Question 4 (20 marks).Robert's utility function is

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

In every graph for this question, setyin the vertical axis andxin the horizontal axis.

1. (a) Obtain the marginal rate of substitutionMRS. Obtain the equation of the indifference curves forU= 2andU= 4(solving foryas a function ofx). Draw these indifference curves, identifying the intersections with each of the axes. (5 marks)
2. (b)Supposepy= 2and Robert's income isM= 6. In a new graph, show the total, income and substitution effects (on the demand ofx) of a drop in the price ofxfrompx= 4topx= 2. (5 marks)
3. (c)Now, the store where Robert goes for shopping has changed the prices topx= 2andpy= 6, however, it also has a new special offer: for any amount ofYthat Robert buys, he gets the same amount ofXfor free. Robert's income isM= 6. Draw Robert's budget line. Write down the equation of this line. Find Robert's utility-maximising bundle. (5 marks)
4. (d)

(d.1) Arthur's income isM= 6and he goes to the same store as Robert. His utility function isU(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 isM= 6and he goes to the same store as well. His utility function isU(x, y) = 2x+ 2y. What is his optimal bundle under the prices of part (c), including the special oer. Provide a graphical argument justifying your answer. (2 marks)