Problem Set 5: Exchange Economies
Lecture 8 - Eciency
Question 8.A
Frank the armadillo and Claire the ant eater consume ants (x) and termites (y ). Frank has
30 ants and 10 termites, while Claire has 20 ants and 20 termites. Frank's utility function
is:
UF (xF , yF ) = x2F yF
While Claire's utility function is:
UC (xC , yC ) = xC yC2
Find the set of interior Pareto ecient allocations.
Question 8.B
Don and Betty consume liquor,
x,
y.
and cigarettes,
Don's utility function is:
yd
Ud (xd , yd ) = min xd ,
3
while Betty's utility is:
yb
Ub (xb , yb ) = min xb ,
3
y
x
(ωd , ωd ) = (10, 40) while
Suppose Don's endowment is
Betty's endowment is
(ωbx , ωby ) =
(10, 20).
1. Draw an Edgeworth box, putting liquor on the horizontal axis and Don's origin on the
bottom left.
On the same graph, draw a typical indierence curve for Don and for
Betty.
2. Find the set of Pareto ecient allocations [Hint: Try to draw two indierence curves,
one for Betty and one for Don, that are tangent to each other. What must be true at
the point of tangency?].
3. Suppose now Betty's endowment changes to
(ωbx , ωby ) = (10, 30).
for Betty and Don to consume their endowment? Explain.
1
Is it Pareto ecient
4. Draw a new Edgeworth box for when Betty's endowment is
(ωbx , ωby ) = (10, 30).
On
the same graph, draw a few indierence curves for Don and for Betty. Mark the set of
Pareto ecient allocations on the graph [Hint: Try to draw two indierence curves,
one for Betty and one for Don, that are tangent to each other. What must be true at
the point of tangency?].
Question 8.C
Richard and Peter also enjoy consuming liquor and cigarettes. Richard's utility function is:
UR (xR , yR ) = 3xR + yR
while Peter's utility function is:
UP (xP , yP ) = xP + 3yP
Richard's endowment is
(30, 30)
while Peter's endowment is
(10, 10).
1. Find an allocation that Pareto dominates Richard and Peter's endowment, but is not
ecient.
2. Find an allocation that is ecient, but that does not Pareto dominate Richard and
Peter's endowment.
3. Draw an Edgeworth box, putting liquor on the horizontal axis and Richard's origin on
the bottom left. On the same graph, draw a typical indierence curve for Richard and
for Peter, marking the slope of each.
4. Find the set of ecient allocations [Hint: Observe rst every ecient allocation must
be exactly feasible. Now, draw Richard and Peter's indierence curves that go through
the same point in the the Edgeworth box's interior. Can you nd a Pareto improvement? What must be true for you not to be able to nd such an improvement?].
Lecture 9 - Competitive Equilibrium
Question 9.A
Frank the armadillo and Claire the ant eater consume ants (x) and termites (y ). Frank has
30 ants and 10 termites, while Claire has 10 ants and 30 termites. Frank's utility function
is:
UF (xF , yF ) = x2F yF
While Claire's utility function is:
UC (xC , yC ) = xC yC2
Find a competitive equilibrium for this economy.
2
Question 9.B
Richard and Bert enjoy consuming liquor and cigarettes. Richard's utility function is:
UR (xR , yR ) = 3xR + yR
while Bert's utility function is:
UB (xB , yB ) = ln xB + ln yB
Richard's endowment is
(40, 0)
and Bert's endowment is
1. Show that if Richard consumes both
that if
px > 3
or
px < 3
x and y
(0, 30).
Normalize
in equilibrium, then
Richard chooses to consume
0
py = 1.
px = 3 (Hint:
Show
of one of the goods).
2. Find the (unique) competitive equilibrium for this economy in which Richard consumes
a positive amount of both goods.
Lecture 10 - First Welfare Theorem and Externalities
Question 10.A (Borrowed from Wolfgang Pesendorfer)
The economy consists of a smoker (S) and an asthma suerer (A). Good
x is cigarettes, good
2 is other stu . S has the utility function:
US (xs , ys , xa , ya ) = xs + ys
where
xs ( xa )
is S's (A's) consumption of cigarettes, and
ys (ya )
is S's (A's) consumption of
other stu. A-types have the utility function:
US (xs , ys , xa , ya ) = ya − 2x̄s
S is endowed with one unit of both goods, while A has 2 units of
y
and no endowment of
x.
1. Is there a an ecient allocation in which S consumes cigarettes?
2. Find a competitive equilibrium of this economy. Show that it is not Pareto ecient.
Question 10.B
The economy consists of two consumers, Chris and Bud. Both consumers are endowed with
one unit of
x
and one unit of
y.
Bud is entirely indierent between all consumption plans.
Chris has the utility function:
UC (xC , yC ) = ln xC + ln yC
1. Find a competitive equilibrium of this economy (Hint: Guess an equilibrium price,
and then check that it works).
2. Find a second competitive equilibrium of this economy, dierent than the one you
found in part 1.
3. Show that both competitive equilibria are not ecient.
3