SOLUTIONS TO 100A, QUIZ 3 (Version A)
1.
U = x
A x
B
. Since P
A
= $1 and P
B
= $2 and the budget was $40, we have x
A
+ 2x
B
= 40; thus, x
A
= 40 - 2x
B
, and so U = (40 - 2x
B
)(x
B
). Differentiating, we get that 40 - 4x
B
= 0, so x
B
= 10, and so x
A
= 20. If the price of apples increased to $1.75 and the price of bananas fell to $1, to be able to just afford the same bundle, his income would have to be 20($1.75) + 10($1) = $35 + $10 = $45 , and so the answer is (a).
2.
If the number of miles she travels by first class is x and the number of miles she travels by second class is y, we have that x + y = 1500 and .4x + .2y = 540. Since x = 1500 – y, we can substitute and get that .4(1500-y) + .2y = 540, so 600 - .4y + .2y = 540, so 60 =
.2y, so y = 300 , and the answer is (a).
3.
As was stated repeatedly in class, with perfect complements, there is no substitution effect (can’t substitute when you consume in fixed proportions), so that the income effect accounts for the entire change in demand , so the answer is (c).
4.
Her endowment is 3 units of x and 6 units of y. At the given prices, this has a value of
$2*3 + $1*3 = $12. Therefore, the budget constraint is 2x + y = 12. Thus, x = 6 – y/2.
Substituting into the utility function, we have U = xy
2
= (6 – y/2)y
2
= 6y
2
- y
3
/2.
Differentiating, we have 12y – 3y
2
/2 = 0, or 12 – 3y/2 = 0, so y = 8 , and the answer is (c).
5.
She only wants to have kumquats and can get 1 kumquat for every 6 quinces that she gives up (since kumquats are 6 times as expensive as quinces). She has 18 quinces, so she can get 3 kumquats in exchange. So she could have as many as 15 + 3 = 18 kumquats , and so the answer is (d).
6.
Her consumption is 5(60-r). Substituting this into the utility function, we have U = cr
4
=
5(60-r)r 4 = 300r 4 – 5r 5 . Differentiating, we have 1200r 3 – 25r 4 = 0, or 1200 = 25r, so r =
48. Thus, she will work 60 – r = 12 hours per week (she really likes leisure), and the answer is (b).
7.
The real rate of interest

= (r -

)/(1 +

), or equivalently, 1 +

= (1 + r)/(1 +

), so
.1 = (r - .8)/1.8, so .18 = r - .8, so r = .98 (equivalently, 1.1 = (1 + r)/(1.8), so 1.98 = 1 + r, so r = .98), and the nominal interest rate is 98%
, and the answer is (a). If you didn’t remember the formulae, you could still do the problem. With 80% inflation, if you have
$1 in the beginning, you must have $1.80 at the end of the period to still have the same value (purchasing power). To make 10% on your money, you need to do 10% better than this, or have $1.98. This is the same as a 98% nominal interest rate.
8.
The present value of the endowment is 1600 + 945((1.05) = 1600 + 900 = 2500 , so the answer is (b).
9.
Since the utility is the minimum of c
1
and c
2
, we will want to equate these. We have that c
1
= 440 + x, where x is how much he borrows from his second-period income, and c
2
=

440 - 1.2x. Thus, x = -1.2x, so 2.2x = 0, so x = 0 and so he consumes 440 in the first period , and the answer is (e). Intuitively, since he has the same income in each period, he really can’t do any better than to consume his income in each period.
10.
One will hold onto this wine until the point in time where the return from holding it is the same (or less) as the interest rate. The return from holding it each period is $3.00 - $0.50
= $2.50. To find the value x at which 5% interest is the same as this $2.50 net return, we have that .05x = $2.50, so x = $50. Since the wine is worth $2 now and this increases by
$3 per year, if y is the number of years until it is worth $50, we have 2 + 3y = 50, so y =
16. Thus, the wine will be worth $50 and be 16 years old when it is (sold and) drunk , so the answer is (b).
11.
Let’s do this by starting with the last period first. If the wine is worth $53 in three years and $51 in two years, there is no sense in holding onto it for that third year, since the interest rate is 10%, and so you could make $5.10 by selling the wine after two years, rather than the $2.00 it went up in value. If you have it after one year, should you hold it for another year? Well, it goes up in value from $44 to $51, and this $7.00 increase is larger than the $4.40 you could earn by selling it for $44 and getting 10% interest.
So figure the price you’d be willing to pay on the basis of holding it for two years and selling it for $51. If you had $x to start with and put it in the bank and received 10% interest, you’d have 1.1x after one year, and 1.21x after two years. So 1.21x = 51, and so x = 42.15
, and the answer is (b).
12.
If the interest rate is 7%, the present value of $5350 in one year and $1144 in two years is
5350/(1.07) + 1144/(1.07)
2
= 5000 + 1000 = $6000 , and the answer is (a).