Question #52109

A firm's total revenue (TR) and total cost (TC) functions are:
TR=110Q-5Q^2 TC=10Q-Q^2+0.33Q^3
Determine: a) The output rate that maximizes total revenue.
b) The output rate that minimizes average cost.
c) The output rate that maximizes profit.
d) Equation for marginal revenue and average revenue.
e) Equation for marginal cost and average cost

Expert's answer

TR=110Q-5Q^2, TC=10Q-Q^2+0.33Q^3

a) The output rate that maximizes total revenue is in the point, where MR = 0. So, MR = TR' = 110 - 10Q = 0

Q = 11 units.

b) The output rate that minimizes average cost will be in the point, where ATC' = (TC/Q)' = 2Q - 2 = 0

Q = 1 unit.

c) The output rate that maximizes profit is in the point, where MR = MC, so:

MR = TR' = 110 - 10Q,

MC = TC' = 10 - 2Q + Q^2

So, 110 - 10Q = 10 - 2Q + Q^2

Q^2 + 8Q - 100 = 0

Q = (-8 + √464)/2

d) Equation for marginal revenue is MR = TR' = 110 - 10Q and average revenue AR = TR/Q = 110 - 5Q

e) Equation for marginal cost is MC = TC' = 10 - 2Q + Q^2 and average cost ATC = TC/Q = 10 - 2Q + 0.33Q^2.

a) The output rate that maximizes total revenue is in the point, where MR = 0. So, MR = TR' = 110 - 10Q = 0

Q = 11 units.

b) The output rate that minimizes average cost will be in the point, where ATC' = (TC/Q)' = 2Q - 2 = 0

Q = 1 unit.

c) The output rate that maximizes profit is in the point, where MR = MC, so:

MR = TR' = 110 - 10Q,

MC = TC' = 10 - 2Q + Q^2

So, 110 - 10Q = 10 - 2Q + Q^2

Q^2 + 8Q - 100 = 0

Q = (-8 + √464)/2

d) Equation for marginal revenue is MR = TR' = 110 - 10Q and average revenue AR = TR/Q = 110 - 5Q

e) Equation for marginal cost is MC = TC' = 10 - 2Q + Q^2 and average cost ATC = TC/Q = 10 - 2Q + 0.33Q^2.

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