In economics, the demand for a product is the amount of that product that consumers are willing to buy at a given price. The quantity demanded of a product usually decreases if the price of that product
increases. Suppose that a company believes there is a linear relationship between the demand for its
product and its price. The company knows that when the price of its product was $3 per unit, the quantity demanded weekly was 500 units, and that when the unit price was raised to $4, the quantity demanded weekly dropped to 300 units. Let D represent the quantity demanded weekly at a unit price of p dollars.
a) Calculate a when p = 5. Interpret your result
b) Find a formula for D in terms of p
c) The company raises the price of the good and the new quantity demanded weekly is 50 units. What
is the new price?
d) Give an economic interpretation of the slope of the function you found in part (b)
e)Find D when p = 0. Find p whenD = 0.Give economic interpretations of both these results.
a) To calculate the quantity demanded when p = 5, we can use the formula for a linear relationship: D = mx + b.
First, let's find the slope, m, of the linear relationship. The slope represents how much the quantity demanded changes for each unit increase in price. Using the given data points (p = 3, D = 500) and (p = 4, D = 300), we can find the slope using the formula:
m = (D2 - D1) / (p2 - p1) = (300 - 500) / (4 - 3) = -200 / 1 = -200
Next, we substitute the slope and one of the data points into the equation to solve for the y-intercept, b:
500 = -200(3) + b
500 = -600 + b
b = 500 + 600
b = 1100
Now that we have the slope (m = -200) and y-intercept (b = 1100), we can calculate the quantity demanded when p = 5:
D = (-200)(5) + 1100
D = -1000 + 1100
D = 100
The interpretation of this result is that when the price increases to $5 per unit, the quantity demanded decreases to 100 units weekly.
b) To find a formula for D in terms of p, we substitute the values of m and b into the equation D = mx + b:
D = -200p + 1100
c) If the new quantity demanded weekly is 50 units, we can use the formula from part (b) to find the new price, denoted by q:
50 = -200q + 1100
Solving this equation for q gives us:
-200q = 50 - 1100
-200q = -1050
q = -1050 / -200
q = 5.25
So, the new price is $5.25.
d) The slope of the function D = -200p + 1100 represents the change in quantity demanded for each unit increase in price. In this case, the slope is -200, indicating that for every $1 increase in price, the quantity demanded decreases by 200 units. This negative slope reflects the inverse relationship between price and quantity demanded, as stated in the given information.
e) When p = 0, we can substitute p = 0 into the formula from part (b):
D = -200(0) + 1100
D = 1100
The economic interpretation of this result is that when the price is $0, the quantity demanded is 1100 units. This implies that there is no price barrier, and consumers are willing to consume 1100 units.
To find the price when D = 0, we set D = 0 in the formula:
0 = -200p + 1100
Solving for p gives us:
-200p = -1100
p = -1100 / -200
p = 5.5
The economic interpretation of this result is that when the quantity demanded is 0, the price of the good is $5.50. This implies that there is no demand for the product at this price point.