log(q i ) pq i + w i, max pq j c 2 q2 j.

Transcription

1 . There are I buyers who take prices as given and each solve q i log(q i ) pq i + w i, and there are sellers who take prices as given and each solve p c. Assume I >. i. In the centralized market, all buyers and sellers trade together. Determine the equilibrium price and quantity traded. Each household s FONC is q i p = 0 q i = p so market demand is Each seller s FONC is and market supply is And market-clearing implies and the market-clearing price is and the market-clearing quantity is Q D (p) = I q i = I p. i= p c = 0 = p c Q S (p) = j= I p = p c p = q = = p c. Ic I c. ii. In the decentralized market, each buyer matches to a seller with probability /I, so that only matches occur. Determine the prices and quantities traded within each match, and the total quantity traded. In each of the mini-markets, demand is q i = p, = p c, so the clearing price is p = c and the clearing quantity is q = / c. The total quantity in the decentralized market then is Q dec = c.

2 iii. Compare the prices and quantities traded in the centralized and decentralized markets. Since I >, I > =. Then q = I/c > / c = Q dec. So there is me traded in the centralized market. 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? If I and increase at the same rate, what happens to price and quantity? In the centralized market, the price is increasing in market tightness: if the ratio of buyers to sellers goes up, so does the price. The reason is that the firms have increasing marginal costs, so that if they produce me, their marginal costs go up. Increasing market tightness means adding me buyers relative to the number of sellers, so this bumps demand up and pushes sellers up their supply curves. If I goes up, the total quantity goes up in the centralized market, since q = I/c.. There are two types of firms: low quality ones that sell goods of quality and occur with probability r, and high quality ones that sell goods of quality and occur with probability r. Low quality firms have production costs q while high quality firms have production costs q + F, where F is a fixed cost of investment in quality. There is a representative consumer that purchases a quantity q to solve r log(q) + ( r) log(q) px + w q where p is the price in the market and w is the household s wealth. i. If types were observable, what would the price and quantity be in the market f low-quality goods? F high-quality goods? In the market f low-quality goods, the household s FONC is and the FONC f a low-quality firm is and the market clears if p = 0 p = q L q L p q L = 0 p = q L /, q L / = q L q L = and p L = /. In the market f high-quality goods, the household s FONC is and the FONC f a high-quality firm is p = 0 p = p = 0 p = /, and the market clears if / = = 8

3 and p H = 8/. ii. Suppose types are not observable, and 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. The household s FONC is now r + ( r) q the FONC f the high-quality firm is and the FONC f the low-quality firm is Total supply then is Market-clearing then requires Q S = Q D, p = 0 Q D = p = 0 p/ =, p q L = 0 p/ = q L. Q S = ( r) + rq L = p/. r + ( r) p = p/ p = r + ( r)8. r + ( r), p The diagram has two deadweight loss triangles: too many low-quality goods are produced, and too few high-quality goods are produced. iii. At what value of r do the high quality producers withdraw from the market? Sketch a graph of the price as a function of r. The high-quality firm s profits are p ( ) F = ( ) r + ( r)8 r + ( r)8 r + ( r)8 F So if r + ( r)8 r + ( r)8 F = r + ( r)8 8 F < 6r F = r + ( r)8 high quality firms withdraw, 8 F r. 6 So if there are too many low-quality firms, the high-quality firms leave the market. The graph looks like the one in the slides. iv. Suppose the low quality firms are trying to impersonate a successful high quality firm. Explain why trademark and copyright laws patents can protect high quality firms. Building a brand reputation allows firms to differentiate themselves from low-quality competits, 3 F.

4 but is costly (advertising and marketing take money). But if low-quality firms can just steal their name trademark and sell their inferi version of the product to unknowing consumers, it destroys the high-quality firms investments. So if copyright and trademark laws exist, it gives firms an incentive to get a reputation f quality. (In early trademark law cases, courts ruled that brands like Bandaid had not defended their turf, so that generics and low-quality knock-offs could use bandaid as a noun f their product. This means that firms have to defend their proper noun name aggressively to keep low quality firms from appropriating it. This explains many lawsuits and cease-and-decist letters.) 3. There is a representative household that picks an amount to consume to solve 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. The goods are produced by j =,,..., profit-imizing firms in Cournot competition, whose costs are C( ) =. i. What amount is produced and consumed in equilibrium? The household s FONC is q p = 0 p = q. Then each firm j solves π j = ( q q... q ), q k = 0. k= Since all the firms face the same demand curve and have the same costs, there s a Cournot/purestrategy Nash equilibrium where every firm does the same thing, q, yielding Then the total quantity produced is ( + )q = 0 q = +. Q = q = + and the price is p = +. ii. What quantity imizes utilitarian social welfare? Social welfare is given by Q Q + w xq Q, since the firms have linear costs and all of the transfers from the household to the firms cancel out. Maximizing yields the FONC, Q x = 0 Q o = x.

5 iii. If me firms entered, does it raise lower social welfare in equilibrium? Explain. What is the optimal number of firms? Let s set the two quantities equal: and solving f yields + = x = + x x x = x = x x. So if < ( x)/x, adding firms raises welfare. But once > ( x)/x, quantity is expanding but the negative effect of the externalities dominates. iv. If the government imposed a tax tq on the household s consumption, what t implements the socially optimal outcome? The household s problem becomes Q Q Q (p + t)q + w X Q p t = 0, and the price is p = Q t. Each firm j then solves π j = ( q k t) k= q k t = 0, and in a symmetric equilibrium in which they all produce q, k= ( + )q t = 0 Q = t +. Now we want to equate this to the socially optimal quantity: which equals So the optimal tax is t + = x, t = ( + )( x) t = ( + )( x). t = ( + )( x). 5