A competitive firm has the following short-run cost function:C(q) =q³ - 8q² + 30q+ 5.

a. Find MC, AC, and AVC and sketch them on a graph.

b. At what range of prices will the firm supply zero output?

c. Identify the firm’s supply curve on your graph.

d. At what price would the firm supply exactly 6 units of output?

• Marginal cost (MC): The cost incurred by the firm on producing one additional unit of output is the marginal cost. It can be calculated by differentiation the cost function.

$$\begin{gathered} MC=\frac{d\left(q^3-16q+30\right)}{dq} \\ =3q^2-16q+30 \end{gathered}$$

• Average cost (AC): It is determined by dividing total cost by output. The division of cost function by q will give the average cost.

$$\begin{gathered} AC=\frac{q3-8q2+30q+5}{q} \\ =q^2-8q+30+\frac{5}{q} \end{gathered}$$

• Average variable cost (AVC): It is determined by dividing the variable cost by the number of units produced. The average variable cost is calculated for the given function by dividing the average cost by q.

$$\begin{gathered} AVC=\frac{q3-8q^2+30q}{q} \\ =q^2-8q+30 \end{gathered}$$

The graphical representation of the MC, AC, and AVC curve is shown below:

In the above graph, the output is on the x-axis, and the price is on the y-axis. The different cost curves are determined by putting the value of q =1, 2, 3, 4, 5, 6, and so on in their respective equations.

The firm will not produce any output for the price level, which will not cover its average variable cost of production. The AVC is lowest at the price level of $15. Below this price level, the firm will not produce any output.

Thus, the price level from $0 to $15 is the price range for which the firm will supply zero output.

The supply curve tells about the quantity of output supplied for each possible price.A perfectively competitive firm will produce output from the point where the marginal cost starts covering the average variable cost. The figure suggests that the MC curve is above the AVC curve from the price level of $15.

The section of the marginal cost curve that lies above the average variable cost curve is the firm's supply curve. It starts from a price level of $15 and increases to infinity.

The part of the marginal cost curve that is above the average variable cost represents the supply curve. The supply curve shows the quantity of output produced at different price levels.

The figure suggests that for 6 units of output, the price is $42. Hence, the price for which the firm would produce exactly 6 units of output is $42.