Intermediate Microeconomics
Problem Set #6
1. Consider the following exchange economy with two consumers and two goods, x and y. Consumer
1 initially has 1 unit of x and no units of y so her initial endowment is (ex , ey ) = (1, 0). Consumer
1
3
1’s preferences are represented by the utility function u(x1 , y1 ) = x14 y14 . Consumer 2 initially has no
units of x and 1 unit of y so her initial endowment is (ex , ey ) = (0, 1). Consumer 2’s preferences are
represented by the utility function v(x, y) = x2 y2 . Assume the price of good x is equal to 1 (px = 1).
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a) Compute the set of Pareto optimal allocations (also called the “contract curve”). This will be
an equation that relates x and y. Which portion of the contract curve represents allocations that
improve consumers’ welfare relative to their initial endowments?
The contract curve traces out all of the feasible allocations where the individuals’ indifference curves are tangent.
At those points, we know that M RS1 for Consumer 1 must equal M RS2 for Consumer 2.
Consumer 1:
1
∂u
∂x1
∂u
∂y1
M RS
Consumer 2:
3
= x14 y14
3
1 y1 4
=
4 x1
1
3 x1 4
=
4 y1
y1
=
3x1
u(x1 , y1 )
v(x2 , y2 ) = x2 y2
∂v
= y2
∂x2
∂v
= x2
∂y2
y2
M RS =
x2
M RS1 = M RS2 :
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y1
3x1
=
y2
x2
We also know that y2 = 1 − y1 and x2 = 1 − x1 , since there are only 1 unit of each good in the economy.
=
1 − y1
1 − x1
3x1 − 3x1 y1
y1 + 2y1 x1
=
3x1
y1 (1 + 2x1 )
=
y1
=
3x1
3x1
1 + 2x1
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y1
3x1
y1 − y1 x1
=
This is the equation for the contract curve. All of the allocations on this line are Pareto optimal. Since both
consumers have utility of zero at their initial endowments, any interior point on this line (where x1 > 0 and
y1 > was
0) will
be welfare
superior tofrom
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initial endowments.
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b) Derive Consumer 1’s demand for goods x and y (i.e. the quantities of x and y that maximize
Consumer 1’s utility subject to her budget constraint).
Since u(x1 , y1 ) is a Cobb-Douglas utility function, the demands for goods x and y can be written as shares of
income dictated by the exponents of the utility function. Since Consumer 1 is endowed with 1 unit of good x
and the price of good x is 1, Consumer 1’s income is equal to 1.
1
4
3
y1 (py ) =
4py
x1 (py ) =
c) Derive Consumer 2’s demand for goods x and y (i.e. the quantities of x and y that maximize
Consumer 2’s utility subject to her budget constraint).
Since v(x2 , y2 ) is a Cobb-Douglas utility function, the demands for goods x and y can be written as shares of
income dictated by the exponents of the utility function. Since Consumer 2 is endowed with 1 unit of good y
and the price of good y is py , Consumer 2’s income is equal to py .
py
2
1
y2 (py ) =
2
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x2 (py ) =
d) Compute the competitive equilibrium for this exchange economy (i.e. with pX = 1, the price py
at which demand for x equals supply of x and demand for y equals supply of y). What are the
trades that lead from the initial endowments to this equilibrium?
The competitive equilibrium price of y is the price that sets x1 = 1 − x2 and y1 = 1 − y2
1
4
=
1−
py
=
3
2
py
2
At this price, x1 = 14 , y1 = 12 , x2 = 34 , and y2 = 21 . So Consumer 1 must trade
2 in exchange for 12 a unit of good y.
3
4
a unit of good x to Consumer
e) Draw the Edgeworth box describing this economy. Include the intial endowments, the contract
curve, and the equilibrium allocation.
2. Consider a pure exchange economy with two traders, Sam and Diane, and two goods, G and H.
Sam’s utility function is Us = (Gs )2 Hs while Diane’s utility function is Ud = Gd Hd . Together, Sam and
Diane own 50 units of G and 150 units of H which are initially divided equally between them.
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a) Compute the set of Pareto optimal allocations (also called the “contract curve”).
The contract curve traces out all of the feasible allocations where the individuals’ indifference curves are tangent.
At those points, we know that M RSs for Sam must equal M RSd for Diane.
Sam:
US (Gs , Hs ) = (Gs )2 Hs
∂Us
= 2Gs Hs
∂Gs
∂Us
= G2s
∂Hs
2Hs
M RS =
Gs
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Diane:
Ud (Gd , Hd ) = Gd Hd
∂Ud
= Hd
∂Gd
∂Ud
= Gd
∂Hd
Hd
M RS =
Gd
M RSs = M RSd :
2Hs
Gs
=
Hd
Gd
We also know that Hd = 150 − Hs and Gd = 50 − Gs , since there are only 150 units of H and 50 units of G in
the economy.
=
150 − Hs
50 − Gs
150Gs − Gs Hs
100Hs − Hs Gs
=
150Gs
Hs (100 − Gs )
=
Hs
=
150Gs
150Gs
100 − Gs
=
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2Hs
Gs
100Hs − 2Hs Gs
This is the equation for the contract curve. All of the allocations on this line are Pareto optimal.
b) Assuming that the price of H is equal to 1 (pH = 1), compute the competitive equilibrium of this
economy (i.e. the price of G at which demand for G equals supply of G and demand for H equals
supply of H).
Again, everything is Cobb-Douglas here so demands are easily written as income shares. All that remains is to
determine incomes. Since we assume that the price of good H is 1, Sam and Diane’s incomes are the same and
equal to 75 + 25pG .
Sam’s demand
Gs (pG )
=
Hs (pG )
=
Gd (pG )
=
Hd (pG )
=
2
150 + 50pG
(75 + 25pG ) =
3pG
3pG
75 + 25pG
1
(75 + 25pG ) =
3
3
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Diane’s demand
1
75 + 25pG
(75 + 25pG ) =
2pG
2pG
1
75 + 25pG
(75 + 25pG ) =
2
2
The competitive equilibrium price of G is the price that sets Gs = 50 − Gd and Hs = 150 − Hd
1
1
(75 + 25pG ) = 150 − (75 + 25pG )
3
2
5
(75 + 25pG ) = 150
6
75 + 25pG = 180
25pG
=
pG
=
105
105
21
=
25
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3. Fry and Leela consume pizza, Z, and Slurm, S. Fry’s utility function is UF = ZF0.6 SF0.4 , and Leela’s is
UL = ZL0.1 SL0.9 . Their initial endowments are ZF = 10, SF = 20, ZL = 20, and SL = 10.
a) Compute the set of Pareto optimal allocations (also called the “contract curve”).
The contract curve traces out all of the feasible allocations where the individuals’ indifference curves are tangent.
At those points, we know that M RSF for Fry must equal M RSL for Leela.
Fry:
= ZF0.6 SF0.4
0.4
SF
= 0.6
ZF
0.6
ZF
= 0.4
SF
3SF
=
2ZF
UF (ZF , SF )
∂UF
∂ZF
∂UF
∂SF
M RS
Leela:
= ZL0.1 SL0.9
0.9
SL
= 0.1
ZL
0.1
ZL
= 0.9
SL
SL
=
9PL
UL (ZL , SL )
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∂UL
∂ZL
∂UL
∂SL
M RS
M RS1 = M RS2 :
3SF
2ZF
=
SL
9ZL
We also know that SL = 30 − SF and ZL = 30 − ZF , since there are only 30 units of each good in the economy.
3SF
=
2ZF
27SF (30 − ZF ) =
30 − SF
9(30 − ZF )
60ZF − 2ZF SF
810SF − 27SF ZF + 2SF ZF
=
60ZF
SF (810 − 25ZF )
=
SF
=
60ZF
12ZF
162 − 5ZF
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This is the equation for the contract curve. All of the allocations on this line are Pareto optimal. Incidentally,
the initial endowment is not Pareto optimal, as you can tell from this equation.
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b) Assuming that the price of Slurm (S) is equal to 1 (pS = 1), compute the competitive equilibrium
of this economy (i.e. the price of pizza (Z) at which demand for Z equals supply of Z and demand
for S equals supply of S).
Again, everything is Cobb-Douglas here so demands are easily written as income shares. All that remains is to
determine incomes. Since we assume that the price of Slurm is 1, Fry’s income is 20 + 10pZ and Leela’s income
is 10 + 20pZ .
Fry’s demand
SF (pZ )
=
ZF (pZ )
=
0.4(20 + 10pZ ) = 8 + 4pZ
12
0.6
(20 + 10pZ ) =
+6
pZ
pZ
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Leela’s demand
SL (pZ )
=
ZL (pZ )
=
0.9(20pZ + 10) = 18pZ + 9
0.1
1
(20pZ + 10) = 2 +
pZ
pZ
The competitive equilibrium price of Z is the price that sets SF = 30 − SL and ZF = 30 − ZL
8 + 4pZ
=
30 − 18pZ − 9
22pZ
=
pZ
=
30 − 17
13
22
At this price, SF ≈ 10.36, ZF ≈ 26.31, SL ≈ 19.64, and ZL ≈ 3.69.
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c) Draw the Edgeworth box describing this economy. Include the intial endowments, the contract
curve, and the equilibrium allocation.
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