Problem 6 (20 points) A consumer purchases two goods, :1: and y. His utility function is: U(:1:, y) = 21n(:1:) + ln(y + 2). Suppose that the price of :1: is 2, the price of y is 1, and the consumer's income is 10. Thus, the consumer chooses a: and y to maximise U(:1:,y), subject to :1: and y satisfying his budget constraint, 2:1:+y=10. a (5 points). Use the Lagrangian method to nd the utility-maximising values of :1: and y. b (5 points). What is the value of the Lagrangian multiplier X? c (5 points). What is the consumer's maximised utility? d (5 points). Now suppose that the price of good :1: increases to 4. That is, the new budget constraint is 451: + y = 10. What are the new utilitymaximising values of :1: and y