Consider an industry where price competition is not very important: all of the action is on advertising budgets. Specifically, total value S(in dollars) gets split between two competitors according to their advertising shares. If a_1 is firm 1's advertising investment (in dollars) then its profit is given by:

[(a_1)/(a_1 + a_2)]S - a_1

(The same applies for firm 2). Both a_1 and a_2 must be non negative. If both firms invest 0 in advertising they split the market.

a.) Determine the symmetric Nash equilibrium of the game whereby firms choose a_i independently and simultaneously.

b.) Determine the jointly optimal level of advertising, that is, the level a* that maximizes joint profits.

c.) given that firm 2 sets a_2 = a*, determine firm 1's optimal advertising level.

d.) suppose that firms compete indefinitely in each period t=1,2,..., and that the discount factor is given by \delta [0,1]. Determine the lowest value of \delta to that, by playing grim strategies, firms can sustain an agreement to set a* in each period.