Chapter 10: Problem 19

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

Expert verified

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.