A firm's total cost function is given by the equation TC=4000+5Q+10Q and marginal cost is given by the equation MC=5+20Q
(A) Write an expression for each of the following cost concepts:
a. Total Fixed Cost
b. Average Fixed Cost
c. Total Variable Cost
d. Average Variable Cost
e. Average Total Cost
(B) Calculate the values of marginal cost and the costs in (a)-(e) above for Q=0,1,2,3.
(C) Determine the quantity that minimizes average total cost. Demonstrate that the predicted relationship between marginal cost and average cost holds.

[tex](a)\ TFC = 4,000\\\\(b)\ AFC = \frac{TFC}{ Q} = \frac{4,000}{ Q}\\\\(c)\ TVC = 5Q + 10\ Q2\\\\(d)\ AVC = \frac{TVC }{Q} = 50 + 10\ Q\\\\(e)\ ATC = \frac{TC }{ Q} = (\frac{4,000}{ Q}) + 50 + 10Q \ \text{Also, ATC = AVC + AFC}\\\\[/tex]

For point B)

TFC remains unchanged at 4,000, regardless of the price of Q.

i)

[tex]\to Q = 0[/tex]

AFC, AVC, and ATC cannot be calculated (division by zero is not possible).

ii)

[tex]Q = 1\\\\AFC =\frac{4,000}{ 1} = 4,000\\\\TVC = (5 \times 1) + (10 \times 1) =5 + 10 = 15\\\\AVC = \frac{TVC}{ Q} = \frac{15}{1} = 15\\\\ATC = 4,000 + 15 = 4,015\\\\MC = 5 + (20 \times 10 = 5 + 20 = 25[/tex]

iii)

[tex]Q = 2\\\\AFC = \frac{4,000}{ 2} = 2,000\\\\TVC = (5 \times 2) + (10 \times 2 \times 2) = 10 + 40 = 50\\\\AVC = \frac{50}{2} = 25\\\\ATC = 2,000 + 25 = 2,025\\\\MC = 5 + (20 \times 2) = 5 + 40 = 45\\\\[/tex]

iv)

[tex]Q = 3\\\\AFC = \frac{4,000}{ 3} = 1,333.33\\\\TVC = (5 \times 3) + (10 \times 3 \times 3) = 15 + 90 = 105\\\\AVC = \frac{105}{3} = 35\\\\ATC = 1,333.33 + 35 = 1,368.33\\\\MC = 5 + (20 \times 3) = 5 + 60 = 65\\\\[/tex]

For point C)

i)

[tex]ATC[/tex] is minimized when [tex]\frac{dATC}{dQ} = 0[/tex]

[tex](- \frac{4,000}{Q2} ) + 10 = 0\\\\\frac{4,000}{Q2} = 10\\\\Q2 = 400\\\\Q = 20\\[/tex]

ii)

Part (B) shows that as MC increases from Q = 0 to Q = 3, ATC decreases, validating the link.

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