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 can use the two points provided: (250, 180) and (1000, 150). We can use the formula for a linear equation, which is given as:
y = mx + b
where y represents the dependent variable (quantity demanded), x represents the independent variable (price), m represents the slope, and b represents the y-intercept.
(a) To find the demand equation:
We can start by finding the slope (m) using the formula:
m = (y2 - y1) / (x2 - x1)
m = (150 - 180) / (1000 - 250)
m = -30 / 750
m = -1/25
Now, we can substitute one of the points and the slope into the equation to find the y-intercept (b). Let's use the point (250, 180):
180 = (-1/25) * 250 + b
Solving for b:
180 = -10 + b
b = 190
Therefore, the demand equation is:
p = (-1/25) * x + 190
p = -x/25 + 190
(b) To find the supply equation:
We can use the two points provided: (750, 80) and (1500, 90). Using the same formula as above, we find:
m = (90 - 80) / (1500 - 750)
m = 10 / 750
m = 1/75
Substituting one of the points and the slope into the equation to find the y-intercept (b), using the point (750, 80):
80 = (1/75) * 750 + b
Solving for b:
80 = 10 + b
b = 70
Therefore, the supply equation is:
p = (1/75) * x + 70
p = x/75 + 70
(c) To find the equilibrium quantity and the equilibrium price:
The equilibrium occurs when the quantity demanded equals the quantity supplied. To find this point, we need to set the demand equation equal to the supply equation:
(-x/25) + 190 = x/75 + 70
Simplifying the equation:
-3x + 570 = x + 525
-4x = -45
x = 11.25
The equilibrium quantity is 11.25 units.
To find the equilibrium price, we can substitute this quantity into either the demand or supply equation. Let's use the demand equation:
p = (-1/25) * 11.25 + 190
p = -0.45 + 190
p = 189.55
The equilibrium price is $189.55.