2. Suppose a household consists of two agents, indexed by i=1,2. Let each agent i have a utility function given by U_i(L_i, c) = a In(c) + (1-a)ln(l_i), i = 1, 2, !3! where a is the preference parameter of each agent, I_i is i's leisure consumption, and c is the household consumption (as public good). The total amount of time available to each agent is 1, so the time constraint is given by 1 = _i +h_i, where h_i is work hours for i. The wage rate of agent i is given by w_i. Non-labor income is Y = y1 +y2, with y_i denoting the non-labor income of agent i. Suppose also each agent behaves non-cooperatively. (3 points) Write down the budget constraint. b. (6 points) Derive the optimal work-hour choice for each agent. a. Suppose the cumulative distribution function is given by F(w)=w, with w takes on values between 0 and 1. (4 points) If agent 1 and agent 2 do not work, i.e. h1 = 0 and h2 = 0, what is the probability that this event occurs? Express it in terms of F(w). (8 points) Suppose Y=0.1. Suppose from the data Pr(h1= 0, h2 = 0) = 0.09. What is the value of a? [leave the answer in terms of a fraction is ok] (5 points) Use the value of a obtained in part (d), and all other values given there, calculate the probability that agent 1 does not work. f. . d. e. (5 points) Suppose there is a preference shock. In particular, the value of a is doubled, and Y remains the same. Recalculate the probability that agent 1 does not work. (5 points) what do you conclude on the effect of a positive preference shock on labor-leisure choice g. (i.e. work more/less)? What is your reasoning?