4. Consider a firm that wants to offer two different qualities of a good marketed to a low type and a high type consumer. The firm cannot tell the difference between the two consumers, but knows that the high type values the good twice as much as the low type, i.e., 0₁ = 1 and ₂ = 2. In order to produce quality in their product, the 01 firm faces the cost function c(s) = s³ where s is the quality level. The proportion of low type consumers is y and the proportion of high type consumers is 1-y where N₁ = 7 = N₁+N₂ = N. (a) Set up the firm's profit maximization problem and list all four constraints. (b) Assume PC₁ and IC₂ bind. Calculate the equilibrium qualities and prices offered to each type of consumer. (c) For what values of y does the firm offer any quality at all to the low type consumer? (Hint: look at st).

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4. Consider a firm that wants to offer two different qualities of a good marketed to a
low type and a high type consumer. The firm cannot tell the difference between the
two consumers, but knows that the high type values the good twice as much as the
low type, i.e., 01 = 1 and 02 = 2. In order to produce quality in their product, the
firm faces the cost function c(s) = s3 where s is the quality level. The proportion
of low type consumers is y and the proportion of high type consumers is 1-y where
Ni+N2
(a) Set up the firm's profit maximization problem and list all four constraints.
(b) Assume PC, and IC2 bind. Calculate the equilibrium qualities and prices offered
to each type of consumer.
(c) For what values of y does the firm offer any quality at all to the low type consumer?
(Hint: look at si).
Transcribed Image Text:4. Consider a firm that wants to offer two different qualities of a good marketed to a low type and a high type consumer. The firm cannot tell the difference between the two consumers, but knows that the high type values the good twice as much as the low type, i.e., 01 = 1 and 02 = 2. In order to produce quality in their product, the firm faces the cost function c(s) = s3 where s is the quality level. The proportion of low type consumers is y and the proportion of high type consumers is 1-y where Ni+N2 (a) Set up the firm's profit maximization problem and list all four constraints. (b) Assume PC, and IC2 bind. Calculate the equilibrium qualities and prices offered to each type of consumer. (c) For what values of y does the firm offer any quality at all to the low type consumer? (Hint: look at si).
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