Industrial Organization or Imperfect Competition Capacity constraint pricing and Consumer Search sta

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Industrial Organization or Imperfect
Competition Capacity constraint pricing
and Consumer Search (start)

• Univ. Prof. dr. Maarten Janssen
• University of Vienna
• Summer semester 2009
• Week 14 (March 30, 31)

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Bertrand competition with capacity constraints

• 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?
• 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

• How to construct it. Suppose p2 gt p1. All
  consumers like to buy from firm 1.
• If D(p1) gt K1, firm 2 gets residual demand
• 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

• Also called parallel rationing
• Efficient, because it would be same outcome if
  consumers would

p 1
K2
K1
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Proportional rationing

• Also called randomized rationing
• Not efficient as some high demand cosumers are
  not or only partially served

p 1
K2
K1
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Analysis under efficient rationing linear
demand, Small capacities

• Linear demand p a- bQ
• Small capacity Ki lt a/3b
• No production cost
• Claim both firms set price pi 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

• Suppose other firm charges p
• It is clear that it does not make sense to
  undercut
• You sell your full capacity and cant serve more
  consumers
• Profit of firm i when pi gt p
• ?i(pi,p) pi(D(pi) Kj) pi(a - pi bKj)/b
• Maximize this wrt pi yields (a - 2pi bKj)/b lt
  - (a - 2i bKj bKj)/b lt 0 (because of small
  capacities
• Thus, for pi gt p firms want to set price as
  small as possible

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Analysis under efficient rationing linear
demand, Large symmetric capacities I

• Numeric example, Linear demand p 10- Q
• No production cost, K1 K2 gt 5
• Claim 1 there is no sym. equilibrium in pure
  strategies
• At any price pgt0, individual firms have incentive
  to undercut as they do not sell up to capacity
• But if p0, 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

• Claim 3 there is an equilibrium in mixed
  strategies, F(p)
• How to construct F(p)?
• Trick in a mixed strategy firm should be
  indifferent between any of the pure strategies
  given that competitor chooses mixed strategy
• ?i(pi,F(p)) pi (1-F(p))K F(p) (10-K-p)
• 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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Types of Consumer Search

• Common consumers have to invest time and
  resources to get information about price and/or
  product
• Sequential
• After each search and information, consumer
  decides whether or not to continue searching
• Simultaneous (fixed sample)
• Have to decide once how many searches you make
  before getting results of any individual search
• Sequential optimal of you get feedback quickly
  otherwise simultaneous search optimal

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Search makes a difference

• Consider Bertrand model
• Each consumer has downward sloping demand
• Add (very) small search cost e gt 0
• What difference does e make?
• All firms charging the (same) monopoly price is
  an equilibrium
• How many times do consumers want to search? (Doi
  they want to deviate?)
• Is firms pricing optimal given strategies others
  (including search strategy consumers)?
• Diamond result! (Diamond 1971)

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Going back in Time

• Stigler (1961) suggested that even for
  homogeneous products, markets seem to be
  characterised by price dispersion
• Suggested this may be due to search costs
• Some firms aim to get many consumers at low
  price, others go for the tourists
• Consumers are also different some search a lot,
  others not at all.

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Varians model of sales (1981) Solution to
Diamond paradox I

• Two types of consumers
• Shoppers compare all prices (fraction ?) and buy
  at shop with lowest price
• Loyal consumers go to only one shop suppose
  every shop has equal number of loyals (fraction
  1-?)
• All have same willingness to pay v
• Firms simultaneously set prices to max profits
• No production cost
• Firms are only strategic decision-makers
• What is an equilibrium?
• Set of prices or price distributions such that no
  firm individually benefits by deviating (Nash)

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Varians model of sales (1981) Solution to
Diamond paradox II

• No equilibrium in pure strategies
• Due to the presence of shoppers
• How to derive sym. equilibrium in mixed
  strategies F(p)?
• No atoms in distribution
• Write down profit equation of individual firm
  given that all other firms charge F(p)
• ?(p) ?(1-F(p))N-1 (1- ?)/N p
• No wholes in the distribution otherwise there
  are prices p1 lt p2 with F(p1) F(p2) implying
  p(p1) ? p(p2)
• If p is max price charged (F(p) 1), then p
  v
• F(p) solves p(p) p(v) (1- ?)v/N