Cournot versus Stackelberg

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Cournot versus Stackelberg

• Cournot duopoly (simultaneous quantity
  competition)
• Stackelberg duopoly (sequential quantity
  competition)

x2
x1
2
Homogeneous duopoly(linear case)

• Two firms (i1,2) produce a homogenous goods.
• Outputs x1 and x2, X x1x2
• Marginal costs MC1 and MC2
• Inverse demand function
• Profit function of firm 1

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Cournot-Nash equilibrium

• Profit functions
• Reaction functions
• Nash equilibrium

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Computing the Cournot equilibrium(accommodation)

• Profit function of firm 1
• Reaction function of firm 1
• Nash equilibrium

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Depicting the Cournot equilibrium
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Exercise (Cournot)

• Find the equilibrium in a Cournot competition.
  Suppose that the demand function is given by p(X)
  24 - X and the costs per unit by c1 3, c2
  2.

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Common interests

• c1, c2 ?obtaining government subsidies and
  negotiating with labor unions
• a ?, b ?advertising by the chemistry industrie

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Exercise (taxes in a duopoly)
Two firms in a duopoly offer petrol. The demand
function is given by p(X)5-0.5X.Unit costs are
c10.2 and c20.5. a) Find the Cournot
equilibrium and calculate the price. b) Now
suppose that the government imposes a quantity
tax t (eco tax). Who ends up paying it?
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Two approaches to cost leadership

• Direct approach (reduction of own marginal
  costs)- change of ratio between fixed and
  variable costs- investments in research and
  development (RD)
• Indirect approach (raising rivals
  costs)- sabotage- minimum wages, enviromental
  legislation

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Direct approach, analytically

• Direct approach (reduction of your own marginal
  costs) lt0 lt0 gt0
  0 direct strategic effect effect

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Direct approach, graphically
x 2
marginal cost reduction of firm 1
equilibria increase in production of firm 1
x 1
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Indirect approach, analytically

• Indirect approach (raising rivals cost)
• 0 lt0 lt0 0
  direct strategic effect effect

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Indirect approach, graphically
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Reaction curve in the linear case
x 2
x 1
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Blockaded entry, graphically
x 2
firm 1 as a monopolist
C
M
x 1
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Blockaded entry

• Entry is blockaded for each firm
• Entry is blockaded for firm 2

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Blockaded entry (overview)
c 2
no supply
firm 1 as a monopolist
firm 2 as a monopolist
duopoly
c 1
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Exercise (Strategic trade policy)

• Two firms, one domestic (d), the other foreign
  (f), engage in Cournot competition on a market in
  a third country. Assume
• The domestic government subsidizes its firms
  exports using a unit subsidy s.
• Which subsidy s maximizes domestic welfare

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The Herfindahl index

• The Herfindahl index measures the concentration
  in an industry.
• ExerciseWhich market is the most concentrated

0.5 0.66 0.44

• 2 firms with equal market shares,
  
• 3 firms with shares of 0.8, 0.1 and 0.1 or
  
• 3 firms with shares of 0.6, 0.2 and 0.2 ?

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n firms in Cournot competition

• Total industry output
• Firm is profit function
• Firm is marginal revenue

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Lerner index of monopoly power

• First order condition
• Lerner index for one firm
• Lerner index for the industry

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Stackelberg equilibrium

• Profit function
• Followers reaction function
• Leaders optimal quantity
• Nash equilibrium

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Finding the profit-maximizing point on the
follower's reaction curve
x 2
Accommodation
Blockade or deterrence
x 1
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Computing the Stackelberg equilibrium
(accommodation)

• Reaction function of firm 2
• Profit function of firm 1
• Nash equilibriumwith and

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Depicting the Stackelberg outcome (both firms
produce)
x 2
quantities in a Stackelberg equilibrium
C
S
x 1
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Exercise (Equilibria)

• Which is an equilibrium in the Stackelberg
  model?
• Are there any additional Nash equilibria ?

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Cournot versus Stackelberg II

• Profit function of firm 1
• First order condition for firm 1 direct
  effect follower effect
  Cournot 0

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Exercise (Stackelberg)

• Find the equilibrium in a Stackelberg
  competition. Suppose that the demand function is
  given by p(X) 24 - X and the costs per unit by
  c1 3, c2 2.
• Possible or not? ?

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Blockaded entry
p
blockaded entry for firm 2
x 1
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Reaction functions in the case of blockaded entry
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Deterring firm 2s entry
p
x 1
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Reaction functions in the case of deterred entry
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Blockaded and deterred entry

• Blockaded entry (firm 2)
  
• Deterred entry (firm 2)

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Blockade and deterrence
c 2
no supply
blockade
firm 1 as a monopolist
deterrence
firm 2 as a monopolist
duopoly
c 1
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Exercise I (entry and deterrence)
1) Suppose a monopolist faces a demand of the
form p(X)10-0.5X. The firms unit costs are
2. a) Find the profit-maximizing quantity and
price. Is entry blockaded for a potential entrant
with unit costs of 8? b) Assume now that the
potential entrants unit costs decrease to 4. Is
entry still blockaded? c) Find the limit output
level and price. Should the incumbent deter?
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Exercise II (entry and deterrence)
2) In addition, the firms are supposed to face
quasi-fixed costs of 1. Hence, the cost functions
are given by a) What is the monopolists
profit-maximizing quantity and price? b) Is
entry blockaded for the potential entrant? c)
Find the incumbents profit-maximizing output
level. (Hint Compare the profits.)
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The quantity cartel

• The firms seek to maximize joint profits
• Optimization conditions

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Cartel quantities
x 2
quantities in a symmetric cartel
C
S
CA
x 1
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The cartel agreement

• The optimization condition is given by
• Each firm will be tempted to increase its profits
  by unilaterally expanding its output.
• In order to maintain a cartel, the firms need a
  way to detect and punish cheating, otherwise the
  temptation to cheat may break the cartel.

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Summary

• The cartel is unstable if the firms meet only
  once. Thus, the cartel agreement is not an
  equilibrium.
• Supposing that the number of repetitions is
  unknown, the cartel agreement can be an
  equilibrium outcome.

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Exercise (cartel quantities)

• Consider a cartel in which each firm has
  identical and constant marginal costs. If the
  cartel maximizes total industry profits, what
  does this imply about the division of output
  between the firms?
• SolutionNothing. Since all firms have the same
  marginal cost, it doesnt matter which of them
  produces the output.

Hal R. Varian, Intermediate Microeconomics
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The outcomes of our models
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Cournot Executive summary I

• A duopoly can only be expected if entry is
  blockaded for other firms. Otherwise industry
  profits would attract potential competitors. An
  entry can be blockaded by cost structures
  or by certain laws or conditions
  (legal/administrative barriers).
• All competing firms have the incentive to lower
  costs in common. Firms profit and the industry
  profit increase. Thus, activities in associations
  which aim at improving cost structures are
  worthwhile (collective agreements, government
  technology promotion,...)

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Cournot Executive summary II

• All competing firms have the incentive to
  increase market demand by appropriate marketing
  activities. Firms profit and the industry profit
  increase. Thus, activities in associations which
  aim at increasing market demand are worthwhile
  (cooperative advertising campaigns,...)
• The attempt to become a cost leader before a
  simultaneous competition is worthwhile. The firm
  with the lower marginal cost or unit cost
  respectively faces higher turnovers and profits.
  Cost leadership even leads to monopolization.

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Cournot Executive summary III

• There are two approaches to cost leadership. The
  direct approach is to lower your own marginal
  cost. The indirect approach is known as raising
  rivals costs.

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Stackelberg Executive Summary I

• Time leadership is worthwhile in a Stackelberg
  equilibrium the leader realizes a profit that is
  higher than the followers and his own in a
  Cournot equilibrium. Even with a slight cost
  disadvantage, the leaders profit might be higher
  than the followers.
• If a firm is both Stackelberg leader and cost
  leader, it can consider deterring the followers
  entry by offering the limit quantity as a
  strategic entry barrier.

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Stackelberg Executive Summary II

• Entry costs make the followers deterrence
  easier. The amount of its costs at which entry is
  deterred increases with the entry costs.
• Since the Stackelberg leader offers a higher
  quantity than the follower, he might enlarge his
  cost advantage by learning curve effects.

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Cartel Executive Summary I

• If all firms keep the cartel agreement, then they
  can increase their profits compared to the
  Cournot competition.
• Nevertheless cartels are unstable. Cheating even
  makes higher profits possible. When all firms
  break up the cartel agreement, their profits
  decrease. Moreover, they might face a worse
  position than in the Cournot competition.

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Cartel Executive Summary II

• Cartels will not continue to exist when potential
  competitors entries, who are attracted by the
  cartel profit, are not blockaded. Therefore, the
  firms have to be able and willing to restrict or
  deter entry. If there are administrative
  barriers, investments in maintenance of these
  barriers are necessary. If the barriers are more
  structural, then the firms have to try to keep
  their cost and technology advantage (cooperation
  in RD).