Transcript No Slide Title

Industrial Organization
Capacity constraint pricing
Univ. Prof. dr. Maarten Janssen
University of Vienna
Summer semester 2013 – Week 15
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Bertrand competition with capacity
constraints
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Same set-up as under homogeneous Bertrand
competition, only difference is that firm i faces
cannot sell more than Ki
Why would this make an important difference?
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Undercutting argument presumes that you can enlarge
your sales (then it can be profitable)
With capacity constraints, highest price firm may also sell
positive amount
Notion of residual demand
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Residual demand
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How to construct it. Suppose p2 > p1. All
consumers like to buy from firm 1.
If D(p1) > K1, firm 2 gets residual demand:
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Proportional rationing: Every consumer is served
a little: K1/D(p1), D2 = D(p2) [1 - K1/D(p1)].
Efficient rationing: (inverse) Residual demand for
firm 2 is given by D2 = D(p2) - K1
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Efficient Rationing
1. Also called parallel
rationing
p
D2(p2;p1)
2. Efficient, because
it would be same
outcome if
consumers would
p1
D(p)
D
K2
K1
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Proportional rationing
1. Also called
randomized rationing
p
D2(p2;p1)
2. Not efficient as some
high demand cosumers
are not or only
partially served
p1
D(p)
D
K2
K1
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Analysis under efficient rationing
linear demand, Small capacities
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Linear demand: p = a- bQ
Small capacity: Ki < a/3b
No production cost
Claim: both firms set price p*i = a – b(K1 +
K2), which is the maximal price they can get
when selling their capacity
Cournot behavior!
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Proof of Claim
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Suppose other firm charges p*
It is clear that it does not make sense to
undercut
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You sell your full capacity and can’t serve more
consumers
Profit of firm i when pi > p*:
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Πi(pi,p*) = pi(D(pi) – Kj) = pi(a - pi – bKj)/b
Maximize this wrt pi yields (a - 2pi – bKj)/b <
- (a - 2 bKi – bKj)/b < 0 (because of small capacities
Thus, for pi > p* firms want to set price as small as
possible
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Analysis under efficient rationing
linear demand, Large symmetric capacities I
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Numeric example, Linear demand: p = 10- Q
No production cost, 5 < K1 = K2 < 10
Claim 1: there is no sym. equilibrium in pure
strategies
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At any price p>0, individual firms have incentive to
undercut as they do not sell up to capacity
But if p=0, then individual firms have incentive to
increase price and make profit
Claim 2: there is no asym. equilibrium in pure
strategies
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Analysis under efficient rationing
linear demand, Large symmetric capacities II
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Claim 3: there is an equilibrium in mixed strategies,
F(p)
How to construct F(p)?
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Trick: in a mixed strategy firm should be indifferent
between any of the pure strategies given that competitor
chooses mixed strategy
Πi(pi,F(p)) = pi { (1-F(pi))K + F(pi) (10-K- pi) }
Compact support: [p , p”]
At p” profit equals (10-K-p”)p”
Deviating to higher price should not be optimal: p” is
monopoly price for residual demand: (10-K)/2
Solve for mixed strategy distribution
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Cournot versus Bertrand I
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Predictions from Cournot and Bertrand
homogeneous product oligopoly models are
strikingly different. Which model of competition
is “correct”?
Kreps and Scheinkman model two stages
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firms invest in capacity installation
then choose prices.
Solution: firms invest exactly the Cournot
equilibrium quantities. In the second stage they
price to sell up to capacity.
We discussed this implicitly when discussing
capacity constraint Bertrand competition
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Cournot versus Bertrand II
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Cournot model is more appropriate in
environments where firms are capacity
constrained and investments in capacity are
slow.
Bertrand model is more appropriate in
situations where there are constant returns to
scale and firms are not capacity constrained
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