Robert's utility function is

U (x, y) = 2px + y.

In every graph for this question, set y in the vertical axis and x in the horizontal axis.

1. (a)Obtain the marginal rate of substitution MRS. Obtain the equation of the indifference curves for U = 2 and U = 4 (solving for y as a function of x). Draw these indifference curves, identifying the intersections with each of the axes.
2. (b)Suppose py = 2 and Robert's income is M = 6. In a new graph, show the total, income and substitution eects (on the demand of x) of a drop in the price of x from px = 4 to px = 2.
3. (c)Now, the store where Robert goes for shopping has changed the prices to p x = 2 and p y = 6, however, it also has a new special oer: for any amount of Y that Robert buys, he gets the same amount of X for free. Robert's income is M = 6. Draw Robert's budget line. Write down the equation of this line. Find Robert's utility-maximizing bundle.

(d)(d.1) Arthur's income is M = 6 and he goes to the same store as Robert. His utility function is U (x, y) = min{x, y}. What is his optimal bundle under the prices of part (c), including the special offer. Provide a graphical argument justifying your answer. (3 marks)

(d.2) Simon's income is M = 6 and he goes to the same store as well. His utility function is U (x, y) = 2x + 2y. What is his optimal bundle under the prices of part (c), including the special offer. Provide a graphical argument justifying your answer.