For a firm, let total cost be $T C ( Q ) = 10 Q ^ { 2 }$ and marginal cost be $M C ( Q ) = 20 Q$. Which of the following is an expression for the output elasticity of total cost?
A) $\epsilon _ { T C , Q } = 10 Q$.
B) $\epsilon _ { I C , Q } = 2 Q$.
C) $\epsilon _ { T C , Q } = 2$.
D) $\epsilon _ { T C , Q } = 20 Q$.

Question

Quizplus · Accepted Answer

The output elasticity of total cost is given by the formula $\epsilon _ { T C , Q } = \frac{d T C / Q}{T C / Q}$. Taking the derivative of $T C ( Q ) = 10 Q ^ { 2 }$ with respect to Q, we get $\frac{d T C}{d Q} = 20 Q$. Plugging this into the formula, we get $\epsilon _ { T C , Q } = \frac{20 Q/Q}{10 Q ^ { 2 }/Q} = 2$. Option (C) matches this result. Therefore, the correct answer is (C). The formula for the output elasticity of total cost is:
$$\epsilon _ { T C , Q }=\frac{\% \Delta T C}{\% \Delta Q}=\frac{\Delta T C / T C}{\Delta Q / Q}=\frac{\Delta (10Q^2) / (10Q^2)}{\Delta Q / Q}=2$$
Therefore, the correct answer is (C). Choices (A) and (D) are incorrect because they do not reflect the correct formula for output elasticity of total cost, and choice (B) refers to an entirely different concept, namely the income elasticity of demand.

For a Firm, Let Total Cost Be TC(Q)=10Q2T C ( Q ) = 10 Q ^ { 2 }TC(Q)=10Q2

For a Firm, Let Total Cost Be $T C ( Q ) = 10 Q ^ { 2 }$