The Consumer - London School of...

Chapter 4

The Consumer

Exercise 4.1 You observe a consumer in two situations: with an income of$100 he buys 5 units of good 1 at a price of $10 per unit and 10 units of good 2at a price of $5 per unit. With an income of $175 he buys 3 units of good 1 ata price of $15 per unit and 13 units of good 2 at a price of $10 per unit. Do theactions of this consumer conform to the basic axioms of consumer behaviour?

x′ = (5,10)

p1 / p2 = 2 p1 / p2 = 1.5

x′′ = (3,13)

x1

x2

Figure 4.1: WARP violated

Outline AnswerAt the original price ratio p1/p2 = 2 the choice is x′ = (5, 10); but at those

prices the and with that budget the consumer could have afforded x′′ = (3, 13):x′ is revealed-preferred to x′′. But at the new price ratio p1/p2 = 1.5 x′′ ischosen, although x′ is still affordable: x′′ is revealed-preferred to x′. Thisviolates WARP —see Figure 4.1.

41

Microeconomics CHAPTER 4. THE CONSUMER

Exercise 4.2 Draw the indifference curves for the following four types of pref-erences:

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. What is the consumer’s cost functionin each case?

x1

C

x2

x1

A

x2

x1

D

x2

x1

B

x2

Figure 4.2: Indifference curves: four cases

Use the fact that expenditure minimisation for the household and cost-minimisation for the firm are essentially the same problem. The indifferencecurves in Figure 4.2 are identical to the isoquants depicted in Exercises 2.4, 2.5.So, substituting the notation in Exercise 2.4 and 2.5we get:

• Case A: C(p, υ) = C(p, υ) = eυ[p1α

]α [ p21−α

]1−α.

• Case B: C(p, υ) = υmin(p1/β, p2)

• Case C: C(p, υ) =√υmin(p1/

√γ, p2)

• Case D: C(p, υ) =[p1δ + p2

]υ.

c©Frank Cowell 2006 42

Microeconomics

Exercise 4.3 Suppose a person has the Cobb-Douglas utility function

n∑i=1

ai log(xi)

where xi is the quantity consumed of good i, and a1, ..., an are non-negativeparameters such that

∑nj=1 aj = 1. If he has a given income y, and faces

prices p1, ..., pn, find the ordinary demand functions. What is special about theexpenditure on each commodity under this set of preferences?

Outline AnswerThe relevant Lagrangean is

n∑i=1

αi log xi + ν

[y −

n∑i=1

pixi

](4.1)

The first-order conditions yield:

x∗i =αiν∗pi

, i = 1, 2, ..., n. (4.2)

y =

n∑i=1

pix∗i (4.3)

From the n + 1 equations (4.2,4.3) we get at the optimum: y =∑ni=1 αi/ν

∗ =1/ν∗. So the demand functions are

x∗i =αiy

pi, i = 1, 2, ..., n. (4.4)

and expenditure on each commodity i is

ei := pix∗i = αiy, (4.5)

—a constant proportion of income.



Exercise 4.4 The elasticity of demand for domestic heating oil is −0.5, andfor gasoline is −1.5. The price of both sorts of fuel is 60c/ per litre: includedin this price is an excise tax of 48c/ per litre. The government wants to reduceenergy consumption in the economy and to increase its tax revenue. Can it dothis (a) by taxing domestic heating oil? (b) by taxing gasoline?

Outline AnswerLet p be the untaxed price, and τ the excise tax. Government revenue

is T = τx, and the purchase price is p + τ . Clearly an increase in τ wouldreduce consumption, and τ/[τ + p] = 0.8. The effect on tax revenue is given by∂T/∂τ = x+ τ∂x/∂τ = x[1 + 0.8ε]. If (a) ε = −0.5 then this is positive. If (b)ε = −1.5 then it is negative.

Exercise 4.5 Define the uncompensated and compensated price elasticities as

εij :=pjx∗i

∂Di(p,y)

∂pj, ε∗ij :=

pjx∗i

∂Hi(p,υ)

∂pj

and the income elasticity

εiy :=y

x∗i

∂Di(p,y)

∂y.

Show how the Slutsky equation can be expressed in terms of these elasticitiesand the expenditure share of each commodity in the total budget.

Use the fact that each demand function Di is homogeneous of degree zeroin all prices and income. Then, using the standard lemma for homogenousfunctions, we have for each i = 1, ..., n :

n∑j=1

pj∂Di(p, y)

∂pj+ y

∂Di(p, y)

∂y= 0 ·Di(p, y)

= 0

which impliesn∑j=1

εij + εiy = 0.

Moreover we can rewrite the Slutsky equation as

εij = ε∗ij − vjεiy

where vj =pjx∗j

yis the expenditure share of commodity j.


Microeconomics

Exercise 4.6 You are planning a study of consumer demand. You have a dataset which gives the expenditure of individual consumers on each of n goods. Itis suggested to you that an appropriate model for consumer expenditure is theLinear Expenditure System:

ei = ξipi + αi

y − n∑j=1

pjξj

where pi is the price of good i, ei is the consumer’s expenditure on good i, yis the consumer’s income, and α1, ..., αn, ξ1, ..., ξn are non-negative parameterssuch that

∑nj=1 αj = 1.

1. Find the effect on xi, the demand for good i, of a change in the consumer’sincome and of an (uncompensated) change in any price pj.

2. Find the substitution effect of a change in price pj on the demand for goodi.

3. Explain how you could check that this demand system is consistent withutility-maximisation and suggest the type of utility function which wouldyield the demand functions implied by the above formula for consumerexpenditure. [Hint: compare this with Exercise 4.3]

Outline Answer

1. We have

x∗i = ξi +αipi

y − n∑j=1

pjξj

(4.6)

Notice that (ξ1, ..., ξn) play the role of “subsistence minima”of the n com-modities, and so y0 :=

∑nj=1 pjξj can be considered as the subsistence

minimum expenditure, and the remaining budget y− y0 as “discretionaryexpenditure”; αi is then the proportion of discretionary expenditure spenton discretionary purchases of commodity i: pi [x∗i − ξi] / [y − y0]. Com-pare this with (4.5). From (4.6) we have:

∂x∗i∂y

=αipi

(4.7)

∂x∗i∂pj

= −αiξjpi

if j 6= i (4.8)

∂x∗i∂pi

= −αipi

[ξi +

y −∑nj=1 pjξj

pi

](4.9)

2. Apply Slutsky equation using (4.7) and (4.8) to establish

dx∗idpj

∣∣∣∣υ=conθtant

=αi[x∗j − ξj

]pi

, if j 6= i (4.10)



3. Check that demand function (4.6) is homogeneous of degree 0 in pricesand income, and that the sum of the right-hand side of the equation in thequestion adds up to total income. Check that cross-substitution effects aresymmetric, and that own-price substitution effects are negative. Using theanalogy with part (b) we can see that the demand system is similar, butwith the commodity origin shifted from 0 to the point (ξ1, ..., ξn);so weexpect the indifference curves from which the demand system was derivedwill look like Cobb-Douglas contours with the origin shifted to the point(ξ1, ..., ξn). The utility function will then be

n∑i=1

αi log(xi − ξi) . (4.11)


Microeconomics

Exercise 4.7 Suppose a consumer has a two-period utility function of the formlabelled type A in Exercise 4.2. where xi is the amount of consumption in periodi. The consumer’s resources consist just of inherited assets A in period 1, whichis partly spent on consumption in period 1 and the remainder invested in anasset paying a rate of interest r.

1. Interpret the parameter α in this case.

2. Obtain the optimal allocation of (x1, x2)

3. Explain how consumption varies with A, r and α.

4. Comment on your results and examine the “income” and “substitution”effects of the interest rate on consumption.

Outline Answer

1. The parameter α captures the consumer’s “impatience”: the higher is αthe more steeply sloped will be the indifference curves in Figure 4.3. Notethat 1

1+r is the price of consumption in period 2 relative to the price ofconsumption in period 1; so the lifetime budget constraint, expressed interms of period-1 prices, is:

x1 +x2

1 + r≤ A (4.12)

and so the Lagrangean is:

α log x1 + [1− α] log x2 + λ

[A− x1 −

x2

1 + r

](4.13)

2. We can be sure an interior maximum will exist (examine the indifferencecurve in Figure 4.3). First-order conditions are

α

x∗1= λ∗

1− αx∗2

= λ∗1

1 + r

x∗1 +x∗2

1 + r= A

From these we find λ∗ = 1A and therefore optimum consumption in each

period is:

x∗1 = αA (4.14)

x∗2 = [1 + r] [1− α]A (4.15)

So we can see that the smaller is α (the lower is the level of impatience), orthe larger is r (the rate of interest), the more consumption will be “tilted”toward period 2.



x*

Ax1

x2

1 + r

Figure 4.3: Equilibrium in 2-period case

3. The effect of an increase in assets is:

∂x∗1∂A

= α (4.16)

∂x∗2∂A

= [1 + r] [1− α] (4.17)

leaving the proportion spent on consumption in each period unaltered.The effect of an increase in the interest rate is:

∂x∗1∂r

= 0 (4.18)

∂x∗2∂r

= [1− α]A (4.19)

4. To find the substitution effect we need to use the Slutsky equation. In aconventional 2-commodity model this would be given by

∂x∗1∂p2

=dx∗1dp2

∣∣∣∣υ=constant

− x∗2∂x∗1∂y

(4.20)

Taking 1/[1 + r] as the “price” p2 of consumption in period 2, withA =lifetime budget y and price of period-1 consumption defined as 1.Noting that in this case dp2 = −1/[1 + r]2dr we can rewrite (4.20) as

∂x∗1∂r

=dx∗1dr


+x∗2

[1 + r]2∂x∗1∂A

(4.21)

Rearranging this, the substitution effect for good 1 of an increase in r may


Microeconomics

then be found (using 4.16 and 4.18) as:

dx∗1dr


=∂x∗1∂r− x∗2

[1 + r]2∂x∗1∂A

= − x∗2[1 + r]2

α < 0 (4.22)



Exercise 4.8 Suppose a consumer is rationed in his consumption of commodity1, so that his consumption is constrained thus x1 ≤ a. Discuss the propertiesof the demand functions for commodities 2, ..., n of a consumer for whom therationing constraint is binding.

Outline AnswerUse the standard analysis on the short-run for the firm (see Chapter 2) to

get insight on the economics of the consumer under rationing. In the case of thefirm has to cope with the side-constraint z3 = z̄3 in the short run; the consumerhas to cope with the rationing constraint x1 ≤ a: if the constraint is slack thenit is irrelevant (the consumer does not use all his ration); if it is binding, thenthe problem is just like that of the firm. The solution is at x′ in Figure 4.4

x′

ax1

x2

Figure 4.4: Ration


Microeconomics

Exercise 4.9 A person has preferences represented by the utility function

U(x) =

n∑i=1

log xi

where xi is the quantity consumed of good i and n > 3.

1. Assuming that the person has a fixed money income y and can buy com-modity i at price pi find the ordinary and compensated demand elasticitiesfor good 1 with respect to pj, j = 1, ..., n.

2. Suppose the consumer is legally precommitted to buying an amount Anof commodity n where pnAn < y. Assuming that there are no additionalconstraints on the choices of the other goods find the ordinary and com-pensated elasticities for good 1 with respect to pj, j = 1, ...n. Compareyour answer to part 1.

3. Suppose the consumer is now legally precommitted to buying an amount Akof commodity k, k = n−r, ..., n where 0 < r < n−2 and

∑nk=n−r pkAk < y.

Use the above argument to explain what will happen to the elasticity of good1 with respect to pj as r increases. Comment on the result.

Outline Answer

1. For the specified utility function it is clear that the indifference curves donot touch the axes for any finite xi, so we cannot have a corner solution.The budget constraint is

n∑i=1

pixi ≤ y.

The problem of maximising utility subject to the budget constraint isequivalent to maximising the Lagrangean

n∑i=1

log xi + λ

[y −

n∑i=1

pixi.

]

The FOC are1

x∗i− λpi = 0, i = 1, ..., n (4.23)

and the (binding) budget constraint. From (4.23) we get

n− λn∑i=1

pix∗i = 0. (4.24)

and so, using the budget constraint, we find λ = n/y. Substituting thevalue of λ into (4.23) we find:

(a) The ordinary demand function for good i is

x∗i =y

npi(4.25)



The indirect utility function V is given by υ = V (p, y) = U(x∗) =∑ni=1 log x∗i . So, from (4.25) we have:

υ = log

(yn

nnp1p2p3...pn

)(4.26)

Inverting the relation (4.26) the cost function C is given by

y = C(p, υ) = [nnp1p2p3...pneυ]

1n = n [p1p2p3...pne

υ]1n (4.27)

Differentiating (4.27) the compensated demand for good 1 is

x∗1 = p1−nn

1 [p2p3p4...pneυ]

1n (4.28)

(b) From (4.25) we have the elasticities

∂ log x∗1∂ log p1

∣∣∣∣y=const

= −1,

∂ log x∗1∂ log pj

∣∣∣∣y=const

= 0, j = 2, ..., n.

(c) From (4.28) we have the compensated elasticities


∣∣∣∣υ=const

=1− nn

< 0,



=1

n> 0, j = 2, ..., n

2. The problem now is equivalent to maximising

n−1∑i=1

log xi + logAn

subject ton−1∑i=1

pixi ≤ y′,

where y′ := y − pnAn. Reusing the method above, the ordinary andcompensated demand functions are, respectively,

x∗1 =y′

[n− 1] p1=y − pnAn[n− 1] p1

(4.29)

x∗1 = p2−nn−11 [p2p3p4...pn−1e

υ]1

n−1 (4.30)

(a) So now, from (4.29) we have


∣∣∣∣y=const

= −1,


∣∣∣∣y=const

= 0, j = 2, ..., n− 1.


Microeconomics

(as before) but

∂ log x∗1∂ log pn

∣∣∣∣y=const

=−pnAny′

< 0

The reason for this last result is that if the person is forced to buy thefixed amount An then changing pn is equivalent to a simple incomeeffect (think about what happens to y′).

(b) From (4.28) we have



=2− nn− 1

< 0,



=1

n− 1, j = 2, ..., n− 1.

∂ log x∗1∂ log pn


= 0.

3. The problem is just a generalisation of part 2. The person is maximisingm∑i=1

log xi + logA′

subject to∑mi=1 pixi ≤ y′, where m := n − r − 1, A′ :=

∏nk=n−r Ak and

y′ := y −∑nk=n−r pkAk. Ordinary and compensated demands are

x∗1 =y′

mp1=y −

∑nk=n−r pkAk

mp1(4.31)

x∗1 = p1−mm

1 [p2p3p4...pmeυ]

1m (4.32)

and so we have .



=1−mm

< 0, (4.33)



=1

m> 0, j = 2, ...,m. (4.34)

∂ log x∗1∂ log pk


= 0, k = n− r, ..., n. (4.35)

Given that m = n−r−1 it is clear that as r increases the elasticity (4.33)decreases in absolute value and (4.34) increases. We also have

∂ log x∗1∂ log pk

∣∣∣∣y=const

=−pkAky′

< 0, k = n− r, ..., n

The model can be used to illustrate in part the comparative statics of some-one who is subject to a quota ration xi ≤ Ai where the rationing constraintis assumed to be binding in the case of goods n − r to n. However, it is notrich enough to allow us to determine which commodities are consumed at aconventional equilibrium with MRS =price ratio, like (4.29), and which will beconstrained by the ration. Parts 2 and 3 show clearly how the compensateddemand becomes “steeper”(less elastic with respect to its own price) the moreexternal constraints are imposed —as in the “short-run”problem of the firm.



Exercise 4.10 Show that if the utility function is homothetic, then ICV = IEV

Outline AnswerLet x0 be optimal for p0 at υ0 and x1 be optimal for p1 at υ0.

0

x2

x1

x0

υ0

x1

υ1αx0

αx1

Figure 4.5: Homothetic preferences

Because of homotheticity, αx0 must be optimal for p0 at υ1 and αx1 beoptimal for p1 at υ1: see Figure 4.5.Hence

C(p0, υ0) =∑

p0ix

0i ,

C(p0, υ1) = α∑

p0ix

0i ,

C(p1, υ0) =∑

p1ix

1i ,

C(p1, υ1) = α∑

p1ix

1i

So in this special case we have

ICV =

∑p1ix

1i∑

p0ix

0i

IEV =α∑p1ix

1i

α∑p0ix

0i

and the result follows.


Microeconomics

Exercise 4.11 Suppose an individual has Cobb-Douglas preferences given bythose in Exercise 4.3.

1. Write down the consumer’s cost function and demand functions.

2. The republic of San Serrife is about to join the European Union. As aconsequence the price of milk will rise to eight times its pre-entry value. butthe price of wine will fall by fifty per cent. Use the compensating variationto evaluate the impact on consumers’welfare of these price changes.

3. San Serrife economists have estimated consumer demand in the republicand have concluded that it is closely approximated by the demand systemderived in part 1. They further estimate that the people of San Serrifespend more than three times as much on wine as on milk. They concludethat entry to the European Union is in the interests of San Serrife. Arethey right?

Outline Answer

1. Using the results from previous exercises we immediately get

C(p, υ) =

[p1

α1

]α1 [ p2

α2

]α2...

[pnαn

]αn.

This is suffi cient. However, it may be useful to see the proof from firstprinciples. The relevant Lagrangean is

n∑i=1

pixi + λ

[υ −

n∑i=1

αi log xi

](4.36)

The first-order conditions are:

xi =αiλ

pi, i = 1, 2, ..., n. (4.37)

υ =

n∑i=1

αi log xi (4.38)

From the n equations (4.37) we get at the optimum:

λ∗ =

∑ni=1 pix

∗i∑n

i=1 αi=

n∑i=1

pix∗i = y (4.39)

where y is the budget, or minimised cost and∑ni=1 αi = 1. From (4.38)

we get, using (4.37):

υ =

n∑i=1

αi logαi + log λ∗n∑i=1

αi −n∑i=1

αi log pi (4.40)

Using (4.39) and writing∑ni=1 αi logαi = − logA, equation (4.40) gives:

y = Aeυpα11 pα22 ...pαnn = C(p, υ). (4.41)



This is the required cost function. The demand functions are known fromExercise 4.2 or are obtained immediately from (4.37) and (4.39):

x∗i =αiy

pi, i = 1, 2, ..., n. (4.42)

2. Let p denote the original price vector, p̂ the price vector after entry.Observe that p̂milk = 8pmilk; p̂wine = 1

2pwine. So, using (4.41):

C(p̂, υ) = Aeυp̂α11 p̂α22 ...p̂αnn = Aeυpα11 pα22 ...pαnn b = bC(p, υ). (4.43)

whereb := [8]αmilk [

1

2]αwine = 23αmilk−αwine (4.44)

Using the definition in the notes the compensating variation is therefore

CV(p→ p̂) := C(p, υ)− C(p̂, υ) = [1− b]C(p, υ) (4.45)

Clearly, the consumer will benefit from p→ p̂ if CV(p→ p̂) > 0: the costof living —interpreted as the cost of hitting the original level of utility —goes down. This condition is satisfied if, and only if, b < 1.

3. Notice that, from (4.42), αi = pix∗i /y —the budget share of commodity

i. So, since we are told that αwine > 3αmilk it is clear that b < 1. Theeconomists are right!


Microeconomics

Exercise 4.12 In a two-commodity world a consumer’s preferences are repre-sented by the utility function

U(x1, x2) = αx121 + x2

where (x1, x2) represent the quantities consumed of the two goods and α is anon-negative parameter.

1. If the consumer’s income y is fixed in money terms find the demandfunctions for both goods, the cost (expenditure) function and the indirectutility function.

2. Show that, if both commodities are consumed in positive amounts, thecompensating variation for a change in the price of good 1 p1 → p′1 isgiven by

α2p22

4

[1

p′1− 1

p1

].

3. In this case, why is the compensating variation equal to the equivalentvariation and consumer’s surplus?

Outline Answer

x2

x1

Figure 4.6: Possible corner solution

First sketch the utility function. Note that the indifference curves touchthe axes — it is possible that one or other commodity is not consumed at theoptimum —See Figure 4.6. In this case it is easiest to substitute directly fromthe budget constraint (binding at the optimum)

p1x1 + p2x2 = y

into the utility function. The consumer will then choose x1 to maximise

αx121 +

y − p1x1

p2



The FOC isα

2[x∗1]− 12 − p1

p2= 0

which suggests that demands are

[x∗1x∗2

]=

[D1(p1, p2, y)D2(p1, p2, y)

]=

[αp22p1

]2yp2− α2

4p2p1

But this neglects the possibility that we may be at a corner. Note that a strictlypositive amount of good 2 requires

p1 > p̄1 :=α2

4

p22

y

So demand functions are given by

x∗1 = D1(p1, p2, y) =

[αp22p1

]2if p1 > p̄1

yp1

otherwise

(4.46)

x∗2 = D2(p1, p2, y) =

yp2− α2

4p2p1

if p1 > p̄1

0 otherwise(4.47)

Also, if p1 > p̄1, maximised utility is

V (p1, p2, y) = U(x∗1, x∗2)

=α2p2

2p1+

y

p2− α2p2

4p1

=α2p2

4p1+

y

p2(4.48)

Otherwise V (p1, p2, y) = α√

yp1. Also note ( for the case p1 > p̄1) that (4.48)

implies:

V1(p1, p2, y) = −α2p2

4p21

= −x∗1

p2< 0

V2(p1, p2, y) =α2

4p1− y

p22

= −x∗2

p2< 0

Vy(p1, p2, y) =1

p2> 0

so that we immediately see that Roy’s identity holds. To find the cost functionwrite υ = V (p, y) and solve for y in terms of p and υ. This gives

υ =α2p2

4p1+C(p, υ)

p2

C(p, υ) = υp2 −α2p2

2

4p1(4.49)


Microeconomics

for the case p1 > p̄1 and

C(p, υ) = p1

[υα

]2otherwise.

2 Using the definition of the compensating variation

V (p1, p2, y) =α2p2

4p1+

y

p2

V (p′1, p2, y − CV) =α2p2

4p′1+y − CV

p2

CV(p→ p′) =α2p2

2

4

[1

p′1− 1

p1

](4.50)

As an alternative method use the consumer’s cost function (4.49). Clearlywe have:

C(p, υ)− C(p′, υ) =α2p2

2

4p′1− α2p2

2

4p1

3 In this case the income effect on commodity 1 is zero if p1 > p̄1 — seeequation (4.46). In the special case of zero income effects CV=EV=CS



Exercise 4.13 Take the model of Exercise 4.12. Commodity 1 is produced bya monopolist with fixed cost C0 and constant marginal cost of production c.Assume that the price of commodity 2 is fixed at 1 and that c > α2/4y.

1. Is the firm a “natural monopoly”?

2. If there are N identical consumers in the market find the monopolist’sdemand curve and hence the monopolist’s equilibrium output and price p∗1.

3. Use the solution to Exercise 4.12 to show the aggregate loss of welfareL(p1) of all consumers’having to accept a price p1 > c rather than beingable to buy good 1 at marginal cost c. Evaluate this loss at the monopolist’sequilibrium price.

4. The government decides to regulate the monopoly. Suppose the governmentpays the monopolist a performance bonus B conditional on the price itcharges where

B = K − L(p1)

and K is a constant. Express this bonus in terms of output. Find themonopolist’s new optimum output and price p∗∗1 . Briefly comment on thesolution.

Outline Answer

1. It is easy to see that the cost function is subadditive and therefore thefirm is a natural monopoly.

2. Because consumers are identical we can just multiply the demand of oneconsumer by N to get the market aggregates. We use this throughout theanswer.

If p2 is normalized to 1 then, given that there are N identical consumersthe market demand curve is given by

q = Nx∗1 = N

[α

2p1

]2

which on rearranging gives

p1 =α

2

√N

q(4.51)

This gives the average revenue curve. So total revenue is α2

√Nq. Given

the structure of costs specified in the question the monopolist’s profits are

p1q − [C0 + cq] =α

2

√Nq − C0 − cq (4.52)

Differentiating (4.52) we find the FOC characterising the monopolist’soptimum as

c =α

4

√N

q∗(4.53)


Microeconomics

where the expression on the right-hand side of (4.53) is marginal revenue.Using (4.53) we find that the monopolist’s equilibrium output is given by

q∗ = N[ α

4c

]2Using (4.51) the price charged will be

p∗1 =α

2

√N

N [α/4c]2

in other wordsp∗1 = 2c (4.54)

Maximised profits are

p∗1q∗ − cq∗ − C0 = 2cN ·

[ α4c

]2− cN

[ α4c

]2− C0

=Nα2

16c− C0

3. Using (4.50) and multiplying by N , the (absolute) loss of welfare of eachconsumer of having to buy at price p1 rather than at marginal cost c isgiven by

L(p1) = −N · CV =Nα2

4

[1

c− 1

p1

](4.55)

Using (4.54) to evaluate (4.55)

L(p∗1) =Nα2

4

[1

c− 1

2c

]=

Nα2

8c> 0

4. Profits, including the performance bonus, are now

p1q − C0 − cq +B =α

2

√Nq − C0 − cq +B (4.56)

where, from the definition in the question and (4.55)

B = K − L(p1)

= K − Nα2

4

[1

c− 1

p1

](4.57)

Given (4.51) we find that (4.57) can be written as

B(q) =α2N

4

[2

α

√q

N− 1

c

]+K

=α

2

√Nq − α2N

4c+K (4.58)

and so, substituting from (4.58) into (4.56) we get

p1q − C0 − cq +B = α√Nq − cq +

[K − C0 −

α2N

4c

](4.59)



The FOC for maximising (4.59)

α

2

√N

q∗∗= c

and so

q∗∗ = N[ α

2c

]2> q∗

p∗∗1 = c < p∗1

Of course this is just the solution “price equal to marginal cost.”The bonusscheme has made the monopolist simulate the outcome of a competitiveindustry.


The Consumer - London School of...

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