Exercise 3.LetU:R2+!Rbe a utility function, i.e.,U(c1; c2)2Rrepresents the utility level associated with a bundle(c1; c2)2R2+. LetUi(c1; c2)denote the partial derivative ofUwith respect to itsithargument, i.e.,Ui(c1; c2)@U(c1; c2)=@cifori= 1;2. Assume thatUis concave, strictly increasing in each argument, and thatlimx!0U1(x; c2) = limx!0U2(c1; x) =1. Consdider the following maximization problem for an individual consumer:

Exercise 3. Let U : R1 > R be a utility function, i.e., U (C1,C2) E R represents the utility level associated with a bundle (61,62) 6 R1. Let U, (C1, C2) denote the partial derivative of U with respect to its ith argument, i.e., U, (C1, 02) E 8U (C1, C2) /5'C,~ fort = 1,2. Assume that U is concave, strictly increasing in each argument, and that limmno U1 (:5, C2) = limm_.o U2 (C1, 3:) = 00. Consdider the following maximization problem for an individual consumer: max U(01,02) 5.1:. p101+p202 S y (5) (c1,cz)E]Ri where pg 6 RH. is the price of good 2', and y E R++ is the consumer's income. 1. Argue that any (c1, 02) that solves (5) must satisfy the budget constraint with equality. 2. Argue that any (C1, C2) that solves (5) must satisfy Ci > 0 fort = 1,2. That is, argue that the constraints C, 2 0 fort = 1, 2 will not bind at an optimum. 3. Given your answers to parts 1 and 2, notice that (5) can be written as max U (01,02) s.t. plcl +p202 = y. (6) (01,02)R2 3.1. Using Lagrangian techniques, derive two conditions that characterize the pair (C1, 02) that solves (6). 3.2. Instead of using Lagrangian techniques, use the budget constraint to turn (6) into a maximization problem in one Choice variable, and derive two conditions that Char- acterize the pair (01,02) that solves (6)