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ProblemSet5-2009

# ProblemSet5-2009 - prices Use this to prove that the cost...

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Econ 6202, Fall 2009 Dmitry Shapiro Problem Set 5 Due Tuesday October 20 1. A Leontief production has the form y = min { αx 1 , βx 2 } , where α > 0 , β > 0. Carefully sketch the isoquant map for this technology. Does this production function exhibit increasing, constant or decreasing returns to scale? 2. Producing output y requires only input x . The production function is y = 70 x. Let w denote the price of x . Compute the marginal cost and the average cost of producing y . Verify that the average cost is less than the marginal cost for all values of y . Explain why it is so. 3. The production function for some good is given by y = 27 x 1 + 5 x 2 . Let w 1 and w 2 denote the prices of inputs 1 and 2, respectively. Derive the conditional input demands. 4. (JR 3.21) A real-valued function is called superadditive if f ( z 1 + z 2 ) f ( z 1 ) + f ( z 2 ) . Show that every cost function is superadditive in input
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Unformatted text preview: prices. Use this to prove that the cost function is non-decreasing in prices. 5. Consider a ﬁrm with the cost function c ( y,w 1 ,w 2 ) = y 2 ( w 1 + w 2 ) , where w i denotes the price of input i,i = 1 , 2 . Let p denote the output price. Derive the output supply function y ( p,w 1 ,w 2 ), and the input de-mand functions x i ( p,w 1 ,w 2 ) ,i = 1 , 2. 6. Consider a ﬁrm with production function y = ( x ρ 1 + x ρ 2 ) α , where 0 < ρ < 1 , and α > 0. For what value of ρ and α are there (i) increasing returns to scale; (ii) constant return to scales; (iii) decreasing returns to scale? Suppose that there are decreasing return to scale. Find the long run cost function. Derive the output supply function and the input demand functions. 1...
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