Chapter 6: Problem 8

Do the following functions exhibit increasing, constant, or decreasing returns to scale? What happens to the marginal product of each individual factor as that factor is increased and the other factor held constant? a. \(q=3 L+2 K\) b. \(q=(2 L+2 K)^{1 / 2}\) c. \(q=3 L K^{2}\) \(\mathbf{d} . q=L^{1 / 2} K^{1 / 2}\) e. \(q=4 L^{1 / 2}+4 K\)

Short Answer

Expert verified

a. Function exhibits increasing returns, both marginal products are constant. \n b. Function exhibits constant returns, marginal products decrease as respective variable increases. \n c. Function exhibits increasing returns, marginal product of L is constant and of K increases as K increases. \n d. Function exhibits increasing returns, marginal products decrease as respective variable increases. \n e. Function exhibits decreasing returns, marginal product of L decreases as L increases, and marginal product of K is constant.

Step by step solution

Analyse the first function

For \(q=3 L+2 K\), doubling both L and K (\(i.e., L=2L', K=2K'\)), we get \(q'=3 * 2L'+2 * 2K'=6L'+4K'=2q\). Thus, the scale is increasing. To find whether marginal product of L decreases or increases, we partially differentiate \(q=3 L+2 K\) with respect to L, which gives us \(3\). Since there is no L in this partial derivative, the marginal product of L is constant. We perform the same process for K to find that the marginal product of K is also constant.

Analyse the second function

For \(q=(2 L+2 K)^{1 / 2}\), doubling both L and K, we have \(q'=(2*2L'+2*2K')^{1/2} = (4L'+4K')^{1/2}=2^{1/2}(2L'+2K')^{1/2}=2^{1/2}q\), and the function exhibits constant returns to scale. The marginal product of L can be found by partially differentiating \(q=(2 L+2 K)^{1 / 2}\) with respect to L. The resulting partial derivative can be implicitly differentiated for L, which provides a means for observing the relationship of the marginal product with L.

Analyse the third function

For \(q=3 L K^{2}\), doubling both L and K gives us \(q'=3 * 2L' *(2K')^{2}= 12L'K'^{2}=4q\), illustrating increasing returns to scale. Taking the partial derivative with respect to L and K independently will provide the marginal products of L and K respectively.

Analyse the fourth function

For \(q=L^{1 / 2} K^{1 / 2}\), doubling both L and K gives us \(q'=(2L')^{1/2} *(2K')^{1/2}=2q\), exhibiting increasing returns to scale. Similarly, the marginal products can be found by partially differentiating for L and K independently.

Analyse the fifth function

For \(q=4 L^{1 / 2}+4 K\), doubling L and K results in \(q'=4*(2L')^{1/2}+4 * 2K'=2^{1/2}4L'^{1/2}+8K'<2q\) for large L' and K', demonstrating decreasing returns to scale. Each partial derivative can be found similarly to determine the marginal product for L and K.

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