Nonsequential searchSequential search modelSearch markets:Hong-Shum paperCaltechEc106(Caltech) Search Feb 2010 1 / 13Why are prices for the same item so different across stores?Can search models explain price dispersion?Modus operandi: estimate search costs which are consistent with observedprice distributions, and see if they are reasonable.Concept: mixed strategy.(Caltech) Search Feb 2010 2 / 13Two search models:Consider two search models:Nonsequential search model: consumer commits to searching n storesbefore buying (from lowest-cost store). “Batch” search strategy.Sequential search model: consumer decides after each search whether tobuy at current store, or continue searching.(Caltech) Search Feb 2010 3 / 13Nonsequential search modelMain assumptions:Infinite number (“continuum”) of firms and consumersObserved price distribution Fpis equilibrium mixed strategy on thepart of firms, with bounds p, ¯p.r: constant per-unit cost (wholesale cost), identical across firmsFirms sell homogeneous productsEach consumer buys one unit of the goodConsumer i incurs cost cito search one store; drawn independentlyfrom search cost distribution FcFirst store is “free”qk: probability that consumer searches k stores before buying(Caltech) Search Feb 2010 4 / 13Nonsequential search modelConsumers in nonsequential modelConsumer with search cost c who searches n stores incurs total costc ∗ (n − 1) + E [min(p1, . . . , pn)]=c ∗ (n − 1) +Z¯ppp · n(1 − Fp(p))n−1fp(p)dp.(1)This is decreasing in c. Search strategies characterized bycutoff-points, where consumer indifferent between n and n + 1 musthave costcn= E [min(p1, . . . , pn)] − E [min(p1, . . . , pn+1)].and c1> c2> c3> · · · .Similarly, define ˜qn= Fc(cn−1) − Fc(cn) (fraction of consumerssearching n stores). Graph.(Caltech) Search Feb 2010 5 / 13Nonsequential search modelFirms in nonsequential modelFirm’s profit from charging p is:Π(p) = (p − r )"∞Xk=1˜qk· k · (1 − Fp(p))k−1#, ∀p ∈ [p, ¯p]For mixed strategy, firms must be indifferent btw all p:(¯p − r )˜q1= (p − r )"∞Xk=1˜qk· k · (1 − Fp(p))k−1#, ∀p ∈ [p, ¯p) (2)(Caltech) Search Feb 2010 6 / 13Nonsequential search modelEstimating search costsObserve data Pn≡ (p1, . . . pn). Sorted in increasing order.Empirical price distributionˆFp= Freq(p ≤ ˜p) =1nPi1(pi≤ ˜p)..Take p = p1and ¯p = pnConsumer cutpoints c1, c2, . . . can be estimated directly by simulatingfrom observed prices Pn. This are “absissae” of search cost CDF.Corresponding “ordinates” recovered from firms’ indifferencecondition. Assume that consumers search at most K (< N − 1)stores. Then can solve for ˜q1, . . . , ˜qKfrom(¯p−r)˜q1= (pi−r)"K −1Xk=1˜qk· k · (1 −ˆFp(pi))k−1#, ∀pi, i = 1, . . . , n−1.n − 1 equations with K unknowns.(Caltech) Search Feb 2010 7 / 13Nonsequential search modelNonsequential model: resultsFigure 3Table 1Table 2(Caltech) Search Feb 2010 8 / 13Sequential search modelSequential modelConsumer decides after each search whether to accept lowest price todate, or continue searching.Optimal “reservation price” policy: accept first price which falls belowsome optimally chosen reservation price.NB: “no recall”(Caltech) Search Feb 2010 9 / 13Sequential search modelConsumers in sequential modelHeterogeneity in search costs leads to heterogeneity in reservationpricesFor consumer with search cost ci, let z∗(ci) denote price z whichsatisfies the following indifference conditionci=Zz0(z − p)f (p)dp =Zz0F (p)dp.Now, for consumer i, her reservation price is:p∗i= min(z∗(ci), ¯p).Let G denote CDF of rese rvation prices, ie. G (˜p) = P(p∗≤ ˜p).(Caltech) Search Feb 2010 10 / 13Sequential search modelFirms in sequential search modelAgain, firms will be indifferent between all pricesLet D(p) denote the demand (number of people buying) from a storecharging price p. Indifference condition is :(¯p − r )D(¯p) = (p − r )D(p) ⇔(¯p − r ) ∗ (1 − G (¯p)) = (p − r ) ∗ (1 − G (p))for each p ∈ [p, ¯p).(Caltech) Search Feb 2010 11 / 13Sequential search modelEstimation: sequential modelObserve prices p1, . . . , pn(order in increasing order, so ¯p = pn).Indifference conditions, evaluated at each price, are:(¯p − r ) ∗ (1 − G (¯p)) = (pi− r) ∗ (1 − G (pi)), i = 1, . . . , n − 1This gives n − 1 equations, but n + 1 unknowns: G (pi) fori = 1, . . . , n as well as r .Define α = 1 − G (¯p): percentage of people who don’t search.Assume that search distribution is Gamma distribution. (Eq. (13) inpaper).Estimate model parameters (δ1, δ2, α, r) by maximum likelihood (Eq.9)(Caltech) Search Feb 2010 12 / 13Sequential search modelResults: sequential search modelFigure 3Table 2Table 3(Caltech) Search Feb 2010 13 /