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 U = x 1 2 1 x 1 4 2 x 1 4 3 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 2 log x h 1 + 1 4 log x h 2 + 1 4 log x h 3 + h [y h p 1 x h 1 p 2 x h 2 p 3 x h 3 ] where h =1; 2; 3 and y 1 = 10p 1 ;y 2 =5p 2 and y 3 = 20p 3 : From the rst- order conditions we nd that for a trader of type h: x h 1 = y h 2p 1 x h 2 = y h 4p 2 x h 3 = y h 4p 3 Excess demand for good 1 and 2 are then: 93
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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.
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
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).
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
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:
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.
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
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
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.
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.
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
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.