21.14 Pricing and Search Theory
21.14.1 Varian and Purohit (1980)
From Stigler’s seminar paper (Stiglitz and Salop 1982; Salop and Stiglitz 1977), model of equilibrium price dispersion is born
Spatial price dispersion: assume uninformed and informed consumers
- Since consumers can learn from experience, the result does not hold over time
Temporal price dispersion: sales
This paper is based on
Stiglitz: assume informed (choose lowest price store) and uninformed consumers (choose stores at random)
Shilony (Shilony 1977): randomized pricing strategies
\(I >0\) is the number of informed consumers
\(M >0\) is the number of uninformed consumers
\(n\) is the number of stores
\(U = M/n\) is the number of uninformed consumers per store
Each store has a density function \(f(p)\) indicating the prob it charges price \(p\)
Stores choose a price based on \(f(p)\)
Succeeds if it has the lowest price among n prices, then it has \(I + U\) customers
Fails then only has \(U\) customers
Stores charge the same lowest price will share equal size of informed customers
\(c(q)\) is the cost curve
\(p^* = \frac{c(I+U)}{(I+U}\) is the average cost with the maximum number of customers a store can get
Prop 1: \(f(p) = 0\) for \(p >r\) or \(p < p^*\)
Prop 2: No symmetric equilibrium when stores charge the same price
Prop 3: No point masses in the equilibrium pricing strategies
Prop 4: If \(f(p) >0\), then
\[ \pi_s(p) (1-F(p))^{n-1} + \pi_f (p) [1-(1-F(p))^{n-1}] =0 \]
Prop 5: \(\pi_f (p) (\pi_f(p) - \pi_s (p))\) is strictly decreasing in \(p\)
Prop 6: \(F(p^* _ \epsilon) >0\) for any \(\epsilon> 0\)
Prop 7: \(F(r- \epsilon) <1\) for any \(\epsilon > 0\)
Prop 8: No gap \((p_1, p_2)\) where \(f(p) \equiv 0\)
Decision to be informed can be endogenous, and depends on the “full price” (search costs + fixed cost)
21.14.2 Lazear (1984)
Retail pricing and clearance sales
Goods’ characteristics affect pricing behaviors
Market’s thinness can affect price volatility
Relationship between uniqueness of a goods and its price
Price reduction policies as a function of shelf time
Set up
Single period model
\(V\) = the price of the only buyer who is willing to purchase the product
\(f(V)\) is the density of V (firm’s prior)
\(F(V)^2\) is its distribution function
Firms try to
\[ \underset{R}{\operatorname{max}} R[1 - F(R)] \]
where \(R\) is the price
\(1 - F(R)\) is the prob that \(V > R\)
Assume \(V\) is uniform \([0,1]\) then
\(F(R) = R\) so that the optimum is \(R = 0.5\) with expected profits of \(0.25\)
Two-period model
Failures in period 1 implies \(V<R_1\).
Hence, based on Bayes’ theorem, the posterior distribution in period 2 is \([0, R_1]\)
\(F_2(V) = V/R_1\) (posterior distribution)
\(R_1\) affect (1) sales in period 1, (2) info in period 2
Then, firms want to choose \(R_1, R_2\) .Firms try to
\[ \underset{R_1, R_2}{\operatorname{max}} R_1[1 - F(R_1)] + R_2 [1-F_2(R_2)]F(R_1) \]
Then, in period 2, the firms try to
\[ \underset{R_2}{\operatorname{max}} R_2[1 - F_2(R_2)] \]
Based on Bayes’ Theorem
\[ F_2(R_2) = \begin{cases} F(R_2)/ F(R_1) & \text{for } R_2 < R_1 \\ 1 & \text{otherwise} \end{cases} \]
Due to FOC, second period price is always lower than first price price
Expected profits are higher than that of one-period due to higher expected probability of a sale in the two-period problem.
But this model assume
no brand recognition
no contagion or network effects
In thin markets and heterogeneous consumers
we have \(N\) customers examine the good with the prior probability \(P\) of being shoppers, and \(1-P\) being buyers who are willing to buy at \(V\)
There are 3 types of people
- customers = all those who inspect the good
- buyers = those whose value equal \(V\)
- shoppers = those who value equal \(0\)
An individual does not know if he or she is a buyer or shopper until he or she is a customer (i.e., inspect the goods)
Then, firms try to
\[ \begin{aligned} \underset{R_1, R_2}{\operatorname{max}} & R_1(\text{prob sale in 1}) + R_2 (\text{Posterior prob sale in 2})\times (\text{Prob no sale in 1}) \\ & R_1 \times (- F(R_1))(1-P^N) + R_2 \{ (1-F_2(R_2))(1- P^N) \} \times \{ 1 - [(1 - F(R_1))(1-P^N)] \} \end{aligned} \]
Based on Bayes’ Theorem, the density for period 2 is
\[ f_2(V) = \begin{cases} \frac{1}{R_1 (1- P^N) + P^N} \text{ for } V \le R_1 \\ \frac{P^N}{R_1 (1- P^N) + P^N} \text{ for } V > R_1 \end{cases} \]
Conclusion:
As \(P^N \to 1\) (almost all customers are shoppers), there is not much info to be gained. Hence, 2-period is no different than 2 independent one-period problems. Hence, the solution in this case is identical to that of one-period problem.
When \(P^N\) is small, prices start higher and fall more rapid as time unsold increases
When \(P^N \to 1\), prices tend to be constant.
\(P^N\) can also be thought of as search cost and info.
Observable Time patterns of price and quantity
Pricing is a function of
The number of customers \(N\)
The proportion of shoppers \(P\)
The firm’s beliefs about the market (parameterized through the prior on \(V\))
Markets where prices fall rapidly as time passes, the probability that the good will go unsold is low.
Goods with high initial price are likely to sell because high initial price reflects low \(P^N\) - low shoppers
Heterogeneity among goods
The more disperse prior leads to a higher expected price for a given mean. And because of longer time on shelf, expected revenues for such a product can be lower.
Fashion, Obsolescence, and discounting the future
The more obsolete, the more anxious is the seller
Goods that are “classic”, have a higher initial price, and its price is less sensitive to inventory (compared to fashion goods)
Discounting is irrelevant to the pricing condition due to constant discount rate (not like increasing obsolescence rate)
For non-unique good, the solution is identical to that of the one-period problem.
Simple model
Customer’s Valuation \(\in [0,1]\)
Firm’s decision is to choose a price \(p\) (label - \(R_1\))
One-period model
Buy if \(V >R_1\) prob = \(1-R_1\)
Not buy if \(V<R_1\) probability = \(R_1\)
\(\underset{R_1}{\operatorname{max}} [R_1][1-R_1]\) hence, FOC \(R_1 = 1/2\), then total \(\pi = 1/2-(1/2)^2 = 1/4\)
Two-period model
Two prices \(R_1, R_2\)
\(R_1 \in [0,1]\)
\(R_2 \in [0, R_1]\)
\[ \underset{R_1}{\operatorname{max}} [R_1][1-R_1] + R_2 (1 - R_2)(R_1) \]
\[ \underset{R_1}{\operatorname{max}} [R_2][\frac{R_1 - R_2}{R_1}] \]
FOC \(R_2 = R_1/2\)
\[ \underset{R_1}{\operatorname{max}} R_1(1-R_1) + \frac{R_1}{2}(1 - \frac{R_1}{2}) (R_1) \]
FOC: \(R_1 = 2/3\) then \(R_2 = 1/3\)
\(N\) customers
Each customer could be a
shopper with probability p with \(V <0\)
buyer with probability \(1-p\) with \(V > \text{price}\)
Modify equation 1 to incorporate types of consumers
\[ R_1(1 - R_1)(1- p^N) + R_2 (1- R_2) R_1 (1-p^N) [ 1 - (1-R_1)(1-p^N)] \]
Reduce costs by
Economy of scale \(c(\text{number of units})\)
Economy of scope \(c(\text{number of types of products})\) (typically, due to the transfer of knowledge)
Experience effect \(c(\text{time})\) (is a superset of economy of scale)
Lal and Sarvary (1999)
Conventional idea: lower search cost (e.g., Internet) will increase price competition.
Assumptions:
Demand side: info about product attributes:
digital attributes (those can be communicated via the Internet)
nondigital attributes (those can’t)
Supply side: firms have both traditional and Internet stores.
Monopoly pricing can happen when
high proportion of Internet users
Not overwhelming nondigital attributes
Favor familiar brands
destination shopping
Monopoly pricing can lead to higher prices and discourage consumer from searching
Stores serve as acquiring tools, while Internet maintain loyal customers.
Kuksov (2004)
For products that cannot be changed easily (design),lower search costs lead to higher price competition
For those that can be easily changed, lower search costs lead to higher product differentiation, which in turn decreases price competition, lower social welfare, higher industry profits.