Search and Price Dispersion Sibo Lu and Yuqian Wang Haas Sibo Lu - - PowerPoint PPT Presentation

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 House Basic Setup The Stigler Model Rothschild Critique and Diamond’s Paradox Sequential Search - Reinganum Model MacMinn Model Burdett and Judd Model

3

Clearing House Basic Setup Rosenthal Model Varian Model Baye 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 competitive markets should be same However, empirical studies reveal that price dispersion is the rule (Varian, 1980, p. 651) Why? Cost of acquiring information about firms/transmitting information 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 time Fixed sample search: PhD interview day Clearinghouse: LinkedIn, Monster, SimplyHired

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No Clearing House Basic Setup

Assumptions and variables

A continuum of price-setting firms compete in homogeneous product market 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 serach cost 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, ¯ p]

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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 ¯

p p

pdF (n)

min(p) + cn

= K

• p +

¯

p 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 preserving spread of a price distribution F.1 Then the expected transactions price of a consumer who obtains n > 1 price quotes is strictly lower under price distribution G than under F Proposition 2 Suppose that an optimizing consumer obtains more than one price quote when prices are distributed according to F, and that price distribution G is a mean preserving spread of F. Then the consumer’s expected total costs under G are strictly less than those under F intuition Consumers pay lower average prices and have lower expected 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 z Sibo 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 of information,advanced the following hypotheses:

• 1. The larger the fraction of the buyer’s expenditures on the

commodity, the greater the savings from search and hence the greater the amount of search

• 2. The larger the fraction of repetitive (experienced) buyers in the

market, the greater the effective amount of search (with positive correlation of successive prices)

• 3. The larger the fraction of repetitive sellers, the higher the

correlation between successive prices, and hence, the larger the amount

• f accumulated search
• 4. The cost of search will be larger, the larger the geographic size of

the 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 marginal benefit of search →more search→low dispersion (Stigler, 1961)Government purchase of coal VS Household purchase of automobile (Pratt, Wise and Zeckhauser, 1979) Regress the standard deviation of (log) price for a given item on the sample (log) mean price for the same item Others

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 of information, such as exceptionally low price from an early search? Only consider the consumers’ effect on distribution of transaction

• price. But what about the firms’ side effect? Is the ex-ante price

distribution 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 quantity according 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 consumer search only once

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

Perfect competition, but monopoly price in equilibrium, the reason is search cost Different from previous model, no price dispersion taking Rothschild’s criticism into account, and increase in search intensity can lead to increases or decreases in the level of equilibrium price dispersion, depending on the model. Since Stigler didn’t consider 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) = Kpǫ with ǫ < −1, K > 0.

q(p) > 0, q′(p) = ǫKpǫ−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, ¯ p].

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, ¯ p].

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)

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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(¯ p) < c and E[v(p)] = ¯

p p v(p)f (p)dp < c

recall consumers start with no price

• ptimal strategy is to not search ⇒ no transactions.

Case 2: B(¯ p) < c and E[v(p)] ≥ c

• ptimal strategy is to search once.

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No Clearing House Sequential Search - Reinganum Model

Optimal Search Strategy

Case 3: B(¯ p) ≥ c

recall B′(z) < 0 consumers search until they obtain a price quote at or below a reservation price r r 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, ¯ m] Each firm anticipates consumers’ search strategy and optimal prices set by other firms. Suppose a fraction 0 ≤ λ < 1 of firms price above r and there are µ consumers 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 a consumer, so: E[π(p)] = (p − m)q(p) µ 1 − λ For p > r?

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No Clearing House Sequential Search - Reinganum Model

Optimal Price Setting

Firm solves: max

p

E[π(p)] Solving the FOC: p∗ = ǫ 1 + ǫm Recall ǫ < −1 so firm’s optimal price is just a constant % markup over cost ⇒ ˆ F(p) = G(p1 + ǫ ǫ ) for p ∈ [m ǫ 1 + ǫ, ¯ m ǫ 1 + ǫ]

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No Clearing House Sequential Search - Reinganum Model

Equilibrium

Additional assumption: v − p∗( ¯ m) > c In response to ˆ F(p), consumers choose an optimal reservation price r. However if r < p∗( ¯ m) then some firms would have no sales ⇒ not NE Instead firms with marginal costs s.t. p∗(m) > r will choose to price at r. So F(p) = ˆ F(p) if p ∈ [m

ǫ 1+ǫ, 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 reservation price: Recall B(r) = c and B(r) = E[v(p) − v(r)|p < r] Prob[p < r] So B(z) is unchanged from ˆ F(p) to F(p) and thus r is still optimal.

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No Clearing House Sequential Search - Reinganum Model

Equilibrium

Need to check that given F(p), r is still consumers’ optimal reservation price: Recall B(r) = c and B(r) = E[v(p) − v(r)|p < r] Prob[p < r] So B(z) is unchanged from ˆ F(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 − ˆ F(r)](r − E[p]) ≥ 0 and inequality holds strictly if r < p∗( ¯ m). And dr

dc > 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 search and 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, ¯ m] ¯ m < 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 effectively competing against each other in an auction. Revenue Equivalence Theorem requires:

1

firms ex-ante symmetric

2

independent private values

3

efficient allocation - consumer buys from firm with lowest m

4

free exit

5

risk neutral

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No Clearing House MacMinn Model

Firm’s problem

Use Revenue Equivalence Theorem and 2nd Price Auction to calculate firms’ expected revenues R(m). Firms bid their private values. So for firm j, if m0 = min{mi}i=j: R(mj) = Prob[mj < m0] E[m0|mj < m0] = mj(1 − G(mj))n−1 +

• ¯

m 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 +

• ¯

m 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) is increasing 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 from n − 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 n m + 1 n ¯ m σ2

p =

n − 1 n 2 σ2

m

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 bias different 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 and firms. 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:

• ptimal price distribution F(p)
• ptimal search distribution < θn >∞

n=1 where θi is fraction of

consumers obtaining i price quotes Price dispersion originates from existence of a mixed search strategy equilibrium.

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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) per customer. 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 a list of prices charged by different firms in the market n firms in the market selling homogeneous product at constant marginal cost m Firm i charge price pi for its product and decide whether list this price at the clearing house at the cost of φ All consumers have unit demand with a maximal willingness to pay of v > m S of consumers are price-sensitive ”shoppers” L of consumers per firm directly purchase if its price doesn’t exceed v

• r do not buy at all

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Clearing House Basic Setup

General Treatment

Proposition 3 Let 0 ≤ φ < n−1

n (v − m)S. Then in a symmetric

equilibrium of the general clearinghouse model, we have

1.Each firm lists its price at the clearinghouse with probability α = 1 −

• n

n−1φ

(v − m)S

• 1

n−1

2.If a firm lists its price at the clearinghouse, it charges a price drawn from the distribution F(p) = 1 α  1 −

• n

n−1φ + (v − p)L

(p − m)S

• 1

n−1

  on [p0, v] , where p0 = m + (v − m)

L L+S +

n n−1

L+S φ

3.If a firm does not list its at the clearinghouse, it charges a price equal to v 4.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 loyal segment But this is not equilibrium because if all firms do so, a firm could just slightly undercut the price and gain all shoppers However, once prices get sufficiently low, a firm is better off by simply charging 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 times pricing fairly high to maintain margins on loyals

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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

• 1

n−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 expected transactions price paid by all consumers go up It is partly because we assume entry brings more loyals into the market. For loyals, they are expected to pay more, for shoppers, the proof need a bit more work

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Clearing House Varian Model

Setup

Environment is similar to previous setup S ”informed consumers” and L = U

n ”uninformed consumers”

φ = 0 We have α = 1 and the equilibrium distribution of prices is F(p) = 1 −

• (v − p)

(p − m)

U n

S

• 1

n−1

• n [p0, v]

where p0 = m + (v − m)

U n U n + S

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Clearing House Varian Model

Questions and Results

What if consumers could make optimal decisions? The equilibrium persist 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 to become informed and all firms price at marginal cost, if consumers’ information costs are sufficiently high, no consumers choose to become informed and all firms charge the monopoly price v. So price dispersion 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 profit φnα + Sκ consumers choose whether or not to access clearinghouse each firm sets its price and chooses whether or not to list on clearinghouse

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Clearing House Baye and Morgan

Equilibrium Results

Baye & Morgan can be seen as a special case of the general clearinghouse

• model. 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 and clearinghouse 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 problem e.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 price information 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) S

2 )(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 all p 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 rule prices leading to higher expected profits more likely to be quoted players have correct beliefs about probability distributions of others’ actions

ǫ - equilibrium

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

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Bounded Rationality / Unobserved Frictions

Bounded Rationality / Unobserved Frictions

Baye and Morgan (2004) use QRE and ǫ-equilibrium to show that only a little bounded rationality is needed to generate patterns of price dispersion seen 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 even with 0 search cost. Increased competition can also increase or decrease price dispersion. Neither firm nor consumer heterogeneities are necessary for price dispersion.

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Conclusion

Recent Work

Price dispersion on Ebay (Levin et al, 2014)

use browsing data to model consumer search accounts for Ebay’s active role: transaction prices fell 5-15% for many products after a search engine re-design price dispersion exists but consumers are highly price sensitive (market elasticity 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 online markets, or why is buying a HDMI cable on Amazon so hard?

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Conclusion

Theory Extensions

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

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

Limited attention Multiple clearinghouses

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