2. Let's examine the consumer maximization problem (and the scenarios) we started at the end of class Thursday. Consider a consumer whose preferences are described by the utility function U(x,y) = 2x; + y and assume that the prices are, Pr = 1 and P), = 4: [a] First off, please the confirm the work we did in class for yourself. Specify an equation that reflects the budget constraint and set up the Iagrangian equation. Use the lagrangian approach to find the optimal bundle of goods (x*,y') when the consumer has an income level, I = 24. [b] Based on your work from part (a), calculate the amount of utility the consumer gets from the optimal bundle. Derive and graph the equation for the indifference curve that corresponds to this utility level. Measuring good x on the horizontal axis, graph the consumer's budget constraint and identify the utility maximizing bundle on your graph. Find and expression for the MRS. How does MRS change as x increases? (c) Now, usingthe same approach as in part (b), find the utility maximizing bundle when the consumer's income falls to level I = 8. Provide a diagram showing the utility maximizing bundle, the indifference curve, and budget constraint. Determine whether MRS equal the ratio of the prices at the utility maximizing bundle - why is your answer significant? [d] Now, usingthe same approach as in part (b), find the utility maximizing bundle when the consumer's income falls to level I = 16. Provide a diagram showing the utility maximizing bundle, the indifference curve, and budget constraint. Determine whether MRS equal the ratio of the prices at the utility maximizing bundle - why is your answer significant? [2] Given the three income level scenarios you worked through above, how would you concisely describe this consumer's preferences over goods x and y