The price of pizza is pz = 20 per unit. 

(a) Donna spends all of her income, 100 dollars, on pizza (z) and tacos (t). Donna’s utility function is U(z, t) = z 0.9 t 0.1 . 

(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 (z, t) = min{z, 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(z, t) = 8z + 2t. Derive Lauren’s demand function for tacos, as a function of the price of tacos, pt .