Consider a market with an incumbent, firm 1, facing the entry of a rival, firm 2. Firm 1 can make an investment _1 (_1 )=((4_1)/3)^2 to accommodate the entry of firm 2. The inverse demand in this market is ()=16 where =1+2. Firms produce a homogeneous good and are Cournot competitors. Firm 1's cost function is _1 (_1,_1)=(4_1)1+((4_1)/3)^2 and firm 2's cost is _2 (_2 )=4_2.

A) Determine 1 and 2 as a function of _1 at the equilibrium of the second-stage, which you can denote _1^ (_1 ) and _2^ (_1 ).

B) Knowing firm 1 chooses its investment _1 to accommodate the entry of firm 2 at the first stage of the game, determine the value taken by _1 to maximize firm 1's profit, which can be written as _1=[16_1^ (_1 )_2^ (_1 )(4_1)] _1^ (_1 )((4_1)/3)^2. (Hint: since firms face a linear demand with a slope of -1 and have a constant marginal cost, the term [16_1^ (_1 )_2^ (_1 )(4_1)] _1^ (_1 ), which represents firm 1's profit before the investment cost, can be simplified quite a bit).