For the cost function $C = q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 }$ consider the following statements:
I.The function exhibits decreasing average cost.
II.The function is homogeneous of degree 1 in v and w.
III.The elasticity of marginal cost with respect to v exceeds the elasticity with respect to w.
A)None is true.
B)All are true.
C)Only I is true.
D)Only I and II are true.

Question

For the cost function $C = q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 }$ consider the following statements: 
I.The function exhibits decreasing average cost.
II.The function is homogeneous of degree 1 in v and w.
III.The elasticity of marginal cost with respect to v exceeds the elasticity with respect to w.
A)None is true.
B)All are true.
C)Only I is true.
D)Only I and II are true.

Quizplus · Accepted Answer

The function is homogeneous of degree 1 in v and w because $C = q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 } = q ^ { \cdot 8 }~~ (v ^ { .4} ~~w ^ { .6 }) ^ { 1 } = q ^ { \cdot 8 }~~ t ^ { 1 }$, where $t = v ^ { .4} ~~w ^ { .6 }$. The elasticity of marginal cost with respect to v is $\frac{d(MC/C)}{d(v/C)} = \frac{d(q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 })/d(q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 })}{d(v ^ { .4} ~~w ^ { .6 })/d(q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 })} = \frac{0.4v ^ { -0.6} ~~w ^ { .6 }}{0.4v ^ { .4} ~~w ^ { -0.6 }} = 1$, and the elasticity of marginal cost with respect to w is $\frac{d(MC/C)}{d(w/C)} = \frac{d(q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 })/d(q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 })}{d(v ^ { .4} ~~w ^ { .6 })/d(q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 })} = \frac{0.6v ^ { .4} ~~w ^ { -0.4 }}{0.4v ^ { .4} ~~w ^ { -0.6 }} = 1.5$. Therefore, the elasticity of marginal cost with respect to v exceeds the elasticity with respect to w.

For the Cost Function C=q⋅8 v.4 w.6C = q ^ { \cdot 8 }~~ v ^ { .4} ~~w ^ { .6 }C=q⋅8 v.4 w.6

For the Cost Function $C = q ^ { \cdot 8 } v ^ { .4} w ^ { .6 }$