3. Suppose there are two individuals, 1' = 1, 2. Each person has wealth in (w '3} 21) and consumes both a public and a private good. The public good is provided by contributions from each individual. If persons 1 contributes 91 and person 2 contributes 92, then total public good provision is G = 9'1 + 92. If person i contributes 9,, is remaining wealth, is g,- is i's private consumption. Each person's utility depends on the amount of the public good, G, and private consump- tion, w;- g, for person i : 1, 2. Preferences are: \"1: 91+H2 +'w1g1 +{THI91}(1+92)= 1W1 + 92 +[1II191)(91+ 92] H2 2 91+92 + \"we 92 + {mg as. + 2) = mg + g1+[w2 92)(91+ 92) Also, note that with G = 511 +92, a1: 2 0+ w;- pa: + [as 9013' = as + 9,; + [as gG. [a] Find the (symmetric) Nash equilibrium levels of g1 and 9-2. [b] Show that total utility U = ul + it; depends only on G and W = a: + w. (c) Find the socially optimal level of the public good the value of Q1 + g2 that maximizes total utility U = u; + 1'12. ((1) Show that the Nash equilibrimn level of the public good is less than the socially optimal level . 4. There are three rms 2' = 0, 1,2. The game is played over two periods. In the rst period, rm I] chooses quantityr rm :3 I]. In the second period, after they observe on, rms 1' = 1 and 2 choose their quantities q,- E II] simultaneously. The payo of each rm i = CI, 1, 2 is given as follows \"0 = (1 Q)qna H1: [1 Qllha and \"2 = [1le2 where Q = on + q1 + 92 stands for the total quantity. (a)I Suppose that rm [I chooses q\" = 1,12 in the rst period. Find the Nash equilibrium in the ensuing subgame in the second period. (b) Find the subgame-perfect equilibrium (of the whole extensive-form game). Now, change the timing as follows: two rms 3' = 1 and 2 choose an and 9'2 simultaneously in the rst period, and then, rm [1 chooses go in the second period after it observes q1 + 112- {c} Find the subgame perfect equilibrium lmder this nevlir timing rule