Industrial Organization - Session 4: The Monopoly

Industrial OrganizationSession 4: The Monopoly

Jiangli Dou

School of Economics

Jiangli Dou (School of Economics) Industrial Organization 1 / 43

Introduction

• In this session, we study a theory of a single seller facing competitive(price-taking) consumers in one or several markets, over one orseveral periods.

• This single seller is facing a downward sloping demand curve.

• Since consumers are always on their demand curve, the monopoly candetermine either the price for the product or the quantity supplied.

• A decision about price implies a decision about quantity produced andvice versa, since price and quantity are related via the demand curve.


Introduction

• In this session, we study a theory of a single seller facing competitive(price-taking) consumers in one or several markets, over one orseveral periods.

• This single seller is facing a downward sloping demand curve.

• Since consumers are always on their demand curve, the monopoly candetermine either the price for the product or the quantity supplied.

• A decision about price implies a decision about quantity produced andvice versa, since price and quantity are related via the demand curve.


The Monopoly’s Profit-Maximization Problem


The monopoly chooses Qm units of output to

maxQπ(Q) = TR(Q) − TC (Q).

A necessary (but not sufficient) condition for Qm > 0 to be themonopoly’s profit maximizing output is

0 =dπ(Qm)

dQ=

dTR(Qm)

dQ− dTC (Qm)

dQ= MR(Qm) − MC (Qm).

This implies that if a profit-maximizing monopoly produces a strictlypositive output level Qm, then the profit output level must satisfy thecondition MR(Qm) = MC (Qm).




Thus, the easiest method for finding the monopoly’s profit-maximizingoutput level is first to solve for Qm from the above equation, and then tosubstitute it into the total profit function to check whether π(Qm) isgreater than or equal to zero. If it is not, then the monopoly sets Qm = 0,and if profit is nonnegtive, then the output level solved from the aboveequation is the profit maximizing output level.After finding the monopoly’s profit-maximizing output, the price chargedby the monopoly can be found by substituting Qm into the demandfunction.



The Monopoly’s Profit-Maximization Problem: AnExample

• For example: TC (Q) = F + cQ2 and p(Q) = a − bQ.

• Then TR(Q) = p(Q)Q = aQ − bQ2

• Hence, MR(Q) = a − 2bQ and MC (Q) = 2cQ. If Qm > 0, it impliesthat

Qm =a

2(b + c)and hence pm = a − bQm =

a(b + 2c)

2(b + c).

• Consequently, π(Qm) = TR(Qm) − TC (Qm) = a2

4(b+c) − F .

• Altogether, the monopoly’s profit-maximizing output is given by

Qm =

{a

2(b+c) if F ≤ a2

4(b+c)

0 otherwise.


Monopoly and Social Welfare




Sherman Antitrust Act of 1890: Every person who shall monopolize, orattempt to monopolize, or combine and conspire with any other person orpersons, to monopolize any part of the trade or commence..., shall bedeemed guilty of a felony. In what follows, we provide two arguments for

why monopolies are discouraged.

• The conventional argument against a monopoly (deadweight loss).

• The social cost of a monopoly (rent seeking).



The conventional argument against monopoly

Comparing the social welfare between the monopoly pricing and marginalcost pricing, we can find that social welfare under marginal-cost pricingassociated with perfectly competitive markets is larger. The difference iscalled deadweight loss.

Computation is needed.



The social cost of a monopoly

Firms wishing to obtain a monopoly status or wishing to maintaining amonopoly position, must allocate resources for that goal.

• Cases should not be considered as reducing welfare include:• R&D leading to a patent monopoly right for seventeen years since the

R&D improves technologies and results in new products.• Bribes to politicians or civil servants for the purpose of getting exclusive

business rights, since this constitutes only a transfer of wealth.

• Cases may be considered as reducing welfare include:• Persuasive advertising, needed to convince consumers that alternative

brands are inferior.• Resources needed to preempt potential entrants from entering the

industry. Also, excessive production or investment in capital for thepurpose of making entry unprofitable for potential competitors.

• Lobbying costs, needed to convince the legislators that a particalarmonopoly is not harmful, provided that these costs divert resourcesfrom productive activities.

• Excessive R&D resulting from a patent race.


Discriminating Monopoly




• Consumers are different: tastes, income, age, location.

• A firm can increases its profit by charging different prices toconsumers with different characteristics.

• However, in order to be able to charge consumers different prices, afirm must possess the means for making arbitrage impossible.

• Examples:

• Firms can charge different prices at different locations. (The marketsshould be isolated by geography, by prohibitive taxes (such as tariffs),or by prohibitive transportation costs such as those resulting fromproduct spoilage while being transported from one location to another.)

• Firms that provide services charge senior citizens lower prices than theycharge younger consumers. (The firm must demand that senior citizenspresent their ID cards.)

• Firms can sell discount tickets to students.• Book publishers manage to charge institutions higher prices than they

charge individuals by selling hardcovers to institutions and softcovers toindividuals.



• Consumers are different: tastes, income, age, location.

• A firm can increases its profit by charging different prices toconsumers with different characteristics.

• However, in order to be able to charge consumers different prices, afirm must possess the means for making arbitrage impossible.

• Examples:• Firms can charge different prices at different locations. (The markets

should be isolated by geography, by prohibitive taxes (such as tariffs),or by prohibitive transportation costs such as those resulting fromproduct spoilage while being transported from one location to another.)

• Firms that provide services charge senior citizens lower prices than theycharge younger consumers. (The firm must demand that senior citizenspresent their ID cards.)

• Firms can sell discount tickets to students.• Book publishers manage to charge institutions higher prices than they

charge individuals by selling hardcovers to institutions and softcovers toindividuals.



• In what follows, we do not analyze how the monopoly manages tosegment the markets so that no arbitrage can take place between twomarkets with different market prices.

• Consider a monopoly selling in two differentiated markets, and assumethat the two markets are isolated in the sense that the monopoly cancharge different prices, and the consumers cannot perform abitrage bybuying in the low price market and selling in the high price market.

• We now seek to investigate how a monopoly determines the outputlevel (hence, the price) in each market.

• The monopoly chooses the output level sold in each market, qm1 and

qm2 , that solves

maxq1,q2

π(q1, q2) = TR1(q1) + TR2(q2) − TC (q1 + q2)



• If the monopoly sells a strictly positive amount in each market, thefollowing two first order conditions are satisfied:

0 =∂π(qm

1 , qm2 )

∂qi= MRi (qm

i ) − MC (qm1 + qm

2 ) for each i = 1, 2.

• Hence, the discriminating monopoly equatesMR1(qm

1 ) = MR2(qm2 ) = MC (qm

1 + qm2 ), when it sells the

profit-maximizing output levels in each market.

• Intuition: If the monopoly chooses qm1 and qm

2 such thatMR1(qm

1 ) > MR2(qm2 ), then it is clear that the monopoly should

tansfer one unit from market 2 to market 1. In this case, thereduction in revenue in market 2 is smaller than the increase inrevenue in market 1.

• To solve for the profit-maximizing output levels qm1 and qm

2 , we needto solve two equations with the two variables given in the equationsabove.



Relationship between the market prices and the demandelasticities

Since MRi (qmi ) = pm

i (1 + 1ηi

), we can get that pm2 > pm

1 if η2 > η1

(recalling that elasticity is a negative number). Hence we can get thefollowing proposition.

Proposition

A discriminating monopoly selling a strictly positive amount in each marketwill charge a higher price at the market with the less elastic demand.



Exercise

Suppose that a monopoly can price discriminate between two markets:market 1, where the demand curve is given by q1 = 2 − p1, and market 2,where the demand curve is given by q2 = 4 − p2. Suppose that once theproduct is sold, it cannot be resold in the other market. That is, assumethat arbitrage is impossible, say, due to strict custom inspections on theborder between the two markets. Assume that the monopoly produceseach unit at a cost of c = 1.

1 Calculate the profit-maximizing output level that the monopoly sellsin each market. Calculate the price charged in each market.

2 Calculate the monopoly’s profit level.

3 Suppose that markets 1 and 2 are now open, and all consumers arefree to trade and to transfer the good costlessly between the markets.Thus, the monopoly can no longer price discriminate and has tocharge a uniform price denoted by p, p = p1 = p2. Find theprofit-maximizing value of p.


The Cartel and the Multiplant Monopoly





• The cartel and the multiplant monopoly are forms of organizationsand contractual agreements among plants, firms, or countries. Forexample:

• If we view the oil-producing countries as plants, the cartel is anorganization that contracts with the contries on how much each wouldproduce and hence on what would be the world price.

• IATA (International Air Transport Association), which regulatesairfares.

• Bar Associations, which regulates attorneys.

• The multiplant monopoly is very similar to the cartel, except that allthe plants are put under a single ownership. Examples ...

• Thus, unlike the cartel, the multiplant monopoly has the power todecide whether to shut down some of its plants or whether to openseveral more.



The Cartel and the Multiplant Monopoly: Technologies

• There are N plants and the output level of plant i = 1, 2, ...,N is qi .

• A line aggregate demand: p = a − bQ, where Q =∑N

i=1 qi .

• Total cost function of plant i is: TCi (qi ) = F + cq2i .

• Hence, plant’s average cost is ACi (qi ) = Fqi

+ cqi , and marginal costis: MCi (qi ) = 2cqi .

• The cartel organizes all the N plants by directing each plant toproduce a certain amount to maximize the sum of the profits of allthe N plants.

maxq1,q2,...,qN

Π(q1, q2, ..., qN) =N∑i=1

πi (qi )

= [a − bN∑i=1

qi ](N∑i=1

qi ) −N∑i=1

TCi (qi )



The Cartel

• The cartel has to solve for N quantities, the N first-order conditionsare given by

0 =∂Π

∂qj= a − 2b

N∑i=1

qi − MCj(qj) = MR(Q) − MCj(qj), j = 1, ...,N

• Since all plants have identical cost functions, we search for asymmetric equilibrium where the cartel directs each plant to producethe same output level. That is, q1 = q2 = ... = qN = q. Hence

a − 2bNq = 2cq implying that q =a

2(bN + c).



The Cartel

• The total cartel’s output and the market price are given by

Q = Nq =Na

2(bN + c)and p = a − bQ =

a(bN + 2c)

2(bN + c).

• When N = 1, the cartel’s output and price coincide with the puremonopoly levels.

• As the number of firms in the cartel increases (N increases), both theoutput level of each firm and the market price fall (q and p decrease).

• Hence, the total revenue and profit of each firm must fall with anincrease in the number of cartel members.

• For this reason, many professional organizations, such as those oflawyers and accountants, impose restrictions on new candidates whowish to practice in their profession.



The multiplant monopoly

• The multiplant monopoly is very similar to the cartel, except that ithas the authority (ownership) to shut down some plants, thereby“saving”variable and fixed costs associated with maintaining theplant.

• If we suppose that the multiplant monopoly can choose the numberof plants, that is, N is a choice variable by the multiplant monopolyowner, then the question is: What is the profit-maximizing number ofplants operated by the multiplant monopoly?



The multiplant monopoly

• The answer is simple: given that the multiplant monopoly can add ordiscard plants, the monopoly would seek to adjust the number ofplants to minimize the cost per unit of production. In other words,the multiplant monopoly will adjust the number of plants to minimizeAC (qi ) = F

qi+ cqi for every plant in operation.

• Hence, we get that qi =√

Fc , then we equating

√Fc = a

2(bN+c) and

solving for N yields that the profit maximizing number of plants is

Nm = a√c

2b√F− c

b .

• The number Nm increases with an increase in the demand parametera, and decreases with the fixed cost parameter of each plant F .


Durable-Goods Monopolies





• Flow goods means goods that are purchased repeatedly and thatperish after usage, for example, food products such as apples andbananas, and many plastic and paper single-use products.

• Durable goods are bought only once in a long time and can be usedfor long time, for example, cars, houses, and land.

• Clearly, with the exception of land, all goods eventually perish, sothese two concepts are relative to a certain time horizon that isrelevant to consumers.

• Coase (1972) first pointed out that a monopoly selling a durable goodwill behave differently from the (familiar) monopoly selling aperishable good analyzed earlier in this chapter.




• A person (monopoly) who owns all the land in the world, and wantsto sell it at the largest discounted profit.

• Suppose that the monopoly charges the monopoly price and sells halfof its land by the end of this year.

• Next year, if population is not growing, the demand for land will belower than the demand for land this year. Thus, the monopoly landprice next year will be lower than the monopoly price this year.

• Given that the monopoly’s next year price will be substantially lowerthan the monopoly land price this year, it is clear that thoseconsumers who do not discount time too heavily would postponebuying land until next year.

• Hence, the current demand facing the monopoly falls, implying thatthe monopoly will charge a lower price than what a monopoly sellinga perishable would charge.



Durable-good monopoly facing a downward slopingdemand

• A continuum of consumers have different valuations for the annualservices of a car that are summarized by the familiar downwardsloping demand curve.

• Consumers live for two periods, denoted by t = 1, 2; and a monopolysells a durable product that lasts for two periods.

• The consumers have different valuations for the product summarizedby the aggregate period t = 1 inverse demand function for one periodof service given by p = 100 − Q.

• In the following, we compare the monopoly’s profit under two types ofcommercial transactions: selling and renting.




Definition

• By selling a product to a consumer, for a price of pS , the firmtransfers all rights of ownership for using the product and getting theproduct back from the consumer, from the time of purchase extendedindefinite.

• By renting a product to a consumer, for a price of pR , the rentermaintains ownership of the product, but contracts with the consumerto allow the consumer to derive services from the product for a givenperiod specified in the renting contract.



Durable-good monopoly facing a downward slopingdemand: A renting monopoly

• Assume that each period the monopoly rents a durable product forone period only.

• Assuming zero product cost, the monopoly would rent an amountdetermined by the condition MR(Qt) = 100 − 2Qt = 0 = MC (Qt),implying that QR

t = 50 and pRt = 50, and πRt = 2500 for t = 1, 2.

• Hence, the life-time sum of profits of the renting monopoly is givenby πR = 5000.



Durable-good monopoly facing a downward slopingdemand: A seller monopoly

• A seller monopoly knows that those consumers who purchase thedurable good in t = 1 will not purchase in period t = 2. That is, int = 2 the monopoly will face a demand for its product that is lowerthan the period t = 1 demand by exactly the amount it sold in t = 1.

• In period 2, the monopoly hence will have to sell at a lower priceresulting from a lower demand, caused by its own earlier sales.

• Formally, we define this two period game as follows:• The payoff to the monopoly is the total revenue generated by period 1

and period 2 sales.• The strategies of the seller are the prices set in period 1, p1, and the

price set in period 2 as a function of the amount purchased in period 1,p2(q̄1).

• The strategies of the buyers are to buy or not to buy as a function offirst period price, and to buy or not to buy as a function of secondperiod price.




We look for a SPE for this simple game. The methodology for solving thisfinite horizon game is to solve it backwards.In the second period:

• The residual demand facing the monopoly in period 2 after it has soldq̄1 units in period 1, given by q2 = 100 − q̄1 − p2, orp2 = 100 − q̄1 − q2.

• Since production was assumed to be costless, in the second period themonopoly sets MR2(q2) = 100 − q̄1 − 2q2 = 0 implying thatq2 = 50 − q̄1

2 .

• Hence, the second period price and profit levels ar given byp2 = 50 − q̄1

2 , and π2 = p2q2 = (50 − q̄12 )2.




In the first period:

• Suppose that the monopolist sells in the first period to q̄1 buyers withthe highest reservation prices. Then, the marginal buyer, with areservation price 100 − q̄1, will be indifferent between purchasing inthe first period (gaining utility of 2(100 − q̄1) − p1) and buying in thesecond period (gaining utility of(100 − q̄1) − p2 = (100 − q̄1) − (50 − q̄1

2 )).

• Thus, 2(100 − q̄1) − p1 = (100 − q̄1) − (50 − q̄12 ), which implies that

p1 = 150 − 3q̄12 . (NOTE)

• In a SPE the selling monopoly chooses a first period output level q̄1

that solves

maxq1

(π1 + π2) = (150 − 3q1

2)q1 + (50 − q1

2)2

yielding a first order condition given by 0 = 100 − 5q12 .




• Denoting the solution values by a superscript S , we have thatqS

1 = 40, qS2 = 30, pS

2 = 30 and pS1 = 90.Hence,

ΠS = pS1 qS

1 + pS2 qS

2 = 4500 < 5000 = ΠR .

• A monopoly selling a durable goods earns a lower profit than arenting monopoly.

• The intuition is that rational consumers are able to calculate that aselling durable good monopoly would lower future prices due to futurefall in the demand resulting from having some consumers purchasingthe durable product in earlier periods. This calculation reduces thewillingness of consumers to pay high prices in the first period themonopoly offers the product for sale. In other words, since themonopoly cannot commit itself not to reduce future prices, themonopoly is induced to lower its first period price.




Then, whether we can claim thatmonopolies have the incentive to produceless than an optimal level of durability(e.g., light bulbs that burn very fast)?




• The following analysis is based on Chapter 12.3.

• A monopoly firm sells light bulbs with variable durability.

• A consumer, who lives for two periods, desires light services for twoperiods. The consumer is willing to pay an amount V (V > 0) pereach period of light services.

• On the supply side, assume that light bulb producing firms possessthe technology for producing two types of light bulbs: a shortdurability light bulb yielding light services for one period only, and along durability light bulb yielding light services for two periods.

• The unit cost of producing the short durability light bulb is denotedby cS , and the unit cost of producing a long durability light bulb isdenoted by cL, where 0 < cS < V , 0 < cL < V , and cS < cL.

• There is no discounting.



Durable-good monopoly facing a downward slopingdemand: Monopoly firm producing light bulbs

• We are going to analyze market equilibria under extreme marketstructures: monopoly and perfect competition.

• The monopoly firm has the option of selling short- or long-durabilitylight bulbs and to charge a monopoly price for either type of bulbs.

• Next let’s analyze when the monopoly would choose to produce short-or long-durability light bulbs. The monopoly needs to compare itsprofit under those two cases.

• Suppose that the monopoly sells short-durability light bulbs. Then,since the consumer is willing to pay V per period of light services, themonopoly would charge pS = V per period and would sell two units(one unit per period). Hence, the profit of a monopoly selling shortdurability light bulbs is given by πS = 2(V − cS).



Monopoly firm producing light bulbs

• Suppose that the monopoly sells long-durability light bulbs. since thelight bulb lasts for two periods, the monopoly would charge pL = 2V .Hence, the profit of a monopoly selling long durability light bulbs isgiven by πL = 2V − cL.

• We would like to know under which condition the monopoly produceslong- or short durability light bulbs. Clearly, the monopoly producesshort-durability bulbs if πS > πL.

Proposition

A monopoly producer of light bulbs would minimize the production costper unit of duration of the light bulbs. Formally, the monopoly wouldproduce short durability light bulbs if 2cS < cL, and would produce longdurability bulbs if 2cS > cL.

• The monopoly’s decision about which type of bulb to producedepends only on cost minimization and not on the market conditions,such as the demand structure.



Competitive light bulb industry

• Under perfect competition, the price of each type of light bulb dropsto its unit cost. Hence, pS = cS and pL = cL.

• The consumer who desires two periods of light services wouldpurchase a short duration light bulb if 2pS < pL, or , if 2cS < cL.Otherwise, consumers purchases long durability light bulbs.

Proposition

• The durability of light bulbs is independent of the market structure.

• The firms would choose the level of durability that minimizes the productioncost per unit of time of the product’s services.

• It is important to note that this analysis assumes that our consumer isonly concerned with the length of time service is provided by theproduct and does not attach any other value for durability per se.



Durable-good monopoly facing a discrete demand

• We now provide an example which demonstrates that Coase’sConjecture is false when the number of consumers is finite.

• Let’s consider an economy with two consumers living only for twoperiods. Both consumers desire car services for the two periods oftheir lives, however, the consumers differ in their willingness to pay forcar services. The maximum amount a consumer denoted by H iswilling to pay for one period of car service is V H , and the maximumamount a consumer denoted by L is willing to pay for one period carservice is V L. We assume that the consumers’ willingness to pay perperiod of car service are substantially different and V H > 2V L > 0.

• Because the product is durable, consumers buy it once in their lifeeither at t = 1 or t = 2.




• The utility functions for consumers type i = H, L are given by

U i =

2V i − p1 if he buys a car in period 1

V i − p2 if he buys a car in period 2

0 if he does not buy a car in any period.

• On the production side, we assume that there is only one firmproducing cars, at zero cost. The monopoly firm lives for two periodsand maximizes the sum of profits from the sales during the twoperiods. We denote by qt the amount produced and sold by themonopoly, and by pt the period t price of a car set by the monopolyin period t = 1, 2.

• The monopoly chooses p1 and p2 to maximize the sum of revenuefrom two periods worth of sales given by π = p1q1 + p2q2. Weimplicitly assume that there is no discounting.



A renting monopoly

• Suppose now that the monopoly firm does not sell cars, but insteadrents car for one period only. Thus, each consumer who rents a car int = 1 has to return the car at the end of the first period and rent itagain in the second period. We denote by pR

t the rental price for oneperiod of renting in period t.

• Since car rentals last for one period only, it is sufficient to calculatethe price for each period separately. Since the renting firm is amonopoly, it has two options: (1) setting pR

t = V H , which inducesonly consumer H to rent a car each period, while consumer L will notrent; (2) setting pR

t = V L, which induces both consumers to rent acar each period. In the first case, the two period profit is πR = 2V H ,and in the second case, πR = 4V L.

• Hence, a renting monopoly would rent cars only to the high valuationconsumers by setting a rental price equal to pR

t = V H and it will earna two period profit of πR = 2V H .



A seller monopoly

• Now suppose that the monopoly sells the car to consumers. Wedenote the selling price by pS

t , t = 1, 2. The period one selling price,pS

1 , means that the consumer pays for two periods of using the car(compared with the renting price pR

1 that entitles the consumer to usethe car only for period 1 only).

• If consumer H purchases in period 1, only consumer L demands a carin the second period; the monopoly will maximize second period profitby setting pS

2 = V L and will earn a second period profit ofπ2 = V L(the monopoly will extract all surplus from consumer L).

• If consumer H does not buy in period 1, then in the second periodthe monopoly faces the entire demand, hence, the monopoly chargespS

2 = V H (selling only to consumer H) yielding a second period profitof π2 = V H .



A seller monopoly

• In the first period, the monopoly sets pS1 , and consumers decide

whether to purchase or not.

• Since consumer L knows that the price in the last period will neverfall below V L, consumer L will buy in the first period at any pricebelow 2V L. Hence, if the seller sets pS

1 = 2V L both consumers wouldpurchase initially.

• Clearly, the monopoly will not set pS1 > 2V H because this price

exceeds the two period sum of consumer H’s valuation. Therefore, wenow check whether pS

1 = 2V H is the profit maximizing first periodprice for the seller monopoly.

• From the second period analysis we conclude that consumer H earnsa utility of zero whether or not he buys the product in the first period.Hence buying the product is an optimal response for consumer H tothe first period price pS

1 = 2V H .




• Thus, in a SPE, pS1 = 2V H , consumer H buys in period 1, pS

2 = V L,consumer L buys in period 2, constitute a SPE equilibrium path forthis game.

Proposition

A durable good selling monopoly facing a discrete demand will (1) chargea first period selling price that is equal to the sum of the per period rentalprices,pS

1 = 2pRt ; (2) earn a higher profit than the renting monopoly, that

is, πS = 2V H + V L > 2V H = πR .

• Thus, in the case of discrete demand, a selling monopoly can extracta higher surplus from consumers than the renting monopoly. Sinceselling enables the monopoly to price discriminate among differentconsumers by setting prices which would induce different consumersto purchase at different time periods.


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Industrial Organization - Session 4: The Monopoly