2005 INTERNATIONAL SYMPOSIUM ON KNOWLEDGE-BASED ECONOMY & GIOBAL MANAGEMENT
Tainan, TAIWAN, 3-4 November 2005
R&D BASED ECONOMIC GROWTH AND WAGE
GAP
Gi-Shian Su, and Hsiao-Ching Chang
Department of International Business, Southern Taiwan University of Technology
Tainan, TAIWAN
E-mail:Drsu@mail.stut.edu.tw
thesmallke@yahoo.com.tw
Abstract. The paper argues acceleration of technological progress contributes to widening wage gap
between R&D workers and laborers along the long-run balanced growth path. We extend Jones’[1] model
through incorporating an aggregate R&D workers supply function that underscores the stock of
knowledge as the entry barrier of the R&D industry. Moreover, the model also narrows down the
permissible ranges of the parameters in Jones’ knowledge accumulation function to generate the
widening wage gap.
Keywords: R&D based economic growth, entry barrier, wage gap, standing on shoulders effect,
duplication effect.
1 INTRODUCTION
[2][3]
Since the seminal work of Solow ,
technological progress has been seen as the source of
economic growth. But the Solow model fails to
explain the technological progress itself. It was not
until Romer’s work[4][5][6] that provides a formal way
of modeling the process of technological progress in
which emphasizes the importance of profit-seeking
economic agents engaging innovation. Jones[7] rejects
Romer’s knowledge accumulation assumption on the
empirical front. Moreover, Jones[1] outlined a model
that captures the essence of Romer’s argument and
modifies the knowledge accumulation function. But
Jones’ model does not take account of the wage
inequality between R&D worker and raw laborer
which is well documented [Blackburn[8]; Bound[9];
Juhn[10]; Katz[[11]; Levy[12]; Murphy[13]; OECD[14];
Gottschalk[15]; Wood[16]; and Esquivel[17]]. It is this
drawback that motivates this paper.
The paper is organized as follows: Section 2
modifies Jones’[1] model, focusing on the behavior of
savings (investment) assumption to pave the way for
the model presented in section 3. Section 4 derives
the permissible parameters in the knowledge
accumulation function, which are crucial in
determining the speed of technological progress.
Section 5 compares the implications of Jones’ and our
model.
(L) is divided into two categories: raw labor ( LY )
involving only hand and eye coordination and human
capital( H A ) or R&D workers.
L  H A  LY
(1)
The labor force is assumed to grow at exogenous rate
of n, i.e.

L
 n
L
(2)
The final goods sector employ raw labor and
machines ( x j ) manufactured by the intermediate
sector to produce output ( Y ), which is the aggregate
of consumption goods and raw capital. The
production function of the final goods sector is
assumed as
Y  L1Y

A
0
xj dj
(3)
where A is the index of existing stock of knowledge
or design available in the economy and  is a
parameter lies between 0 and 1. The firms in the final
goods sector are price takers with operation objective
of profit-maximization
2 R&D BASED MODEL OF
ECONOMIC GROWTH WITH
CLASSICAL SAVINGS ASSUMPTION
max L1Y
The economy consists of three sectors: the final
goods, the intermediate goods and R&D. Labor force
where p j is the price of jth intermediate goods
LY , x j

A
0
x j dj  wY LY 

A
0
p j x j dj
(4)
Gi-Shian Su, and Hsiao-Ching Chang.
(machine) and wY is the wage of raw labor. The first
order conditions are
We are to study steady-state solution of the
model. Define the share of R&D worker ( s A ) as
wY  1   
sA 
Y
LY
and
p j   L1Y xj 1
(5)
.
(6)
Equations (5) and (6) are the demand function of raw
labor and x j respectively. The aggregate production
function of the final goods sector can be shown as
Y  K   A LY
 1
HA
L
(12)
At the steady state, the variable s A is a constant.
Taking logarithm over both side of (12 and
differentiating with respect to time, we have


H
L
0  A 
HA
L
(13)
(7)
Substitute (2) into (13) to obtain
The firm in the intermediate sector has to get patent
from R&D sector and rent raw capital from the final
goods sector to produce machines. One unit of raw
capital can be transformed into one unit of new
machine and vice versa. The evolution of total raw
capital or designed machines follows from the
classical savings assumption.

HA
n
HA
(14)
From (10) and (11), we have

A   A H A

K
Y
 sK
 d
K
K
or
where sK is the exogenous savings rate and d is the
depreciation rate, also assumed to be exogenous.
Finally, a firm with patent enjoys monopolistic profit
(  ) as follows:
   1   
Y
A
(9)
The R&D sector is composed of many R&D workers
who utilize the existing stock of knowledge to
develop new knowledge or design. Once a new idea
has been developed, the R&D worker obtains patent
and sells or rents it to the intermediate sector. The
patent never expires. The production function of this
sector is assumed as

A   HA
where
 H
(15)
(8)
(10)
λ-1
A

A
H
  1A
A
A
(16)

The growth rate of knowledge ( g A  A / A ) must be
a constant in the steady state. From (16) and (14), we
determine the rate of technological progress as
gA 
n
1
(17)
Just as Solow’s model, the steady state is characterrized by the constancy of capital-effective labor ratio,
defined as
~
K
k
k 

AL
A
(18)

A
(11)
 is the arrival rate of R&D effort and  is the
productivity of R&D workers, which is an
exogenous variable. The parameter  reflects the
duplication effect in research and is assumed to
be 0    1 in Jones model. Moreover, the
parameter   0 is to capture the standing on
shoulders effect, i.e. the existing knowledge is
h e l p fu l i n d e v e l o p i n g n e w kno wled ge.
Taking logarithm on both side of (18) and
differentiating with respect to time with equation (2),
the definition of g A and the growth rate of labor force
in mind, we have
~
~
k  sK k   1  s A
 1   d
~
 gA  n k
~
(19)
At steady state we have k  0 . Substitute this con-
~
*
dition into (20) to obtain steady-state k as
Gi-Shian Su, and Hsiao-Ching Chang.
 s  1  s A  1  11
~
k   K

 d  gA  n 
(20)
where  is the arrival rate of R&D given in (15). The
price of patent ( PA ) is the present value of future
earnings generated from holding the patent. At the
steady state, we have g y  g A . Because y L  Y , we
A
To determine the growth rate of per capita income
y ) as
growth, define output per effective labor ( ~
Y
y
~
y 

AL
A
(21)
From (1) and (12), we derive

0
(22)
Then substitute (22) into (7) to get
Y  K   A  1  s A  L  1
(23)
Substitute (23) into (21) to have
~y  k~  1  s
A

PA 
LY   1  sA  L
 1
(24)
Substituting (20) into (24), we have

 1
sK
~
y*  

 d  gA  n 
1 
(25)
e
rt
dt 

r  n
(29)
With the help of equations (11), (9), (29), equation
(30) can be written as

rn

 ent
Equating the wage of raw labor and the payoff of
R&D worker, we solve for s *A , the equilibrium share
of R&D workers
.
(30)
wH   PA  1    Y  wY
LY
 gA
sA
A
conclude that the growth rate of ( Y / A ) is n . In
addition, we can see from (9) that  and Y / A are
proportional. Therefore, the growth rate of  is n .
Because the patent last forever and the discount rate
(interest rate) is constant (we prove that in the
appendix), the price of patent is
HA
LY
(31)
Finally, substitute (1), (12), (27), and (A7) into (31)
to obtain
From the definition of (21), we determine steady-state
income per capita,
y * , as


sK
y*  

 d  gA  n 
s *A 

1
1 
sA
 At 
(26)
Income per capita is proportional to A because
s A , g A , sK ,  , d , and n are constant. Therefore, we
have
n
gY  g A 
1
To close the model, Jones assumes a person can
provide raw labor or do R&D at his/her will.
Therefore, the payoff of the two activities should be
equal and sA can be calculated from this condition.
We will modify this assumption in the next section
The wage of raw labor is given in (5). The wage
of R&D worker ( wH ) is governed by the following
equation
wH   PA
(28)
 2





1 



n

 d 
 n 
1



  n 1   

sK



α λn
(32)
3
(27)
1
THE MODEL
The model differentiates the market of R&D
workers and raw labor. Through the force of supply
and demand, we determine the associated equilibrium
wage and quantity from which we derive the wage
ratio ( wR ) and the share of R&D workers.
The demand function of R&D workers is derived
from Jones’ model. Substitute (29) into (28) and take
into consideration of (11) and (9) to get
wH 
gA   1
rn

Y
HA
Assume the supply function of R&D workers as
(33)
Gi-Shian Su, and Hsiao-Ching Chang.
H A  F  wH , Y , A  
wH Y
A2
(34)
and supply of R&D workers. Solving (33) and (34)
simultaneously, we determine the momentary
equilibrium wage of as
H A is positively related with wH because people tend
to be attracted to high- pay jobs. The higher the
existing knowledge level is, the higher the barrier of
entry because it is necessary to study the existing
stock of knowledge before the person can start to
innovate. Therefore, H A and A are negatively
correlated. This is why A appears in the denominator
in (34). Finally, a bigger economy should be able to
supply more R&D workers ceteris paribusi. Another
rationale in formulating the supply function as
equation (34) as follows: rewrite (34) as
and the momentary equilibrium number of R&D
workers
w Y
HA  H
A A
 g  1
s*A   A
 rn 
(35)
From the experience gain from Jones’ model, Y / A
and L should grow at the same rate at steady state.
To have steady-state solution, H A should be related
with the growth rate of L. This is why ( Y / A ) appears
in the right hand side of (35). The first A appears in
the denominator to reflect the entry barrier of R&D
sector.
Methodologically speaking, we directly impose
the assumption of aggregate supply function of R&D
workers in (34). In other words, we do not provide
micro foundation of (34). This should not be the
drawback of the model. Many prominent economists
do not attach much importance on providing micro
foundation in formulating macroeconomic model. In a
sequence of interviews conducted by Snowdon[18]ii ,
several economists were asked the following question
“How important do you think it is for macroeconomic
models to have choice-theoretic micro foundations?”
To give an example, we quote the reply from Gregory
Mankiw:
”It is certainly true that all macro phenomena are the
aggregate of many micro phenomena; in that sense
macroeconomics
is
inevitably founded
on
microeconomics. Yet I am not sure that all
macroeconomics necessarily has to start off with
microeconomic building-blocks. We have a lot of
models like the IS－LM model which are very useful,
even though those models don’t start off with the
individual unit and build up from there .”(p.332
italics added)
Next, we consider the interaction of the demand
i
Two possibilities for the case of bigger economy: (1) the
greater is the population, the more people can become
R&D workers and (2) the higher the average income, the
more affordable to become an R&D worker.
ii In addition to Gregory Mankiw, the authors also
interviewed Stanley Fischer (p34), James Tobin (p129),
David Laidler (p180) and Robert Lucas (p221). They all
carried similar attitude.
*
H
w
1
 g  1
  A
 rn 
 g  1
H A*   A
 rn 
 2
(36)
 A

1
 2
(37)
Y
A


Substitute (37) into (12), we have
1
 2


Y
AL
(38)
Finally, we solve the steady-state share of R&D
*
*
worker, s A . At the steady state, s A must be a
constant. Following the logic presented in section 1,
we conclude
gY  g A 
n
1
(27)
, exactly the same as Jones’ model. Substituting
equations (17), (21), and (25) into equation (38), we
have
s*A 












1













1 





λn
 1 
1


λn
 2  n 
 d 
1

 n
sK







2













sK


λn
n

d


1


1





1







λn
 1 
1


λn
 2  n 
 d 
1

 n
sK







2













sK


λn
nd 
1


1





(39)
Next , we determine the equilibrium wage and the
quantity of raw labor.
Under the assumption of full employment, the
quantity of raw labor is simply the labor force minus
the equilibrium number of R&D workers:

L*Y  L 1  s*A

(40)
Solving (5) and (40) simultaneously to determine the
equilibrium wage of raw labor as
Gi-Shian Su, and Hsiao-Ching Chang.
w*Y 
y  1 α
1  s*A


(41)

*
and d into (44), we obtain the functional relation wR
and (  ,  ) demonstrated in Fig. 1.
Define the wage ratio between R&D worker and raw
labor ( wR ) as
wR 
wH
wY
(42)
Substitute the equilibrium wage described by (36) and
(41) into (42) to get
wR 
1 s 
~
y
Fig.1 the relation between wR* and ( , )
1

2
gA 
 r  n 1    


*
A
(43)
where ~y is given in (12). Using (17) and (25),
equation (43) gives the steady-state wage ratio as
From Fig. 1 we infer wR* is an increasing function
of  and  . Therefore, the permissible ranges should
lie within the upper contour set UC f (wr'  1) where the
contour is defined as
λn

nd 
1
wR*  

sK



1















λnα


 



λn
2
   n 


 d 

 
1
  n  1   1    



sK





 


1
2
1
2
0.015317 λ  3

 0.015317  0.1 

1 

1 

0
.
181







2










0
.
4
*
0
.
015317
λ







0.15317 λ

  0.4 2  0.015317 



0
.
1



1




 0.15317  1  0.41   

0.181









(45a)
(44)
or
According to (44), wR* is a function of parameters
only and hence is a constant. Therefore, wH and wY
should grow at the same speed. In fact, (36)
demonstrates that wH should grow at the rate of
g A and (41) reveals that the growth rate of wY is g y .
Here we obtain the same result as Jones' model,
specifically, equation (27).
4
THE PERMISSIBLE RANGES OF

AND

Jones’ model already points out the range of
parameters  and  must be 0    1 and 0    1 .
Our model adds another feature—wage differentials
and hence the range of parameters should be
narrower. The existence of wage gap implies the
  12.414108586945368896 1   
(45b)
Combining the restriction from Jones’ model, we
deduce the permissible ranges lie within the triangle
area in Fig. 2 enclosed by the two north-east borders
of the box and the straight line which is the locus of
(45b).The slope of the straight line is negative. The
smaller the duplication effect (associated with larger
 ) is, the faster technological progress is, which
*
*
causes wR to increase. To keep wR  1 ,  (standing
on shoulders effect) has to decline.
*
wage ratio wR must be greater than 1. Due to the
complexity of (44), we use computer simulation to
pin down the permissible range of parameters. The
data is adopted from Barro & Sala-i-Martin[19].
n  1.5317 % is the U.S. average population growth
rate between 1890 and 1990. sK  18.1 % is the
average gross savings rate from 1870 to1989.   0.4
and d  0.1 . Substituting the values of n , sK ,  ,
Fig. 2 the permissible ranges of  and 
Gi-Shian Su, and Hsiao-Ching Chang.
5
COMPARATIVE STEADY STATE
We focus on the effect of a change in the
parameters on the endogenous variables s *A and wR*
and compare our model with Jones’ model.
An increase in  i.e. the smaller the number of
R&D workers simultaneously engaging the same
research (duplication effect) will accelerate the speed
of technological progress and hence raise the entry
barrier of doing research. Therefore s *A is lower. This
fact is demonstrated in Fig.3. A decrease in s *A pushes
up the wage of R&D workers and causes wR* to
increase, as reported in Fig. 4. These are the
implication of our model.
Fig. 5 relation between  and
s *A in Jones’ model
The value of parameters are
n  0.015317 , d  0.1 ,   0.98 , sK  0.181 ,   0.4
and   0.01 ~ 0.99 .
An increase in  (greater standing on shoulders
effect) accelerates the accumulation of knowledge
and consequently increases the entry barrier of R&D
industry. This, in turn, reduces the number of R&D
workers and s *A as described in Fig. 6. Just as Fig. 4.
a decrease in s *A pushes up the wage of R&D workers
and causes wR* to increase, as reported in Fig. 7..
Fig. 3 relation between  and
s *A
Fig. 6 relation between  and s A
*
Fig.4 relation between
 and wR*
The value of parameters are
n  0.015317 , d  0.1 ,   0.99 , sK  0.181 ,   0.4 and
  0.124 ~ 0.99 。
In contrast, Jones’ model assumes no wage
inequality since there is no entry barrier in R&D
*
industry. Therefore wR is always equal to 1. An
increase in  also boosts the wage of R&D workers
and attracts more people doing research. Therefore,
s *A increases as reported in Fig. 5.
Fig. 7 relation between
 and wR*
The value of parameters are
n  0.015317 , d  0.1 , sK  0.181 ,   0.4 and   0.99
Jones’ model reveals different implication again
because there is no entry barrier. An increase in 
induces faster wage increases in R&D that attracts
more people engaging research and hence increases
s *A as depicted in Fig. 8
Gi-Shian Su, and Hsiao-Ching Chang.
Fig. 8 relation between  and s *A in Jones’ model
The values of parameters are
n  0.015317 , d  0.1 , sK  0.181 ,   0.4 and

 0.99 .
An increase in the population growth rate n
raises the speed of the increase in the number of R&D
workers, quickens the process of knowledge
accumulation and eventually heightens the entry
barrier that causes s *A to decline as demonstrated in
Fig. 9. As explained previously, a decrease in s *A
causes wR* to increase (see Fig. 10)
Fig. 9 relation between n and s *A
Fig. 10 relation between
n and wR*
Fig. 11 relation between
model
n and s *A in Jones’s
The values of the parameters are
n  0.01 ~ 0.1, d  0.1 ,   0.98 , sK  0.181 ,
  0.4 and   0.5 。
We are in a position to draw the main conclusion
of this paper. An increase in the population growth
rate, an increase in the standing on shoulders effect
and a decrease in the duplication effect contribute to a
faster technological progress [see equation (27)].
These changes also increase wage inequality, as
evident from Fig. 4, 7, and 10. Therefore, our model
predicts that a faster technological progress goes hand
in hand with wage inequality because of the change in
underlying forces of population growth and the
improvement in the efficiency in knowledge
accumulation. Our conclusion is consistent with the
empirical findings of Esquivel[[17]] and Bound[[9]] who
argue wage differential is driven by technological
change.
Finally, an increase in the savings rate sK raises
the size of the economy. A bigger economy will churn
out more R&D workers and hence increase s *A . This
fact is reported in Fig. 12. From previous experience,
an increase in s *A depressed wR* as demonstrated in
Fig. 13. The impact of sK on s *A in Jones’s model is
the same as our model (see Fig. 14). But the
underlying adjustment mechanism is different. An
increase in sK raises the raw capital to labor ration,
which in turn cheapen the price of raw capital. Lower
price of raw capital increases the profitability of
intermediate goods sector and eventually the value of
patent (and the wage of R&D workers). Higher pay
attracts more people join the R&D sector and in
equilibrium, raises s *A .
The values of the parameters are
n  0.01 ~ 0.1, d  0.1 ,   0.98 , sK  0.181 ,
  0.4 and   0.5 。
However, Jones’ model also generates different
result from our model. As argued above, an increase
in population growth rate hastens technological
progress and boosts the wage of R&D workers.
Because there is no entry barrier, the number of
researchers increases and raises s*A . This result can be
found in Fig. 11.
Fig. 12 relation between
s K and s *A
Gi-Shian Su, and Hsiao-Ching Chang.
APPENDIX : THE PROOF OF
INTEREST RATE R IS A CONSTANT
From the text, we have (with suppression of subscript
j)
p   L1Y x 1
(A1)
and it can be shown that
Fig. 13 relation between
s K and wR*
The values of the parametersare
n  0.015317 , d  0.1 ,   0.96 , sK  0.01 ~ 0.6 ,
  0.4 and   0.8 。
p 
1

r
(A2)
Combining (A1)and (A2), we have
r   2 L1Y x 1
(A3)
With (7) and the fact that
x
K
A
, we have
Fig. 14 relation between
model
sK and s *A in Jones’
The values of the parameters are
n  0.015317 , d  0.1 ,   0.99 , sK  0.01 ~ 0.6 ,
  0.4 and   0.99 .
6
CONCLUDING REMARKS
We extend Jones’ model to generate wage
differentials by emphasizing the existence of entry
barrier in the R&D sector and consequently narrows
down the ranges of key parameters (  and  ) in the
knowledge accumulation function. We also
demonstrate that faster technological progress
induced by faster population growth, less duplication
in R&D effort and/or more efficient use of the
existing knowledge in R&D could widen the wage
gap between skilled(R&D) and unskilled workers(raw
laborer).
Our model is subject to the following limitations:
1. The technological progress is assumed to be
horizontal innovation not vertical innovation i.e.
technological progress only increases the variety of
intermediate durable goods instead of raising the
productivity of it. 2. Since the return of old patent
does not decrease because of the advent of new one,
the model fails to capture Schumpeterian-style
process of creative destruction[20].
 1
K
1
r   2 L1Y     2  ALY  K  1
 A
 ALY 1 K    2 Y
 2
K
K
(A4)
From (8), we obtain
 

K d
K

Y 


K
sK
(A5)
~
From k  constant , we derive



K A L
 
K A L
(A6)
Substitute (A5) and (A6) into (A3) to have
r
 2 n  g A  d 
(A7)
sK
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