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) The average variable cost for 100,000 units of goods is $55; c) The marginal cost of production is $55; d) The average fixed cost for 100,000 units of goods is $2; e) The new cost equation is TC = (200 + 50 + 3i) + 45q.

Step-by-step solution

Fixed and Variable Costs

The total cost equation is given as TC = 200 + 55q. Here, 200 represents the fixed costs of the company (in thousands). The variable cost is represented by the 55q term, where q is the quantity of output.

Average Variable Cost

To find the average variable cost(AVC), the equation is AVC = Variable Cost / Quantity. Here, the variable cost is 55q, and for 100,000 units (or 100 in thousands), the AVC = 55q / q = 55.

Marginal Cost

The marginal cost is obtained by differentiating the total cost function with respect to quantity. Here, the derivative of 55q is 55. Hence the marginal cost is 55.

Average Fixed Cost

The average fixed cost(AFC) is calculated as AFC = Fixed Cost / Quantity. Here, the fixed cost is 200 so AFC for 100,000 units of goods (or 100 in thousands) is AFC = 200/100 = 2.

New cost equation

It is mentioned that fixed cost rises by $50,000 (or 50 in thousands) and variable cost falls to $45,000 per 1000 units (or 45). The new cost equation includes interest rate (i) which increases costs by $3000 per unit increase in 'i'. Therefore, the new total cost function is TC = (200 + 50 + 3i) + 45q.

Key concepts

Fixed Costs

Fixed costs are expenses that do not change with the quantity of goods or services produced. They are incurred even if no output is produced and remain constant over time. For a company, this might include costs like rent, salaries of permanent staff, or equipment depreciation.
In our exercise, the equation for the total cost is given by \( TC = 200 + 55q \). Here, 200 is the fixed cost. This means the company incurs a fixed expense of $200,000 regardless of how many goods it produces. This amount is a constant component of the total cost equation and essential for understanding cost behavior in the short run.

Variable Costs

Variable costs change with the level of output. These costs increase as more units are produced and decrease when fewer units are made. Common examples include material costs, labor directly associated with the production, and utilities costs that vary with production volume.
In the cost function \( TC = 200 + 55q \), the term \( 55q \) represents the variable costs. This implies that for each additional thousand units produced, the cost increases by $55,000. Since variable costs are directly tied to output, they play a crucial role in determining how total costs vary with production levels.

Marginal Cost

The marginal cost is the additional cost of producing one more unit of output. It is a helpful measure for businesses to decide whether increasing production is worth the additional cost.
To calculate marginal cost in this exercise, we look at the derivative of the total cost function with respect to quantity \( q \). Given the cost function \( TC = 200 + 55q \), the derivative is simply 55. This means the marginal cost is \(55 per thousand units. This tells us that producing one more thousand units will increase the cost by \)55,000, guiding decisions on scaling production.

Average Cost

Average cost is obtained by dividing the total cost by the quantity of output. It helps to understand the cost per unit of production.
The original exercise calculates average fixed cost (AFC) and average variable cost (AVC), which separate fixed and variable components. For 100,000 units, the AFC is calculated as \( \frac{200}{100} = 2 \), meaning the fixed cost per thousand units is \(2,000. Similarly, the AVC with the given data is a constant \)55 per thousand units, matching the variable cost component 55 in the equation.

• AFC focuses on spreading fixed costs over many units, reducing cost per unit as output increases.
• AVC captures the variable part, crucial for maintaining efficiency in production.

Overall, average cost metrics are vital for pricing strategies and profitability analysis.