Mention steps.

There are two individuals, A and B, with quasilinear preferences over a private good, xi, and over a public good, G, represented by the following utility function, Ulixi, G) = x.- + In G. Each of them has inoome, mi, which can be either spent on the private good, 17:, or contributed to the public good, G. 1 unit of individual oontributions, 3:, from either individual is used to produoe 1 unit of the public good, G. The social planner has a Rawlsian social welfare function given by W(UA, HR) = min[5UA, SUB]. a) [3 pts.] Derive the Lagrangean corresponding to the optimization problem faced by individual B. b} [3 pts.] Calculate the firstorder conditions. :1 [2 pts.] Derive the best response function of individual B, 93. d} [4 pts.] Calculate the demand functions for x; and G". e] [2 [313.] State the conceptual difference between the best response function, 33, and the function, 3;. f] [2 pm] Brieflv explain whether the bundle Lg; {1.5, 95) constitutes a Nash equilibrium. g} [4 pts.] Assuming that m = mg, calculate for what income range 6* = 1. h} [2 pm] State whether the individual contributions, 91-, would differ across two individuals who earn different but sufficiently low income levels, mg. Justify your answer using your ndings from parts d] and g]. i) (2 pts.) Write down the social planner's problem. j) (2 pts.) Show that the social welfare function implies that XA = XB. k) (2 pts.) Briefly explain why the Rawlsian welfare function captures only equity considerations. I) (2 pts.) Assume for the rest of the question that mA = 2 and mp = 1. Derive an expression for individual A's utility function where G is the only endogenous variable contained in it. m) (2 pts.) Use your result from part I) to calculate the socially efficient quantity of the public good. Hint: Take a first-order condition with respect to G. n) (2 pts.) Verify that the Samuelson condition yields the same G* as in part m). o) (2 pts.) Briefly explain why the decentralized provision of the public good is (in)efficient for this economy