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?

Price	Quantity	TC
$\backslash (25.00$	$0$	$\backslash )130$
$\backslash (24.00$	$10$	$\backslash )275$
$\backslash (23.00$	$20$	$\backslash )435$
$\backslash (22.50$	$30$	$\backslash )610$
$\backslash (22.00$	$40$	$\backslash )800$
$\backslash (21.60$	$50$	$\backslash )1,005$
$\backslash (21.20$	$60$	$\backslash )1,225$

Short answer

Total Revenue	Marginal Revenue	Average Cost	Marginal Cost
$0$	$0$	$0$	$0$
$240$	$24$	$27.5$	$14.5$
$460$	$22$	$21.75$	$16$
$675$	$21.5$	$20.33$	$17.5$
$880$	$20.5$	$20$	$19$
$1080$	$20$	$20.1$	$20.5$
$1272$	$19.2$	$20.41$	$22$

• $40$will be the profit maximizing output
• The firm will earn a profit of$\$80$

Step-by-step solution

Step 1. Introduction

In a monopolistic competition, the profit maximizing level of output is determined by the point where marginal revenue and marginal cost are equal.

Step 2. Calculating cost and revenue

For output level 0 -

$$\begin{gathered} TR=Q*P=0*25=0 \\ MR=dTR/dQ=0/0=0 \\ AC=TC/Q=130/0=0 \\ MC=dTC/dQ=130/0=0 \end{gathered}$$

For output level 10 -

$$\begin{gathered} TR=Q*P=10*24=240 \\ MR=dTR/dQ=240/10=24 \\ AC=TC/Q=275/10=27.5 \\ MC=dTC/dQ=145/10=14.5 \end{gathered}$$

For output level 20 -

$$\begin{gathered} TR=Q*P=20*23=460 \\ MR=dTR/dQ=220/10=22 \\ AC=TC/Q=435/20=21.75 \\ MC=dTC/dQ=160/10=16 \end{gathered}$$

For output level 30 -

$$\begin{gathered} TR=Q*P=30*22.50=675 \\ MR=dTR/dQ=215/10=21.5 \\ AC=TC/Q=610/30=20.33 \\ MC=dTC/dQ=175/10=17.5 \end{gathered}$$

For output level 40 -

$$\begin{gathered} TR=Q*P=40*22=880 \\ MR=dTR/dQ=205/10=20.5 \\ AC=TC/Q=800/40=20 \\ MC=dTC/dQ=190/10=19 \end{gathered}$$

For output level 50 -

$$\begin{gathered} TR=Q*P=50*21.60=1080 \\ MR=dTR/dQ=200/10=20 \\ AC=TC/Q=1005/50=20.1 \\ MC=dTC/dQ=205/10=20.5 \end{gathered}$$

For output level 60 -

$$\begin{gathered} TR=Q*P=60*21.20=1272 \\ MR=dTR/dQ=192/10=19.2 \\ AC=TC/Q=1225/60=20.41 \\ MC=dTC/dQ=220/10=22 \end{gathered}$$

Step 3. Profit maximizing output

The profit maximizing output for this treatment is $40$because if the firm increases the output at $50$, marginal revenue will be less than marginal cost which cause negative economic profit. On the other hand the firm decrease its output at $30$, there will be an incentive to increase its output as marginal revenue exceeds marginal cost.

Step 4. Profit

At the output level $40$,

Profit = TR - TC

or, Profit =$880-800$

Therefore, Profit =$80$