Question 1 (22 marks). Adam consumes only beer (B) and whisky (W). His utility function is U(B, W) =^B + 5^W ,

where B is the quantity of beer and W is the quantity of whisky. The price of beer is PB  = 5, the

price of whisky is PW  = 40, and Adam’s income is 疝 = 100 dollars.

(a) Obtain the equation of Adam’s indifference curve for the utility level U = 6, with the quantity of whisky as a function of the quantity of beer. (2 marks)

(b) Obtain the amount of whisky required to attain the utility level U = 6 if the amount of beer is B = 1?  Compute the marginal rate of substitution (MRS) at that bundle and explain how you interpret that magnitude (of the MRS). (4 marks)

(c) Obtain the equation of Adam’s budget line.  What is the magnitude of the slope of the budget line? What is the economic interpretation of this magnitude? (4 marks)

(d) Adam’s cousin, John, has a utility function given by   V (B, W) = 3B2 + W2 .

(d.1) Are John’s preferences monotone?  Are they strictly monotone?  Justify your answers. (3 marks)

(d.2) Are John’s preferences convex?  Are they strictly convex?  Justify your answers.   (3 marks)

(d.3) Do John’s preferences satisfy the diminishing marginal rate of substitution property?

Explain. (2 marks)

(d.4) Assume that the prices are as indicated above (PB   = 5 and PW   = 40), and John’s income is Ⅰ > 0. Obtain John’s optimal bundle. (4 marks)

Question 2 (22 marks).  Sofia consumes only clothes (good α) and books (good 夕).  Her utility function is given by

U (α, 夕) = 3α + 2夕 .

The price of α is px  per unit and the price of 夕 is py  per unit.

(a) Are Sofia’s preferences monotone?  Are they strictly monotone?  Does this utility function satisfy diminishing MRS? (5 marks)

(b) Find the expenditure-minimising bundle (α, 夕) as a function of the prices px , py , and the target utility U .  Find the corresponding expenditure-minimising function E* (px , py , U).  (5 marks)

(c) Sofia’s girlfriend, Jingshuang, shares her passion for fashion and reading; her utility function is

V (α, 夕) = min(α, 夕}.

(c.1) Find the expenditure-minimising bundle (α, 夕) as a function of the prices px , py , and the target utility V . Find the corresponding expenditure-minimising function E* (px , py , V). (3 marks)

(c.2) The owner of the store where Jingshuang  buys clothes and  books offers  a special deal:  customers who  buy α pieces of clothing get the same  number of books for free.   Customers can buy additional books at py   per book.   Find Jingshuang’s new expenditure minimising bundle and the corresponding expenditure-minimising function E* (px , py , U). (2 marks)

(d) Excited about this offer, Jingshuang calls Sofia to tell her. Find Sofia’s expenditure-minimising function E* (px , py , V) with the special deal of this store. (7 marks)

Question 3 (26 marks). The price of pizza is pz  = 20 per unit.

(a) Donna spends all of her income, 100 dollars, on pizza (2) and tacos (t).  Donna’s utility

function is

U (2, t) = 20.9t0.l .

(a.1) Derive Donna’s demand function for tacos, as a function of the price of tacos, pt  (as

throughout the rest of the test, show your derivations, step by step). (5 marks)

(a.2) Derive Donna’s inverse demand function for tacos (writing the price of tacos, pt , as a

function of the quantity). (2 marks)

(a.3) Using derivatives, determine whether the inverse demand function for tacos is decreasing,

and convex or concave. (4 marks)

(a.4) Draw a graph of Donna’s inverse demand function for tacos.  Make sure to illustrate

clearly what happens when t goes to zero and when t goes to infinity. (5 marks)         (b) Just like Donna, Alison spends all of her income, 50 dollars, on pizza and tacos.  Alison’s

utility function is

V (2, t) = min(2, t}.

(b.1) Derive Alison’s demand function for tacos, as a function of the price of tacos, pt .  (4

marks)

(b.2) Derive Alison’s inverse demand function for tacos (writing the price of tacos, pt , as a

function of the quantity). (2 marks)

(c) Just like Donna and Alison, Lauren spends all of her income, 40 dollars, on pizza and tacos. Lauren’s utility function is

W (2, t) = 82 + 2t.

Derive Lauren’s demand function for tacos, as a function of the price of tacos, pt . (4 marks)