Industrial Organization or Imperfect Competition Consumer Search

1
Industrial Organization or Imperfect
Competition Consumer Search

• Univ. Prof. dr. Maarten Janssen
• University of Vienna
• Summer semester 2008
• Week 6 (April 22)

2
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

3
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)
• Any price above monopoly price can be sustained
  as a pure strategy equilibrium!

4
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.

5
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)

6
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

7
First, search is exogenous I JR 2001

• Consumer wants to have house painted
• May ask one or several (N) firms to do job
• Each firm is asked (active) with prob a
• An active firm thinks that with probability
  (1-a)N-1 it is a monopolist
• Firms have cost c
• Consumer has willingness to pay v (suppose this
  is known).
• Which price will a firm charge?

8
First, search is exogenous II JR 2001

• No price equilibrium in pure strategies
• How to construct equilibrium in mixed strategies?
• F(p) cum. symmetric equilibrium distribution
  function
• Write down expected profits ind. firm given F(p)
• Equate with certain profits of charging upper
  bound what is upper bound?
• Derive mixed strategy distribution
• Mixed strategies interpreted as price dispersion
  identical goods are sold at different prices

9
Comparative statics wrt N

• Individual profits declining in N
• As ind. profits are a(1-a)N-1 v
• Industry profits Na(1-a)N-1 v are also declining
  in N
• By choosing a can be made to mimic empirical
  relation of industry profits to number of firms

10
Endogenous Fixed Sample Search BJ 1983, JM
2004

• Consumers can decide how many firms to search 0,
  1, 2,
• each search has cost s
• willingness to pay v
• N Firms choose prices as before
• Symmetric Nash equilibrium where
• Consumer search behaviour is optimal given
  strategy of firms
• Firm pricing behaviour is optimal given strategy
  of other firms and consumers

11
Endogenous Fixed sample Search Ruling out
equilibria

• Can there be a sym. equilibrium in which firms
  charge pure strategies?
• No. Suppose they did. Consumers will only search
  once. But if they do, firms have incentive to set
  pv and then consumers wont search
• Due to unit demand assumption cf., Diamond
  result
• Can there be a sym. equilibrium where all
  consumers search at least two times?
• No. Suppose they did. No firms would like to
  charge highest price in F(p). All price equal
  marginal cost, but then consumers would like to
  search only once
• Same argument if consumers choose either at least
  two searches or not to search at all
• Mixed strategy eq where some consumers search
  only once

12
Endogenous Fixed sample Search Ruling out
equilibria

• If consumers search one time, their exp pay-off
  is v - E(p) c
• If they dont search exp pay-off is 0
• If they search k times v- E(minp1,.., pk) kc
• Consumers cant randomize between one search and
  no search at all (firms would price at v)
• Consumers cant randomize between 1 and 3 or more
  firms v - E(p) c v- E(minp1,.., pk) kc.
  But E(minp1,.., pk) is decreasing in k and at a
  decreasing rate. Searching more than once and
  less than three times would be better
• Consumers have to randomize between 1 and 2
  times.

13
Endogenous Fixed sample Search mixed strategy
equilibria

• If consumers search once or twice (with prod. a,
  resp 1- a
• ?(p) 2(1- a)(1-F(p))/N a /N p
• ?(v) av /N
• F(p) 1- a(v-p) / 2(1- a)p
• E(p) ? pdF(p) (av / 2(1- a)) ln (2- a)/a
• How to determine a?
• v - E(p) c v- E(minp1,p2) 2c
• E(minp1,p2) ? pdF(minp1,p2) 2 ?
  p(1-F(p))f(p) dp
• No explicit solution for a (only implicitly
  defined, or numerically)
• Equilibrium solution independent of N