Assume A Firm Operating Under a short run production period with a total cost function given as TC=200+5Q+2Q^2.Then What Must be the out put size to minimize average cost of production and show if cost of production is increasing or decreasing at this point.
The output size to minimize average cost of production is Q = 10.
The average cost of production is AC = (200 + 5Q + 2Q^2)/Q = 20 + 5 + 2Q.
At Q = 10, AC = 20 + 5 + 20 = 45.
The cost of production is decreasing at this point, since the derivative of AC with respect to Q is dAC/dQ = 2Q - 5, which is negative at Q = 10.
To minimize the average cost of production, we need to find the output size where the average cost is at its lowest point. The average cost is calculated by dividing the total cost by the quantity of output produced.
The total cost function in this case is given as TC = 200 + 5Q + 2Q^2, where Q represents the quantity of output.
To find the output size that minimizes the average cost, we need to differentiate the total cost function with respect to Q and set it equal to zero. This will help us find the critical points, where the average cost could be minimized.
Let's calculate the average cost first. The average cost (AC) is given by AC = TC / Q.
We can rewrite the total cost function as TC = 200Q + 5Q^2 + 2Q^3.
Substituting this value of TC into the formula for average cost, we get:
AC = (200Q + 5Q^2 + 2Q^3) / Q
AC = 200 + 5Q + 2Q^2
Now, to find the output size that minimizes the average cost, we differentiate the average cost function with respect to Q and set it equal to zero.
d(AC) / dQ = 5 + 4Q = 0
Solving this equation, we find:
4Q = -5
Q = -5/4
Since the output size cannot be negative, we can discard the negative value. Therefore, the output size that minimizes the average cost of production is Q = -5/4.
To determine whether the cost of production is increasing or decreasing at this point, we need to analyze the second derivative of the average cost function. If the second derivative is positive, the cost of production is increasing, and if the second derivative is negative, the cost of production is decreasing.
Taking the second derivative of the average cost function, we get:
d^2(AC)/dQ^2 = 4
Since the second derivative is a constant value of 4, and it is positive, the cost of production is increasing at the output size Q = -5/4.
To find the output size that minimizes the average cost of production, we need to differentiate the total cost function with respect to output (Q) and set it equal to zero.
Given the total cost function: TC = 200 + 5Q + 2Q^2
First, differentiate TC with respect to Q:
dTC/dQ = 5 + 4Q
Setting this derivative equal to zero:
5 + 4Q = 0
4Q = -5
Q = -5/4
Since the output size cannot be negative, we ignore the negative solution.
Therefore, the output size that minimizes the average cost of production is Q = -5/4, which is equivalent to -1.25 units.
To determine if the cost of production is increasing or decreasing at this point, we need to examine the second derivative of the total cost function.
Taking the second derivative of TC with respect to Q:
d^2TC/dQ^2 = 4
Since the second derivative d^2TC/dQ^2 is positive (4 > 0), the cost of production is increasing at the output size that minimizes the average cost.