ECON 157 Midterm Solution
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Question
1 (45 points)
Consider
a situation where a population of consumers is deciding whether or not to buy a
street
drug
called Bearoin. A researcher observes these consumers’ purchases over two
years, 1 and 2. The
price
of Bearoin changes from year 1 to year 2, as a result of an increased number of
sellers in the
market.
The researcher observes the following prices of the drug, and quantities sold,
in year 1 and
year
2:
The
year 1 price of the drug for consumers is P1 = $80. The year 1 quantity consumed at that
price
is Q1 = 400
1. (5
points) Write down the formula for arc price elasticity of demand in terms of P1, Q1, P2,
and Q2.
2.
(10 points) What is the arc price elasticity of demand for Beroin?
3.
(10 points) Now consider a second, new population of consumers buying the same
drug in
another
location. In this new market:
The
year 1 price of the drug for consumers is P1 = $80. The year 1 quantity consumed
at
that price is Q1 = 240
The
year 2 price of the drug for consumers is P2 = $20. The year 2 quantity consumed
at
that price is Q2 = 1000
What
is the arc price elasticity of demand for consumers in this second market?
4.
(10 points) Now, there is a music festival where both of these populations come
together and
form
one population. They have the same demand for Bearoin at the prices listed
above in
each
market. What is the arc elasticity of demand for this combined population of
consumers?
5.
(10 points) The researcher now wants to use the arc elasticity computed for
this combined
population
to determine what the quantity sold of the drug would be if she could enact a
government
crackdown so that the price of the drug at the festival rose from $20 to $120.
Using
the arc elasticity computed above, and the quantities given above for P = 20, compute
the
quantity that will be purchased under this new price.
Question
2 (35 points)
1. (5
points) In the Grossman model, consumers face a time budget constraint as well
as the
usual
financial budget constraint. Write out the time budget constraint for one day,
and
clearly
list each type of time use that enters the model.
2.
(10 points) Assume the consumer can do at most one of these exercises, and
could also choose
to do
none of them (in which case there is no change to H or Z relative to
baseline). If the
consumer
only considers his/her current period utility, which activity will she choose,
if any?
3.
(10 points) Now, assume that that consumer has interactions between health and
the home
good
in their utility function, because being healthier specifically allows him /
her to better
enjoy
other things being consumed. The new utility function is:
U(H,Z) = 3Z + 5H + 3(H ⇤ Z)
Assume
that the consumer would have H = 5 and Z = 5 if they chose not to do any
exercise.
Assume
that the changes from any exercise chosen for H and Z are made relative to this
baseline.
Now which exercise will they choose, if any (5 points)? What is his/her utility
for
the
current period, after making that choice (5 points)?
4.
(10 points) Now, assume that instead of caring about static utility, consumers
care about
utility
for the current period and the next period. The period-by-period utility
function is
the
same in each period, and is as specified in the previous part of this question
(part 3, with
interactions).
Assume that both periods are valued the same, i.e. there is no time discounting
( = 1).
The
consumer chooses which exercise to pick (if any) in period 1 only, and there is
no choice
made
for period 2 (i.e. no exercise is done in period 2). Assume that base
consumption is
H = 5 and Z = 5 in each period
(i.e. this is what happens if no exercise is chosen in period
1,
and this is the base o↵ which changes to H and Z from exercise are made relative to).
Assume
that the benefits from choosing an exercise for period 1 H and Z are as specified
at
the beginning of the question. Assume that there is no benefit from exercising
in period 1
for
the period 2 level of Z. Assume that the health benefits of exercising have some persistence
for
period 2, where they equal 50% of the incremental health benefits specified for
period 1.
Which
exercise choice do consumers make (if any) (5 points)? What is their combined
utility
over
both periods with that choice (don’t forget to account for their base
consumption each
period)
(5 points)?
Question
3 (50 points)
Consider
a consumer with the following utility function for income I:
U(I) = pI
Think
about the expected utility model that we studied in class and in chapter 7.
This consumer
has a
probability of p = 0.25 of becoming sick and (1-p) = 0.75 of being healthy. If the consumer
is
uninsured,
when sick she has income of IS = 16. If uninsured, but healthy, the consumer has income
IH = 144.
1. (5
points) Write down the formula for expected utility, in terms of p, 1−p, IS, and IH. Don’t
insert
the specific numbers yet, just give the general formula given these variables
2.
(10 points) Using the exact numbers given in this problem for these variables,
compute this
consumer’s
expected utility if uninsured.
3.
(10 points) Using the exact numbers given in this problem, compute the utility
gain to consumers
of a
fair and full insurance contract, relative to being uninsured. This means:
compute
the
utility this consumer has under a fair and full contract, and subtract from
this their expected
utility
if uninsured.
4.
(10 points) Assume that a monopolist insurer o↵ers the consumer a full
insurance contract,
but
as an unfair contract that extracts the most possible money from the consumer.
What is
the
premium of this insurance contract?
5. (5
points) If no insurance contracts are available, i.e the consumer will remain uninsured
apart
from
the government subsidy, what is the increase in utility provided by the subsidy
for this
consumer?
6.
(10 points) Assume that an insurer can o↵er a partial insurance contract with a
payout of
IH−Is
2 . Assume that
insurance payouts occur before the government allocates subsidies to consumers:
if
your income with the insurance contract considered is greater than 36, you get
no
subsidy. If the partial
insurance contract is fairly priced, what is total consumer expected
utility
if they buy this contract (5 points)? Will they buy the contract in this
environment
with
the government subsidy (5 points)?
Question
4 (50 points)
Consider
the Akerlof ’Lemons’ model discussed in class and discussed in Chapter 8 of the
textbook.
Make
the standard assumptions in that model that (i) sellers have private
information and know
the
quality of their cars and (ii) buyers don’t know the quality of any specific
car, and just know
the
probability distribution of car quality overall.
Buyers
(who are risk-neutral, as in the standard model) have the following utility
functions over
quality
Xj and money M:
1. (5
points) Consider a potential price in the market of P = 150. Don’t worry
about how this
price
arises, just assume this is the price. What is the range of car quality that
sellers will
o↵er
up for sale given this price? I.e. write down the lowest quality XL and highest quality
XH that
sellers will o↵er for sale.
2.
(10 points) Given this same potential price, P = 150, and the cars that sellers o↵er in this
market, will buyers buy any
cars at this price? Answer this question by providing (i) buyer
expected
utility for buying a car in this market given this price (5 points) and (ii) a
yes or no
answer
about whether they will purchase any cars (5 points).
3.
(10 points) Assume that a regulator is considering a quality standards policy
that removes
cars
of quality less than some value X⇤ from the market. What is the minimum value of
this
minimum
quality standard X⇤ such that any cars will be sold in the market if the price (for
whatever
reason) is P = 180?
4.
(15 points) Assume that for whatever reason, the price in the market with these
sellers is
P = 180. Specify the
quality range(s) of cars that sellers will o↵er for sale in the market (10
points).
What is the average quality of cars sold in the market at this price (5
points)?
5.
(10 points, HARD) Assume that the cars not sold in the last part of this question (two types
of
sellers, P = 180) are able to break away from that market and form their own
distinct
market.
These cars not sold in the overall market are the only ones that break away and
the
only
ones that can be o↵ered for sale in this new market. Assume that seller utility
functions
remain
the same as they were in the two-type market, and the buyer utility functions
remain
as
have been throughout the question.
What
is the minimum price at which at least half of the cars in this breakaway
market are sold
(5
points)? What is the average surplus for buyers who actually buy cars in this breakaway
market,
at this minimum price where at least half of the cars are sold (5 points)?
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