nominal rigidities in the wages of workers, who are quantitatively
dominant in the economy. By contrast, if the returns to research are low in idea production, nominal
rigidities in researcher wages fail to produce a substantial real effect of money. Indeed, the model’s dynamics
are effectively the same as a model without researchers.
Our results imply that the shape of the idea production function is key to understanding monetarybusiness cycle models with ideas and knowledge capital. In his survey of idea-based growth theory, Jones(2019a) concludes that “[t]he shape of the idea production function remains an intriguing subject of study.”
Our analysis suggests that this is indeed so, not only for economic growth but also monetary business cycles.
The analysis in this paper could be extended in many important directions. First, it would be interesting
to incorporate productivity and fiscal policy shocks into the model. While we focus on the real effects ofmonetary shock, the literature on business cycles emphasizes productivity shocks as an important driver offluctuations. Likewise, introducing fiscal policies in this model would allow us to see how a fiscal policy shock
influences R&D sectors. Second, although we did not introduce physical and human capital, incorporatingthese elements into our model would allow us to consider how our sticky-wage model is related to the growthmodels originally developed by Romer (1990) and Jones (1995). Finally, examining alternative specificationsof monetary policy (e.g., the presence of an effective lower bound on the nominal interest rate) would also
be an important direction for future research.

35

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38

Appendix
Equilibrium conditions
In what follows, we describe the log-linearized equilibrium conditions.
From the first-order conditions for households, together with (17) and (18),\M
RS h,t=bh,t + σ C
bt ,
ψh N

(35)

c∗
W
h,t

=

c∗
βθh Et Wh,t+1 + (1 − βθh )






1
ch,t + Pbt + W
ch,t ,\
M
RS h,t − W
1 + ψh εh(36)c ∗ represents the log deviation of the optimal nominal wage for h = W, R. Also, M\
where W
RS h,t is theh,taverage marginal rate of substitution between consumption and labor. Here, the hatted lowercase letters
are the log deviations from the initial steady state before a monetary shock materializes. Because of ourassumption of sticky wages, we have log-linearized laws of motion for nominal wages:∗
ch,t = θh W
ch,t−1 + (1 − θh ) W
ch,t
W
,

(37)

for h = W, R.
We next turn to the log-linearized equations for firms. Based on our assumption of the Cobb–Douglas
production function, (12), (13), and (27) are linearized as

cW,t − Pbt − C
bt
W
bt
Q
b t − Pbt − σ Cbt
QbW,t ,= −N(38)

cR,t + (1 − φ) N
bR,t − λKb t−1 ,
= W

(39)

=

h
i
bt − Kb t + β (1 − δ) Et [Qb t+1 − Pbt+1 − σ C
bt+1 ].
[1 − β (1 − δ)] (1 − σ) C

(40)

In terms of the firm’s technology, we have the log-linearized law of motion for knowledge capital:b t = (1 − δ) K
b t−1 + δ X
bt .
K

39

(41)

Goods production and idea production functions are

bW,t + αK
bt,(1 − α) N

bt
C

=

bt
X

bR,t + λK
b t−1 .
= φN

(42)
(43)

Using the definitions of nominal and real GDP, we derive the equations for GDP. The log deviation ofreal GDP is derived from (16):bt + ω(Xbt + Qb t − Pbt ),YbtGDP = (1 − ω) C

(44)

where
−1

ω=

αδ [1 − β (1 − δ)](Q/P )X=−1
C + (Q/P )XµP + αδ [1 − β (1 − δ)]is the steady-state expenditure share of R&D to GDP.
The remaining equations used for simulations are associated with the money market equilibrium. The
money demand from firms can be approximated bycW,t + N
bW,t + κ W
cR,t + N
bR,t ,
ct = (1 − κ) W
M

(45)

where
κ=

W R NRW W NW + W R NR

is the steady-state wage share of researcher labor as calculated from the labor cost ratio of researchers toworkers. Finally, money supply follows a random walkct = M
ct−1 + ξt ,M

(46)

ct = Mcts .
where we use the money market equilibrium condition: M
Equations (35) – (46) consist of the log-linearized system of equations characterizing the equilibrium40

ct , WcW,t , W
cR,t , Kbt, Wc∗ , Wc∗ , NbW,t , NbR,t , C
bt , Mbt , Xbt ,\\conditions for 15 variables M
RS W,t , M
RS R,t , Pbt , Q
W,tR,tand YbtGDP .

Data
For the labor cost ratio of researchers to workers, we take the data from the 2018 OES compiled by the
Bureau of Labor Statistics.24 The OES reports the number employed and annual mean wages of various
occupations, among other things. We classify these occupations into our definition of “researchers” and“workers.”
Our definition of “researchers” includes some research-related managers, computer and mathematicaloccupations, engineers, scientists and researchers, and doctors. The remaining occupations are considered“workers.” All occupations included in our definition of “researchers” are presented in Table 4.
Using the mean annual wage and the number of employed, this classification results in a labor cost ratioof researchers to workers of (WR NR )/(WW NW ) = 0.08.

24 The data are available at https://www.bls.gov/oes/2018/may/oes nat.htm.

41

Table 4: List of occupations classified as “researchers”

Medical and Health Services Managers
Natural Sciences Managers
Computer and Information Research Scientists
Mathematicians
Operations Research Analysts
Statisticians
Misc. Mathematical Science Occupations
Architects, Except Naval
Surveyors, Cartographers, and Photogrammetrists
Aerospace Engineers
Agricultural Engineers
Biomedical Engineers
Chemical Engineers
Civil Engineers
Computer Hardware Engineers
Electrical and Electronics Engineers
Environmental Engineers
Industrial Engineers, Incl. Health and Safety
Marine Engineers and Naval Architects
Materials Engineers
Mechanical Engineers
Mining and Geological Engineers, Incl. Mining Safety Engineers
Nuclear Engineers
Petroleum Engineers
Misc. Engineers
Agricultural and Food Scientists
Biological Scientists
Conservation Scientists and Foresters
Medical Scientists

42

Misc. Life Scientists
Astronomers and Physicists
Atmospheric and Space Scientists
Chemists and Materials Scientists
Environmental Scientists and Geoscientists
Misc. Physical Scientists
Economists
Survey Researchers
Psychologists
Sociologists
Urban and Regional Planners
Misc. Social Scientists and Related Workers
Agricultural and Food Science Technicians
Biological Technicians
Chemical TechniciansGeological and Petroleum Technicians
Nuclear TechniciansSocial Science Research Assistants
Misc. Life, Physical, and Social Science Technicians
Dentists
Optometrists
Pharmacists
Physicians and Surgeons
Veterinarians
Audiologists