Problem 2. Screening: In class, we considered price discrimination via quality. The same model also works for price discrimination via lump sum discount (buy more at a lower price per unit). Consider a small yogurt shop that sells fresh-made yogurt to customers. The shop owner is facing two types of customers and is currently designing a price menu and want to max. expected revenue by giving a larger discount to the customers who buy more.

 Buyer’s utility function is given by: u(v, q, t) = v ⋅ q − p ⋅ q

 In which:

o Her willingness to pay for quality: v ∈ {1,2}.

o q: quantity of yogurt she is going to buy

o p: the price she has to pay for each unit she buys

 Seller has cost of production for each customer: c(q) = q2

 The profit is given by: p ⋅ q − c(q)

 Seller does not know buyer’s willingness to pay but knows the distribution

Prob(v = 1) = 4 / 5 ; Prob(v = 2) = 1/ 5 .

Solve for the second degree price discrimination pricing strategy. [Hint: Seller wants the low type to participate and does not want high type to pretend to be low type.]