Suppose a firm's demand function is Q = 5 – P/8. What is the firm's revenue function?

Answer 1

Revenue (R) = Price * Quantity = P*Q

If R = P*Q then P = R/Q and Q = R/P

Now we have 2 variables and 3 equations, we can use substitutions

If P = R/Q then we can replace P in the original equation so instead of

Q = 5 - P/8

it can be rewritten as

Q = 5 - ((R/Q)/8)

From there, we multiply both sides of the equation by 8

8Q = 40 - (R/Q)

Then multiply both sides by Q

8Q² = 40Q - R

Then we can rearrange the equation so R is on one side and everything else is on the other

R = 40Q - 8Q²

The same can be done by substituting for Q

Q = R/P (From above)

R/P = 5 - P/8 (Substitute 'R/P' for 'Q' in the original equation)

R= 5P - (P² / 8) (Multiply Both Sides by 'P')

Check:

If Price = $8 then

Quantity = 5 - P/8 = 5 - (8/8) = 5 - 1 = 4

Revenue = Price * Quantity = $8 * 4 = $32

Revenue = 40Q - 8Q² = 40(4) - 8(4²) = 160 - 8(16) = 160 - 128 = $32

Revenue = 5P - (P² / 8) = 5($8) - ($8² / 8) = 40 - (64 / 8) = 40 - 8 = $32