All questions are weighted equally, and each roman numeral is weighted equally within each question. Good luck!. There are I buyers who take prices as given and each solve max q i log(q i ) pq i + w i, and there are sellers who take prices as given and each solve Assume I >. max q j pq j c q j. i. In the centralized market, all buyers and sellers trade together. Determine the equilibrium price and quantity traded. Each buyer i s demand is characterized by so that total demand is Each seller j s supply is characterized by q i p = 0 q i = p, I q i = I p. i= p cq j = 0 q j = p c, so that total supply is Setting supply and demand equal yields j= q j = p c. p c = I p p = I c and I q = c. ii. In the decentralized market, matching is one-to-one, so only of the I buyers get to match to a seller to trade. Determine the prices and quantities traded within each match, and the total quantity traded. In this case, within each mini-market, we set supply equal to demand to get and quantity in each mini-market is p = p c p = c, q = c.
Then the price in the decentralized market is just c and the quantity is / c. iii. Compare the prices and quantities traded in the centralized and decentralized markets. The price in the centralized market is I/c, and since I >, this implies I/c > /c = /c, which is the price in the decentralized market. So prices are higher. The quantity in the centralized market is I/c > /c = / c, which is the quantity in the decentralized market. So prices are higher and quantities are higher. iv. In the centralized market, how does the price depend on market tightness, I/? If there are I buyers and sellers, there are ways to match the first buyer to each seller, ways to match the second buyer to each seller, and so on, f a total of I connections. How does the quantity in the centralized market depend on the total number of connections? As the number of sellers grows large in the decentralized market, does price get closer to marginal cost as would happen in a centralized perfectly competitive market not? The price is increasing in market tightness. The quantity is increasing in the number of connections. As the number of sellers in the centralized market gets large, the price never varies from c, so no, it does not converge to the perfectly competitive outcome.. While many people enjoy consuming alcohol, it has a number of negative externalities: lowered inhibitions lead to fights and drunk driving, and life-long consumption can lead to cirrhosis of the liver various kinds of cancer. To model this, suppose there is a representative household that takes prices as given and decides how much alcohol to consume, Q, solving max Q Q Q pq + w X where X = xq is an externality that the household does not take into account when making its consumption decision. Alcohol is produced by j =,,..., profit-maximizing firms in Cournot competition, whose costs are C(q j ) = q j. i. What amount is produced and consumed in equilibrium? The household s demand curve is Each firm k then maximizes with FONC Q p = 0 p = Q. π k = ( ) q j q k q k j= q j q k = 0. j= Since the firms face the same demand curve and have the same costs, let q = q =... = q, and we ll look f a symmetric solution to the game. This implies ( + )q = 0 q = +
and the total quantity produced is Q = q = +. ii. What quantity maximizes utilitarian social welfare? Utilitarian social welfare is W = Q Q pq + w xq + pq Q where Q is the total amount produced. The transfers cancel, and maximizing yields Q x = 0 Q o = x. iii. If me firms entered the market, does it raise lower social welfare in equilibrium? Explain. Is the amount in the market greater less than the optimal amount? + x ( x)( + ) x x + x x So if the number of firms is too large relative to ( x)/x, the externality dominates and social welfare is harmed by entry. iv. If the government imposed a tax tq on the household s consumption, what t implements the socially optimal outcome? The household s FONC then becomes Q p t = 0 p = t Q, and the total quantity produced in the market is q (t) = ( t) +. In der to match the social efficient quantity, this means and ( t) = ( x) + t = + ( x). 3
v. Briefly explain in wds how when there are multiple market failures, crecting one might reduce social welfare overall. In this case, market power reduces quantity, which can be beneficial in the presence of a negative externality. In general, when there are multiple market failures wking in opposite directions, society might benefit on net relative to a market where someone intervenes to solve one market failure but not another. 4
3. Consider a lending market, where entrepreneurs with different levels of talent take out loans f businesses. There are two kinds of browers: low risk, L, who occur with probability r and succeed with probability, and high risk, H, who occur with probability r, and succeed with probability /. The profit of a successful business is π, and it costs c to start a business. Banks can loan c to the brower in der to fund the business, not. Successful businesses produce profits π and repay t, while those that fail produce no profits and do not repay the loan. Call t the price of the loan. The payoff to the low risk type from getting a loan is π t, while getting no loan gives a payoff of zero; the payoff to high risk types from getting a loan is (π t) + 0, while getting no loan gives a payoff of zero. Assume π > c. i. Suppose the lending market is perfectly competitive, so that the expected return on the loan is equal to the cost of a loan (remember, loans ending in default repay nothing). If types are observable, what are the prices of loans f low and high risk browers? Under what conditions do the low risk browers take out loans? What about high-risk browers? If types are observable, the expected profit on a loan to a low-risk firm is t c so that t L = c. The expected profit to a high-risk firm is t + 0 c = 0, so that t H = c. Low risk browers always take out loans since π > c, but high-risk browers select out of the market on their own if π < c. ii. Suppose the lending market is perfectly competitive, so that the expected return on the loan is set equal to the cost of a loan. If types are unobservable, what is the price of a loan? How does the price depend on r? Suppose π = and c = 3/4: at what value of r do the low-risk types withdraw from the market? At what value of r do the high-risk types withdraw from the market? The expected profit on a loan is now rt + ( r) t c = 0, so that c t I = r + ( r) = c + r. The price is decreasing in r, since the presence of me low-risk people in the market reduces the cost of capital/lending. With these numbers, both types brow if 3/ + r 0 + r 3 r. iii. Suppose, instead, that a monopolist bank was selling loans to maximize its profits but could not observe browers types. What price would it charge? Suppose π = and c = 3/4: f what 5
values of r is lending profitable? The monopolist faces the constraints π t 0 and (π t) 0, so he can charge all the way up to π = t. Then his profits are F those parameters, his profits are positive if πr + ( r) π c. r + ( r) 3 4 0 r + ( r) 3 r. iv. Explain in wds how the presence of bad lending risks in the market can cause it to collapse, and no one ends up with loans. (This is an imptant part of the 008 financial crisis that started in the market f mtgages.) The presence of the high types with incomplete infmation drives up the price of a loan. If there are too many bad risks, the price is too high f anyone to want to brow in the perfectly competitive market, and under the same conditions, it is not profitable f a monopolist to lend. This causes the market to unravel, and no credit is available f anyone. 6