3012 Words Oct 24th, 2014 13 Pages
University of Minnesota
Department of Economics
Econ 4331w: Economic Development
Homework Assignment 2 - Answer Key
Exercise 1
a) (5 points) Describe the …nal goods producing sector in the Romer model. Is there perfect or imperfect competition? Write down the problem of the …rm in this sector and derive the …rst order conditions for optimality. Answer: This sector produces the …nal good using intermediate goods and sells that good to consumers. There
RA
are a large number of perfectly competitive …rms using the CRS production function Y = L1 xj dj. The
Y
0
…rm’ problem is s Z A
Z A
1
pj xj , max LY xj dj wLY
LY ;xj

0

0

and the …rst order conditions are

[LY ]
[xj ]

: (1
:

) LY

L1
Y

A

xj dj
Now, suppose time is discrete, t = 0; 1; 2; ::: and that if you sell your patent today (t = 0) whoever buys it will start receiving the pro…ts tomorrow (t = 1). For how much would you be able to sell your patent for? (Hint 1: you would sell it for the present value of the future
P1
pro…ts; Hint 2: if a 2 (0; 1), then t=0 at = 1= (1 a)). Answer this question for the following possibilities:
a) (5 points) The monopoly pro…ts are constant. That is,

t

=

for all t.

Answer: The price of the patent, PA , will be equal to the present value of future pro…ts, that is
PA =

1
X
t=1

t t (1 + r)

where the summation starts at t = 1 because that is when whoever buys the patent will start to receive the pro…ts. Notice that since t = for all t,
PA =

1
X
t=1

1
1+r

t

,

and that since r > 0, it follows that 1= (1 + r) < 1 and we can solve this using the simple geometric series formula !
!
1 t X
1
1
1+r
PA =
1 =
1 =
1
1
1+r
r
1 1+r t=0 and …nally, we get

PA =

3

r

:

b) (5 points) The monopoly pro…ts are grow at a constant rate n. That is, and suppose that n < r.
1 =

t+1

= (1 + n)

t

for all t. Let

1
X

PA =

t=1

Since

t+1

= (1 + n)

t

for all t with

1

=

t

t:

(1 + r)

it follows that t 1

t

hence,
PA =

1
X (1 + n)t t=1 = (1

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