Assume a firm operating under a short run production period with a total cost function given as TC=200+5Q+2Q^2.
What must be the output size to minimize the average cost of production, and show if marginal cost of production is increasing or decreasing at this point
To find the output size that minimizes the average cost of production, we first need to find the average cost function. The average cost (AC) is calculated by dividing the total cost (TC) by the output level (Q):
AC = TC / Q
Given that the total cost function is TC = 200 + 5Q + 2Q^2, we substitute this into the formula for average cost:
AC = (200 + 5Q + 2Q^2) / Q
To minimize the average cost, we need to find the critical point where the derivative of the average cost function with respect to Q is equal to zero.
To find the derivative, we differentiate the average cost function with respect to Q:
d(AC) / dQ = (d(200 + 5Q + 2Q^2) / dQ) / Q
= (5 + 4Q) / Q
Setting this equal to zero:
(5 + 4Q) / Q = 0
Solving for Q, we get:
5 + 4Q = 0
4Q = -5
Q = -5 / 4
Since the output level cannot be negative, we discard the negative solution.
Therefore, the output size that minimizes the average cost of production is Q = -5/4.
Next, we need to determine if the marginal cost of production is increasing or decreasing at this point.
The marginal cost (MC) can be found by taking the derivative of the total cost function with respect to Q:
MC = d(TC) / dQ
= 5 + 4Q
Substituting the output size that minimizes the average cost (Q = -5/4) into the marginal cost equation, we get:
MC = 5 + 4(-5/4)
= 5 - 5
= 0
At the output size that minimizes the average cost, the marginal cost of production is equal to zero. This indicates that the marginal cost is neither increasing nor decreasing at this point.
To minimize the average cost of production, we need to find the output size at which the average cost (AC) is at its lowest point. The average cost can be calculated by dividing the total cost (TC) by the output quantity (Q):
AC = TC / Q
First, let's differentiate the total cost function with respect to Q to find the marginal cost (MC).
TC = 200 + 5Q + 2Q^2
To find MC, we differentiate TC with respect to Q:
MC = d(TC) / dQ = 5 + 4Q
Next, we equate AC and MC to find the output size (Q) where the average cost is minimized:
AC = MC
TC / Q = 5 + 4Q
Multiplying both sides by Q:
TC = (5 + 4Q)Q
Expanding the equation:
200 + 5Q + 2Q^2 = 5Q + 4Q^2
Rearranging the terms:
4Q^2 - 3Q - 200 = 0
Now, we can solve this quadratic equation to find the value of Q.
Using the quadratic formula:
Q = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4, b = -3, and c = -200.
Plugging in these values:
Q = (-(-3) ± √((-3)^2 - 4(4)(-200))) / (2 * 4)
Simplifying:
Q = (3 ± √(9 + 3200)) / 8
Q = (3 ± √3209) / 8
Since we are operating under a short-run production period, we are primarily concerned with the positive root value. Hence:
Q ≈ (3 + √3209) / 8
Using a calculator or math software, we find that Q ≈ 7.95.
Therefore, the output size that minimizes the average cost of production is approximately 7.95 units.
To determine whether the marginal cost of production is increasing or decreasing at this point, we need to analyze the value of MC at Q = 7.95. Plugging this value into the MC equation:
MC = 5 + 4Q
MC ≈ 5 + 4(7.95)
MC ≈ 5 + 31.8
MC ≈ 36.8
Since the value of MC at Q = 7.95 is positive, the marginal cost of production is increasing at this point.