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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. T
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