Problem 21 - Applied Calculus

Question

Production costs for manufacturing running shoes consist of a fixed overhead of $$ 650,000\( plus variable costs of $$ 20 per pair of shoes. Each pair of shoes sells for $$ 70 .\) (a) Find the total cost, \(C(q),\) the total revenue, \(R(q)\) and the total profit, \(\pi(q),\) as a function of the number of pairs of shoes produced, \(q\) (b) Find the marginal cost, marginal revenue, and marginal profit. (c) How many pairs of shoes must be produced and sold for the company to make a profit?

Step-by-Step Solution

Answer

Profit starts with more than 13,000 pairs. Marginal costs, revenue, and profit are 20, 70, and 50, respectively.

Step 1: Define Total Cost Function

Total cost consists of a fixed cost and a variable cost. The fixed cost is \( 650,000 \) and the variable cost is \( 20 \) per pair. Therefore, the total cost function \( C(q) \) is given by: \[ C(q) = 650,000 + 20q \] where \( q \) is the number of pairs of shoes.

Step 2: Define Total Revenue Function

Revenue is calculated by multiplying the selling price per unit by the quantity sold. Given that each pair of shoes sells for \( 70 \), the total revenue function \( R(q) \) is: \[ R(q) = 70q \]

Step 3: Define Total Profit Function

Profit is the difference between total revenue and total cost, so the total profit function \( \pi(q) \) is: \[ \pi(q) = R(q) - C(q) \] Substituting from previous steps, we get \[ \pi(q) = 70q - (650,000 + 20q) = 50q - 650,000 \]

Step 4: Calculate Marginal Cost

The marginal cost is the derivative of the total cost function \( C(q) \). Since \( C(q) = 650,000 + 20q \), the marginal cost is the coefficient of \( q \), which is \( 20 \). Therefore, the marginal cost is \( 20 \) per pair of shoes.

Step 5: Calculate Marginal Revenue

The marginal revenue is the derivative of the total revenue function \( R(q) \). Since \( R(q) = 70q \), the marginal revenue is the coefficient of \( q \), which is \( 70 \). Therefore, the marginal revenue is \( 70 \) per pair of shoes.

Step 6: Calculate Marginal Profit

The marginal profit is the derivative of the total profit function \( \pi(q) \). Since \( \pi(q) = 50q - 650,000 \), the marginal profit is the coefficient of \( q \), which is \( 50 \). Therefore, the marginal profit is \( 50 \) per pair of shoes.

Step 7: Find Break-Even Point for Profit

To determine when the company makes a profit, set the profit function greater than zero: \[ \pi(q) = 50q - 650,000 > 0 \] Solving for \( q \): \[ 50q > 650,000 \] \[ q > 13,000 \] Hence, the company must produce and sell more than 13,000 pairs of shoes to make a profit.

Key Concepts

Fixed Costs

Fixed costs play a crucial role in understanding overall production costs. These are expenses that remain constant regardless of the level of production or sales volume. In the shoe production example, we see that the fixed cost is $650,000. This means that whether one pair or thousands of pairs of shoes are made, this cost does not change.

Fixed costs typically include:

• Rent or lease payments for facilities
• Salaries of permanent staff
• Insurance premiums
• Depreciation of equipment

Understanding fixed costs is vital for management to determine how many units must be sold to cover these costs, ultimately guiding financial planning and pricing strategies.

Variable Costs

Variable costs fluctuate based on the level of production or the volume of output. Unlike fixed costs, these expenses increase directly as production ramps up. In our shoe production scenario, the variable cost is $20 per pair of shoes produced. Hence, if more shoes are made, the variable costs will proportionately rise.

Examples of variable costs include:

• Raw materials (like fabric for shoes)
• Packaging supplies
• Direct labor costs (wages of workers directly involved in production)
• Utility expenses depending on usage

Identifying and calculating variable costs is essential because it helps to determine the total cost of producing any number of units. This helps in analyzing the production efficiency and setting the right pricing strategy for products.

Marginal Cost

Marginal cost is a key concept in economic analysis and decision-making. It refers to the additional cost incurred for producing an additional unit of output. In the context of our scenario, the marginal cost is determined by the variable cost per unit, which is \(20 per pair of shoes.

The formula typically used for calculating marginal cost from the total cost function is:\[ MC = \frac{dC(q)}{dq} \]where \( C(q) \) is the total cost function. For our example, since \( C(q) = 650,000 + 20q \), the derivative \( \frac{dC(q)}{dq} \) gives us the marginal cost of \)20.

Understanding marginal cost is important for making efficient production decisions. It helps in identifying the tipping point where producing additional units may increase costs disproportionately compared to revenues, thus impacting profitability.