4 Consider a market with inverse demand function p 14 Q Firms have constant

4 consider a market with inverse demand function p 14

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4. Consider a market with inverse demand function p = 14 ° Q . Firms have constant marginal cost 2 and °xed cost 2. Firms compete by simultaneously choosing quantities. (a) Suppose there are n °rms in this market. Derive the Nash equilibrium prices, quantities and pro°ts. Firm i maximizes ° i = p ( Q ) q i ° 2 q i ° 2 : The FOC is ² 14 ° X j 6 = i q j ³ ° 2 q i ° 2 = 0 : We are looking for a symmetric equilibrium, and so this implies q j = 12 n + 1 for all j: 2
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We also have p = 14 ° Q = 14 ° 12 n n + 1 = 2 n + 14 n + 1 and ° i = 2 n + 14 ( n + 1) 12 ( n + 1) ° 2 12 ( n + 1) ° 2 = ° 12 n + 1 ± 2 ° 2 : (b) Treating n as a continuous variable (i.e. ignoring the integer constraint), solve for the equilibrium number of °rms when there is free entry. Entry occurs until the pro°t equals to zero: ° i = ° 12 n + 1 ± 2 ° 2 = 0 , n fe = 6 p 2 ° 1 ² 7 : 49 : (c) What is the e¢ cient number of °rms? How does this compare to the equilibrium number of °rms? Provide some intuition for this discrepancy. For given n , the consumer surplus is CS ( n ) = 1 2 (14 ° p ) Q = 1 2 ° 12 n n + 1 ± 2 : The total surplus is TS ( n ) = CS ( n ) + n X i =1 ° i = 1 2 ° 12 n n + 1 ± 2 + ° 12 n + 1 ± 2 n ° 2 n: Di/erentiate this with respect to n to get ° 12 n n + 1 ± 12 ( n + 1) ° 12 n ( n + 1) 2 + 144 ( n + 1) 2 ° 288 n ( n + 1) ( n + 1) 4 ° 2 = 0 , n ° = 72 1 3 ° 1 ² 3 : 16 : When making an entry decision, °rms do not take into account the e/ect of entry on other °rms± pro°t. Without °xed costs, entry leads to a higher total surplus, since an increase in consumer surplus associated with °rm±s entry is always larger than the decrease in °rm±s pro°t. However, with the °xed cost, the increase in consumer surplus gets smaller as n gets bigger, but the °xed cost incurred is the same for any additional entry.
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