general equilibrium - the subjective approach to...
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Chapter 7
General Equilibrium
Exercise 7.1 Suppose there are 200 traders in a market all of whom behaveas price takers. Suppose there are three goods and the traders own initially thefollowing 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
U = x121 x
142 x
143
What are the equilibrium relative prices of the three goods? Which group oftraders has members who are best off ?
Outline Answer:For each group of traders the Lagrangean may be written
1
2log xh1 +
1
4log xh2 +
1
4log xh3 + υh[yh − p1x
h1 − p2x
h2 − p3x
h3 ]
where h = 1, 2, 3 and y1 = 10p1, y2 = 5p2 and y3 = 20p3. From the first-
order conditions we find that for a trader of type h:
xh1 =yh
2p1
xh2 =yh
4p2
xh3 =yh
4p3
Excess demand for good 1 and 2 are then:
93
Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
E1 = 100x11 + 50x2
1 + 50x31 − 1000
E2 = 100x12 + 50x2
2 + 50x32 − 250
Substituting in for xhi and yh and putting E1 = E2 = 0 we find
−500 + 125p2
p1+ 500
p3
p1= 0
250p1
p2− 750
4+ 250
p3
p2= 0
that impliesp2
p1= 2 and
p3
p1= 1
2 .
Using good 1 as numeraire we immediately see that y1 = y2 = y3 = 10. Allare equally well off.
c©Frank Cowell 2006 94
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 group1 is always twice his expenditure on good 2. Charlie never uses good 2.
1. Describe the indifference maps of the three individuals and suggest utilityfunctions consistent with their behaviour.
2. If the original endowments are respectively (5, 0), (3, 6) and (0, 4), computethe equilibrium price ratio. What would be the effect on equilibrium pricesand 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 ρ =p1
p2so that values are measured in terms of good 2.
(a) Alf’s (binding) budget constraint is
ρxa1 + xa2 = 5ρ
Therefore, given the information in the question, the demand func-tions are
xa1 = xa2 =5ρ
ρ+ 1.
The utility function consistent with this behaviour is
Ua (xa1 , xa2) = min {xa1 , xa2}
—see Figure 7.1.
(b) Bill’s budget constraint is
ρxb1 + xb2 = 3ρ+ 6
From the question we have
ρxb1 = 2xb2
Therefore:
xb1 = 2 +4
ρ
xb2 = ρ+ 2.
The utility function is Cobb-Douglas:
U b = 2 log xb1 + log xb2
—see Figure 7.2.
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Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
x1
x2
(5,0)
•
Figure 7.1: Alf’s preferences and demand
x1
x2
x1
x2
• •(3,6)
Figure 7.2: Bill’s preferences and demand
c©Frank Cowell 2006 96
Microeconomics
(c) Charlie’s budget constraint is
ρxa1 + xa2 = 4
Given the information in the question we have
xc1 = 4/ρ
xc2 = 0
and utility isU c = xc1
—see Figure 7.3.
x1
x2
•
•
(4,0)
Figure 7.3: Charlie’s preferences and demand
2. Excess demand for good 2 is :
E2 =5ρ
ρ+ 1+ ρ+ 2− 10.
Putting E2 = 0 yields ρ = −2 or 4. Hence the equilibrium price ratio is4. Utility levels are Ua = 4, U b = log(54) and U c = 1.
(a) Excess demand is now
9ρ
ρ+ 1+ ρ+ 2− 10
and the equilibrium price ratio is 2. Utility levels are Ua = 6, U b =log(64) and U c = 2.
(b) Excess demand for good 2 is :
E2 =5ρ
ρ+ 1+ ρ+ 2− 10.
Putting E2 = 0 yields ρ = −2 or 4. Hence the equilibrium price ratiois 4. Utility levels are Ua = 4, U b = log(54) and U c = 4 + 1 = 5.
c©Frank Cowell 2006 97
Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.3 In a two-commodity economy assume a person has the endow-ment (0, 20).
1. Find the person’s demand function for the two goods if his preferences arerepresented by each of the types A to D in Exercise 4.2. In each caseexplain what the offer curve must look like.
2. Assume that there are in fact two equal sized groups of people, each withpreferences of type A, where everyone in group 1 has the endowment (10, 0)with α = 1
2 and everyone in group 2 an endowment (0, 20) with α = 34 . Use
the offer curves to find 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
α log xh1 + [1− α] log xh2 + λ[20− ρxh1 − xh2
](7.1)
First order conditions for an interior maximum of (7.1) areα
xh1− λρ = 0
1− αxh2
− λ = 0
20− ρxh1 − xh2 = 0
Solving these we find λ = 120 and so the demands will be
xh=
[ 20αρ
20 [1− α]
](7.2)
and the offer curve will simply be a horizontal straight line at xh2 =20 [1− α].
(b) If h has preferences of type B then demand will be
xh = x′, if ρ > β
xh ∈ [x′,x′′], if ρ = β
xh = (20/ρ, 0), if ρ < β
where x′ := (0, 20), x′′ := (20/β, 0), and their offer curve will consistof the union of the line segment [x′,x′′] and the line segment fromx′′ to (∞, 0).
(c) If group-2 persons have preferences of type C then their demands willbe
xh = x′, if ρ >√γ
xh = x′ or x′′, if ρ =√γ
xh = (20/ρ, 0), if ρ <√γ
where x′ := (0, 20), x′′ := (20/√γ, 0), and their offer curve will
consist of the union of the point x′ and the line segment from x′′ to(∞, 0).
c©Frank Cowell 2006 98
Microeconomics
x1
20[1α]
20
x1
β
x'
x''indifference
curve
x1
x2
x'
x''x1
x2
δ
A B
DC
√ γ
Figure 7.4: Offer Curves for Four Cases
(d) If group-2 persons have preferences of type D then their demands willbe
xh=
[20ρ+δ20δρ+δ
]
and their offer curve is just the straight line xh2 = δxh1 . These areillustrated in Figure 7.4.
2. If a type-A person had an income of 10 units of commodity 1 then, byanalogy with part 1, demand would be
x1=
[10α10ρ [1− α]
](7.3)
and the offer curve will simply be a vertical straight line at xh1 = 10α.From (7.2) and (7.3) we have x1
1 = 10 · 12 = 5, x2
2 = 20[1− 3
4
]= 5.
Given that there are 10 units per person of commodity 1 and 20 units perperson of commodity 2 the materials balance condition then means that
c©Frank Cowell 2006 99
Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
the equilibrium allocation must be
x1 =
[515
]x2 =
[55
]Solving for ρ from (7.2) and (7.3) we find that the equilibrium price ratiomust be 3.
c©Frank Cowell 2006 100
Microeconomics
Exercise 7.4 The agents in a two-commodity exchange economy have utilityfunctions
Ua(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 propertydistribution 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 fixed at R := Ra+ Rb =(21, 9), derive the contract curve.
Outline Answer:
1. To get the demand functions for each person we need to find the utility-maximising solution. The Lagrangean for person a is
La(xa, νa) := log (xa1) + 2 log (xa2) + νa [9p1 + 3p2 − p1xa1 − p2x
a2 ]
First-order conditions are
1x∗1− ν∗ap1 = 0
2x∗2− ν∗ap2 = 0
9p1 + 3p2 − p1x∗a1 − p2x
∗a2 = 0
Define ρ := p1/p2 and normalise p2 arbitrarily at 1. Then, rearrangingthe FOC we get
1ν∗a = ρx∗12ν∗a = x∗29ρ+ 3 = ρx∗a1 + x∗a2
(7.4)
Subtracting the first two equations from the third in (7.4) we can see thatν∗a = 1
1+3ρ . Substituting back for the Lagrange multiplier ν∗a into the
first two parts of (7.4) we see that the first-order conditions imply:[x∗a1x∗a2
]=
[3 + 1
ρ
6ρ+ 2
](7.5)
Using exactly the same method for person b we would find[x∗b1x∗b2
]=
[8 + 4
ρ
4ρ+ 2
](7.6)
Using the definition we can then find the excess demand functions byevaluating:
Ei := x∗ai + x∗bi −Rai −Rbi (7.7)
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Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
Doing this we get [E1
E2
]=
[ 5ρ − 10
10ρ− 5
](7.8)
Now construct the weighted sum of excess demands. It is obvious that
ρE1 + E2 = 0 (7.9)
thus confirming Walras’Law. In equilibrium the materials’balance con-dition must hold and so excess demand for each good must be zero, unlessthe corresponding equilibrium price is zero (markets clear).
2. Solving for E1 = 0 in (7.8) we find ρ∗ = 12 for the (normalised) equilibrium
prices.
3. The allocation is[x∗b1x∗b2
]=
[164
]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 [xb1xb2
]=
[21− xa19− xa2
](7.12)
Applying (7.10) we then get
2xa1xa2
=21− xa1
2 [9− xa2 ](7.13)
which implies that the equation of the contract curve is:
xa2 =12xa1xa1 + 7
. (7.14)
c©Frank Cowell 2006 102
Microeconomics
Exercise 7.5 Which of the following sets of functions are legitimate excess de-mand functions?
E1 (p) = −p2 + 10p1
E2 (p) = p1
E3 (p) = − 10p3
(7.15)
E1 (p) = p2+p3p1
E2 (p) = p1+p3p2
E3 (p) = p1+p2p3
(7.16)
E1 (p) = p3p1
E2 (p) = p3p2
E3 (p) = −2
(7.17)
Outline Answer:The first system is not homogeneous of degree zero in prices. The second
violates Walras’Law. The third one is both homogeneous and satisfies Walras’Law.
c©Frank Cowell 2006 103
Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.6 In a two-commodity economy let ρ be the price of commodity 1in terms of commodity 2. Suppose the excess demand function for commodity 1is given by
1− 4ρ+ 5ρ2 − 2ρ3.
How many equilibria are there? Are they stable or unstable? How might youranswer be affected if there were an increase in the stock of commodity 1 in theeconomy?
Outline Answer:The excess demand for commodity 1 at relative price ρ can be written
E(ρ) := 1− 4ρ+ 5ρ2 − 2ρ3 = [1− ρ]2[1− 2ρ].
So thatdE(ρ)/dρ = −4 + 10ρ− 6ρ2
—see Figure 7.5. From this we see that there are two equilibria as follows:
1. • ρ = 0.5. Here dE(ρ)/dρ < 0 and so it is clear that the equilibrium islocally stable
• ρ = 1. Here dE(ρ)/dρ = 0. But the graph of the function revealsthat it is locally stable from “above” (where ρ > 1) and unstablefrom “below”(where ρ < 1).
If there were an increase in the stock of commodity 1 the excess demandfunction would be shifted to the left in Figure 7.5 —then there is only one,stable equilibrium.
0.5
1
1.5
ρ
1 0.5 0 0.5 1
°••
E
Figure 7.5: Excess demand
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Microeconomics
Exercise 7.7 Consider the following four types of preferences:
Type A : α log x1 + [1− α] log x2
Type B : βx1 + x2
Type C : γ [x1]2
+ [x2]2
Type D : min {δx1, x2}
where x1, x2 denote respectively consumption of goods 1 and 2 and α, β, γ, δ arestrictly positive parameters with α < 1.
1. Draw the indifference curves for each type.
2. Assume that a person has an endowment of 10 units of commodity 1 andzero of commodity 2. Show that, if his preferences are of type A, then hisdemand for the two commodities can be represented as
x :=
[x1
x2
]=
[10α10ρ [1− α]
]where ρ is the price of good 1 in terms of good 2. What is the person’soffer curve in this case?
3. Assume now that a person has an endowment of 20 units of commodity2 (and zero units of commodity 1) find the person’s demand for the twogoods if his preferences are represented by each of the types A to D. Ineach case explain what the offer 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 peoplein group 2 own all of commodity 2 (20 units per person). If Group 1 haspreferences of type A with α = 1
2 find the competitive equilibrium pricesand allocations in each of the following cases:
(a) Group 2 have preferences of type A with α = 34
(b) Group 2 have preferences of type B with β = 3.
(c) Group 2 have preferences of type D with δ = 1.
5. What problem might arise if group 2 had preferences of type C? Comparethis case with case 4b
Outline Answer:
1. Indifference curves have the shape shown in the figure 7.6.
2. The income of a group-1 person is 10ρ. If group-1 persons have preferencesof type A then the Lagrangean is
α log x11 + [1− α] log x1
2 + λ[10ρ− ρx1
1 − x12
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Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
z1
(3)
z2
z1
(1)
z2
z1
(4)
z2
z1
(2)
z2
Figure 7.6: Indifference Curves
First order conditions for an interior maximum areα
x11
− λρ = 0
1− αx1
2
− λ = 0
10ρ− ρx11 − x1
2 = 0
Solving these we find λ = 110ρ and so the demands will be
x1=
[10α10ρ [1− α]
]and the offer curve will simply be a vertical straight line at x1
1 = 10α.
3. The income of a group-2 person is 20. So, if group-2 persons have prefer-ences of type A, then their demands will be
x2=
[ 20αρ
20 [1− α]
]and their offer curve will simply be a horizontal straight line at x2
2 =20 [1− α]. If group-2 persons have preferences of type B then their de-mands will be
x2 = x′, if ρ > β
x2 ∈ [x′,x′′], if ρ = β
x2 = (20/ρ, 0), if ρ < β
c©Frank Cowell 2006 106
Microeconomics
where x′ := (0, 20), x′′ := (20/β, 0), and their offer curve will consist ofthe union of the line segment [x′,x′′] and the line segment from x′′ to(∞, 0). If group-2 persons have preferences of type C then their demandswill be
x2 = x′, if ρ >√γ
x2 = x′ or x′′, if ρ =√γ
x2 = (20/ρ, 0), if ρ <√γ
where x′ := (0, 20), x′′ := (20/√γ, 0), and their offer curve will consist
of the union of the point x′ and the line segment from x′′ to (∞, 0). Ifgroup-2 persons have preferences of type D then their demands will be
x2=
[20ρ+δ20δρ+δ
]
and their offer curve is just the straight line x22 = δx2
1.
4. In each case below we could work out the excess demand function, setexcess demand equal to zero, find the equilibrium price and then the equi-librium allocation. However, we can get to the result more quickly byusing an equivalent approach. Given that an equilibrium allocation mustlie at the intersection of the offer curves of the two parties the answer ineach case is immediate.
(a) From the above computations we have x11 = 10 · 1
2 = 5, x22 =
20[1− 3
4
]= 5. Given that there are 10 units per person of commod-
ity 1 and 20 units per person of commodity 2 the materials balancecondition then means that the equilibrium allocation must be
x1 =
[515
]x2 =
[55
]Solving for ρ we find that the equilibrium price ratio must be 3.
(b) We have x11 = 5, and so x2
1 = 5. Using the fact that the equilibriummust lie on the group-2 offer curve we see that the solution must lieon the straight line from (0, 20) to (20/3, 0) we find that x2
2 = 5 and,from the materials balance condition x1
2 = 20 − 5 = 15 (as in theprevious case). By the same reasoning as in the previous case theequilibrium price must be ρ = 3.
(c) Once again we have x11 = 5, and so x2
1 = 5. Given that the group-2 offer curve in this case is such that the person always consume’sequal quantities of the two goods we must have x2
2 = 5 and so againx1
2 = 20−5 = 15 (as in the previous cases). As before the equilibriumprice must be 3.
5. Note that the demand function and the offer curve for the group-2 peopleis 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 offer curves do not intersect). In thelarge numbers case we could appeal to a continuity argument and havean equilibrium with proportion of group 2 at point x′ and the rest at x′′.The equilibrium would then look very much like case 4b.
c©Frank Cowell 2006 108
Microeconomics
Exercise 7.8 In a two-commodity exchange economy there are two equal-sizedgroups of people. Those of type a have the utility function
Ua (xa) = −1
2[xa1 ]
−2 − 1
2[xa2 ]
−2
and a resource endowment of (R1, 0); those of type b have the utility function
U b(xb)
= xb1xb2
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
2[xa1 ]
−2 − 1
2[xa2 ]
−2+ λ [pR1 − pxa1 − xa2 ]
where p is the price of good 1 in terms of good 2. The FOC for a maximum are
[xa1 ]−3 − pλ = 0
[xa2 ]−3 − λ = 0
Rearranging and using the budget constraint we get
xa1 = p−1/3λ−1/3
xa2 = λ−1/3
pxa1 + xa2 =[p2/3 + 1
]λ−1/3 = pR1
So
xa2 = λ−1/3 =pR1
p2/3 + 1
For consumers of type b the Lagrangean is
log xb1 + log xb2 + µ[R2 − pxb1 − xb2
]The FOC for a maximum are[
xb1]−1 − pµ = 0[xb2]−1 − µ = 0
Rearranging and using the budget constraint we get
xb1 =1
pµ
xb2 =1
µ
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Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
pxb1 + xb2 =2
µ= R2
So
xb2 =1
µ=R2
2
The excess-demand function for good 2 is therefore
pR1
p2/3 + 1− R2
2
1. Excess demand is 0 where
pθ − p2/3 − 1 = 0
where θ := 2R1/R2. This is equivalent to requiring
p2/3 = pθ − 1
The expression p2/3 is an increasing concave function through the origin.It is clear that the straight line given by pθ − 1 can cut this just once.There is one equilibrium.
2. If (R1, R2) = (10, 32) then θ := 20/32 = 5/8 and p = 8.
c©Frank Cowell 2006 110
Microeconomics
Exercise 7.9 In a two-person, private-ownership economy persons a and beach have utility functions of the form
V h(p, yh
)= log
(yh − p1β
h1 − p2β
h2
)− 1
2log (p1p2)
where h = a, b and βh1 , βh2 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 :
xhi = −Vhi
V hy.
Now we have
V hi = −βhi [yh − p1βh1 − p2β
h2 ]−1 − 1/2pi,
V hy = [yh − p1βh1 − p2β
h2 ]−1.
Combining the two results we find for each h:
xhi = βhi +1
2pi
[p1
[Rh1 − βh1
]+ p2
[Rh2 − βh2
]]Defining βi = βai + βbi and Ri = Rai + Rbi we obtain the excess demand for
good 2:
E2 = β2 −R2 +1
2p2[p1 [R1 − β1] + p2 [R2 − β2]]
Hence putting E2 = 0 we get the equilibrium price ratio thus:
p1
p2=R2 − β2
R1 − β1
.
c©Frank Cowell 2006 111
Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
Exercise 7.10 In an economy there are large equal-sized groups of capitalistsand workers. Production is organised as in the model of Exercises 2.14 and 6.4.Capitalists’ income consists solely of the profits from the production process;workers’ income comes solely from the sale of labour. Capitalists and workershave the utility functions xc1x
c2 and x
w1 − [R3 − xw3 ]
2 respectively, where xhi de-notes the consumption of good i by a person of type h and R3 is the stock ofcommodity 3.
1. If capitalists and workers act as price takers find the optimal demands forthe consumption goods by each group, and the optimal supply of labourR3 − xw3 .
2. Show that the excess demand functions for goods 1,2 can be written as
Π
2p1+
1
2 [p1]2 −
A
2p1
Π
2p2− A
2p2
where Π is the expression for profits found in Exercise 6.4. Show that inequilibrium p1/p2 =
√3 and hence show that the equilibrium price of good
1 (in terms of good 3) is given by
p1 =
[3
2A
]1/3
3. What is the ratio of the money incomes of workers and capitalists in equi-librium?
Outline Answer:
1. Given that the capitalist utility function is
xc1xc2
it is immediate that in the optimum the capitalists spend an equal shareof their income on the two consumption goods and so
xci =Π
2pi.
Worker utility isxw1 − [R3 − xw3 ]
2 (7.18)
and the budget constraint is
p1xw1 ≤ R3 − xw3 (7.19)
Maximising (7.18) subject to (7.19) is equivalent to maximising
1
p1[R3 − xw3 ]− [R3 − xw3 ]
2. (7.20)
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Microeconomics
The FOC is1
p1− 2 [R3 − xw3 ] = 0 (7.21)
which gives optimal labour supply as:
R3 − xw3 =1
2p1(7.22)
and, from (7.19), the workers’optimal consumption of good 1 is
xw1 =1
2 [p1]2 . (7.23)
2. The economy has no stock of good 1 or good 2; workers do not consumegood 2; so excess demand for the two goods is, respectively:
xc1 + xw1 − y1 =Π
2p1+
1
2 [p1]2 −
A
2p1 (7.24)
xc2 − y2 =Π
2p2− A
2p2 (7.25)
To find the equilibrium set each of (7.24) and (7.25) equal to zero. Thisgives
Πp1 + 1 = A [p1]3 (7.26)
Π = A [p2]2 (7.27)
Substituting in for profits in (7.27) we have
[p1]2
+ [p2]2
4= [p2]
2
and sop1
p2=√
3. (7.28)
Substituting for Π and p2 we get
p1A[p1]
2+ 1
3 [p1]2
4+ 1 = A [p1]
3
and, on rearranging, this gives
p1 =
[3
2A
]1/3
(7.29)
3. Profits in equilibrium are
A1 + 1
3
4[p1]
2=A
3[p1]
2=A
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 · 1
2p1=
1
2
[3
2A
]−1/3
=
[A
3
]1/3
2−2/3
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Microeconomics CHAPTER 7. GENERAL EQUILIBRIUM
So, workers and capitalists get the same money income in equilibrium!Note that this is unaffected by the value of A; increases in A could beinterpreted as technical progress and so the income distribution remainsunchanged by such progress.
c©Frank Cowell 2006 114