) The demand curve for haircuts at Terry barnyards Hair Design is P=20-0.20Q
Where Q is the number of cuts per week and P is the price of a haircut. Terry is considered raising her price above the current price of $15. Terry is unwilling to raise price if the price hike will cause revenues to fall.
a) Should Terry raise the price of haircuts above $15? Why or why not?
b) Suppose demand for Terry’s haircuts increases to P=40-0.40Q. At a price of $15, should Terry raise the price of her haircuts? Why or Why not?
To answer both questions, we need to analyze the relationship between price, quantity, and revenue. We'll start with the given demand curve, which is P = 20 - 0.20Q, where P is the price and Q is the quantity.
a) Should Terry raise the price of haircuts above $15?
To determine this, we need to assess the impact on revenue. Revenue (R) is calculated by multiplying price (P) by quantity (Q), so we can express it as R = P * Q.
Currently, the price (P) is $15. Let's substitute this into the demand equation to find the corresponding quantity (Q):
15 = 20 - 0.20Q
0.20Q = 20 - 15
0.20Q = 5
Q = 5 / 0.20
Q = 25
So, at the current price of $15, Terry's haircuts have a demand of 25 per week. Now, let's calculate the total revenue:
R = P * Q
R = 15 * 25
R = $375
To assess if Terry should raise the price, let's consider two scenarios: increasing the price to $16 and $17.
For a price of $16:
The new demand equation becomes 16 = 20 - 0.20Q.
0.20Q = 20 - 16
0.20Q = 4
Q = 4 / 0.20
Q = 20
The new total revenue is:
R = P * Q
R = 16 * 20
R = $320
For a price of $17:
The new demand equation becomes 17 = 20 - 0.20Q.
0.20Q = 20 - 17
0.20Q = 3
Q = 3 / 0.20
Q = 15
The new total revenue is:
R = P * Q
R = 17 * 15
R = $255
Comparing the total revenues for each price, we see that raising the price to $16 results in lower revenue ($320) compared to the current revenue of $375. Similarly, raising the price to $17 gives even lower revenue ($255). Therefore, Terry should not raise the price above $15 as it would cause a decrease in revenue.
b) Suppose demand for Terry’s haircuts increases to P = 40 - 0.40Q. At a price of $15, should Terry raise the price of her haircuts?
To answer this question, we'll follow a similar process as in part a.
Substituting the price of $15 into the new demand equation:
15 = 40 - 0.40Q
0.40Q = 40 - 15
0.40Q = 25
Q = 25 / 0.40
Q = 62.5
The demand at a price of $15 is 62.5 haircuts per week. Now, let's calculate revenue:
R = P * Q
R = 15 * 62.5
R = $937.5
Considering the same two scenarios, let's calculate the revenues for prices of $16 and $17.
For a price of $16:
The new demand equation becomes 16 = 40 - 0.40Q.
0.40Q = 40 - 16
0.40Q = 24
Q = 24 / 0.40
Q = 60
The new total revenue is:
R = P * Q
R = 16 * 60
R = $960
For a price of $17:
The new demand equation becomes 17 = 40 - 0.40Q.
0.40Q = 40 - 17
0.40Q = 23
Q = 23 / 0.40
Q = 57.5
The new total revenue is:
R = P * Q
R = 17 * 57.5
R = $977.5
Comparing the total revenues, we see that at a price of $15, the revenue is $937.5, while at prices of $16 and $17, the revenues are higher ($960 and $977.5, respectively). Therefore, if the demand increases as given, Terry should raise the price of her haircuts from $15 to maximize revenue.