HKKK TMP 38E050 © Markku Stenborg 2005 1 4. Oligopoly Topics Review of basic models of oligopolistic competition How can firms change the rules of game

HKKK TMP 38E050

© Markku Stenborg 20051

4. Oligopoly

Topics• Review of basic models of oligopolistic competition• How can firms change the rules of game to their advantage?• How can firms avoid intensive rivalry?Read• or review Oligopoly chapter in any modern Micro or IO

textbook to make sure you are comfortable with game theoretic reasoning and Nash equilibrium

• Europe Economics reportNote: topics following oligopoly (Collusion and Mergers) will be based on oligopoly theory

HKKK TMP 38E050


3.1 Cournot or Quantity Competition

• Assumptions– Market demand: price function of total quantity produced,

p = p(q), eg. p = a - bq– Assume 2 firms on relevant market denoted by i and j

– Firms produce quantities qi and qj

– Firms have total costs ci(qi,qj)

– No threat of entry– Profits for firm i = Total Revenue - Total Costs

= pi(qi,qj) = p(qi+qj)qi - ci(qi,qj)

• Note: i's profit depends on what rival j does, unlike in monopoly or perfect competition

• Firm faces a problem of strategic interaction or plays a game

HKKK TMP 38E050


3.1 Cournot or Quantity Competition• How much will i want to produce?

– Depends on how much i expects j to produce, qje

• How much will j want to produce?

– Depends on how much j expects i to produce, qie

• Note, for each qje, there is an optimal output

qi*(qje) = argmax i(qi,qj

e)

• qi*(qje) is called i’s reaction function

• Compare with monopoly profit max

Problem• i needs to put himself on j’s position and try to predict how j

will behave• j needs to put himself on i’s position and try to predict how i

will behave• i needs to to put himself on j’s position and try to predict how j

will think how i will behave

HKKK TMP 38E050


3.1 Cournot or Quantity Competition• j needs to ... predict how i will think how j will behave• etc. ad inf.

Solution

• Suppose both i and j know p(q), ci(qi,qj), and cj(qj,qi), and also expect that rival will produce profit-maximizing quantity qi*(q-i

e) = argmax j(qi,q-ie)

• Now i chooses qi* = argmax i(qi,qj*) and j chooses qj* = argmax j(qj,qi*)

• Each firm chooses its strategy taking rivals equilibrium strategy as given

• Firm i needs to predict j’s equilibrium production

• To solve, simplify further: ci(qi,qj) = ciqi, ci = constant MC

• Now i = p(qi+qj)qi - cqi

HKKK TMP 38E050


3.1 Cournot or Quantity Competition• Simultaneously but individually

max i = p(qi+qj)qi - cqi max j = p(qi+qj)qi - cqj di/dqi = q(dp/dqi) + p(qi+qj) - ci = 0 dj/dqj = q(dp/dqj) + p(qi+qj) - cj = 0

These are familiar 1st order conditions MR - MC = 0 Compare with monopoly profit max Plug in p(qi+qj) = a - b(qi+qj) and solve for qi*(qj

e) and qj*(qie),

you get reaction fns:

(1) qi*(qj) = (a - ci)/2b - qj/2

(2) qj*(qi) = (a - cj)/2b - qi/2

Solve simultaneously [eg, insert qj*(qi) from (2) into (1)] to get Cournot-Nash equilibrium quantities

HKKK TMP 38E050


3.1 Cournot or Quantity Competition(3) qi* = (a + cj - 2ci)/3b

(4) qj* = (a + ci - 2cj)/3b

Note: Each firms is on her reaction function In equil, no firm has incentive to alter her strategy choice

unilaterally Insert qi* and qj* to demand fn to get equil price p*, and then

plug these to profit fn to get equilibrium profits Note: reaction fns (1) and (2) are downward-sloping:

dqi*(qi)/dqj = -1/2 < 0

• This also applies to more general Cournot games • If j increases her production (eg, due to reduction in marginal

cost cj), i will want to reduce his output

• Lower action from one firm induces higher reaction from her rivals

HKKK TMP 38E050


3.1 Cournot or Quantity Competition• Strategies qi are here strategic substitutes

• Downward-sloping rfs strategic substitutes

Properties of Cournot-Nash Equil• Go back to reaction functions (1) and (2), and rewrite as

(5) p(q) - ci = -qi dp/dqi |:p

(6) Li = si/e,

si = qi/q is i’s market share, q = iq,Li = (p - ci)/p is firm i’s mark-up or Lerner Indexe = -p(q)/qp’(q) is elasticity of market demand

• (6) is basic Cournot pricing formula– In Cournot-Nash equilibrium, market share determined by

firm’s relative cost efficiency and demand elasticity

HKKK TMP 38E050


3.1 Cournot or Quantity Competition• Each firm has limited mkt power:

– i’s marginal revenue MRi = p + qip’, so

– p - MRi = qip’(q) > 0 MR > M

• Smaller mkt shares s (or more rivals) smaller mark-up, ie. competion more vigorous

• Greater demand elasticity larger mark-up, less competitive equil

• Mark-up is proportional to firm mkt share

• Mkt shares are directly related to firms cost-efficiency ci

• Less efficient firms are able to survive

– sj > 0 even if cj >> min c

• Average industry-wide mark-up i si (p - ci)/p = MU

• In Cournot-Nash equil, MU = i si2/e = HHI/e, where HHI is

the Herfindahl-Hirschman Index• Performance negatively related to HHI

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3.1 Cournot or Quantity CompetitionAside• What if there are more than 2 firms?

– Interpert j as vector of all other firms and proceed as above

– Each firm takes the actions of other firms as given, and assumes all firms are maximizing profits

• What if i doesn’t know j’s costs cj(qj,qi) exactly?

– Just assume i is Bayesian decision-maker, who makes subjective probability assesment for cj(qj,qi), uses expected costs Eicj(qj,qi), and then proceed as above

• Existence of equil? Uniqueness of equil?– Not guaranteed for general p(q) and c(.)– Coordination problem if more than one equil strategy

combination

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3.2 Bertrand or Price Competition

• In reality, firms choose and compete w/ prices• Often prices are easier to adjust than quantities• Who chooses prices in Cournot game?• Cournot unrealistic model?• Naive thought: firms select prices as in (6) above:

pi* st. (pi* - ci)/pi* = si/e?

• Bertrand paradox: No• Model: identical product, mkt demand q = q(p), eg q = a-bp• Demand for firm i:

pi > pj i cannot sell at all, qi = 0pi = pj i and j split demand, qi = q(p)/2pi = pj i sells total mkt demand, qi = q(p)

HKKK TMP 38E050


3.1 Cournot or Quantity Competition• Note: small change in rival’s price causes huge change in

firm’s demand

• Suppose cj = ci = c

• If i charges pi > c, j can increase her profits by undercutting i slightly

• If i charges pi < c, i is making losses but j can guarantee j = 0 by staying out of mkt

Only equil price can be pi = pj = c

• Duopoly enough for perfect competition!• Depends crucially on

– firms able and willing to serve all customers at announced price

– identical products– customers have complete information eg on prices Then firms have no bargaining power

HKKK TMP 38E050


3.1 Cournot or Quantity CompetitionProduct Differentation and Price Competition• Simple example only• Products are imperfect substitutes, demands are symmetric

qi = a - fpi + gpj

• Assume constant marginal costs ci

• Product differentation is assumed fact, not designed by firms– g/f measures degree of product differentation (how?)

• Profit for i herei = pi qi(pi,pj) - ci(q(pi,pj)) = (pi - ci)(a - fpi + gpj)

• Bertrand-Nash equil found similarly as above:

– max profit wrt to strategy variable pi

– solve for rfs– find where rfs intersect– solve for prices, quantities, and profits

HKKK TMP 38E050


3.1 Cournot or Quantity Competition• Rfs slope up the higher price i charges, the higher the price

rival j wants to charge• Prices are strategic complements• Higher strategy draws a higher reaction from rivals• Upward-sloping rfs strategic complements

Homework

• Assume qi = a - fpi + gpj and ci = 0

• Prove: In price competition with differentiated products, reaction functions slope up

• Solve for Bertrand-Nash equil prices, quantities and profits for game above

• Solve for Cournot-Nash equil quantities, prices, and profits for game w/ same demands and c = 0

HKKK TMP 38E050


3.1 Cournot or Quantity CompetitionCapacity Constraints and Price Competition• What if firms first choose capacities q and then, knowing all

q’s, select prices p?– We have a 2-stage game (more on this later)– In equil, higher price than w/o capacity constraints– Intuition: limited capacity business stealing not

attractive option want to price less aggressively rival prices less aggressively higher profits

• Cournot outcome possible w/ price competition– Interpret: Cournot = capacity competition

• Cournot– mkts where production desicions in advance, flexible

price, high storage costs– consistent w/ empirical evidence

• Bertrand– more realistic assumptions?

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2.4 Two-Stage Competition

• Simplest way to model dynamic rivalry; to model j’s reactions to strategic moves by firm i

• Simplest way of allowing firms to change game they are playing

• Idea: Choose a strategy now that affects game you play tomorrow st your expected profits increase

• Capacity-Price -model above an example: Smaller capacity now reduce ability to compete aggressively in future draw less aggressive reactions from rivals higher profit

Stackelberg Oligopoly• Stackelberg-Cournot game: Firm i chooses its output first,

and j after i’s choice• Precommitment by i is relevant, not physical timing of moves

HKKK TMP 38E050


2.4 Two-Stage Competition• Solve by backward induction:– First look at last possible moves of the game– What is optimal last move?– Then work backward to beginning of game, as in dynamic

programming– Given last move will optimal, what is optimal penultimate

move?– Game tree (compare to decision tree)

• Simple mode: Demand p = a - b(qi + qj), c = 0Last move

• When j chooses her capacity, she knows i’s capacity qiS

• j's optimal capacity determined by her reaction function (2)qj*(qi) = a/2b - qi/2

HKKK TMP 38E050


2.4 Two-Stage CompetitionPenultimate move• To design good strategy, i must put himself on j’s shoes and

try to think how he would behave were he the last to move

• i chooses qiS to max profits, taking as given i’s reaction

function, not equilibrium output as in Cournot game• i chooses best point from rival’s reaction function

• Plug (2) into i’s profit function (a - b(qi+qj))qi and solve for qiS

• Plug qiS back to (2) and solve for qj*, and then solve for

prices and profits • In Stackelberg game, i’s profits higher and j’s lower than in

Cournot game• First-mover advantage• Intuition: Commit to flood the market induces rival to lower

output increases your profit

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2.4 Two-Stage Competition• Crucial reasons: 1) commitment, 2) strategies substitutes• Equil above “subgame perfect Nash equilibrium”• Also other Nash equil possible:

• i announces to produce qi s.t. p(qi) < cj if j enters

• This is not be credible: i will not want to undertake threat should j enter (more on this later)

Homework• Solve Stackelberg equilibrium capacities, prices and profits

• You can assume symmetric demands qi = a - fpi + gpj,qj = a - fpj + gpi and c = 0

• Show: In Stackelberg-Bertrand duopoly, there is second mover advantage.

• You can assume symmetric demands as above

HKKK TMP 38E050


2.4 Dynamic Competition

• 2 time periods, denoted by 1 and 2• Firm i can take strategic action k on period 1• Strategic action measured by its cost• Strategy k is sunk on 2nd period, i cannot revoke it• k is investment, precommitment• On period 2, i and j compete• Assume k does not affect j’s demand or costs directly

• i’s 2nd period profits are i(qi,qj,k)

• i’s 1st period profits are i (qi,qj,k) - k

• k shifts i’s 2nd period profit fn• Strategic move k alters i’s own incentives to choose later 2nd

period tactics• To find equil, solve for by backward induction starting from 2nd

period game

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2nd period• For given k, equil again given by

di/dqi = 0di/dqj = 0

• 2nd period reaction functions qi(qj,k) and qj(qi,k), and optimal tactics qi*(k) are now functions of k

• Equil profits are i(qi*(k),qj*(k),k)

1st period• How to choose k?

• Profits i(qi*(k),qj*(k),k) - k

• To find max profit, differentiate i wrt k to get MR – MC = 0; this gives

01 =++dk

d

dk

dq

dq

d

dk

dq

dq

dij

j

ii

i

i

HKKK TMP 38E050


2.4 Two-Stage Competition• First term is zero because i will choose tactic qi stdi/dqi = 0; we have:

• LHS 2nd term: direct effect• LHS 1st term: strategic effect• RHS: Direct cost of commitment• How can k alter j’s 2nd period tactics since k does not directly

affect j’s profits?• Strategic move k alters i’s own incentives to choose alters

j’s incentives to react changes i’s profits• Sign of strategic effect is equal to sign of

1=dkd

dk

dqdq

)k,q,q(d i*j

j

*j

*ii

j

i

i

i

i

i

dqd

dqjdqd

dkdqd 22

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2.4 Two-Stage CompetitionThree effects• How commitment k changes i’s own optimal tactics• How j reacts to changes in i’s incentives• How i’s profits are affected by changes in j’s tactics

Strategic effect > 0 overinvest in kStrategic effect < 0 underinvest in k

• Example: Cost reduction in Cournot and Bertrand games– How reaction functions shift as marginal costs of j are

decreased?• Example: Increased marketing in Cournot and Bertrand

games– How reaction functions shift as j increases her marketing

expenses?

HKKK TMP 38E050


2.4 Two-Stage CompetitionTaxonomy for Strategies• Strategic substitutes vs complements

– Cournot game = strategic substitutes– Bertrand game = strategic complements

• Commitment makes firm tough vs soft• Investment k makes i tough

– i will produce more or price below– k shifts i’s rf right and up in Cournot game– k shifts i’s rf right and down in Bertrand game

• Investment k makes i soft– i will produce less or price above – k shifts i’s rf left and down in Cournot– k shifts i’s rf left and up in Bertrand

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Commitment makes firm

Stage 2 variables are Tough Soft

StrategicComplements

(eg, prices)

Puppy Dog PloyStrategic effect < 0Commitment cause rivals behave more

aggressively

Fat Cat EffectStrategic effect > 0Commitment cause rivals behave less

aggressively

StrategicSubstitutes

(eg, capacities)

Top-Dog StrategyStrategic effect > 0Commitment cause rivals behave less

aggressively

Lean and Hungry LookStrategic effect < 0Commitment cause rival behave more

aggressively

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2.4 Two-Stage CompetitionStrategic Incentives to Commit in Cournot• Commitment makes firm tough

– Firm will produce more for all given rivals’ output– Reaction function shifts outward– Example: Marginal cost reducing innovation– Beneficial side-effect– Strategic effect might outweigh direct effect

Invest even if NPV < 0!– Top-Dog: Big or strong to become aggressive

• Commitment makes firm soft– Firm will produce less for all given rivals’ output– Reaction function shifts inward– Example: Marginal cost increasing entry into other mkt

Even monopoly might not be enough– Negative side-effect– Lean and Hungry Look: Refrain from expanding to avoid

weakness

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2.4 Two-Stage CompetitionStrategic Incentives to Commit in Bertrand• Commitment makes firm tough

– Firm will underprice– Reaction function shifts inward– Example: MC-reducing innovation– Negative side-effect– Puppy-Dog Ploy: stay small or weak to avoid agressive

competition Do not lower costs!• Commitment makes firm soft

– Firm will overprice– Reaction function shifts outward– Beneficial side-effect– Example: Target small niche, Product differentation– Fat-Cat Effect: Become soft to attract only weak

competition Sumo-strategy

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2.4 Two-Stage Competition• Need to look more than just direct effects of irreversible decisions

• Nature of future competition affects incentives to make investments or commitments now

Examples of 1st Stage Commitments• Build excess capacity deter entry• Enter and underinvest avoid attracting tough competition• R&D: reduce costs price aggressively / gain mkt share• Networks: build large customer base, costly to switch less

competition in future• Patent licensing withhold or exchange key info or patents

with rivals• Underinvest in marketing less loyal customers become

aggressive in 2nd stage• Overinvest in marketing loyal customers become soft in

2nd stage

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2.4 Two-Stage Competition• Merger: profitable under Bertrand, unprofitable under Cournot competition

• Make products less similar soften price competition• Financial structure: overleverage to make managers more

aggressive• Managerial compensation: Tie managers’s compensation on

sales Stackelberg equil• Long-term contracts with customers, Most favorite nation

clause reduce incentive to cut prices• Customer Swithcing Costs: lock in customers less

incentives to go after new customers draw less aggressive reactions from rivals

• Multimarket Contact strategic effects from mkt 1 to mkt 2

Documents

HKKK TMP 38E050 © Markku Stenborg 2005 1 4. Oligopoly Topics Review of basic models of oligopolistic competition How can firms change the rules of game