Question 3 (26 marks). The price of pizza is p, = 20 per unit. (a) Donna spends all of her income, 100 dollars, on pizza (=) and tacos (t). Donna's utility function is U(2, t) = 20.3/0.1 (a.1) Derive Donna's demand function for tacos, as a function of the price of tacos, p: (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, p,, 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, p. (4 marks)