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# Determining the optimal number of plants

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Suppose that any firm intending to produce SOMA must build an integer number of plants: 0,1,2,... Building Q plants costs each firm 3.5 x Q dollars. Each plant produces one unit of SOMA. If firm 1 builds Q1 plants and firm 2 builds Q2 plants, the market price p for one unit of SOMA will be 9 - (Q1 + Q2). For example, if firm 1 builds 2 plants and firm 2 builds 4 plants, the market price will be 9 - (2 + 4) = 3 per unit. At this price firm 1 will make a profit of 2 x 3 - 2 x 3.5 = -1 while firm 2 will make a profit of 4 x 3 - 4 x 3.5 = -2. Assume, no firm will build more than 4 plants.

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1) Set up a 4-by-4 table that records the profits of each firm for each possible choice of the number of plants to build. Then, use your table to answer the questions below.

2) If there was a single firm capable of producing SOMA, how many plants would it install to maximize profit?

3) Suppose not there are two firms (aware of each other) capable of producing SOMA with the technology described above. Each chooses the number of plants it must build independently of the other and at the same time. What is the equilibrium number of plants that will be built by each?

4) Suppose that firm 1 builds before firm 2 (firm 1 is aware that firm 2 will decide after firm 1 decides and firm 2 will see how many plants firm 1 has built before having to decide). How many plants should firm 1 build and what will firm 2's reaction be?