The manager of caribbean glass company estimates that total revenue from the sale of her firm's product is given by the equation TR=300Q-Qsquared/2. The total cost equation is TC=5,000 + 60Q + Qsquared
What are the profit maximizing price and output rate? What is the amount of Economic profit?
At what output rate is is average cost at a minimum? At this output rate what is the amount of economic profit?
Maximize where Marginal cost = Marginal revenue.
MC is the first derivitive of TC. Same same for MR. So
MR = 300 - Q
MC = 60 + 2Q
Set MC=MR and solve for Q. Plug this Q back into TC and TR to calculate profit.
AC = TC/Q = 5000/Q + 60 +Q
For a minimum, take the derivitive of this, then set the equation = zero and solve for Q.
AC = TC/Q = 5000/Q + 60 +Q
To find the profit-maximizing price and output rate, we need to determine the quantity at which Total Revenue (TR) minus Total Cost (TC) is maximized. This can be done by calculating the profit function and finding its maximum point.
The profit function (π) is given by subtracting the total cost equation from the total revenue equation:
π = TR - TC
Substituting the given equations, we get:
π = (300Q - Q^2/2) - (5,000 + 60Q + Q^2)
Simplifying further:
π = 300Q - Q^2/2 - 5,000 - 60Q - Q^2
π = -1.5Q^2 + 240Q - 5,000
To find the profit-maximizing output rate, we need to take the derivative of the profit function with respect to Q and set it equal to zero:
dπ/dQ = -3Q + 240 = 0
Solving for Q gives:
Q = 80
Now, we can substitute this value of Q back into either the total revenue or total cost equation to find the corresponding price. Let's use the total revenue equation:
TR = 300Q - Q^2/2
TR = 300(80) - (80)^2/2
TR = 24,000 - 3,200 = 20,800
Therefore, the profit-maximizing output rate is 80 units and the price is $20,800.
To calculate the economic profit, we need to subtract the total cost at the profit-maximizing output rate from the total revenue:
Total Cost at Q = 80:
TC = 5,000 + 60(80) + (80)^2
TC = 5,000 + 4,800 + 6,400 = 16,200
Economic Profit = Total Revenue - Total Cost
Economic Profit = 20,800 - 16,200
Economic Profit = $4,600
At what output rate is the average cost at a minimum? At this output rate, what is the amount of economic profit?
To find the output rate at which average cost is at a minimum, we need to calculate the average cost function. Average cost (AC) is calculated by dividing the total cost (TC) by the quantity (Q):
AC = TC / Q
Substituting the total cost equation:
AC = (5,000 + 60Q + Q^2) / Q
AC = 5,000/Q + 60 + Q
To minimize the average cost, we need to take the derivative of the average cost function with respect to Q and set it equal to zero:
dAC/dQ = -5000/Q^2 + 1 = 0
Solving for Q gives:
Q = √5000 = 70.7 (approx.)
At this output rate, the amount of economic profit can be determined by substituting Q = 70.7 into the profit function π:
π = -1.5Q^2 + 240Q - 5,000
π = -1.5(70.7)^2 + 240(70.7) - 5,000
π ≈ $3,400 (approx.)
Therefore, at the output rate of approximately 70.7 units, the average cost is at a minimum and the amount of economic profit is approximately $3,400.