Question

Andrea's Day Spa began to offer a relaxing aromatherapy treatment. The firm asks you how much to charge to maximize profits. The first two columns in Table 10.5 provide the price and quantity for the demand curve for treatments. The third column shows its total costs. For each level of output, calculate total revenue, marginal revenue, average cost, and marginal cost. What is the profit-maximizing level of output for the treatments and how much will the firm earn in profits? $$\begin{array}{l|l|l}\hline {\text { Price }} & {\text { Quantity }} & {\text { TC }} \\\\\hline \$ 25.00 & 0 & \$ 130 \\\\\hline \$ 24.00 & 10 & \$ 275 \\\\\hline \$ 23.00 & 20 & \$ 435 \\\\\hline \$ 22.50 & 30 & \$ 610 \\ \hline \$ 22.00 & 40 & \$ 800 \\\\\hline \$ 21.60 & 50 & \$ 1,005 \\\\\hline \$ 21.20 & 60 & \$ 1,225 \\ \hline\end{array}$$

Short answer

The profit-maximizing level of output for Andrea's Day Spa is when 50 treatments are offered, with a price of $21.60 per treatment. At this point, the Marginal Revenue (MR) equals Marginal Cost (MC). The total revenue at this level is $1,080 ($21.60 × 50), and the total costs are $1,005. Therefore, the firm will earn a profit of $75 ($1,080 - $1,005) by offering 50 aromatherapy treatments.

Step-by-step solution

Calculate Total Revenue (TR) for each level of output

Calculate total revenue at each quantity level. Total Revenue (TR) is given by the product of Price (P) and Quantity (Q). Thus, for each row in the table, multiply the Price and Quantity values to get the Total Revenue.

Calculate Marginal Revenue (MR) for each level of output

Marginal Revenue (MR) is the addition to total revenue resulting from selling one more unit of output. Calculate the change in total revenue (∆TR) for each additional unit of quantity (∆Q). Hence, MR = ∆TR/∆Q.

Calculate Average Cost (AC) for each level of output

Average cost is the total cost (TC) divided by the quantity of output (Q). Hence, for each output level, calculate AC = TC/Q.

Calculate Marginal Cost (MC) for each level of output

Marginal Cost (MC) is the cost of producing one more unit of output. It is calculated as the change in total cost (∆TC) per unit change in quantity of output (∆Q). Hence, MC = ∆TC/∆Q.

Identify the Profit-maximizing level of output

Look for the output level where Marginal Revenue (MR) equals Marginal Cost (MC). This is the profit-maximizing level of output, as any deviation from this level would either lower profits or raise losses.

Calculate the firm's total profit at the profit-maximizing level of output

The total profit is calculated by subtracting the total cost (TC) from the total revenue (TR) at the profit-maximizing level of output. Hence, Profit = TR - TC. In the context of business, understanding and utilizing these concepts can help optimize a firm's production and pricing strategy for profitability. This exercise is an application of the fundamental economic principle that firms aim to maximize profits.

Key concepts

Marginal Revenue

Marginal Revenue (MR) is a crucial concept in understanding profit maximization for businesses. It represents the additional revenue that a company earns when it sells one more unit of its product. Calculating MR helps businesses to determine the impact of increasing output on their income. In this exercise, Marginal Revenue is determined by analyzing the change in Total Revenue as more aromatherapy treatments are sold.

To calculate MR, follow these steps:

• Identify the Total Revenue (TR) for each level of output by multiplying the price by the quantity sold.
• Determine the change in Total Revenue (ΔTR) as you move from one quantity level to the next.
• Divide this change (ΔTR) by the change in quantity (ΔQ) to find Marginal Revenue using the formula: \( MR = \frac{\Delta TR}{\Delta Q} \).

Understanding Marginal Revenue assists firms in deciding the optimal quantity to produce. If the MR exceeds marginal cost (the cost of producing another unit), increasing production will boost profits. Conversely, if MR is lower, the firm should reduce production.

Marginal Cost

Marginal Cost (MC) is the additional expense a company incurs when producing one additional unit of output. It is a critical factor in determining the optimal level of production for achieving maximum profit. The calculation of MC is essential for deciding whether to increase or decrease production levels.

The steps to calculate MC are straightforward:

• Observe the Total Cost (TC) for each level of output.
• Calculate the change in Total Cost (ΔTC) between different levels of output.
• Divide this change (ΔTC) by the change in quantity (ΔQ) to calculate Marginal Cost with the formula: \( MC = \frac{\Delta TC}{\Delta Q} \).

Understanding Marginal Cost is vital. If the MC is lower than the Marginal Revenue (MR), producing more units will likely result in increased profit. Matching production where MR equals MC is ideal, as producing beyond this will increase costs more than revenues, leading to reduced profits.

Average Cost

Average Cost (AC) provides insight into the efficiency of production at various levels of output by illustrating the cost per unit produced. It is computed by dividing Total Cost (TC) by the quantity of output (Q). Knowing the AC is essential for pricing strategies and profit calculations.

To compute the Average Cost:

• Use the formula \( AC = \frac{TC}{Q} \).
• Calculate for each level of output by dividing the Total Cost reported by the corresponding Quantity.

Tracking Average Cost helps businesses identify economies of scale, which occur when AC decreases as production increases. It assists in pricing decisions to ensure that selling price covers costs and contributes to profit. When combined with marginal analyses, understanding AC allows firms to align production levels closely with market demand and cost structures.