The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $180; the quantity demanded each month is 1000 when the unit price is $150. The suppliers will market 750 espresso makers if the unit price is $80. At a unit price of $90, they are willing to market 1500 units. Both the demand and supply equations are known to be linear.
(a) Find the demand equation.
p = _________
(b) Find the supply equation.
p = _______
(c) Find the equilibrium quantity and the equilibrium price.
equilibrium quantity _______ units
equilibrium price $ _____
To find the demand equation, we need to determine the relationship between the quantity demanded and the unit price. We are given two data points: when the quantity demanded is 250, the unit price is $180, and when the quantity demanded is 1000, the unit price is $150.
Step 1: Use the formula for a linear equation, y = mx + b, where y represents the quantity demanded (x) and b represents the y-intercept.
Step 2: Determine the slope (m) using the two data points. The slope is calculated as (change in y / change in x).
m = (250 - 1000) / ($180 - $150)
= -750 / 30
= -25
Step 3: Use one of the data points to solve for the y-intercept (b).
Using the point (x, y) = (250, 180):
180 = -25(250) + b
180 = -6250 + b
b = 6430
So the demand equation is:
p = -25x + 6430
To find the supply equation, we need to determine the relationship between the quantity supplied and the unit price. We are given two data points: when the unit price is $80, the quantity supplied is 750, and when the unit price is $90, the quantity supplied is 1500.
Step 1: Use the formula for a linear equation, y = mx + b, where y represents the quantity supplied (x) and b represents the y-intercept.
Step 2: Determine the slope (m) using the two data points. The slope is calculated as (change in y / change in x).
m = (750 - 1500) / ($80 - $90)
= -750 / (-10)
= 75
Step 3: Use one of the data points to solve for the y-intercept (b).
Using the point (x, y) = (750, 80):
80 = 75(750) + b
80 = 56250 + b
b = -56170
So the supply equation is:
p = 75x - 56170
To find the equilibrium quantity and price, we need to set the demand and supply equations equal to each other (since equilibrium occurs when quantity demanded equals quantity supplied) and solve for x (quantity) and p (price).
Setting the demand equation equal to the supply equation:
-25x + 6430 = 75x - 56170
Simplifying the equation:
100x = 62500
x = 625
Substituting x back into either the demand or supply equation to calculate p:
p = -25(625) + 6430
p = -15625 + 6430
p = $9865
Therefore, the equilibrium quantity is 625 units and the equilibrium price is $9865.