Question

The production function 9 = k1.270.5 exhibits: a. increasing returns to scale but no diminishing marginal productivities. b. decreasing returns to scale. C. increasing returns to scale and diminishing marginal product for / only. d. increasing returns to scale and diminishing marginal products for both k and I.

Answer 1

Answer: C. Increasing returns to scale and diminishing marginal product of L only.

The production function is as follows;

^{Let us assume that both K and L are increased by '$\lambda$'. Now. let us assume that for the increase of the inputs, the output is changed by ' $\Delta$ '.}

^{$\therefore$$\Delta Q = (\lambda K)^{1.2}(\lambda L)^{0.5}$}

^{$Or, \Delta Q = (\lambda)^{1.2} (K)^{1.2} * (\lambda)^{0.5} (L)^{0.5}$}

^{$Or, \Delta Q = (\lambda)^{1.7} (K^{1.2} L^{0.5})$}

^{$Or, \Delta Q = (\lambda)^{1.7} Q$}

From above, we see that Q is increased by  $(\lambda)^{1.7}$, which is more than '$\lambda$'. So, the production function exhibits the increasing returns to scale.

Let us now see the marginal products of K and whether it is diminishing. We will keep 'L' constant.

Or,

Now,

From above, we see that,the value of  'd(MPK) /dK' is positive.So, the marginal product of last unit of employed capital is rising.

Now, let us see the marginal products of L and whether it is diminishing. We will keep 'K' constant.

Now,

From above, we see that,the value of  'd(MPL) /dL' is negative. So, the marginal product of last unit of employed labor is diminishing.

Thus, the production function shows an increasing returns to scale and diminishing marginal product of labor(L) only.

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