Suppose Y Company estimates the following total cost function from cost output data:
TC = $135,000 + $250Q + $1.5Q2.
Find the optimum level of output that makes the company efficient and per unit cost (AC) at the rate of output.
The optimum level of output that makes the company efficient is found by taking the derivative of the total cost function and setting it equal to zero. This gives us:
dTC/dQ = 250 + 3Q = 0
Solving for Q, we get Q = -83.33.
The per unit cost (AC) at the rate of output is found by substituting the optimum level of output into the total cost function and dividing by the output. This gives us:
AC = (135,000 - 83.33*250 + 1.5*(-83.33)2) / -83.33 = $2,917.
To find the optimum level of output that makes the company efficient and per unit cost (AC) at the rate of output, we need to determine the minimum point on the average cost curve.
The average cost (AC) is calculated by dividing the total cost (TC) by the quantity of output (Q):
AC = TC / Q
For the given total cost function TC = $135,000 + $250Q + $1.5Q^2, we can substitute this into the formula for AC:
AC = ($135,000 + $250Q + $1.5Q^2) / Q
To find the minimum, we can take the derivative of AC with respect to Q and set it equal to zero:
d(AC) / dQ = (d/dQ) (($135,000 + $250Q + $1.5Q^2) / Q) = 0
Simplifying the equation, we get:
[$135,000 + $250Q + $1.5Q^2)' * Q - ($135,000 + $250Q + $1.5Q^2) * Q'] / Q^2 = 0
[$250 + 3Q - ($135,000 + $250Q + $1.5Q^2) * Q'] / Q^2 = 0
Simplifying further, we have:
$250 + 3Q - ($135,000 + $250Q + $1.5Q^2) * Q' = 0
Now, to find the value of Q that makes this equation true, we need to solve for Q.