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# The idea of this problem is to analyze whether this way to signal about the quality will work in a market equilibrium.

Please help me solve this problem with explanation. The idea of this problem is to analyze whether this way to signal about the quality will work in a market equilibrium.

There are 2 mobile phone producers good and bad quality producers.The only way a quality manifests itself is through durability of the device. Good-quality phones last longer than bad-quality ones. Within 2 years of sale, 10% of good-quality phones fail, whereas 50% of bad-quality phones fail. A good-quality phone costs \$200 to manufacture, bad-quality costs \$150 to manufacture. If a particular brand of a phone is known to be of the good quality, the price of the phone is ptt. If a particular brand of a phone is known to be of the bad quality, the price of the phone is pB, with ptt > pB. These prices are given by the market and cannot be changed by producers. The issue is that it is hard for consumers to tell which brand is good and which one is bad. The only way for the producers to signal good quality is by offering a warranty to replace any phone that fails within two years of purchase by a new product of the same quality.

I need to answer the following: a) What is the expected profit per phone for a bad producer who does not offer the waranty? What is the expected profit per phone for a bad producer who does offer the waranty? What condition needs to be satisfied for the bad producers to not offer the warranty? What does this condition imply for the size of the price differential ptt − pB? b) What is the expected profit per phone for a good producer who does not offer the warranty? What is the expected profit per phone for a good producer who does offer the waranty? What condition needs to be satisfied for the good producers to offer the warranty? What does this condition imply for the size of the price differential ptt − pB? c) What is the range of values for the price differential ptt pB for which the bad producers do not provide the warranty whereas the good producers do? d) (a)  when manufacturers think about the long term production strategy, they can choose whether to produce good or bad quality phones. Suppose that, in equilibrium, both quality levels are produced, the bad producers do not provide the warranty and the good producers do. This implies that choosing one or the other strategy leads to the same expected profit per phone. What does this condition imply for the size of the price differential ptt pB? For this price differential, do the bad producers indeed have an incentive to not provide the warranty and do the good producers indeed have an incentive to provide the warranty? e) Now assume that the good phones fail with probability 20%. Repeat parts (a)-(d) for this changed setup? Will the equilibrium with quality signaling through warranty work in this case?

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