The quantity demanded each month of Russo Expresso Makers is 250 when the unit price is $136.oo. The quantity demanded each month is 1,2oo when the unit price is $98.00. The suppliers will market 700 espresso makers if the unit price is $58.00 or lower. At a unit price of $74.50, they are willing to make available 2,350 units in the market. Both the demand and supply equations are known to be linear. find the equilibrium quantity and the equilibrium price.
if d(p) and s(p) are the demand and supply functions at price p,
d(136) = 250
d(98) = 1200
d(p) = 3650-25p
s(58) = 700
s(74.50) = 2350
s(p) = 100p - 5100
d(p) = s(p) when p = 70
d(70) = s(70) = 1900
To find the equilibrium quantity and price, we need to set the quantity demanded equal to the quantity supplied.
First, let's write the demand equation using the given points:
Point 1: (Price1, Quantity1) = ($136.00, 250)
Point 2: (Price2, Quantity2) = ($98.00, 1200)
The demand equation is given by:
quantity demanded = m * price + b
To find the slope (m) and the y-intercept (b), we can use the two points. Let's substitute Point 1 into the equation:
250 = m * 136 + b
And Point 2:
1200 = m * 98 + b
We now have a system of equations:
1: 250 = m * 136 + b
2: 1200 = m * 98 + b
Next, we need to find the supply equation. Using the given points:
Point 3: (Price3, Quantity3) = ($58.00, 700)
Point 4: (Price4, Quantity4) = ($74.50, 2350)
The supply equation is also linear and given by:
quantity supplied = m * price + b
Substituting Point 3:
700 = m * 58 + b
And Point 4:
2350 = m * 74.50 + b
We have another system of equations:
3: 700 = m * 58 + b
4: 2350 = m * 74.50 + b
Now, we can solve the system of equations simultaneously to find the equilibrium quantity and price.
Solution:
Solving equations 1 and 2, we find:
m ≈ 4.375
b ≈ -425
Substituting these values into equations 3 and 4:
3: 700 = 4.375 * 58 + (-425)
4: 2350 = 4.375 * 74.50 + (-425)
Simplifying these equations will give us the values of m and b for the supply equation.
Solving equations 3 and 4, we find:
m ≈ 37.625
b ≈ -1190
Now, we have the demand and supply equations:
Demand: Quantity Demanded = 4.375 * Price - 425
Supply: Quantity Supplied = 37.625 * Price - 1190
To find the equilibrium quantity and price, we set the quantity demanded equal to the quantity supplied:
4.375 * Price - 425 = 37.625 * Price - 1190
Simplifying the equation will give us the equilibrium price:
33.25 * Price = 765
Dividing both sides by 33.25:
Price ≈ $22.99
To find the equilibrium quantity, we substitute the equilibrium price into either the demand or supply equation. Let's use the supply equation:
Quantity = 37.625 * 22.99 - 1190
Quantity ≈ 527.9
Therefore, the equilibrium quantity is approximately 528 units, and the equilibrium price is approximately $22.99.