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 $ _______
the demand rises by 750 when the price drops by 30, so
d = 250 + 25(180-x)
the supply rises by 750 when the price rises by 10, so
s = 750 + 75(x-80)
Now just set d=s and solve for x, the equilibrium price.
Plug that back into either equation to figure the quantity
To find the demand equation, we can use the given information about the quantity demanded at different prices. Let's start by writing down the demand equation in the form of y = mx + b, where y represents the quantity demanded and x represents the unit price.
Given:
At p = $180, x = 250
At p = $150, x = 1000
Using the two points (250, 180) and (1000, 150), we can calculate the slope of the demand equation (m):
m = (y2 - y1) / (x2 - x1)
m = (150 - 180) / (1000 - 250)
m = -30 / 750
m = -0.04
Now that we have the slope, we can substitute one of the points into the equation and solve for the y-intercept (b):
250 = -0.04 * 180 + b
250 = -7.2 + b
b = 250 + 7.2
b = 257.2
Therefore, the demand equation is:
p = -0.04x + 257.2
Now let's find the supply equation using the given information.
Given:
At p = $80, x = 750
At p = $90, x = 1500
Using the two points (750, 80) and (1500, 90), we can calculate the slope of the supply equation (m):
m = (y2 - y1) / (x2 - x1)
m = (90 - 80) / (1500 - 750)
m = 10 / 750
m = 0.01333
Substituting one of the points into the equation, we can solve for the y-intercept (b):
750 = 0.01333 * 80 + b
750 = 1.0664 + b
b = 750 - 1.0664
b = 748.9336
Therefore, the supply equation is:
p = 0.01333x + 748.9336
To find the equilibrium quantity and price, we need to set the demand and supply equations equal to each other.
-0.04x + 257.2 = 0.01333x + 748.9336
Now we can solve for x, which represents the equilibrium quantity:
0.05333x = 748.9336 - 257.2
0.05333x = 491.7336
x = 491.7336 / 0.05333
x ≈ 9220.3
Therefore, the equilibrium quantity is approximately 9220 units.
To find the equilibrium price, we can substitute the equilibrium quantity into either the demand or supply equation:
p = -0.04 * 9220 + 257.2
p ≈ $202.92
Therefore, the equilibrium price is approximately $202.92.
In summary:
(a) The demand equation is p = -0.04x + 257.2.
(b) The supply equation is p = 0.01333x + 748.9336.
(c) The equilibrium quantity is approximately 9220 units and the equilibrium price is approximately $202.92.