Transcript No Slide Title

Consumer Search
Prof. dr. Maarten Janssen
University of Vienna
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Types of Consumer Search
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Common: consumers have to invest time and
resources to get information about price and/or
product
Sequential
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Simultaneous (fixed sample)
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After each search and information, consumer decides
whether or not to continue searching
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
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Consider Bertrand model
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Each consumer has downward sloping demand
Add (very) small search cost ε > 0
What difference does ε make?
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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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Endogenous Sequential Search
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Two types of consumers: fraction λ fully informed, fraction 1-λ bears
search cost s for 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?
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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
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Optimal search rule I
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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– (1–F(p’))p’ - F(p’)E(p│p < p’)
 Price ρ that makes consumer indifferent between two options is ρ =
E(p│p < ρ) + s/F(ρ)
 Claim: largest price in support of F(p) cannot be above min (ρ, v)
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Suppose it were, consumers will continue to search; will find lower price
with probability 1
Thus, F(ρ) =1 and ρ = s + E(p)
In last period, consumer buys iff price is at or below min (ρ, v)
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Optimal search rule II
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So, in last period, consumer buys iff price is at or below
ρ (if it is smaller than v)
Consider penultimate period
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Stationary process: optimal search is characterized by
reservation price ρ: buy iff p ≤ min (ρ, v)
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Buying yields v – p’
Continue searching yields v – s– Ep (given that all firms charge
below ρ
Price ρ that makes consumer indifferent between two options is
ρ = s+ E(p)
Due to perfect recall
This reservation price is equal to maximum price in
support of F(p)
Similar for case where ρ > v.
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Characterization of F(p) and ρ when ρ ≤ v
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Write down profit function for p < ρ ≤ v
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Π(p) = { λ(1-F(p))N-1 + (1- λ)/N } p
Π(ρ) = (1- λ)ρ/N
F(p) = 1 – [ (1- λ)(ρ-p)/λNp ]1/(N-1)
E(p) = ∫ pf(p) dp = ∫ p dy (by using the change
of variables y = 1 - F(p))
Ep = ρ ∫ dy/[1+bNyN-1], where b = λ / (1- λ)
Reservation price ρ = s/ {1 - ∫ dy/[1+bNyN-1] }
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Can be larger than v if s is large enough.
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First Search (and last Search)
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When do consumers want to start searching?
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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 - ρ.
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Thus, if ρ ≤ 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
(ρ, v), consumers buy immediately
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Comparative statics
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What is impact of increase in s on Ep?
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For small c, ρ ≤ v and Ep increases in s
When sis close to 0, then model close to Bertrand
competition and Ep is almost 0
For larger s, ρ > v and Ep decreases in s (as v - Ep – s= 0)
Non-monotonic
What is impact of increase in N on Ep?
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In partial participation equilibrium: none
In full participation equilibrium: increasing
When N increases transition from full to partial participation
equilibrium
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Conclusions
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With consumer search, prices above
marginal cost
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Consumer search can explain price
dispersion in homogeneous goods markets
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When s becomes small, convergence to Bertrand
model
Involves calculation of mixed strategy distributions
Mathematical complications
Interesting comparative statics
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