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# EC3313 coursework

1. Consider the market for a homogeneous good, with market demand given by

P(Q) = 200 − 16Q.

Let there be a single incumbent firm (I) and a single potential entrant (E). Let both firms have access to the same technology, characterized by constant marginal cost c = 8 and constant fixed cost F. The fixed cost F is avoidable, i.e., the total cost for a non-producing firm is zero. In addition, assume that F < 100.

• 7 marks Solve for the Stackelberg equilibrium in this market. What are the two firms’ profits in this equilibrium?
• Now, suppose that instead of accommodating entry (engaging in Stackelberg competition) the incumbent decides to deter entry. The timing of the events is as follows. First, the incumbent sets its quantity qI and commits to that. Next, the entrant observes qI and decides whether or not to enter (it believes that the incumbent will not revise qI afterwards). If the entrant does not enter, the incumbent produces the qI it set in the first stage, profits are realized, and the game ends. If the entrant enters, it decides its own quantity qE. The two firms then produce the quantities they have selected, profits are realized, and the game ends. Assume that a potential entrant who expects to obtain zero profit prefers to stay out of the market.
1. 7 marks What is the minimum quantity that the incumbent needs to produce to keep the entrant out of the market (the limit quantity)?
2. 3 marks What is the profit of the incumbent if it chooses to deter entry by producing the limit quantity?
• 10 marks For what valid values of F (recall that we already know that F < 100) does the incumbent prefer entry deterrence (producing the limit quantity) to accommodating entry (competing in Stackelberg competition)?
• 3 marks Explain why the assumption we made about the beliefs of the firms in part (b) may not be very realistic. Illustrate your argument by a game tree.
• 3 marks Discuss how you could augment the model to solve the problem you identified in part (c). Another game tree would be helpful.
1. Consider a real estate developer who has built a new apartment block and needs to decidewhether to sell or rent out the new apartments to consumers. Because the apartments have already been built, they can be released to the market (sold or rented out) at zero marginal cost. Assume that all apartments are identical and that the apartment block is large enough that demand can be fully met without any additional construction. Further assume that consumers derive utility from owning an apartment only indirectly, via the service of living in the apartment. Thus everyone is indifferent between owning the apartment for a given amount of time and renting it for the same time. Finally, assume that the apartments do not depreciate over time.

Let there be two time periods, period 1 and period 2, and let both the developer and the consumers discount period 2 at rate δ, where 0 < δ < 1. Let the rental price in period t be given by Pt = 50−Qt, where Qt is the total number of occupied apartments in period t (i.e., the number of apartments sold up to period t or rented in period t). Note that Pt is the price that consumers are willing to pay to live in an apartment just for period t. Assume that consumers are forward-looking (i.e., they can correctly anticipate future prices).

Please answer the following questions. Make sure that you explain all the steps of your analysis and that you define any new notation that you use.

• 5 marks Suppose the developer does not sell any apartments, but instead rents them out on a period-by-period basis. Determine the profit-maximizing quantities, Q1 and Q2, that the developer will rent out in each of the two periods. Compute the corresponding equilibrium rental prices in each period. Calculate the developer’s total (discounted) profit (in period-1 pounds).
• Now, suppose the developer does not rent out any apartments, but sells theminstead. Assume that all apartments sold up to period t remain occupied in period t (for any t). Let QSt be the number of apartments sold in period t and let PtS be the selling price in period t.
1. 6 marks Derive the indirect demand functions ) and

(note that both prices should depend on both quantities!). Make sure you explain the logic behind your calculations.

1. 10 marks Suppose the developer cannot commit to a second-period quantity in period 1, that is, he sets only the first-period quantity in period one, postponing the determination of the second-period quantity to period two. Solve for the quantities and prices in both periods and find the total discounted profit of the developer under this scenario.
• 5 marks Does the developer obtain higher profits when he rents the apartments out or when he sells them? Why does this happen?
• 8 marks Denote the rental profit from part (a) by Πrent and the sales profit from part (d) by Πsales. Now compute the percentage difference between these two profits, ∆ = (Πrent−Πsales)/Πrent×100, and plot this difference as a function of the discount factor, δ. Is this relative profit difference increasing or decreasing in the discount factor? Why?
1. Consider a monopolist selling candy to two consumers (assume candy is infinitely divisible, so that the monopolist can sell any nonnegative, real quantity).

The demand function of consumer 1 is p1(q1) = 9 − 3q1.

Consumer 2’s demand function is p2(q2) = 8 − 5q2.

All quantities are expressed in pounds (weight) and prices in pounds (currency). The monopolist can produce candy at no cost.

• 5 marks Suppose the monopolist can distinguish between the two consumers and is also able to offer them fixed-quantity packages, so that they cannot continuously choose any amount they want. What packages will the monopolist offer to the two consumers (state quantities and fees)? What profit will the monopolist earn?
• 5 marks Suppose the monopolist can still distinguish between the two consumers, but now, because of regulation, it must supply any amount of candy that the consumers choose (i.e., it cannot restrict consumers’ choice to a menu of fixed packages). Thus, the monopolist will charge each consumer a different constant price per pound of candy. Furthermore, the monopolist is not allowed to charge consumers any fixed fees. What price will the monopolist set for each consumer? What quantities will be consumed, and what will be the monopolist’s profit?
• 10 marks Now suppose that the monopolist cannot distinguish anymore between the two consumers, but is allowed to offer fixed packages. How can the monopolist achieve the highest profit (state quantities and fees for each package)? What is the monopolist’s profit?
• 5 marks Suppose consumer 2’s mum tells her to cut down on her candy consumption. As a result, this consumer’s demand function collapses to p2(q2) = 4 − 5q2. The monopolist still cannot distinguish the consumers, but can offer fixed packages. What are the optimal packages now?
• 8 marks For this part, suppose consumer 2’s demand function has returned to normal, i.e., p2(q2) = 8 − 5q2. Like in part (b), the monopolist can distinguish between the two consumers, but is not allowed to restrict the consumers to fixed packages. However, unlike part (b), the monopolist is now allowed to ask the consumers for a fixed payment to enter the store. What kind of tariff will the monopolist use for each consumer to maximize profit?
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