Question 2: Lemons Problem - Challenging! (5 points) Consider the following version of the lemons problem. There is a continuum of buyers and sellers in the market; the total mass of each group is 1. Each seller has one car to sell and each buyer wishes to buy at most one car, but only sellers know the quality of their cars before trading. It is common knowledge however that the quality of cars, denoted s, is drawn from a uniform distribution on the interval [0,1] (hence, the probability that a car's quality is below some number x is equal to w if 0 1). It is also common knowledge that a fraction a the sellers are of type 1 and have a payoff U1 = p-8/8 if they sell their cars and 0 otherwise, and a fraction 1 - a of the sellers are of type 2 and their payoff is U2 = p - 8/4 if they sell their cars and 0 otherwise, where p is the price of the car (note that the two types of sellers differ only with respect to their payoffs but not with respect to the quality of cars they have to sell). There is a continuum of buyer types: the payoff of a type- 0 buyer if she buys a car whose quality is s is U (0) = 0s - p, where 0 is distributed uniformly on the unit interval. If a buyer does not buy a car her payoff is 0. The buyers cannot observe the quality of cars before they buy nor can they observe the type of seller they face. (a) Compute the supply of cars by type 1 sellers, S.(p), and type 2 sellers, S. (p), and the aggregate supply of cars, S(p) (i.e., compute the fraction of cars that will be supplied at a given price by each type of sellers and then add the two to obtain the aggregate supply). Show that this aggregate supply is S(p) = 4p(1 + a) and illustrate your answer in a figure. (b) In the previous question you should have obtained that S(p) = 8ap and S2(P) = 4(1 - a)p leading to S(p) = S(p) + S2(P) = 4p(1 + a). Using this, show that if you observe an offer of any seller at price p, the probability that the seller is of type 1 is 2 and the probability that the seller is of type 2 is: 1+ (c) Let s(p) denote the average quality of cars supplied on the market as a function of p. Using your answers to (a) and (b), derive that $(p) = 2(1 + 3a)p/(1+a). How does s(p) vary with p and with a? Explain the intuition for this. (d) Assume that buyers correctly anticipate s(p) and compute the demand for cars (i.e., the fraction of buyers that will wish to buy a car at a given price) and show your answer in the figure you drew in subquestion (a). Explain the shape of the demand function. (e) Assume that the market is perfectly competitive and solve for the equilibrium price, p*, given that S(p) = 4p(1 + a) from (a). 1- Question 2: Lemons Problem - Challenging! (5 points) Consider the following version of the lemous problem. There is a continuum of buyers and sellers in the market; the total mass of each group is Fach seller has one car to sell and cach buyer wishes to buy at most one car, but only sellers know the quality of their cars before trwling. It is common knowledge however that the quality of cars, denoted s, is drawn from a uniform distribution on the interval [0,1] (hence, the probability that a car's quality is below some number is equal to 1 and is equal to 1 if 1). It is also common knowledge that a fraction or the sellers are of wpe I and have a payoff U =p-8/8 if they sell their cars and 0 otherwise, and a fraction 1 -a of the sellers are f type 2 and their payoff is U2 = p-/4 if they sell their cars and otherwise, where p is the prices of the car (note that the two types of sellers diffor only with respect to their pavoffs but not with respect to the quality of cars they have to sell). There is a continuum of buver types: the pavoff of a type o buver if she buys a car whose quality is s is U (0) = 85 - p, where A is distributed uniformly on the mit interval. Ta buyer does not buy a car her payoff is 0. The buyers cannot observe the quality of cars before they buy nor can they observe the type of seller they face. (a) Compute the supply of cars by type 1 sellers. Sixp), and type 2 sellers. Sip and the aggregate supply of cars, S(p) (ie., compute the fraction of cars that will be supplied at a given price by each type of sellers and then add the two to obtain the aggregate supply). Show that this aggregate supply is S(p) = 4p(1 + a) and illustrate your answer in a figure. (b) In the previous question you should have obtained that Sip) = 8p aud Sap) = 101 - op loading to S(p) = S(p) + Sp) = 4p(1 + a). Using this, show that if you observe an oficr of any sellor at price p. the probability that the seller is of type lis . and the probability that the seller is of type 2 is: 1 (c) Lct (p) denote the average quality of cars supplied on the market as a function of p. Using your answers to (a) and (b), derive that slp) = 2(14 3a)p/(1+How does sr) vary with p and with ex? Explain the intuition for this. (d) Assume that buyers correctly anticipate sip) and compute the demand for cars i.e. the fraction of buyers that will wish to buy a car at a given price) and show your answer in the figure you drew in subquestion (a). Explain the shape of the demand function. (e) Assume that the market is perfectly competitive and solve for the equilibrium price, p, given that SOP) = 4p(1 + a) from (a). Question 2: Lemons Problem - Challenging! (5 points) Consider the following version of the lemons problem. There is a continuum of buyers and sellers in the market; the total mass of each group is 1. Each seller has one car to sell and each buyer wishes to buy at most one car, but only sellers know the quality of their cars before trading. It is common knowledge however that the quality of cars, denoted s, is drawn from a uniform distribution on the interval [0,1] (hence, the probability that a car's quality is below some number x is equal to w if 0 1). It is also common knowledge that a fraction a the sellers are of type 1 and have a payoff U1 = p-8/8 if they sell their cars and 0 otherwise, and a fraction 1 - a of the sellers are of type 2 and their payoff is U2 = p - 8/4 if they sell their cars and 0 otherwise, where p is the price of the car (note that the two types of sellers differ only with respect to their payoffs but not with respect to the quality of cars they have to sell). There is a continuum of buyer types: the payoff of a type- 0 buyer if she buys a car whose quality is s is U (0) = 0s - p, where 0 is distributed uniformly on the unit interval. If a buyer does not buy a car her payoff is 0. The buyers cannot observe the quality of cars before they buy nor can they observe the type of seller they face. (a) Compute the supply of cars by type 1 sellers, S.(p), and type 2 sellers, S. (p), and the aggregate supply of cars, S(p) (i.e., compute the fraction of cars that will be supplied at a given price by each type of sellers and then add the two to obtain the aggregate supply). Show that this aggregate supply is S(p) = 4p(1 + a) and illustrate your answer in a figure. (b) In the previous question you should have obtained that S(p) = 8ap and S2(P) = 4(1 - a)p leading to S(p) = S(p) + S2(P) = 4p(1 + a). Using this, show that if you observe an offer of any seller at price p, the probability that the seller is of type 1 is 2 and the probability that the seller is of type 2 is: 1+ (c) Let s(p) denote the average quality of cars supplied on the market as a function of p. Using your answers to (a) and (b), derive that $(p) = 2(1 + 3a)p/(1+a). How does s(p) vary with p and with a? Explain the intuition for this. (d) Assume that buyers correctly anticipate s(p) and compute the demand for cars (i.e., the fraction of buyers that will wish to buy a car at a given price) and show your answer in the figure you drew in subquestion (a). Explain the shape of the demand function. (e) Assume that the market is perfectly competitive and solve for the equilibrium price, p*, given that S(p) = 4p(1 + a) from (a). 1- Question 2: Lemons Problem - Challenging! (5 points) Consider the following version of the lemous problem. There is a continuum of buyers and sellers in the market; the total mass of each group is Fach seller has one car to sell and cach buyer wishes to buy at most one car, but only sellers know the quality of their cars before trwling. It is common knowledge however that the quality of cars, denoted s, is drawn from a uniform distribution on the interval [0,1] (hence, the probability that a car's quality is below some number is equal to 1 and is equal to 1 if 1). It is also common knowledge that a fraction or the sellers are of wpe I and have a payoff U =p-8/8 if they sell their cars and 0 otherwise, and a fraction 1 -a of the sellers are f type 2 and their payoff is U2 = p-/4 if they sell their cars and otherwise, where p is the prices of the car (note that the two types of sellers diffor only with respect to their pavoffs but not with respect to the quality of cars they have to sell). There is a continuum of buver types: the pavoff of a type o buver if she buys a car whose quality is s is U (0) = 85 - p, where A is distributed uniformly on the mit interval. Ta buyer does not buy a car her payoff is 0. The buyers cannot observe the quality of cars before they buy nor can they observe the type of seller they face. (a) Compute the supply of cars by type 1 sellers. Sixp), and type 2 sellers. Sip and the aggregate supply of cars, S(p) (ie., compute the fraction of cars that will be supplied at a given price by each type of sellers and then add the two to obtain the aggregate supply). Show that this aggregate supply is S(p) = 4p(1 + a) and illustrate your answer in a figure. (b) In the previous question you should have obtained that Sip) = 8p aud Sap) = 101 - op loading to S(p) = S(p) + Sp) = 4p(1 + a). Using this, show that if you observe an oficr of any sellor at price p. the probability that the seller is of type lis . and the probability that the seller is of type 2 is: 1 (c) Lct (p) denote the average quality of cars supplied on the market as a function of p. Using your answers to (a) and (b), derive that slp) = 2(14 3a)p/(1+How does sr) vary with p and with ex? Explain the intuition for this. (d) Assume that buyers correctly anticipate sip) and compute the demand for cars i.e. the fraction of buyers that will wish to buy a car at a given price) and show your answer in the figure you drew in subquestion (a). Explain the shape of the demand function. (e) Assume that the market is perfectly competitive and solve for the equilibrium price, p, given that SOP) = 4p(1 + a) from (a)