final_f09solutions

final_f09solutions - Economics 105 Final Exam SOLUTIONS Dec...

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Unformatted text preview: Economics 105 Final Exam SOLUTIONS Dec. 11, 2009 1. Mark whether the statement is True or False (a) For second degree price discrimination, the incentive compatibility constraint holds with equality for the low demanders. (2 pts) False, the low demands never have an incentive to pretend to be the high demanders. (b) A normal good can never be a Giffen good. (2 pts) True, substitution effects are always positive, and since normal goods have a positive income effect, then if prices falls, demand must go up. (c) The production function ( L 2 + K 2 ) 1 / 3 is quasi-concave. (2 pts) False, this production function does not have the property that averages are at least preferred to extremes. (d) First degree price discrimination is Pareto efficient. (2 pts) True, The monopolist extracts all of the consumer surplus, and the individual who’s marginal benefit equals marginal cost is receiving the good. (e) For third degree price discrimination, the monopolist sets marginal revenue to be the same in both markets. (2 pts) True, profits are maximized in both markets. 1 2. Firms producing x have profit functions given by p 2 2 w + r- 200 . (a) Derive the individual firm’s input demand for labor and capital as well as the supply function. Verify that the supply function is homogenous of degree in . (5 pts) The input demands and supply function can all be derived using Hotelling’s lemma and the profit function. l ( w,r,p ) =- ∂π ∂w =-- p 2 2 (2 w + r ) 2 = 2 p 2 (2 w + r ) 2 k ( w,r,p ) =- ∂π ∂r =-- p 2 (2 w + r ) 2 = p 2 (2 w + r ) 2 x ( w,r,p ) = ∂π ∂p = 2 p (2 w + r ) The supply function is HD0 in prices x ( τw,τr,τp ) = 2 τp (2 τw + τr ) = 2 p (2 w + r ) = x ( w,r,p ) (b) Derive the firm’s cost function and conditional input demands. (4 pts) The conditional inputs can be found by inverting the supply function, and plugging the value of p into the unconditional demands. p = x (2 w + r ) 2 l ( w,r,x ) = 2 p 2 (2 w + r ) 2 = 2 x (2 w + r ) 2 2 (2 w + r ) 2 = (1 / 2) x 2 k ( w,r,x ) = p 2 (2 w + r ) 2 = x (2 w + r ) 2 2 (2 w + r ) 2 = (1 / 4) x 2 The cost function is then: c ( w,r,x ) = (1 / 2) x 2 w + (1 / 4) x 2 r (c) Suppose demand is given by X = 1400- 10 p , w = 3 , and r = 2 . Find the long run equilibrium price, the number of firms, and how much each firm produces. (6 pts) Since the firms have a decreasing returns to scale technology with fixed costs, the the equilib- rium price is set by the zero profit condition. π = 0 = p 2 2 w + r- 200 = p 2 2(3) + 2- 200 0 = p 2 8- 200 p LR = 40 2 Aggregate demand is X LR = 1400- 10 p = 1400- 10(40) = 1000 Plugging price in to the individual supply functions gives the amount each firm produces, x ( w = 3 ,r = 2 ,p = 40) = 2(40) (2(3) + (2)) = 10 The number of the firms in the market it then,...
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This note was uploaded on 05/22/2010 for the course ECON 105D taught by Professor Cur during the Fall '09 term at Duke.

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final_f09solutions - Economics 105 Final Exam SOLUTIONS Dec...

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