Question 3 & 4 please

1222 all '5' C} E newclasses.nyu.edu , J associated with a bundle (c1,C2) E R1. Assume U is differentiable, with (9U (1,62)/ac; > 0 for i : 1, 2. Show that the marginal rate of substitution is equal to the absolute value of the slope of the indifference curve. Exercise 3. Let U : 1R?+ R be a utility function, i.e., U (c1,cz) E R represents the utility level associated with a bundle (1,82) E R1. Let U, (C1,C2) denote the partial derivative ofU with respect to its 2'\" argument, i.e., U1: (c1,C2) E 6U (c1, (32) [66,- for 2' : 1, 2. Assume that U is concave, strictly increasing in each argument, and that limwo U1 (3:, 02) : \"mm,0 U2 (1,515) : 00. Consdider the following maximization problem for an individual consumer: max U(C1,C2l S-t- P101 +P22 S y (5) (51:52)ER1 where p.- E 1R++ is the price of good i, and y E R++ is the consumer's income. 1. Argue that any (01,62) that solves ( 5) must satisfy the budget constraint with equality. 2. Argue that any (c1, C2) that solves (5) must satisfy c1: > 0 for i = 1, 2. That is, argue that the constraints 0, 2 0 for i : 1, 2 will not bind at an optimum. 3. Given your answers to parts I and 2, notice that (5) can be written as max U(cl, C2) s.t. p161 +p2C2 : y. (6) (EJVEZlERE 3.1. Using Lagrangian techniques, derive two conditions that characterize the pair (61,62) that solves (6) lDue by 10:30 am on September 14. 32. 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 (c1, c2) that solves (6). 4. Prove that the \"unit ofaccount\" in which we choose to denominate prices and income (e.g., euro, dollar, peso, yuan, apples, bitcoin, etc.) is irrelevant for the individual consumer. That is, prove that multiplying income, 1;, and both prices, p1 and p2, by any strictly positive constant does not change the solution to