Page 353 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes
Example: If the demand law is x = e p , , 0 , express marginal revenue as a function
of x. At what levels of output and price the total revenue is maximum? Also find maximum total
revenue.
Solution:

Taking log of both sides of the demand function, we get
1
log x = log p or p log log x

1
TR = p.x = x .log x .logx

d TR 1
Now MR = = log logx 1
dx
1
= log 1 0 , for maximum TR
x

log 1 = 0 or log 1
x x

log = loge e or x
x x e
d TR 1
2
Further, 2 = < 0, the second order condition is satisfied. Also price, when x , is
dx x e
1 1 1
given by p log logx log x

1
Hence, maximum TR = e .

10
Example: A firm’s demand function is : x 400ln . Find the price and quantity where
p
total revenue is maximum. Also find price elasticity of demand at that price.
Solution:

Note: ln denotes log with base e.
Here it will lie convenient to express total revenue as a function of p.
p
p
p
TR = p . . 400 ln10 ln p 400 .ln10 400 .ln p
x
( d TR ) 400ln10 400ln p 400p 1
dp = p
= 400 ln10 lnp 1 0 for maxima.

ln10 lnp = 1

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