Chapter 7
General Equilibrium
Exercise 7.1 Suppose there are 200 traders in a market all of whom behave
as price takers. Suppose there are three goods and the traders own initially the
following quantities:
100 of the traders own 10 units of good 1 each
50 of the traders own 5 units of good 2 each
50 of the traders own 20 units of good 3 each
All the traders have the utility function
1
1
1
U = x12 x24 x34
What are the equilibrium relative prices of the three goods? Which group of
traders has members who are best o¤ ?
Outline Answer:
For each group of traders the Lagrangean may be written
1
1
1
log xh1 + log xh2 + log xh3 +
2
4
4
h
[y h
p1 xh1
p2 xh2
p3 xh3 ]
where h = 1; 2; 3 and y 1 = 10p1 ; y 2 = 5p2 and y 3 = 20p3 : From the …rstorder conditions we …nd that for a trader of type h:
xh1
=
xh2
=
xh3
=
yh
2p1
yh
4p2
yh
4p3
Excess demand for good 1 and 2 are then:
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Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
E1
=
100x11 + 50x21 + 50x31
1000
E2
=
100x12 + 50x22 + 50x32
250
Substituting in for xhi and y h and putting E1 = E2 = 0 we …nd
p2
p3
+ 500
p1
p1
750
p3
+ 250
4
p2
500 + 125
=
0
p1
p2
=
0
250
p3
p2
= 2 and
= 21 .
p1
p1
Using good 1 as numeraire we immediately see that y 1 = y 2 = y 3 = 10: All
are equally well o¤.
that implies
c Frank Cowell 2006
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Microeconomics
Exercise 7.2 Consider an exchange economy with two goods and three persons.
Alf always demands equal quantities of the two goods. Bill’s expenditure on group
1 is always twice his expenditure on good 2. Charlie never uses good 2.
1. Describe the indi¤ erence maps of the three individuals and suggest utility
functions consistent with their behaviour.
2. If the original endowments are respectively (5; 0), (3; 6) and (0; 4), compute
the equilibrium price ratio. What would be the e¤ ect on equilibrium prices
and utility levels if
(a) 4 extra units of good 1 were given to Alf;
(b) 4 units of good 1 were given to Charlie?
Outline Answer:
1. Let
=
p1
so that values are measured in terms of good 2.
p2
(a) Alf’s (binding) budget constraint is
xa1 + xa2 = 5
Therefore, given the information in the question, the demand functions are
5
:
xa1 = xa2 =
+1
The utility function consistent with this behaviour is
U a (xa1 ; xa2 ) = min fxa1 ; xa2 g
–see Figure 7.1.
(b) Bill’s budget constraint is
xb1 + xb2 = 3 + 6
From the question we have
xb1 = 2xb2
Therefore:
xb1
=
xb2
=
2+
4
+ 2:
The utility function is Cobb-Douglas:
U b = 2 log xb1 + log xb2
–see Figure 7.2.
c Frank Cowell 2006
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Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
x2
•
x1
(5,0)
Figure 7.1: Alf’s preferences and demand
x2
•
(3,6)
•
x1
Figure 7.2: Bill’s preferences and demand
c Frank Cowell 2006
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Microeconomics
(c) Charlie’s budget constraint is
xa1 + xa2 = 4
Given the information in the question we have
xc1
=
4=
xc2
=
0
and utility is
U c = xc1
–see Figure 7.3.
x2
•
(4,0)
•
x1
Figure 7.3: Charlie’s preferences and demand
2. Excess demand for good 2 is :
E2 =
5
+ +2
+1
10:
Putting E2 = 0 yields = 2 or 4. Hence the equilibrium price ratio is
4. Utility levels are U a = 4, U b = log(54) and U c = 1:
(a) Excess demand is now
9
+ +2
+1
10
and the equilibrium price ratio is 2. Utility levels are U a = 6, U b =
log(64) and U c = 2:
(b) Excess demand for good 2 is :
E2 =
5
+ +2
+1
10:
Putting E2 = 0 yields = 2 or 4. Hence the equilibrium price ratio
is 4. Utility levels are U a = 4, U b = log(54) and U c = 4 + 1 = 5:
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Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.3 In a two-commodity economy assume a person has the endowment (0; 20).
1. Find the person’s demand function for the two goods if his preferences are
represented by each of the types A to D in Exercise 4.2. In each case
explain what the o¤ er curve must look like.
2. Assume that there are in fact two equal sized groups of people, each with
preferences of type A, where everyone in group 1 has the endowment (10; 0)
with = 12 and everyone in group 2 an endowment (0; 20) with = 34 . Use
the o¤ er curves to …nd the competitive equilibrium price and allocation.
Outline Answer:
1. The income of person h is 20.
(a) If he has preferences of type A then the Lagrangean is
log xh1 + [1
] log xh2 +
xh1
20
xh2
(7.1)
First order conditions for an interior maximum of (7.1) are
xh1
1
xh2
xh1
20
Solving these we …nd
=
1
20
xh2
=
0
=
0
=
0
and so the demands will be
20
xh =
20 [1
(7.2)
]
and the o¤er curve will simply be a horizontal straight line at xh2 =
20 [1
].
(b) If h has preferences of type B then demand will be
xh
h
x
h
x
=
x0 , if
2
0
=
>
00
[x ; x ], if
(20= ; 0), if
=
<
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will consist
of the union of the line segment [x0 ; x00 ] and the line segment from
x00 to (1; 0).
(c) If group-2 persons have preferences of type C then their demands will
be
p
xh = x0 , if >
p
xh = x0 or x00 , if =
p
xh = (20= ; 0), if <
p
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will
consist of the union of the point x0 and the line segment from x00 to
(1; 0).
c Frank Cowell 2006
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Microeconomics
x'
20
B
ce
ren
iffe
ind curve
A
20[1-α]
β
x1
x1
x''
x2
x2
x'
C
D
√γ
δ
x1
x1
x''
Figure 7.4: O¤er Curves for Four Cases
(d) If group-2 persons have preferences of type D then their demands will
be
"
#
xh =
20
+
20
+
and their o¤er curve is just the straight line xh2 = xh1 . These are
illustrated in Figure 7.4.
2. If a type-A person had an income of 10 units of commodity 1 then, by
analogy with part 1, demand would be
x1 =
10
10 [1
]
(7.3)
and the o¤er curve will simply be a vertical straight line at xh1 = 10 .
From (7.2) and (7.3) we have x11 = 10 21 = 5, x22 = 20 1 43 = 5.
Given that there are 10 units per person of commodity 1 and 20 units per
person of commodity 2 the materials balance condition then means that
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Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
the equilibrium allocation must be
Solving for
must be 3.
x1
=
5
15
x2
=
5
5
from (7.2) and (7.3) we …nd that the equilibrium price ratio
c Frank Cowell 2006
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Microeconomics
Exercise 7.4 The agents in a two-commodity exchange economy have utility
functions
U a (xa ) = log(xa1 ) + 2 log(xa2 )
U b (xb ) = 2 log(xb1 ) + log(xb2 )
where xhi is the consumption by agent h of good i, h = a; b; i = 1; 2. The property
distribution is given by the endowments Ra = (9; 3) and Rb = (12; 6).
1. Obtain the excess demand function for each good and verify that Walras’
Law is true.
2. Find the equilibrium price ratio.
3. What is the equilibrium allocation?
4. Given that total resources available remain …xed at R := Ra + Rb =
(21; 9), derive the contract curve.
Outline Answer:
1. To get the demand functions for each person we need to …nd the utilitymaximising solution. The Lagrangean for person a is
La (xa ;
a
) := log (xa1 ) + 2 log (xa2 ) +
a
[9p1 + 3p2
p1 xa1
p2 xa2 ]
First-order conditions are
1
x1
2
x2
a
p1 = 0
a
p2 = 0
9p1 + 3p2 p1 x1a
p2 x2a = 0
9
>
=
>
;
De…ne := p1 =p2 and normalise p2 arbitrarily at 1. Then, rearranging
the FOC we get
9
1
x1
=
a =
2
(7.4)
a = x2
;
9 + 3 = x 1 a + x2 a
Subtracting the …rst two equations from the third in (7.4) we can see that
1
a
= 1+3
. Substituting back for the Lagrange multiplier a into the
…rst two parts of (7.4) we see that the …rst-order conditions imply:
x1 a
x2 a
=
3+ 1
6 +2
(7.5)
Using exactly the same method for person b we would …nd
x1 b
x2 b
=
8+ 4
4 +2
(7.6)
Using the de…nition we can then …nd the excess demand functions by
evaluating:
Ei := xi a + xi b Ria Rib
(7.7)
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CHAPTER 7. GENERAL EQUILIBRIUM
Doing this we get
5
E1
E2
=
10
10
5
(7.8)
Now construct the weighted sum of excess demands. It is obvious that
E1 + E2 = 0
(7.9)
thus con…rming Walras’Law. In equilibrium the materials’balance condition must hold and so excess demand for each good must be zero, unless
the corresponding equilibrium price is zero (markets clear).
2. Solving for E1 = 0 in (7.8) we …nd
prices.
3. The allocation is
x1 b
x2 b
=
1
2
for the (normalised) equilibrium
16
4
=
4. The contract curve is traced out by the MRS condition
MRSa12 = MRSb12
(7.10)
and the materials balance condition
E=0
(7.11)
From (7.11) we have
xb1
xb2
=
21 xa1
9 xa2
(7.12)
Applying (7.10) we then get
21
2xa1
=
xa2
2 [9
xa1
xa2 ]
(7.13)
which implies that the equation of the contract curve is:
xa2 =
c Frank Cowell 2006
102
12xa1
:
xa1 + 7
(7.14)
Microeconomics
Exercise 7.5 Which of the following sets of functions are legitimate excess demand functions?
9
E1 (p) = p2 + p101 =
E2 (p) = p1
(7.15)
;
E3 (p) = p103
9
3
>
E1 (p) = p2p+p
=
1
3
E2 (p) = p1p+p
(7.16)
2
>
2 ;
E3 (p) = p1p+p
3
p3 9
E1 (p) = p1 =
E2 (p) = pp32
(7.17)
;
E3 (p) = 2
Outline Answer:
The …rst system is not homogeneous of degree zero in prices. The second
violates Walras’Law. The third one is both homogeneous and satis…es Walras’
Law.
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CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.6 In a two-commodity economy let be the price of commodity 1
in terms of commodity 2. Suppose the excess demand function for commodity 1
is given by
1 4 + 5 2 2 3:
How many equilibria are there? Are they stable or unstable? How might your
answer be a¤ ected if there were an increase in the stock of commodity 1 in the
economy?
Outline Answer:
The excess demand for commodity 1 at relative price
E( ) := 1
4 +5
2
2
3
]2 [1
= [1
can be written
2 ]:
So that
dE( )=d =
4 + 10
6
2
–see Figure 7.5. From this we see that there are two equilibria as follows:
1.
= 0:5. Here dE( )=d < 0 and so it is clear that the equilibrium is
locally stable
= 1. Here dE( )=d = 0. But the graph of the function reveals
that it is locally stable from “above” (where > 1) and unstable
from “below” (where < 1).
If there were an increase in the stock of commodity 1 the excess demand
function would be shifted to the left in Figure 7.5 –then there is only one,
stable equilibrium.
ρ
1.5
1
0.5
-1
-0.5
•
•
°
0
E
0.5
Figure 7.5: Excess demand
c Frank Cowell 2006
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1
Microeconomics
Exercise 7.7 Consider the following four types of preferences:
Type A
:
log x1 + [1
] log x2
Type B
:
x1 + x2
Type C
:
[x1 ] + [x2 ]
Type D
:
min f x1 ; x2 g
2
2
where x1 ; x2 denote respectively consumption of goods 1 and 2 and ; ; ; are
strictly positive parameters with < 1.
1. Draw the indi¤ erence curves for each type.
2. Assume that a person has an endowment of 10 units of commodity 1 and
zero of commodity 2. Show that, if his preferences are of type A, then his
demand for the two commodities can be represented as
x :=
x1
x2
=
10
10 [1
]
where is the price of good 1 in terms of good 2. What is the person’s
o¤ er curve in this case?
3. Assume now that a person has an endowment of 20 units of commodity
2 (and zero units of commodity 1) …nd the person’s demand for the two
goods if his preferences are represented by each of the types A to D. In
each case explain what the o¤ er curve must look like.
4. In a two-commodity economy there are two equal-sized groups of people.
People in group 1 own all of commodity 1 (10 units per person) and people
in group 2 own all of commodity 2 (20 units per person). If Group 1 has
preferences of type A with = 12 …nd the competitive equilibrium prices
and allocations in each of the following cases:
3
4
(a) Group 2 have preferences of type A with
=
(b) Group 2 have preferences of type B with
= 3.
(c) Group 2 have preferences of type D with
= 1.
5. What problem might arise if group 2 had preferences of type C? Compare
this case with case 4b
Outline Answer:
1. Indi¤erence curves have the shape shown in the …gure 7.6.
2. The income of a group-1 person is 10 . If group-1 persons have preferences
of type A then the Lagrangean is
log x11 + [1
c Frank Cowell 2006
] log x12 +
105
10
x11
x12
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
z2
z2
(1)
(2)
z1
z1
z2
z2
(4)
(3)
z1
z1
Figure 7.6: Indi¤erence Curves
First order conditions for an interior maximum are
x11
1
x12
x11
10
Solving these we …nd
=
1
10
x12
=
0
=
0
=
0
and so the demands will be
x1 =
10
10 [1
]
and the o¤er curve will simply be a vertical straight line at x11 = 10 .
3. The income of a group-2 person is 20. So, if group-2 persons have preferences of type A, then their demands will be
20
x2 =
20 [1
]
and their o¤er curve will simply be a horizontal straight line at x22 =
20 [1
]. If group-2 persons have preferences of type B then their demands will be
x2
c Frank Cowell 2006
x
2
x
2
= x0 , if
2
=
0
>
00
[x ; x ], if
(20= ; 0), if
106
=
<
Microeconomics
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will consist of
the union of the line segment [x0 ; x00 ] and the line segment from x00 to
(1; 0). If group-2 persons have preferences of type C then their demands
will be
p
x2 = x0 , if >
p
x2 = x0 or x00 , if =
p
x2 = (20= ; 0), if <
p
where x0 := (0; 20), x00 := (20= ; 0), and their o¤er curve will consist
of the union of the point x0 and the line segment from x00 to (1; 0). If
group-2 persons have preferences of type D then their demands will be
"
#
20
+
20
+
x2 =
and their o¤er curve is just the straight line x22 = x21 .
4. In each case below we could work out the excess demand function, set
excess demand equal to zero, …nd the equilibrium price and then the equilibrium allocation. However, we can get to the result more quickly by
using an equivalent approach. Given that an equilibrium allocation must
lie at the intersection of the o¤er curves of the two parties the answer in
each case is immediate.
(a) From the above computations we have x11 = 10 21 = 5, x22 =
20 1 34 = 5. Given that there are 10 units per person of commodity 1 and 20 units per person of commodity 2 the materials balance
condition then means that the equilibrium allocation must be
Solving for
x1
=
5
15
x2
=
5
5
we …nd that the equilibrium price ratio must be 3.
x11
(b) We have
= 5, and so x21 = 5. Using the fact that the equilibrium
must lie on the group-2 o¤er curve we see that the solution must lie
on the straight line from (0; 20) to (20=3; 0) we …nd that x22 = 5 and,
from the materials balance condition x12 = 20 5 = 15 (as in the
previous case). By the same reasoning as in the previous case the
equilibrium price must be = 3.
(c) Once again we have x11 = 5, and so x21 = 5. Given that the group2 o¤er curve in this case is such that the person always consume’s
equal quantities of the two goods we must have x22 = 5 and so again
x12 = 20 5 = 15 (as in the previous cases). As before the equilibrium
price must be 3.
5. Note that the demand function and the o¤er curve for the group-2 people
is discontinuous. So, if there are relatively small numbers in each group
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Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
there may be no equilibrium (the two o¤er curves do not intersect). In the
large numbers case we could appeal to a continuity argument and have
an equilibrium with proportion of group 2 at point x0 and the rest at x00 .
The equilibrium would then look very much like case 4b.
c Frank Cowell 2006
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Microeconomics
Exercise 7.8 In a two-commodity exchange economy there are two equal-sized
groups of people. Those of type a have the utility function
1 a
[x ]
2 1
U a (xa ) =
1 a
[x ]
2 2
2
2
and a resource endowment of (R1 ; 0); those of type b have the utility function
U b xb = xb1 xb2
and a resource endowment of (0; R2 ).
1. How many equilibria does this system have?
2. Find the equilibrium price ratio if R1 = 5, R2 = 16.
Outline Answer:
For consumers of type a the relevant Lagrangean is
1 a
[x ]
2 1
2
1 a
[x ]
2 2
2
pxa1
+ [pR1
xa2 ]
where p is the price of good 1 in terms of good 2. The FOC for a maximum are
[xa1 ]
3
p
3
[xa2 ]
=
0
=
0
Rearranging and using the budget constraint we get
xa1
1=3
1=3
= p
1=3
=
h
i
pxa1 + xa2 = p2=3 + 1
xa2
So
1=3
xa2 =
1=3
= pR1
pR1
p2=3 + 1
=
For consumers of type b the Lagrangean is
log xb1 + log xb2 +
R2
pxb1
The FOC for a maximum are
1
xb1
xb2
p
1
=
0
=
0
Rearranging and using the budget constraint we get
c Frank Cowell 2006
xb1
=
xb2
=
109
1
p
1
xb2
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
pxb1 + xb2 =
So
xb2 =
1
2
=
= R2
R2
2
The excess-demand function for good 2 is therefore
pR1
p2=3 + 1
R2
2
1. Excess demand is 0 where
p
where
p2=3
1=0
:= 2R1 =R2 . This is equivalent to requiring
p2=3 = p
1
The expression p2=3 is an increasing concave function through the origin.
It is clear that the straight line given by p
1 can cut this just once.
There is one equilibrium.
2. If (R1 ; R2 ) = (10; 32) then
c Frank Cowell 2006
:= 20=32 = 5=8 and p = 8:
110
Microeconomics
Exercise 7.9 In a two-person, private-ownership economy persons a and b
each have utility functions of the form
V h p; y h = log y h
p1
h
1
h
2
p2
1
log (p1 p2 )
2
where h = a; b and h1 , h2 are parameters. Find the equilibrium price ratio as
a function of the property distribution [R].
Outline Answer:
Using Roy’s identity we have, for each h and each i :
Vih
:
Vyh
xhi =
Now we have
Vih
h h
i [y
=
Vyh
=
[y
h
p1
p1 h1
h
1
h 1
2]
p2
1=2pi ;
p2 h2 ] 1 :
Combining the two results we …nd for each h:
h
i
xhi =
De…ning
good 2:
i
a
i
+
E2 =
2
=
+
b
i
1 h h h
p 1 R1
2pi
h
1
i
h
+ p2 R2h
h
2
ii
and Ri = Ria + Rib we obtain the excess demand for
R2 +
1
[p1 [R1
2p2
1]
+ p2 [R2
2 ]]
Hence putting E2 = 0 we get the equilibrium price ratio thus:
p1
R2
=
p2
R1
c Frank Cowell 2006
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2
1
:
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.10 In an economy there are large equal-sized groups of capitalists
and workers. Production is organised as in the model of Exercises 2.14 and 6.4.
Capitalists’ income consists solely of the pro…ts from the production process;
workers’ income comes solely from the sale of labour. Capitalists and workers
2
h
have the utility functions xc1 xc2 and xw
[R3 xw
1
3 ] respectively, where xi denotes the consumption of good i by a person of type h and R3 is the stock of
commodity 3.
1. If capitalists and workers act as price takers …nd the optimal demands for
the consumption goods by each group, and the optimal supply of labour
R3 xw
3.
2. Show that the excess demand functions for goods 1,2 can be written as
2p1
+
2p2
A
p1
2
1
2 [p1 ]
A
p2
2
2
where
is the expression
for pro…ts found in Exercise 6.4. Show that in
p
equilibrium p1 =p2 = 3 and hence show that the equilibrium price of good
1 (in terms of good 3) is given by
1=3
3
2A
p1 =
3. What is the ratio of the money incomes of workers and capitalists in equilibrium?
Outline Answer:
1. Given that the capitalist utility function is
xc1 xc2
it is immediate that in the optimum the capitalists spend an equal share
of their income on the two consumption goods and so
xci =
2pi
:
Worker utility is
xw
1
2
(7.18)
xw
3
(7.19)
xw
3]
[R3
and the budget constraint is
p1 xw
1
R3
Maximising (7.18) subject to (7.19) is equivalent to maximising
1
[R3
p1
c Frank Cowell 2006
xw
3]
112
[R3
2
xw
3] :
(7.20)
Microeconomics
The FOC is
1
2 [R3 xw
3]=0
p1
which gives optimal labour supply as:
xw
3 =
R3
(7.21)
1
2p1
(7.22)
and, from (7.19), the workers’optimal consumption of good 1 is
xw
1 =
1
2:
(7.23)
2 [p1 ]
2. The economy has no stock of good 1 or good 2; workers do not consume
good 2; so excess demand for the two goods is, respectively:
xc1 + xw
1
y1
=
xc2
y2
=
2p1
+
2p2
A
p1
2
1
2
2 [p1 ]
A
p2
2
(7.24)
(7.25)
To …nd the equilibrium set each of (7.24) and (7.25) equal to zero. This
gives
p1 + 1
A [p1 ]
3
(7.26)
= A [p2 ]
2
(7.27)
=
Substituting in for pro…ts in (7.27) we have
2
2
[p1 ] + [p2 ]
2
= [p2 ]
4
and so
p
p1
= 3:
p2
and p2 we get
Substituting for
(7.28)
2
2
[p1 ] + 31 [p1 ]
3
+ 1 = A [p1 ]
4
and, on rearranging, this gives
p1 A
p1 =
3
2A
1=3
(7.29)
3. Pro…ts in equilibrium are
A
1+
4
1
3
2
[p1 ] =
A
A 3
2
[p1 ] =
3
3 2A
2=3
=
A
3
1=3
2
2=3
:
Given that the price of good 3 is normalised to 1, using (7.22)and (7.29)
total labour income in equilibrium is
1
c Frank Cowell 2006
1
1 3
=
2p1
2 2A
113
1=3
=
A
3
1=3
2
2=3
Microeconomics
CHAPTER 7. GENERAL EQUILIBRIUM
So, workers and capitalists get the same money income in equilibrium!
Note that this is una¤ected by the value of A; increases in A could be
interpreted as technical progress and so the income distribution remains
unchanged by such progress.
c Frank Cowell 2006
114