Question

A firm has two plants that produce identical output. The cost functions are Upper C 1 equals 309 q minus 8 q squared plus 0.5 q cubed and Upper C 2 equals 309 q minus 14 q squared plus 1.0 q cubed. ​First, note that if AC 1equals309minus8qplus0.5q squared​, then StartFraction dAC 1 Over dq EndFraction equals negative 8 plus 2 (0.5 )q . ​Similarly, if AC 2equals309minus14qplus1.0q squared​, then StartFraction dAC 2 Over dq EndFraction equals negative 14 plus 2 (1.0 )q . At what output level does the average cost curve of each plant reach its​ minimum? The first plant reaches minimum average cost at nothing units of output. ​(E

Answer 1

Complete and clearly written Question:

A firm has two plants that produce identical output. The cost functions are:

$C_{1} = 309q - 8q^{2} + 0.5q^{3}$ and $C_{2} = 309q - 14q^{2} + 1.0q^{3}$. At what output level does the average cost curve of each plant reach its minimum?

Answer:

The first price reaches minimum average at 8 units of outputs

The second price reaches minimum average at 7 units of outputs

Explanation:

$C_{1} = 309q - 8q^{2} + 0.5q^{3}$

$A_{C_1} = C_1 /q = (309q - 8q^{2} + 0.5q^{3})/q\\A_{C_1} =309 - 8q + 0.5q^{2}$

$\frac{dA_{C_{1} } }{dq} = -8 + q$

At minimum price, $\frac{dA_{C_{1} } }{dq} = 0$

-8 + q = 0

q = 8

$C_{2} = 309q - 14q^{2} + 1.0q^{3}$

$A_{C_1} = C_1 /q = (309q - 14q^{2} + 1.0q^{3})/q\\A_{C_1} =309 - 14q + 1.0q^2}$

$\frac{dA_{C_{1} } }{dq} = -14 + 2q$

At minimum price, $\frac{dA_{C_{1} } }{dq} = 0$

-14 + 2q = 0

2q = 14

q = 7