q S pq S cq S. where q is the total amount produced/consumed. There are two representative firms that maximizes their profits, max pq 1 c 2 q2 1 q 1

Transcription

1 Market design, problem set 2 (externalities, incomplete information, search) Do five of nine. 1. There is a representative consumer who picks a quantity q D to imize his utility, b log(q D ) pq D + w X q D where X = xq is an externality which the consumer ignores at the time they make their decision, where q is the total amount produced/consumed. There is a representative firm that imizes its profits, q S pq S cq S. i. What is the outcome in the perfectly competitive market? ii. If a social planner takes the externality into account, what is the optimal quantity to trade? Is it greater or less than the perfectly competitive market? Is the perfectly competitive outcome efficient? Sketch the supply-and-demand diagram that illustrates the externality. iii. If the government wanted to impose a quota on production or consumption of this good, what would the optimal quota be? iv. If the government wanted to impose a tax on consumers to implement the optimal outcome, what would the tax be? If the government wanted to impose a tax on firms to implement the optimal outcome, what would the tax be? Sketch the supply-and-demand diagram that illustrates how the optimal tax works. 2. There is a representative consumer who picks a quantity q D to imize his utility, q D 1 q D 2 (qd ) 2 pq D + w X where X = xq is an externality which the consumer ignores at the time they make their decision, where q is the total amount produced/consumed. There are two representative firms that imizes their profits, q 1 pq 1 c 2 q2 1 and q 2 pq 2 c 2 q2 2. i. What is the outcome in the perfectly competitive market? ii. If a social planner takes the externality into account, what is the optimal quantity to trade and for each firm to produce? iii. Suppose the government used a cap-and-trade program, where the total number of permits allotted is set equal to the optimal quantity from part ii. Solve for the price that prevails in the goods market, then determine each firm s demand for permits, where each permit costs π to purchase. iv. Set total demand for permits equal to total supply and derive the equilibrium price for permits. What does it equal? v. Explain one reason a government might prefer cap-and-trade to a tax or quota, and show how you would adjust the model to reflect this motivation. (You don t have to resolve for the equilibrium, just explain clearly how your adjustment to the model reflects the motivation you are proposing.) 1

2 3. There are two kinds of firms: low quality ones that sell goods of quality x L = 0 and occur with probability r, and high quality ones that sell goods of quality x H = 9 and occur with probability 1 r. Low quality firms have production costs (c/2)q 2 while high quality firms have production costs (c/2)q 2 + F, where F = 9/4 is a fixed cost of investment in quality. There is a representative consumer that purchases a quantity q to solve rx L log(q) + (1 r)x H log(q) px + w q where p is the price in the market and w is the household s wealth. i. Suppose both high and low quality firms operate in the market. What is the market clearing price and quantity, and what are the profits of the firms? Draw a supply-and-demand diagram and show where the inefficiencies occur. ii. At what value of r do the high quality producers withdraw from the market? Sketch a graph of the quantity traded as a function of r. iii. Suppose firms offered refunds: if the customer didn t like the product, the customer can return it and get their money back. (For example, in 1999, Hyundai extended their warranty from 5 years and 60,000 miles to 10 years and 100,000 miles after criticisms that their cars were low quality.) Explain how such a signal changes competition in this market. Is the refund signaling or screening? Is it actually costly for the high quality firms to adopt this behavior? 4. The classic adverse selection model is a bit simpler than the one we have been studying. There are buyers and sellers. Sellers have a lemon with probability r with value x L, and a peach with probability 1 r with value x H, where x H > x L. If a buyer pays a price t to get a good of quality x, his payoff is x t, and zero if no trade occurs. If a seller sells her good of quality x at a price of t, her payoff is t, while if she keeps her good, her payoff is 1 gx, where 0 < g < 1. i. Under what conditions does a high-quality seller sell her good? Under what conditions does a low-quality seller sell her good? ii. Suppose there are many more buyers than sellers, so that buyers bid the price in the market up to the expected quality of the good. What is the expected quality/price when peaches and lemons trade? When only lemons trade? Is it possible for peaches to trade but not lemons? iii. For what values of r do both peaches and lemons trade in the market? Sketch a graph of the market price as a function of r. At what value of r do peaches withdraw from the market, leading to unraveling? 5. There are two kinds of people: low risk, L, who occur with probability r and get sick with probability x, and high risk, H, who occur with probability 1 r and get sick with probability 1. If someone is healthy, they get a benefit v H, and if sick, they get a benefit of v S = 0. People can buy insurance which costs p, and ensures that if they get sick, they are treated and become healthy again. Thus, the payoff to the low risk types from getting insured is v H p, while going uninsured gives a payoff of xv S + (1 x)v H ; high risk types get v H p from getting insured and v S from going uninsured. Treating someone costs c < v H v S, so that treating people is socially optimal once they are sick. i. Assume the insurance market is perfectly competitive, so that the price is equal to the expected cost of care. If types are observable, what are the prices of insurance for low and high 1 The variable g creates a motivation for sellers to be willing to part with their goods: the initial allocation isn t Pareto optimal, since buyers value goods more than sellers. Think of a market for used cars, where some people are selling because they have a legitimate reason, like moving to a new city across the country and not wanting to drive, while other people have a lemon and want to dump it. 2

3 risk people? Under what conditions do the low risk types refuse to purchase insurance? Do high risk types ever refuse to purchase insurance? ii. Assume the insurance market is perfectly competitive, so that the price is equal to the expected cost of care. If types are unobservable, what is the market price of insurance? How does the price depend on r? When do the high types withdraw from the perfectly competitive market? iii. If a profit-imizing monopolist were setting prices assuming that both high and low risk types purchase insurance, what price would it pick for insurance? Is that price profitable? If that price isn t profitable, what happens in the market? iv. Explain why firms wish to withdraw from the ACA exchanges. 6. Suppose there are two types of people: low productivity, who occur with probability r and produce profits π L for a firm, and high productivity, who occur with probability 1 r and produce profits π H for a firm, π H > π L. The market for labor is perfectly competitive, so the wage is equal to the expected productivity of the worker. A worker s payoff equals his wage, and being unemployed yields a payoff of zero. i. If productivity is observable, what are the wages for the different types of workers? ii. If productivity is unobservable, what is the market wage for labor? Now suppose a high productivity person can pay a cost c to go to college rather than just high school, but a low productivity person has to pay c + e, since college imposes a higher effort cost, e, on low productivity people. A worker s payoff now equals his wage minus any education costs, and being unemployed yields a payoff of zero. iii. If the high productivity people could spend money on education to separate themselves from low productivity people, the wages in part i would prevail rather than part ii. If productivities are unobservable as in part ii, what are high productivity people willing to spend on education in order to signal their types? How high must the payment be in order to deter low types from trying to act as if they are high types? How much can universities charge the high types? iv. Now, assume that everyone believes high types signal by going to college while low types do not. What are high productivity people willing to spend on education in order to signal their types? How high must the payment be in order to deter low types from trying to act as if they are high types? How much can universities charge the high types? v. Comment on (a) what parts ii and iii say about the way our beliefs affect the value of education and how we evaluate college and high school graduates and (b) the benefits and downsides of universities in American society as brokers of talent. 7. There are I buyers who take prices as given and each solve q i 2 q i pq i + w i, and there are J sellers who take prices as given and each solve q j pq j cq j. Assume I > J. i. In the centralized market, all buyers and sellers trade together. Determine the equilibrium price and quantity traded. 3

4 ii. In the decentralized market, each buyer matches to a seller with probability J/I, so that only J matches occur. Determine the prices and quantities traded within each match, and the total quantity traded. iii. Compare the prices and quantities traded in the centralized and decentralized markets. iv. The firm s cost curves are C(q j ) = cq j, known formally as constant returns to scale or CRS. Sketch a supply-and-demand diagram for this market and show where the inefficiencies arise. Explain how the model and outcomes differ from the one we looked at in class. v. Suppose an entrepreneur created a platform on which these agents could trade. What profits could he make by improving efficiency in the market, in expectation? Once all trade is going through the platform and the decentralized market is abandoned, what kinds of profits can he make? 8. When the trade of a good is made illegal, it creates decentralized, unregulated black markets (presumably because of externalities, but let s keep it simple). Each consumer i = 1,..., I takes prices as given and solves q i log(q i ) pq i + w i. The government decides to make production of the good illegal, so it cannot be traded in a centralized market. In addition, the government imposes penalties on firms caught selling the good, equal to t 1 2 q2 j per unit sold (more production is penalized more severely); firm are caught with probability 0 < e 1. Thus, each firm j = 1, 2,..., J takes prices as given and solves q j c q j 2 q2 j et 2 q2 j (1 e)0. Assume I > J. i. In a centralized market without the penalty, what would be the equilibrium price and quantity? In the decentralized market with the penalty where only J consumers and firms trade, what is the price and quantity? ii. How does the price and quantity in the decentralized market depend on the expected penalty, et? iii. If the government sets et high enough, does the decentralized market shut down? Explain why or why not. iv. Explain why decentralization and penalties create an environment in which entry by new firms is attractive relative to the decentralized market, and how this partially undoes efforts to reduce production. 9. Why does money exist? It is a technology that solves search problems. There are three types of people: A, B, and C. Type A people want A goods but produce B goods, type B people want B goods but produce C goods, and type C people want C goods but produce A goods. Finding a good you want is worth u and it costs c to produce a unit of a good, and u > c. Each day, people go to the market to trade. The probability of meeting another type is uniformly random, so the likelihood an A meets an A is 1/3, that an A meets a B is 1/3, that an A meets a C is 1/3, and likewise for B and C. i. Show that in the absence of money, no one can trade with one another, since there are no double coincidences of wants (this is called Autarky). If there was a centralized market where barter is at the level of goods instead of individuals, how much trade would occur? 4

5 ii. Suppose now that some fraction of the population, m hold money rather than produce (call these people shoppers), and a fraction 1 m hold goods (call these people producers). When a shopper encounters a producer with a good they want, they trade: the producer now holds the money and the shopper becomes a producer; if a shopper meets a producer whose good they don t want, they don t trade, and if two shoppers or two producers meet, they don t trade. This implies there are two states (shopper or producer), and the values of being in either state are characterized by the asset pricing equations: J shopper = 1 m 3 ((u c) + δj producer ) Consume and go back to work Meet a producer of preferred type J producer = m 3 δj shopper Get paid, go shop Meet a shopper of preferred type ( m ( + ) } 3 {{ } Fail to trade 1 m ) }{{ 3 } Fail to trade δj producer Still waiting δj shopper Still shopping where 0 < δ < 1 is the discount factor, and captures their impatience to consume. Solve the above equations to find the expected values of being in the two states. Are the values of these two states bigger than zero? This implies that it is better to trade for the money than never take it, so that money has value in the economy. iii. If δ =.9, m = 1/2, and u c = 1, what is the value of being a shopper and producer? iv. If δ =.9 and u c = 1, can you compute expected welfare as a function of the money in the economy, m, and determine the optimal quantity of money? This is a very basic theory of optimal monetary policy, like what the Fed does. I would be super impressed. Hint: welfare is W = mj shopper(m) + (1 m)j producer (m) m 5