Question 1. Consider the following utility functions:
(1) U (P, B) = 2P
(2) U (P, B) = 2min {P, 1/3B)
(3) U (P, B) = 4BP
(4) U ( P, B) + ln P + B
Let Pp = 4$, Pb = 2$, I = 100$
(Put pizzas (P) on the horizontal axis and beer (B) on the vertical axis all the time). Please answer all the following questions for each utility function).
Q1. Determine the demand functions for pizza and beer (mathematically). How many pizzas and how many beers are consumed in equilibrium for each of the utility functions? Determine the total utility associated with each utility function at the optimum.
Q2. Illustrate your results in part 2) graphically. (Sketch the graph).
Q3. Suppose the I = 20$, Pp = 4 and Pb = 2. Find the values of B and P that arise in consumer equilibrium. What is the total level of utility generated at this consumer equilibrium? Determine whether P and B are normal or inferior goods.
Q4. Derive the equation for the income expansion path. Derive the Engel curve and its slope. (Both graphically and mathematically).
Q5. Suppose I = 100$, Pp = 4 and Pb = 4, graph the price offer curve. Derive the corresponding demand functions, what is its slope? Determine whether P and B are compliments, substitutes or independent goods.
Q6. Referring to the initial conditions and question e) only. Then for each utility function decompose the total effect into substitution and income effect. Compute the actual numbers and sketch the graphs.