Question

Assume the quantity level is 1500 where the average revenue function is zero. Further assume firms can effectively control their own prices. Total revenue is maximized where the quantity level equals zero for marginal revenue. Find the quantity level where total revenue is maximized.  Show your work.

Answer 1

We know that, the Average Revenue (AR) curve and the Demand curve is equivalent. Hence, we suppose that an inverse demand function is given as

P = a - b.Q

Now, This demand curve can be written as average revenue curve i.e.

AR = a - b.Q

Where, AR = Average Revenue Function

Now, When AR = 0, Q = 1500

Hence, AR = 0

or, a - b×1500 = 0

or, a = 1500.b.........(1)

Now, Total Revenue function is

TR = AR×Q = a.Q - b.Q^{​​​​2}

Hence, Marginal Revenue function is

MR = dTR/dQ = a - 2b.Q

Now, when Marginal Revenue is zero, Total Revenue is maximized. Hence,

MR = 0

or, a - 2b.Q = 0

or, 1500.b = 2b.Q [as a = 1500.b]

or, Q = 750

The marginal revenue is zero when quantity level is 750.

Hence, total revenue is maximized at quantity level 750.

Hope the solution is clear to you my friend.