Please show all necessary work. Use paper and pencil please do not type out the answer.

Question 1 (Consumer optimization problem) Suppose the utility function is given as: U = clz. Answer the following questions: 1. Write down the problem ofthe consumer (Hint: Write down what the consumer is trying to maximize or minimize and what is the constraint) Solve for the optimal values of c* and 1* (Hint: The final values should be in terms of the known parameters w, h, 1: and T. You can use either the Lagrangian approach or the substitution method) What happens to the optimal values of c'\" and l* when a. it increases b. T increases c. w increases (explain this in terms of income and substitution effects) Now instead of U = 12 assume that U = CI and derive the Optimal values of c" and 1*. Compare the new and old values of c" and 1*. Is consumption higher with this utility function compared to the previous one? Is leisure higher with this utility function compared to the previous one? Explain. Now instead of a lump-sum tax T assume that there is an income tax. Write down the problem of the consumer using the utility function U = 12 (hint: the budget constraint will change) and derive the optimal values of 6* and I". What happens to c" and 1* when w increases