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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
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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
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Introduction
Models and Approaches
Search Theoretical Model/Marginal Search Cost
Information Clearinghouse
Others, e.g. limited rationality, asymmetric consumers
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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
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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
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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]
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No Clearing House The Stigler Model
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)
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No Clearing House The Stigler Model
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
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No Clearing House The Stigler Model
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
No Clearing House The Stigler Model
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
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No Clearing House The Stigler Model
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
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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
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No Clearing House Rothschild Critique and Diamond’s Paradox
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
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No Clearing House Rothschild Critique and Diamond’s Paradox
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.
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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
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No Clearing House Sequential Search - Reinganum Model
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.
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No Clearing House Sequential Search - Reinganum Model
Consumer’s Problem
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.
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No Clearing House Sequential Search - Reinganum Model
Consumer’s Problem
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)
Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 16 / 56
No Clearing House Sequential Search - Reinganum Model
Consumer’s Problem
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.
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No Clearing House Sequential Search - Reinganum Model
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.
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No Clearing House Sequential Search - Reinganum Model
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
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No Clearing House Sequential Search - Reinganum Model
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?
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No Clearing House Sequential Search - Reinganum Model
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]
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No Clearing House Sequential Search - Reinganum Model
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
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No Clearing House Sequential Search - Reinganum Model
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.
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No Clearing House Sequential Search - Reinganum Model
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.
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No Clearing House Sequential Search - Reinganum Model
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.
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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
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No Clearing House MacMinn Model
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
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No Clearing House MacMinn Model
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
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No Clearing House MacMinn Model
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.
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No Clearing House MacMinn Model
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.
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No Clearing House MacMinn 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
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No Clearing House MacMinn Model
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
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No Clearing House MacMinn Model
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)
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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
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No Clearing House Burdett and Judd Model
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.
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No Clearing House Burdett and Judd Model
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).
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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
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Clearing House Basic Setup
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
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Clearing House Basic Setup
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
Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 38 / 56
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
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Clearing House Rosenthal Model
Results
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Clearing House Rosenthal Model
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
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Clearing House Rosenthal Model
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
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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
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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
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Clearing House Baye and Morgan
Clearinghouse with listing cost
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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
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Clearing House Baye and Morgan
Clearinghouse with listing cost
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
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Clearing House Baye and Morgan
Equilibrium Results
Baye & Morgan can be seen as a special case of the general clearinghousemodel. In equilibirium:
φ > 0
κ = 0⇒ L = 0
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Clearing House Baye and Morgan
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
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Clearing House Baye and Morgan
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.
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Clearing House Baye and Morgan
Empirical Evidence - Competition
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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
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Asymmetric Consumers
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).
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Asymmetric Consumers
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)]
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Bounded Rationality / Unobserved Frictions
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
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Bounded Rationality / Unobserved Frictions
Bounded Rationality / Unobserved Frictions
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.
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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.
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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
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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?
Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 55 / 56
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
Sibo Lu and Yuqian Wang (Haas) Search and Price Dispersion 56 / 56