University of Warwick
EC9A2 Advanced Macroeconomic Analysis
Problem Set 4: Endogenous Growth
Jorge F. Chavez∗
December 3, 2012
1
Question 1: Quality ladder
Consider the quality ladder R&D model. Suppose β = 1.
(a) Suppose the economy exists only one period, (period 1). Is the competitive allocation of
R&D workers, H, efficient?
Solution. Production is yt = At ⇒ Yt = Lt At . Ideas accumulate according to:
At = At+1 + it
where it = µAt−1 Ht .
Recall that β is the fraction of the surplus from innovation that is allocated to R&D workers. Then
incomes in both sectors will be:
ItH
= βµAt−1 Lt
ItL
= At−1 + (1 − β) µAt−1 Ht
Equilibrium in the labor market imply that ItH = ItL which implies:
βµLt = 1 + (1 − β) µHt
When β = 1 the above condition reduces to µLt = 1. Then, since Lt = Ht + Nt , equilibrium in the
labor market implies:
Ht = Nt − 1/µ
In a one period economy, we need to assume an initial exogenous non-zero endowment of ideas
A0 > 0. Production will be:
Y1
= L1 (1 + µH1 )A0
= (N − H1 )(1 + µH1 )A0
∗
e-mail:j.chavez-cotrado@warwick.ac.uk
1
EC9A2 (Fall 2012)
Problem Set # 4
To check efficiency we need to check whether the market allocation maximize output:
max
0≤H1 ≤L1
{(N − H1 ) (1 + µH1 ) A0 }
FONC is:
∂Y1
= − (1 + µH1 ) A0 + µ (N − H1 ) A0
∂H1
i. Corner 1:
∂Y1 = −A0 + µN A0 = (µN − 1) A0 < 0 (⇒⇐)
| {z } |{z}
∂H1 H1 =0
>0
>0
ii. Corner 2:
∂Y1 = − (1 + µN ) A0 > 0 (⇒⇐)
| {z } |{z}
∂H1 H1 =N
>0
>0
iii. For interior:
∂Y1 = − (1 + µH1 ) A0 + µ (N − H1 ) A0 = 0
∂H1 H1 ∈(0,L1 )
Solving:
H1planner =
µN − 1
N
1
=
−
2µ
2
2µ
Equilibrium in the labor market implied that H1ce = N − 1µ. To claim that it is efficient it must
be the case that it solves the planner’s problem and therefore it must satisfy the FOCN:
∂Y1 = −1 − µ (N − 1/µ) + µ (N − N − 1/µ)
∂H1 H ce =N −1/µ
1
)
(
µN − 1
= 1 − µN ̸= 0
= −µ
µ
Therefore market equilibrium is not efficient. In fact is easy to show that H1ce > H1planner . That
is, market equilibrium provides too much R&D (over-investment). The reason is the assumption
β = 1: the reward to workers in the R&D workers in this case is too high.
(b) Suppose the economy exists for two periods, (periods 1 and 2) and the interest rate between
the two periods is zero. Is the competitive allocation in the first period, H1 efficient? Hint:
consider the effect of a small change in H1 on total output in both periods, holding H2 in
its equilibrium level.
Solution. As before, for t = 1, 2 equilibrium in labor markets imply ItH = ItN and N = Ht + Lt .
These conditions imply:
H1 = N − 1/µ
H2 = N − 1/µ
Jorge F. Chávez
2
EC9A2 (Fall 2012)
Problem Set # 4
In the two-period economy, total output (given that there is no discounting) Y = Y1 + Y2 where:
Y1 = (N − H1 ) (1 + µH1 ) A0
Y2 = (N − H2 ) (1 + µH2 ) A1
Recall that A1 = A0 + it = A0 + µA0 H1 = (1 + µH1 ) A0 . Then:
Y = (N − H1 ) (1 + µH1 ) A0 + (N − H2 ) (1 + µH2 ) (1 + µH1 ) A0
Planner solves:
max
(H1 ,H2 )∈[0,N ]2
{(N − H1 ) (1 + µH1 ) A0 + (N − H2 ) (1 + µH2 ) (1 + µH1 ) A0 }
FONCs for interior solution:
∂Y
∂H1
∂Y
∂H2
= (−1) (1 + µH1 ) + µ (N − H1 ) + µ (N − H2 ) (1 + µH2 ) = 0
= (−1) (1 + µH2 ) + (N − H2 ) µ = 0
Note that H2planner =
N
2
−
1
2µ .
Following the hint:
∂Y = −1 + 2µN − 2µH1
∂H1 H2 =N −1/µ
∂Y =1>0
∂H1 H1 =H2 =N −1/µ
which implies that competitive level of R&D is not optimal. In fact, now we have under-investment.
(c) Explain the different results for parts a and b.
Solution. An economy with an R&D sector is always characterized as having externality problems.
In the one-period economy, there is an externality across sectors: hiring an additional worker in the
R&D sector implies reducing the number of workers in the productive sector, which intern implies
that there less workers using the new technology. This results in a reduction in output. Because the
competitive allocation does not take that into account, the resulting outcome is overinvestment in
R&D.
In turn, in the two-period economy there are externalities across periods as investing in R&D today
produces a positive externality on the level of knowledge next period. Once again, this externality is
not taken into account by the competitive allocation outcome, resulting in underinvestment in the
first period.
Jorge F. Chávez
3
EC9A2 (Fall 2012)
Problem Set # 4
Question 2
Consider the quality ladder R&D model. Suppose that the number of new inventions (new
√
knowledge) produced in period t is At−1 Ht , where each of the R&D workers develops an equal
number of inventions. Therefore,
At = At−1 + At−1
√
Ht = At−1 (1 +
√
Ht ).
(a) For β = 1 find an implicit function defining Ht as a function of N .
Solution. Income in the R&D sector:
(
ItH
(because β = 1 )
= β
1/2
At−1 Ht
Ht
)
Lt
1/2
=
At−1 Lt Ht
Income in the production sector:
(
ItL
(because β = 1 )
= At−1 + (1 − β)
=
1/2
At−1 Ht
Ht
)
Ht
At−1
Equilibrium in the labor market implies:
−1/2
ItH = ItL ⇒ At−1 Lt Ht
−1/2
= At−1 ⇔ (N − Ht ) Ht
=1
Finally, the implicit function we were looking for is:
−1/2
g (N, H) ≡ (N − Ht ) Ht
−1=0
Note that this implies that N (H) = H + H 1/2
(b) For β = 1 show (using a figure) that there exists a unique H > 0 for any N, and that H is
increasing with N. What is the economic intuition behind these results? (i.e., the uniqueness
of H, that H is strictly positive, and that it is increasing in N ).
Solution. Figure 1 plots the N (H) function. We want to show (WTS) several things:
i. WTS H is increasing with N
Suppose not. Then exists:
1/2
N1
= H1 + H1
N2
= H2 + H1
1/2
where without loss of generality (WLOG) N1 > N2 which imply H1 < H2 . Then:
1/2
H1 + H1
1/2
> H2 + H2
1/2
⇒ H1 − H2 > H2
| {z } |
<0
Jorge F. Chávez
1/2
(⇒⇐)
−H
{z 1 }
>0
4
EC9A2 (Fall 2012)
Problem Set # 4
N
N (H) = H + H 1/2
H
Figure 1: Plot of N (H) = H + H 1/2
The economic intuition is as follows. As N increases, the market for technology increases as well
and so does H.
ii. WTS ∃H unique for any N
Note that N (H) is a strictly increasing and strictly concave function. Therefore there is a unique
N for any H. Since N (·) is also one-to-one (an injection), N −1 will also be one-to-one, which
implies that for any N , there is a unique H.
More formally, suppose H1 = H(N1 ) and H2 = H(N1 ). WLOG we can assume H1 > H2 . Then
H −1 (H1 ) = N1 and H −1 (H2 ) = N1 . But then:
H −1 (H1 ) = N1 = H1 + H1
1/2
1/2
= H2 + H2
= N1 = H −1 (H2 )
which contradicts H1 > H2 .
The intuition is as follows. The equilibrium condition is ItH = ItL . The LHS is a function of
H (the income of workers in the R&D sector declines with the number of such workers) while
the RHS is not a function of H nor L. Therefore for every N there is a unique value of H that
guarantees equilibrium.
iii. WTS H is strictly positive
We can get H from the roots of the quadratic function defined by N (H):
f(
√
√
(H)) ≡ H + (H) − N = 0
Using the standard general formula to get the roots of a f (·):
√
√
−1 ± 1 − 4 (1) (−N )
H =
2 (1)
√
1 + 4N − 1
=
2
Jorge F. Chávez
5
EC9A2 (Fall 2012)
Problem Set # 4
√
where we can rule out the negative root√because the (H) is defined as a positive number (recall
that new knowledge produced is At−1 (Ht )). Therefore:
H=
(√
)2
1 + 4N − 1
>0
2
The economic intuition here is that as H → 0 the marginal productivity
of H goes to +∞.
(
)
1/2
Recall that the production function of inventions is it = At−1 Ht
. Then, the marginal
productivity of Ht is
lim
Ht →0
∂it
∂Ht
=
−1/2
At−1
.
2 Ht
Thus:
∂it
= +∞
∂Ht
This is the same reason why we ruled out corner solutions.
(c) For β < 1 find an implicit function defining H as a function of N .
Solution. Now β < 1. Then:
−1/2
ItH
= βAt−1 Lt Ht
ItL
= At−1 + (1 − β) At−1 Ht
+1/2
Equilibrium implies:
−1/2
β (N − Ht ) Ht
1/2
Then βN = Ht
+ Ht so that the implicit function we are looking for is:
1/2
g (N, Ht ) = Ht
Jorge F. Chávez
+1/2
= 1 + (1 − β) Ht
+ Ht − βN = 0
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