ECO 301
Fall 2010
Prof. Sullivan
Homework:
You may submit this homework by October 13 for bonus credit. Even if you choose not to submit
for bonus credit, you should review these problems as you will be asked to perform similar tasks
on the midterm.
1.
Suppose there are 3 products, a coffee, a pizza, and a hat. None of the goods can be
divided or shared. There are 2 consumers – Alan and Betty. Each gains the following amounts of
satisfaction from the various possible combinations:
Alan:
Utility
0
5
4
8
8
10
10
13
Coffee
x
yes
x
x
yes
yes
x
yes
Pizza
x
x
yes
x
yes
x
yes
yes
Hat
x
x
x
yes
x
yes
yes
yes
Betty:
Utility
0
15
24
18
38
20
30
53
Coffee
x
yes
x
x
yes
yes
x
yes
Pizza
x
x
yes
x
yes
x
yes
yes
Hat
x
x
x
yes
x
yes
yes
yes
In the tables above, “yes” means the consumer has that product, while “x” means that he/she does
not have that product. What are the Pareto efficient allocations of goods? (Hint: whatever Alan
does not consume can be consumed by Betty, and vice-versa.)
2.
Suppose a restaurant has 2 pizzas. There are 3 possible customers, A, B, and C. The value
of a pizza to the restaurant owner is $1 each (this is the utility of a pizza to the restaurant owner.)
A values a pizza at $8, B values a pizza at $4, and C values a pizza at $2.00. The pizzas are sold
for $1 each to the person who stands in line longest outside the restaurant. The cost to each
customer of time spent waiting in line is $2 per hour for A and B, and $0.50 per hour for C.
a.
Which customers will get the pizza?
b.
is this a Pareto efficient allocation of pizza (and time)?
c.
Suppose the restaurant owner sells the “right to be first in line” and the “right to be
second in line” to the highest bidder. Assume A, B, and C each bids the highest amount he would
be willing to pay. Now there is no waiting, since the places are reserved. Is this Pareto efficient?
3.
Suppose a consumer has an indifference curves between goods x1 and x2 which is:
(1/2)
(1/2)
x1
+ x2
= N, where N represents the level of utility.
a.
b.
c.
Are these preferences monotonic (weakly or strictly)?
Are these preferences convex (weakly or strictly)?
What is the marginal rate of substitution where N = 8 and x1 = 4?
4.
Bob says his utility function is U = x1x2, so his utility level is 4 if x1 = 2 and x2. Carol
says her utility function is U = x1x2 + 10. Ted says his utility function is U = 2x1x2 + 20.
2
Alice says her utility function is U = [2x1x2 + 20] .
Is it possible to tell which person is choosing bundles of x1 and x2 for a given budget just by
seeing what choices are being made? How?
5.
Below are some indifference curves. For each graph, identify whether the preferences are:
a.
b.
non-monotonic (NM), weakly monotonic (WM), or strongly monotonic (SM)
non-convex (NC), weakly convex (WC), or strongly convex (SC)
6.
Suppose U = x1
(1/2)
+ x2
Both p1 and p2 equal 1.
a.
Show the draw a graph showing the indifference curves and show the income expansion
path (income offer curve).
b.
Draw the Engel curve for x1. (Hint: be careful to think about what happens when M gets
small).
c.
Suppose M = 5, p2 = 1, and p1 = 1/4. (Hint: x1 = 4, x2 = 4 is a good point to take a close
look at.) Then p1 rises to p1 = 1/2. Find the change in x1 demanded, and de-compose this change
in x1 into the substitution effect and the income effect using the Slutsky method.
d.
For the same change in p1 as part c, decompose the change in x1 into substitution and
income effects using the Hicks method.
(For parts c and d, you should be able to find the quantity of x1 demanded after the compensation.)
7.
Suppose U = x1 + x2 and the consumer’s budget is p1x1 + p2x2 ≤ 12 where p2 = 4.
Hint, this requires no computation or calculation, just drawing a good diagram and a little thought.
a.
b.
c.
Draw (accurately) the ordinary demand function for x1.
Draw (accurately) the Slutsky compensated demand function for x1 if p1 is initially 3.
Draw (accurately) the Hicks compensated demand function for x1 if p1 is initially 3.