ECON UN3211 - INTERMEDIATE MICROECONOMICS
Columbia University - Department of Economics
Spring 2024
Problem Set 3
1. Justin spends most of his time in Just Coffee shop. Justine has $12 a week to spend on coffee and muffins.
Just Coffee sells muffins for $2 each and coffee for $1.2 per cup. Justin consumes xc cups of coffee per
week and xm muffins per week. His utility function for coffee and muffins is u(xc , xm ) = xkc xkm , where
k > 0.
(a) Find Justin’s optimal consumption bundle. Does it depend on k?
(b) Now Just Coffee has introduced a frequent-buyer card: For every five cups of coffee purchased at
the regular price of $1.2 per cup, Justin receives a free sixth cup. Draw Justin’s new budget set.
(c) With frequent-buyer card, find the new optimal consumption bundle.
2. A consumer can buy two goods: good 1 denoted by x1 and good 2 denoted by x2 . Her utility function is
x1 x2
given by u(x1 , x2 ) = x1 +x2 , and p1 and p2 are the prices of good 1 and good 2, respectively, and m is the
consumer’s income level.
(a) Is this utility function well-behaved? (Hint: It is continuous and differentiable. How about mono-
tonicity and DMRS?)
(b) Solve for her demand for x1 and x2 both as a function of p1 , p2 and m, that is, x1 (p1 , p2 , m) and
x2 (p1 , p2 , m).
3. Leo consumes only nuts and berries. Fortunately, he likes both goods. The consumption bundle where
Leo consumes x1 units of nuts per week and x2 units of berries per week is written as (x1 , x2 ).
(a) The set of consumption bundles (x1 , x2 ) such that Leo is indifferent between (x1 , x2 ) and (16, 4) is
√
the set of bundles such that x1 ≥ 0, x2 ≥ 0, and x2 = 20 − 4 x1 . Plot several points that lie on the
indifference curve that passes through the point (16, 4) and sketch this curve.
√
(b) In fact, Leo’s preferences can be represented by the utility function u(x1 , x2 ) = 4 x1 + x2 . The
price for nuts is $1 per unit and the price for berries is $2 per unit. Leo has $24 to spend on the
two goods. Write down Leo’s budget constraint and solve for his optimal consumption bundle.
(c) Now Leo has $10 more to spend on the two goods ($34 in total), and what is his optimal consumption
bundle now? Compare your solution with the previous one. Can you say anything interesting?
(d) Now Leo has only $9 in total to spend. Is he still able to consume the same amount of nuts as in
part b? Explain.
4. Leo’s birthday is coming up and all he has in his possession is one bottle of soda. His friends get together
and decide to buy him a gift. They agree on getting him some bottles of soda (s) and some fancy colorful
soda making powder/shaker contraptions (f ). Leo’s utility function is given by U (s, f ) = s + f s − f 2 .
The soda costs ps = 1 and the powder shaker contraptions cost pf = 2. His friends decide to buy him
two bottles of soda (in addition to the one he already has) and one powder shaker thing, spending a
1
total of $4. (Throughout this question, you can assume that both soda and the fancy powder things can
be consumed in any fractional amount. When drawing diagrams, put soda on the horizontal axis and
the powder shaker things on the vertical axis. Also, you should only be concerned about the negatively
sloped portions of the indifference curves. Do not worry about portions which are positively sloped.)
(a) At Leo’s initial position prior to receiving the gifts, is he optimized?
(b) What is Leo’s utility level before (U0 ) and after (U1 ) the gifts?
(c) Could Leo’s friends have spent their money more wisely in terms of making Leo happier? Explain.
If so, what would have been a better way to spend the $4?
5. Leo consumes only goods 1 and 2. Their prices are p1 and p2 , respectively, and we let p = (p1 , p2 ). Leo’s
√
income is m. His utility function is u(x) = x1 + x2 .
(a) Suppose p1 = 1. Find the general form of his demand functions xL L
1 (1, p2 , m) and x2 (1, p2 , m), which
solve the problem
max u(x1 , x2 ) subject to 1 · x1 + p2 · x2 = m
x1 ,x2 ≥0
(b) Now find the demand functions for a general price vector (p1 , p2 ). (Hint: Use the homogeneity
property of the demand: for any α > 0 and prices and income, (p1 , p2 , m), we have xL (p1 , p2 , m) =
xL (αp1 , αp2 , αm), that is, when all prices and income change proportionately the demand does
not change. This is because the budget set does not change when all prices and income change
proportionately.)
(c) Find the marginal utility of income in terms of prices and income.
