Formulate profit fuction

Question

The Total Revenue and Total Costs of a company is given by the following formulae

TR=600q-2q²+600 and TC= 3q²-200q+400

Formulate:

a). The profit function.

b). The Marginal Cost and Marginal Revenue functions, hence calculate the profit maximizing output.

Answer 1

TR = -2q^2 +600q +600

TC= 3q^2 -200q + 400

Profit, P, = Total Revnue minus Total Cost

= TR-TC =-2q^2 +600q +600 -[3q^2 -200q +400]

-5q^2 +800q -200

P'(q)) = -10q +800 =0

q = 800/10 = 80 = profit maximizing output

MR= -4q +600

MC =6q -200

MR=MC

-4q+600 = 6q-200

10q = 800

q = 80

P=-5q^2 + 800q -200

P(80) =-5(80)^2 +800(80) -200

= -32000 + 64000 -200

= 32000-200

= 31,800 = maximum profit

or you might try a graphical approach

each curve is a parabola, one upward opening, the other downward opening

their difference is another parabola, downward opening

P=-5x^2 + 800x -200

=-5(x^2 -160x + 80^2) -200 +5(80^2)

= -5(x-80)^2 +31,800 which has vertex = maximum = (80,31,800)

TR =-2q^2 +600x +600 is a downward opening parabola

= -2(q^2 -300x + 150^2) + 600 +2(150^2)

= -2(q-150)^2 + 44600

vertex = max = (150,44,600)

TC =3q^2 -200q +400

=3(q^2 -200q/3 + 100^2/3) + 400 -100^2/3

= 3(q-100/3)^2 - 9,800/3

vertex = minimum = (33 1/3, -3,266 2/3)

graph MR and MC. they're two lines

where they intersect is the profit maximizing price and quantity