Question

The director of a theater company in a small college town is considering changing the way he prices tickets. He has hired an economic consulting firm to estimate the demand for tickets. The firm has classified people who go to the theater into two groups and has come up with two demand functions. The demand curves for the general public (Q_gp) and students (Qs)

are given below:

Q_gp = 500 - 5P

Qs = 200 - 4P

a. Graph the two demand curves on one graph, withon the vertical axis andQon the horizontal axis. If the current price of tickets is \(35, identify the quantity demanded by each group.

b. Find the price elasticity of demand for each group at the current price and quantity.

c. Is the director maximizing the revenue he collects from ticket sales by charging \)35 for each ticket? Explain.

d. What price should he charge each group if he wants to maximize revenue collected from ticket sales?

Short answer

a. The demand curve is given below:

The quantity demanded by each general public will be 325, and by students will be 60 tickets.

b. The price elasticity for the general public will be 0.54, and for students will be 2.33.

c. The director is not maximizing revenue.

d. The director must charge $50 from the general public and $25 from students to maximize revenue.

Step-by-step solution

Explanation for part (a)

The prices are taken on the y-axis, and the quantity of tickets on the x-axis.

Given the demand curve equation for the general public, the vertical intercept is calculated by keeping the demand as zero:

$$\begin{gathered} Q_{gp}=500-5P \\ 0=500-5P \\ P=\frac{500}{5}=100 \end{gathered}$$

The horizontal intercept is calculated by keeping P=0. Thus:

Q_gp= 500 - 5P

= 500 - 5(0)

= 500

The quantity demanded when P=35 is calculated below:

Q_gp= 500 - 5(35)

= 325

Therefore, the quantity of tickets demanded by the general public would be 325 at a price of $35.

Given the demand curve equation for students, the vertical intercept will be calculated by keeping demand as zero:

$$\begin{gathered} Q_S=200-4P \\ 0=200-4P \\ P=\frac{200}{4}=50 \end{gathered}$$

And the horizontal intercept will be calculated by keeping P=0.

Q_s= 200 - 4P

= 200 - 4(0)

= 200

Now, at P=35, demand is calculated below:

Q_s= 200 - 4(35)

= 60

Therefore, the quantity of tickets demanded at the price of $35 would be 60.

Explanation for part (b)

The price elasticity is equal to:

$E=\frac{\partial q}{\partial p}\times \frac{p}{q}$

Therefore; the price elasticity for the general public would be:

$E_{gp}=\frac{-5\left(35\right)}{325}=0.54$

The price elasticity for students would be:

$E_s=\frac{-4\left(35\right)}{60}=2.33$

Explanation for part (c)

No, the director is not maximizing the revenue. It is because the general public’s demand for tickets is inelastic, and the demand of students is elastic;therefore, whatever revenue he makes from the general public would be compensated for the revenue he loses from students. Therefore, his revenue is not maximized.

Explanation for part (d)

The elasticities are given as:

The formula for the elasticity of demand is

$E=\frac{\partial q}{\partial p}\times \frac{p}{q}$

Therefore, for the general public:

$$\begin{gathered} E_{gp}=\frac{-5P}{Q}=-1 \\ 5P=Q=500-5P \\ P=50 \\ Q=250 \end{gathered}$$

For students;

$$\begin{gathered} E_S=\frac{-4P}{Q}=-1 \\ 4P=Q=200-4P \\ P=25 \\ Q=100 \end{gathered}$$

The elasticities will determine if a change in price will increase the revenue or not. Therefore, based on elasticities, the director charges $50 from the general public and $25 from students; he will maximize revenue:

(P_gpx Q_gp)+ (P_sx Q_s)

= (50 x 250) + (25 x 100)

= 15000

This is more than the revenue that is generated by charging $35.

35(Q_gp+Q_s )

= 35(325 + 60)

=13,475

Therefore, the director must charge $50 and $25 from the general public and students, respectively, to maximize profits.