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Consumer Search Prof. dr. Maarten Janssen University of Vienna 1 Types of Consumer Search Common: consumers have to invest time and resources to get information about price and/or product Sequential Simultaneous (fixed sample) 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 2 Search makes a difference Consider Bertrand model Each consumer has downward sloping demand Add (very) small search cost ε > 0 What difference does ε 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) 3 Endogenous Sequential Search 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? 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 4 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– (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) 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) 5 Optimal search rule II So, in last period, consumer buys iff price is at or below ρ (if it is smaller than v) Consider penultimate period Stationary process: optimal search is characterized by reservation price ρ: buy iff p ≤ min (ρ, v) 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. 6 Characterization of F(p) and ρ when ρ ≤ v Write down profit function for p < ρ ≤ v 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] } Can be larger than v if s is large enough. 7 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 - ρ. 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 8 Comparative statics What is impact of increase in s on Ep? 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? In partial participation equilibrium: none In full participation equilibrium: increasing When N increases transition from full to partial participation equilibrium 9 Conclusions With consumer search, prices above marginal cost Consumer search can explain price dispersion in homogeneous goods markets When s becomes small, convergence to Bertrand model Involves calculation of mixed strategy distributions Mathematical complications Interesting comparative statics 10