I am a student in DSGE

I am confuse about the implecation of the elasticity of substitution, as the picture as follows:

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.

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.

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

Best wishes to you ^ ^

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

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!