Econ questions

2007-05-06 5:16 am

Please help+_+

1)
The demand equation of a good is given by:
P+2Q=20

and the total cost function is
Q^3-8Q^2+20Q+2

(a) Find the level of output which maximizes total revenue.

(b)Find the maximum profit and the value of Q at which it is achieved. Verify that, at this value of Q, MR=MC

回答 (1)

Danny

2007-05-06 8:20 am

✔ 最佳答案

(a) P + 2Q = 20
P = 20 - 2Q
TR = P * Q = 20Q - 2Q^2
TR' = 20 - 4Q
Set TR' = 0 we get Q* = 5
TR'' = -4 < 0
So TR is maximized at Q = 5

(b) Profit = TR - TC
= ( 20Q - 2Q^2 ) - ( Q^3 - 8Q^2 + 20Q + 2 )
= - Q^3 + 6Q^2 - 2
Profit' = - 3Q^2 + 12Q
Set Profit' = 0 we get Q* = 4 or 0
Profit'' = - 6Q + 12 which is smaller than zero when Q = 4 (not at Q = 0)
So profit is maximized when Q = 4, at which the profit is - 4^3 + 6*4^2 - 2 = 30

MR = TR' = 20 - 4Q, which is 4 when Q = 4
MC = TC' = 3Q^2 - 16Q + 20 which is also 4 when Q = 4

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