EEP101/ECON125 Spring 99
Prof D. Zilberman
TA's: Malick/Marceau
Suggested Solutions to Problem Set 4
*
1.
Assume you receive a flow of income (V) at the end of every year for (N) years. This income
grows at a rate of (G) percent per year, and the discount factor is given by the nominal
interest rate (I). The present value (PV) of this flow of income is given by
PV =
V(1
G)
(1
I)
j 1
j
j 1
N
+
+

=
∑
(1)
Let's multiply both sides of equation (1) by (1+ G)
PV (1 + G)
=
V
1
G
1
I
j
j 1
N
+
+
=
∑
=
V
j
j 1
N
β
=
∑
, where
β =
+
+
1
G
1
I
(2)
Note that this geometric series converge if and only if
β <
1 , which means that
G
I
<
Let's multiply again both sides of equation (2) by
β
PV (1+ G)
β
=
V
j+1
j 1
N
β
=
∑
(3)
Subtracting equation (3) from equation (2), we obtain
PV (1+ G)(
)
1
β
= V (1
N
β
β

)
(4)
Upon dividing both sides of (4) by (1+G)(1
β
) , and after substituting back the value of
β
, we
finally obtain
V
I  G
1
1+ G
1+ I
N

(5)
Note that if
N
→ ∞
, then
PV =
V
I
G

.
*
Solution to question 1 provided
by G. Malick. Solution to question 2 provided by S. Marceau.
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Prof D. Zilberman
TA's: Malick/Marceau
The nominal interest rate (I) is approximately equal to the sum of the real interest rate (r) and the
inflation rate (
π
)
1
I = r +
π
(6)
Similarly, the growth in the value of
income (G) is explained by both the biological growth in the
volume of timber (Gb) and the growth in the price of timber (Gp)
2
.
G = Gb + Gp
(7)
Substituting (6) and (7) into (5)
PV =
V
(r +
)  (Gb + Gp)
1
1 + Gb + Gp
1 + r +
N
π
π

(8)
With equation (8) in hand, we can now proceed to answer the question.
(a) N = 10,000/400 = 25 ;
V = 400P; r =3%;
π
= 3%; Gb = 2%; Gp = 0. Hence, the present
value (or what the government should pay to prevent cutting) is given by
PV =
400P
.04
1
1
02
1
06
25
+
+
.
.
=
$6,177 P
(b) A year has passed (so we have 24 years left) and the inflation rate is now only 2%.
N = 24;
V = 400P; r = 3%; Gb = 2%; Gp = 0 and
π
= 2%. The present
value as of that moment (a
year later) is
PV =
400P
.03
1
1
02
1
05
24
+
+
.
.
=
$6, 684 P
A reduction in the inflation rate reduces the nominal interest rate, so the present value of the
flow of income increases (the government needs to pay him a little bit more to prevent him
from cutting).
(c) N = 25; V = 400P; r=3%;
π
= 3%; Gb = 2%;
but Gp = 1%. The present value is
1
This is not entirely correct.
The nominal interest rate is given by I = r +
π
+ r
π
. Since r
π
is very small for a
low real interest rate and for a low inflation rate, then we sometimes ignore this.
2
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 Spring '09
 Prof D. Zilberman, D. Zilberman TA, Prof D. Zilberman TA

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