Search and Price Dispersion - Berkeley-Haasfaculty.haas.berkeley.edu/rjmorgan/phdba279b/Search Presentation... · Search and Price Dispersion Sibo Lu and Yuqian Wang Haas Sibo Lu

Search and Price Dispersion

Sibo Lu and Yuqian Wang

Haas

Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 1 / 56

Outline

1 Introduction

2 No Clearing HouseBasic SetupThe Stigler ModelRothschild Critique and Diamond’s ParadoxSequential Search - Reinganum ModelMacMinn ModelBurdett and Judd Model

3 Clearing HouseBasic SetupRosenthal ModelVarian ModelBaye and Morgan

4 Asymmetric Consumers

5 Bounded Rationality / Unobserved Frictions

6 Conclusion


Introduction

Questions

Simple textbook models price of homogeneous product in competitivemarkets should be same

However, empirical studies reveal that price dispersion is the rule(Varian, 1980, p. 651)

Why? Cost of acquiring information about firms/transmittinginformation to consumers

Search cost and other problem


Introduction

Models and Approaches

Search Theoretical Model/Marginal Search Cost

Information Clearinghouse

Others, e.g. limited rationality, asymmetric consumers


Introduction

Motivating Examples

Online Shopping

Sequential search: Nike, then Reebox, then Addidas...Clearinghouse: Zappos, Amazon

Labor Market

Sequential search: worker looking for jobs over timeFixed sample search: PhD interview dayClearinghouse: LinkedIn, Monster, SimplyHired


No Clearing House Basic Setup

Assumptions and variables

A continuum of price-setting firms compete in homogeneous productmarket

Unlimited capacity to supply, marginal cost m

Mass of consumers be µ, indirect utility V (p,M) = v(p) + M

Roy’s identity, we have q(p) = −v ′(p)

Consumer’s (indirect) utility V = v(p) + M − cn where c is serachcost per price quote if obtaining n price quotes


No Clearing House The Stigler Model

Assumptions and Setups

The Stigler Model

For each consumer, K = q(p) = −v ′(p)

Fixed sample search, size n which is pre-determined

Observed exogenous distribution of price, cdf F (p) on [p, p̄]



Calculation

Consumer Minimize the expected total cost

E [C ] = KE [p(n)min] + cn, since F

(n)min = 1− [1− F (p)]n

We have

E [C ] = K

∫ p̄

ppdF

(n)min(p) + cn

= K

[p +

∫ p̄

p[1− F (p)]ndp

]+ cn

Consumer choose optimal n∗ to minimize E [C ]

So the distribution of transaction price should be F(n∗)min (p)



Calculation

Marginal benefit of increasing sample size from n − 1 to n is

[E [B(n)] = (E [p(n−1)min ]− E [p

(n)min])× K

The above is increasing in K and decreasing in n

n∗ is increasing in K

A firm’s expected demand at price p is

Q(p) = µn∗(1− F (p))n∗−1



Propositions and Results

Proposition 1 Suppose that a price distribution is a mean preservingspread of a price distribution F .1 Then the expected transactionsprice of a consumer who obtains n > 1 price quotes is strictly lowerunder price distribution G than under F

Proposition 2 Suppose that an optimizing consumer obtains morethan one price quote when prices are distributed according to F , andthat price distribution G is a mean preserving spread of F . Then theconsumer’s expected total costs under G are strictly less than thoseunder F

intuition Consumers pay lower average prices and have lowerexpected total cost if prices are more dispersed

1G is a mean preserving spread of F if (a)∫ +∞−∞ [G(p)− F (p)]dp = 0 and (b)∫ z

−∞[G(p)− F (p)] ≥ 0 for all z and strict for some zSibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 9 / 56


Empirical Works

George Stigler, in his seminal article on the economics ofinformation,advanced the following hypotheses:

1. The larger the fraction of the buyer’s expenditures on thecommodity, the greater the savings from search and hence the greaterthe amount of search2. The larger the fraction of repetitive (experienced) buyers in themarket, the greater the effective amount of search (with positivecorrelation of successive prices)3. The larger the fraction of repetitive sellers, the higher thecorrelation between successive prices, and hence, the larger the amountof accumulated search4. The cost of search will be larger, the larger the geographic size ofthe market



Empirical Works

Dispersion for ”Cheap” versus ”Expensive” Items

”Expensive”(Large K in his model or high price)→high marginalbenefit of search →more search→low dispersion(Stigler, 1961)Government purchase of coal VS Household purchase ofautomobile(Pratt, Wise and Zeckhauser, 1979) Regress the standard deviation of(log) price for a given item on the sample (log) mean price for thesame itemOthers

Dispersion and Purchase Frequency

(Sorensen, 2000) Market for priscription drug


No Clearing House Rothschild Critique and Diamond’s Paradox

Problems

(Rothschild, 1973)Why consumers have fixed sample size search?exogenous? Optimal? Do we need to consider the update ofinformation, such as exceptionally low price from an early search?

Only consider the consumers’ effect on distribution of transactionprice. But what about the firms’ side effect? Is the ex-ante pricedistribution F really exogenous?

Why firms do not optimize their profits by setting price p?

”partial-partial equilibrium” approach



Diamond’s Paradox

Demand Curve: −v ′(p) = q(p) and −v ′′(p) = q′(p) < 0

Sequential Search

Monopoly Price p∗, here we assume that consumers buy quantityaccording to p, not a constant K

v(p∗) > c



Diamond’s Paradox

It is a unique equilibrium for all firms to set price p∗, and consumersearch only once

(Existence)It is an equilibrium(Uniqueness)For any firm, setting price above p∗ is a dominatedstrategy(Uniqueness)If lowest price p′ < p∗, it has incentive to deviate tominp∗, p′ + c

Perfect competition, but monopoly price in equilibrium, the reason issearch cost

Different from previous model, no price dispersion

taking Rothschild’s criticism into account, and increase in searchintensity can lead to increases or decreases in the level of equilibriumprice dispersion, depending on the model. Since Stigler didn’tconsider firm’s optimization behavior, it challenges his hypotheses.


No Clearing House Sequential Search - Reinganum Model

Sequential Search - Reinganum Model

Identical consumers search firm by firm and choose a stopping rule;search is costly

Firms have heterogeneous marginal costs and set prices

Aim: show existence of a dispersed price equilibrium.

Stopping rule is optimal given firms’ optimal prices and vice versa



Consumer’s Problem

Identical demand: −v ′(p) = q(p) = Kpε with ε < −1,K > 0.

q(p) > 0, q′(p) = εKpε−1 < 0

Search costs c > 0 per additional firm.

Free-recall i.e. customers can always go back.




Assume for now a given distribution of prices F (p).

F (p) is atomless with support [p, p̄].

Let z = min(p1, p2, ..., pn) be the lowest price found after n searches.




Assume for now a given distribution of prices F (p).

F (p) is atomless with support [p, p̄].

Let z = min(p1, p2, ..., pn) be the lowest price found after n searches.

Then the expected benefit of one more search is:

B(z) = E [v(p)− v(z)|p < z ] Prob[p < z ]

=

∫ z

p(v(p)− v(z))f (p)dp

=

∫ z

p−v ′(p)F (p)dp (int. by parts)




How does B(z) vary with z?

By Fundamental Theorem of Calculus:

B ′(z) = −v ′(z)F (z) = q(z)F (z) > 0 ∀z > p

So lower z ⇒ lower benefit of additional search.



Optimal Search Strategy

Case 1: B(p̄) < c and E [v(p)] =∫ p̄p v(p)f (p)dp < c

recall consumers start with no priceoptimal strategy is to not search ⇒ no transactions.

Case 2: B(p̄) < c and E [v(p)] ≥ c

optimal strategy is to search once.



Optimal Search Strategy

Case 3: B(p̄) ≥ c

recall B ′(z) < 0consumers search until they obtain a price quote at or below areservation price rr satisfies B(r) = c ⇔

∫ r

p(v(p)− v(r))f (p)dp = c

Effect of search costs on r :

dr

dc=

1

q(r)F (r)> 0



Firm’s Problem

Each firm has a marginal cost of production m

m is drawn from an atomless distribution G (m) with support [m, m̄]

Each firm anticipates consumers’ search strategy and optimal pricesset by other firms.

Suppose a fraction 0 ≤ λ < 1 of firms price above r and there are µconsumers on average per firm.

Let E [π(p)] be a firm’s profit as a function of the price it sets.

All firms with p ≤ r have an equal chance of being picked by aconsumer, so:

E [π(p)] = (p −m)q(p)µ

1− λFor p > r?



Optimal Price Setting

Firm solves:maxp

E [π(p)]

Solving the FOC:

p∗ =ε

1 + εm

Recall ε < −1 so firm’s optimal price is just a constant % markup over cost

⇒ F̂ (p) = G (p1 + ε

ε) for p ∈ [m

ε

1 + ε, m̄

ε

1 + ε]



Equilibrium

Additional assumption: v − p∗(m̄) > c

In response to F̂ (p), consumers choose an optimal reservation price r .

However if r < p∗(m̄) then some firms would have no sales ⇒ not NE

Instead firms with marginal costs s.t. p∗(m) > r will choose to priceat r .

So F (p) = F̂ (p) if p ∈ [m ε1+ε , r) and F (r) = 1



Equilibrium

Need to check that given F (p), r is still consumers’ optimal reservationprice:

Recall B(r) = c and

B(r) = E [v(p)− v(r)|p < r ] Prob[p < r ]

So B(z) is unchanged from F̂ (p) to F (p) and thus r is still optimal.

Recall also that p∗(m) is independent of λ, µ.



Equilibrium

Need to check that given F (p), r is still consumers’ optimal reservationprice:

Recall B(r) = c and

B(r) = E [v(p)− v(r)|p < r ] Prob[p < r ]

So B(z) is unchanged from F̂ (p) to F (p) and thus r is still optimal.

Recall also that p∗(m) is independent of λ, µ.



Comparative Statics

Variance in prices:

σ2 = E [p2]− E [p]2

Effect of reservation price on variance in prices:

dσ2

dr= 2[1− F̂ (r)](r − E [p]) ≥ 0

and inequality holds strictly if r < p∗(m̄).

And drdc > 0 so an increase in search costs increases the variance of

equilibirium prices.


No Clearing House MacMinn Model

MacMinn Model

Aim: show price dispersion when consumers conduct fixed sample searchand firms optimally set prices.

identical consumers demand 1 unit of a good with valuation v

marginal cost of search c > 0 per price quote

firms have private marginal costs m ∼ G (m), atomless with support[m, m̄]

m̄ < v



Firm’s problem

When a consumer obtains n > 1 price quotes, the n firms are effectivelycompeting against each other in an auction.

Revenue Equivalence Theorem requires:1 firms ex-ante symmetric2 independent private values3 efficient allocation - consumer buys from firm with lowest m4 free exit5 risk neutral



Firm’s problem

Use Revenue Equivalence Theorem and 2nd Price Auction to calculatefirms’ expected revenues R(m).

Firms bid their private values. So for firm j , if m0 = min{mi}i 6=j :

R(mj) = Prob[mj < m0] E [m0|mj < m0]

= mj(1− G (mj))n−1 +

∫ m̄

mj

(1− G (t))n−1dt



Firm’s problem

For a given price pj , expected revenue is:

R(mj) = pj Prob[mj < m0] = pj(1− G (mj))n−1

We can therefore solve for equilibrium price pj as a function of mj :

pj(mj) = E [m0|mj < m0]

= mj +

∫ m̄

mj

(1− G (t)

1− G (mj)

)n−1

dt

Thus G (m) results in distribution of prices F (p(m)). Notice that p(m) isincreasing in m so allocation is efficient.



Consumer’s problem & Equilibrium

Optimal sample size n is set by:

E [B(n+1)] < c ≤ E [B(n)]

where E [B(n)] is the expected benefit from increasing sample size fromn − 1 to n, as in the Stigler Model.



Comparative Statics

Special case when G (m) is uniform:

p(m) =n − 1

nm +

1

nm̄

σ2p =

(n − 1

n

)2

σ2m

higher variance in m ⇒ higher variance in p

larger sample size n⇒ higher variance in p



MacMinn vs. Reinganum

Sequential search:

lower search costs ⇒ lower reservation price, e.g. from r to r ′.

firms with p ≤ r ′ do not change their prices

firms with p > r ′ lower their prices to r ′ so dispersion decreases.

Fixed sample search:

increase in n increases competition faced by all firms

E [m0|mj < m0]−mj decreasing in n



Empirical Evidence - Search Cost

Online vs. Offline

selection biasdifferent search behaviors⇒ mixed results re price dispersion

Geographic distance

Estimates of search costs from structural estimation:

$1.31 to $29.40 for online listings of economics and stats textbooks(Hong and Shum 2006)


No Clearing House Burdett and Judd Model

Burdett & Judd

Aim: show equilibirium price dispersion with identical consumers andfirms.

Consumers demand 1 unit of valuation v > 0

Fixed sample search

Firms have identical marginal cost c < v

v −max{p} ≥ c



Burdett & Judd

Equilibrium characterized by:

optimal price distribution F (p)

optimal search distribution < θn >∞n=1 where θi is fraction of

consumers obtaining i price quotes

Price dispersion originates from existence of a mixed search strategyequilibrium.



Burdett & Judd

No pure search strategy:

θ1 ⇒ all firms price at identical monopoly price p = v

θ1 = 0⇒ multiple identical firms compete so p = c

Therefore firms face no competition with probability θ1 ∈ (0, 1) percustomer.

Firms randomize prices so that each is indifferent between charging p or v ,for p in support of F (p).


Clearing House Basic Setup

Assumptions

An information clearinghouse provides a subset of consumers with alist of prices charged by different firms in the market

n firms in the market selling homogeneous product at constantmarginal cost m

Firm i charge price pi for its product and decide whether list this priceat the clearing house at the cost of φ

All consumers have unit demand with a maximal willingness to pay ofv > m

S of consumers are price-sensitive ”shoppers”

L of consumers per firm directly purchase if its price doesn’t exceed vor do not buy at all



General Treatment

Proposition 3 Let 0 ≤ φ < n−1n (v −m)S . Then in a symmetric

equilibrium of the general clearinghouse model, we have1.Each firm lists its price at the clearinghouse with probability

α = 1−( n

n−1φ

(v −m)S

) 1n−1

2.If a firm lists its price at the clearinghouse, it charges a price drawnfrom the distribution

F (p) =1

α

1−( n

n−1φ+ (v − p)L

(p −m)S

) 1n−1

on [p0, v ]

,

where p0 = m + (v −m) LL+S +

nn−1

L+S φ3.If a firm does not list its at the clearinghouse, it charges a price equalto v4.Each firm earns equilibrium expected profits equal to

Eπ = (v −m)L +1

n − 1φ



Explanation

Several forces influence firm’s strategy

Firms wish to charge v to extract maximal profits from the loyalsegment

But this is not equilibrium because if all firms do so, a firm could justslightly undercut the price and gain all shoppers

However, once prices get sufficiently low, a firm is better off by simplycharging v and giving up their on shoppers

The only equilibrium is mixed strategy, firms randomize their prices,sometimes pricing relatively low to attract shoppers and other timespricing fairly high to maintain margins on loyals


Clearing House Rosenthal Model

Assumptions

Environment is similar to previous setup

But each firm enjoys a mass L of ”loyal” consumers

Costless to list prices on the clearing house: φ = 0



Results



Results

Follows from Proposition 3 and set φ = 0 and get α = 1

The equilibrium distribution of price is

F (p) = 1−(

(v − p)

(p −m)

L

S

) 1n−1

where

p0 = m + (v −m)L

L + S

Mixed Strategy equilibrium



Results

Loyal customers expect to pay the price

E [p] =

∫ v

p0

pdF (p)

Shoppers expect to pay

E[p

(n)min

]=

∫ v

p0

pdF(n)min(p)

As the number of competing firms increases, the expectedtransactions price paid by all consumers go up

It is partly because we assume entry brings more loyals into themarket.

For loyals, they are expected to pay more, for shoppers, the proofneed a bit more work


Clearing House Varian Model

Setup

Environment is similar to previous setup

S ”informed consumers” and L = Un ”uninformed consumers”

φ = 0

We have α = 1 and the equilibrium distribution of prices is

F (p) = 1−

((v − p)

(p −m)

Un

S

) 1n−1

on [p0, v ]where

p0 = m + (v −m)Un

Un + S


Clearing House Varian Model

Questions and Results

What if consumers could make optimal decisions? The equilibriumpersist if consumers have different cost of accessing the clearinghouse.And the value of information is

VOI (n) = E [p]− E [p(n)min]

If consumers’ information costs are zero, all consumers choose tobecome informed and all firms price at marginal cost, if consumers’information costs are sufficiently high, no consumers choose tobecome informed and all firms charge the monopoly price v . So pricedispersion is not a monotonic function of consumers’ information cost


Clearing House Baye and Morgan

Clearinghouse with listing cost




Clearinghouse enters to serve all markets:

each firm can pay φ ≥ 0 to list on clearinghouse

each consumer can pay κ ≥ 0 to shop at clearinghouse




In equilibrium:

clearinghouse optimally sets φ, κ to maximize expected profitφnα + Sκ

consumers choose whether or not to access clearinghouse

each firm sets its price and chooses whether or not to list onclearinghouse



Equilibrium Results

Baye & Morgan can be seen as a special case of the general clearinghousemodel. In equilibirium:

φ > 0

κ = 0⇒ L = 0



Equilibrium Results

Apply Proposition 3 to obtain equilibrium listing probability α andclearinghouse price distribution F (p).

F (p) atomless with support [p0, v ] , p0 < v

clearinghouse ⇒ higher competition ⇒ lower prices

κ = 0, φ > 0

lower κ⇒ more customers on clearinghouse & fewer local customers⇒ higher incentives for firm to list

lower φ⇒ pricing is more competitive ⇒ lower local prices ⇒free-rider probleme.g. Amazon, Zappos, Ebay



Equilibrium Results

Price dispersion persists even when search costs are 0.

Rather price dispersion exists because it is costly for firms to transmit priceinformation to consumers i.e. list on clearinghouse.



Empirical Evidence - Competition


Asymmetric Consumers

Asymmetric Consumers in Duopoly Market

Two firms competing in the market i = 1, 2

Customers demand 1 unit

Mass Li customers are loyal to firm i , with L1 ≥ L2

Mass S customers buy at lowest price on clearinghouse



Firm’s Problem

Let Ai = 1 if firm i lists on clearinghouse and 0 otherwise.

Expected profits if firm i posts price p, given firm j ’s actions:

E [πi (p|Ai = 0)] = (Li + (1− αj)S2 )(p −m)

E [πi (p|Ai = 1)] = [Li + S(1− αj) + Sαj(1− Fj(p)](p −m)

Mixed strategy equilibirum: E [πi (p|Ai = 0)] = E [πi (p|Ai = 0)] for allp in support of F (p).



Equilibrium Pricing

Fi (p) =1

α

[1−

2φ+ (v − p)Lj(p −m)S

]∀p ∈ [p0, v ]

Notice Fi (p) is decreasing in Lj . Moreover,

F1(p)− F2(p) =1

α

v − p

(p −m)S[L1 − L2] > 0

Due to mixed strategy: randomize to keep other player indifferent, i.e.E [πi (p|Ai = 0)] = E [πi (p|Ai = 0)]


Bounded Rationality / Unobserved Frictions


Relaxing best response assumption of Nash Equilibirium:

Quantal Response Equilibrium (QRE)

firm’s price determined by stochastic decision ruleprices leading to higher expected profits more likely to be quotedplayers have correct beliefs about probability distributions of others’actions

ε - equilibrium

in equilibrium no firm can gain more than ε > 0 by changing its price




Baye and Morgan (2004) use QRE and ε-equilibrium to show that only alittle bounded rationality is needed to generate patterns of price dispersionseen in lab experiments and online.


Conclusion

Remarks on Theory

Not one size fits all

Reductions in search costs may lead to more or less price dispersion,depending on the market structure. Price dispersion can exist evenwith 0 search cost.

Increased competition can also increase or decrease price dispersion.

Neither firm nor consumer heterogeneities are necessary for pricedispersion.


Conclusion

Recent Work

Price dispersion on Ebay (Levin et al, 2014)

use browsing data to model consumer searchaccounts for Ebay’s active role: transaction prices fell 5-15% for manyproducts after a search engine re-designprice dispersion exists but consumers are highly price sensitive (marketelasticity on order of -10)

Obfuscation by online retailers (Ellison & Ellison 2008)

Parallel line of research in comp sci/ information systems

TED talk on filter bubbles: http://bit.ly/19yDhxc


Conclusion

Other Applications

Search for information, e.g. individuals choosing news sources,policymaker consulting lobbyists

Jesse Shapiro and Matthew Gentzkow

Start-up looking for VC investment

Dispersion in quality among near perfect substitutes in onlinemarkets, or why is buying a HDMI cable on Amazon so hard?


Conclusion

Theory Extensions

Sequential search where distribution of prices F (p) is not known byconsumers.

Instead they have ex-ante priorsUpdate their beliefs about F (p) as they search

Limited attention

Multiple clearinghouses


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Search and Price Dispersion - Berkeley-Haasfaculty.haas.berkeley.edu/rjmorgan/phdba279b/Search Presentation... · Search and Price Dispersion Sibo Lu and Yuqian Wang Haas Sibo Lu