Question

The short-run cost function of a company is given by the equation \(\mathrm{TC}=200+55 q\), where \(\mathrm{TC}\) is the total cost and \(q\) is the total quantity of output, both measured in thousands. a. What is the company's fixed cost? b. If the company produced 100,000 units of goods, what would be its average variable cost? c. What would be its marginal cost of production? d. What would be its average fixed cost? e. Suppose the company borrows money and expands its factory. Its fixed cost rises by \(\$ 50,000,\) but its variable cost falls to \(\$ 45,000\) per 1000 units. The cost of interest ( \(i\) ) also enters into the equation. Each 1-point increase in the interest rate raises costs by \(\$ 3000 .\) Write the new cost equation.

Short answer

a. The company's fixed cost is \$200,000. b. Its average variable cost per unit for production of 100,000 units is \$55. c. Its marginal cost of production is \$55. d. Its average fixed cost per unit is \$2. e. The new cost equation is \( TC = 250 + 45q + 3i\).

Step-by-step solution

Identify the Fixed Cost

From the total cost function \(TC=200+55q\), it is evident that the constant term not associated with \(q\) represents the fixed cost. Therefore, the fixed cost is \( \$200,000\)

Calculate the Variable cost per unit

Variable cost (VC) is the portion of the total cost that is associated with the quantity, \(q\) in our equation. Here the variable cost per unit is 55. So, if the company produced 100,000 units of goods, its average variable cost (AVC) would be \$55 per unit.

Calculate the Marginal Cost

Marginal cost (MC) is the change in total cost when an additional unit of output is produced. It is the coefficient of \(q\) in our cost function. Therefore, the marginal cost of production is \$55.

Calculate Average Fixed Cost

Average fixed cost (AFC) is the total fixed cost divided by the quantity of output. Assuming production of 100,000 units, AFC is \( 200,000 / 100,000 = \$2 \) per unit.

Write the new cost equation

Given that the fixed cost rises by \$50,000 and variable cost falls to \$45 per 1000 units, and each 1-point increase in the interest rate raises costs by \$3000, the new cost function will be \(TC = 250 + 45q + 3i\). This reflects the increased fixed costs, lower variable costs, and the impact of interest rates.

Key concepts

Fixed Cost

Understanding fixed costs is crucial when analyzing a company's expenses. These costs do not change with the level of goods or services produced within a certain range of activity for a given period.

Fixed costs are necessary to keep a business operating and are often considered sunk costs, meaning they cannot be recovered once incurred. In our exercise, the fixed cost is derived from the total cost equation \(TC=200+55q\) as the component independent of output quantity (\(q\)), so the fixed cost is \$200,000\. This would remain constant regardless of the production level and includes expenses such as rent, salaries of permanent staff, and machinery.

Average Variable Cost

Average variable cost (AVC) plays a vital role in the production and pricing decisions of a company. It signifies the per-unit variable cost at a certain level of output. Variable costs change directly with the level of production, including materials and labor costs directly tied to the production volume.

Using the formula \(AVC = VC / q\), where \(VC\) is the total variable cost, and \(q\) is the total quantity of output, we can calculate the AVC. From the problem, we're given the total cost function's variable segment as \(55q\), making the AVC \$55\ when producing 100,000 units. This information is helpful for the firm considers pricing their product to cover variable costs when setting the selling price.

Marginal Cost

Marginal cost (MC) is the additional cost incurred by producing one more unit of a product. It's an important concept for businesses when considering how to scale production and maximize profits.

In practice, MC helps in determining the optimal production level where profits are maximized. It's calculated as the change in total cost divided by the change in quantity (\(MC = \Delta TC/\Delta q\)). From our function \(TC=200+55q\), MC is the coefficient of the output level \(q\), which in this case is consistently \$55 per additional unit produced. In the situation where marginal cost starts to exceed marginal revenue, it indicates that it may not be profitable to increase production further.

Average Fixed Cost

Average fixed cost (AFC) provides insight into how fixed costs can be spread over different production levels.

As production quantity increases, the AFC decreases since the fixed cost is allocated over more units. Calculated by the formula \(AFC = FC / q\), where \(FC\) is the total fixed cost, AFC helps businesses understand the advantages of economies of scale. In the provided exercise, with a fixed cost of \$200,000 and a production level of 100,000 units, the AFC is \$2 per unit. This reduction in AFC with increased production volume incentivizes businesses to produce more to lower per-unit costs.

Total Cost Equation

The total cost (TC) equation is a fundamental formula representing the overall cost a business incurs to produce a given level of output. It encapsulates all expenses, both fixed and variable. The general formula is \(TC = FC + VC\), where \(FC\) is the fixed cost and \(VC\) is the variable cost that depends on the production quantity.

In the modification presented in the exercise, the new cost equation after adjustments for increased fixed costs, decreased variable costs, and the introduction of an interest component becomes \(TC = 250 + 45q + 3i\). This updated equation will guide the company in adjusting to new financial circumstances, like borrowing costs, and in making strategic decisions about pricing and production.