Question #59111

If the fixed cost were $25 and the variable cost per unit were $ 2 and the demand function as follows: P=20-Q
(a) Get π in terms of Q and plot the graph of him .
(b) Find the value of Q in break even point.
(c) Find the production level which gives $ 31 as revenue.
(d) Find the maximum profit and value of Q where the maximum profit is achieved.
Thank You...

Expert's answer

FC = $25, AVC = $2, P = 20 - Q.

(a) π = TR - TC = P*Q - (FC + VC) = (20 - Q)*Q - FC - AVC*Q = -Q^2 + 20Q - 2Q - 25 = -Q^2 + 18Q - 25.

(b) In break even point π = 0, so:

-Q^2 + 18Q - 25 = 0

Q^2 - 18Q + 25 = 0

Q1 = (18 + (324 - 4*1*25)^0.5)/2 = (18 + 14.97)/2 = 16.5 units.

Q2 = (18 - (324 - 4*1*25)^0.5)/2 = (18 - 14.97)/2 = 1.5 units.

(c) At the production level which gives $ 31 as revenue TR = 31, so:

P*Q = 31

(20 - Q)*Q = 31

-Q^2 + 20Q - 31 = 0

Q^2 - 20Q + 31 = 0

Q1 = (20 + (400 - 4*1*31)^0.5)/2 = (20 + 16.6)/2 = 18.3 units.

Q2 = (20 - (400 - 4*1*31)^0.5)/2 = (20 - 16.6)/2 = 1.7 units.

(d) The profit is maximized, when π' = 0, so:

(-Q^2 + 18Q - 25)' = 0

-2Q + 18 = 0

2Q = 18

Q = 9 units.

(a) π = TR - TC = P*Q - (FC + VC) = (20 - Q)*Q - FC - AVC*Q = -Q^2 + 20Q - 2Q - 25 = -Q^2 + 18Q - 25.

(b) In break even point π = 0, so:

-Q^2 + 18Q - 25 = 0

Q^2 - 18Q + 25 = 0

Q1 = (18 + (324 - 4*1*25)^0.5)/2 = (18 + 14.97)/2 = 16.5 units.

Q2 = (18 - (324 - 4*1*25)^0.5)/2 = (18 - 14.97)/2 = 1.5 units.

(c) At the production level which gives $ 31 as revenue TR = 31, so:

P*Q = 31

(20 - Q)*Q = 31

-Q^2 + 20Q - 31 = 0

Q^2 - 20Q + 31 = 0

Q1 = (20 + (400 - 4*1*31)^0.5)/2 = (20 + 16.6)/2 = 18.3 units.

Q2 = (20 - (400 - 4*1*31)^0.5)/2 = (20 - 16.6)/2 = 1.7 units.

(d) The profit is maximized, when π' = 0, so:

(-Q^2 + 18Q - 25)' = 0

-2Q + 18 = 0

2Q = 18

Q = 9 units.

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