Consider the consumer optimization problem discussed in class where the representative consumer's preference are represented by the utility function U(c,l) =ceiie (1) where 6 is a preference parameter and is positive. The consumer's budget constraint is C=wN+7n-T. (2) The consumer also faces the time constraint [+ N* =h. (3) The notation is borrowed from the textbook. a) Maximize the consumer's utility function subject to constraints (2) and (3) (use the Lagrange method used in the class). Solve for consumption in terms of 6, h, w, and T. [10 pts] b) Solve for leisure and labour supply in terms of 8, h, m and T'. [5 pts] c) According to your solution to part (a), how does real wage income affect consumption? Briefly explain.[5 pts] a) To maximize the consumer's utility function subject to constraints (2) and (3), we set up the Lagrangian: C = Cn . 1 1 - n + X . ( wN* + n - T - C)+ M . (1 + N* - h) Taking partial derivatives with respect to (C), ((), and (N"*) and setting them to zero, we get the following equations: n . Cin-1 . /l-n = 1 1 = M . w Solving these equations, we find expressions for (C) in terms of (0), (h). (w), and (T). b) Using the results from part (a), we can now solve for leisure ((()) and labor supply ((N")). Substitute the expressions for (C) and (\\lambda) into the time constraint equation (3) to obtain the values of (() and (N) in terms of (0), (h), and (T). c) From the solution in part (a), we find that real wage income ((w)) affects consumption positively. An increase in (w) leads to a higher budget constraint, allowing the consumer to allocate more resources to consumption, thereby increasing overall utility