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Section II. Economic Growth
C. Endogenous Growth Theory
Endogenous Growth Theory
The A-K Model
In the neo-classical model, long-term economic growth is determined outside of the
model and none of the fundamentals of the economy matter for long-term growth.
Without technology growth, which is assumed to fall freely from the sky, long term
growth will not exist because capital has diminishing returns and cannot replicate
itself at growing rates forever. A simple way to show an alternative type of growth is
two simply assume that technology is fixed, At = A, capital does not have diminishing
returns.
(1.1)
Yt  AKt  yt  Akt
Here the marginal and average product of capital are the same MPKt  APKt  A and
are constant. We can also see that
ln yt  ln A  ln kt
ln yt 1  ln A  ln kt 1
 ty

(1.2)
0   tk
So the growth rate of output is equal to the growth rate of capital per worker unit. As
in the previous neo-classsical model without capital the growth rate of the capital
labor ratio is equal to the investment capacity per unit of machine less the replacement
rate.
y
 tk1  s t  (n   )  sA  (n   )
(1.3)
kt
In this case, investment capacity per unit of machines is constant. Therefore, if sA >
(n+δ), average investment capacity is always greater than replacement rates and the
capital stock per worker unit will continue to grow at a constant rate. Moreover, this
growth rate will be a function of capital fundamentals like the level of s and n.
The difficulty with this production function is that it displays no role for workers.
The marginal product of labor is zero and presumably then wages should be zero.
Thus, this cannot be a realistic model to explain labor productivity. However, it does
show that the key to modeling long-run growth is some sort of diminishing returns. It
the accumulation of input can cause output to grow at a non-diminishing rate in
perpetuity, it stands to reason that input cannot display returns that diminish below the
point at which that input needs to be replaced.
Human Capital
Human capital is sometimes suggested as a potential engine of growth. To study this
we assume that human capital (i.e. education and experience) can be used for two
things. It can be used by workers to produce goods and it can be used by teachers to
produce even more human capital.
H t  H tE  H tY
(1.4)
Where Ht is human capital, H tE , is human capital used in the education sector and
H tY is human capital used in the production sector. Assume that a constant proportion
is used in each sector so that
H tE  s E  H t & H tY  (1  s E )  H t
(1.5)
Consider the production function where output is a function of capital and skill
adjusted labor supply and technology is assumed to be constant.
Yt  Kt  L*t 
1
 Kt  H tY Lt 
1
 yt  kt  (1  s E )  H t 
1
(1.6)
With this production function the growth rate of labor productivity will be a weighted
average of the growth rate of capital per worker unit and the growth rate of human
capital.

 ty    gtk  1     gtH     s 


yt
   n   1     gtH
kt

(1.7)
In this model, we also assume a model for education in the production sector.
Educationt  b  H tE  b  s E  H t
(1.8)
Where b is a parameter measuring the efficiency of the education sector. Education
adds to the stock of human capital which does not depreciate:
Ht 1  Ht  Educationt
(1.9)
Combine (1.8) and (1.9) to get
 tH1  b  s E
(1.10)
Note that the creation of human capital is a non-diminishing function of itself.
Therefore, human capital can grow in perpetuity at a constant rate. With human
capital perpetually growing, labor productivity can also grow at the same rate in the
long run. We can write the growth rate of labor productivity as:

 ty     s 


yt
   n   1     b  s E
kt

(1.11)
The dynamics of the capital-labor ratio can be described by the following graph. We
see that whenever the capital labor ratio is higher than APKBGP, the growth rate of the
capital labor ratio will be faster than the growth rate of education and faster than the
growth rate of labor productivity and will thereby be falling.
g tk
gty
g tH  b  s E
y
0
k
APKBGP
When the growth rate of capital per worker exactly reaches b·sE, then two things are
happening with capital productivity. The growing amount of capital used by each
worker is pushing down the average productivity of the machines. But at the same
time, workers are getting smarter and more educated. This makes the tools they work
with more productive. When these two effects exactly cancel out, capital productivity
will be stable and the economy will be on its balanced growth path.
[Work in Progress] ey for understanding long-term growth in this economy is
that human capital has non-diminishing returns. Not in the creation of output, but in
its own recreation. Consider the case where output productivity is non-diminishing in
human capital.
Yt  AL*t  AH tY Lt  yt  s E H t
(1.12)
While the accumulation of human capital is subject to depreciation
Technology Accumulation
Learning by Doing
In the neo-classical model, technology falls freely from the sky for anyone to use. The
technology appears like magic without any effort at all needed to acquire it. This
seems unrealistic, but we might have to consider that some advances in knowledge of
production techniques come not from careful study or costly research, but comes only
from experience in producing goods. Naturally, as people use industrial machinery to
produce goods they will come up with ways to use their capital stock more efficiently.
Technological advance might simply be the natural by-product of the production of
goods. Further, we might gather that once one person begins using a new technique to
produce goods, others will pick up on it and freely use it in their own production.
Thinking about this, however, we might think that an economy in which technology
appears as a by-product of the actual experience of producing goods might be quite
different than an economy in which capital falls freely from the sky.
An economy in which knowledge is created as a by-product or spillover from
production is sometimes referred to as a learning-by-doing economy. In this economy,
there are a fixed number of production locations indexed by n = 1….n. The
production function of goods at location n is:
Yt ,n  Kt ,n ( At Lt ,n )1  yt ,n  kt ,n At1
(1.13)
Assume that each firm is able to rent capital and labor in a competitive market at rate
Rt and Wt. The first order conditions are:
1
 1 Rt   1
MPK t   kt ,i  At 
 kt , i  
At
Pt
  Pt 
Y
WL
W
MPLt  (1   ) t ,i  t  t t ,i  (1   )
Pt
Lt ,i
PY
t t ,i
 1
1
Rt
(1.14)
The implication is that all firms will have the same capital labor ratio kt,i = kt and the
same labor share of income.
In this economy, knowledge comes from working with machines and the more
people work with machinery, the greater will be the stock of knowledge. The
technology level that is generated through the experienced of working with machines
is assumed to be proportional to the the average capital to labor ratio of the economy
as a whole.
1
1
At  A
 kt
1
K  K t ,2  ...  K t , N
Kt
1
kt 
 A  t ,1
Lt
Lt ,1  Lt ,2  ...  Lt , N
(1.15)
1
L
L
L

 A1   t ,1 kt ,1  t ,2 kt ,2  ...  t , N kt , N 
Lt
Lt
 Lt

So technology is a weighted average of the capital-labor ratio at each of the N firms.
So:
1
1
L
L
L

At  A1   t ,1 kt  t ,2 kt  ...  t , N kt   A1  kt
Lt
Lt 
 Lt
(1.16)
Substitute (1.16) into (1.13) to get
1
 1

yt  kt  A1  kt 



 A  kt
(1.17)
After we consider the externality effect of aggregate capital on technological advance,
the total impact of an increase in the capital labor ratio on labor productivity is
positive and non-diminishing.
The growth rate of labor productivity will then keep up with the growth rate of
the capital-labor ratio.
 ty1   tk1  s
yt
 (n   )  sA  (n   )
kt
(1.18)
So this is equivalent to the AK model, but the real wage rate is positive and keeps up
with labor productivity so labor share is constant.
However, one questionable effect of this model is that it divides the impact of
capital accumulation into two impacts, a direct effect and an indirect effect.
IndirectEffect
}
}
k
   t 1  (1   )  tk1
DirectEffect

y
t 1
(1.19)
But note, we usually estimate α ≈ ⅓ and (1-α) ≈ ⅔. So in this model, the indirect
effect of capital accumulation is more powerful than the direct effect. This may be
hard to believe.
Research and Development
There are two sectors of the economy that use labor. One sector of the economy
produces output goods using a Cobb Douglas technology.
Yt  Kt   At LYt 
1
(1.20)
The remainder of labor is used in the research and development industry,
LYt  LtA  Lt
(1.21)
Assume that a constant fraction of labor is used for research and development
LtA  s R  Lt where 0 < sR < 1 so LYt  (1  s R )  Lt
We can write labor productivity as
yt  kt   At (1  s R ) 
1
(1.22)
We can say that, as previously, growth in labor productivity is a weighted average of
growth in the capital labor ratio and technology
 ty     tk  (1   )   tA
(1.23)
Again as previously, growth in the capital labor ratio is a function of capital
productivity
y
 tk  s t  (n   )
kt
(1.24)
So, along the balanced growth path,  ty   tk   tA . The only difference between this
model and the previous neo-classical model is that we explore the creation of
technology.
There is a research and development industry that creates new inventions. The
number of new inventions are proportional to the number of workers working on R &
D
New Inventionst  Bt  LtA  Bt  s R  Lt
(1.25)
Where Bt is the efficiency of the R & D sector. For now assume that this efficiency is
a fixed parameter Bt=B. Technology advances with new inventions.
At 1  At  New Inventionst  B  s R  Lt
(1.26)
Divide both sides of the equation by At to derive the growth rate of technology
 tA1  gtA1 
At 1  At
L
 B  sR  t
At
At
(1.27)
If technology growth does go to a long-term balanced growth level, η, what would it
be? If technology growth remains at a stable level,   B  s R 
Lt
, then the ratio
At
(which represents R&D capacity per unit of technology) must also remain stable. But
R & D capacity is proportional to the population: the more people you have, the more
inventors you have and the more inventions you have. Since the population is growing
at rate n, R&D capacity is also growing at rate n. Therefore, technology growth is
stable when technology (the denominator of the ratio
population (the numerator of the ratio
Lt
) grows at the same rate as the
At
Lt
); i.e. η=n.
At
It is interesting that along the balanced growth path, none of the R & D
fundamentals (i.e. the efficiency of the R & D sector, B, or the proportion of
investment in R & D, sR) matter for the growth rate of technology. We can think about
this by examining a phase diagram of
Lt
. We see that when the growth rate of the
At
numerator, n, is greater than the growth rate of the denominator  A
 
A LA n
 
A LA n
0
L
A
k
We can draw these two growth rates as a function of the ratio
 
 A L A  sR  B  L A
n
L
0
L
A
k
BGP
A
Note that there will be one level of the labor-technology ratio L
BGP
A
where
technology growth will be equal to n.
The level of L
A
along the balanced growth path is L
BGP
A

n
so the level
B  sR
of technology is proportional to the population along the balance growth path The
factor of proportion is increased by the R& D fundamentals, B and sR. So when the
sR B
gLt |BGP . Thus, although better
economy is on the long run technology path At 
n
R&D fundamentals does not influence growth rates in the long run, it does impact the
level.
Imagine if for whatever reason, we saw the population technology level hit that
level. Then, at time t0 there was an increase in sR which shifts workers from the
prdocution sector to the R & D sector. The number of new inventions goes up. At any
level of population to technology, the technology growth rate should rise as R & D
capacity rises. But since technology goes up faster than population, the R & D
capacity relative to the stock of existing inventions will decline. The growth rate of
technology will fall until it is growing no faster than growth in R & D capacity which
in the long run can be no faster than growth in the population rate.
 
A LA
sR ↑
 
A LA
n
L
0
L
BGP
A
L
BGP*
A
k
A
gA
n
time
The temporary acceleration in technology growth leads technology to a new higher
path.
ln(A)
time
What is the impact of this rise in R&D investment on the level of labor productivity?
In the short-run, the transfer of workers from the production sector to the R&D sector
reduces output and the country’s level of labor productivity. The level of output will
start to rise as technology rises, offsetting the loss of workers. What happens to labor
productivity in the long run? Rewriting (1.22)

 1
y
yt   t  (1  s R ) At
 kt 
Along the balanced growth path.
(1.28)
yt
n   
 APK BGP 
so that
kt
s
yt   APK


BGP  1
(1  s R ) At
(1.29)
Along the balanced growth path


sR B
(1.30)
yt |   APK  (1  s ) At |   APK  (1  s )
gLt
n
So an increased level of sR may have positive or negative long-run effects on labor
BGP
BGP  1
R
BGP
BGP  1
R
productivity on the balanced growth path depending on how high the level of
investment already is.
We might also think about what this would mean for advances in productivity in
the very long run. Through human history, the number of people in existence has been
growing. However, demographic projections suggest that in the future, global
population will roughly peak. We see that in some countries birth rates are already
lower than replacement level and one country, Japan, may be shrinking in population.
What will happen in the future when global population rate reaches a steady state, or
n=0.
In the long run, the technology growth rate will go to zero if population growth
stops This might be hard to understand. After all, new inventions are proportional to
to the level of the populations. When population reaches its steady state and Lt = LSS
then there will be a fixed quantity of new inventions in every period
New InventionsSS  B  s R  LSS . So new technology will be developed in the long run.
However, in percentage terms, the new inventions generated in every period will be
infintessimally small relative to the overall quantity of inventions that would have
New InventionsSS
0
been developed through the past of human history. Therefore,
At
and growth in technology or labor productivity would be imperceptibly low.
One channel might prevent technology growth from slowing down in the far
future which me might think of as the “Shoulder of Giants” effect. Isaac Newton once
said “If I have seen farther than other men, it is because I have stood on the shoulders
of giants.” The giants in this case being those innovators who developed ideas in the
past. One might imagine such an effect is part of the development of new inventions.
If so the body of knowledge that has been developed in the past would make
developing new inventions either, making the R & D sector more efficient. Imagine if
the efficiency of the technology sector were a function of the existing stock of
technology, Bt = B•At, We might think of this spillover that is an externality of
creating knowledge. If you as an inventor create an invention, you will also generate
knowledge that will naturally help future inventors. Consider what that means for
technology growth
At 1  At  Bt  s R  B  At  s R  Lt 
(1.31)
At 1  At
 B  s R  Lt  gtA1   tA1
At
In this model, technology growth rates do not fall to the population growth rate,
since the knowledge spillovers of invemtion keep the technology process from
diminishing. Then, long-term economic growth is a function of itself. However, this
brings up another question. In this framework, long run technology growth is strictly
proportional to the number of inventors. However, we have observed over time that
the number of engineers and scientists has been growing, but growth has not been
accelerating.
Microfoundations of Technology Accumulation
In this model, technology is costly to produce. Therefore, it will only be produced if
the producers of technology (i.e. the inventors) receive some benefit from their work.
Consider if competitive, price taking firms simply rent technology in competitive
markets from the inventors. So, we might imagine that inventors license firms to use
their technology at rate Qt just as capitalists rent capital at rate Rt and workers rent
their labor at rate Wt. The profit function of the representative firm would be:
Y
PY
t t  Wt Lt  Rt Kt  Qt At
 1 Y 1
PK
Lt  Wt LYt  Rt Kt  Qt At
t t At
(1.32)
If the firm chooses a quantity of capital, labor and technology to rent in order to
maximize profits, the first order conditions will be
1 1 Y 1
Pt  
 K t  42
A4444444
Lt 3  Rt  0  
t
1444444
MPKt
Yt Rt
  Rt K t   PY
t t
K t Pt
Y W
Pt  (1   )  K t  At1 LYt   Wt  0  (1   )  Yt  t  Wt LYt  (1   )  PY
t t (1.33)
1444444442 444444443
Lt
Pt
MPL
t
Y
Q
Pt  (1   )  K t  At  LYt 1  Qt  0  (1   )  t  t  Qt At  (1   )  PY
t t
1444444442 444444443
At
Pt
MPA
t
So if we add up total costs:
(1 ) PY
t t
(1 ) PY
 PY
t t
}
}
}t t
Y
Total Costst  Wt Lt  Rt Kt  Qt At  (1   )  PY
t t    PY
t t  (1   )  PY
t t (1.34)
 [1  (1   )]  PY
t t  PY
t t  Total Revenues
So if the firm pays labor, capital, and technology its marginal cost, it will end up
earning negative profits even if it chooses quantities of inputs that will maximize
profits. The reason is that the Cobb-Douglas production function is constant returns to
scale in capital and labor, but is increasing returns to scale in capital, labor, and
technology. Any increasing returns to scale firm that operates in a competitive market
will lose money. For this reason, many industries thought to operate according to
increasing returns to scale (such as power distribution or other utilities) often operate
in a monopoly environment.
It stands to reason that to explain how an economy operates when it accumulates
technology that we could deviate in some way from perfect competition. Rather than
consider the implausible case that all national production was conducted by a single
monopolistic firm. Instead, we consider an intermediate case of monopolistically
competition.
So far, we have assumed that aggregate production is a simple function of
aggregate capital. By this idea, all that matters for knowing how much output we will
have is the total market value of capital. By this logic, it does not matter how the this
value of capital is allocated. If you have $1 trillion worth of trucks, then you will
produce the same as if you have $1 trillion worth of machine tools or $1 trillion worth
of computrers. Each of these, under the standard theory, would generate the same
amount of output as a well-balanced set of different types of capital valued at a total
of $1 trillion. This seems implausible, so lets imagine that capital might be imperfect
substitutes.
 At

Yt     xi ,t   ( LYt )1
 i 1


Yt   x1,t    x2,t    x3,t   .....  x At ,t




(1.35)


 ( LY )1
 t
In this production function, there are different types of capital goods indexed by i.
These might include capital of different types including trucks, machine tools,
computers etc. The total number of types of capital at time t is represented by At. The
quantity of each type of capital that is used is xi,t.
We can say a fow things about this production function. First, it has a constant returns
to scale form.
Imagine if we were to multiply all of the inputs by a constantκ,
including each type of capital, x1,t, x2,t,….,xA,t and labor LYt .
 At
 At 

  1
Y 1
Y 1


x
(


L
)



i ,t
t


    xi ,t    ( Lt ) 
 i 1

 i 1

   x      x      x   .....    x    1 ( LY )1 
1,t
2,t
3,t
At ,t
t








   x1,t    x2,t    x3,t   .....   x A ,t    1 ( LYt )1


t
 At

      xi ,t    1 ( LYt )1 
 i 1


At
(1.36)



 i 1
At

     xi ,t    1 ( LYt )1     1    xi ,t   ( LYt )1

 i 1




     xi ,t   ( LYt )1   Yt
 i 1

So if you increase each of the inputs by a proportionκthen you will increase output
At
by the same proportion.
The second point is that we have decreasing returns to scale in terms of each type
input. We can write this production function so that we see that each type of capital
helps labor to produce goods separatey.
 
Yt   x1,t  ( LYt )1   x2,t  ( LYt )1   x3,t  ( LYt )1  .....  xAt ,t



Draw the production function in terms of each type of capital.

( LYt )1 (1.37)
Figure 4
Y
x 

i ,t
( LYt )1
MPxi
1
xi
The slope of the production function is always positive but gets flatter as x gets
bigger. We would see that the marginal product of each type of capital is
 1
Y
MPxi 
    xi ,t  ( LYt )1  0
xi
 2
MPxi
     1   xi ,t  ( LYt )1  0
14
42
44
3
xi
0
(1.38)
Though this production function may seem very different than the standard
Cobb-Douglas production function, it actually is very close. Consider there is a total
market value of capital, Kt, this should be equal to sum of the market value of each
At
type of capital,
x
i 1
i ,t
 Kt . If the economy divides its capital evenly amongst the
various types so that xi ,t  xt , then At xt  Kt . The production function is
 At


Yt    xt   ( LYt )1   At   xt   ( LYt )1


 i 1

Substitute xt 
Kt
At
(1.39)
into (1.39) to get





K
Yt   At   t   ( LYt )1  At   At    Kt  ( LYt )1
At 
 

  At 
1
(1.40)
  Kt  ( LYt )1  Yt   Kt  ( At LYt )1


So when capital is divided evenly, this production function has the exact same form
as the Cobb-Douglas production function. Notice, that in this function, the number of
types of capital goods, At, augments the productivity of labor. Why does having many
types of capital goods make workers more productive? The reason is that there is a
diminishing returns to each type of capital as shown in Figure 4. Thus, the average
productivity of each type of capital is highest when the amount allocated to each
capital is stock is low. Given a total quantity of capital, Kt, the amount allocated to
each type can only be reduced if there are more types of capital. Therefore, with many
types of capital, producers can have each one operating at a high level of productivity
without facing too many diminishing returns. This, in turn, will make the workers
productive as they will be using many types of tools each one operating at a high level
of productivity rather just putting all the capital to a few uses whose productivity will
have been pushed down by diminishing returns. This type of technology function is
called a Returns to Variety production function.
In this model, we assume that firms hire workers in a competitive labor market
and also rent each type of capital from a firm that owns the franchise for capital of a
particular sort. Franchise i is the only firm that has a license to own capital good of
type i which he will rent to production firm at time t at price Qt,i. Assume the
production firm is a price taker. Profits of the production firm will be the revenues
minus the cost of hiring workers and the cost of renting all the capital stock from the
different franchises.
At
Y
Pr ofit  PY
t t    Qt ,i xt ,i   Wt Lt
i 1
At


 Pt    xt   ( LYt )1    Qt ,i xt ,i   Wt LYt
i 1
 i 1

At
(1.41)

 Pt  x1,t  ( LYt )1  ..   xi ,t  ( LYt )1  .....  x At ,t





( LYt )1 

 Qt ,1 xt ,1  ..  Qt ,i xt ,i  .....  Qt , At xt , At   Wt LYt
The first order conditions will be: 1) that marginal product of labor equals real
wage; 2) the marginal product of each type of capital should equal the real capital
rental cost:
1) (1   )
Yt Wt

LYt
Pt
i 2)    xi ,t 
 1
Y 1
t
(L )

Qi ,t
(1.42)
Pt
The first condition is familiar and no more needs to be said. However, the second
condition can easily be re-written as a demand curve for capital of type i. First raise
both sides of part 2) of (1.42) by the power
1
  1
1

1
 1
Y 1
t
 xi ,t ( L ) 
Qi ,t  1
Pt
1
 1
1
 xi ,t  Zt  Qi ,t  1
(1.43)
Where the variable Zt is just simplifying notation Zt   Pt 1 LYt . Note the exponent
1
1
  1
<0 is negative, so this condition is saying that the optimal quantity of xi,t is a
negative function of the price Qi,t. We can think of this as a demand curve in which
the demand for an object is a negative function of the price xi ,t  D(Qi ,t ) meaning

that the slope of the demand curve is negative
dxi ,t
dQi ,t
 D '(Qi ,t ) 
1 1
1
 Zt  Qi ,t  1
 1
Qi
D(Qi)
xi
Now lets turn to the problem of franchise i, the firm that controls i. This firm has a
license to construct capital of type i. We assume that he has obtained this license from
the inventor who invented the capital of type i. For simplicity, we assume that
franchise has a special way of constructing the capital that they rent. We imagine there
are capitalists who have a accumulated a mass of undifferentiated capital, Kt, which is
in the form of some sort of putty that can be molded into capital of any type i by
someone who has access to the instructions/blueprints for making that type of capital.
Only franchise i has the blueprints for capital of type i. To construct a quantity of
capital, xt,i they will need to rent an equal amount of undifferentiated capital from
capitalists at rate Rt . Their profits will be:
Profitst ,i  Qt ,i xt ,i  Rt xt ,i
(1.44)
If the franchise were a price taker, the first order condition would be Qt ,i  Rt .
However, the franchise is a monopolist that can set their own price. They know that
the quantity of goods they will sell will depend on the price they set. In particular, if
they want to increase the quantity of the goods they sell, they will have to cut their
price. Insert the demand curve xi ,t  D(Qi ,t ) into (1.44) to get

Qt ,i  D(Qi ,t )  Rt D(Qi ,t )
(1.45)
We see that profits are a function of the price charged by the monopolist franchise.
Pr ofit (Qt ,i )  Re venue(Qt ,i )  Cost (Qt ,i )
The price that generates the highest profit would be to set
dProfit
dRevenue dCost
0

dQ
dQ
dQ
(1.46)
(1.47)
The right hand side of this equation is simple to derive:.
dCost d  Rt D(Qi ,t )  Rt dD(Qi ,t )


 Rt D '(Qi ,t )
dQ
dQ
dQ
(1.48)
Revenue is a product of two functions of Qi,t. Therefore we can derive the slope of the
revenue curve with the product rule.
df
Product Rule If f(Q) = h(Q)∙g(Q) then f '(Q) 
 h '(Q)  g (Q)  h(Q)  g '(Q)
dQ
In this case, f(Q) = Revenue(Q), h(Q) = Q and g(Q) = D(Q), so we would write
f '(Q)  D(Q)  Q  D '(Q) and
dRevenue
 D(Qt ,i )  Qt ,i  D '(Qt ,i )
dQ
(1.49)
D(Qt ,i )  Qt ,i  D '(Qt ,i )  Rt  D '(Qt ,i )
(1.50)
Combining we get
Divide both sides of this equation by D '(Qt ,i )
D(Qt ,i )
D '(Qt ,i )
 Qt ,i  Rt
(1.51)
Factor out a Q from the left of the equation to get
 D(Qt ,i )

 1  Qt ,i  Rt  Qt ,i 

 D '(Qt ,i )Qt ,i 
D(Qt ,i )
Note that since D '(Qt ,i ) /<0,
<0 so 1+
D '(Qt ,i )Qt ,i
Rt
D(Qt ,i )
1
D '(Qt ,i )Qt ,i
D(Qt ,i )
D '(Qt ,i )Qt ,i
(1.52)
<1, so Qt,i > Rt,i. A
monopolist charges a price greater than the marginal cost of their inputs in order to
achieve profits. This markup is a function of the price sensitivity of the import.
dx
Rewrite (1.52) using the definition D '(Qt ,i )  t ,i and D(Qt,i) = xt,i
dQt ,i
Qt ,i 
1
Rt
xt ,i
Rt
dQt ,i

dxt ,i
dQt ,i
Qt ,i
1
(1.53)
Qt ,i
dxt ,i
xt ,i
Define the price elasticity of demand as the % reduction in demand resulting from a
1% increase in the price level  D  
dxt ,i
dQt ,i
xt ,i
Qt ,i

D '(Qt ,i )  Qt ,i
D(Qt ,i )
;
price
elasticity is the most commonly used measure of price sensitivity in microeconomics.
The markup of the monopolists is an inverse function of this price sensitivity. When
consumers are not very price sensitive, monopolists can take advantage by sharply
raising their price.
Qt ,i 
1
1
1
(1.54)
Rt
D
In this model,
1 1
1
1
 Zt  Qt ,i  1  Qt ,i
1 Zt  Qt ,i  1
1
D '(Qt ,i )  Qt ,i


1



1
1
D(Qt ,i )
  1 Zt  Qt ,i  1
1
Zt  Qt ,i  1
So,
(1.55)
1
is the elasticity of demand which is conveniently constant. The markup is
1
Qt ,i 
1
1
1
1
Rt 
1
1   1
Rt 
1

Rt
(1.56)
(1   )
This means a number of things. First, all franchises charge the same price. Since, the
demand curve for capital is the same for each type and all have the same type, the
production firm will indeed set xt,i = xt. So, the production function (1.40) applies.
Second, the monopoly franchise earns profits
Pr ofitt  Qt xt  Rt xt   Qt  Rt   xt   1 Rt  Rt   xt 
1
Rx
(1.57)
 t t
These profits can be used to pay for the license of the blueprint from the inventor.
Thus, creating an invention creates a cash flow of rents which could in turn be used to
pay for the resources necessary for R & D. Third, the existence of the markup means
that the rental price of capital is less than the marginal product of capital. This means
that those who invest in building K will receive a low return.
So we see a trade-off. Monopoly profits may be necessary to generate funds
necessary to pay for R & D while these profits will act as a tax on the accumulation of
capital equipment. A higher technology level may come at the cost of a lower capital
level. Most monopoly power for inventors comes from