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

• This page is part of a series under PHDBA279B

Baye, Michael R., John Morgan, and Patrick Scholten (2006), "Information, Search, and Price Dispersion," Handbook of Economics and Information Systems (T. Hendershott, ed.), Elsevier Press, Amsterdam.

Key Reference(s)

• Diamond, P. (1971), "A Model of Price Adjustment", Journal of Economic Theory, 3, 156-168.
• Reinganum, J.F. (1979), "A Simple Model of Equilibrium Price Dispersion", Journal of Political Economy, 87, 851-858.
• Rothschild, M. (1974), "Searching for the Lowest Price When the Distribution of Prices is Unknown", Journal of Political Economy, 82(4), 689-711
• Stigler, G. (1961), "The Economics of Information", Journal of Political Economy, 69 (3), 213-225.

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 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 Reinganum (1979) Model

Reinganum (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]

Reinganum (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.

In sequential search models the number of searches is a random variable, and the expected number of searches is:

[math]\mathbb{E}(n) = \frac{1}{F(r)} \,[/math]

Whereas fixed search commits to [math]n\,[/math] searches up front. There is a trade-off between flexibility (i.e. economizing on information costs) in sequential search and speed (and certainty) in fixed search.

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]

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 firms 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]

Given this distribution of prices consumers will set their reservation price using:

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

However, a firm's demand is zero above [math]r\,[/math], so firms will have no sales in the interval <math\left (r,\frac{\overline{m}\epsilon}{1+\epsilon} \right ]\,</math>, and will set their price at [math]r\,[/math] (as the elasticity of demand is constant).

Therefore:

[math] F(p) = \begin{cases} \hat{F}(p) & \mbox{if}\; p_j \lt r \\ 1 & \mbox{if}\; p_j = r \end{cases} [/math]

To verify that this is an equilibrium, we must check that it is a best response for consumers to set their reservation price as before. The reservation price is:

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

[math]h(r) = \int_{\underline{p}}^r (v(p) -v(r)d\hat{F}(p) + \left [1-\hat{F}(r) \right ] \left[ v(r)-v(r) \right ] -c\,[/math]

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

Therefore we can see that downward sloping demand and cost heterogeneity together give rise to price dispersion with optimizing consumers and firms. In fact both conditions are required. This is shown in detail in the paper. However, briefly, if consumers only want a single unit of the product then it is easy to show that [math]r=v\,[/math], and if firms all had the same marginal cost then the Diamond Paradox would re-assert itself.

A final result in the Reinganum (1979) model is that a reduction in search costs leads to a reduction in equilibrium prices. However, this is not a general result.

The MacMinn (1980) Model

The MacMinn (1980) Model provides an example in which fixed search is optimal. The constraints in MacMinn (1980) are:

• Customers have unit valuation [math]v\,[/math]
• Customers engage in optimal fixed sample search
• Firms have privately observed marginal costs drawn from [math]G(m)\,[/math] on [math][\underline{m},\overline{m}]\,[/math], where [math]\overline{m} \lt v\,[/math].

The MacMinn model can be solved using the Revenue Equivalence Theorem. Each [math]n^*\,[/math] firms competes with [math]n^*-1\,[/math] firms. The firm offering the lowest price 'wins' the 'auction'.

The revenue to any auction where firms have a marginal cost [math]m\,[/math], the lowests price firm cost wins and the the firms with highest marginal cost earns zero surplus is:

[math]R(m) = m( 1-G(m))^{n^*-1} + \int_{m}^{\overline{m}} (1-G(t))^{n^*-1}dt\,[/math]

In the MacMinn model, the firm's expected revenues are:

[math]p(m) \times ( 1-G(m))^{n^*-1}.\,[/math]

This allows use to solve (using integration by parts) to find:

[math]p(m) = \mathbb{E} \left [ m_{min}^{n^*-1} | m_{min}^{n^*-1} \ge m \right ]\,[/math]

where [math]m_{min}^{n^*-1}\,[/math] is the lowest [math]n^*-1\,[/math] draws from [math]G\,[/math].

This gives rise to a distribution of posted [math]F(p) = G(p(m))\,[/math].

For this to be optimal it must be optimal for consumers to sample [math]n^*\,[/math] firms, so that:

[math]\mathbb{E}(B^{n^*+1}) \lt c \le \mathbb{E}(B^{n^*})\,[/math]

So, if the search costs are low enough, a dispersed price equilibrium can exist! In this model there are ex-post differences in consumers information sets (they observe different prices). Furthermore, the greater the variance in marginal costs, the greater the variance in prices. Somewhat counter-intuitively the dispersion in prices also increases as the sample size increases.

Information Clearinghouses

In information clearinghouse models a subset of the consumers consults a clearinghouse that displays prices for a sub-set of firms. In these models, for simplicity, [math]n\,[/math] will denote the number of firms in the market. The general environment is:

• a finite [math]n\gt 1\,[/math] number of price-setting firms competes in a homogeneous product market.
• Firms have unlimited capacity and marginal cost [math]m\,[/math]
• It costs firms [math]\phi \ge 0\,[/math] to list their prices in the clearinghouse
• Consumers have unit demand with a willingness to pay of [math]v \gt m\,[/math]
• [math]S \gt 0\,[/math] are shoppers that consult the clearinghouse (if no prices are listed below [math]v\,[/math], the shopper chooses at random and buys if [math]p \le v\,[/math])
• [math]L \ge 0\,[/math] are loyal customers that buy if [math]p \le v\,[/math]

Information Clearinghouse General Result
If [math]L \gt  0\,[/math] or if [math]\phi \gt  0\,[/math] then equilibrium price dispersion exists
providing [math]\phi\,[/math] is not so large that all firms refuse to list their products

Assume that:

[math]0 \le \phi \lt \frac{n-1}{n}(v-m)S\,[/math]

Then each firm lists with probability:

[math]\alpha = 1 - \left ( \frac{\frac{n-1}{n}\phi}{v-m)S} \right )^{\frac{1}{n-1}}\,[/math]

If a firm lists then its price is drawn from:

[math]F(p) = \frac{1}{\alpha} \left ( 1 - \left ( \frac{\frac{n-1}{n}\phi + (v-p)L}{(p-m)S} \right )^{\frac{1}{n-1}} \right )\,[/math] on [math][p_0,v]\,[/math]

where:

[math]p_0 = m + (v-m)\frac{L}{L+S} + \frac{\frac{n-1}{n}}{L+S}\phi\,[/math]

(so [math]p_0 \gt 0\,[/math] if [math]L \gt 0\,[/math] or [math]\phi \gt 0\,[/math], and [math]p_0 = m\,[/math] otherwise fulfilling the Bertrand Paradox)

If a firm does not list it charges [math]v[/math] (this is a dominant strategy) and each firm's expected profits are:

[math]\mathbb{E}\phi = (v-m)L + \frac{1}{n-1}\phi\,[/math]

The condition on [math]\phi\,[/math] implies that [math]\alpha \in \left ( 0,1 \right]\,[/math].

We must show that a firm can do no better than pricing accoring to [math]F\,[/math]. Pricing outside the support of [math]F\,[/math] is dominanted and it will transpire that pricing in the support of [math]F\,[/math] leads to constant profits through-out the support.

The expected profits of the firm pricing in [math]F\,[/math] are:

[math]\mathbb{E}\pi(p) = (p-m) \left ( L + \left ( \sum_{i=0}^{n-1} \binom{n-1}{i} \alpha^i (1-\alpha)^{n-1-i}(1-F(p))^i \right ) S \right ) - \phi\,[/math]

Using the binomial theorem:

[math]\mathbb{E}\pi(p) = (p-m) \left ( L + \left ((1-\alpha F(p))^n-1 \right ) S \right) - \phi\,[/math]

[math]\mathbb{E}\pi(p) = (v-m)L + \frac{\phi}{n-1}\,[/math]

Therefore the firm's profits are constant on the support and so must be a best-response. When [math]\phi = 0\,[/math] it is weakly dominant to list. When [math]\phi \gt 0\,[/math] and [math]\alpha \in (0,1)\,[/math] a firm's expected profits when it doesn't list are:

[math]\mathbb{E}\pi(p) = (v-m) \left ( L + \frac{S}{n}(1-\alpha)^n-1 \right )\,[/math]

[math]\mathbb{E}\pi(p) = (v-m)L + \frac{\phi}{n-1}\,[/math]

Therefore the firm earns the same expected profit whether it lists or not. This leads to "Ed's observation".

Ed's observation
In clearinghouse models, the use of mixed strategies by firms drives the price-dispersion results.

The Rosenthal (1980) Model

In the Rosenthal (1980) model we suppose:

• [math]\phi = 0\,[/math] (i.e. costless listing)
• [math]\L \gt 0\,[/math] (i.e. some loyal customers)

Since [math]\phi=0\,[/math],  [math]\alpha=1\,[/math] and all firms list at the clearinghouse. The equilibrium distribution of prices is therefore:

[math]F(p) = \frac{1}{\alpha} \left ( 1 - \left ( \frac{\frac{n-1}{n}\phi + (v-p)L}{(p-m)S} \right )^{\frac{1}{n-1}} \right )\,[/math] on [math][p_0,v]\,[/math]

[math]F(p) = \left ( 1 - \left ( \frac{(v-p)L}{(p-m)S} \right )^{\frac{1}{n-1}} \right )\,[/math] on [math][p_0,v]\,[/math]

where:

[math]p_0 = m + (v-m)\frac{L}{L+S} + \frac{\frac{n-1}{n}}{L+S}\phi\,[/math]

[math]p_0 = m + (v-m)\frac{L}{L+S}\,[/math]

[math] \,[/math]

[math] \,[/math]