Econ 301
Summer 2003
Asinski
Problem Set 3 Suggested Solution.
1. Problem 1.
a. A popular novel and gossip magazine can regarded as (imperfect) substitutes. To
consider these items as perfect substitutes a person should not care at all what he reads,
which is not likely.
b. A camera and film are complements.
c. A gun and stick of butter are not related.
d. A Panasonic CD player and a JVC CD player are substitutes. It is likely that some
people would regard them as perfect substitutes.
2. Problem 6. The consumer’s optimum will not change. The reason is that the budget
constraint will be exactly the same if we double (triple or multiply by any other positive
constant) all price and income. Suppose that initially my budget constraint was
PB*B+PZ*Z=Y. Where B and Z are consumption levels of Burritos and Pizzas,
respectively, PB, PZ are the prices, Y is income. Now suppose that all prices and income
go up by factor σ, i.e. PBnew= σ PB, PZnew= σ PZ, Ynew= σ Y. Then, the new budget
constraint is
PBnew *B + PZnew *Z = Ynew,
which can be rewritten as
σ PB *B + σ PZ *Z = σ Y,
we can divide both sides by σ>0 to get the original budget equation. Consumer has to
choose the same bundle because she faces exactly the same optimization problem as
before. You can check that the budget line on a graph is exactly the same as before
prices/income change.
3. Suppose that the Utility function is U=B2Z, where B is the (monthly) consumption of
Burritos and Z is the (monthly) consumption of Pizzas. Price of one burrito is $2 and
price of one pizza is $1. The monthly income is $30.
We will determine the optimal consumption bundle using step methodology outlined in
the hint to the problem:
„ Suppose you start off with the bundle consisting of one unit of each good (which
gives you U=1 and you spend $1+$2=$3);
„ Now you have to decide whether to buy one burrito or one pizza. The decision is
based on which one gives you more additional (marginal) utility per dollar of
investment, i.e. if you buy second pizza for $1 it will increase your utility from
U=1 to U=2, therefore the Marginal Utility is 1 per dollar of investment. Now, if
you decide to buy one burrito for $2 instead, you would get an increase in Utility
equal to 3 (U=1 to U=4), which gives 1.5 (=3/2) of Marginal Utility per dollar
spent. Therefore, you decide to buy one Burrito because it brings more additional
utility per dollar spent on it.
„ Now you start with the bundle consisting of one pizza and two burritos (which
gives you U=4 and you spend $1+2*$2=$5) and face the same decision: whether
to buy one pizza or one burrito.
MUB/PB=MUB/2 MUZ/PZ=MUZ Choice: B or Z
1.5 = (4-1)/2
2.5 = (9-4)/2
5 = (18-8)/2
7 = (32-18)/2
10.5 = (48-27)/2
13.5 = (75-48)/2
18 = (100-64)/2
22 =(144-100)/2
(this unit is not
even affordable)
1
4 = (8-4)
4 = 12-8
9 = 27-18
9 = 36 – 27
16 = 64 - 48
16 = 80 – 64
25 = 125-100
2nd B (1.5>1)
2nd Z (4>2.5)
3rd B (5>4)
3rd Z (9>7)
4th B (10.5>9)
4th Z (16>13.5)
5th B (18>16)
5th Z (25>22)
Total utility U(B,Z)
1 (U(1,1))
4 (U(2,1))
8 (U(2,2))
18 (U(3,2))
27 (U(3,3))
48 (U(4,3))
64 (U(4,4))
100 (U(5.4))
125 (U(5,5))
Income spent
3
5
6
8
9
11
12
14
15
So, optimal bundle is B=5, Z=5, which gives utility of U=125 and all $15 of income are
exhausted. Observe that technically our utility maximization result (MUB/PB) = (MUZ/PZ)
is not satisfied, MU of the fifth pizza when we already consume 5 Burritos and 4 Pizzas
is 25 (from the table). The MU per dollar spent of the fifth Burrito when we already
consume 4 Burritos and 5 Pizzas is 22.5 (=(125-80)/2) – not in the table. The reason is
purely technical – we use rough linear approximation to the true MU, which can be found
using derivatives. In fact, the requirement (MUB/PB) = (MUZ/PZ) is exactly satisfied at
our optimal bundle.
4. Problem 12. In order to be a Giffen good leisure needs to be inferior good in the first
place. However, leisure is a normal good despite the fact that, in the text, the author
hypothesizes about leisure being an inferior good. Most importantly, leisure inferiority is
not necessary to achieve backward bending labor supply curve, but it does have some
connotations of the kind ‘I don’t really like having time off’, which are hard to explain.
5. Problem 17. Alice is not maximizing her total utility. Here’s how she can improve on
her wellbeing (get higher utility) while keeping expenses within the same budget. Let
Alice sell (not buy) her last book, her total utility goes down by 10 utils, but she’s able to
free $10 to spend on cookies. Now, she can buy 5 cookies for $10, each of which gives
her 5 utils, which makes it 5*5=25 utils in total. So, having spent the same amount of
money, she’s got 15 more utils (25 gained from cookies minus 10 lost due to one fewer
book). Therefore, her original allocation wasn’t optimal.