Question

Suppose the same firm’s cost function is C(q) = 4q2 + 16.

a. Find variable cost, fixed cost, average cost, average variable cost, and average fixed cost. (Hint: Marginal cost is given by MC = 8q.)

b. Show the average cost, marginal cost, and average variable cost curves on a graph.

c. Find the output that minimizes average cost.

d. At what range of prices will the firm produce a positive output?

e. At what range of prices will the firm earn a negative profit?

f. At what range of prices will the firm earn a positive profit?

Short answer

a. Variable cost is 4q2, fixed cost is 16, the average cost is 4q+16/q, average variable cost 4q, and average fixed cost 16/q.

b. The graph is shared below:

c. The firm will minimize its average cost at 2 units.

d. The firm produces positive output for prices greater than zero.

e. The firm will produce a negative profit between the price range of $0 and $16.

f. The firm will produce a positive profit from a price range of $16 and above.

Step-by-step solution

Finding different types of costs

Variable cost:The variable cost (VC) is the cost that changes with a change in the quantity of output. In the function, the expression which changes with the change in the value of q represents the variable cost. Thus, the value of variable cost is:

VC = 4q²

• Fixed cost:It is the cost that does not change with the change in the number of output. In the function, the expression which does not change with the change in the value of q represents fixed cost. Therefore, the value of fixed cost is:

FC = 16

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

$$\begin{gathered} AC=\frac{4q^2+16}{q} \\ =4q+\frac{16}{q} \end{gathered}$$

• Average variable cost: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.

$AVC=\frac{4q^2}{q}=4q$

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

$AFC=\frac{16}{q}$

Graphical representation of average cost, marginal cost, and average variable cost

The following figure shows the graphical representation of average cost (AC), marginal cost (MC), and average variable cost (AVC).

The x-axis of the graph shows the quantity of output, and the y-axis shows the cost. The cost curve is determined by putting the value of q = 1, 2, 3, 4, 5, and so on. It can be seen from the graph that the AC curve cuts the MC curve at output 2, and the AVC curve lies below the AC curve.

Determining the output that minimizes average cost

The figure suggests that the average cost is lowest at 2 units of output. Thus, the output that minimizes the average cost is 2 units.

The range of prices for which the firm will produce positive output

The price range for which the marginal cost is greater than the average variable cost is the price range for which the firm will produce positive output.It is clear from the figure that marginal cost and average variable cost are equal at a price level of $0. The marginal cost is greater than the average variable cost for all price levels greater than $0, as MC lies above the AVC curve in the figure.

Hence, the firm's price range to produce positive output is $0 and above.

The range of prices for which the firm will produce a negative profit

Any price level for which the marginal cost cannot cover the average cost will provide negative profit. It is clear from the figure that the average cost is lowest at $16. After this price level, the marginal cost is greater than the average cost. Thus, any price level for the product lower than $16 will provide negative profit to the firm.

Hence, the price range at which the firm will earn a negative profit is $0 to $16.

The range of prices at which the firm will produce a positive profit

Any price level for which the marginal cost is greater than the average cost will provide positive profit.It is clear from the figure that the average cost is lowest at $16. After this price level, the marginal cost is greater than the average cost. Thus, any price level for the product greater than $16 will provide positive profit to the firm.

Hence, the firm's price range to earn a positive profit is $16 and above.