1. Take an economy with 2 consumers, 1 private good and 1 public good. Let each consumer have an income of M. The prices of public and private goods are both 1. Let the consumers have utility functions U4 = log(x4) + log(G) UB = log(xf) + log(G) a. Assume that the public good is privately provided so that G = g4 + g&. Eliminating A from the utility function using the budget constraint, show that along an indifference curve 1 1 | Al __+ B~ | _ ag [gf'+gf" Mg*\"]+d'g [g*l+g3] 0 dg _9+g> _ dg8 M gA4 and hence: b. Consider A choosing g4 to maximize utility. Show that the optimal choice satisties: a_M_g 2 2 c. Repeat part b for B, and calculate the level of private provision od the public good. d. Calculate the optimal level of provision for the welfare function W=U4+U8 Contrast this with the private provision level