Econ 300 Section 1
Take-home Homework 3
Due Date: Friday, February 22nd, 2013 by 4pm
You can turn your homework in during class or into the economics department where they will
time stamp your homework and place it in my mailbox. Make sure you show your work!
1. (4 points) What information is contained in the slope of an indifference curve? Why are
these curves typically convex to the origin?
The slope of an indifference curve indicates the rate at which the consumer is willing to exchange
one commodity for the other. This is the Marginal Rate of Substitution (MRS). Indifference
curves are typically convex because the consumer places greater value on the last unit of the
commodity that is more scarce.
2. (4 points) Andy purchases only two goods, apples ( ) and kumquats ( ). He has income
of $400 and can buy apples at $10 per pound and kumquats at $8 per pound. His utility
function is
(
)
That is, his (constant) marginal utility for apples is 5 and his marginal utility for
kumquats is 5.
a. (4 points) What bundle of apples and kumquats should he purchase to maximize his
utility?
Because the
consume all kumquats.
0 apples
50 kumquats
b. (4 points) Graph Andy’s utility maximizing bundle, making sure to graph his
indifference curve and his budget constraint.
Indifference curve:
Budget constraint:
Budget line
(
)
3. Vasco’s utility function is
(
The price of
is
, the price of
)
is
, and his income is
a. (4 points) What is his optimal bundle (his utility maximizing bundle)?
Plug into budget constraint:
Solve for
.
Plug
into
and solve for
(
)
.
b. (4 points) Graph Vasco’s utility maximizing bundle, making sure to graph his
indifference curve and his budget constraint.
Budget line
4. (4 points) Confirm that if a consumer’s utility function is described by
(
)
where the
,
, and prices are
and
, there is no unique
utility-maximizing solution regardless of income level. What does this tell you about X
and Z as commodities? (Hint: draw a graph showing a budget constraint and indifference
curve using the information provided.)
Because the commodities are perfect substitutes, and the price ratio is equal to the MRS, any budget
line drawn would be coincident with some indifference curve, resulting in no unique solution. The
consumer is equally happy choosing any point on the budget line.
5. (4 points) Suppose Carmela’s income is $100 per week, which she allocates between
sandwiches and books. Sandwiches cost $2 each. Books cost $10 each if she purchases
between 1 and 5 books. If she purchases more than 5 books in a week, the price falls to
$5 for the 6th book and all subsequent books. Draw the budget constraint. Is it possible
that Carmela might have more than one utility-maximizing solution?
6. Because the price of books falls when Carmela purchases more than 5 books in one week, the
budget constraint is non-linear. As Figure 4.5 shows, the budget line is kinked at a. This nonlinearity makes it possible that a single indifference curve could be tangent to the constraint in
two places. In this case, the consumer is indifferent between purchasing 3 books per week at full
price and purchasing 10 books per week with the discount.
Or 15 intercept, depending on how
they interpreted the question.