PROBLEM (1) A risk neutral consumer considers buying a used car of quality 0 == {G, B}. The quality is not directly observable for the consumer. She has prior beliefs p on the quality; p = P[0 = G] = 4/5 The consumer can take one of two actions a EA = {buy, don't buy). The price of the car is denoted by pc. The utility of the consumer is given by: v(buy, 0) = 5,000-pc if 0 = G and v(buy, 0) = -pc, otherwise. v(don't buy, 0) = 0 regardless of 0. a) What price pe would make the consumer indifferent between buying and not buying the car? Assume that the seller asks a price pc = 4, 500. Will the consumer buy the car? b) A mechanic offers a pre-purchase car test to the consumer. The test outcome can be good (g) or bad(b); t E{g, b}. Assume first that the test is fully informative; P[t=g 10=G]=P[t=b |0=B]=1 What price pm should the mechanic charge for his test in order to maximize his revenue? (Still assuming that the seller of the used car charges pc = 4, 500.) c) Assume now that the test is not fully informative: P[t=g |0=G] = P[t=b |0=B] = 3/4. What price pm maximizes the mechanic's revenue? (Still assuming that the seller charges pc = 4,500.) d) (Harder, but important !!) Let: V (p) = max Eo-p[v(a, 0)] for aЄA. Show that V (p) is convex (To consider a convex combination of beliefs, it may be helpful to write out the expectation as a sum). e) Consider the following information structures: For both, the signal space is T = {g, b), and for the signal (information) structure Ii with i E{1,2} we have: P[t=g 10 = G] =1, P[t=b 10=B] = r with r₂ > ri > 1/2 Prove that the consumer would prefer the signal (information) structure I2 over Ii.
PROBLEM (1) A risk neutral consumer considers buying a used car of quality 0 == {G, B}. The quality is not directly observable for the consumer. She has prior beliefs p on the quality; p = P[0 = G] = 4/5 The consumer can take one of two actions a EA = {buy, don't buy). The price of the car is denoted by pc. The utility of the consumer is given by: v(buy, 0) = 5,000-pc if 0 = G and v(buy, 0) = -pc, otherwise. v(don't buy, 0) = 0 regardless of 0. a) What price pe would make the consumer indifferent between buying and not buying the car? Assume that the seller asks a price pc = 4, 500. Will the consumer buy the car? b) A mechanic offers a pre-purchase car test to the consumer. The test outcome can be good (g) or bad(b); t E{g, b}. Assume first that the test is fully informative; P[t=g 10=G]=P[t=b |0=B]=1 What price pm should the mechanic charge for his test in order to maximize his revenue? (Still assuming that the seller of the used car charges pc = 4, 500.) c) Assume now that the test is not fully informative: P[t=g |0=G] = P[t=b |0=B] = 3/4. What price pm maximizes the mechanic's revenue? (Still assuming that the seller charges pc = 4,500.) d) (Harder, but important !!) Let: V (p) = max Eo-p[v(a, 0)] for aЄA. Show that V (p) is convex (To consider a convex combination of beliefs, it may be helpful to write out the expectation as a sum). e) Consider the following information structures: For both, the signal space is T = {g, b), and for the signal (information) structure Ii with i E{1,2} we have: P[t=g 10 = G] =1, P[t=b 10=B] = r with r₂ > ri > 1/2 Prove that the consumer would prefer the signal (information) structure I2 over Ii.
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.3P
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![PROBLEM (1) A risk neutral consumer considers buying a used car of quality 0 == {G, B}. The quality
is not directly observable for the consumer. She has prior beliefs p on the quality; p = P[0 = G] = 4/5
The consumer can take one of two actions a EA = {buy, don't buy). The price of the car is denoted by pc. The
utility of the consumer is given by: v(buy, 0) = 5,000-pc if 0 = G and v(buy, 0) = -pc, otherwise.
v(don't buy, 0) = 0 regardless of 0.
a) What price pe would make the consumer indifferent between buying and not buying the car?
Assume that the seller asks a price pc = 4, 500. Will the consumer buy the car?
b) A mechanic offers a pre-purchase car test to the consumer. The test outcome can be good (g) or bad(b);
t E{g, b}. Assume first that the test is fully informative; P[t=g 10=G]=P[t=b |0=B]=1
What price pm should the mechanic charge for his test in order to maximize his revenue? (Still assuming
that the seller of the used car charges pc = 4, 500.)
c) Assume now that the test is not fully informative: P[t=g |0=G] = P[t=b |0=B] = 3/4. What price pm
maximizes the mechanic's revenue? (Still assuming that the seller charges pc = 4,500.)
d) (Harder, but important !!) Let: V (p) = max Eo-p[v(a, 0)] for aЄA. Show that V (p) is convex (To consider
a convex combination of beliefs, it may be helpful to write out the expectation as a sum).
e) Consider the following information structures: For both, the signal space is T = {g, b), and for the signal
(information) structure Ii with i E{1,2} we have: P[t=g 10 = G] =1, P[t=b 10=B] = r with r₂ > ri > 1/2
Prove that the consumer would prefer the signal (information) structure I2 over Ii.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b242623-39cc-484b-9f1f-99c1a2828400%2Fc30159df-9be0-4839-9c78-47a722e08ffc%2F4xn0j1l_processed.jpeg&w=3840&q=75)
Transcribed Image Text:PROBLEM (1) A risk neutral consumer considers buying a used car of quality 0 == {G, B}. The quality
is not directly observable for the consumer. She has prior beliefs p on the quality; p = P[0 = G] = 4/5
The consumer can take one of two actions a EA = {buy, don't buy). The price of the car is denoted by pc. The
utility of the consumer is given by: v(buy, 0) = 5,000-pc if 0 = G and v(buy, 0) = -pc, otherwise.
v(don't buy, 0) = 0 regardless of 0.
a) What price pe would make the consumer indifferent between buying and not buying the car?
Assume that the seller asks a price pc = 4, 500. Will the consumer buy the car?
b) A mechanic offers a pre-purchase car test to the consumer. The test outcome can be good (g) or bad(b);
t E{g, b}. Assume first that the test is fully informative; P[t=g 10=G]=P[t=b |0=B]=1
What price pm should the mechanic charge for his test in order to maximize his revenue? (Still assuming
that the seller of the used car charges pc = 4, 500.)
c) Assume now that the test is not fully informative: P[t=g |0=G] = P[t=b |0=B] = 3/4. What price pm
maximizes the mechanic's revenue? (Still assuming that the seller charges pc = 4,500.)
d) (Harder, but important !!) Let: V (p) = max Eo-p[v(a, 0)] for aЄA. Show that V (p) is convex (To consider
a convex combination of beliefs, it may be helpful to write out the expectation as a sum).
e) Consider the following information structures: For both, the signal space is T = {g, b), and for the signal
(information) structure Ii with i E{1,2} we have: P[t=g 10 = G] =1, P[t=b 10=B] = r with r₂ > ri > 1/2
Prove that the consumer would prefer the signal (information) structure I2 over Ii.
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