q S pq S cq S. b q D p = 0, firms maximize profits taking prices as given, so they which to produce 0, p < c , p > c

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? Households imize utility taking prices as given (but ignoring their contribution to the externality), yielding a FONC b q D p = 0, firms imize profits taking prices as given, so they which to produce 0, p < c q S = [0, ), p = c., p > c So the only case in which markets can clear is when p = c, implying that q = b/c. (This is the case where the supply curve is infinitely elastic at a price of c.) 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. The social planner imizes b log(q) + w xq cq, q with FONC or b x c = 0, (1) q q 0 = b x + c < b c = q, so that more is produced in the perfectly competitive market. Therefore, the perfectly competitive outcome is not efficient. iii. If the government wanted to impose a quota on production or consumption of this good, what would the optimal quota be? The quota is... q 0 = b/(x + c). 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 1

2 how the optimal tax works. The household would then solve where tq is the tax. The FONC is b log(q) pq tq + w X q b q p t = 0, and we know that the firm s solution and market-clearing implying p = c, leaving b q c t = 0. Now look at equation (1) above: what s missing? Well, t should be equal to x in order for the household to be induced to behave the way the firm does. Therefore, the optimal tax is t = x. 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? The household takes prices as given and solves q D 1 q D 2 (qd ) 2 pq D + w X with FONC while firm j solves with FONC so the supply curve for both firms is 1 q D p = 0, pq j c q j 2 q2 j p cq j = 0, q S = q 1 + q 2 = 2p c. For the market to clear, q D = q S, or 1 p = 2p /c, 2

3 yielding p = c 2 + c, q = c. ii. If a social planner takes the externality into account, what is the optimal quantity to trade and for each firm to produce? We sum household utilities and firm profits to get the social planner s problem (remember, payments to and from the firm and household cancel out), with FONCs q 1,q 2 (q 1 + q 2 ) 1 2 (q 1 + q 2 ) 2 + w x(q 1 + q 2 ) c 2 q2 1 c 2 q2 2 1 (q 1 + q 2 ) x cq 1 = 0 1 (q 1 + q 2 ) x cq 2 = 0. This is a linear system of two equations in two unknowns which you can solve by substitution, or just notice that the two firms are identical, so that the solution should involve q 1 = q 2 = q, so that the quantity for each firm is 1 x = (x + c)q q = 1 x 2 + c, and the total optimal quantity is 2(1 x) q = 2 + c. So it is (1 x) times the perfectly competitive quantity. 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. Since q < q, less will be sold in the cap-and-trade market than in the perfectly competitive market. Since there is a cap, the quantity traded will be q, and the household s marginal benefit will be the price, or Then each firm solves with FONC p = 1 2(1 x) 2 + c = c + 2x 2 + c. m j pm j c 2 m2 j πm j, p cm j π = 0, so total demand for permits is m 1 + m 2 = 2 p π. c iv. Set total demand for permits equal to total supply and derive the equilibrium price for permits. What does it equal? If total supply for permits equals total demand, we get q = m 1 + m 2 3

4 or or or (doing a few more rounds of algebra), 2 1 x 2 + c = 2 p π, c c(1 x) 2 + c = c + 2x 2 + c π, π = x. So the equilibrium price of a permit is the size of the externality! It s like magic! We implemented the same outcome as the quota or the tax with a market! 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.) Suppose the government were unsure of the size of the externality, x, so that it might be high with probability p and take the value x, or low with probability 1 p and take the value zero. In the quota or the tax regime, it might be difficult to fix the policy once people learn what value the externality really takes. In the cap-and-trade version, we could simply plan for x being high, and if it turns out to be zero, we then flood the market with permits for free. This way, the policy maker can do open market operations to control behavior, rather than legislating it. 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. If both firms are in the market, the buyer solves rx L log(q) + (1 r)x H log(q) px + w q with FONC so demand is Low quality firms solve rx L + (1 r)x H q p = 0, Q D (p) = rx L + (1 r)x H p q L pq L c 2 q2 L 4

5 with FONC and high quality firms solve p cq L = 0 q L = p/c q H pq H c 2 q2 H with FONC so market supply is p cq H = 0 q H = p/c, Q S (p) = rq L + (1 r)q H = p c. Market clearing requires Q D (p ) = Q S (p ), or so that p c = rx L + (1 r)x H, p p = c(rx L + (1 r)x H and q rxl + (1 r)x H =. c There are two deadweight loss triangles in the graph, one representing over-production of lowquality goods and one representing under-production of high-quality goods. 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. The profits for the high-quality firm are: so that if π H = p p c c p 2 2 c F 2 = rx L + (1 r)x H 1 2 (rx L + (1 r)x H ) F = 1 2 (rx L + (1 r)x H ) F x H 2F x H x L r, the high types withdraw, and the market collapses, leaving only low quality goods. 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? For the high types, there will be no refunds (because they are high quality), while the low types will have to refund all their customers. So there is no consequence to the high types of signaling in this way: if you never have to pay out on the warranty because your cars really are high quality, making this promise is costless and you can convince people of your type. 5

6 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? A high-quality seller sells if t gx H and a low quality seller sells if t gx L. 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? If lemons and peaches are both in the market, the expected price/quality is t = rx L + (1 r)x H while if only lemons are in the market, the price is t = x L, since everyone knows that only lemons will trade at such a low price. Remember: being willing to participate in a trade reveals something about you. It is not possible for peaches to trade but not lemons. If only peaches traded, the price would be t = x H, and x H > gx L, so lemons would definitely join the market. 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? Peaches are in the market only if rx L + (1 r)x H gx H, or (1 g)x H r, x H x L so that there aren t too many lemons. At this particular r, the peaches withdraw, and the price crashes to x L, resulting in market unravelling. 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. 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. 6

7 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 risk people? Under what conditions do the low risk types refuse to purchase insurance? Do high risk types ever refuse to purchase insurance? High risk people always get sick, so the price in the high risk market is c. Low risk people face a price of t = xc + (1 x)0. The low risk people buy insurance if or or v H t L (1 x)v H + xv S, v H xc (1 x)v H + xv S, v H c, so high types always buy insurance if the markets are separate. High risk types buy insurance if v H c 0, which is, again, always true. 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? If types are unobservable the price/expected cost of care is t = r(xc + (1 x)0) + (1 r)c = rxc + (1 r)c. The price is decreasing in r, so that as you add more low risk people, the price of insurance comes down. High types stay in the market if or v H rxc (1 r)c > (1 x)v H xv H c > rc(x 1) So if xv H c, they always stay in, since x 1 < 0, so the right-hand side is negative. If xv H c < 0, then we can re-arrange to get r > c xv H c(1 x), so that there need to be enough low types for the insurance to be valuable. Otherwise, if r falls too low, the high types exit the market and it crashes. 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? There are two cases: the monopolist sells to the high types only, or sells to both high and low types. i. High types only: The high types will buy if v H t 0, so the monopolist can raise the price as high as v H before the high types withdraw. That means the monopolist s profits are π 1 = (1 r)(v H c). 7

8 ii. High and low types: The high types will buy if v H t 0 and the low types will buy if v H t (1 x)v H, or xv H t. So the highest the monopolist can raise the price is xv H. This gives profits of π 2 = r(xv H xc) + (1 r)(xv H c) = xv H rxc (1 r)c. Now, π 2 0 if x is sufficiently high, so that the low risk types are actually pretty sickly; if x = 0, pi 2 = (1 r)c, while π 1 0. If x is close to 1, so that the low risk types rarely get sick, then profits are close to v H c, which is better than π 1 = (1 r)(v H c). So it depends on x what is profitable or more profitable. iv. Explain why firms wish to withdraw from the ACA exchanges. r is too low or x is too high or both. The patients on the exchanges are more costly than actuaries expected, so that they cannot cover their costs. 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? If productivity is observable, the wage to high-productivity workers is w H = π H, and the wage to low-productivity workers is w L = π L. ii. If productivity is unobservable, what is the market wage for labor? The average productivity is w = rπ L + (1 r)π H. 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? High productivity people are willing to signal in order to separate themselves if π H c rπ L + (1 r)π H r(π H π L ) c while low productivity people are not willing to signal if π H c e rπ L + (1 r)π H r(π H π L ) e c, so that the range of c the university can charge for this service is r(π H π L ) c r(π H π L ) e. So the upper bound is the gain to the high types over the wage in a pooled market where everyone is paid the average productivity. 8

9 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? In this world, there is a rat race where people go to college simply to prove their type. So high types go if π H c π L π H π L c and low types do not if π H c e π L π H π L e c, so universities can charge anything in the range of π H π L c π H π L e. So universities can now charge a c as high as the difference between π H π L : they can get the entire difference in productivity between the high and low wage worker. 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. As universities become central to the way talent is allocated, they can charge much higher prices. Where part iii just says that high talent people might be willing to go to college to prove their type, part iv says that once society recognizes a college education as a necessity for signaling, universities can charge much higher prices. It s a benefit that we allocate talent well, but it s a potentially large downside that these institutions can extract so much value without necessarily providing much benefit. 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. Each household s FONC is 1 qi p = 0 q i = 1 p 2, so market demand is Each firm s solution is I Q D (p) = q i = I p. 2 i=1 0, p < c q j = [0, ), p = c, p > c, 9

10 so firms are happy to collectively produce any quantity at a price of c. This means the market clearing price is p = c and the market-clearing quantity is q = I c 2. 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. In each of the J mini-markets, there is one buyer and one seller, each clear at a price of p = c, and q = 1 c 2. Since there are J such mini-markets, the total quantity is Q dec = J c 2. iii. Compare the prices and quantities traded in the centralized and decentralized markets. Q dec = J/c 2 < I/c 2 = q, so strictly less is traded in the decentralized market: there are a bunch of missed opportunities for the buyers who fail to find a partner to trade with. 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. Here, supply is infinitely elastic at a price of c. In class, the supply curves were upwards sloping, so marginal cost was increasing. When we centralized the market, prices also went up, because the new buyers drove up aggregate demand, so sellers operated at higher costs. In this problem, the only thing that happens when we centralize the market is that more households get to trade. 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? Initially, he could get the deadweight loss triangle, since the outside option of the search market still exists. Once the search market is gone, no one has an outside option, and the platform is a monopolist. So he buys from sellers and sells to buyers, solving q I q cq = I q cq, q q with FONC I 2 q c = 0, or q m = I 4c. 2 So the monopoly platform might be even worse than the decentralized market: if I/4c 2 < J/c 2, or J > I/4, then a monopoly platform actually reduces welfare. Theory of the second best: if you solve one market failure, it might make another worse. 10

11 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 pq 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? Each household s FONC is 1 q i p = 0 q i = 1 p so that market demand in a centralized market is Q D (p) = I q i = I p. i=1 Each firm s FONC is p cq j etq j = 0 q j = so that market supply in a centralized market is Q S (p) = Jp c + et. p c + et So in the centralized market with no penalty, the clearing price is I p = Jp Ic c p = J and the clearing quantity is IJ q = c. In the centralized market with the penalty, within each of the J mini-markets, the market clears where and the price is 1 q = p = (c + et)q q = p = c + et. 1 c + et 11

12 So the total quantity traded in the black market is 1 q dec = J c + et. ii. How does the price and quantity in the decentralized market depend on the expected penalty, et? et enters the denominator of q dec positively, so if it goes up, quantity goes down. iii. If the government sets et high enough, does the decentralized market shut down? Explain why or why not. No, there will always be a black market. As et goes up, the price skyrockets, since there are consumers who have unbounded high value for the good. For the black market to shut down, you would have to make the penalty infinitely awful. 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. If prices are skyrocketing in this way, it becomes attractive for new firms to enter, even if there is a risk of getting caught. If entry is allowed, firms operate at lower marginal costs and face a lower risk of getting caught (they all have smaller scale operations) but still make plenty of money due to high prices. It looks a lot like the drug war in the 80 s and 90 s in the US. 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? When an A meets a B, B wants A s good but A does not want B s good. When a B meets a C, C wants B s good but B doesn t want C s good. When a C meets an A, A wants C s good but C doesn t want A s good. When two types of the same agent meet, neither want to trade. This leads to no trade occurring in equilibrium, despite it being socially optimal for As to give their goods to Bs, Bs to give their goods to Cs, and Cs to give their goods to As. 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 m ) 3 }{{} Fail to trade ( + δj shopper }{{} Still shopping

13 J producer = m 3 }{{} δj shopper }{{} Get paid, go shop Meet a shopper of preferred type ( + ) 1 m }{{ 3 } Fail to trade δj producer }{{} Still waiting 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. The second equation implies δm J pr = 3(1 δ) + mδ J sh and substituting this into the first equation implies ( u c + J sh = 1 m 3 and solving for J sh yields J sh = δ 2 m 3(1 δ) + mδ J sh ) + 3(1 δ) + mδ (u c). 3(1 δ)(3 + (1 δ)(2 m)) ( 1 1 m ) δj sh 3 Since J sh is positive, so is J pr, and money has value in equilibrium. iii. If δ =.9, m = 1/2, and u c = 1, what is the value of being a shopper and producer? Jsh = , J pr = Producers get lower utility because they have to wait to become a shopper again. 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 W grid 13

14 The imum is at m =.42: so about 60% of society should be producing goods, and the other 40% consuming/shopping. 14