ECN 445 SPRING 2016 PROBLEM SET 3

ECN 445 SPRING 2016
PROBLEM SET 3
Answers due by the end of class on 3-31-2016

See Lecture 2-12 Welfare evaluation
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1. A consumer has a Cobb-Douglas utility function given by u x10.2 x2 . The corresponding

ordinary demand functions are: x1 0.2( y / p1 ) and x2 0.8( y / p2 ) .
(a) If his income is $1200 and prices are p1 = $2.50 and p2 = $1, how much does he buy of each
good? What is his expenditure on each good?
(b) The indirect utility function corresponding to the above direct utility function is given by

u 0.606 y p10.2 p2 0.8 . Give a definition of the indirect utility function: what relationship does it

portray?
(c) Suppose the price of good 1 becomes p1 = $1.50 while, as before his income is $1200 and p2
= $1. Is the consumer better off as a result of this price change, worse off, or equally well off as
before? Explain the reason for your answer.
(d) How does his expenditure on x1 change?
(e) Draw a diagram showing the change in Marshallian consumer’s surplus associated with this
particular price change (you need not actually calculate the change in consumer’s surplus).
(f) The maximum amount that the consumer would have been willing to pay for this price
reduction is given by the compensating variation measure. Write down the algebraic equation
which defines the compensating variation in this case.
(g) Now solve that equation – calculate the numerical value of the compensating variation in this
case.
(h) The minimum amount of compensation that the consumer would have wanted to forego the
price decrease is given by the equivalent variation. Write down the algebraic equation which
defines the compensating variation in this case.
(i) Now solve that equation – calculate the numerical value of the equivalent variation in this
case.
(j) Are you surprised that the two numerical values – WTP and WTA – that you calculated in (g)
and (i) are not the same? Are you surprised that they are not the same as the numerical value you
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calculated in (d)? Would you be surprised to learn that they are not the same as the change in
consumer’s surplus you graphed in (e)? Explain the reason for your answers.
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(k) If the utility function were given by u 105 x10.2 x2 instead of the formula given above, would

this cause a change in the formula for the demand functions stated in part (a)? Explain the reason
for your answer?
(o) If your answer to (k) was that the new utility function would not cause a change to your the
demand function formula given in part (a), would it then cause a change to your answers to parts
(g) and (i)?
2. A study of people’s choice of transportation mode for commuting to work develops the
following empirical estimate of the indirect utility function for using a given mode of
transportation: u = -4 T – 10C
where u is the (dis)utility associated with travelling by that mode, T is the total time required for
the trip, in minutes, and C is the total cost of the trip in dollars.
(a) Suppose that a particular consumer who has this indirect utility function faces a choice
between two alternative travel modes: taking the bus to work takes 50 minutes and costs $4.75;
driving to work takes 30 minutes but (with gas and tolls) costs $11.50. Which mode will that
person choose? Explain the reasoning underlying your answer.
(b) What is the value of travel time for this consumer (the rate at which he would be willing to
trade-off an extra minute of travel time for a reduced travel cost)?
(c) Suppose this indirect utility function is in fact representative for most commuters in the San
Francisco Bay area. The San Francisco Metropolitan Transport Agency is thinking of investing
in drilling an additional tunnel through the hills separating Walnut Creek from the East Bay
Bridge. 25,000 commuters travel along this route on a typical day. The new tunnel will save the
average driver 15 minutes each way on his daily commute. Assume a typical commuter makes
this commute 225 days a year. The annualized cost of the tunnel amounts to about $1,000 per
commuter per year. Based on this information, would you recommend that the tunnel be built -do the benefits of the new tunnel outweigh its cost? Explain the reason for your answer.
For the next two questions, see the lecture on 3-24-2016.
3. The table below gives information on visitors to Jellybear Park (distance is in miles). A
researcher keeps careful track of visitors to Jellybear Park and learns that there is a total of 3,750
visits to the park per year. There is no admission fee. The only cost of visiting Jellybear is the
travel cost, which costs $1/mile. The researcher estimates the relationship between the number of
visits per capita (per person), x, and the one-way travel cost from the town in which people live,
p. He finds this demand function to be: x = 4 – 0.4*p .
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(a) What is the cutoff price (the price at which demand falls to zero) with this demand function?
(b) Add columns to the table above identifying the cost of a visit to Jellybear from each town, the
predicted number of visits per capita from that town, and the total number of visits from the
town (i.e., the visits per capita multiplied by the population).
(c) Why doesn’t the researcher observe anyone coming from Delight to Jellybear?
(d) Because of management costs, park managers are considering charging for admission to
Jellybear. They are considering prices ranging from $4/visit to $6/visit. For each whole dollar
value between $4 and $6 per visit, figure out (i) the new cost of visiting for each town, (ii) the
new number of visits per capita from each town, and (iii) the new total number of visits from
each town (remember that negative visits count as zero).
(e) Estimate the consumer surplus’s for residents of each town (both the consumer’s surplus per
capita and the total consumer’s surplus across all residents of the town) associated with visiting
Jellybear Park when the admission charge is $0.
(f) Assume there is a $1 admission charge at Jellybear. The County authorities are planning to
open a new park at South Bend. South Bend is located 6 miles from Alabaster, 8 miles from
Beautiful, 2 miles from Cornucopia, and 3 miles from Delight. For now, there will be no
admission charge at South Bend Park. Which people will benefit from the opening of South
Bend? How many visits will it attract annually? What is the aggregate gain in consumer’s
surplus associated with the opening of South Bend Park?
4. Joe is an avid surfer – he surfs regularly at Huntington Beach. His monthly demand function
for surfing is: x = 40 - 2p, where p is the cost of a surfing trip to Huntington Beach and x is the
number of surfing trips he takes per month.
(a) What is the cutoff price at which he would stop surfing at Huntington Beach altogether?
(b) The cost of a surfing trip to Huntington Beach is $6. How many trips does Joe take per
month?
(c) What is Joe’s monthly consumer’s surplus from being able to surf at Huntington Beach?

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(d) What is his consumer’s surplus per trip from surfing at Huntington Beach?
(e) An oil spill leads to the closure of Huntington Beach for the months of March and April. Joe
cannot go surfing there during those two months. What is the economic measure of Joe’s loss
from the closure of Huntington Beach?
(f) The state of California successfully sues the company that caused the oil spill for damages
suffered by the public because of the Huntington Beach closure. The jury awards the state $15
million. It uses these funds to renovate the beach and improve the quality of surfing there. As a
result, Joe’s demand function for surfing at Huntington Beach changes and becomes x = 52-1.5p.
What is Joe’s gain in consumer’s surplus as a result of the quality improvement?

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