The demand plan for hotel rooms is as follows: Price (dollars per night) Quantity demanded (millions of rooms per night) 200 100 250 80 400 50 500 40 800 25 1000 20 a. What happens to the total income if the price drops from $400 to $250 per night? Calculate the price elasticity of demand for that price range and indicate what type of elasticity occurs.
To calculate the total income, we need to multiply the price by the quantity demanded.
Before the price drops from $400 to $250 per night:
Total income = price x quantity demanded
= $400 x 50 million rooms per night
= $20,000 million
After the price drops to $250 per night:
Total income = price x quantity demanded
= $250 x 80 million rooms per night
= $20,000 million
There is no change in the total income when the price drops from $400 to $250 per night.
To calculate the price elasticity of demand for that price range, we use the formula:
Elasticity = Percentage change in quantity demanded / Percentage change in price
Percentage change in quantity demanded:
= (New quantity demanded - Old quantity demanded) / Old quantity demanded
= (80 - 50) / 50
= 30 / 50
= 0.6
Percentage change in price:
= (New price - Old price) / Old price
= (250 - 400) / 400
= -150 / 400
= -0.375
Elasticity = 0.6 / -0.375
= -1.6
The price elasticity of demand for the price range from $400 to $250 per night is -1.6. This indicates that the demand is elastic, meaning that a decrease in price leads to a more than proportionate increase in quantity demanded, resulting in increased total income.
To calculate the change in total income when the price drops from $400 to $250 per night, we first need to calculate the total income before and after the price change.
Before the price change:
Price: $400 per night
Quantity demanded: 50 million rooms per night
Total income = Price * Quantity demanded
Total income before = $400 * 50 million rooms = $20 billion
After the price change:
Price: $250 per night
Quantity demanded: 80 million rooms per night
Total income after = $250 * 80 million rooms = $20 billion
The total income remains unchanged at $20 billion.
To calculate the price elasticity of demand, we use the formula:
Elasticity = (% change in quantity demanded) / (% change in price)
Using the price and quantity demanded from before and after the price change:
% change in quantity demanded = (80 million - 50 million) / 50 million * 100%
= 30 million / 50 million * 100%
= 60%
% change in price = ($250 - $400) / $400 * 100%
= -$150 / $400 * 100%
= -37.5%
Elasticity = 60% / -37.5%
= -1.6
The price elasticity of demand in this price range is -1.6. Since the elasticity value is negative, it indicates that the demand is elastic, meaning that a decrease in price leads to a proportionally larger increase in quantity demanded and total income remains unchanged.
To calculate the total income when the price drops from $400 to $250 per night, we need to determine the quantity demanded at each price point and then multiply it by the respective price.
Given the demand plan, we have the following price-quantity pairs:
Price (dollars per night): 200, 250, 400, 500, 800, 1000
Quantity demanded (millions of rooms per night): 100, 80, 50, 40, 25, 20
To find the quantity demanded at a price of $400, we look at the corresponding quantity value in the demand plan, which is 50 million rooms per night.
Now, let's calculate the total income at a price of $400 per night:
Total income at $400 per night = Price ($400) * Quantity demanded (50 million rooms per night)
Total income at $400 per night = $400 * 50 = $20,000 million
Next, we want to calculate the total income at a price of $250 per night. So, we need to find the quantity demanded at that price.
To calculate the quantity demanded at $250, we can use a technique called interpolation. We determine the two price points that bracket the desired price and then find the corresponding quantity values. In this case, the two price points are $200 and $400.
Using linear interpolation, we can find the quantity demanded at $250 as follows:
Quantity demanded at $250 = ((Quantity at $400 - Quantity at $200) / (Price at $400 - Price at $200)) * (Price at $250 - Price at $200) + Quantity at $200
Quantity demanded at $250 = ((50 - 100) / (400 - 200)) * (250 - 200) + 100
Quantity demanded at $250 = (-50 / 200) * 50 + 100
Quantity demanded at $250 = -12.5 + 100
Quantity demanded at $250 = 87.5 million rooms per night (rounded to one decimal place)
Now, let's calculate the total income at a price of $250 per night:
Total income at $250 per night = Price ($250) * Quantity demanded (87.5 million rooms per night)
Total income at $250 per night = $250 * 87.5 = $21,875 million
Therefore, if the price drops from $400 to $250 per night, the total income increases from $20,000 million to $21,875 million.
To calculate the price elasticity of demand (PED) for that price range, we can use the following formula:
PED = (Percentage change in quantity demanded) / (Percentage change in price)
First, let's calculate the percentage change in quantity demanded:
Percentage change in quantity demanded = (Quantity demanded at $250 - Quantity demanded at $400) / Quantity demanded at $400
Percentage change in quantity demanded = (87.5 - 50) / 50
Percentage change in quantity demanded = 75%
Next, let's calculate the percentage change in price:
Percentage change in price = (Price at $250 - Price at $400) / Price at $400
Percentage change in price = (250 - 400) / 400
Percentage change in price = -37.5%
Now, let's calculate the price elasticity of demand:
PED = (Percentage change in quantity demanded) / (Percentage change in price)
PED = 0.75 / -0.375
PED = -2
The price elasticity of demand for the price range of $400 to $250 per night is -2. This indicates that the demand is elastic because the absolute value of PED is greater than 1. A price elastic demand means that a change in price will lead to a relatively larger change in quantity demanded.