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Econ 200 Problem Set 1 Solution

Econ 200 Problem Set 1 Solution - Answer Key Problem Set 1...

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Answer Key: Problem Set 1 January 26, 2005 Problem 1 (a) Demand: P = α - βq Revenue: Pq = ( α - βq ) q Marginal Cost: MC = c ; no fixed cost The monopolist maximizes: π = Pq - cq = ( α - βq ) q - cq The first order condition with respect to q is α - 2 βq - c = 0 Solving for q : q * = ( α - c ) 2 β Substituting this into demand function yields P * = α + c 2 1
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b) Profits: π * = P * q * - cq * = ( α - c )( α + c ) 4 β - c α - c 2 β = ( α - c ) 2 4 β c) Consumer Surplus (see Figure at end of text): CS = ( α - α + c 2 ) α - c 2 β 1 2 = α - c 2 α - c 2 β 1 2 = ( α - c ) 2 8 β d)Deadweight loss: DWL = ( q e - q * )( P * - c ) 2 = ( α - c β - α - c 2 β )( α + c 2 - c ) 1 2 = ( α - c ) 2 8 β e) P = 10 - q , hence q = 10 - P . Market Demand is calculated by summing over all individual demands: Q = 100 * (10 - P ) P = 10 - Q/ 100 We have α = 10 and β = 1 100 . Using your results from part (b) and (d) you get π * = 25(10 - c ) 2 DWL = 12 . 5(10 - c ) 2 2
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f) The monopolist would set price equal to marginal cost c and set a fixed fee equal to the consumer surplus q e ( α - c ) 2 that would result were there no fixed fee. The deadweight loss equals zero. Profits then equal the fixed fee π * = ( α - c ) 2 2 β = 50(10 - c ) 2 Problem 2 First we derive consumer i’s demand. The monopolist sets a two part tariff T = a + bq .Each consumer maximizes u i ( q, T ) = θ i v ( q ) - ( a + b ) q The first order condition yields (1 - q ) θ i - b = 0 and hence q * i ( b ) = 1 - b θ i . The monopolist sets a to extract all consumer surplus from the first consumer: a ( b ) = θ 1 v ( q ) - bq * 1 ( b ) = θ 1 1 - [1 - (1 - b θ 1 )] 2 2 - b (1 - b θ 1 ) = ( θ 1 - b ) 2 2 θ 1 She then maximizes aggregate profits : π = 2 a ( b ) + ( b - c ) ( q * 1 ( b ) + q * 2 ( b )) = 2 ( θ 1 - b ) 2 2 θ 1 + ( b - c )[(1 - b θ 1 ) + (1 - b θ 2 )] The first order condition with respect to b yields: 3
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- 2 ( θ 1 - b ) θ 1 + [(1 - b θ 1 ) + (1 - b θ 2 )] - ( b - c ) θ 1 - ( b - c ) θ 2 = 0 Solving for b yields: b * = c 2 θ 1 +
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