2. Constructing an Equilibrium Households live two periods and have preferences U(C₁) + BU (C₂), where 0 < 3 < 1 and U is the utility function and satisfies our usual assumptions. There are Ñ households in the economy. N₁ of these households have endowment y in the first period and no endowment in the second - these agents are called "Type 1". The remaining N₂ have no endowment in the first period and y2 in the second period - these agents are called "Type 2." Hence the resources of the economy are N131 in the first period and N₂92

2. Constructing an Equilibrium Households live two periods and have preferences U(c) + BU(c2), where 0 0. The type 1 agent faces budget constraints Y1 c+s' rs' where consumption for the type i agent in period j is denoted c. The type 2 agent faces budget constraints G + s² Y2 +rs? = The resource constraints are N1c + N2c Nịc+ N2c (a) State the maximization problem solved by each type of agent and derive the first- order and second-order conditions. Derive the solution using the implicit function theorem.

In a two-good market, a consumer starts with an initial endowment of (x₁, x2) = (15.00, 5.00), while the market prices for these goods are given by (P1, P2) = (7.00, 3.00). The consumer has the following utility function: U 0.52 0.48 - Given this information, what will this consumer's final choice of quantity for each good be? x1 = x2 =

9. Consider an Edgeworth box economy with two consumers, whose utility func- tions and endowments are u'(x},x}) = (x}Xx})} (5,5) In the following, use the normalization p2 = 1. (a) Find the competitive equilibrium price. (b) State the first fundamental theorem of welfare and verify that it holds in this economy. (e) Consider the allocation ĩ = (x',) = (2,3), (8, 7). Show whether this allo- cation can supported as an equilibrium with transfers. (d) State the second fundamental theorem of welfare, and briefly discuss whether the result in part (c) conform with or violate this theorem. 3.