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xj
Solving for pj (xj ) from equation (2) we get the inverse demand function pj (xj ) = L1
Y
xj
1
max L1
Y
rxj ,
and the problem can be rewritten as xj xj
and the …rst order condition is
[xj ] :
2
L1
Y
xj
1
r = 0.
It follows that xj =
r
2
1
1
LY
and it is clear that xj = x for all j since the right hand side does not depend on j. Plugging this on the inverse demand function we obtain r (3) pj = , and again it is clear that pj = p for all j.
1
Exercise 2
Use the results from the previous exercise and the fact that in equilibrium Ax = K to calculate:
a) (5 points) The equilibrium interest rates, r, as a function of Y =K.
Answer: From equations (2) and (3) it follows that
L1
Y
1
x
r
=
and, therefore
2
r=
L1
Y
1
x
.
Multiplying the right hand side by Ax=Ax we get
1
2 LY
r= then, using the fact that Y = L1
Y
Ax
,
Ax
Ax and K = Ax, it follows that
2
r=
Y
.
K
(4)
b) (5 points) The equilibrium wage in the …nal goods market, w, as a function of Y =LY .
Answer: From equations (1) we have that w = (1
Z
) LY
A
xj dj. …show more content…
Answer: Since distortions 1 and 3 imply underprovision of ideas, to …x the distortion the government should increase sA . The opposite is true for distortion 2.
c) (5 points) Would this type of government intervention a¤ect the long-run growth rate of the economy?
Answer: No, when
2 (0; 1), and
2 (0; 1) the long-run growth rate of output per capita is given by equation (7). Notice that, the long-run growth rate is exogenous and does not depend on sA .
d) (5 points) Is the long-run growth rate e¢ cient?
Answer: Yes, the long-run growth rate is exogenous. The government might intervene by providing incentives for agents to go to the research sector for instance, e¤ ectively increasing sA . This policy would increase welfare, provided that distortions 1 and 3 dominate distortion 2. However, this intervention would have no e¤ ect on the long-run growth rate of output, it would have only a level e¤ ect. In other words, because of the distortions described in part a, the e¢ cient allocation is di¤ erent than the equilibrium, but the long-run growth rate of the e¢ cient allocation is the same as the equilibrium one.
e) (5 points) Now suppose that
Solving for pj (xj ) from equation (2) we get the inverse demand function pj (xj ) = L1
Y
xj
1
max L1
Y
rxj ,
and the problem can be rewritten as xj xj
and the …rst order condition is
[xj ] :
2
L1
Y
xj
1
r = 0.
It follows that xj =
r
2
1
1
LY
and it is clear that xj = x for all j since the right hand side does not depend on j. Plugging this on the inverse demand function we obtain r (3) pj = , and again it is clear that pj = p for all j.
1
Exercise 2
Use the results from the previous exercise and the fact that in equilibrium Ax = K to calculate:
a) (5 points) The equilibrium interest rates, r, as a function of Y =K.
Answer: From equations (2) and (3) it follows that
L1
Y
1
x
r
=
and, therefore
2
r=
L1
Y
1
x
.
Multiplying the right hand side by Ax=Ax we get
1
2 LY
r= then, using the fact that Y = L1
Y
Ax
,
Ax
Ax and K = Ax, it follows that
2
r=
Y
.
K
(4)
b) (5 points) The equilibrium wage in the …nal goods market, w, as a function of Y =LY .
Answer: From equations (1) we have that w = (1
Z
) LY
A
xj dj. …show more content…
Answer: Since distortions 1 and 3 imply underprovision of ideas, to …x the distortion the government should increase sA . The opposite is true for distortion 2.
c) (5 points) Would this type of government intervention a¤ect the long-run growth rate of the economy?
Answer: No, when
2 (0; 1), and
2 (0; 1) the long-run growth rate of output per capita is given by equation (7). Notice that, the long-run growth rate is exogenous and does not depend on sA .
d) (5 points) Is the long-run growth rate e¢ cient?
Answer: Yes, the long-run growth rate is exogenous. The government might intervene by providing incentives for agents to go to the research sector for instance, e¤ ectively increasing sA . This policy would increase welfare, provided that distortions 1 and 3 dominate distortion 2. However, this intervention would have no e¤ ect on the long-run growth rate of output, it would have only a level e¤ ect. In other words, because of the distortions described in part a, the e¢ cient allocation is di¤ erent than the equilibrium, but the long-run growth rate of the e¢ cient allocation is the same as the equilibrium one.
e) (5 points) Now suppose that