Question

Suppose the same firm's cost function is \(C(q)=4 q^{2}+16\) a. Find variable cost, fixed cost, average cost, average variable cost, and average fixed cost. (Hint: Marginal cost is given by \(\mathrm{MC}=8 q\).) 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

The variable cost is \(4q^2\), the fixed cost is 16, the average cost is \(4q + 16/q\), the average variable cost is \(4q\) and the average fixed cost is \(16/q\). The output that minimizes average cost is 2 units. The firm will produce positive output when the price is equal or higher than 8, have negative profits when the price is less than 12, and have positive profits when the price is greater than 12.

Step-by-step solution

Define the cost function

The cost function of the firm is given as \(C(q) = 4q^2 + 16\) where \(q\) represents the quantity of goods produced.

Find the variable cost and fixed cost

The variable cost is the cost that changes with the quantity produced. As seen in the cost function, the variable cost would be \(4q^2\). On the other hand, the fixed cost is cost which does not change with the quantity produced. In the cost function, the fixed cost would be 16.

Calculate mean costs

The Average Cost (AC) is computed as total cost divided by the quantity produced i.e. \(AC = C(q) / q = (4q^2 + 16) / q = 4q + 16/q\). The Average Variable Cost (AVC) is the variable cost divided by the quantity i.e. \(AVC = VC / q = 4q\). The Average Fixed Cost (AFC) is the fixed cost divided by the quantity i.e. \(AFC = FC / q = 16 / q\).

Show the cost curves on a graph

Plot average cost (AC), marginal cost (MC), and average variable cost (AVC) on the graph with quantity on the x-axis and cost on the y-axis.

Find minimum average cost output

To find the output level that minimizes AC we can take the derivative of AC with respect to q and set equal to zero. The derivative of \(4q + (16/q)\) is \(4 - (16/q^2)\). Setting this equal to zero we find \(q = 2\). So the output that minimizes average cost is 2 units.

Find the range of prices for positive output

The firm will produce positive output when the price is greater than or equal to the minimum point of the average variable cost. So, \(P >= 4q\), when \(q > 2\). Therefore, \(P >= 8\).

Find the range of prices for negative profit

The firm will face a negative profit when the price is less than the average cost. So, \(P < 4q + 16/q\), when \(q > 2\). Therefore, \(P < 8 + 8/2 = 12\).

Find the range of prices for positive profit

The firm will face a positive profit when the price is greater than average cost. So, \(P > 4q + 16/q\), when \(q = 2\). Therefore, \(P > 8 + 8/2 = 12\).

Key concepts

Variable Cost

Variable cost refers to the expenses that change with the level of output produced. In our given cost function, which is expressed as \(C(q) = 4q^2 + 16\), the variable component is \(4q^2\). This part demonstrates that as you produce more, your variable cost increases proportionally to the square of the quantity \(q\).
Think of it as costs for raw materials, hourly wages, or utilities that rise as production ramps up.
The behavior of variable cost is crucial for understanding the firm's responses to changes in production scale. When analyzing, visualize that the \(4q^2\) grows significantly larger with increments in \(q\), indicating how rapidly costs can escalate with higher output.

• Variable cost (\(VC\)) is given by \(4q^2\).
• It varies with output quantity \(q\).

Keep in mind, variable costs are integral in pricing strategies and determining production levels to ensure they don't outpace revenue.

Fixed Cost

Fixed cost refers to those expenses that remain constant regardless of the output level. In our function \(C(q) = 4q^2 + 16\), the fixed cost is embodied by the constant term \(16\).
This represents expenditures that do not fluctuate, like rent, salaries, or administrative expenses, which must be paid no matter the quantity produced.
Understanding fixed costs is essential for managing budgeting and predicting break-even points.

• Fixed cost (\(FC\)) is a constant: \(16\).
• It does not change as the output varies, \(q\).

Even if production is zero, fixed costs must be covered, emphasizing their critical role in financial planning and decision-making for a business.

Average Cost

The average cost is a measure of the per-unit cost of production. It helps in evaluating the efficiency of manufacturing and pricing strategies. Here, the average cost (AC) can be calculated by dividing the total cost \(C(q)\) by \(q\). In our example, the formula results in \(AC = \frac{4q^2 + 16}{q} = 4q + \frac{16}{q}\), showing the individual contribution of variable and fixed costs to the total average cost per unit.
Notice that at low levels of \(q\), the fixed cost component, \(\frac{16}{q}\), is significant, but as \(q\) grows, its impact diminishes.

• Average cost (\(AC\)) is calculated as \(4q + \frac{16}{q}\).
• Assessing per unit cost helps in optimizing production levels.

Determining the scale of operations that minimizes average cost is key to achieving operational efficiency. This occurs when smaller average and marginal costs align, possibly setting the stage for competitive pricing in the market.