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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Verify for the Cobb-Douglas production function

P(L, K) = 1.01L0.75L0.25

discussed in Example 3 that the production will be doubled if both the amount of labor and the amount of capital are doubled. Determine whether this is also true for the general production function

P(L, K) = bLαK1−α

To determine

Whether this result, “The Cobb-Douglas production function P(L,K)=1.01L0.75K0.25 will be doubled if both the amount of labor and the amount of capital are doubled”,  is true for the general production function P(L,K)=bLαK1α .

Explanation

Given:

The Cobb-Douglas production function, P(L,K)=1.01L0.75K0.25 (1)

The general production function, P(L,K)=bLαK1α (2)

Calculation:

Suppose the amount of labour and capital are doubled find the production function as follows.

Substitute 2L for L and 2K for K in the equation (1),

P(2L,2K)=1.01(2L)0.75(2K)0.25=1.01(20.75)L0.75(20.25)K0.25=1.01(20.75)(20.25)L0.75K0.25=(20.75+0.25)1.01L0.75K0.25

Simplify further as follows.

P(2L,2K)=(21)1.01L0.75K0.25=(2)1.01L0.75K0.25=2P(L,K)

Thus, identify that the production is doubled when the amount of labour and capital are doubled

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P-14RECh-14 P-15RECh-14 P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-58RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-1PCh-14 P-2PCh-14 P-3PCh-14 P-4PCh-14 P-5PCh-14 P-6PCh-14 P-7PCh-14 P-8P

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