Assume that KPM’s demand function for coffee is (1) Q = 1 – p, where p is the pr

Question

Assume that KPM’s demand function for coffee is
(1) Q = 1 – p, where p is the price of a cup of espresso.
(a) Assume that the marginal cost of a cup of coffee is 10 cents. Assume that the seller can use a two-part tariff coffee purchasing plan (where plan has a price per cup and an entry fee). Concretely, the amount of money that consumer spends on X cups of espresso is given by T = F + p*X, where F is the entry fee in dollars. Under that plan, what is the profit-maximizing price of a cup of coffee? How many cups will KPM drink? What is the maximum entry fee that the seller can charge KPM to participate in the plan? [Hint: Use the fact the that entry fee should extract consumer’s surplus, when possible. Consumer surplus is the area of the triangle above the equilibrium price and the price at which demand equals zero. Given the demand function, that price is equal to p^=1.]
(b) Assume that CAD’s demand for coffee is given by
(2) Z = 0.5 – p.
What is the profit-maximizing plan (price plus entry fee) that the seller can charge CAD? How many cups of coffee will CAD drink under the plan designed for her?
(c) Assume that the seller cannot differentiate between CAD and KPM. In such a situation, would you expect that KPM will purchase under the plan designed for him (as in point (a), above) or under the plan designed for CAD (as in point (b), above).
(d) What can the seller do to make CAD’s plan unattractive to KPM? In answering this question discuss the following ideas (i) participation constraint; (ii) self-selection or individual rationality constraint; (iii) informational rent.

Explanation / Answer

(a) "Under that plan, what is the profit-maximizing price of a cup of coffee?" The profit to the seller is V=F + p*X - 0.10X. The marginal profit (derivative with respect to quantity) is MV. MV=p-0.10=0 (MV=0 at max profit) So, p=0.10 The seller does not behave as a regular monopoly, for example, because it charges a tariff. This makes p=mc the optimal price because it maximizes what would normally be consumer surplus. But, in this case, the firm will extract all consumer surplus in the form of the tariff. "How many cups will KPM drink?" Plug your p into the demand function. Q = 1 – p Q = 1 – 0.1 Q = 0.9 "What is the maximum entry fee that the seller can charge KPM to participate in the plan?" This is equal to consumer surplus. In this case, it is a triangle with the base of the quantity and the height of (1-mc) CS=(1/2)*0.9*0.9 CS=0.405 (b) Assume that CAD’s demand for coffee is given by (2) Z = 0.5 – p. "What is the profit-maximizing plan (price plus entry fee) that the seller can charge CAD?" V=F + p*X - 0.10X MV=p-0.10=0 p=0.10 Z = 0.5 – p Z = 0.5 – 0.10 Z = 0.4 F = (1/2)*0.4*0.9 F = 0.18 "How many cups of coffee will CAD drink under the plan designed for her?" We already found Z = 0.4 "(c) Assume that the seller cannot differentiate between CAD and KPM. In such a situation, would you expect that KPM will purchase under the plan designed for him (as in point (a), above) or under the plan designed for CAD (as in point (b), above)." KPM will choose the CAD option only if his consumer surplus is greater than the tariff for CAD. At a quantity of Q=0.4, he would be willing to pay: Q = 1 – p 0.4 = 1 - p p = 0.6 So, his consumer surplus would be given by the area of a trapezoid. CS=(0.6-0.1)*0.4 + (1/2)*(1-0.6)*0.4 CS=0.28 The tariff for CAD was only 0.18. So, KPM will prefer CAD's offer to the KPM offer. "(d) What can the seller do to make CAD’s plan unattractive to KPM?" (i) participation constraint If the seller can restrict KPM from purchasing CAD's plan, that works great. (ii) self-selection or individual rationality constraint If the seller and charge KPM a fine of $0.10 or more for choosing CAD's plan, then KPM will not choose CAD's plan. (iii) informational rent If the seller only advertises the CAD plan to CAD and it costs more than $0.10 for KPM to discover this information, then KPM will not choose the CAD plan.

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