02(5)
= $110,517.09.
6. At an interest rate of 5%/year, the present discounted value of the 3 payments is
8000[(1.05)
–1
+ (1.05)
–2
+ (1.05)
–3
] = $21,785.98. At an interest rate of 15%/year, the
present discounted value of the 3 payments is 8000[(1.15)
–1
+ (1.15)
–2
+ (1.15)
–3
] =
$18,265.80. As the interest rate increases, the discount factor attached to future net
returns increases, and thus the present discounted value of the opportunity falls.
7. In 2010, Exponentia’s GDP/capita is 200,000/800 = 250 zlotniks/person. The
growth rate of
real
per capita GDP (measured in 2010 dollars) is simply the rate of
growth of
nominal
GDP (= 10%/year) minus the rate of inﬂation (= 5%/year) minus
the rate of growth of population (= 2%/year), or 3%/year. Denoting per capita
income by
X
, we have the following relation:
X
t
= 500 =
X
0
e
g
t
= 250
e
.03
t
, from which
M
ATH
M
ODULE
S
olutions to
Exercises
9
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it is possible to determine that
t
= [
ln
(500/250)]/.03 = 23.105 years, so that
Exponentia will have doubled its GDP per capita in about 2033.
8.
There are several ways to calculate the average annual rate of increase in the price
of Wedgios relative to other consumer goods. The simplest exact method is the fol
lowing. The relative price of Wedgios
P
W
in 2010 was 12/80 = 0.15, and in 2013 it
was 13/96 = .13541667.
P
W
therefore declined over the 3year period to 0.902778 of
its 2010 level. Letting
g
Pw
be the average annual relative price increase, and solving
(1 +
g
Pw
)
3
= 0.902778, we have (1 +
g
Pw
) = (0.902778)
1/3
= 0.96648167, and so
g
Pw
=
0.96648167 – 1 = –0.03351833, or an average rate of –3.35%/year. The relative price
of Wedgios
declined
over the period by 3.35%/year, on average.
[Note that in this example, we did
not
round off the intermediate results, but saved
the rounding off until the final stage. This is a good practice to follow in calculating
with growth rates, since when compounding occurs, minor initial discrepancies can
be considerably magnified by the end of a problem. To the nearest tenth, for exam
ple, 1/3 = 0.3, but calculate and compare the values for (4/3)
50
and (1.3)
50
.]
M92
MATH MODULE 9: SOLUTIONS TO EXERCISES
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 Fall '12
 Danvo
 gross domestic product, per capita, Value added, National accounts, Final goods

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