As in the lemons model, suppose that there is one seller and one buyer who may exchange a good of quality v U[0, 1]. The seller, who values the good at v dollars, knows the value of v. The buyer, who values the good at 3v/2 dollars, does not know the value of v (only that it is uniformly distributed on [0,1]). There is a fixed price of p = 1/2 at which trade may occur, which happens if and only if both the buyer and the seller agree to trade. Before deciding whether to trade, the buyer can pay a cost c € (0, 1) to learn the value of u. The seller's payoff is p if trade occurs and vif trade does not occur. The buyer's payoff is 3v/2-p-c if trade occurs and he first learned the value of v; 3v/2-p if trade occurs and he did not learn the value of u; -c if trade does not occur and he first learned the value of v; and 0 if trade does not occur and he did not learn the value of u. Find the probability that trade occurs for each c € (0, 1). Solution: If the buyer chooses not to learn the value, we know from the lemons model that trade will not occur for any v> 0 (since 3/2 < 2). If the buyer pays the cost to learn the value, he is willing to buy if and only if 3v/2≥ 1/2, that is, if and only if v≥ 1/3. The seller is willing to sell if and only if v ≤ 1/2. The buyer's expected payoff is P² ( 137 ≤ 0 ≤ 1/2) Pr VI VI E 3v 1 2 1 SU -c= 11 68 ==C.
As in the lemons model, suppose that there is one seller and one buyer who may exchange a good of quality v U[0, 1]. The seller, who values the good at v dollars, knows the value of v. The buyer, who values the good at 3v/2 dollars, does not know the value of v (only that it is uniformly distributed on [0,1]). There is a fixed price of p = 1/2 at which trade may occur, which happens if and only if both the buyer and the seller agree to trade. Before deciding whether to trade, the buyer can pay a cost c € (0, 1) to learn the value of u. The seller's payoff is p if trade occurs and vif trade does not occur. The buyer's payoff is 3v/2-p-c if trade occurs and he first learned the value of v; 3v/2-p if trade occurs and he did not learn the value of u; -c if trade does not occur and he first learned the value of v; and 0 if trade does not occur and he did not learn the value of u. Find the probability that trade occurs for each c € (0, 1). Solution: If the buyer chooses not to learn the value, we know from the lemons model that trade will not occur for any v> 0 (since 3/2 < 2). If the buyer pays the cost to learn the value, he is willing to buy if and only if 3v/2≥ 1/2, that is, if and only if v≥ 1/3. The seller is willing to sell if and only if v ≤ 1/2. The buyer's expected payoff is P² ( 137 ≤ 0 ≤ 1/2) Pr VI VI E 3v 1 2 1 SU -c= 11 68 ==C.
Chapter18: Asymmetric Information
Section: Chapter Questions
Problem 18.8P
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![As in the lemons model, suppose that there is one seller and one buyer who may exchange a
good of quality v~ U[0, 1]. The seller, who values the good at v dollars, knows the value of
v. The buyer, who values the good at 3v/2 dollars, does not know the value of v (only that it
is uniformly distributed on [0,1]). There is a fixed price of p = 1/2 at which trade may occur,
which happens if and only if both the buyer and the seller agree to trade. Before deciding
whether to trade, the buyer can pay a cost c € (0, 1) to learn the value of u. The seller's
payoff is p if trade occurs and vif trade does not occur. The buyer's payoff is 3v/2-p-c if
trade occurs and he first learned the value of v; 3v/2-p if trade occurs and he did not learn
the value of v; -c if trade does not occur and he first learned the value of v; and 0 if trade
does not occur and he did not learn the value of u. Find the probability that trade occurs for
each c € (0, 1).
Solution: If the buyer chooses not to learn the value, we know from the lemons model that
trade will not occur for any v> 0 (since 3/2 < 2). If the buyer pays the cost to learn the
value, he is willing to buy if and only if 3v/2≥ 1/2, that is, if and only if v≥ 1/3. The seller
is willing to sell if and only if v ≤ 1/2. The buyer's expected payoff is
3v
1
Pr
P² ( 1 ≤ 0 ≤ 12 ) ² [2/07 - 12/1
E
<<
SUS-C²
-c=
11
68
==C.
If e≤ 1/48, the buyer will learn and trade will occur with probability 1/6. Otherwise, trade
occurs with probability 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0542aa5b-cdbd-4921-99ff-0dadb1c30f45%2F10f07af2-d623-44ae-8ac2-333827f9ed88%2Fywzlyj_processed.png&w=3840&q=75)
Transcribed Image Text:As in the lemons model, suppose that there is one seller and one buyer who may exchange a
good of quality v~ U[0, 1]. The seller, who values the good at v dollars, knows the value of
v. The buyer, who values the good at 3v/2 dollars, does not know the value of v (only that it
is uniformly distributed on [0,1]). There is a fixed price of p = 1/2 at which trade may occur,
which happens if and only if both the buyer and the seller agree to trade. Before deciding
whether to trade, the buyer can pay a cost c € (0, 1) to learn the value of u. The seller's
payoff is p if trade occurs and vif trade does not occur. The buyer's payoff is 3v/2-p-c if
trade occurs and he first learned the value of v; 3v/2-p if trade occurs and he did not learn
the value of v; -c if trade does not occur and he first learned the value of v; and 0 if trade
does not occur and he did not learn the value of u. Find the probability that trade occurs for
each c € (0, 1).
Solution: If the buyer chooses not to learn the value, we know from the lemons model that
trade will not occur for any v> 0 (since 3/2 < 2). If the buyer pays the cost to learn the
value, he is willing to buy if and only if 3v/2≥ 1/2, that is, if and only if v≥ 1/3. The seller
is willing to sell if and only if v ≤ 1/2. The buyer's expected payoff is
3v
1
Pr
P² ( 1 ≤ 0 ≤ 12 ) ² [2/07 - 12/1
E
<<
SUS-C²
-c=
11
68
==C.
If e≤ 1/48, the buyer will learn and trade will occur with probability 1/6. Otherwise, trade
occurs with probability 0.
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