Finding the linear demand function

In summary, the manager of a 120-room resort hotel is trying to determine the optimal price to charge in order to maximize revenue. Based on the given information, it can be assumed that the demand function is linear. By plotting the points from the problem on a graph with price on the vertical axis and output on the horizontal axis, the manager can use the point-slope formula to construct the demand function and determine the optimal price.
  • #1
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Homework Statement


The manager of a 120-room resort hotel has found that, on average, 50 rooms are booked when the price is $100 per night and 80 rooms are booked when the price is $80 per night. Find the demand function, assuming that it is linear, and what price should he charge to maximize revenue?


Homework Equations


So I thought of it as two linear equations, but I don't understand how to obtain the demand function. Is it 100=50-x and 80=80-x? Doesn't seem right to me...


The Attempt at a Solution


Sadly, all the other questions involve a single price, and I don't know how to work with two. Any help would be awesome!
 
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  • #2
Hey!

Draw a figure with price on the vertical axis and output on the horizontal axis. Plot the points from the problem. You'll notice that you have four points, you can use the point slope formula to construct your demand function

Good luck!
 
Last edited:

FAQ: Finding the linear demand function

What is a linear demand function?

A linear demand function is a mathematical equation that represents the relationship between the quantity demanded of a product and its price. It is a straight line that shows how the quantity demanded changes as the price of the product changes.

How do you find the linear demand function?

The linear demand function can be found by using the formula Q = a - bP, where Q represents the quantity demanded, a is the intercept of the demand line, and b is the slope of the line. This formula can be derived from the basic economic law of demand, which states that as the price of a product increases, the demand for that product decreases.

What information is needed to find the linear demand function?

To find the linear demand function, you will need the price and quantity data for the product. This data can be collected through market research or by analyzing past sales data. You will also need to choose a starting point for the demand line, which can be determined by estimating the demand at a given price.

Why is it important to find the linear demand function?

Finding the linear demand function is important for businesses to understand the relationship between price and demand for their product. It can help them make pricing decisions, forecast demand, and determine the optimal price point for maximizing profits. It is also a useful tool for understanding the market demand for a product and predicting consumer behavior.

Are there any limitations to using a linear demand function?

Yes, there are limitations to using a linear demand function. It assumes that the relationship between price and demand is a straight line, which may not always be the case in real-world scenarios. It also does not account for external factors that may influence demand, such as changes in consumer preferences or economic conditions. Additionally, it is based on historical data and may not accurately predict future demand if there are significant changes in the market. Therefore, it is important to regularly review and update the linear demand function to account for any changes in the market.

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