About the implecation of the elasticity of substitution

General DSGE Modeling
1

I am a student in DSGE
I am confuse about the implecation of the elasticity of substitution, as the picture as follows:
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if eta become larger ,will the elasticity of substitution will become larger or smaller? and why? I guess is related to the first order condition but I am not sure.
thanks for any advice! best wishes to you.

2

Start with
Y=\left[\alpha K^\frac{\sigma-1}{\sigma}+(1-\alpha)N^\frac{\sigma-1}{\sigma}\right]^{\frac{\sigma }{\sigma - 1}}
The elasticity of substitution can then be computed by noticing that the ratio of marginal products is
\frac{F_N}{F_K} = \frac{1 - \alpha}{\alpha}{\left( {\frac{N}{K}} \right)^{ - \frac{1}{\sigma }}} = \frac{1 - \alpha}{\alpha}{\left( {\frac{K}{N}} \right)^{\frac{1}{\sigma }}} \\
and therefore
\begin{align} \ln \left( {\frac{{{F_N}}}{{{F_K}}}} \right) &= \ln \left( \frac{1 - \alpha}{\alpha} \right) + \frac{1}{\sigma }\ln \left( {\frac{K}{N}} \right)\\ \ln \left( {\frac{K}{N}} \right) &= \sigma \ln \left( {\frac{{{F_N}}}{{{F_K}}}} \right) - \sigma \ln \left( \frac{1 - \alpha}{\alpha} \right)\\ {\sigma _{K,N}} &= \frac{{\partial \ln \left( {\frac{K}{N}} \right)}}{{\partial \ln \left( {\frac{{{F_N}}}{{{F_K}}}} \right)}} = \sigma \end{align}
This expression gives the percentage change in inputs for a given change in marginal products (which corresponds to a change in the relative input prices). If \sigma becomes smaller, the factor ratio will respond less to changes in input prices: factors are less substitutable, i.e. complements. The limit for \sigma\to \infty is Leontieff.

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3

Thank you very much! DSGE seems complicated and I cannot get some details about it.
Best wishes to you ^ ^

4

This is not at all about DSGE , but simply about CES production functions.

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5

I always thought CES production function was part of DSGE, which was my mistake.
I need to work harder to learn more about it.
Your advice was greatly appreciated！Thanks for your help!