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Question 1: There is a mass of voters who can be lined up over the interval [0; 1].The interpretation is that these voters di
er according to their ideological positions. So,0 represents the extreme left whereas 1 the extreme right. Assume the distribution is uniform.There are two political parties, D and R. Assume the only thing the parties care about are winning the election. They choose party platforms, which are two numbers xD [0; 1]and xR [0; 1].

Assume each voter votes for the party whose chosen platform is closest to their own ideological position. So, for example, consider the voter with ideological position rep-resented by the point 0.3. If xD was 0.2 and xR was 0.45 then this voter will vote for party D since |0.3 - 0.2| < |0.3 - 0.45| If a voter is indi erent then he votes for either

party with equal chance.

(i) Set this up as a game. [Mention who are the players, what are their action sets and what are the payoffs.]

(ii) Show one Nash Equilibrium of the game.

(iii) Is the Nash Equilibrium unique? If not, show another than the one you showed in part (ii). If it is unique, then prove uniqueness.

Question 2: There are N individuals in a society. Assume N is a positive integer strictly greater than 1. Individual i has income mi > 0. This income can be spent on getting private good ci  0, or contributing to a public good, gi  0. Satisfaction of

budget constraint requires, as usual, that ci + gi  <=mi By de nition, total public good G is given by

G =n∑i gi   

Each person's utility function ui(G; ci) is given by

ui(G; ci) = ln(G) + ci

where remember that \ln" stands for natural logarithm.

(i) Argue rst that budget constraint can be rewritten as ci + gi = mi. Hence utility function can be rewritten more conveniently as ui(gi; g-i) = ln(G) + mi -gi.

(ii) Find a symmetric Nash equilibrium, bgi and hence the total public good G  produced in this Nash equilibrium.

(iii) Solve for the socially ecient level of public good G. (Recall that socially ecient level of public good G is the level of G that maximizes the expression PN i=1 ui(gi; g-i).)

(iv) Give the economic rationale behind the fact that G < G. What happens to the

ratio

Question 3: Consider the following asymmetric Cournot duopoly. As usual, inverse

demand P(Q) is given by P(Q) = a - Q with Q = q1 + q2 where q1  >0 and q2  >0 are

the outputs of rms 1 and 2 respectively.

The asymmetry is with respect to costs. Assume marginal costs of rms 1 and 2 are

given by c1 > 0 and c2 > 0 where c1 6= c2. Assume also that there are no xed costs.

Solve for the Nash equilibrium (show all your workings). Consider in particular the

following two di
erent scenarios.

(i) ci < a=2; for both i = 1; 2.

(ii) c1 < c2 < a and 2c2 > c1 + a.