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midterm2_f09solutions

# midterm2_f09solutions - Economics 105 Midterm 2 SOLUTIONS...

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Unformatted text preview: Economics 105 Midterm 2 SOLUTIONS Oct. 30, 2009 1. Mark whether the statement is True or False (a) Third degree price discrimination is Pareto efficient (2 pts) FALSE: Third degree price discrimination still has monopolies charging prices higher than marginal cost, which means some consumers are not getting the good who would be willing to pay more than marginal cost. (b) The supply function can be found by substituting conditional input demands into the production function (2 pts) FALSE: the supply function is found by substituting unconditional input demands into the production function (c) Hotelling’s Lemma allows us to go from profit functions to supply functions and input demands (2 pts) TRUE: ∂π ∂w =- l ( p,w,r ), ∂π ∂r =- k ( p,w,r ), and ∂π ∂p = x ( p,w,r ) (d) The production function min { 2 L 2 , 3 K 2 } is quasi-concave but not concave (2 pts) TRUE: Since the production function is increasing returns to scale, f ( τL,τK ) = min { 2( τL ) 2 , 3( τK ) 2 } = τ 2 min { 2 L 2 , 3 K 2 } = τ 2 f ( L,K ) the production function is convex (not concave). Also, since the isoquants are Leotieff (L- shaped), the production function has a convex upper contour set, so it is also quasi-concave. 1 2. A firm’s production function is given by x = l 1 / 2 k 1 / 2 . (a) Find the firm’s conditional input demands. Verify that they are homogeneous of degree in (5 pts) To find the conditional inputs we need to do expenditure minimization. min wl + rk s.t. x = l 1 / 2 k 1 / 2 L = wl + rk + λ ( x- l 1 / 2 k 1 / 2 ) ∂ L ∂w = w- λ (1 / 2) l- 1 / 2 k 1 / 2 = 0 ∂ L ∂r = r- λ (1 / 2) l 1 / 2 k- 1 / 2 = 0 ∂ L ∂λ = x- l 1 / 2 k 1 / 2 = 0 Solving for the conditional inputs demands gives k ( x,w,r ) = x w r 1 / 2 l ( x,w,r ) = x r w 1 / 2 Conditional input demands are homogeneous of degree 0 in prices. l ( x,τw,τr ) = x τr τw 1 / 2 = x r w 1 / 2 = l ( x,w,r ) k ( x,τw,τr ) = x τw τr 1 / 2 = x w r 1 / 2 = k ( x,w,r ) (b) Find the firm’s cost function. Verify that it is homogeneous of degree in (2 pts) The cost function is: C ( x,w,r ) = wx r w 1 / 2 + rx w r 1 / 2 = 2 x ( rw ) 1 / 2 Cost function are homogeneous of degree 1 in prices....
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midterm2_f09solutions - Economics 105 Midterm 2 SOLUTIONS...

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