Class 24 (answers) General equilibrium in a pure exchange economy

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Microeconomics II 
Undergraduate degree in Economics 
 
Class nr. 24 
 
Subject: 4. General equilibrium theory 
 4.1. General equilibrium in a pure exchange economy 
 
Exercise 31.1. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, p. 377-378 
Morris Zapp and Philip Swallow consume wine and books. Morris has an initial 
endowment of 60 books and 10 bottles of wine. Philip has an initial endowment of 20 
books and 30 bottles of wine. They have no other assets and make no trades with 
anyone other than each other. For Morris, a book and a bottle of wine are perfect 
substitutes. His utility function is ( ) MMMMM wbwbU +=, , where “ Mb ” is the number of 
books Morris consumes and “ Mw ” is the number of bottles of wine he consumes. 
Philip’s preferences are more subtle and convex. He has a Cobb-Douglas utility 
function, ( ) PPPPP wbwbU ×=, , where “ Pb ” is the number of books Phillips consumes 
and “ Pw ” is the number of bottles of wine he consumes 
a) Sketch an Edgeworth box and mark the initial endowment of the two consumers 
as well as their indifference curves that pass through that point. 
b) At any Pareto optimum, where both people consume some of each good, it must 
be that their marginal rates of substitution are equal. What is Morris’s marginal 
rate of substitution? What is Morris’s marginal rate of substitution when he 
consumes at the optimal Pareto point ( )PP wb , ? Determine the contract curve 
equation, that is, the set of all optimal Pareto points and sketch it in the 
Edgeworth box. 
c) At a competitive equilibrium, it will have to be that Morris consumes some 
books and some wine. But in order for him to do so, what value must the ratio of 
the price of wine to the price of books assume? If we make books the numeraire, 
what will be the price of wine in a competitive equilibrium? 
d) At the equilibrium prices you found in the last question, what is the value of 
Philip Swallow’s initial endowment? How many books and bottles of wine will 
Phillip consume at these prices? If Morris Zapp consumes all of the books and 
all of the wine that Philip doesn’t consume, what is his consumption bundle 
going to be? 
e) At the competitive equilibrium prices that you found above, what is Morris’ 
income? What is the cost that Morris faces from consuming books and the 
bottles of wine that Phillip doesn’t consume? At these prices, can Morris afford 
a bundle that he likes better than the bundle (55, 15)? 
f) Suppose that an economy consisted of 1,000 people just like Morris and 1,000 
people just like Philip. Each of the Morris types had the same endowment and 
the same tastes as Morris. Each of the Philip types had the same endowment and 
tastes as Philip. Would the prices that you found to be equilibrium prices for 
Morris and Philip still be competitive equilibrium prices? If each of the Morris 
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types and each of the Philip types behaved in the same way as Morris and Philip 
did above, would supply equal demand for both wine and books? 
 
Answers: 
 
a) 
 Edgeworth box chart and initial endowment: 
802060 =+=+ ωω PM bb 
403010 =+=+ ωω PM ww 
( ) ( )
( ) ( )⎪⎩
⎪⎨⎧ ==Ω
==ΩΩ
20,30,
10,60,
: ωω
ωω
PPP
MMM
wb
wb
 
 
Indifference curves: 
( ) MMMMM wbwbU +=, 
( ) 70106010,60 =+=MU 
( )
70
0
70, =⇒
⎩⎨
⎧
=
=
M
M
MMM b
w
wbU
 We found the point where Morris’s indifference curve 
reaches the lower edge of the box. 
( )
30
40
70, =⇒
⎩⎨
⎧
=
=
M
M
MMM b
w
wbU
 We found the point where Morris’s indifference curve 
reaches the upper edge of the box. 
Morris 
Philip
0 
10 
20 
30 
40 
40
30
20
10
0
0 10 20 30 40 50 60 70 80 
80 70 60 50 40 30 20 10 0 
Mb
Mw 
Pw 
W
Pb 
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 ( ) PPPPP wbwbU ×=, 
( ) 600302030,20 =×=PU 
( )
15
40
600, =⇒
⎩⎨
⎧
=
=
P
P
PPP b
w
wbU
 We found the point where Phillip’s indifference curve 
reaches the lower edge of the box. 
( )
5,7
80
600, =⇒
⎩⎨
⎧
=
=
P
P
PPP w
b
wbU
 We found the point where Phillip’s indifference curve 
reaches the left edge of the box. 
 
Technical Note: (not necessary for this exercise): How could you determine where the 
indifference curves cross each other. 
Note that, 
⎩⎨
⎧
−=
−=⇔
⎩⎨
⎧
=+
=+
MP
MP
MP
MP
ww
bb
ww
bb
40
80
40
80
 
Intersection of the indifference curves: 
( )
( ) ( ) ( )
, 70 7070
80 40 600600, 600
M M M M MM M
M MP PP P P
U b w b wb w
b wb wU b w
⎧ = + =⎧+ =⎧⎪ ⎪⇔ ⇔ ⇔⎨ ⎨ ⎨ − × − =× == ⎪⎩⎪ ⎩⎩
 
( ) ( ) ( ) ( )
( ) ( )2 2
70 70
80 70 40 600 10 40 600
70 70
400 30 600 30 200 0
M M M M
M M M M
M M M M
M M M M
b w b w
w w w w
b w b w
w w w w
= −⎧ = −⎧⎪ ⎪⇔ ⇔ ⇔⎨ ⎨⎡ ⎤− − × − = + × − =⎪⎪⎣ ⎦ ⎩⎩
= − = −⎧ ⎧⎪ ⎪⇔ ⇔ ⇔⎨ ⎨+ − = − + =⎪ ⎪⎩ ⎩
 
Morris 
Philip
0 
10 
20 
30 
40 
40
30
20
10
0
0 10 20 30 40 50 60 70 80 
80 70 60 50 40 30 20 10 0 
Mb
Mw 
Pw 
W
Morris’s indifference curve 
Philip’s indifference curve 
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Undergraduate degree in Economics 
 
( ) ( ) ⎩⎨
⎧
=
=∨
⎩⎨
⎧
=
=⇔
⎪⎩
⎪⎨
⎧
±=
−=
⇔
⎪⎩
⎪⎨
⎧
×
××−−±−−=
−=
⇔
20
50
10
60
2
1030
70
12
200143030
70
2
M
M
M
M
M
MM
M
MM
w
b
w
b
w
wb
w
wb
 
By using the quadratic formula: 
a
cabbxcbxax ×
××−±−=⇔=++
2
40
2
2 
 
b) 
Morris’s marginal rate of substitution: 
( )
( ) 11
1
,
,
, −=−=
∂
∂
∂
∂
−=
M
MMM
M
MMM
M
wb
w
wbU
b
wbU
MRS 
Philip’s marginal rate of substitution at the Pareto optimal point must also be -1, 
because we know that, at the optimal point both marginal rates of substitution must be 
equal. 
The contract curve (or Pareto set) is given by the equality of both consumer’s marginal 
rates of substitution. 
 
Morris 
Philip
0 
10 
20 
30 
40 
40
30
20
10
0
0 10 20 30 40 50 60 70 80 
80 70 60 50 40 30 20 10 0 
Mb
Mw 
Pw 
W
Morris’s indifference curve 
Philip’s indifference curve 
Pb 
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Undergraduate degree in Economics 
 ( )
( ) P
P
P
PPP
P
PPP
P
wb b
w
w
wbU
b
wbU
MRS −=
∂
∂
∂
∂
−= ,
,
, 
PP
P
PP
wb
M
wb bwb
wMRSMRS =⇔−=−⇔= 1,, 
 
Technical Note (not necessary for this exercise): 
Remember that: 
⎩⎨
⎧
−=
−=⇔
⎩⎨
⎧
=+
=+
MP
MP
MP
MP
ww
bb
ww
bb
40
80
40
80
 
Now, rewrite the contract curve from Morris point of view: 
408040 −=⇔−=−⇔= MMMMPP bwbwbw , in line with the graph drawn. 
 
c) 
At the competitive equilibrium, the price ratio must be equal to both agents’ marginal 
rates of substitution; therefore, it must be equal to -1. 
1−=−
w
b
p
p
 
If the books are working as numeraire, it means that by multiplying all numbers for a 
constant,
bp
k 1= , we get 1=bp . 
Morris 
Philip
0 
10 
20 
30 
40 
40
30
20
10
0
0 10 20 30 40 50 60 70 80 
80 70 60 50 40 30 20 10 0 
Mb
Mw 
Pw 
W
Morris’ indifference curve 
Philip’s indifference curve 
Contract curve
Pb 
Microeconomics II 
Undergraduate degree in Economics 
 
d) 
By the price ratio we already know that 1−=−
w
b
p
p
, therefore 111 =⇔−=− w
w
p
p
 
The value of Philip’s initial endowment is: 
50301201 =×+×=+ ωω PwPb wpbp 
3 ways of solving Philip’s optimization problem: 
1) Solving the constrained optimization problem by usingthe Lagrange function. 
( )
( )
⎪⎩
⎪⎨
⎧
=
=
=
⇔
⎪⎩
⎪⎨
⎧
+=
−
=
⇔
⎪⎩
⎪⎨
⎧
+=
=
=
⇔
⎪⎩
⎪⎨
⎧
=−−
=−
=−
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
=∂
∂
=∂
∂
=∂
∂
−−+×=
⎪⎩
⎪⎨
⎧
=×+×
×=
25
25
25
5050050
0
0
0
0
0
50
5011..
,max
,
P
P
PP
PP
PP
P
P
PP
P
P
P
P
PPPP
PP
PPPPPwb
b
w
bb
bw
wb
b
w
wb
b
w
Lg
w
Lg
b
Lg
wbwbLg
wbts
wbwbU
PP
λλ
λ
λ
λ
λ
λ 
or 
2) By replacing the budget constraint into the objective function, we can transform the 
constrained optimization problem into an unconstrained optimization problem: 
( ) ( ) ( ) ( )
( )[ ] ( ) ( )
25255050
2505001501050
50,max
50..
,max
5011..
,max
,,
=−=−=
=⇔=−−⇔=−×+−×⇔=−×∂
∂
−×=⇔⎪⎩
⎪⎨
⎧
−=
×=⇔⎪⎩
⎪⎨
⎧
=×+×
×=
PP
PPPPPPP
P
PPPPPb
PP
PPPPPwb
PP
PPPPPwb
bw
bbbbbbb
b
bbwbU
bwts
wbwbU
wbts
wbwbU
P
PPPP
or 
3) By using directly Philips’ optimum condition (MRS = price ratio, i.e., the 
indifference curve is tangent to the budget constraint; equal slopes), together with the 
budget constraint. 
⎩⎨
⎧
=
=⇔
⎩⎨
⎧
=+
=⇔
⎪⎩
⎪⎨
⎧
=+
−=−⇔⎪⎩
⎪⎨⎧ =×+×
−=
25
25
50
50
1
5011
1,
P
P
PP
PP
PP
P
P
PP
P
wb
b
w
bb
bw
wb
b
w
wb
MRS 
Note: Apparently we get the impression that we can solve Philip’s optimization problem 
regardless of Morris’s, however that’s not true. This happens because the prices used 
came from the equality of both marginal rates of substitution. That’s why this last 
method is equivalent to finding out the intersection between the contract curve and 
Philip’s budget constraint. 
Philip Swallow consumes 25 books and 25 bottles of wine. 
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Undergraduate degree in Economics 
 
Morris Zapp consumes the remaining quantities that Philip Swallow doesn’t: 
152540
552580
=−=
=−=
M
M
w
b
 
 
Technical Note: 
Although it’s not required, let’s see the results for Morris optimization problem. 
Morris’s initial endowment: 
70101601 =×+×=+ ωω MwMb wpbp 
 
Neither Phillip nor Morris’s optimization problem can be solved using the first or the 
second method used above. This occurs because the system of equations is 
undetermined due to the match between the indifference curve and the budget 
constraint. Therefore, any point from these two functions is an intersection point. 
 
 
Example: Solving the conditional optimization problem by using the Lagrangian 
function. 
( )
( )MMMM
MM
MMMMMwb
wbwbLg
wbts
wbwbU
MM
−−++=
⎪⎩
⎪⎨
⎧
=×+×
+=
70
7011..
,max
,
λ
 
Morris 
Philip
0 
10 
20 
30 
40 
40
30
20
10
0
0 10 20 30 40 50 60 70 80 
80 70 60 50 40 30 20 10 0 
Mb
Mw 
Pw 
W
Morris’s indifference curve (coincide, on this case, with the BC)
Philip’s indifference curve
Contract curve
X
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Undergraduate degree in Economics 
 
⎩⎨
⎧
−=
=⇔
⎪⎩
⎪⎨
⎧
=−−
=−
=−
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
=∂
∂
=∂
∂
=∂
∂
MM
MM
M
M
wb
wb
Lg
w
Lg
b
Lg
70
1
070
01
01
0
0
0
λλ
λ
λ
 
MM wb −= 70 This expression is, at the same time, Morris’s budget constraint and his 
indifference curve. 
So, we should use the third method (intersection between Morris’s budget constraint 
and the contract curve): 
( )
⎩⎨
⎧
=
=⇔
⎩⎨
⎧
−=
=⇔
⇔
⎩⎨
⎧
−=
−=−−⇔
⎩⎨
⎧
=+
−=−⇔
⎩⎨
⎧
=×+×
=
15
55
70
1102
70
807040
70
8040
7011
M
M
MM
M
MM
MM
MM
MM
MM
PP
w
b
bw
b
bw
bb
wb
bw
wb
bw
 
 
e) 
Morris’s income is given by his initial endowment, which is equal to: 
70101601 =×+×=+ ωω MwMb wpbp 
By consuming what Philip does not consume, Morris has a cost of: 
70151551 =×+×=+ MwMb wpbp 
The value of Morris's initial endowment must be equal to the cost of what he consumes 
(his expenditure). 
With these prices Morris cannot consume another bundle which he prefers to the bundle 
(55, 15), because this one is already the result of his own optimization problem subject 
to his budget constraint for these prices. 
 
f) 
Yes, we just need to interpret the units of books and bottles of wine consumed in the 
previous problem as thousands (we would have a different scale) and the behavior of 
each consumer as the aggregate of all who belong to their group. We can also think 
about Philip Swallow and Morris Zapp as the representative consumers of each 
consumer group. 
 
 
 
 
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Exercise 31.3. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, p. 380-381 
Dean Foster Z. Interface and Professor J. Fetid Nightsoil exchange platitudes1 and 
bromides2. When Dean Interface consumes TI platitudes and BI bromides, his utility is 
given by 
( ) IIIII TBTBU 2, += 
When Professor Nightsoil consumes TN platitudes and BN bromide, his utility is given 
by 
( ) NNNNN TBTBU 4, += 
Dean Interface’s initial endowment is 12 platitudes and 8 bromides. Professor 
Nightsoil’s initial endowment is 4 platitudes and 8 bromides. 
 
a) If Dean Interface consumes TI platitudes and BI bromides, what will be his 
marginal rate of substitution? If professor Nightsoil consumes TN platitudes and 
BN bromides, what will be his marginal rate of substitution? 
b) On the contract curve, Dean Interface’s marginal rate of substitution equals 
Professor Nightsoil. Write an equation that states this condition. 
c) Along the contract curve, what is the ratio between the quantities of platitudes 
consumed by Dean Interface and the quantities of platitudes consumed by 
professor Nightsoil? 
d) Along the contract curve, what is the aggregate consumption of platitudes? 
e) Along the contract curve, what is the individual consumption of platitudes by 
both of them? 
f) Sketch an Edgeworth box and represent the initial endowment, the indifference 
curves which pass through the initial endowment and the contract curve. 
g) Find the competitive equilibrium prices and quantities (hint: what is the 
relationship between the prices and marginal rates of substitution? Use a 
numeraire). 
Answers: 
a) 
( ) IIIII TBTBU 2, += 
( )
( ) I
II
III
I
III
I T
TT
TBU
B
TBU
MRS −=
××
−=
∂
∂
∂
∂
−=
1
2
12
1
,
,
 
 
1 Platitudes in portuguese means “trivialidades”. 
2 Bromides in portuguese means “banalidades”. Bromide is also a chemical used in medicine as a 
sedative. 
Microeconomics II 
Undergraduate degree in Economics 
 
( ) NNNNN TBTBU 4, += 
( )
( ) N
NN
NNN
N
NNN
N T
TT
TBU
B
TBU
MRS
2
1
1
2
14
1
,
,
−=
××
−=
∂
∂
∂
∂
−= 
 
b) ( ) NINININI TTTTTTMRSMRS 412121
2
2 =⇔⎟⎠
⎞⎜⎝
⎛−=−⇔−=−⇔= 
 
c) 
4
1
4
1 =⇔=
N
I
NI T
TTT 
 
d) The aggregated consumption of platitudes must equal the sum of platitudes in the 
initial endowment allocation 
16412 =+=+ NI TT 
 
e) 
⎩⎨
⎧
=
=⇔
⎪⎩
⎪⎨
⎧
=
−
⇔
⎪⎩
⎪⎨
⎧
=+
−
⇔
⎪⎩
⎪⎨
⎧
=+
=
8,12
2,3
16
4
516
4
1
16
4
1
N
I
NNN
NI
NI
T
T
TTTTT
TT
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Undergraduate degree in Economics 
 
f) Initial endowment allocation: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Indifference curves: 
( ) IIIII TBTBU 2, += 
( ) 93,14348122812,8≈+=+=IU 
( ), 8 4 3 2 8 4 3 16 2 8 4 3
1616
2 8 4 3 4 2 3
I I I I I
I
II
I I
U B T B T
T
BB
T T
⎧ ⎧= + + = +⎪ ⎪⇔ ⇔ + = + ⇔⎨ ⎨ == ⎪⎪ ⎩⎩
⇔ = − + ⇔ = − +
 
Note that this equation has no solution because 4 2 3 0.54 0− + ≈ − < (the square root 
function must return a positive value, not a negative one). 
We couldn’t find a point where Dean Interface’s indifference curve reaches the upper 
edge of the box, so let’s see if it reaches the left edge of the box instead. 
2 
0 
4 
8 
6 
10 
12 
14 
16 
16 
16 
16 
14 
14 
14 
12 
12 
12 
10 
10 
10 
8 
8 
8 
6 
6 
6 4 
4 
2 
2 
0 
0 
0 
4 2 
Interface 
Nightsoil 
TI 
TN 
BI 
BN 
Ω 
Microeconomics II 
Undergraduate degree in Economics 
 
( ), 8 4 3 2 0 8 4 3 8 4 3 14,93
00
I I I I
I
II
U B T B B
TT
⎧ ⎧= + + = +⎪ ⎪⇔ ⇔ = + ≈⎨ ⎨ == ⎪⎪ ⎩⎩
 
We have found the point where the Dean Interface’s indifference curve reaches the left 
edge of the box. 
( )
93,634348162
16
93,14, ≈=⇔+=+⇔
⎩⎨
⎧
=
≈
II
I
III BB
T
TBU
 
We have found the point where the Dean Interface’s indifference curve reaches the right 
edge of the box. 
( ) NNNNN TBTBU 4, += 
( ) 164484,8 =+=NU 
( )
016416
16
16, =⇔=+⇔
⎩⎨
⎧
=
=
NN
N
NNN TT
B
TBU
 
We have found the point where professor Nightsoil’s indifference curve reaches the 
lower edge of the box (specifically on the bottom right corner). 
( )
016164
16
16, =⇔=+⇔
⎩⎨
⎧
=
=
NN
N
NNN BB
T
TBU
 
We have found the point where the professor Nightsoil’s indifference curve reaches the 
left edge of the box (specifically on the upper left corner). 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
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Undergraduate degree in Economics 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
g) 
2 
0 
4 
8 
6 
10 
12 
14 
16 
16 
16 
16 
14 
14 
14 
12 
12 
12 
10 
10 
10 
8 
8 
8 
6 
6 
6 4 
4 
2 
2 
0 
0 
0 
4 2 
Interface 
Nightsoil 
TI 
TN 
BI 
BN 
Ω 
Dean Interface’s indifference curve: ( ) 93,142, ≈+= IIIII TBTBU 
Professor Nightsoil’s indifference curve: ( ) 164, =+= NNNNN TBTBU 
Contract Curve: 2,3=IT 
( ) 93,14, ≈III TBU
( ) 16, =NNN TBU
2,3=IT 
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In a perfect competition market with convex preferences, as in our case, a Pareto 
optimal point is a competitive equilibrium (according to the second welfare theorem). 
At that point both consumers’ marginal rate of substitution is equal to the price ratio. 
T
B
T
B
I
T
B
I
T
B
NII
T
B
NI
P
P
P
P
P
P
P
PTT
P
PMRSMRS
≈=⇔==⇔
⇔−=−=−⇔−=−=−⇔−==
79,179,18,12
2
12,3
8,12
2
12,3
2
1
 
By using platitudes (T) as a numeraire (the only thing that matters is the relative price; if 
we multiply both prices by a constant, the final result doesn’t change): 
79,11 =⇒= BT PP 
Using the budget constraint we are able to determine IB and NB . 
92,12
79,1
2,312879,1
121879,12,3179,1128
=−+×=⇔
⇔×+×=×+×⇔×+×=+
I
ITBITIB
B
BPPTPBP
 
08,392,121616 =−=−= IN BB 
Actually, what we did, once again, was to find the intersection between the contract 
curve and the Dean Interface’s budget constraint (which is the same for Professor 
Nightsoil). 
⎩⎨
⎧
=
=⇔
⎩⎨
⎧
−=
=⇔
⇔
⎪⎩
⎪⎨
⎧
−=
=
⇔
⎩⎨
⎧
×+×=×+×
=⇔
⇔
⎪⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
×+×=×+
≈
=
⇔
⎪⎪⎩
⎪⎪⎨
⎧
×+×=+
≈
=
92,12
2,3
56,070,14
2,3
79,1
32,26
2,3
121879,1179,1
2,3
12181
79,1
2,3
128
79,1
2,3
I
I
II
I
I
I
I
II
I
T
B
II
T
B
T
B
I
TBITIB
T
B
I
B
T
TB
T
TB
T
TB
T
P
PTB
P
P
P
P
T
PPTPBP
P
P
T
 
Deans Interface’s level of utility reached at the optimal point: 
( ) 50,162,3292,122, ≈+=+= IIIII TBTBU 
Professor Nightsoil’s level of utility reached at the optimal point: 
( ) 39,178,12408,34, ≈+=+= NNNNN TBTBU 
 
 
Microeconomics II 
Undergraduate degree in Economics 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2 
0 
4 
8 
6 
10 
12 
14 
16 
16 
16 
16 
14 
14 
14 
12 
12 
12 
10 
10 
10 
8 
8 
8 
6 
6 
6 4 
4 
2 
2 
0 
0 
0 
4 2 
Interface 
Nightsoil 
TI 
TN 
BI 
BN 
Ω 
Dean Interface’s indifference curve: ( ) 50,162, ≈+= IIIII TBTBU 
Professor Nightsoil’s indifference curve: ( ) 39,174, =+= NNNNN TBTBU 
Contract curve: 2,3=IT 
( ) 50,16, ≈III TBU 
( ) 39,17, =NNN TBU
2,3=IT 
X 
Budget constraint: II TB 56,070,14 −=