Chapter 1: Monopoly I

Home , Monopoly price, Profit maximization

Notes on

Microeconomic Theory IV

3º - LE-: 2008-2009

Iñaki Aguirre

Departamento de Fundamentos del Análisis Económico I

Universidad del País Vasco

Microeconomic Theory IV Monopoly I

Introduction

1.1. Profit maximization by a monopolistic firm.

1.2. Linear demand and constant elasticity demand.

1.3. Comparative statics.

1.4. Welfare and output.

Introduction

We say that a firm is a monopoly if it is the only seller of a good (or goods) in a market.

Problem: it is not easy to define good and market.

A firm may become a monopoly by various reasons:

- Control over raw materials.

- Acquisition of the exclusive selling rights (by a patent, by a public auction etc.).

- Better access to the capital market.

- Increasing returns of scale etc.

In contrast with a perfectly competitive firm which faces a perfectly elastic demand (taking price as given), a monopolist faces the market demand. Therefore, a firm with monopolistic power in a market it is aware of the amount of output that it is be able to sell it is a continuous function of the price charged. Put differently, the monopolistic firm takes into account that a reduction in output will increase the price that can be charged. In consequence, a monopolist has the power to set the market price. While we can consider a competitive firm as a “price taker”, a monopolist is price decision-maker or price setter.

Microeconomic Theory IV Monopoly I

1.1. Profit maximization

(i) The problem of profit maximization in prices and in quantities. First order conditions.

Second order conditions. A graphical interpretation of the profit maximization problem.

(ii) Interpretation of marginal revenue.

(iii) Marginal revenue equals marginal cost condition.

(iv) Output and demand elasticity.

(v) Lerner Index of monopolistic power.

(vi) Graphical analysis.

(vii) Second order conditions.

(i) The problem of profit maximization in prices and in quantities

There are two types of constraint that restrict the behaviour of a monopolist: a) Technological constraints summarized in the cost function C(x). b) Demand constraints: x(p).

We can write the profit function of the monopolist in two alternative ways:

- Π=()ppxpCxp () − (()) by using the demand function.

- Π=()x pxx () − Cx () by using the inverse demand function.

The demand, x (p), and the inverse demand, p(x), represent the same relationship between price and demanded quantity from different points of view. The demand function is a complete description of demanded quantity at each price whereas the inverse demand gives us the maximum price at which a given output x may be sold in the market.

maxΠ (p ) maxΠ (x ) p x≥0 mm ⇓≡⇓px xmm==xp() p mm px ()

Microeconomic Theory IV Monopoly I

The problem of profit maximization as a function of price maxΠ≡ (ppxpCxp ) max ( ) − ( ( )) pp

Π=''''()pxppxpCxpxp () + () − (())() = 0

'' ' '' '' '2 ' '' Π=()p 2() xp + pxpCxp () − (())()⎣⎦⎡⎤ xp − Cxpxp (())() < 0

The problem of profit maximization as a function of the output maxΠ≡ (xpxxCx ) max ( ) − ( ) xx≥≥00

Π=''(0)pC (0) − (0) >⇒> 0 pC(0) ' (0)

Π=''''()xpxxpxCx () + () − () =⇔Π= 0 ( xm ) 0 First order condition.

Π=''()xpxxpxCx 2 ' () + '' () − '' () < 0 Strictly concave profit function (regular case).

Π' ()0xm =

Π()x Π>' (0) 0

m x x

Microeconomic Theory IV Monopoly I

(ii) Interpretation of marginal revenue

Marginal revenue, rx' (), is:

'' rx()=+N px ()N xpx () (1)

Additional revenue from selling Loss of revenue from selling units an additional unit. already produced at a lower price.

(iii) Marginal revenue equals marginal cost condition

The profit-maximizing output level (interior solution) satisfies:

Π=''()xrxCxpxxpxCxmm () − ' () mm = () + ' () m − ' ()0 m = (2)

At the monopolistic optimal output the marginal profit is zero, Π' ()0;xm = that is, an infinitesimal change in the level of output maintains profit unchanged. An output level such that Π>' (.) 0 does not maximize profits: an (infinitesimal) increase in output would increase profits. In a similar way, a level of output such that Π' (.)< 0 does not maximize profits: a (infinitesimal) decrease in output would increase profits.

At the profit-maximizing level of output marginal revenue equals marginal cost, rx''()mm= Cx (); that is, an infinitesimal change in the level of output changes revenue and cost equally. (In other words, an infinitesimal increase in the level of output increases revenue and cost by the same amount and an infinitesimal decrease in the level of output reduces revenue and cost by the same amount). An output level such that rC''(.)> (.) does not maximize profits: an (infinitesimal) increase in output would increase revenue more than cost (therefore increasing profits). Likewise, a level of output such that rC''(.)< (.) does not

Microeconomic Theory IV Monopoly I maximize profits: a (infinitesimal) decrease in output would reduce cost more than revenue

(therefore increasing profits).

(iv) Output and elasticity: ε ()x ≥ 1

We seek to show that at the monopoly output the price-elasticity of demand is 1 or more.

First, we define the price-elasticity of demand in absolute value:

p - as a function of price: ε ()pxp=− ' () , (3) x()p

1()p x - as a function of output : ε ()x =− . (4) p' ()xx

We next represent marginal revenue as a function of the price-elasticity of demand:

rx''()=+ px () xpx () (5)

' ' ⎡ p ()x ⎤ rx()=+ px ()1⎢ x ⎥ (6) ⎣ p()x ⎦

' ⎡ 1 ⎤ rx()=− px ()1⎢ ⎥ (7) ⎣⎢ ε ()x ⎦⎥

In the monopoly output marginal revenue and marginal cost are equal:

⎡⎤1 rx''()=−= px ()1⎢⎥ Cx (). (8) ⎣⎦⎢⎥ε ()x

Given that the marginal cost is by definition non-negative (zero or more) then the marginal revenue must be non-negative. This occurs when the price-elasticity of demand in absolute value is 1 or more. That is:

⎡⎤1 Cx' ()≥⇒ 0 px ()1⎢⎥ − ≥ 0 ⇒ε () x ≥ 1. ⎣⎦⎢⎥ε ()x px()0≥

Microeconomic Theory IV Monopoly I

(v) Lerner index of market power

Now we obtain the Lerner index of monopoly power (or market power) also called the relative price-marginal cost margin. From condition (8) we obtain:

px() p()xCx−=' (). ε ()x

By rearranging we get:

px()− C' () x 1 = . (9) p()xxε ()

Therefore, the Lerner index is a decreasing function of the price-elasticity of demand in

px()− C' () x absolute value. In particular, when ε ()x = 0 monopoly power would be =∞ px() and when ε ()x =∞ (as it would occur if the firm behaved as a perfectly competitive firm)

px()− C' () x market power would be zero, = 0. . px()

Microeconomic Theory IV Monopoly I

(vi) Graphical representation

Cx' () p

m p

m Π rx' () px()

m x x Marginal revenue, rx''()=+ px () xpx (), is located below inverse demand given that the inverse demand function is downward sloping, px' ()< 0. That is, rx' ()< px () for x > 0, but both functions have the same intercept, rp' (0)= (0). The profit of the monopolist (when there is no fixed cost) is given by:

m x ⎡ Cx()m ⎤ Π=Πmmmmmmm()x =px − Cx () = px − Czdz' () = p m − x m ∫ ⎢ m ⎥ 0 ⎣ x ⎦

(vii) Second order conditions

Interpretation

We assume for the sake of simplicity that the profit function is strictly concave.

Π=''()xrxCxpxxpxCx '' () − '' () = 2 ' () + '' () − '' () < 0 (10)

Condition (10) is equivalent to saying that the slope of the marginal revenue has to be lower than the slope of the marginal cost:

dr(())'' x dC ( ()) x < dx dx

In other words, the marginal revenue curve must cross marginal cost from above.

Microeconomic Theory IV Monopoly I

dr('' ( x )) dC ( ( x )) > dx dx

dr(())'' x dC ( ()) x '' < rC, dx dx

Cx' ()

' rx()

m x x

Cases

1. Strictly convex cost or linear cost: Cx'' ()≥ 0 (increasing or constant marginal cost)

a) Strictly concave demand or liner demand: px'' ()≤ 0

'' ' '' '' Π ()xpxxpxCx=+−< 2N ()N () () 0 <≤00≤0

b) Strictly convex demand: px'' ()> 0

'' ' '' '' '' rx()=+ 2N px () xpxN (). We need to check rx()< Cx (). <>00

2. Strictly concave cost: Cx'' ()< 0 (decreasing marginal cost)

We always have to check whether rx''()< Cx '' ().

Microeconomic Theory IV Monopoly I

1.2. Linear demand, constant elasticity demand and constant marginal cost

(i) Linear demand and constant marginal cost

Inverse demand: p (xabx ) =− (ab>> 0, 0 ).

Production cost: Cx ( ) = cx ( c ≥ 0 ). ( ac> )

Marginal revenue: rx' ()=− a 2 bx.

Slope of inverse demand: p' ()xb=−

dr(())' x Slope of marginal revenue: =−2b dx

Strictly concave profit function: Π=''()xrx '' () =−< 2 b 0.

Marginal profit at zero: Π=''(0)pC (0) − (0) =−> ac 0.

ac− Profit maximization: rx''()mm=⇒−=⇒ Cx () a 2 bx m c x m = 2b

ac+ Monopoly price: ppxpabxpmmm=⇒=−⇒() mm = 2

acacac−−() −2 Monopoly profits: Π=Πmm()[()xpxcxpcx = mmmm − ] = [ − ] = = 22bb 4

(ii) Constant elasticity demand and constant marginal cost

Demand: x(pAp ) = −b ( A >>0,b 1).

Production cost: Cx ( ) = cx ( c > 0 ).

pp Price-elasticity of demand: ε ()p =−xp'(1) () = bAp−+b = b . xp() Ap−b

11 − Inverse demand: p()xAx= bb.

11 (1)b − − Marginal revenue: rx' ()= Abb x . b

Microeconomic Theory IV Monopoly I

1(1)+b (1)b − − Slope of marginal revenue: rx'' ()=− Abb x . b2

Strictly concave profit function:

1(1)+b (1)b − − Π=''()xrxA '' () =−bb x <⇔> 0 b 1. b2

Marginal profit at zero: Π=∞>' (0) 0.

Profit maximization:

11 11 (1)bb− −−− rx''()mm=⇒= Cx () rx ' () Abb () x m =⇒ c ()= x m bb A c bb(1)−

11−−bb −b −− ⎛⎞⎛mmbbb ⎞ ⎛⎞− ⎜⎟⎜ ()x bb =AcxAc ⎟⇒=⎜⎟ ⎝⎠⎝(1)bb−− ⎠ ⎝⎠ (1)

Monopoly price:

1 − 111 −b b − ⎛⎞ mmm m⎛⎞bb− b m p =⇒=px() p Abbb () x = A⎜⎟ A⎜⎟ c ⇒ p= c ⎜⎟(1)bb−− (1) ⎝⎠⎝⎠

Monopoly profits:

−b −b mm mmmmcb⎛⎞− b b−−(1) b Π=Π()[()xpxcxpcxA = − ] = [ − ] =⎜⎟ cA = −−(1)b c bb−−1(1)⎝⎠ (1) b −

By solving the problem of profit maximization as a function of price, we obtain the Lerner index:

pcm − 1 = pm ε ()p

pcm − 1 Under constant elasticity demand the Lerner index becomes = and it is pm b straightforward to obtain the monopoly price. Then it is easy to obtain the monopoly output and the monopoly profits.

Microeconomic Theory IV Monopoly I

1.3. Comparative statics

We now study how the monopoly price and output respond to a change in production costs.

Economic intuition tells us that an increase in marginal cost should entail a reduction in output and an increase in price. We assume that the marginal cost is constant (and there is no fixed cost). The cost function is given by Cx ( ) = cx. maxΠ≡ (xpxxCx ) max ( ) − ( ) xx≥≥00

Π=''(0)pC (0) − (0) >⇒> 0 pC(0) ' (0)

Π=''()xpxxpxc () + () −= 0 (11) ⇒ xcm () → the monopoly output is an implicit function of the marginal cost.

Π=''()xpxxpxCx 2 ' () + '' () − '' () < 0 Strictly concave profit function (regular case).

We can analyze the change in monopoly output due to a change in marginal cost in two equivalent ways:

(i) By completely differentiating condition (11) with respect to x and c.

''' ⎣⎦⎡⎤2px ( )+−= xpx ( ) dxdc 0

We get:

dx 1 = '''<0 (12) dc2() p x+ xp () x <0 C2ºO

Therefore, an infinitesimal increase in marginal cost reduces output and an infinitesimal reduction in marginal cost increases output.

(ii) By using the fact that the optimal output for the monopolist xm (c ) is an implicit function of marginal cost. Therefore, by definition, xm (c ) satisfies the first order condition; that is,

Microeconomic Theory IV Monopoly I

px(())()(())mmm c+−= x cp' x c c 0

By differentiating with respect to marginal cost:

2px'' (mm ()) cx () c+= x m () cpx ''' ( mm ()) cx () c 1

''''mm mm ⎣⎦⎡⎤2px ( ()) c+= x () cpx ( ()) c x () c 1

1 Rearranging: xcm' ()=< 0 '''mm m ⎣⎦⎡⎤2px ( ()) c+ x () cpx ( ()) c

Finally the change in price due to the change in marginal cost is:

P<0 dp dp dx p' () x =='''>0 (13) dc dx dc2() p x+ xp () x <0 C2ºO

Examples

(i) Linear demand

ac+ dpm 1 pm = →= 22dc

dp p' () x 1 = '''= dc2() p x+ xN p ()2 x =0

1 Under linear demand the change in price is half the change in marginal cost: dp= dc 2

(ii) Constant elasticity demand

bdpbm pcm =→=>1 bdcb−−11

11 1 (1)++bb 1 (12) −−1(1)+ b − px()=→=−→= Axbb p''' () x A b x b p () x A b x b bb2

Microeconomic Theory IV Monopoly I

dp111 b =='' 1 (1+ 2b ) ==>1 dcpx() (1+ b ) − (1+ b ) b −1 2 + x Axbb2 − px' () b2 b 2 + x 1(1)+b 1 − −Axbb b

Under constant elasticity demand the increase in the monopoly price is greater than the increase in marginal cost: dp> dc .

Microeconomic Theory IV Monopoly I

1.4. Welfare and output

(i) The representative consumer approach. Quasi-linear utility.

(ii) Maximum willingness to pay. Marginal willingness to pay.

(iii) The demand function is independent of income.

(iv) Social welfare function and social welfare maximizing output.

(v) Total surplus, consumer surplus and producer surplus.

(vi) Efficiency conditions in the presence of several consumers or markets.

(vii) A comparison between monopoly output and efficient output by using the profit maximization problem.

(viii) A comparison between monopoly output and efficient output by using the social welfare maximization problem.

(ix) Irrecoverable efficiency loss.

(i) The representative consumer approach. Quasi-linear utility

We will follow the representative consumer approach to analyze welfare and evaluate monopoly from a social welfare point of view. Under this approach, it is assumed that market demand x(p) is generated by maximizing the (quasi-linear) utility of a representative consumer.

Consider an economy with two goods, x and y. Good x is produced in the monopolistic market while we can interpret the good as the amount of money to be spent on the other good by the consumer once he/she has spent the optimal amount of money on good x. We assume that the representative consumer has a Quasi-linear Utility Function:

Uxyuxy( , )=+ ( ) ( u (0) = 0; u''' (.) > 0; u (.) < 0)

Microeconomic Theory IV Monopoly I

(ii) Maximum willingness to pay and marginal willingness to pay

Maximum willingness to pay, R(x ) : the maximum amount of money that the consumer is willing to pay for x units of the good. He/she pays the maximum if he/she is indifferent between consuming x units by payingR (x ) and not consuming the good, thus using all his/her income endowment to consume the other goods.That is:

Uxm(,− Rx ())= U (0,) m

Note that the consumer must be indifferent and, therefore, the above condition must be satisfied with equality. If for example Uxm(,−> Rxi ()) U (0,) m then the consumer would wish to pay a greater amount to Ri()x and if Uxm(,−< Rxi ()) U (0,) m then Ri()x would be higher than his/her maximum willingness to pay.

Given that the utility function is quasi-linear then:

Uxm(,− Rx ())= U (0,) m

ux()+− m Rx () = u (0) + m

R()xux= ()

Therefore, under quasi-linear utility:

ux()→ Maximum willingness to pay

Marginal willingness to pay: this is the change in maximum willingness to pay due to an infinitesimal change in the quantity consumed.

ux' ()→Marginal willingness to pay

Microeconomic Theory IV Monopoly I

(iii) The demand function is independent of income

maxux ( ) + y Lxy(,,)λ xy, ≡++−−maxux ( ) yλ [] m y px sa . y+= px m xy,,λ

∂L ⎫ =−=ux' ()λ p 0 ∂x ⎪ ∂L ⎬→=pux' ( ) → Inverse demand funtion =−10λ = ⎪ ∂y ⎭⎪ ∂L =−−mypx =0 ∂λ

The demand function x(p) is the inverse of this function and therefore satisfies the first order condition:

puxp=→' (( )) Demand function

Property of the quasi-linear utility function: the demand function is independent of income.

By differentiating with respect to p we get:

1(())()= uxpxp'' '

' 1 xp( )=<→'' 0 negative sloping demand uxp (( )) <0

(iii) Social welfare function and social welfare maximizing output

In this subsection we justify the use of Wx()= ux ()− Cx () as the social welfare function.

We consider the problem of obtaining the allocation that maximizes the utility of the representative consumer with a resources constraint: we interpret the production cost of good x as the amount of good y to which must be given up in order to have the good x.

maxux ( ) + y xy, sa . y=− m C ( x )

Microeconomic Theory IV Monopoly I

By replacing y in the objective function we get:

max()ux+−≡ m Cx () max() ux − Cx () xxN constan te

Therefore the social welfare maximizing problem becomes: max()Wx≡− max() ux Cx () xx≥≥00

WuC'''(0)=− (0) (0) > 0

Wx'''()=− ux () Cx () =⇔= 0 Wx '(e ) 0 (13) First order condition.

Wx''()=− ux '' () Cx '' () < 0 Strictly concave welfare function (regular case).

Therefore, the welfare maximizing output or efficient output satisfies

Wx'''()0eee=⇔ ux () = Cx (). Under constant marginal cost the efficiency condition becomes:

ux' ()e = c ,

That is, at the efficient output marginal willingness to pay equals marginal cost.

(v) Total surplus, consumer surplus and producer surplus

The function Wx ( )=− ux ( ) Cx ( ) can be interpreted as the total surplus; that is, the difference between maximum willingness to pay and production cost. By definition the following is satisfied:

x x ux()−= u (0) u' () zdz Cx()−= C (0) C' () zdz N ∫0 N ∫0 =0 ==F 0

Therefore maximizing ux ( )− Cx ( ) is equivalent to chosing the level of output that maximizes the area below the inverse demand and above the marginal cost.

Microeconomic Theory IV Monopoly I

' ' p Cx() p Cx()

e e ux() Cx()

px() px()

e e x x x x

Cx' () p

e Wx()

px()

e x x

By adding and subtracting px we can rewrite the total surplus as:

W() x=− u () x C () x = u () x −+− px px cx [ ] [ ] EC() x EP () x

The consumer surplus, CS(x), measures the difference between maximum willingness to pay

and the amount of money actually paid. The producer surplus, PS(x), measures the profits of

the firm (when there are no fixed costs). Therefore, efficient production also maximizes the

addition of the consumer surplus and the producer surplus.

Microeconomic Theory IV Monopoly I

Cx' () p CS() xe

PSx()e

px()

e x x

(vi) Efficiency conditions in the presence of several consumers or markets

We next analyze the problem of obtaining a Pareto efficient allocation when we consider an

economy with two consumers under quasi-linear utility, uxii ( ) + y i, and an endowment of

mii , =1,2.. We maximize the utility of one agent (for example consumer 1) while maintaining constant the utility of the other (consumer 2), given a resource constraint

(marginal cost, c, is assumed to be constant).

maxux11 ( ) + y 1 xyx112,,, y 2

sa. u22 ( x )+= y 2 u2

y12+=+−ymmcxx 1 2 .( 12 + )

By substituting y1 and y2 in the objective function the problem becomes:

maxux11 ( )+−+++− ux 2 ( 2 ) cx .( 1 x 2 ) m 1 m 2 u2 xx12,

From the first order conditions we get:

' e ux11()−= c 0⎫ ⎪ ''ee ⎬ →==→ux11( ) ux 2 ( 2 ) c Efficiency condition (14) ' e ⎪ ux22()−= c 0⎭

Microeconomic Theory IV Monopoly I

(vii) A comparison between monopoly output and efficient output by using the profit maximization problem. maxΠ≡ (xpxxCx ) max ( ) − ( ) xx≥≥00

Π=''(0)pC (0) − (0) >⇒> 0 pC(0) ' (0)

Π=''''()xpxxpxCx () + () − () =⇔Π= 0 ( xm ) 0 First order condition.

Π=''()xpxxpxCx 2 ' () + '' () − '' () < 0 Strictly concave profit function (regular case).

⎧⎫Π=' ()0xm ⎪⎪' e ⎨⎬Π ()?x ⎪⎪'' ⎩⎭Π<()x 0

''''''eeeee⎡⎤ e eee Π=()xpxxpxCxuxCxxpxN () + () − () =⎣⎦ () − () + ()0 < ' e =ux() =0 <0

By definition of efficient output.

⎧⎫Π=' ()0xm ⎪⎪'''eemem ⎨⎬Π<→Π<Π→>()0x ()xxxx ( ) ⎪⎪'' ⎩⎭Π<()x 0

dxΠ' () Π<⇔''()x 0 <→↑↓Π 0xx ' () dx

Π=' ()0xm

' e Π ()0x <

xm xe x

Microeconomic Theory IV Monopoly I

(vii) A comparison between monopoly output and efficient output by using the social welfare maximization problem. maxWx ( )≡− max ux ( ) Cx ( ) xx≥≥00

WuC'''(0)=− (0) (0) >⇒> 0 pC(0) ' (0)

Wx'''()=− ux () Cx () =⇔= 0 Wx '(e ) 0 First order condition.

Wx''()=− ux '' () Cx '' () < 0 Strictly concave welfare function.

⎧⎫Wx' ()0e = ⎪⎪' m ⎨⎬Wx()? ⎪⎪'' ⎩⎭Wx()< 0

ux'' ()m '''mmmmm ' Wx()=− ux () Cx () =−> xpx ()0 px()m <0

By definition of monopoly output.

⎧⎫Wx' ()0e = ⎪⎪'''memem ⎨⎬Wx()0>→ Wx () < Wx () →> x x ⎪⎪'' ⎩⎭Wx()< 0

dW' () x Wx''()<⇔ 0 <→↑↓ 0 x Wx ' () dx

' e Wx()0=

Wx' ()0m >

m e x x 22 x

Microeconomic Theory IV Monopoly I

(vii) Irrecoverable efficiency loss (IEL).

xxem x e IELWx=−=−−−=−(em ) Wx ( ) [() uz'' Czdz ()] [() uz '' Czdz ()] [() uz '' Czdz ()] ∫∫00 ∫xm

Cx' () Cx' () p e p m CS() x CS() x

m p

e m PSx() PSx( )

px() px()

e m x x x x

Cx' () p

m p

IEL

px()

m e x x x

Microeconomic Theory IV Monopoly I

Basic Bibliography

Varian, H. R., 1992, Microeconomic Analysis, 3th edition, Norton. Chapter 14, sections: introduction, 14.1, 14.2 and 14.3. Chapter 13, sections 13.6, 13.7, 13.9 and 13.10.

Complementary Bibliography

Kreps, D. M., 1994, A course in microeconomic theory, Harvester Wheatsheaf.

Tirole, J., 1990, The Theory of Industrial Organization, MIT Press.

Varian, H. R., 1998, Intermediate Microeconomics: A Modern Approach, Norton.