Lecture 12: Search and price dispersionmshum/ec105/matt12.pdf · But still doesn’t explain price...

$: Lecture 12: Search and price dispersionmshum/ec105/matt12.pdf · But still doesn’t explain price dispersion Assume consumers di er in search costs Two types of consumers: \natives"$
Lecture 12: Search and price dispersion

EC 105. Industrial Organization.

Matt ShumHSS, California Institute of Technology

EC 105. Industrial Organization. (Matt Shum HSS, California Institute of Technology)Lecture 12: Search and price dispersion 1 / 26

Information costs

In many markets, consumers or firms may lack important information abouteach other

Two types of information costs:

1 Search costs:

Consumers are not fully aware of firms’ prices in the marketgasoline, furniture

2 Verification costs:Consumers are not fully aware of product quality: how reliable the product is

automobiles, apartments, electronics

Firms are not fully aware of consumers’ characteristics, actions

insurance: are people driving safely, do they have unreported health risks?

We focus on search costs in this lecture.


Why are prices for the same item so different across stores?

mss # Hong & Shum; art. # 2; RAND Journal of Economics vol. 37(2)

HONG AND SHUM / 259

FIGURE 1

RAW HISTOGRAMS OF ONLINE PRICES

homogeneous products, so that only search frictions (arising from consumers’ imperfect informa-tion about stores’ prices) and heterogeneity in search costs in the consumer population generateprice dispersion in this market.

As a motivating example, we consider 20 online prices for the classic mathematical statisticstextbook Probability and Measure (Billingsley, 1992), retrieved by the MySimon (www. mysi-mon.com) and Pricescan (www.pricescan.com) search engines, on February 5, 2002. A histogramof these prices is given in the third panel of Figure 1.5

The long left tail of the histogram suggests that consumers may have an incentive to search,because the potential cost savings can be over $15 (the lowest price, $85.58, is over $15 lessthan the highest price of $100.87). On the other hand, the large spike in the histogram around$100 suggests that despite the low prices, consumers may not be searching very much, becauseotherwise firms would not find it optimal to “pile up” at a relatively high price. These argumentsillustrate that the search models may imply conflicting interpretations of observed prices, if onlythe consumer or firm side is considered in isolation. For this reason, in this article we imposethe optimality conditions for both consumer search and firm pricing in obtaining our estimates ofsearch costs, thus rationalizing the observed prices as equilibrium outcomes for a given theoreticalmodel.

5 These prices include shipping costs.© RAND 2006.

A puzzle considering basic economic theory: review this.Consider the benchmark of perfect competition


Leaving the PC world

One important implicit assumption of PC paradigm is that consumers areaware of prices at all stores. This implies an infinitely elastic demand curvefacing firms. (ie. if one firm raises prices slightly, he will lose all demand).

Obviously, this assumption is not realistic. Here we consider what happens, ifwe relax just this assumption, but maintain other assumptions of PCparadigm: large #firms, perfect substitutes, etc.


Search model

Each consumer demands one unit;

Starts out at one store, incurs cost c > 0 to search at any other store.

Consumer only knows prices at stores that she has been to, and buys fromthe canvassed store with the lowest price. “free recall”

Utility u from purchasing product: demand function is{purchase if p ≤ udon’t purchase otherwise

(1)

What is equilibrium in this market?


Diamond paradox

Claim: a nonzero search cost c > 0 leads to equilibrium price equal to u(“monopoly price”)

Assume that marginal cost=0, so that under PC, p = 0

n firms, with n large. Consumers equally distributed initially among all firms.

Start out with all firms at PC outcome. What happens if one firm deviates,and charges some p1 such that 0 < p1 < c?

Consumers at this store?Consumers at other stores?How will other stores respond?By iterating this reasoning ....


Now start at “monopoly outcome”, where all firms are charging u.

What are consumers’ purchase rules?

Do firms want to undercut? Given consumer behavior, what do they gain?

Role for advertising?

P. Diamond (1971), “A Theory of Price Adjustment”, Journal of EconomicTheory


Remarks

Diamond result quite astounding, since it suggests PC result is “knife-edge”case.

But still doesn’t explain price dispersion

Assume consumers differ in search costs

Two types of consumers: “natives” are perfectly informed about prices, but“tourists” are not.


Tourist-natives model

Tourists and natives, in proportions 1− α and α. L total consumers (so αLnatives, and (1− α)L tourists).

Tourists buy one unit as long as p ≤ u, but natives always shop at thecheapest store.

Each of n identical firms has U-shaped AC curve

Each firm gets equal number of tourists(

(1−α)Ln

); natives always go to

cheapest store.

Consider world in which all firms start by setting pc = minq AC (q).

Note that deviant store always wants to price higher. Demand curve for adeviant firm is kinked (graph). Deviant firm sells exclusively to tourists.


Deviant firm will always charge u. Only tourists shop at this store. If chargeabove u, no demand. If below u, then profits increase by charing u.First case: many informed consumers (α large)

Number qu ≡ (1−α)Ln of tourists at each store so small that u < AC (qu).

In free-entry equilibrium, then, all firms charge pc , and produce the samequantity L/n.

If enough informed consumers, competitive equilibrium can obtain (notsurprising)


Second case: few informed consumers (α small)

Assume enough tourists so that u > AC (qu).

But now: hi-price firms making positive profits, while lo-price firms making(at most) zero profits. Not stable.

In order to have equilibrium: ensure that given a set of high-price firms(charing u) and low-price firms (charging pc), no individual firm wants todeviate. Free entry ensures this.

Let β denote proportion of lo-price firms.

Each high-price firm charges u and sells an amount

qu =(1− α)L(1− β)

n(1− β)=

(1− α)L

n(2)

Each low-price firm charges pc and sells

qc =αL + (1− α)Lβ

nβ(3)


In equilibrium, enough firms of each type enter such that each firm makeszero profits. Define quantities qa, qc such that (graph):

AC (qa) = u; AC (qc) = pc .

(Quantities at which both hi- and lo-price firms make zero profits.)

With free entry, n and β must satisfy

qa =(1− α)L

n; qc =

αL + (1− α)Lβ

nβ(4)

Solving the two equations for n and β yields

n =(1− α)L

qa; β =

αqa

(1− α)(qc − qa)(5)

N.B: arbitrary which firms become high or low price. Doesn’t specify processwhereby price dispersion develops.

As α→ 0, then β → 0 (Diamond result)


Nonsequential vs. sequential search

Two search models:

Consider two search models:

1 Nonsequential search model: consumer commits to searching n storesbefore buying (from lowest-cost store). “Batch” search strategy.

2 Sequential search model: consumer decides after each search whether tobuy at current store, or continue searching.

Question:

from observed prices (as above), can we infer what consumers’ search costs in amarket are?

Consider data for the four textbooks.


Nonsequential search model

Main assumptions:

Infinite number (“continuum”) of firms and consumers

Observed price distribution Fp is equilibrium mixed strategy on the part offirms, with bounds p, p̄.

r : constant per-unit cost (wholesale cost), identical across firms

Firms sell homogeneous products

Each consumer buys one unit of the good

Consumer i incurs cost ci to search one store; drawn independently fromsearch cost distribution Fc

First store is “free”

q̃k : probability that consumer searches k stores before buying



Consumers in nonsequential model

Consumer with search cost c who searches n stores incurs total cost

c ∗ (n − 1) + E [min(p1, . . . , pn)]

=c ∗ (n − 1) +

∫ p̄

p

p · n(1− Fp(p))n−1fp(p)dp.(6)

This is decreasing in c . Search strategies characterized by cutoff-points,where consumer indifferent between n and n + 1 must have cost

cn = E [min(p1, . . . , pn)]− E [min(p1, . . . , pn+1)].

and c1 > c2 > c3 > · · · .Similarly, define q̃n = Fc(cn−1)− Fc(cn) (fraction of consumers searching nstores). Graph.




HONG AND SHUM / 261

FIGURE 2

IDENTIFICATION SCHEME FOR SEARCH-COST DISTRIBUTION INNONSEQUENTIAL-SEARCH MODEL

!4 are the search costs of the indifferent consumers, where a consumer with search costs !k isindifferent between obtaining k and k + 1 price quotes.

Let F̂p denote the empirical distribution of the observed prices. First, we note that we canobtain estimates of these indifference points from the empirical price distribution F̂p, via therelation (2). Second, define

q̃1 ! 1" Fc(!1) : the proportion of consumers with one price quote;q̃2 ! Fc(!1)" Fc(!2) : the proportion of consumers with two price quotes;q̃3 ! Fc(!2)" Fc(!3) : the proportion of consumers with three price quotes. (3)

We can estimate q̃1, q̃2, . . . by exploiting the firms’ equilibrium pricing conditions. To seethis, note that a firm’s profits from following the mixed pricing strategy Fp(·) are (see Burdett andJudd, 1983)

"(p) = (p " r )! !"

k=1q̃kk(1" Fp(p))k"1

#

for all p # [p, p]. The characterization of the equilibrium price distribution starts with the mixed-strategy condition that firms be indifferent between charging the monopoly price p (and sellingonly to people who never search but receive an initial free draw equal to p) and any other pricep in the equilibrium support [p, p]:

(p " r )q̃1 = (p " r )! !"

k=1q̃kk(1" Fp(p))k"1

#

. (4)

The optimality equation (4) allows us to recover a nonparametric estimate of the search-cost distribution Fc from F̂p alone, as we now show. Let p̂ and p̂ denote the lowest and highestobserved prices, respectively. For convenience, we index the n observed prices in ascending order,so that

p̂ = p1 $ p2 $ · · · $ pn"1 $ pn = p̂.

Let K ($ n " 1) denote the maximum number of firms from which a consumer obtains pricequotes in this market. Given this condition, the indifference condition (equation (4)) for each of© RAND 2006.



Firms in nonsequential model

Firm’s profit from charging p is:

Π(p) = (p − r)

[ ∞∑k=1

q̃k · k · (1− Fp(p))k−1

], ∀p ∈ [p, p̄]

Since firms use mixed streategy, they must be indifferent btw all p:

(p̄ − r)q̃1 = (p − r)

[ ∞∑k=1

q̃k · k · (1− Fp(p))k−1

], ∀p ∈ [p, p̄) (7)



Reconsider this datamss # Hong & Shum; art. # 2; RAND Journal of Economics vol. 37(2)

HONG AND SHUM / 259

FIGURE 1

RAW HISTOGRAMS OF ONLINE PRICES

homogeneous products, so that only search frictions (arising from consumers’ imperfect informa-tion about stores’ prices) and heterogeneity in search costs in the consumer population generateprice dispersion in this market.

As a motivating example, we consider 20 online prices for the classic mathematical statisticstextbook Probability and Measure (Billingsley, 1992), retrieved by the MySimon (www. mysi-mon.com) and Pricescan (www.pricescan.com) search engines, on February 5, 2002. A histogramof these prices is given in the third panel of Figure 1.5

The long left tail of the histogram suggests that consumers may have an incentive to search,because the potential cost savings can be over $15 (the lowest price, $85.58, is over $15 lessthan the highest price of $100.87). On the other hand, the large spike in the histogram around$100 suggests that despite the low prices, consumers may not be searching very much, becauseotherwise firms would not find it optimal to “pile up” at a relatively high price. These argumentsillustrate that the search models may imply conflicting interpretations of observed prices, if onlythe consumer or firm side is considered in isolation. For this reason, in this article we imposethe optimality conditions for both consumer search and firm pricing in obtaining our estimates ofsearch costs, thus rationalizing the observed prices as equilibrium outcomes for a given theoreticalmodel.

5 These prices include shipping costs.© RAND 2006.



Estimating search costs

We observe data Pn ≡ (p1, . . . pn). Sorted in increasing order.

Empirical price distribution F̂p = Freq(p ≤ p̃) = 1n

∑i 1(pi ≤ p̃)..

Take p = p1 and p̄ = pn

Consumer cutpoints c1, c2, . . . can be estimated directly by simulating fromobserved prices Pn. This are “absissae” of search cost CDF.

Corresponding “ordinates” recovered from firms’ indifference condition.Assume that consumers search at most K (< N − 1) stores. Then can solvefor q̃1, . . . , q̃K from

(p̄ − r)q̃1 = (pi − r)

[K−1∑k=1

q̃k · k · (1− F̂p(pi ))k−1

], ∀pi , i = 1, . . . , n − 1.

n − 1 equations with K unknowns.



Nonsequential model: results


HONG AND SHUM / 267

TABLE 1 Summary Statistics on Prices for Different Products

StandardProduct n List Mean Deviation Median p p

Stokey-Lucas 19 60.50 66.60 5.64 64.98 59.75 86.80Lazear 17 31.95 34.73 2.48 35.27 29.51 37.70Billingsley 20 99.95 95.48 5.87 98.90 83.58 100.87Duffie 15 65.00 62.71 4.91 63.48 50.58 69.95

Note: Including shipping and handling costs. Price data for all products downloaded from Pricescan.com and MySimon.com: February5, 2002. Summary price including S&H costs may not exceed the corresponding summary price without S&H costs, since we could notdetermine the shipping and handling charges from some of the websites.

the standard deviation of the prices with shipping costs was roughly the same magnitude as theprices without shipping costs. Moreover, the Spearman rank correlation statistics between the twosets of prices was around .90, indicating strong correlation. This evidence might cast some doubton the possibility that online retailers are engaging in bait-and-switch strategies with regard toshipping costs.

Table 2 contains estimates for the nonsequential-search model. The estimates of the q̃’sindicate that in most markets, about half of the consumers never search (more precisely, theyshop at the store where they received their initial “free” price). For the Stokey-Lucas text, 52%(=100 ! (1 " .480)) of purchasers have search costs exceeding !1 = $2.32, while over 60% ofthe Lazear book purchasers have search costs exceeding !1 = $1.31. As was the case with theexample considered above, the substantial proportion of people who don’t search implies thatwe cannot identify the shape of the distribution for these people. For example, we know nothingabout the shape of the search-cost distribution above the 52nd quantile for the Stokey-Lucas book,and above the 65th quantile for the Lazear book.

Table 3 presents the maximum likelihood estimates for the sequential-search model. Com-paring these results to those obtained from the nonsequential-search models, we see that thesequential-search model predicts higher magnitudes for search costs: as an example, the median

TABLE 2 Search-Cost Distribution Estimates for Nonsequential-Search Model

Selling MELProduct K a Mb q̃c1 q̃2 q̃3 Cost r Value

Parameter estimates and standard errors: nonsequential-search modelStokey-Lucas 3 5 .480 (.170) .288 (.433) 49.52 (12.45) 102.62Lazear 4 5 .364 (.926) .351 (.660) .135 (.692) 27.76 (8.50) 84.70Billingsley 3 5 .633 (.944) .309 (.310) 69.73 (68.12) 199.70Duffie 3 5 .627 (1.248) .314 (.195) 35.48 (96.30) 109.13

Search-cost distribution estimates!1 Fc(!1) !2 Fc(!2) !3 Fc(!3)

Stokey-Lucas 2.32 .520 .68 .232Lazear 1.31 .636 .83 .285 .57 .150Billingsley 2.90 .367 2.00 .058Duffie 2.41 .373 1.42 .059

a Number of quantiles of search cost Fc that are estimated (see equation (5)). In practice, we set K and M to the largest possiblevalues for which the parameter estimates converge. All combinations of larger K and/or larger M resulted in estimates that either did notconverge or did not move from their starting values (suggesting that the parameters were badly identified).

b Number of moment conditions used in the empirical likelihood estimation procedure (see equation (17)).c For each product, only estimates for q̃1, . . . , q̃K!1 are reported; q̃K = 1!

!K!1k=1

q̃k .d Indifferent points !k computed as Ep(1:k) ! Ep(1:k+1) (the expected price difference from having k versus k + 1 price quotes),

using the empirical price distribution. Including shipping and handling charges.

© RAND 2006.


Sequential search model

Sequential model

Consumer decides after each search whether to accept lowest price to date,or continue searching.

Optimal “reservation price” policy: accept first price which falls below someoptimally chosen reservation price.

NB: “no recall”



Consumers in sequential model

Heterogeneity in search costs leads to heterogeneity in reservation prices

For consumer with search cost ci , let z∗(ci ) denote price z which satisfies thefollowing indifference condition

ci =

∫ z

0

(z − p)f (p)dp =

∫ z

0

F (p)dp.

Now, for consumer i , her reservation price is:

p∗i = min(z∗(ci ), p̄).

Let G denote CDF of reservation prices, ie. G (p̃) = P(p∗ ≤ p̃).



Firms in sequential search model

Again, firms will be indifferent between all prices

Let D(p) denote the demand (number of people buying) from a storecharging price p. Indifference condition is:

(p̄ − r)D(p̄) = (p − r)D(p)⇔(p̄ − r) ∗ (1− G (p̄)) = (p − r) ∗ (1− G (p))

for each p ∈ [p, p̄).



Estimation: sequential model

Observe prices p1, . . . , pn (order in increasing order, so p̄ = pn).

Indifference conditions, evaluated at each price, are:

(p̄ − r) ∗ (1− G (p̄)) = (pi − r) ∗ (1− G (pi )), i = 1, . . . , n − 1

This gives n − 1 equations, but n + 1 unknowns: G (pi ) for i = 1, . . . , n aswell as r .

Define α = 1− G (p̄): percentage of people who don’t search.

Assume that search costs are distributed according to Gamma distribution:

f (f ) =1

δδ12 Γ(δ1)

cδ1−1 exp(−c/δ2), δ1, δ2 > 0.

Estimate model parameters (δ1, δ2, α, r) by maximum likelihood (Eq. 9)



Results: sequential search modelmss # Hong & Shum; art. # 2; RAND Journal of Economics vol. 37(2)

268 / THE RAND JOURNAL OF ECONOMICS

TABLE 3 Estimates of Sequential-Search Model

Mediana Selling Log-LProduct !1 !2 Search Cost Cost r "b F!1

c (1! "; ! ) Value

Stokey-Lucas .46 (.02) 1.55 (.03) 29.40 (1.45) 22.90 (1.31) .58 19.19 31.13Lazear .40 (.01) 1.15 (.01) 16.37 (1.00) 11.31 (.79) .69 4.56 34.35Billingsley .25 (.01) 2.01 (.04) 9.22 (.94) 65.37 (.83) .51 8.43 23.73Duffie .21 (.02) 4.57 (.29) 10.57 (2.01) 28.24 (1.63) .54 7.00 18.93

Note: Including shipping and handling charges. Standard errors in parentheses. !1 and !2 are parameters of the gamma distribution; seeequation (13).

a As implied by estimates of the parameters of the gamma search-cost distribution.b Proportion of consumers with reservation price equal to p, implied by estimate of r (see equation (11)).

search cost for the Stokey-Lucas text consistent with the nonsequential-search model (as reportedin the bottom panel of Table 2) is roughly $2.32, whereas the corresponding number for thesequential-search model (as reported in Table 3) is $29.40, over 10 times higher. Moreover, theestimates of r , the selling costs, are uniformly lower across the four books than the correspondingestimates in the nonsequential-search model.19

At first glance, the larger search-cost magnitudes for the sequential-search model mightlead one to support the nonsequential-search model as a better descriptor of search behaviorin the markets we consider. As pointed out by Morgan and Manning (1985), nonsequential-search strategies may be optimal when consumers face nonzero fixed costs of initiating a search,regardless of howmany prices one obtains. Such a situation may describe the online market, sincethere may be nonzero search costs associated with, say, seeking out a computer, logging on, andso forth. Moreover, due to search engines, price quotes might tend to be obtained in groups, whichmay fit better with the nonsequential-search assumption.

However, as we remarked earlier, the large median search-cost estimates for the sequentialmodel are due in part to parametric extrapolation. In the fourth column of Table 3, we report theimplied estimates of ", the proportion of consumers with reservation price equal to p and, hence,the proportion of consumers who never search.20 " exceeds .50 for all four products, suggestingthat the high median search costs estimated in this specification are due in part to extrapolationbased on the gamma functional form for the search-cost distribution. To see if this is true, we alsocomputed, in the fifth column, implied estimates of F!1

c (1 ! "), which are the search costs forthe consumer who has a reservation price just equal to p. We see that for three out of the fourcases (excepting the Lucas-Stokey book), the value of F!1

c (1 ! ") is comparable in magnitudeto the search costs from the nonsequential model (and reported in the bottom panel of Table 2).21

! Specification checks for the sequential-searchmodel. On the other hand, we might worrythat the sequential-search model may be biased toward large search-cost estimates, due to thenecessary condition (remarked above) that the slope of the search-cost density fc(· · ·) be negativein order for the price density (9) to constitute a valid equilibrium price density. For the gammadensity function in (13), we found that when !1 " 1 (which was required for the density to bestrictly decreasing), the hazard rate parameter !2 must be increased so that the tail of the densitydoes not die off too quickly, which we need in order to fit the price data (which do not have thin

19 As remarked above, we confirmed that at the reported estimates, the implied search-cost distribution had a strictlydecreasing density. This was true for all four books. Furthermore, to verify the robustness of these estimates, we alsoreestimated the sequential-search models using a variety of starting values, but the reported results were very robust.

20 " in the sequential model corresponds to q̃1 in the nonsequential model.21 For the Lucas-Stokey text, the high search costs implied by the sequential-search model appear to be driven

by the outlier price of $86.80 (more than $20 above the list price) charged by opengroup.com. We have confirmed thatthis price was not a temporary oversight: on February 7, 2004, the price on this site was still $83.70 (without shippingcosts). However, we note again that our modelling approach assumes that each price is “real,” in the sense that it generatespositive sales in equilibrium. This assumption may not apply to this website.© RAND 2006.

Generally, nonsequential (batch) search model generates smaller (morereasonable) search cost estimates

Batch search is feature of online search engines?



Additional empirical evidence

Online book markets (redux)

Do consumer recall? (Return to earlier searched stores)

Gasoline markets


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