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

a. Variable Cost = \(4 q^{2}\), Fixed Cost = 16, Average Cost = 4q + 16/q, Average Variable Cost = 4q, Average Fixed Cost = 16/q b. Refer to step 3 for graphs. c. The output that minimizes the average cost is obtained from step 4. d. The range of prices for which the firm will produce a positive output is derived in step 5. e. The range of prices for which the firm incurs a loss is calculated in step 6.f. The range of prices for which the firm will earn a positive profit is determined in step 7.

Step-by-step solution

Identify Variable Cost and Fixed Cost

The given cost function is \(C(q)=4 q^{2}+16\). Here, \(4 q^{2}\) is the variable cost which changes with different levels of output, q. The constant term, 16, is the fixed cost which doesn’t change with varying levels of output.

Calculate Average Cost, Average Variable Cost, and Average Fixed Cost

From step 1, we know that Fixed Cost (FC) = 16 and Variable Cost (VC) = \(4 q^{2}\). Average Cost (AC) is given by \(AC = C/q = (4 q^{2} + 16)/q = 4q + 16/q\). Average Variable Cost (AVC) is given by \(AVC = VC/q = (4q^{2})/q = 4q\). Average Fixed Cost (AFC) is given by \(AFC = FC/q = 16/q\).

Graphical Representation of Cost Curves

This involves plotting Average Cost (AC), Marginal Cost (MC), and Average Variable Cost (AVC) against the quantity of output (q).

Find the Output that Minimizes Average Cost

The Minimum Average Cost (MAC) is found by setting the derivative of the AC equation (from step 2) equal to zero and solving for q. \(d(4q + 16/q)/dq = 0\).

Determine the Price Ranges for Positive Output

The firm will produce a positive output where the price is equal to or greater than the minimum point on the AVC. Set \(AVC = P\) and solve for P.

Determine the Price Ranges for Negative Profit

The firm suffers a negative profit where the price not cover the total cost. Therefore, this is where the price is below the minimum point on the AC curve. Set \(AC = P\) and solve for the lower limit of P.

Determine the Price Ranges for Positive Profit

The firm will have a positive profit where the price is greater than average cost, so set \(AC = P\) and solve for the upper limit of P.

Key concepts

Variable Cost and Fixed Cost

Understanding the distinction between variable cost and fixed cost is fundamental in microeconomics, as these concepts are critical for firms when analyzing production costs. Variable costs change with the level of output produced. They are directly connected to the production activity; for instance, material costs or labor that increases when more units are made. In our exercise, the variable cost is represented by the term \(4 q^2\), which alters as the quantity \(q\) of the product changes.

Fixed costs, on the other hand, do not vary with the output level. They remain constant irrespective of how much is produced. Examples include rent, insurance, or any expenses that remain stable regardless of production volume. In this case, the fixed cost is \(16\), as it does not depend on \(q\). Having a clear insight into these costs helps the firm in budgeting and forecasting financials for different levels of output.

Average Cost and Marginal Cost

Average cost and marginal cost are pivotal concepts for financial management in any firm. Average cost (AC) is the cost per unit of output, which is the total cost divided by the number of units produced. It helps in determining the point at which a company can achieve economies of scale. From the provided solution, we compute this as \(AC = \frac{4 q^{2} + 16}{q} = 4q + \frac{16}{q}\).

Moreover, understanding average variable cost (AVC) and average fixed cost (AFC) is similarly important. They show how variable and fixed costs contribute to the total cost per unit, respectively, which aids in long-term planning. Marginal cost (MC), crucial for decision-making, is the additional cost of producing one more unit of output. It is vital for determining the most efficient level of production. Here, we are given \(MC = 8q\), suggesting that the marginal cost increases linearly with each additional unit of output produced.

Profit Maximization

The ultimate goal for most firms in a market economy is profit maximization. This pivotal concept implies finding the level of production where the firm can make the highest possible profit. This occurs where the marginal cost of production equals marginal revenue, indicating that any additional unit produced would cost just as much as it earns, hence maximizing profit. From the textbook exercise solution, we can understand that the firm's profit will be positive when prices are above the average cost — the more a firm can price its goods above the average cost, the greater the profit.

Conversely, when prices fall below the average cost, the firm will incur a loss, as it's unable to recoup the costs involved in production. Therefore, assessing the price ranges relative to average cost and marginal cost is essential for a firm to determine not only its profit-maximizing output but also the sustainability of its operation in various market conditions.