Download Monopoly and Oligopoly

Monopoly and Oligopoly
ECON 212 Lecture 15
Tianyi Wang
Queen’s Univeristy
Winter 2013
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
1 / 16
Introduction
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Firm takes price as given in a perfectly competitive industry.
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The opposite is monopoly, a market with only one …rm.
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Note monopoly cannot chooses price and quantity independently.
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Monopoly can choose price and let quantity determined by market
demand. Or the other way around.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
2 / 16
Monopoly Problem
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Monopoly faces market constraint in the form of market demand.
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Let p (q ) be the inverse market demand curve. Let c (q ) be the cost
function. Then monopolist’s problem is
max p (q )q
q
c (q )
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Note monopoly must choose output level that MR=MC.
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Solve by taking the FOC,
p (q ) + qp 0 (q ) = c 0 (q )
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Marginal revenue consists of two parts: one from selling addtional
unit and one from reduced price for all existing units.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
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Markup
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We can rewrite the FOC,
p (q ) =
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1
MC (q )
(1 + e(1q ) )
where e(q ) is the price elasticity of demand and the term
1
(1 + e(1q ) )
is
known as markup for the monopoly.
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Note for perfect competition FOC is p (q ) = MC (q ). Thus it is a
special case where demand elasticity is negative in…nity.
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Also monopoly will never operate on inelastic part of demand curve
( 1 < e 0), for pro…t can be increased by reducting output.
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See classnotes for graph.
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See classnotes for linear demand.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
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Extensions
1. E¢ ciency of monopoly.
2. Price Discriminations.
3. Natural monopoly.
4. Quality choices.
5. Group discount, bundling...
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
5 / 16
Oligopoly
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Oligopoly lies between perfect competitive and monopoly.
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There are a small number of …rms and entry is di¢ cult.
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Strategic consideration: …rms recognize the interaction of decisions.
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We consider a range of models.
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sequential vs simultaneous
quantity vs price
Limit the number of …rms to two (duopoly) for simplicity.
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ECON 212 Lecture 15
Winter 2013
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Simultaneous Quantity
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Firms forcast each other’s decision when make decisions
simultaneously.
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Cournot model - quantity competition
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Consider an industry with 2 identical …rms. Firm 1 expects …rm 2 to
produce q2e .
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Then if …rm 1 produces q1 , he expects the total output to be
q = q1 + q2e .
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Each …rm knows that equilibrium price depends on total output by
inverse demand function p (q ).
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
7 / 16
Simultaneous Quantity
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Firm 1’s problem is to choose q1 to max pro…t
max p (q1 + q2e )q1
q1
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c ( q1 )
FOC gives
q1 p 0 (q1 + q2e ) + p (q1 + q2e ) = c 0 (q1 )
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This implies …rm 1 set quantity as a function of …rm 2’s choice (in
expectation).
q1 = f1 (q2e )
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This equation is known as a reaction function; it describes how …rm
1 reacts to …rm 2’s choice.
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Similarly …rm 2’s reaction function is
q2 = f2 (q1e )
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
8 / 16
Simultaneous Quantity
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In equilibrium expectations are con…rmed so that no one has incentive
to revise choices.
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An equilibrium is two output levels q1 and q2 that solve the reaction
functions simultaneously:
q1
q2
= f1 ( q 2 )
= f2 ( q 1 )
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Note f 0 < 0. (strategic substitutes)
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See class notes for graph.
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See calss notes for …x-point theorem.
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See class notes for example.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
9 / 16
Simultaneous Price
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Alternatively …rm can set price and take as given the price it expects
other …rm to set.
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Bertrand competition - price competition.
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We will get the same result as perfect competitive equilibrium.
1. Firm will not set a price below MC
2. If both set a price above MC, each has incentive to decrease by a bit.
3. So in equilibrium both set price at MC.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
10 / 16
Leadership - Quantity
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One …rm makes quantity choice before the other.
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Stakelberg model.
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Assume …rm 1 is the leader and …rm 2 is the follower.
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Equilibrium price determined by total output q = q1 + q2 according
to inverse demand function p (q ).
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In equilibrium, leader chooses output given expectation on follower’s
output.
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Follower produces the expected amount, so that no one has incentive
to change.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
11 / 16
Leadership - Quantity
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Start with follower’s problem, known as backward induction.
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Given leader’s choice q1 , follower’s problem is
max p (q1 + q2 )q2
q2
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c2 ( q 2 )
FOC is
q2 p 0 (q1 + q2 ) + p (q1 + q2 ) = c20 (q2 )
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This implies follower’s reaction function q2 = f (q1 ).
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Note f 0 < 0
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ECON 212 Lecture 15
Winter 2013
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Leadership - Quantity
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The Leader’s problem can be stated as follows
max p (q1 + f (q1 ))q1
q1
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c1 ( q 1 )
Solve by taking FOC.
q1 p 0 (q1 + q2 )(1 + f 0 (q1 )) + p (q1 + q2 ) = c10 (q1 )
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Output is higer than monopoly, since f 0 < 0 makes …rst term less
negative.
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In other words, competitiion increases output.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
13 / 16
Leadership - Price
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Leader may set a price given his expection on follower’s choice.
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Both will set the same price. Otherwise consumer buys from lower
price.
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Suppose leader set price p, and follower takes this as given, follower’s
problem is
max pq2 c2 (q2 )
q2
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FOC gives
p = c 0 ( q2 )
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which is essentially a supply curve for follower S (p ).
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Leader may calculate the residual demand RD (p ) = D (p )
which has an inverse residual demand curve of prd (q1 ).
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
S (p ),
Winter 2013
14 / 16
Leadership - Price
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The leader’s problem is
max prd (q1 )q1
q1
c 1 ( q1 )
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Output can be found when MR=MC.
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Price decreases and output increases compared with monopoly.
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ECON 212 Lecture 15
Winter 2013
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Cartel
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Firms can collude on production.
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Cartel maximize total industry pro…t and allocate output to each …rm.
max p (q1 + q2 )(q1 + q2 )
q 1 ,q 2
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c1 ( q 1 )
c2 ( q 2 )
FOCs are
p (q1 + q2 ) + (q1 + q2 )p 0 (q1 + q2 ) = c10 (q1 )
p (q1 + q2 ) + (q1 + q2 )p 0 (q1 + q2 ) = c20 (q1 )
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Similar to monoply’s. Equate marginal cost.
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However each member has incentive to cheat.
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Punish cheating by reverting to competitive equilibrium. Game theory.
Tianyi Wang (Queen’s Univeristy)
ECON 212 Lecture 15
Winter 2013
16 / 16