National University of Singapore EC2101 Microeconomic Analysis I

Department of Economics Semester 1 AY 2014/2015

1
Practice Problems 2 Solutions
Consumer Choice, Revealed Preference, and Demand

Presentation schedule: two student presentations; one student presents Question 1 and
Question 2, the other student presents Question 3 and Question 4

Question 1 Connie has a monthly income of $200 that she allocates among two
goods: potatoes and meat. Her utility function is given by the equation
U(P, M) = P+2M .

a) Suppose potatoes cost $2 per pound and meat $4 per pound. Draw her budget line.
What combination of meat and potatoes should she buy to maximize her utility?

Her budget line is 4M+2P=200, or M=50-0.5P. See graph below.



The marginal utility of potato is 1, and the marginal utility of meat is 2. Therefore
MRS
P,M
=1/2. Note that meat is exactly twice as expensive as potatoes, thus the
indifference curve has the same slope as the budget line. Another way to see this is to
write down the equation of the indifference curve. For any utility level U, the
equation of the indifference curve is P+2M=U, or M=U/2-0.5P. The indifference
curve for U=100 coincides with her budget line. Any combination of meat and
potatoes along this indifference curve, which is also the budget line, will provide her
with maximum utility.

b) Connies supermarket has a special promotion. If she buys 20 pounds of potatoes
(at $2 per pound), she gets the next 10 pounds for free. This offer applies only to the
first 20 pounds she buys. All potatoes in excess of the first 20 pounds (excluding
bonus potatoes) are still $2 per pound. Draw her budget line. What combination of
meat and potatoes maximizes her utility? Is Connie better off with the promotion, i.e.,
does Connie receive higher utility when there is promotion?

With potatoes on the horizontal axis, Connies budget constraint has a slope of !1/2
until Connie has purchased 20 pounds of potatoes. This is because nothing changes if
Connie buys less than 20 pounds of potatoes. If Connie buys 20 pounds of potatoes,
she can afford 40 pounds of meat.

Her budget line is then flat from 20 to 30 pounds of potatoes. This is because these
additional 10 pounds of potatoes are free, hence she does not have to give up any
meat to get these extra potatoes. Plus, Connie can throw away any potatoes that she
National University of Singapore EC2101 Microeconomic Analysis I
Department of Economics Semester 1 AY 2014/2015
2
does not want. Thus any basket that contains 40 pounds of meat and an amount of
potatoes between 20 to 30 costs Connie the same amount of money -- her entire
income. (For example, how much money does Connie need to spend to get 25 pounds
of potatoes and 40 pounds of meat? 40 pounds of meat costs $160. To get 25 pounds
of potatoes, Connie only needs to purchase 20 pounds of potatoes at a price of $2 per
pound, and the supermarket will give her 10 pounds free. To get 25 pounds, Connie
can simply throw away 5 pounds. Thus 25 pounds of potatoes cost Connie $40. The
total cost for Connie is $160+$40=$200, which is her entire income. )

After 30 pounds of potatoes, there is no more promotion, the slope of her budget line
becomes !1/2 again. The horizontal intercept is 110, since if Connie spends all her
money on potatoes, she can now get 110 pounds of potatoes (because of the 10
pounds free).

Any basket on the budget line in which there are at least 30 pounds of potatoes
maximize her utility, for example, the basket with 110 pounds of potatoes and 0 meat.
At the optimal basket, Connies utility is 110. At the optimal basket in part b),
Connies utility is 100. So the promotion makes Connie better off.



c) An outbreak of potato rot raises the price of potatoes to $4 per pound. The
supermarket ends its promotion. What does her budget line look like now? What
combination of meat and potatoes maximizes her utility?

With the price of potatoes at $4, Connie may buy either 50 pounds of meat or 50
pounds of potatoes, or any combination in between. Since
MU
P
P
P
=
1
4
<
MU
M
P
M
=
2
4
,
meat gives her higher marginal utility on a per dollar base than potatoes. Thus she
will only buy meat. She maximizes utility when she consumes 50 pounds of meat and
no potatoes. This is a corner solution.

National University of Singapore EC2101 Microeconomic Analysis I
Department of Economics Semester 1 AY 2014/2015
3


Question 2 A consumers utility function is . His monthly income is I
dollars, the price of x is P
x
and the price of y is P
y
. (x and y do not have to be
integers.)

a) Show that the consumer always spends one third of his income on x, regardless of
the size of his income. (Hint: write down the tangency condition.)

The budget line is P
x
x+P
y
y=I. We also have MU
x
= y
2
and MU
y
= 2xy . Thus the
optimal basket must also satisfy
MU
x
MU
y
=
y
2x
=
P
x
P
y
. This implies 2P
x
x=P
y
y. At the
optimal basket, the expenditure on y is twice as high as the expenditure on x. Thus the
consumer will spend one third of his income on x. We can also see this by substituting
the tangency condition into the budget constraint, which gives us P
x
x=I/3. The left-
hand side is the total expenditure on x, therefore, at the optimal basket, the consumer
always spends one third of his income on x.

b) Draw the Engel curve for y. Is y a normal good or an inferior good? (Hint: derive
the demand function for y.)

Since we already know that P
x
x=I/3, it follows that P
y
y=2I/3. Since y=2I/3P
y
, we have
I=3P
y
y/2. Keeping the price of y fixed and increase the income, quantity demanded
for y will increase at the rate of 2/3P
y
. See graph below. As the Engel curve is upward
sloping, y is a normal good.



U(x, y) = xy
2
National University of Singapore EC2101 Microeconomic Analysis I
Department of Economics Semester 1 AY 2014/2015
4
Question 3 As shown in the following figure, a consumer buys two goods, food and
housing. Assume the consumers preference satisfy the three assumptions
(completeness, transitivity, and more is better). When she has budget line BL
1
, her
optimal choice is basket A. Given budget line BL
2
, her optimal choice is basket B, and
with BL
3
, her optimal choice is basket C.



a) What can you infer about how the consumer ranks baskets A, B, C, and D?

A is strictly preferred to B because given BL
1
, B is cheaper than A, but the consumer
chooses A instead of B, so it must be the case that A is strictly preferred to B.

When the budget line becomes BL
2
, the consumer chooses B, not C, but C is cheaper
than B. So the consumer strictly prefers B to C.

Both C and D are the budget line BL
3
, but C is the optimal choice given BL
3
, thus C
must be at least as good as D, that is, C is weakly preferred to D.

By transitivity, A>B>C>=D.

b) On the graph, shade in the areas that are strictly less preferred to basket B.

c) On the graph, shade in the areas that are strictly preferred to basket B.

See graph below. B is weakly preferred to all other baskets on BL
2
and strictly
preferred to all baskets below BL
2
including point C. C is strictly preferred to all
baskets below BL
3
and weakly preferred to all baskets on BL
3
. By transitivity, B is
strictly preferred to all baskets below BL
2
and all baskets on and below BL
3
. In graph,
this is the orange shaded area, including the solid blue segment on BL
3
.

B is weakly preferred to any other basket on the dashed blue segment on BL
2
.

By more is better, all baskets that have more food and/or more housing than B are
strictly preferred to B. All baskets that have more food and/or more housing than A
are strictly preferred to A. By transitivity, the baskets that are strictly preferred to A
are also strictly preferred to B. In the graph, this is the blue shaded area.


National University of Singapore EC2101 Microeconomic Analysis I
Department of Economics Semester 1 AY 2014/2015
5


Question 4 A consumer has utility function U(x, y) = min(2x, y) . His monthly
income is I dollars, the price of x is P
x
and the price of y is P
y
. (x and y do not have to
be integers.)

a) Suppose the consumers income is 24, the price of x is 2, and the price of y is 5.
What is the optimal basket for the consumer? (Hint: draw the budget line and several
indifference curves in a graph. Which point on the budget line gives the consumer the
highest utility?)

Based on the utility function, the two goods are perfect complements, so the
indifference curves will be L-shaped. For any indifference curve, the corner (or kink)
of the indifference curve is where 2x=y. For example, the corner of the indifference
curve U=2 is when 2x=y=2, thus x=1, y=2.

The consumers optimal choice will be at the kink of the indifference curve, which is
point A in the graph below. Any other point on the budget line, such as point B, lies
on a lower indifference curve than point A. Thus the optimal basket must satisfy 2x =
y. The budget line is 2x+5y=24. Solving the two equations together we get x=2 and
y=4.



b) Derive the equation of the demand function for x and y (x and y as a function of I,
P
x
and P
y
). Suppose x becomes more expensive while the price of y and the
consumers income remain the same. Will the consumer buy more x? Will the
National University of Singapore EC2101 Microeconomic Analysis I
Department of Economics Semester 1 AY 2014/2015
6
consumer buy more y? Will the consumers utility increase, decrease, or stay the
same?

The optimal basket will always be such that 2x = y. Also, at the optimal basket, it
must be true that P
x
x + P
y
y = I . Substituting the first condition into the second we get
P
x
x +2P
y
x = I which implies that the demand function for x is given by x =
I
P
x
+2P
y
.
Therefore the demand function for y is y =
2I
P
x
+2P
y
.
If x becomes more expensive, according to the demand functions, the consumer will
buy less x and less y. The consumers utility will decrease because the consumer buys
less x and less y.