6005+Chptr+8+HWAns - Mgt 6005 Chptr 8 HW Answers P8.4...

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Mgt 6005 Chptr 8 HW Answers P8.4 Returns to Scale . Determine whether the following production functions exhibit constant, increasing, or decreasing returns to scale. A. Q = 0.5X + 2Y + 40Z B. Q = 3L + 10K + 500 C. Q = 4A + 6B + 8AB D. Q = 7L 2 + 5LK + 2K 2 E. Q = 10L 0.5 K 0.3 P8.4 SOLUTION A. Initially, let X = Y = Z = 100, so output is: Q = 0.5(100) + 2(100) + 40(100) = 4,250 Increasing all inputs by an arbitrary percentage, say 2%, leads to: Q = 0.5(102) + 2(102) + 40(102) = 4,335 Because a 2% increase in all inputs results in a 2% increase in output (Q 2 /Q 1 = 4,335/4,250 = 1.02), the output elasticity is 1 and the production system exhibits constant returns to scale. B. Initially, let L = K = 100, so output is: Q = 3(100) + 10(100) + 500 = 1,800 Increasing both inputs by an arbitrary percentage, say 3%, leads to: Q = 3(103) + 10(103) + 500 = 1,839 Because a 3% increase in both inputs results in a 2.2% increase in output (Q 2 /Q 1 = 1,839/1,800 = 1.022), the output elasticity is less than 1 and the production system exhibits diminishing returns to scale. C. Initially, let A = B = 100, so output is: 1
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Q = 4(100) + 6(100) + 8(100)(100) = 81,000 Increasing both inputs by an arbitrary percentage, say, 1%, leads to: Q = 4(101) + 6(101) + 8(101)(101) = 82,618 Because a 1% increase in both inputs results in a 2% increase in output (Q 2 /Q 1 = 82,618/81,000 = 1.02), the output elasticity is greater than 1 and the production system exhibits increasing returns to scale. D. Initially, let L = K = 100, so output is: Q = 7(100 2 ) + 5(100)(100) + 2(100 2 ) = 140,000 Increasing both inputs by an arbitrary percentage, say, 2%, leads to: Q = 7(102 2 ) + 5(102)(102) + 2(102 2 ) = 145,656 Because a 2% increase in both inputs results in a 4% increase in output (Q 2 /Q 1
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