General Equilibrium - London School of...

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



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.



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


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.



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).


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



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.


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)



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)


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.



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


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

]c©Frank Cowell 2006 105


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


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



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.


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



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.


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

.



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)


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



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.


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