An assertion is made in the lemons discussion in this chapter that demand may well be backward bending in a market in which quality is unobservable prior to purchase. Consider the following model: demand, X, is a function of price, P, and average quality, A. In particular, X(P, A) = 16000 ∙ P-2A4.
a. What type of demand function is this? What is the price elasticity of demand and the quality elasticity of demand?
Average quality depends on price: A higher price in the market will create a higher average quality level in the market. Let A(P) = 100 ∙ (P/[P+ 10,000]).
b. What is average quality at P = $0, $5,000, $10,000, $15,000, and $20,000?
c. Based on your answer to part b, what can you say about the shape of the average quality function? In particular, fill in the following sentence.
Average quality __________ at a(n) __________ rate as price increases.
d. What is demand X at P = $5,000, $10,000, $15,000, and $20,000?
e. Based on your answer to part d, demand must become backward bending at a price somewhere in what range of prices? In particular, fill in the following sentence.
The price must be more than___________ and less than __________ at the point where demand turns from being upward sloping to downward sloping (as price increases).
f. This part of Problem 9 requires calculus. Determine the price at which demand turns from being upward sloping to downward sloping (as price increases). Hint: Apply the total derivative discussed in footnote 9.
g. This part of Problem 9 requires Excel, but you need not complete part f. Create two graphs: (1) average quality as a function of price in {P, A) space, and (2) demand as a function of price in (X, P) space. Make sure your average quality and demand functions have the shapes and hit the (P, A) and (X, P) values you derived in parts a to e above.