working and detailed explanation

2. A consumer must choose how many units of a composite good to consume in each of three periods t = 0, 1, 2. Denote the amount of the composite good consumed in period t by act, and assume that each unit of the good costs exactly $1 at the time of purchase. The consumer has no income in periods t = 0 and t = 1, so 100 = 0 and wl = 0, but receives a dollar-denominated income 102 = w > 0 in period t = 2. To purchase any goods in periods 15 = O and t = 1, the consumer can borrow against his period 2 income at a per-period interest rate r E (O, 1), but must be able to repay all his debts out of his period 2 income 11:. Hence, if the consumer chooses a consumption stream (m0, 321, 51:2) 2 D, the repayment on the incurred debt at t = 2 is equal to (1 + 702330 + (1 + r):(:1, and therefore, the consumer's budget constraint is given by (1 + 102330 + (1 + T)931 + 332 S w. Extend the domain of the natural logarithm In to R... by setting ln(0) := 00, so that the codomain of In is given by R U {00}, and In : lR+ > R U {00}. Assume that the consumer's preferences over consumption streams (:60, .171, :32) E R: can be represented by the utility function u : IR'E'Ir > RU {00} dened by u(:ro, :61, m2) := ln $0 + [3'61nm1 + 662 In 932, where B and 6 are xed parameters that satisfy (2 (0, 1] and 6 E (0, 1). (Note that this utility function shows an example of quasi-hyperbolic discounting, Where the relative discounting of the logarithmic per-period utilities may be greater between t = O and t = 1, than between t = 1 and t = 2.) Assume furthermore that the consumer chooses his complete consumption plan (:50, :31, :62) at t = 0, in order to maximise the utility function u subject to the budget constraint and non-negativity constraints for all {L}. (i) Summarise the consumer's utility maximisation problem, making sure to include all inequality and non-negativity constraints. Explain which of the constraints, if any, can never bind at an optimum, and which constraints, if any, must always bind at an optimum. Show that all the assumptions of Theorem 10.11 are satised for the consumer's maximisation problem, so that the Kuhn-Tucker rstorder conditions are necessary and sufcient to charac- terise a maximiser. Write down the Lagrangian for the consumer's utility maximisation problem, and derive the associated Kuhn-Tucker necessary conditions. If you know that any constraints cannot bind at an optimum, you can ignore them in your Lagrangian and necessary conditions; if you know that any constraints must bind at an optimum, you can treat them as equality constraints. Find all critical points of the Lagrangian from part (iii). Explain which of the critical points you found must correspond to a solution to the consumer's utility maximisation problem. The maximum value function for the consumer's utility maximisation problem is a function of the parameters 7', B, 6 and w. Denoting this function by 11(7', , 6, w), explain carefully how the Envelope Theorem can be used to derive the partial derivative of v with respect to ,8, and use the Envelope Theorem approach to nd 3; at an interior solution (933,323 233) >> 0. Make sure to express 3; as a function of the parameters 1', , 5 and w only, and not as a function of the consumption levels and/ or Lagrangian multipliers