Assume that JAO\'s demand function for coffee is Q = 10 - p, where p is the pric

Question

Assume that JAO's demand function for coffee is Q = 10 - p, where p is the price of a cup of coffee.
(a) Assume that the marginal cost of a cup of coffee is one dollar. Assume that the seller can use a two-part tariff coffee purchasing plan (where plan has a price per cup and an entry fee). Specifically, the amount of money that consumer spends on X cups of espresso is given by T = F +pX, 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 JAO drink? What is the maximum entry fee that the seller can charge JAO to participate in the plan? [Hint: Use the fact the that entry fee should extract consumer's surplus, when possible]
(b) Assume that LLO's demand for coffee is given by Z = 5 - p. What is the profit-maximizing plan (price plus entry fee) that the seller can charge LLO? How many cups of co§ee will LLO drink under the plan designed for her?
(c) Assume that the seller cannot di§erentiate between LLO and JAO. In such a situation, would you expect that JAO will purchase under the plan designed for him (as in point (a), above) or under the plan designed for LLO (as in point (b), above).
(d)What can the seller do to make LLO's plan unattractive to JAO? In answering this question discuss the following ideas (i) participation constraint; (ii) self-selection or individual rationality constraint; (iii) informational rent.

Explanation / Answer

a. Marginal Costs and Revenues are the first derivatives of Revenue and Cost functions Marginal Revenue T = F + pX dt/dx = p = MR (this should make sense; the revenue of each additional coffee sold is the price of the coffee!) Marginal Cost dt/dx = 1 = MC (given in the problem) Set marginal cost and marginal revenue equal to each other Q = 10 - p p = 10 - Q p = 10 - Q = 1 Q = 9 P = 1 Profit is maximized at a price of 1 and 9 cups of coffee will be sold Now to Find out how much consumer surplus is: NOTE*** a company's supply function is also its marginal cost Set the demand function equal to the supply function BUT we first must use the demand function in respect to P because that is what our marginal cost is annotated in Demand Function = MC 10 - Q = 1 9 = Q 1 = P This should not be surprising. Demand and Supply meet at the same place where marginal cost and marginal revenue meet. Consumer Surplus is the triangular area above equilibrium price and below the demand curve. It is the benefit that people who would be willing to pay more for the coffee, but are still getting it at the lower price. for this we can either use the fundamental theorem of calculus or the triangle area formula 1/2 base * height base = 9 (our equilibrium quantity) height equal height if the price were zero minus the equilibrium price = 10 - 1 = 9 1/2(9) * 9 = 40.5 total consumer surplus the maximum you can charge for entry though is for that one customer who will pay the most for the coffee which is 9*** b. Once again we have MC = 1 MR = p Now we plug in the new p p = 5 - Z = 1 Z = 4 P = 1 LLO will drink 4 coffees at a dollar a piece. c. The coffee shop should sum up both their demand functions and do the same as before to find the point of optimization d. you will need to look these terms up on the professors notes or in the book.

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