Question

A firms demand function is x=200log(20/p).Find the price and quantity where total revenue is maximum

Answer 1

To solve this question let us list the necessary logarithmic laws that are relevant to this question given that the equation I logarithmic.

log m/n = log m - log n

log₁₀b = c

10^c = b

x = 200 log 20/p

x = 200 [ log 20 - log P]

Revenue function = R(x)

R(x) = p [ 200(log 20 - log P)]

200p log 20 - 200 log p

At maximum revenue :

d(x) / d(p) = 0

d(x) / d(p) = 200 log20 — [ 200 log p + 200p × 1 / pln10]

200 log 20 — 200 log p — 200 / ln10 = 0


Divide through by 200

log 20 — log p —1/ln 10 = 0

log 20 — 1/ln 10 = log p

1.3010 — 0.4343 = log p

0.8667 = log p

p = 10⁰°⁸⁶⁶⁷

P = 7.36

x = 200 × log 20/7.39

200 × 0.4342 = 86.84

The price = 7.36

The quantity = 86.84