Simplifying an expression using a logarithm I have the following expression 1

res2bfitjq

res2bfitjq

Answered question

2022-05-21

Simplifying an expression using a logarithm
I have the following expression
1 1 + ρ ( 1 + n ) ( 1 σ ) ( 1 + γ A ) 1 σ < 1
and have to use logarithms to get the following
( 1 σ ) ( n + γ A ) < ρ
Could someone please offer me some help on how to do it? That's a homework question in Macroeconomics - we got the solutions and were told to use the logarithms to get them. However, I'm struggling with the complete procedure, so any help is more than welcome.
Thanks!

Answer & Explanation

Austin Solis

Austin Solis

Beginner2022-05-22Added 12 answers

Taking the logarithms brings the exponent down and turns all the multiplies into adds, so we have
log ( 1 + ρ ) + ( 1 σ ) log ( 1 + n ) + ( 1 σ ) log ( 1 + γ A ) < 0.
We can move the first term across and factor to get
( 1 σ ) ( log ( 1 + n ) + log ( 1 + γ A ) ) < log ( 1 + ρ ) .
If ρ < 1, then log ( 1 + ρ ) < ρ from the series expansion, so that will get us ρ on the right. Unfortunately the same trick doesn't quite work for the left side, since I don't want to replace the left side with something bigger. Perhaps you could do something with changing the sum of logs into log ( 1 + n + γ A + n γ A ) and showing that that is bigger than n + γ A because of the extra bit at the end (again assuming n and γ A are relatively small).

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