Question

Problem 7-1 Use a cell reference or a single formula where appropriate in order to receive full credit. Do not copy and paste values, as you will not receive full credit for your answers. Labor, L, and capital, K, are the only inputs in each of the following production functions below. For each production function, find the output associated with the following output combinations: L-2,K-2L-4K-4; and L -8, K-8. a. 4. - (L+K) b.q2 - (sqrt(L.) + sqrt(K))? 6.4 - (20 + sqrt(L) + sqrt(K)) I 2 I 2 1 44 Determine the return to scale of each production function The production function, exhibits The production function, exhibits The production function q, exhibits returns to scale. returns to scale. returns to scale. Problem instructions +

Answer 1

1. $\dpi{80} \fn_phv \small q=(L+K)^2$
$\dpi{80} \fn_phv \small f(2,2)=(2+2)^2=16$
$\dpi{80} \fn_phv \small f(4,4)=(4+4)^2=64$
$\dpi{80} \fn_phv \small f(8,8)=(8+8)^2=256$
To check for returns to scale, increase inputs by $\dpi{80} \fn_phv \small \alpha$ times and observe the change in output. $\dpi{80} \fn_phv \small f(\alpha L,\alpha K)=(\alpha L+\alpha K)^2=\alpha(L+K)^2=\alpha^2 f(L,K)$
Increasing inputs by $\dpi{80} \fn_phv \small \alpha$ times increases output by more than $\dpi{80} \fn_phv \small \alpha$ times. Therefore, this production function exhibits increasing returns to scale.

2. $\dpi{80} \fn_phv \small q=(\sqrt{L}+\sqrt{K})^2$
$\dpi{80} \fn_phv \small f(2,2)=(\sqrt{2}+\sqrt{2})^2=8$
$\dpi{80} \fn_phv \small f(4,4)=(\sqrt{4}+\sqrt{4})^2=16$
$\dpi{80} \fn_phv \small f(8,8)=(\sqrt{8}+\sqrt{8})^2=32$
$\dpi{80} \fn_phv \small f(\alpha L,\alpha K)=(\sqrt{\alpha L}+\sqrt{\alpha K})^2=\alpha(\sqrt{L}+\sqrt{K})^2=\alpha f(L,K)$
Increasing inputs by $\dpi{80} \fn_phv \small \alpha$ times increases output by $\dpi{80} \fn_phv \small \alpha$ times. Therefore, this production function exhibits constant returns to scale.

3. $\dpi{80} \fn_phv \small q=(20+\sqrt{L}+\sqrt{K})^2$
$\dpi{80} \fn_phv \small f(2,2)=(20+\sqrt{2}+\sqrt{2})^2=(2\sqrt{2}+20)^2$
$\dpi{80} \fn_phv \small f(4,4)=(20+\sqrt{4}+\sqrt{4})^2=576$
$\dpi{80} \fn_phv \small f(8,8)=(20+\sqrt{8}+\sqrt{8})^2=(4\sqrt{2}+20)^2$
$\dpi{80} \fn_phv \small f(\alpha L,\alpha K)=(20&plus;\sqrt{\alpha L}&plus;\sqrt{\alpha K})^2<\alpha f(L,K)$
Increasing inputs by $\dpi{80} \fn_phv \small \alpha$ times increases output by less than $\dpi{80} \fn_phv \small \alpha$ times. Therefore, this production function exhibits decreasing returns to scale.