Dale Stahl : "Oligopolistic Pricing with Sequential Consumer Search"
Program
1. Introduction & Definitions 2. Bertrand & Diamond models 3. Introduction Stahl model 4. Analysis Stahl model 5. Welfare Analysis 6. Conclusions 7. Critique on the model 8. Questions
1. Introduction & Definitions Consumers are trying to buy products at the lowest price. They have to search to find this information Search: spending time to find information Time: is money (for some people)
Information: Where can you find the particular product at the lowest price?
2 Bertrand & Diamond models 2 models already available: All information available: price competition: Bertrand Model Marginal revenue = Marginal Costs Bertrand paradox: only two firms, no profit! |:Nash Equilibrium No information available: firms can charge what they want. Firms charge the monopoly price. Firms produce until the last product which generates profit (Marginal Revenue = 0) This is called the Diamond paradox. Again a Nash Equilibrium
3 Introduction Stahl model In his article Stahl is looking for a connection between these two models. (See picture on sheet) In his model there are two main variables: Consumers, who like to shop, so called funshoppers; they don’t have any costs: µ Then there are people who don’t like to shop, the so-called non-shoppers. (1-µ) They have costs for every time they search. c: search costs; cost the (1-µ) part of the population have for each time they search.
Basic Assumptions in Stahl Model First search is for "free" All products are identical (homogenous)
Firms are trying to prevent consumers to search another time. At the same time they want to make as much as possible profit. “For consumers with search costs c continuous search is profitable, if the expected benefits from continued search exceed the cost” (page 702 Stahl) If consumers are indifferent than the gains are equal to the costs of searching. In the literature this conjecture is called the reservation price by consumers
At the same time the firms also make a conjecture about the consumers price:
4 Analysis Stahl model
Bertrand Paradox/ Diamond Paradox Illustration ( not rigorous ) 6P This paper tries to bridge the gap between the two striking results
No
Diamond
! Information
Full
Bertrand
How do prices move with the level of informed people? 7 P
(not rigorous)
P= MC
0
1
Optimal Search behavior: Illustration: (example) i (1,2,3….,10) • 10 firms: Pi • Firms are identical P 10 9 8 6
7
5 4
1
2
3
1 2 3 4 5 6 7 8 9 10 What are the gains from searching an other price? P7/P10: No gains k-1 Gains are: V(pk) = (pk-pi)1/N i=1
Optimal search behavior: More general: • You observe a price Pk F(p) Gains are
Gains are zero Pk V(Pk) = 3kP)*f(p)dp
F(p)
_
P -
Pk
P
Consumers: Assume they conjecture that the price distribution in the market is a function: G(p) , and prices are between the lower bound and upper bound. G(p) G(p)
P -
_ P
A consumer ( non-shopper) optimal behavior is characterized by a reservation price: rf rf-p)g(p)dp =c p -
What are the gains from searching an other price? • P7,P8,P9,P10: gains are zero • P5: gains: (P6-P5)*1/10 • P4: gains: (P6-P4)*1/10 • P3: gains: (P6-P3)*1/10 • P2: gains: (P6-P2)*1/10 • P1: gains: (P6-P1)*1/10 In general: • N prices: piL 1 • Suppose as a consumer you observe price Pk Gains are: V(pk) =
k −1
∑(p i =1
k
− pi ) * 1
N
Or by using the propability function f(p); Gains are: V(pk) = ∑ ( p − p) * f ( p)dp pk
k
P −
−
Result 1: No firm will choose a price above rc Result 2: Diamond type of result p1=p2=rc is not a NE! “The intuition for this fact is that profits could be discretely increased by undercutting atoms, so price distribution with atoms cannot be optimal”(page 703, left column ) Result 3: Is a Bertrand type of result a NE, p1=p2=0? No by charging in between p o, r you create a larger profit. c
Result 4: There is no NE in which firms charge some price p1=p2=p, with propability 1.There is no NE in pure strategie.
Firm pricing rule in a mixed strategy: • 2 firms • S o, r c
Consider the problem faced by firm 1. Firm 1 conjectures that firm 2 adopts a price according to a distribution function F(p) i S
