Question

Suppose that the inverse demand function for a​ monopolist's product is: p=7-(Q/20)

Its cost function is: C=10+14Q-4Q^2+(2Q^3/3)

Marginal revenue equals marginal cost when output equals: q=_ units and q=_units (Enter numeric responses using real numbers rounded to three decimal​places.)

What is the profit maximizing output?

Answer 1

In order to maximize profit a firm should produce that quantity at which Marginal Revenue(MR) = Marginal Cost(MC) and MC is rising.

MC = dC/dQ = d(10+14Q-4Q²+(2Q³/3))/dQ = 14 - 8Q + (2Q²)

MR = d(TR)/dQ = d(PQ)/dQ = d(7-(Q/20))/dQ = 7 - Q/10

Marginal revenue = Marginal cost when 7 - Q/10 = 14 - 8Q + (2Q²)

=> 2Q² - 7.9Q +7 = 0

Formula:

When aQ² + bQ + c = 0 then Q = (-b + (b² - 4ac)^1/2)/(2a) and Q = (-b - (b² - 4ac)^1/2)/(2a)

Here a = 2 , b = -7.9 and c = 7

Hence Q =  (-(-7.9) + ((7.9)² - 4*2*7)^1/2)/(2*2) = 2.608

and Q = (-(-7.9) - ((7.9)² - 4*2*7)^1/2)/(2*2) = 1.342

Hence, Marginal revenue equals marginal cost when output equals: q= 1.342 units and q= 2.608 units

As discussed In order to maximize profit a firm should produce that quantity at which Marginal Revenue(MR) = Marginal Cost(MC) and MC is rising. As MR = MC is for Q = 1.342 units and Q = 2.608 units and hence we have to find Q for which MC is rising.

MCis rising when d(MC)/dQ > 0 =>d(14 - 8Q + (2Q²))/dQ > 0 => -8 + 4Q > 0 => Q > 2

So, MC is rising when Q > 2 and among our options Q . 2 when Q = 2.608.

Therefore our profit maximizing output is 2.608 units