4 1 General equilibrium in a pure exchange economy (answers)

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Microeconomics II 
Undergraduate degree in Economics 
 
Review Exercises - 4-1 General equilibrium in a pure exchange economy 
 
Exercise 31.2. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 379-380 
 
Consider a small exchange economy with two consumers, Astrid and Birger, and two 
commodities, herring1 and cheese. Astrid’s initial endowment is 4 units of herring and 1 
unit of cheese. Birger’s initial endowment has no herring and 7 units of cheese. Astrid’s 
utility function is ( ) AAAA CHCHU =, . Birger is a more inflexible person. His utility 
function is ( ) { }BBBB CHCHU ,min, = . Here AH and AC are the amounts of herring and 
cheese for Astrid, and BH and BC are amounts of herring and cheese for Birger. 
a) Draw an Edgeworth box, showing the initial allocation and sketching in two 
indifference curves. Measure Astrid’s consumption from the lower left and 
Birger’s from the upper right. 
b) Mark the Pareto optimal allocations in your Edgeworth box. (Hint: Since Birger 
is kinky, calculus won’t be of much help. But notice that because of the rigidity 
of the proportions in which he demands the two goods, it would be inefficient to 
give Birger a positive amount of either good if he had less than that amount of 
the other good.) 
c) Let cheese be the numeraire (with price 1) and let p denote the price of herring. 
Write an expression for the amount of herring that Birger will demand at these 
prices. (Hint: Since Birger initially owns 7 units of cheese and no herring and 
since cheese is the numeraire, the value of his initial endowment is 7. If the price 
of herring is p, how many units of herring will he choose to maximize his utility 
subject to his budget constraint?) 
d) Where the price of cheese is 1 and p is the price of herring, what is the value of 
Astrid’s initial endowment? How much herring will Astrid demand at price p? 
 
Answers: 
 
a) 
Edgeworth box chart and graphical representation of the initial endowment allocation: 
404 =+=+ ωω BA HH 
871 =+=+ ωω BA CC 
( ) ( )
( ) ( )⎪⎩
⎪⎨⎧ ==Ω
==ΩΩ
7,0,
1,4,
: ωω
ωω
BBB
AAA
CH
CH
 
 
1 In Portuguese it means “arenque”. 
Microeconomics II 
Undergraduate degree in Economics 
 
 
Determining Astrid’s indifference curve that goes through the endowment: 
( ) AAAAA CHCHU =, 
( ) 4141,4 =×=AU 
( )
2
1
8
4, =⇒
⎩⎨
⎧
=
=
A
A
AAA H
C
CHU
 We found the point where Astrid’s indifference curve, 
reaches the right edge of the box (we already had the point where it reached the upper 
edge) 
Another Astrid’s indifference curve: 
( )
2
4
8, =⇒
⎩⎨
⎧
=
=
A
A
AAA C
H
CHU
 
( )
1
8
8, =⇒
⎩⎨
⎧
=
=
A
A
AAA H
C
CHU
 
Determining Birger’s indifference curve: 
( ) { }BBBBB CHCHU ,min, = 
( ) { } 07,0min7,0 ==BU 
( ) 0707,0 =×=BU 
( ) [ ]8;0
0
0, ∈⇒
⎩⎨
⎧
=
=
B
B
BBB C
H
CHU
 
Astrid 
Birger
0 
1 
2 
3 
4 
4
3
2
1
0
0 1 2 3 4 5 6 7 8 
8 7 6 5 4 3 2 1 0 
AC
AH 
BH 
W 
BC 
Microeconomics II 
Undergraduate degree in Economics 
 ( ) [ ]8;0
0
0, ∈⇒
⎩⎨
⎧
=
=
B
B
BBB H
C
CHU
 
Another of Birger’s indifference curves: 
( ) [ ]8;1
1
1, ∈⇒
⎩⎨
⎧
=
=
B
B
BBB C
H
CHU
 
( ) [ ]8;1
1
1, ∈⇒
⎩⎨
⎧
=
=
B
B
BBB H
C
CHU
 
 
b) 
In this case, the optimum rule that states that marginal rates of substitution (slope of the 
indifference curves) must be equal at the optimal allocation does not apply because 
Birger is kinky. Nevertheless, by following the hint we know that Birger will always 
consume both goods together, in a one-to-one proportion at the optimal point, therefore 
the expression of the contract curve (or Pareto set) must be BB CH = . 
Astrid 
Birger
0 
1 
2 
3 
4 
4
3
2
1
0
0 1 2 3 4 5 6 7 8 
8 7 6 5 4 3 2 1 0 
AC
AH 
BH 
W 
BC 
Astrid’s indifference curves 
Birger’s indifference curves 
4=AU 
8=AU 1=BU 
0=BU 
Microeconomics II 
Undergraduate degree in Economics 
 
 
c) 
BB CH = is Birger’s optimal condition. We can combine it with his budget constraint in 
order to determine his optimal consumption bundle. 
 
( )
1
717
171011
+==⇔⎩⎨
⎧
=
+=⇔
⇔
⎩⎨
⎧
=
×+×=×+×⇔
⎩⎨
⎧
=
×+×=×+×
p
CH
CH
Hp
CH
HHpp
CH
CHpCHp
BB
BB
B
BB
BB
BB
BBBB
ωω
 
 
d) 
Astrid’s initial endowment: 141 +=×+× pCHp AA ωω 
Because Astrid’s preferences are well-behaved we can determine her optimal 
consumption bundle by using the traditional methods: 
 
Astrid’s optimization problem: 
( ) ( )
, ,
max , max ,
. . 4 1. . 1 1
A A A A
A A A A A A A A A AH C H C
A AA A A A
U H C H C U H C H C
s t p p H Cs t p H C p H Cω ω
⎧ = ⎧ =⎪ ⎪⇔⎨ ⎨ + = × +× + × = × + ×⎪ ⎪⎩⎩
 
( )max 4 1
A
A AH
H p p H⇔ × + − × 
Astrid 
Birger
0 
1 
2 
3 
4 
4
3
2
1
0
0 1 2 3 4 5 6 7 8 
8 7 6 5 4 3 2 1 0 
AC
AH 
BH 
W 
BC 
Astrid’s indifference curves 
Birge’s indifference curves 
Contract curve (Pareto Set) 
Microeconomics II 
Undergraduate degree in Economics 
 
( )( ) ( ) ( )4 1 0 1 4 1 0 4 1 2 0
4 1 12
2 2
A A A A A
A A
H p p H p p H H p p pH
R
pH H
p p
∂ × + − × = ⇔ × + − × + × − = ⇔ + − = ⇔∂
+⇔ = ⇔ = +
( 
2
12
2
1214
2
141414 +=−−+=⎟⎟⎠
⎞⎜⎜⎝
⎛ +×−+=×−+= ppp
p
pppHppC AA ) 
or 
[ ]
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
+=
+=
+=
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
+=
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛ +×=
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
+=
=
×=
⇔
⇔
⎪⎩
⎪⎨
⎧
×+×=+
=
×=
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
+×=+
=
=
⇔
⎪⎩
⎪⎨
⎧
=−×−+
=−
=−
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
=
=
=
−×−++=
p
H
p
pC
p
H
p
p
pC
p
pH
H
HpC
HpHpp
H
HpC
CHpp
H
p
C
CHpp
H
pC
d
dLg
dC
dLg
dH
dLg
CHppCHLg
A
A
A
A
A
A
AA
AA
A
AA
AA
A
A
AA
A
A
A
A
AAAA
2
12
2
12
2
12
2
12
2
12
2
12
2
14
1414014
0
0
0
0
0
14
λλλ
λλ
λ
λ
λ
λ
λ
or 
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
+×=×+×
−=
∂
∂
∂
∂
−⇔
⎪⎩
⎪⎨
⎧
×+×=×+×
−=
AA
A
A
A
A
AAAA
C
H
CH
CHpp
p
C
U
H
U
CHpCHp
p
pMRS
AA
114
1
11
,
ωω
 
⎪⎪⎩
⎪⎪⎨
⎧
+=
+=
⇔
⎪⎩
⎪⎨
⎧
+=
×=
⇔
⎩⎨
⎧
×+×=+
×=⇔
⎪⎩
⎪⎨
⎧
+×=+
−=−⇔
p
H
pC
p
pH
HpC
HpHpp
HpC
CHpp
p
H
C
A
A
A
AA
AA
AA
AA
A
A
2
12
2
12
2
14
14
14