Economics 211 Assignment 2 Must be handed in, in class, Thursday

Economics 211
Assignment 2
Must be handed in, in class, Thursday 21st May 2015.
Please print this assignment and then answer all questions in the space provided. Please write clearly and concisely.
1. Let u : <2+ → < be a utility function for some consumer who always prefers more to
less of either good.
(a) Define what it means to say that u is “strictly concave”.
(b) Let x0 ∈ <2+ . Use this utility function to define the upper contour set (or better set)
of x0 .
(c) Define what is means to say that a set is “strictly convex”.
(d) Prove that if u is strictly concave the upper contour set of x0 must be strictly convex,
that is, u is strictly quasi-concave. (See page 55 in Hoy et al.)
ANSWER
(a) Let a, b ∈ <2+ . u is strictly concave if for t ∈ (0, 1)
u (ta + (1 − t)b) > tu(a) + (1 − t)u(b).
(b) The upper contour set (or better set) of x0 , call this B (x0 ), for this utility function
is
B x0 = {x ∈ <2+ : u(x) ≥ u x0 }
(c) A set, say A, is strictly convex if for all a, b ∈ A and for all t ∈ (0, 1), ta + (1 − t)b ∈
the interior of A, which is the largest open set contained in set A.
(d) Let a and b be any two elements of B (x0 ). Given that u is strictly concave we know
that for t ∈ (0, 1)
u (ta + (1 − t)b) > tu(a) + (1 − t)u(b).
But a, b ∈ B (x0 ) so u (a) ≥ u (x0 ), u (b) ≥ u (x0 ) and then
1
tu(a) + (1 − t)u(b) ≥ tu(x0 ) + (1 − t)u(x0 ) = u(x0 ),
Therefore
u (ta + (1 − t)b) > u(x0 ).
The boundary of B (x0 ) is given by {x ∈ <2+ : u(x) = u (x0 )} so ta + (1 − t)b must lie in the
interior of the better set. Thus the better set must be strictly convex.
2. Let f : <+ → <+ be f (x) = x1/2 . Prove the function f is concave.
ANSWER
f is concave if and only if (use iff for this term), for all a, b ∈ <+ and t ∈ [0, 1] we know
f (ta + (1 − t)b) ≥ tf (a) + (1 − t)f (b).
Observe that
(ta + (1 − t)b)1/2 ≥ ta1/2 + (1 − t)b1/2 iff
2
2
1/2
≥ ta1/2 + (1 − t)b1/2 iff
(ta + (1 − t)b)
ta + (1 − t)b ≥ t2 a + (1 − t)2 b + 2t(1 − t)a1/2 b1/2 iff
t(1 − t)a + t(1 − t)b − 2t(1 − t)a1/2 b1/2 ≥ 0 iff
2
t(1 − t) a1/2 − b1/2
≥ 0,
which is true.
2
3. Let {an }∞
n=1 be a sequence in < .
(a) What does the following notation mean?
lim
a = a.
n→∞ n
∞
(b) What does it mean to say that {sn }∞
n=1 is a series based on {an }n=1 ?
ANSWER
(a) It means that for any > 0, there is a natural number K which depends on such
that
{an }∞
n=K ⊂ N (a) .
(b) It means that for n ∈ {1, 2, 3, . . .},
sn ≡
j=n
X
j=1
2
aj .
4. State as precisely as you can what each of the following equals, and justify your
answer.
(a) limx→∞
(b)
(c)
3x2 − 6x + 1
,
9x2 + 2x + 7
x∈<
∞
X
1
j=1
n
X
j
aβ j−1
j=1
(d) limn→∞
i
1+
n
n
ANSWER
(a)
limx→∞
3 − 6/x + 1/x2
3
1
3x2 − 6x + 1
=
lim
= = .
x→∞
2
2
9x + 2x + 7
9 + 2/x + 7/x
9
3
(b)
1 1 1 1
+ + + +
2 3 4 5
1 1 1 1
> 1+ + + + +
2 4 4 8
1 1 1
= 1 + + + + ...
2 2 2
= ∞
1+
1 1 1 1
1
1
1
1
1
1
1
+ + + +
+
+
+
+
+
+
+ ...
6 7 8 9 10 11 12 13 14 15 16
1 1 1
1
1
1
1
1
1
1
1
+ + +
+
+
+
+
+
+
+
+ ...
8 8 8 16 16 16 16 16 16 16 16
So
∞
X
1
j=1
j
= ∞.
(c) Let
S = a + aβ + aβ 2 + ... + aβ n−1
Then
βS = aβ + aβ 2 + aβ 3 + ... + aβ n
Subtracting
3
(1 − β) S = a − aβ n = a (1 − β n )
So
S=
a (1 − β n )
.
1−β
(d) Let 1/s = i/n or cross-multiplying n = is. Then
limn→∞
i
1+
n
n
is
1
= lims→∞ 1 +
s
s i
1
= lims→∞ 1 +
s
i
= e,
where
s
1
.
e ≡ lims→∞ 1 +
s
5. (This question is closely related to problem number 6, page 96 in Hoy et al.)
(a) Derive a formula for the sum of the first n terms of the following geometric series.
a + ar + ar2 + ar3 + . . .
(b) Show that if 0 < r < 1 the limit, as n → ∞, of your answer to part (a) is a/(1 − r).
(c) Write the formula for the net present value as of the beginning of year 0 for the
following project. The project will cost 6 billion dollars with half of this amount to be paid
at the beginning of year 0 and the other half to be paid at the beginning of year 1. The
revenues are in two parts. There will be an infinite sequence of payments of 200 million
dollars per year with the first payment at the end of year 2. And there will be another
infinite sequence of payments of 300 million dollars per year with the first payment at the
end of year 10. Assume an annual interest rate of i with annual compounding.
ANSWER
(a) Write the sum of the first n terms of this geometric series as
sn = a + ar + ar2 + ar3 + . . . + arn−1 .
Multiplying this equation by r yields
rsn = ar + ar2 + ar3 + . . . + arn .
Now subtracting the second equation from the first equation obtain
4