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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 14:21, 26 January 2010
, 14:21, 26 January 2010no edit summary
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:
<center><math>\mathbb{E}(n) = \frac{1}{F(r)} \,</math></center>
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===
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 '''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} < v\,</math>. The MacMinn model can be solved using the [http://en.wikipedia.org/wiki/Revenue_equivalence_theorem#Revenue_equivalence 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: <center><math>R(m) = m( 1-G(m))^{n^*-1} + \int_{m}^{\overline{m}} (1-G(t))^{n^*-1}dt\,</math></center> In the MacMinn model, the firm's expected revenues are: <center><math>p(m) \times ( 1-G(m))^{n^*-1}.\,</math></center> This allows use to solve (using integration by parts) to find: <center><math>p(m) = \mathbb{E} \left [ m_{min}^{n^*-1} | m_{min}^{n^*-1} \ge m \right ]\,</math></center> 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: <center><math>\mathbb{E}(B^{n^*+1}) < c \le \mathbb{E}(B^{n^*})\,</math></center> 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.
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