Download Problem Set 4

Macroeconomics I: Economic Growth
Problem Set 4
Solutions
1
Expanding Variety: Romer (JPE 1990).
1. The economy in the Romer model consists of three activities. The final good sector uses
labour and intermediate goods as inputs. Each firm in this sector is characterized by a
Cobb-Douglas production function and acts under perfect competition (takes prices as
given). The intermediate goods sector uses final goods alone as inputs (no labour needed),
each firm in this sector has constant marginal productivity and produces a specialized
intermediate good i for which it is the monopoly producer. Finally, the R&D sector uses
either final goods or labour as the sole input (depending on the exact specification that
we will employ) and is characterized by constant returns. The output of the R&D sector
is a Blueprint (or patent) which is sold to an intermediate good producer.
To prove that the production function exhibits constant returns to scale we apply the
definition for CRS:
F (λX1 , ..., λXn ) = λF (X1 , ..., Xn )
with Xi representing the 1, ..., n input factors. The single steps are:
� nt
F (λxt (i) , λNt ) =
(λxt (i))1−α di (λNt )α
�0 nt
=
λ1−α [xt (i)]1−α di λα Ntα
0
� nt
= λ
[xt (i)]1−α di Ntα
0
= λF (xt (i) , Nt )
nt represents the number of varieties of the intermediate sector, or in other words the
number of different types of intermediate goods that are produced. When using an integral,
it determines the interval length on which the intermediate goods are distributed.
2. The problem of the representative firm in the final goods sector is to maximize profits. As
there is no capital in this version, the firm maximizes its profit statically in every period:
�
�
� nt
max Yt − wt Nt −
pt (i) xt (i) di
Nt ,xt (i)
0
�� nt
�
1−α
s.t.
Yt =
xt (i)
di Ntα
0
To obtain the demand for inputs we need to determine the FOCs with respect to Nt and
each xt (i) and solve for the input factor. The demand is then a function of the price of
1
the input factor.
Nt = αYt /wt
�
�1
(1 − α) α
D
xt [pt (i)] =
Nt
pt (i)
where the fact has been exploited that
� nt
∂
xt (j)1−α dj = (1 − α) xt (i)−α .
∂xt (i) 0
The inverse demand is
pt (i) = Ntα (1 − α) xt (i)−α
3. The problem of the monopolistic firm is to maximize profits by setting the price pt (i) for
good i taking the demand function (or the inverse demand) from the final goods sector as
given. The profit maximizing price level may be obtained by
max [pt (i) xt (i) − φxt (i)]
pt (i)
FOC
∂xt [pt (i)] !
=0
∂pt (i)
�
�1
�
�1
(1 − α) α
(1 − α) α Nt
0 =
Nt + [pt (i) − φ]
pt (i)
pt (i)
αpt (i)
�
�1 �
�
(1 − α) α
pt (i) − φ
0 =
1−
pt (i)
αpt (i)
φ
pt [xt (i)] =
=p
1−α
∂πi
∂pt (i)
= xt (i) + [pt (i) − φ]
Alternatively one can maximize w.r.t the quantity and taking the inverse demand as given.
max [pt (i) xt (i) − φxt (i)]
xt (i)
FOC
∂πi
∂xt (i)
=
∂pt (xt (i))
!
xt (i) + pt [xt (i)] − φ = 0
∂xt (i)
!
= −αNtα (1 − α) xt (i)−α xt (i) + pt [xt (i)] − φ = 0
�
��
�
pt [xt (i)]
φ
pt [xt (i)] =
=p
1−α
Due to the symmetry of all products, the price in all sectors is identical. The only two
parameters affecting the price are the production costs φ and the elasticity α of the final
production function.
4. Using the profit maximizing price of the intermediate firm and plugging it into the demand
function, we obtain the equilibrium amount of intermediate goods:

1
α
(1
−
α)
�  Nt
xt [pt (i)] =  �
φ
1−α
xt (i) =
2
�
(1 − α)2
φ
�1
α
Nt
which is linear in the total amount of labour in the economy. It follows that total output
becomes
� 1−α
� nt �
(1 − α)2 α
Yt =
Nt1−α di Ntα
φ
0
�
� 1−α
(1 − α)2 α
= nt
Nt
(1)
φ
The production function for the final output after using the optimality conditions becomes
linear in the number of intermediate goods nt and the population size Nt . Since we
assume no population growth, the growth rate of output will be equal to the growth rate
of varieties:
Ẏt
ṅt
=
Yt
nt
5. Ouput Yt can be used for consumption Ct and as an input for the intermediate and R&D
sector. Total expenditure must therefore equate the costs ψ for all new product inventions
ṅ and the costs φ for intermediate good production x of all nt sectors. At the BGP with
constant savings rate:
Γt = sYt = ṅt ψ + φxnt .
By using equation (1), the equation for intermediate good production, and dividing by nt
yields
ṅt
nt
=
=
=
1
[sYt − φxnt ]
nt ψ
 �
� 1−α
�
�1 
1
(1 − α)2 α
(1 − α)2 α 
s
N −φ
N
ψ
φ
φ
�
�1 �
N � α−1
s
α
φ
(1 − α)2
−
1
ψ
(1 − α)2
As we know from question 4, output grows at the rate varieties growth and we obtain
�
�1 �
Ẏt
N � α−1
s
2 α
=
φ
(1 − α)
−1
Yt
ψ
(1 − α)2
To obtain a positive per capita growth rate, the ”savings rate” needs to be large enough
in comparison with α: s > (1 − α)2 . An increase in this amount increases the growth rate
of output permanently both in absolute and per capita terms. The level of the population
N affects the growth rate as well (scale effect), larger economies would grow faster in
this setup. This is due to the fact that larger economies have larger amounts of savings
(but not a higher saving rate) which allow them to generate more new varieties every
period. Finally, the model will only reach a BGP with constant positive output growth if
the constant saving rate fullfills s > (1 − α)2 . Otherwise output growth either accelerates
towards infinity or declines to zero.
6. R&D production now requires labor input (constant at BGP) and the growth rate of
varieties is given by
ṅt
H
=
.
nt
ψnt
3
Already from this equation it becomes clear that as the number of varieties nt increases,
the growth rate of the economy (given by growth rate of varieties) decreases towards zero.
Hence the long-run growth rate tends to zero. By integrating the equation ṅt = H
ψ we can
obtain the exact evolution of varieties
� T
� T
H
ṅt dt =
dt
0
0 ψ
H
nt =
t + n0
ψ
The growth rate of the economy evolves according to
ṅt
=
nt
H
ψ
H
ψt
+ n0
=
1
t+
ψ
H n0
which depends on the initial number of varieties and decreases as time evolves. The longrun growth rate is obviously zero:
ṅt
lim
=0
t→∞ nt
7. In this case the R&D sector is characterized by externalities from past R&D output. The
BGP growth rate is constant and characterized by
ṅt
H
=
nt
ψ
When comparing questions 5-7, we observe long-run growth in questions 5 and 7, but
no growth in question 6. In (5), there are no direct externalities in R&D technology.
However, the input into R&D production (final output) depends positively on the number
of varieties. Investing a fraction of final output into R&D causes an increase in varieties,
but this in turn increases output and thus raises the input into R&D. This ”feedback”
effect results in endogenous growth. In (7), input in the R&D production (research labor)
is constant, but there are direct externalities in the R&D technology. More varieties imply
higher change in the number of varieties, ṅt (the more inventions we have, the smarter we
become). If new varieties are instead generated by a factor which does not accumulate
and there are no externalities, as in (6), the constant inputs into research generate only a
constant number of new varieties and not an increasing number that is needed to generate
constant varieties growth. This is why we do not observe endogenous growth in question
6, although the R&D production function is similar to the one in question 5.
2
Opening Up the Romer Model
1. Zero profit condition in R&D sector is imposed by free entry into R&D and requires that
the value of a new patent equals the cost incurred to invent a new intermediate good.
With the price of patent denoted by Pt , and the cost of employing labor in R&D being
wt , zero profit condition reads
Pt = ψ
wt
nt
Or, stating this condition as an arbitrage condition between manufacturing and research
labor:
4
One unit of labor withdrawn from manufacturing produces an increase in varieties of
ṅt = nψt , whit the value of this increase being nψt Pt . In equilibrium, this has to be equal
to the value which the same unit of labor would have produced had it stayed in the
manufacturing sector, wt (value of marginal product of labor in manufacturing).
Substituting for wt by expression found in question (2) of part one
Pt = ψα
�
(1 − α)2
φ
� 1−α
α
from which we can see that the equilibrium price of patent is constant,
Pt = πrtt , so that
Ṗt
Pt
= 0. Therefore
�
� 1−α
1
(1 − α)2 α
[pt (i) xt (i) − φxt (i)] = ψα
rt
φ
1
�
�
�
� 1−α
�
�
1 φ − (1 − α)φ (1 − α)2 α
(1 − α)2 α
Lt = ψα
rt
1−α
φ
φ
Lt =
ψrt
.
1−α
In the steady state with constant interest rate, labor allocation between manufacturing
ψr
and research sectors will also be constant so that L = 1−α
. Imposing labor market clearing
condition, labor has to be fully employed, N = L + H, implying that labor employed in
the research sector is given by
H=N−
ψr
.
1−α
2. Since final output is used for consumption and as input into intermediate-good production
at the rate φ,
Yt = Ct + φxnt
we can use the expression above, the equilibrium amount of intermediate input and the
expression for output derived in question (4) of part one to examine the relation of output
and consumption per capita growth rates.
Ct = nt
�
(1 − α)2
φ
� 1−α
α
�
(1 − α)2
L−φ
φ
�1
α
Lnt
and in per capita terms,
ct
�
� 1−α
�
�1
(1 − α)2 A α L
(1 − α)2 A α L
= nt
−φ
nt
φ
N
φ
N
�
� 1−α
�
�1
α
2
2 α
L
(1 − α)
(1 − α)
.
= nt 
−φ
N
φ
φ
5
With constant steady-state labor allocation, it follows that consumption per capita grows
at the same rate as output (and output per capita as population does not grow):
ċt
ẏt
ṅt
=
=
≡g
ct
yt
nt
The Euler equation can be written as g = σ(rt − ρ), where r, and thus g, will be constant
at the BGP. With steady-state research labor defined in the previous question, the growth
rate of output per capita at the BGP then reads
�
�
ṅt
1
ψr
g=
=
N−
,
nt
ψ
1−α
and using the Euler equation to substitute for interest rate,
g=
1
ψ (1
− α)N − ρ
(1 − α) +
1
σ
.
BGP growth rate increases with higher productivity of research activities (lower ψ) and
with lower rate of time preference, ρ. Furthermore, bigger economies will also grow faster
at the BGP - scale effect.
3. To be formal, let us use ”ˆ” to denote the variables in the open economy, except for those
that are given, such as all parameters and population size. To derive the equilibrium
equations for the open economy, we follow the same procedure, taking into account the
import and export of intermediate inputs.
The problem of the representative firm in the final goods sector becomes
max
L̂t ,x̂t (i),x̂∗t (i)
s.t.
Ŷt =
��
0
n̂t
�
Ŷt − ŵt L̂t −
x̂t (i)1−α di +
�
n̂∗t
0
�
0
n̂t
p̂t (i) x̂t (i) di −
�
�
0
n̂∗t
�
p̂∗t (i) x̂∗t (i) di
x̂∗t (i)1−α di L̂αt
To obtain the demand for inputs we need to determine the FOCs with respect to L̂t and
each x̂t (i) and x̂∗t (i), and solve for the input factor.
L̂t = αŶt /ŵt
�
�1
(1 − α) α
D
x̂t [p̂t (i)] =
L̂t
p̂t (i)
�
�1
(1 − α) α
∗D ∗
x̂t [p̂t (i)] =
L̂t
p̂∗t (i)
The inverse demand is
p̂t (i) = L̂αt (1 − α) x̂t (i)−α
p̂∗t (i) = L̂αt (1 − α) x̂∗t (i)−α
Since the two economies are identical, the demands and prices of any particular intermediate input will be the same, except that intermediate goods are itself different. Knowing
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this, the problem of the monopolistic firm in each economy is to maximize profits by setting
the price p̂t (i) for good i taking the demand function (or the inverse demand) from the
final goods sector as given, but now the revenue is coming from both markets, domestic
and foreign, and the cost of production is related to production for both domestic and
foreign market. The profit maximizing price level may be obtained by
max [p̂t (i) 2x̂t (i) − φ2x̂t (i)]
p̂t (i)
which yields exactly the same solution for optimal price as in the closed economy
φ
= p,
1−α
p̂t [xt (i)] =
true also for the foreign economy, p̂∗t (i) = p. Using the profit maximizing price of the
intermediate firm and plugging it into the demand function, we obtain the equilibrium
amount of intermediate goods:
�
(1 − α)2
x̂t = x̂∗t =
φ
�1
α
L̂t
It follows that total output becomes

� 1−α � ∗ �
� 1−α 
� n̂t �
α
α
2
2
n̂t
(1 − α)
(1 − α)
 L̂1−α
Ŷt = 
+
di L̂αt
t
φ
φ
0
0
� 1−α
(1 − α)2 α
= (n̂t +
L̂t
φ
�
� 1−α
(1 − α)2 α
= 2n̂t
L̂t
φ
n̂∗t )
�
(2)
(3)
with n̂∗t being of the same measure as n̂t . Therefore, if the labor allocated to final good
sector is the same as in closed economy, the output in the open economy is doubled
compared to the autarky. The growth rate of output per capita is still given by the growth
rate of varieties:
g(ŷt ) = g(n̂t ),
which is to be derived as following. The equilibrium wage is given by
ŵt = αŶt /L̂t
�
= 2αn̂t
(1 − α)2
φ
� 1−α
α
and thus the cost of innovation is double compared to autarky
ŵt
ψ
= 2ψα
n̂t
7
�
(1 − α)2
φ
� 1−α
α
Now lets look at the benefit side of doing research. There are no capital gains/losses,
since the price of patent, P̂A is constant (in equilibrium it is equal to the cost given above,
thus constant), so what determines the price of patent from the ”benefit” side is only the
discounted flow of profits in the intermediate sector, associated to this particular patent
π̂t
r̂t
=
=
1
2[p̂t (i) x̂t (i) − φx̂t (i)]
r̂t
�1
�
��
1
φ − (1 − α)φ (1 − α)2 α
2
L̂t
r̂t
1−α
φ
Finally, we know that in equilibrium, both cost and discounted profit have to be equal
to the price of patent (one force coming from R&D sector, the other from intermediate),
so we have the following key condition from which we derive the steady-state equilibrium
allocation of labor (remember, steady-state interest rate is constant):
�1
�
� 1−α
�
��
1
φ − (1 − α)φ (1 − α)2 α
(1 − α)2 α
2
L̂t = 2ψα
=⇒
r̂t
1−α
φ
φ
ψr̂
1−α
L̂ =
Ĥ = N −
ψr̂
1−α
where we use labor market clearing to get the last row. Now, knowing the steady state
labor in research and using the Euler equation, we derive the growth rate of open economy.
g=
1
ψ (1
− α)N − ρ
(1 − α) +
1
σ
.
As derived above, the growth rate does not change compared to the autarky environment.
But are there any level effects? Lets go back to Euler equation on BGP. Having concluded
that the growth of consumption is unchanged in the open economy, from the Euler equation
it must be that the BGP interest rate is the same. Therefore, labor allocation is exactly
the same, and assuming without loss of generality that n̂t was the same as nt when BGP
was reached, at every time instant open economy has double BGP output, profit and
wages compared to its closed counterpart.
Why is the growth rate not changing? Intuitively, with double measure of varieties, the
marginal product of labor is doubled, implying that the equilibrium wage has to duplicate
and this in turn duplicates the cost of innovation. However, due to duplication of profits in
the intermediate sector, it follows that the cost and return on research change in the same
proportion and the equilibrium allocation of labor is unchanged. There are no forces that
tend to move labor between sectors. At the same time, same R&D labor can not invent
varieties faster unless its productivity or externalities are increased, which is not the case
here. Merely doubling the size of the market does not have any effect on the BGP growth,
since the engine of growth comes from R&D where no changes to technology occurred.
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4. Let’s be very formal here as well and denote this open economy variables by ”˜”. We can
use the results from the previous question, but also allow for the exchange of ideas between
the two economies which transforms the R&D production function as following
dñt
H̃t
=
(ñt + ñ∗t ).
dt
ψ
With equal measure of varieties their growth rate becomes
g(ñt ) = 2
H̃
.
ψ
Zero profit condition in R&D sector now reads
P̃t = ψ
w̃t
w̃t
=ψ
∗
(ñt + ñt )
2ñt
Following the same procedure as in previous question, the equilibrium wage can be derived
w̃t = 2αñt
�
(1 − α)2
φ
� 1−α
α
and it follows that the price of patent is unchanged compared to the closed economy,
P̃t = Pt = ψα
�
(1 − α)2
φ
� 1−α
α
.
Why? It is true that marginal product of labor has doubled and so the cost that R&D
sector has to pay for it is double, but with innovation technology used here, externalities
in the R&D production are double, as well. In other words, for the same labor (which is
now twice as expensive), we get twice as many new patents. It is clear why per unit patent
cost, and thus it’s price, is unaffected.
However, the benefits of doing research change as the profits in the intermediate sector
associated to a patent increase. Higher profits on one side, but still low cost on the other
has to imply steady-state equilibrium labor reallocation towards the research sector as
doing research is now relatively more profitable. The equilibrium is reached when benefits
and cost are equal (both equal to the price of patent).
1
2[p̃t (i) x̃t (i) − φx̃t (i)] = ψα
r̃t
�
(1 − α)2
φ
1 ψr̃
.
21−α
ψr̃
H̃ = N −
2(1 − α)
L̃ =
Growth rate of output per capita at the BGP is now given by
9
� 1−α
α
�
�
1
ψr̃
g = 2
N−
ψ
2(1 − α)
1
ψ (1 − α)2N − ρ
=
(1 − α) + σ1
which is clearly bigger than the growth rate in a closed economy. The result comes from
the fact that opening to both flow of inputs and ideas is not merely doubling the market
size, but implies magnified research externalities. What previously meant ”standing on
shoulders”, now means ”standing on shoulders of somebody twice as tall”. However, this
positive effect of openness to the flow of goods and ideas depends crucially on how we
model research technology.
Note: You can also go back and use Euler equation to find exactly what r̃ is and how it
compares to r, BGP interest rate in the closed economy. You should find that it is higher
ψr̃
but not twice as high, so from L̃ = 12 1−α
it necessarily follows that L̃ < L, and the research
labor is higher.
5. Assume now the growth rate of varieties depends not on absolute measure of labor devoted
to research, but on its share in total population instead.
ṅt =
1 Ht
nt
ψN
Zero profit condition in R&D sector becomes
wt
N
nt
�
� 1−α
(1 − α)2 α
= ψα
N
φ
Pt = ψ
Now, the equilibrium labor allocation is derived from
�
� 1−α
1
(1 − α)2 α
[pt (i) xt (i) − φxt (i)] = ψα
N
rt
φ
�1
�
� 1−α
�
��
1 φ − (1 − α)φ (1 − α)2 α
(1 − α)2 α
Lt = ψα
N
rt
1−α
φ
φ
Lt
N
=
ψrt
.
1−α
so, the condition that previously determined the equilibrium labor in research, now determines its relative measure. Imposing labor market clearing condition at the BGP,
L = N + H, and the share of steady-state labor employed in the research sector is given
by
�
�
H
ψr
= 1−
N
1−α
10
Growth rate of output per capita at the BGP is now given by
�
�
ṅt
1
ψr
g=
=
1−
nt
ψ
(1 − α)
and it becomes clear that there is no scale effect. In the open economy, the growth rate of
varieties depends on the share of research labor in the total population of both economies,
and it is immediately evident that with symmetric economies, law of motion for varieties
is identical to its counterpart in the closed economy:
ṅt =
=
=
1
H
(nt + n∗t )
ψ N + N∗
1 H
2nt
ψ 2N
1H
nt
ψN
Since with trade in intermediate goods wage and profits change in equal proportion, the
cost - benefits ratio of research activities is unchanged compared to the closed economy
result, and thus, equilibrium labor allocation is unaffected. So, no change in the expression
for varieties growth rate and no change in research labor result in no change in the growth
rate of output per capita compared to the autarky environment (though level effect
exists!).
11