Question

A competitive firm has the following short-run cost function: \(C(q)=q^{3}-8 q^{2}+30 q+5\) a. Find \(\mathrm{MC}, \mathrm{AC}\), 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?

Short answer

MC: \(3q^{2}-16q+30\), AC: \(q^{2}-8q+30+ \frac{5}{q}\), AVC: \(q^{2}-8q+30\). Zero output is supplied at price ranges below 14. The supply curve is represented by MC above its intersection with AVC. The firm supplies exactly 6 units of output when the price is 86.

Step-by-step solution

- Find MC, AC, and AVC

First, find the derivatives of the cost function to derive MC, AC and AVC. MC is the derivative of the total cost function. Therefore \(\mathrm{MC} = \frac{dC}{dq} = 3q^{2}-16q+30\).AC is total cost divided by quantity. Therefore, \(\mathrm{AC} = \frac{C}{q} = q^{2}-8q+30+ \frac{5}{q}\).AVC is total variable cost divided by quantity. The variable cost here is \(q^{3}-8q^{2}+30q\). So, \(AVC = \frac{q^{3}-8q^{2}+30q}{q} = q^{2}-8q+30\)

- Graph MC, AC and AVC

For graphing MC, AC and AVC, one can make use of softwares like Desmos or a graphing calculator. Input the equations for MC, AC and AVC to produce the graph, remembering to specify a reasonable domain. Pay attention to the intercepts and intersections between these curves.

- Identify Zero Output Range

The firm will supply zero output when the price is less than the minimum point on the AVC curve. Solve for q when \(AVC = q^{2}-8q+30\) is at its minimum. This is achieved by taking the derivative of AVC and solving for q equal to zero. Here, the minimum AVC is obtained at \(q = 4\), and substituting back to AVC, we find that when AVC = 14, the firm will supply zero output. Thus, at prices less than 14, the firm won't supply any output.

- Supply Curve Identification

The firm's supply curve can be identified in the graph from Step 2. It occurs where the price curve (a horizontal line at any given price level) intersects with the MC curve and is above AVC. Remember that a firm will only produce if the price is more than AVC.

- Find Specific Output Level

To find the price at which the firm would supply exactly 6 units of output, substitute \(q = 6\) in the MC function. This is because MC also represents the supply curve for perfect competition in the short run above the shutdown point. Therefore, \(P = 3*6^{2}-16*6+30 = 86\)

Key concepts

Marginal Cost (MC)

Marginal Cost (MC) is a crucial concept in economics that helps firms understand how their costs change with each additional unit of output.
In simple terms, it's the cost of producing one more unit. For a firm, knowing these costs is essential for making profit-maximizing decisions.
The MC is derived from the derivative of the total cost function. In our exercise, the cost function is given as \(C(q) = q^{3} - 8q^{2} + 30q + 5\). Therefore, the marginal cost is calculated as follows:

• Take the derivative of the total cost function: \(MC = \frac{dC}{dq} = 3q^{2} - 16q + 30\).

Understanding MC helps firms decide whether producing additional units will bring additional costs that the selling price may not cover. This decision ensures that the firm stays in business and doesn't incur losses. The firm's supply curve in a perfectly competitive market is essentially the MC curve above the average variable cost (AVC). This is because, below AVC, the firm would generate losses and may shut down.

Average Cost (AC)

Average Cost (AC) represents the total cost per unit of output. It provides a big picture of production efficiency and profitability for any firm. The average cost is calculated by dividing the total cost by the quantity of output produced. This helps determine how spreading out fixed costs over more units can reduce the cost per unit.For the given cost function, \(C(q) = q^{3} - 8q^{2} + 30q + 5\), the average cost formula is:

• \(AC = \frac{C}{q} = q^{2} - 8q + 30 + \frac{5}{q}\)

Understanding the AC can help firms find the most efficient level of production and pricing, where the cost per unit is minimized. The U-shape of the AC curve is typical due to decreasing average costs at low levels of production (economies of scale) and increasing average costs at higher levels (diseconomies of scale).

Average Variable Cost (AVC)

Average Variable Cost (AVC) is another vital concept that focuses on the variable portion of production costs. Unlike AC, AVC does not consider fixed costs, only variable ones, which change with the level of output. For businesses, keeping track of AVC helps in decision-making, especially when determining the lowest selling price per unit to avoid losses.For our exercise using \(q^{3} - 8q^{2} + 30q\) as the variable component of the cost function, the AVC is:

• \(AVC = \frac{q^{3} - 8q^{2} + 30q}{q} = q^{2} - 8q + 30\)

Firms often use AVC to decide the shutdown point. A firm will continue to operate in the short run as long as the market price is equal to or exceeds the AVC. The shutdown point is the point at which the price is low enough that the firm isn't covering its variable costs. For example, if the AVC minimum is \(14\), and the price falls below this, the firm should cease operations to prevent losses.