Question 2 (20 marks). Jack consumes only bread (good x) and cheese (good y). His utility function is given by

U (x, y) = 4 ln x + ln y.

(a) Consider only bundles with both x > 0 and y > 0. Are Jack's preferences monotone? Are they strictly monotone? Does this utility function satisfy diminishing MRS? (5 marks)

(b) Find the expenditure-minimising bundle (x,y) as a function of the prices px and py, and the target utility U. (5 marks)

(c)(c.1) Find the expenditure necessary to attain the utility U as a function of the prices px and py, and the target utility U itself. If the prices are px = 1, py = 1, obtain the expenditure necessary to attain the utility U = ln 32. (3 marks)

(c.2) What is Jack's utility-maximising bundle if his income is M = 5 and prices are px = 1 and py = 1? In a well labelled graph, represent Jack's budget line and his utility-maximising consumption bundle. (2 marks)

(d) Jack's sister, Elena has a utility function given by

V (x, y) = max{2x, y},

where the function max{x, y} gives the maximum of x and y (for instance, max{1, 2} = 2,

max{6, 3} = 6, max{5, 5} = 5, etc.)

(d.1) Provide the utility-maximising consumption bundle as a function of the prices px, py,

and Elena's income, denoted by I. (2 marks)

(d.2) Provide the expenditure-minimising consumption bundle as a function of the prices px,

py , and the target utility V . (2 marks)

(d.3) Suppose that px < py. What is the income elasticity of x? (1 mark)