Determine the price and quantity of toasters, Operation Management

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Determine the price and quantity of toasters:

There are 100 producers of toasters. Half of the producers are "low quality," i.e. their products will break with probability of 0.8, and their cost of production is $8 per toaster. The other half of the producers are "high quality," i.e., their products will break with probability of 0.1, and their cost of production is $40 per toaster. (A producer's type is fixed, so if you are low (high) quality then you only know how to manufacture low (high) quality toasters.) For simplicity, we assume that each producer only produces 1 toaster if it chooses to produce. For a buyer, a toaster that works has a value of $50, and a toaster that breaks has a value of 0. There is an infinite number of buyers. a. If a buyer knows for sure that a toaster is of low quality, what is its expected value to him?

b. If a buyer knows for sure that a toaster is of high quality, what is its expected value to him?

c. Suppose buyers cannot tell the difference between high and low quality toasters. Draw the supply and demand curves for toasters in a graph. Label everything clearly.
 
d. In equilibrium, what is the price and quantity of toasters? What kind(s) of toasters are bought and sold?

This is a follow-up to Problem. Suppose that all the producers sell toasters through Wal-Mart, and Wal-Mart lets producers choose from 2 options. With Option A, a producer sells a toaster at $49 but has to offer a free warranty to the consumer. With Option B, a producer sells a toaster at $10 without any warranty. A free warranty means that if a customer's toaster breaks, then she can return it and the producer must give her a full refund. (The returned good is of no use to anyone and will be dumped.)


a. Which option will a low-quality producer choose, A or B? Justify numerically.
 
b. Which option will a high-quality producer choose, A or B? Justify numerically.
 
c. Will offering a free warranty send a useful signal to the consumer in this case? Why?


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