Baye Morgan Scholten (2006) - Information Search and Price Dispersion

• This page is part of a series under PHDBA279B

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.

• By Roy's identity:

[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 graphical proof of this. 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) [math] \,[/math]

[math] \,[/math]