A manufacturer of lapel buttons test marketed a new item at a university. It was found that 1,000 items could be sold if they were priced at $4, but only 600 items could be sold if the price were raised to $8. On the other hand, the manufacturer finds that 400 items can be supplied at $1 each. If they were supplied at $9 each, overtime shifts could be used to raise the supply to 800 items. What is the optimum price for the items?
To find the optimum price for the items, we need to determine the price at which the demand and supply are balanced. The optimal price occurs when the quantity demanded equals the quantity supplied.
In this case, we have two sets of information: one for demand and one for supply.
Demand Information:
Price: $4
Quantity: 1,000
Price: $8
Quantity: 600
Supply Information:
Price: $1
Quantity: 400
Price: $9
Quantity: 800
Let's analyze the demand and supply information separately and then find the point where they intersect.
Demand Analysis:
Using the demand information, let's calculate the demand curve's slope:
Slope = (Change in Quantity Demanded) / (Change in Price)
Slope = (600 - 1000) / ($8 - $4)
Slope = -400 / $4
Slope = -100
We know the equation of a linear demand curve is: Qd = a - bP, where Qd represents the quantity demanded, P is the price, a is the y-intercept, and b is the slope. We have the slope (-100), and to find the y-intercept (a), we can substitute the quantity and price values from one of the demand points into the equation:
1,000 = a - (-100) x $4
1,000 = a + $400
a = 1,000 - $400
a = $600
The demand curve equation now becomes: Qd = 600 - 100P
Supply Analysis:
Using the supply information, let's calculate the supply curve's slope:
Slope = (Change in Quantity Supplied) / (Change in Price)
Slope = (800 - 400) / ($9 - $1)
Slope = 400 / $8
Slope = 50
Similar to the demand curve, the equation of a linear supply curve is: Qs = c + dP, where Qs represents the quantity supplied, P is the price, c is the y-intercept, and d is the slope. We have the slope (50), and to find the y-intercept (c), we can substitute the quantity and price values from one of the supply points into the equation:
400 = c + 50 x $1
400 = c + $50
c = 400 - $50
c = $350
The supply curve equation now becomes: Qs = 350 + 50P
Now, we can find the equilibrium point by setting the quantity demanded equal to the quantity supplied and solving for the price:
Qd = Qs
600 - 100P = 350 + 50P
Combine like terms:
150P = 250
Divide both sides by 150:
P = 250 / 150
P ≈ $1.67
Therefore, the optimum price for the lapel buttons is approximately $1.67.