1. Consider the following utility function U = 1 + 2. Prices for 1 and T2 are p1 and p2 respectively. Consumer's income is m. (a) Derive the optimal bundle. (5 points) (b) Is the solution to the consumer's problem unique (need proof)? (5 points) (c) What are the demands when p1 = p2 = 1 and m = 4? (5 points) (d) Suppose the price of commodity 1 goes up to 2 (but p2 remains unchanged) and the consumer succeeds in convincing her employer that her income should be raised to m' so that even after pi goes up from 1 to 2, she can still just afford to buy her original demand bundle. What is the value of m'? (10 points) (e) Although the consumer can now continue to consume the original commodity bundle (before the change in p1 and m), is it rational to do so? (Compute the optimal demands on the new budget line, and use these to prove that the consumer is better/worse-off on the new line.) (10 pointsy