4. Assume a firm operating under a short run production period with a total cost function given
as Tc=200Q+5QTHE POWER 2+2QTHE POWER 3
4.1. 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 (1.5pts)
To find the output size that minimizes the average cost of production, we need to first find the expression for the average cost (AC). The average cost is calculated by dividing the total cost (TC) by the quantity produced (Q):
AC = TC / Q
Given that the total cost function is Tc = 200Q + 5Q^2 + 2Q^3, we can substitute this expression back into the average cost formula:
AC = (200Q + 5Q^2 + 2Q^3) / Q
Now, let's simplify this expression:
AC = 200 + 5Q + 2Q^2
To minimize the average cost, we need to find the critical point where the derivative of the average cost function is equal to zero. Let's differentiate the average cost with respect to Q:
dAC/dQ = 5 + 4Q
Setting this derivative equal to zero and solving for Q:
5 + 4Q = 0
4Q = -5
Q = -5/4
However, since production quantity cannot be negative, we discard this solution. Therefore, there is no output size that minimizes the average cost of production in this case.
To determine if the marginal cost of production is increasing or decreasing at this point, we need to find the derivative of the total cost function. Let's differentiate Tc = 200Q + 5Q^2 + 2Q^3 with respect to Q:
dTC/dQ = 200 + 10Q + 6Q^2
Evaluating the derivative at the given output quantity of Q = -5/4:
dTC/dQ = 200 + 10*(-5/4) + 6*(-5/4)^2
= 200 - 12.5 + 9.375
= 196.875
Since the derivative is positive (196.875 > 0), the marginal cost of production is increasing at this output quantity.
To find the output size that minimizes the average cost of production, we need to differentiate the total cost function (TC) with respect to the quantity (Q), set it equal to zero, and solve for Q.
Given:
TC = 200Q + 5Q^2 + 2Q^3
First, let's differentiate TC with respect to Q:
dTC/dQ = 200 + 10Q + 6Q^2
To find the output size that minimizes the average cost of production, we need to find the point where the marginal cost (MC) equals the average cost (AC).
Marginal Cost (MC) is the derivative of the total cost function (TC) with respect to the quantity (Q):
MC = dTC/dQ
Average Cost (AC) is the total cost (TC) divided by the quantity (Q):
AC = TC/Q
Setting MC equal to AC:
dTC/dQ = TC/Q
Substituting the values for MC and TC:
200 + 10Q + 6Q^2 = (200Q + 5Q^2 + 2Q^3)/Q
Simplifying the equation:
200Q + 10Q^2 + 6Q^3 = 200 + 5Q + 2Q^2
Now, let's solve this equation for Q by putting it in standard form:
6Q^3 + 8Q^2 - 5Q + 200 = 0
To find the output size that minimizes the average cost of production, we need to solve this equation. However, since this is a cubic equation, it may not have a simple analytical solution. You can use numerical methods or a graphing calculator to approximate the value of Q that minimizes the average cost and determine if the marginal cost at that point is increasing or decreasing.