THE SHORT RUN COST FUNCTION OF THE FIRM IS OF THE FORM .TC=300+50Q-10Q2+Q3. WHAT IS THE VALUE OF FIXED COST ? WRITE EXPRESSIONS FOR ATC,AVC,AFC AND MC . CALCULATE THE OUTPUT AT WICH AVERAGE TOTAL COST ARE MIMIZED?
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To calculate the fixed cost (FC), we need to identify the term in the cost function that does not depend on the level of output (Q). In this case, the term that remains constant regardless of the level of output is 300. Therefore, the fixed cost (FC) is 300.
To calculate the Average Total Cost (ATC), we need to divide the Total Cost (TC) by the quantity of output (Q):
ATC = TC / Q
Given that the Total Cost (TC) is 300 + 50Q - 10Q^2 + Q^3, the expression for the Average Total Cost (ATC) would be:
ATC = (300 + 50Q - 10Q^2 + Q^3) / Q
To calculate the Average Variable Cost (AVC), we need to subtract the fixed cost (FC) from the Total Cost (TC) and then divide it by the quantity of output (Q):
AVC = (TC - FC) / Q
Given that the Fixed Cost (FC) is 300, the expression for the Average Variable Cost (AVC) would be:
AVC = (300 + 50Q - 10Q^2 + Q^3 - 300) / Q
To calculate the Average Fixed Cost (AFC), we need to divide the fixed cost (FC) by the quantity of output (Q):
AFC = FC / Q
Given that the Fixed Cost (FC) is 300, the expression for the Average Fixed Cost (AFC) would be:
AFC = 300 / Q
To calculate the Marginal Cost (MC), we need to take the derivative of the Total Cost (TC) function with respect to output (Q):
MC = d(TC) / dQ
Given that the Total Cost (TC) function is 300 + 50Q - 10Q^2 + Q^3, the expression for the Marginal Cost (MC) would be:
MC = 50 - 20Q + 3Q^2
To find the output level at which the Average Total Cost (ATC) is minimized, we need to differentiate the Average Total Cost (ATC) function with respect to output (Q) and set it equal to zero, and then solve for Q:
d(ATC) / dQ = 0
By simplifying the expression for the Average Total Cost (ATC), we have:
(ATC = (300 + 50Q - 10Q^2 + Q^3) / Q)
(ATC = (300 / Q) + 50 - 10Q + Q^2)
Differentiating the expression for ATC with respect to Q:
d(ATC) / dQ = -300 / Q^2 - 10 + 2Q
Setting d(ATC) / dQ = 0 and solving for Q:
-300 / Q^2 - 10 + 2Q = 0
-300 - 10Q^2 + 2Q^3 = 0
2Q^3 - 10Q + 300 = 0
To solve this equation for Q, you can use numerical methods or graphing calculators.
Once you find the value of Q at which ATC is minimized, you can substitute that value back into the ATC equation to find the minimum Average Total Cost (ATC).