Suppose a firm has a revenue function described by Total Revenue (TR) = 24Q – 3Q². What is the marginal revenue gained from producing the 4th unit, and from producing the 5th unit? What is the maximum revenue the firm can earn?

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  To find the marginal revenue of the 4th unit, first find the total revenue generated from selling 4 units, then subtract the total revenue generated from selling 3 units. This can be written as one equation like the following:

  MR of the 4th unit = TR(4) – TR(3) = 24(4) – 3(4²) – (24(3) – 3(3²)) = 96 – 36 – (72 – 27) = 60 – 45 = 15.

  We conduct the same process to find the marginal revenue of the 5th unit:

  MR of the 5th unit = TR(5) – TR(4) = 24(5) – 3(5²) – (24(4) – 3(4²)) = 120 – 75 – (96 – 36) = 45 – 60 = -15.

  To maximize total revenue, we must take the derivative, set it equal to zero, and solve:

  dTR/dq = 24 – 6q = 0. Solving, we find that q = 4.

  We have already calculated the total revenue earned at 4 units; it is TR(4) = 24(4) – 3(4²) = 96 – 36 = 60. Therefore, the maximum revenue the firm can earn is earned at 4 units sold. The marginal revenue from producing this 4th unit is 15 monetary units.

  Note that we discovered the marginal revenue from the 5th unit is negative; so, the firm should stop producing at 4 units, because producing more units means that MR < MC, and the firm would have a lower revenue from selling the 5th unit.

   

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