Question

An economist estimated that the cost function of a single-product firm is Based on this information,

$\mathbf{C}\mathbf{\left(}\mathit{Q}\mathbf{\right)}\mathbf{=}\mathbf{100}\mathbf{+}\mathbf{20}\mathit{Q}\mathbf{+}\mathbf{15}{\mathit{Q}}^{\mathbf{2}}\mathbf{+}\mathbf{10}{\mathit{Q}}^{\mathbf{3}}$determine:

a. The fixed cost of producing $\mathbf{10}$units of output .

b. The variable cost of producing $\mathbf{10}$units of output .

c. The total cost of producing $\mathbf{10}$units of output .

d. The average fixed cost of producing $\mathbf{10}$units of output .

e. The average variable cost of producing $\mathbf{10}$units of output .

f. The average total cost of producing $\mathbf{10}$units of output .

g. The marginal cost when $\mathit{Q}\mathbf{=}\mathbf{10}$

Short answer

1. The fixed cost of producing $10$units of output is $\$100$

2. The variable cost of producing $10$units of output is $\$11700$

3. The total cost of producing $10$units of output is $\$11800$

4. The average fixed cost of producing $10$units of output is $\$10$

5. The average variable cost of producing $10$units of output is $\$1170$

6. The average total cost of producing $10$units of output is $\$1180$

7. The marginal cost when $Q=10$is $\$3320$

Step-by-step solution

 The equation

This equation represents the cost function of the single product firm.

$\mathrm{C}\left(Q\right)=100+20Q+15Q^2+10Q^3$

Final Solution

a. The fixed cost is 100 which does not depend on the units of output.

$FC=100$

b. The variable cost part from the given equation can be determined by,

$\mathrm{VC}=20Q+15Q^2+10Q^3$

Thus, putting the $10$units of output in the equation we get the variable cost as,

role="math" localid="1657887712661" $$\begin{gathered} \mathrm{VC}=(20*10)+(15*10*10)+(10*10*10) \\ =200+500+10000 \\ =11700 \end{gathered}$$

Hence, the variable cost is$\$11700$

c. The total cost of $10$units of output is,

role="math" localid="1657887729250" $$\begin{gathered} TC=FC+VC \\ =100+11700 \\ =11800 \end{gathered}$$

Thus, the total cost of producing $10$units of output is$\$11800$

d.The average fixed cost is calculated as,

$$\begin{gathered} AFC=\frac{FC}{Q} \\ =\frac{100}{10} \\ =10 \end{gathered}$$

Hence the average fixed cost$AFC=\$10$

e.The average variable cost can be calculated as follows,

$$\begin{gathered} AVC=\frac{VC}{Q} \\ =\frac{11700}{10} \\ =1170 \end{gathered}$$

Therefore, the average variable cost$AVC=\$1170$

f.The average total cost is calculated as,

$$\begin{gathered} ATC=\frac{TC}{Q} \\ =\frac{11800}{10} \\ =1180 \end{gathered}$$

Thus, the average total cost for units of output is$\$1180$

g. The marginal cost is calculated as,

$$\begin{gathered} \mathrm{MC}=\frac{\partial \mathrm{TC}}{\partial Q} \\ =20+(15*2)Q+(10*3)Q^2 \\ =20+30Q+30Q^2 \end{gathered}$$

After putting the value of$Q=10$ we get,

$$\begin{gathered} MC=20+(30*10)+(30*10*10) \\ =20+300+3000 \\ =3320 \end{gathered}$$

Hence, the marginal cost MC is found to be$\$3320$