The quantity demanded x each month of Russo Espresso Makers is 250 when the unit price p is $170; the quantity demanded each month is 1000 when the unit price is $140. The suppliers will market 750 espresso makers if the unit price is $90. At a unit price of $110, they are willing to market 2250 units. Both the demand and supply equations are known to be linear.
What is the supply equation? Equilibrium quantity? equilibrium price?
To find the supply equation, we need to find the equation of the supply curve, which represents the quantity supplied at different prices. Since both the supply and demand equations are linear, we can use the slope-intercept form of a linear equation, which is y = mx + b.
Let's start by finding the supply equation using the given information:
At a unit price of $90, the suppliers will market 750 espresso makers. This point can be represented as (x1, p1) = (750, 90).
At a unit price of $110, the suppliers are willing to market 2250 espresso makers. This point can be represented as (x2, p2) = (2250, 110).
We can use these two points to find the slope (m) of the supply curve:
m = (p2 - p1) / (x2 - x1)
= (110 - 90) / (2250 - 750)
= 20 / 1500
= 1/75
Now that we have the slope, we can use one of the given points to find the y-intercept (b). Let's use the first point (750, 90):
90 = (1/75)(750) + b
90 = 10 + b
b = 90 - 10
b = 80
Therefore, the supply equation is:
p = (1/75)x + 80
To find the equilibrium quantity and price, we need to find the point where the supply and demand curves intersect. This point represents the quantity demanded and supplied at the same price.
Now, let's find the demand equation:
At a unit price of $170, the quantity demanded is 250 espresso makers. This point can be represented as (x3, p3) = (250, 170).
At a unit price of $140, the quantity demanded is 1000 espresso makers. This point can be represented as (x4, p4) = (1000, 140).
Using these two points, we can find the slope of the demand curve:
m = (p4 - p3) / (x4 - x3)
= (140 - 170) / (1000 - 250)
= -30 / 750
= -1/25
Using the first point (250, 170), we can find the y-intercept (b):
170 = (-1/25)(250) + b
170 = -10 + b
b = 170 + 10
b = 180
Therefore, the demand equation is:
p = (-1/25)x + 180
To find the equilibrium quantity and price, we need to set the supply and demand equations equal to each other and solve for x (quantity):
(1/75)x + 80 = (-1/25)x + 180
Multiplying the equation by 75 to eliminate the fractions gives:
x + 6000 = -3x + 13500
Adding 3x to both sides and subtracting 6000 from both sides:
4x = 7500
Dividing both sides by 4:
x = 1875
So, the equilibrium quantity (x) is 1875.
Now, substitute this value back into either the supply or demand equation to find the equilibrium price (p). Let's use the demand equation:
p = (-1/25)(1875) + 180
p = -75 + 180
p = 105
Therefore, the equilibrium price (p) is $105.
To summarize:
- The supply equation is p = (1/75)x + 80.
- The equilibrium quantity (x) is 1875 espresso makers.
- The equilibrium price (p) is $105.