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Approved PART I - Answer all the questions in this part Question 1 (24 points) Let a consumer's utility be given by U (x, y, z) = In(x) + In( y) + In(2 + z). The consumer has a budget constraint of x + 3y + 2z = 18. a) What are the optimal choices for this consumer? (7 marks) b) What would the optimal choices be if income were increased from 18 to 19? Compute the optimal bundle that maximises the utility function and show the changes in the optimal bundle from a) to b) in a graph. (7 marks) c) What is the consumer's utility in each case? (4 marks) d) How do you interpret the Lagrange multiplier in b)? Show how you could use the Lagrange multiplier to estimate the change in consumer utility caused by increasing her budget from 18 to 19 without having to resolve for x, y, and z in (c). (6 marks) Question 2 (13 points) Consider the following game Player 2 A B C D Player 1 Up 1, 1 2, 2 3, 4 9, 3 Down 2, 5 3, 3 1, 2 7, 1 a) Find all the pure-strategy Nash Equilibria. (6 Marks) b) Find a mixed-strategy Nash Equilibrium of the game. (7 Marks)