Industrial Organization or Imperfect Competition Consumer Search II

1
Industrial Organization or Imperfect
Competition Consumer Search II

• Univ. Prof. dr. Maarten Janssen
• University of Vienna
• Summer semester 2008
• Week 7 (April 28, 29)

2
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?

3
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

4
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

5
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

6
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

7
Endogenous Fixed sample Search Ruling out
equilibria

• If consumers search one time, their exp pay-off
  is v - E(p) s
• If they dont search exp pay-off is 0
• If they search k times v- E(minp1,.., pk) ks
• 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) s v- E(minp1,.., pk) ks.
  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.

8
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) s v- E(minp1,p2) 2s
• 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

9
Endogenous Sequential Search

• Two types of consumers fraction ? fully
  informed, fraction 1-? bears search cost sfor
  each additional search Max. willingness to pay v
  for both groups
• After each search, consumers can decide whether
  or not to continue searching
• Perfect recall of prices
• How to decide whether to start searching?
• First search is for free or not
• N Firms choose prices as before
• Symmetric Nash equilibrium where
• Static game, despite sequential search
• Consumer search behaviour is optimal given
  strategy of firms
• Firm pricing behaviour is optimal given strategy
  of other firms and consumers

10
Optimal search rule I

• Suppose F(p) is firms pricing strategy and p is
  lowest price consumers have observed so far.
• Buy now yields v-p
• Continue searching yields ??? (at least v Ep
  s) but take into account optimal behaviour after
  search
• Start at possible end when consumer has observed
  N-1 prices.
• Continue search v s (1F(p))p - F(p)E(pp lt
  p)
• Price ? that makes consumer indifferent between
  two options is ? s F(?)E(pp lt ?)
• Claim largest price in support of F(p) cannot be
  above min (?, v)
• Suppose it were, consumers will continue to
  search will find lower price with probability 1
• Thus, F(?) 1 and ? s E(p)
• In last period, consumer buys iff price is at or
  below min (?, v)

11
Optimal search rule II

• So, in last period, consumer buys iff price is at
  or below ?
• Consider penultimate period
• Buying yields v p
• Continue searching yields v s Ep (given that
  all firms charge below ?
• Price ? that makes consumer indifferent between
  two options is ? s E(p)
• Stationary process optimal search is
  characterized by reservation price ? buy iff p
  min (?, v)
• Due to perfect recall
• This reservation price is equal to maximum price
  in support of F(p)

12
Characterization of F(p) and ? when ? v

• Write down profit function for p lt ? v
• ?(p) ?(1-F(p))N-1 (1- ?)/N p
• ?(?) (1- ?)?/N
• F(p) 1 (1- ?)(?-p)/?Np 1/(N-1)
• E(p) ? pf(p) dp ? p dy (by using the change
  of variables y 1 - F(p))
• Ep ? ? dy/1bNyN-1, where b ? / (1- ?)
• Reservation price ? s/ 1 - ? dy/1bNyN-1
• Can be larger than v if s is large enough.

13
First Search (and last Search)

• When do consumers want to start searching?
• When first search is for free (Stahl 1989),
  dominant strategy to search at least once.
• When first search costs s, pay-off of first
  search is v Ep s v - ?.
• Thus, if ? v, uninformed consumers want to
  search
• Otherwise, they prefer not to search, but this
  cannot be an equilibrium (as with only active
  informed consumers prices would be equal to 0)
• In both cases, as no firm charges above min (?,
  v), consumers buy immediately

14
Characterization of F(p) and ? when ? gt v

• When first price quotation is for free, write
  down profit function for p v
• ?(p) ?(1-F(p))N-1 (1- ?)/N p
• ?(v) (1- ?)v/N
• F(p) 1 (1- ?)(v-p)/?Np 1/(N-1)
• When first price quotation is not for free, only
  part µ of uninformed consumers are active
• ?(p) ?(1-F(p))N-1 µ(1- ?)/N p
• ?(v) µ(1- ?)v/N
• F(p) 1 µ(1- ?)(v-p)/?Np 1/(N-1)
• Ep v ? dy/1bNyN-1/µ and µ such that vEps
  0

15
When which equilibrium?

• When sis relatively large
• Partial consumer participation equilibrium
• When many fully informed consumers (? large)
• Full consumer participation equilibrium

16
Comparative statics

• What is impact of increase in son Ep?
• For small c, ? v and Ep increases in c
• When sis close to 0, then model close to Bertrand
  competition and Ep is almost 0
• For larger s, ? gt v and Ep decreases in s(as v -
  Ep s 0)
• Non-monotonic
• What is impact of increase in N on Ep?
• In partial participation equilibrium none
• In full participation equilibrium increasing
• When N increases transition from full to partial
  participation equilibrium

17
Conclusions

• With consumer search, prices above marginal cost
• When s becomes small, convergence to Bertrand
  model
• Consumer search can explain price dispersion in
  homogeneous goods markets
• Involves calculation of mixed strategy
  distributions
• Mathematical complications
• Interesting comparative statics