consider a game between two firms. the timing of the game is as follows:
-firm 1 chooses to make a costly investment in capital. denote the level chosen by K. The cost of investment is r per unit of capital.
-firm 1 and firm 2 decide to make (additional) investments and then produce. each unit of output requires on unit of capital and one unit of labor. labor costs w per unit.
the market demand curve is given by: P=A-B(q1+q2)
there are no other variables costs of production outside of capital and labor. however, each firm must incur a fixed cost of F to produce any positive quantity.
1. what are the marginal costs of firm 2 in period 2?
2. what are the marginal costs of firm 1 in period 2? do they depend on k?
3. if k=0, how is the game different than a standard cournout game?
4. derive the best response function for firm 2 in period 2 assuming that firm 2 produces
5. derive the best response function for firm 1 as a function of q2 and k.
6. Draw a graph showing the best response functions for firm 1 and firm 2 in (q1,q2) space. Show the two resulting equilibrium allocations should the firms produce.
7.Write out the profit function for firm 2 assuming that K = 0. Repeat for the case where K = ∞. Which profit is higher?
8. Work out parts a-e from the numerical example Practice Problem 12.2 on p. 279 of the book (this is mostly just plugging in numbers into what you've derived above).
9. Explain, in plain English, why the incumbent's firm choice to set K = 32 induces social welfare loss.