Difference between revisions of "Baye Morgan Scholten (2006) - Information Search and Price Dispersion"

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<center><math>B(z) = \int_{\underline{p}}^{z} (v(p) - v(z))dF(p) =  \int_{\underline{p}}^{z} v'(p)dF(p)\,</math></center>
 
<center><math>B(z) = \int_{\underline{p}}^{z} (v(p) - v(z))dF(p) =  \int_{\underline{p}}^{z} v'(p)dF(p)\,</math></center>
  
By Liebnitz' rule:  
+
By [http://en.wikipedia.org/wiki/Leibniz's_rule Liebnitz' rule]:  
  
 
<center><math>B'(z) = -v'(z)dF(z) = Kz^{\epsilon}F(z) > 0\,</math></center>
 
<center><math>B'(z) = -v'(z)dF(z) = Kz^{\epsilon}F(z) > 0\,</math></center>
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Therefore a the reservation price is increasing in search costs. Note the special case where <math>q(r)=1\,</math> leads to a magnification effect, but attenuation effects are also possible.
 
Therefore a the reservation price is increasing in search costs. Note the special case where <math>q(r)=1\,</math> leads to a magnification effect, but attenuation effects are also possible.
 
  
 
===Reinganum (1979) Revisited===
 
===Reinganum (1979) Revisited===

Revision as of 04:45, 26 January 2010

  • This page is part of a series under PHDBA279B

Key Reference(s)

Introduction

Baye et al. (2006) provides a survey of models of search and clearinghouse that exhibit price dispersion. The survey is undertaken through two specializable frameworks, one for search and one for cleaninghouses, which are then adapted to show the key results from the literature. There are a number of equivalent results across the two frameworks.

Search Theoretic Models of Price Dispersion

The general framework used through-out is as follows:

  • A continuum of price-setting firms with unit measure compete selling an homogenous product
  • A mass [math]\mu[/math] is interested in purchasing the product
  • Consumers have quasi-linear utility (i.e. additively-seperable in income):
[math]u(q) + y\,[/math] where [math]y\,[/math] is a numeraire good
  • The indirect utility of consumers is:
[math]V(p,M) = v(p) + M\,[/math] where [math]v(\cdot)\,[/math] in nonincreasing in [math]p\,[/math], and [math]M\,[/math] is income.
[math]q(p) \equiv -v'(p)\,[/math].
  • There is a search cost [math]c\,[/math] per price quote
  • The customer purchases after [math]n\,[/math] price quotes
  • The final indirect utility of the customer is:
    [math]V(p,M) = v(p) + M - cn\,[/math]

A note on the derivation of demand

Recall that [math]M=e(p,u)\,[/math],


so that [math]v(e(p,u),p)=u\,[/math] when the expenditure function is evaluated at [math]p\,[/math] and [math]u\,[/math].


Taking the derivitive with respect to [math]p\,[/math]:

[math]\frac{d(v(M,p))}{dp} = \frac{\partial v(M,p)}{\partial m} \cdot \frac{\partial M}{\partial p} + \frac{\partial v(M,p)}{\partial p} = 0,\,[/math] where


[math]\frac{\partial M}{\partial p} = \frac{\partial e(p,u)}{\partial p}\,[/math].


[math]\therefore q(m,p) = \frac{\partial e(p,u)}{\partial p} = -\frac{\frac{\partial v(M,p)}{\partial p}}{\frac{\partial v(M,p)}{\partial m}}\,[/math]


[math]\therefore q(m,p) = -\frac{d}{dp(v(p))}\quad[/math] in our case.

The Stigler (1961) Model

The first special case examined in the general framework is that of Stigler (1961):

  1. Each consumer purchases [math]K \ge 1\,[/math] units, so that [math]q(p) = -v'(p) = K\,[/math]
  2. Fixed sample search is used
  3. The distribution of firms' prices is given exogenously by the non-degenerate CDF [math]F(p)\,[/math] on [math][\underline{p}, \overline{p}]\,[/math].
Fixed sample search
In a fixed sample search the consumer commits to conducting [math]n\,[/math]  searches
and then buys from the firm offering the lowest price

The consumer seeks to minimize the expected cost (purchase + search) given by:


[math]\mathbb{E}(C) = K \mathbb{E}(p_{min}^{(n)}) + cn\,[/math]


where [math]\mathbb{E}(p_{min}^{(n)}) = \mathbb{E}(min\{p_1,p_2,\ldots,p_n\}) \,[/math]


The distribution of the lowest [math]n\,[/math] draws is:

[math]F_{min}^{(n)}(p) = 1 - (1-F(p))^n\,[/math]


[math]\therefore \mathbb{E}(C) = K \int_{\underline{p}}^{\overline{p}} p \; dF_{min}^{(n)}(p) + cn\,[/math]


[math]\therefore \mathbb{E}(C) = K \left [ \underline{p} + \int_{\underline{p}}^{\overline{p}} (1-F(p))^n \; dp \right ] + cn\,[/math]


To see this, first recall that

[math]\mathbb{E}(X) = \int_{\underline{x}}^{\overline{x}} x f(x) dx \,[/math]


Then use the integration by parts formula

[math]\int u\, \frac{dv}{dx}\; dx=uv-\int v\, \frac{du}{dx} \; dx\![/math]


Observe that as the expected purchase price [math]\left [ \underline{p} + \int_{\underline{p}}^{\overline{p}} (1-F(p))^n \; dp \right ]\,[/math] is decreasing in [math]n\,[/math] and that the cost of search is positive ([math]c \gt 0\,[/math]) the optimum will be finite.


The expected benefit for a customer to increase their sample size from [math]n-1\,[/math] to [math]n\,[/math] is:


[math]\mathbb{E}(B^{(n)})=\left ( \mathbb{E} (p_{min}^{n-1}) - \mathbb{E} (p_{min}^{n}) \right ) \times K\,[/math]


This is decreasing in [math]n\,[/math] and increasing in [math]K\,[/math]. Also as the cost of the [math]n,[/math]th search is independent of [math]K\,[/math], the equilibrium search intensity is increasing in [math]K\,[/math]. Note that [math]K\,[/math] may refer to either purchases in greater quantities or more frequent purchases.


Although customers inelastically purchase [math]K\,[/math] units, a version of the law of demand holds: Each firm's expected demand is a non-increasing function of its price.


A firm charging price [math]p\,[/math] is visited by [math]\mu n^*\,[/math] customers


A firm charging price [math]p\,[/math] offers the lowest price with probability [math](1-F(p))^{n^*-1}\,[/math]


A firm's expected demand is: [math] Q(p) - \mu n^* K (1-F(p))^{n^*-1}\,[/math]


A mean preserving spread on a Normal distribution

The Stigler model also implies that expected transaction prices will be lower when prices have the same mean but are more dispersed. There is a simple intuition for this, that can be demonstrated graphically. Suppose that the customer is drawing from one of the two distributions pictured - a draw from the green distribution (that has the higher variance) would be more likely to yield a lower price and have lower total costs.

These results are proved formally (as propositions 1 and 2), and the essence of the proofs are as follows:


Let

[math] \Delta = \mathbb{E}_F \left [ p_{min}^{(n)} \right ] - \mathbb{E}_G \left [ p_{min}^{(n)} \right ]\,[/math]


So that [math]\Delta\,[/math] represents the difference in expected transaction prices. Then show that for [math]n\gt 1\,[/math] that [math]\Delta \gt 0\,[/math].


Then suppose that the expected number of searches under [math]F\,[/math] is [math]n^*\,[/math]. The consumer's expected total cost under [math]F\,[/math] is:


[math]\mathbb{E}(C_F) = \mathbb{E}_F \left [ p_{min}^{(n^*)} \right ] \times K - c n^* \, \gt \mathbb{E}_G \left [ p_{min}^{(n^*)} \right ]\, \ge \mathbb{E}(C_G)\,[/math]


Note that the strick inequality follows from the proof that [math]\Delta \gt 0\,[/math], and the weak inequality follows as [math]n^*\,[/math] may not be optimal under [math]G\,[/math].


The Rothschild Critique

Rothschild (1973) pointed out that the search procedure used in Stigler (1961) may not be optimal. Specifically the customer's commitment to a fixed number of searches may not be 'credible'. The strategy fails to incorporate new information as it becomes available: Once a sufficiently low price quote has been obtained, the benefit of additional searches may drop below the marginal cost.

Sequential search
In sequential search there is an optimal stopping rule. 
Once search results have fallen below some threshold, called the reservation price, search stops.
Consumers know the distribution of prices, which is exogenously specified. 
The reservation price is endogenously determined. 
Recall is free.


Recall that a firm's expected demand is: [math] Q(p) - \mu n^* K (1-F(p))^{n^*-1}\,[/math]


Then a firm with constant marginal cost ([math]m\,[/math]) has expected profits:

[math]\pi(p) = (p-m)Q(p)\,[/math]


All firms have the same marginal cost and so the same expected profit function. So then why would profit-maximizing firms choose the same distribution of prices?

The Rothschild Critique 
The Stigler (1961) model has only optimizing consumers, and not optimizing firms
This is therefore a 'partial-partial equilibrium' approach.
Why would profit-maximizing firms choose the same profit-maximizing price?

Diamond's Paradox

Diamond (1971) provides conditions under which the unique equilibrium in undominated strategies has firms all charging the monopoly price, with costly search by the consumers.

To see Diamond's result, suppose:

  • Consumers have identical downward sloping demand:
[math]-v''(p)=q'(p)\lt 0\,[/math]
  • Consumers engage in optimal sequential search
  • Firms all charge the unique monopoly price [math]p^*\,[/math]
  • A consumer earns enough surplus at the monopoly price to cover one price search: [math]v(p^*)\gt c\,[/math]

In Diamond's model all firms charge the monopoly price and all consumers visit exactly one store. This is a Nash equilibrium: Given the stopping rule of consumers the firm's best response is to charge the monopoly price; Given the monopoly price, the consumers' best response is to search just once and then buy. This is unique as a firm posting below the monopoly price would want to deviate up by an epsilon - the consumers would still buy and the firm would make more.

Diamond's Paradox
Even though there is a continuum of competing firms (i.e. perfect competition)
in the presence of any search frictions whatsoever the monopoly price is the equilibrium.

The Reiganum (1979) Model

Reiganum (1979) showed that price dispersion could exist with sequential search as well as with optimizing consumers and optimizing firms.

Consider the following special case of our environment:

  • Consumers have identical demands:
[math]-v'(p)=q(p)=Kp^{\epsilon}\,[/math], where [math]\epsilon \lt -1\,[/math] and [math]K \gt 0\,[/math]
  • Consumers engage in optimal sequential search
  • Firms have heterogeneous (private) marginal costs drawn from [math]G(m)\,[/math] on [math][\underline{m},\overline{m}]\,[/math]
  • The consumer with the highest marginal cost [math]\overline{m}\,[/math] still earns sufficient surplus to cover the cost of one price quote:
[math]v \left ( \frac{\epsilon}{1+\epsilon}\overline{m} \right ) \gt c\,[/math]

Reiganum (1979) shows that under these assumptions there is an equilibrium in which firms optimally set prices, consumers engage in optimal sequential search, and yet there is still price dispersion. We return to the Reiganum (1979) model after a discussion of sequential search.


Sequential Search Models

The first step to solving sequential search models is determining the optimum reservation price. Suppose that following n searches the consumer has found a best price (to date) of [math]z\,[/math]. Then the benefit of an additional search is:


[math]B(z) = \int_{\underline{p}}^{z} (v(p) - v(z))dF(p) = \int_{\underline{p}}^{z} v'(p)dF(p)\,[/math]

By Liebnitz' rule:

[math]B'(z) = -v'(z)dF(z) = Kz^{\epsilon}F(z) \gt 0\,[/math]


This is smaller when [math]z\,[/math] is small, that is the benefits are lower when the best price already identified is lower. Furthermore, search is costly, so consumers must make a trade-off. The expected net benefit of an additional search is:


[math]h(z) \equiv B(z) - c\,[/math]


The optimal consumer strategy is:


Case 1)   [math]h(\overline{p}) \lt 0\,\,[/math]  and  [math] \int_{\underline{p}}^{\overline{p}} v(p)dF(p) \lt c\,[/math]. In this case it is better not to search.


Case 2)   [math]h(\overline{p}) \lt 0\,[/math]  and  [math]\int_{\underline{p}}^{\overline{p}} v(p)dF(p) \ge c\,[/math]. In this case the net benefit at the current price is negative, but the consumer is best off by searching until they get a price quote at (or below) [math]\underline{p}\,[/math].


Case 3)   [math]h(\overline{p}) \gt 0\,[/math]. This is the interior solution and the interesting case. The customer should search until they get a reservation price [math]r\,[/math] (or below) which makes them exactly indifferent between buying now and making another search. This is given by:

[math]h(r) = \int_{\underline{p}}^{r} (v(p) - v(r))dF(p) - c = 0\,[/math]


Note that this is uniquely defined because:


[math]h(\underline{p}) = -c \lt 0\,[/math]


[math]h(\overline(p)) \ge 0\,[/math]


[math]h'(z) = B'(z) \gt 0\,[/math]


Using the equation above to find the optimal [math]r\,[/math] (i.e. taking the first order condition), and then differentiating with respect to [math]c\,[/math], we can determine an interesting comparative static:


[math]\frac{\partial r}{\partial c} = \frac{1}{q(r)F(r)} = \frac{1}{Kr^{\epsilon}F(r)} \gt 0\,[/math]


Therefore a the reservation price is increasing in search costs. Note the special case where [math]q(r)=1\,[/math] leads to a magnification effect, but attenuation effects are also possible.

Reinganum (1979) Revisited

Recall that Reinganum (1979) has firms with marginal costs drawn from a distribution [math]G(m)\,[/math]. Suppose that an individual firm's cost is [math]m_j\,[/math], and that a fraction [math]\lambda\,[/math], where [math]\lambda \in \left [0,1 \right )\,[/math], of firms price above [math]r\,[/math]. Then, with a mass [math]\mu \,[/math] of consumers as before:


[math] \mathbb{E} \pi_j = \begin{cases} (p_j-m_j)q(p_j)\left ( \frac{\mu}{1 - \lambda} \right ) & \mbox{if}\; p_j \le r \\ 0 & \mbox{if}\; p_j \gt r \end{cases} [/math]

It is instrumental to temporarily ignore that a firm's demand is zero above [math]r\,[/math]. The profit maximizing price from above is (from the first order condition):


[math]\left [ (p_j - m_j) q'(p_j) + q(p_j) \right ] \left ( \frac{\mu}{1 - \lambda} \right ) = 0\,[/math]

Which implies (given the consumer's demand function above):


[math]p_j=\left ( \frac{\epsilon}{1+\epsilon} \right ) m_j\,[/math]

If firm's were to do this then consumers would face a distribution of prices:


[math]\hat{F}(p)=G \left (p \left ( \frac{\epsilon}{1+\epsilon} \right ) \right)\,[/math] on the interval [math]\left [ \frac{\underline{m}\epsilon}{1+\epsilon} , \frac{\overline{m}\epsilon}{1+\epsilon} \right ]\,[/math]

[math] \,[/math]

[math] \,[/math]