PS4
Due Friday, Nov. 11, on Brightspace at midnight
This problem set contains 4 regular problems and a challenge problem.
• If you do not need an extra challenge, simply do the 4 regular problems.
You can get full credit by doing this.
• If you need an extra challenge, attempt the challenge problem. Your score
will be the maximum of your score for the challenge problem and your score
for the 4 regular problems. So if you are confident you got the challenge
problem right, you do not need to do the regular problems. (However, it
is probably a good idea to make sure that you know how to do them.)
1 Regular problems
1. Consider a consumer whose preferences are represented by
u(x1, x2) =
√
x1 +
√
x2
and has budget constraint
m ≥ p1x1 + p2x2.
(a) Are the consumer’s preferences monotone? Are they convex?
(b) Compute the consumer’s optimal bundle as a function of prices and
income.
(c) Compute the elasticity of the consumer’s demand for good 1 with
respect to income. Is good 1 a luxury good, a necessary good, or
neither? Are preferences homothetic?
(d) Compute the elasticity of the consumer’s demand for good 1 with
respect to p2 (the price of the other good). Are the two goods com-
plements, substitutes, or neither? If p1 and p2 are initially equal, and
then p2 goes up by 1%, by (approximately) what percentage would
demand for good 1 rise or fall?
2. Consider a consumer whose preferences are represented by
u(x1, x2) =
√
x2 + x1,
1
and who has budget constraint
m ≥ p1x1 + p2x2.
(a) Compute the equation of an indifference curve corresponding to util-
ity level u¯.
(b) Draw some indifference curves.
(c) How are the indifference curves related to one another?
(d) Compute the consumer’s optimal bundle as a function of prices and
income. (Hint: don’t forget about boundary bundles.)
(e) Plot how the demand for good 2 changes with income. Put m on the
y-axis and x2 on the x-axis.
3. There are 2 periods. The consumer must determine how much to spend
in each period. Her preferences are given by
u(c1, c2) = ln c1 + δ ln c2.
The consumer can borrow at interest rate rb and save at interest rate
rs ≤ rb.
(a) Write down the budget constraint. (Hint: you will need to handle
the cases c1 > m1 and c1 < m1 separately.)
(b) Assuming the consumer borrows, how much will she consume in each
period? (Hint: once you assume borrowing, the problem looks like
a normal utility maximization problem with a normal budget con-
straint.)
(c) Under what conditions on income and interest rate will the consumer
choose the bundle from (b)? (Hint: you want to find the condition
under which the c1 you found in (b) is indeed greater than m1.)
4. There are 2 states. The probability of state i is πi, where π1 + π2 = 1.
There is one risky asset that pays out xi in state i, and one safe asset that
pays out R in both states, where x1 > R > x2. Consider a consumer who
has $m to divide between the risky asset and the safe asset. The consumer
maximizes expected utility and is risk averse (u is strictly concave).
(a) Write down the expected utility that the consumer will get if he
spends fraction α of his income on the risky asset and the rest on the
safe asset.
(b) Write down an equation that characterizes the optimal level of α.
(Hint: take the derivative of expected utility and set it to 0.)
(c) If R equals the expected value of the risky asset, what level of α is
optimal?
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(d) Suppose that
u(c) =
c1−γ
1− γ .
Using your answer to (b) (but not using anything from (c)), show
that the optimal proportion of income spent on the risky asset does
not depend on the level of income.
(e) Now suppose that
u(c) = −e−γc
for some γ > 0. Using your answer to (b) (but not using anything
from (c)), show that the optimal amount of money spent on the risky
asset does not depend on the level of income. (Hint: the amount of
money spent on the risky asset is αm.)
(f) The utility specifications in (d) and (e) are frequently used in eco-
nomic models. Which one do you think delivers predictions more in
line with reality?
2 Challenge problem
1. This problem investigates the relationship between preferences and utility
functions.
(a) Consider an arbitrary preference ≿ over a finite set of objects:
{x1, x2, . . . , xn}. Propose a method for constructing a utility function
that represents ≿.
(b) Now consider an arbitrary preference ≿ over a countably infinite set
of objects: {x1, x2, . . .}. Propose a method for constructing a utility
function that represents ≿.
(c) Consider the set of pairs (x1, x2), where x1 is a real number and x2
is 0 or 1. Consider a preference that ranks x strictly higher than
y if one of two conditions is met: (1) x1 > y1, or (2) x1 = y1 and
x2 > y2. Argue that there is no utility function that represents these
preferences. (Hint: suppose there is one. Use it to assign a different
rational number to each real number. But since the cardinality of
the real numbers is higher than that of the rational numbers, this is
impossible.)
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