Market with friction problem set
Problem Set #3 Econ 366-Markets with Frictions Prof. Guido Menzio New York University Spring 2024 Instructions: There are four questions worth a total of 100 points. You are encouraged to work in groups of up to 4 members. One submission per group. The problem set is due by midnight of April 10. Please email your solutions to the TA Nicolo’Ceneri at [email protected] . Good Luck! 1. (25) The Blue Razor/ Pink Razor Puzzle. Even thogh identical in all aspects, except color, pink razors are sold at higher prices than blue razors. Using the search-theoretic model of price dispersion of Burdett and Judd (1983), we want to provide a possible explanation for this puzzling phenomenon. Consider a market populated by a measure 1=2 of women and a measure 1=2 of men. Each woman has a unit demand for a pink razor, and gets utility uw p from purchasing it at the price p. Each man has a unit demand for a blue razor, and gets utility um p from purchasing it at the price p. Each seller has the technology to produce any kind of razor at the unit cost c. Buyers can only shop from a small number of sellers on a given day. A female buyer can shop from 1 randomly-selected seller with probability w;1 2 (0; 1) and from 2 randomly-selected sellers with probability w;2 = 1 aw;1 . A male buyer can shop from 1 randomly-selected seller with probability m;1 2 (0; 1) and from 2 randomly-selected sellers with probability m;2 = 1 am;1 . a. Write down the equilibrium distribution of posted prices Fw (p) for pink razors. Why are sellers indi¤erent between posting di¤erent prices on the support of Fw ? b. Write down the equilibrium distribution of transaction prices Gw (p) for pink razors. Explain the formula. c. What is the sign of the di¤erence between Fw and Gw ? Explain. d. Suppose that women have a higher opportunity cost of spending time shopping than men and, hence, w;1 > m;1 and w;2 < m;2 . Also suppose that women and men value razors equally, i.e. uw = um . Under these assumptions compute and sign the di¤erence between Fm and Fw . Interpret your …ndings. e. Compute and sign the di¤erence between Gm and Gw . Interpret your …ndings. f. Is the average posted price for a pink razor higher or lower than for a blue razor? What about transation prices for pink razors and blue razors?. 1 g. Now suppose that the women value pink razors more than men value blue razors, i.e. uw > um . Compute and sign the di¤erence between Fm and Fw assuming that w;1 = m;1 and w;2 = m;2 . What explains the di¤erence between Fm and Fw .now? 2. (25) Gains from trade and adverse selection. Consider the product market model of Akerlof (1970). The market is populated by a continuum of sellers with measure S = 1. Each seller has one unit of the good. Goods di¤er in quality . The distribution of quality across sellers is a uniform with support [ ; ] with = 1, = 2. The seller’s valuation of the good is v( ) = , with 2 (2=3; 1). The market is also populated by a continuum of buyers with measure B > 1. Each buyer gets utility p from purchasing a good of quality at the price p. Information however is asymmetric. Sellers know the quality of their own good. Buyers only know the average quality of the goods that are put up for sale in equilibrium. a. Write down and solve the problem of a seller with a good of quality . b. Compute the reservation quality function R(p), i.e. the highest quality of the goods that sellers choose to put on the market when the price is p. c. Compute the average quality function (p), i.e. the average quality of the goods that sellers choose to put on the market when the price is p. d. Plot the reservation quality function, R(p), the average quality function, equilibrium of the economy. e. Compute the reservation quality, R , the average quality, assume that R 2 (1; 2)]: (p), and identify the , and the price, p . [You can safely f. What is the reservation quality when = 3=4 and how does the equilibrium look like? What is the reservation quality when = 1 and how does the equilibrium look like? Why does the equilibrium change so much as increases? 3. (25) Participation mandate in the market for lemons. Consider the product market model of Akerlof (1970). The market is populated by a continuum of sellers. Each seller is endowed with a unit of the good. If the quality of his good is 2 [1; 2], the seller gets utility v( ) = 45 from consuming the good and utility p from selling the good at the price p. The fraction of sellers endowed with a good of quality ~ is F ( ) = 1. The market is also populated by a continuum of buyers. Each buyer gets utility p from purchasing a good of quality at the price p. Information is asymmetric. Sellers know the quality of their own good. Buyers only know the average quality of the goods that are traded in equilibrium. a. Write down and solve the problem of the seller. 2 b. Compute the reservation quality function R(p), i.e. the highest quality of the goods that sellers choose to trade when the price is p. c. Compute the average quality function (p), i.e. the average quality of the goods that sellers choose to trade when the price is p. d. Plot the reservation quality function, R(p), the average quality function, equilibria of the economy. e. Compute the reservation quality, R , the average quality, traded for all of the equilibria of the economy. (p), and identify the set of , the price, p , and the number of goods f. Suppose that the government introduces a mandate that forces all sellers to put their goods up for sale. With the mandate in place, compute the average quality of the goods traded, ^ , and the equilibrium price, p^. 4. (25) IT revolution and product markets. Consider the product market model of Burdett and Judd (1983). The buyers’utility from consuming a unit of the good is u = 1 and the disutility from paying p dollars for the good is p. The sellers can produce the good at the unit cost c, where c = 1=2. Each buyer meets one seller with probability 1 > 0 and two sellers with probability 2 > 0. a. Let 1 = 1=4 and 2 = 1=4. Given these values for 1 and 2 , compute the equilibrium distribution of sellers’prices, F1 (p), the highest equilibrium price, p1 , and the lowest equilibrium price, p1 . b. Suppose that the IT revolution increases the probability that a buyer …nds multiple sellers. Speci…cally, the IT revolution increases 2 from 1=4 to 1=2. Given this value for 2 , compute the equilibrium distribution of sellers’prices, F2 (p), the highest equilibrium price, p2 , and the lowest equilibrium price, p2 . c. Does the price distribution increase or decrease (in the sense of …rst order stochastic dominance) in response to the IT revolution? Provide an intuition for your …ndings. d. Compute the sellers’pro…ts before and after the IT revolution. Provide an intuition for your …ndings. e. Compute the amount of goods traded in the product market before and after the IT revolution. Provide an intuition for your …ndings. 3
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