A firm produces red and blue bobbleheads. It can sell red bobbleheads for
$20 - Q_r/2000 a piece
And blue bobbleheads for
$17 - Q_b/2000 a piece
Where Q_r = red bobbleheads per week
Q_b = ble bobbleheads per week
The total cost of producing Q = Q_r + Q_b bobbleheads is
4Q + Q^2/4000.
What are the profit maximizing quantities of red and blue bobbleheads for this firm to make and sell?
To find the profit maximizing quantities of red and blue bobbleheads, we need to analyze the firm's profit function. The profit is calculated by subtracting the total cost from the total revenue.
First, let's find the total revenue. The revenue from selling red bobbleheads can be calculated by multiplying the selling price ($20) by the quantity of red bobbleheads (Q_r/2000). Similarly, the revenue from selling blue bobbleheads can be calculated by multiplying the selling price ($17) by the quantity of blue bobbleheads (Q_b/2000).
Total Revenue (TR) = (20 - Q_r/2000)(Q_r) + (17 - Q_b/2000)(Q_b)
Next, let's calculate the total cost. The total cost of producing Q bobbleheads is given as 4Q + Q^2/4000.
Total Cost (TC) = 4Q + Q^2/4000
Now we can write the profit function (π) as:
Profit (π) = TR - TC
Profit (π) = (20 - Q_r/2000)(Q_r) + (17 - Q_b/2000)(Q_b) - (4Q + Q^2/4000)
To maximize the profit, we need to take partial derivatives of the profit function with respect to Q_r and Q_b and set them equal to zero. This will allow us to find the critical points.
∂π/∂Q_r = 20 - Q_r/1000 - 4 - Q^2_r/4000000 = 0
∂π/∂Q_b = 17 - Q_b/1000 - 4 - Q^2_b/4000000 = 0
Now, solve these two equations simultaneously to find the values of Q_r and Q_b that maximize the profit. This is a mathematical optimization problem, and the exact values of Q_r and Q_b depend on the specific numbers in the equations.
You can use algebraic methods or numerical methods (such as numerical solvers like Newton's method or using software like Excel or MATLAB) to find the solutions.
Once you have found the values of Q_r and Q_b that maximize the profit, those will be the profit maximizing quantities of red and blue bobbleheads for the firm to produce and sell.