Consider the following production function: $$ Y=A\left[\alpha K^{\rho}+\left(1-\alpha\right)L^{\rho}\right]^{\frac{1}{\rho}} $$ In the above, $\rho$ is the substitution parameter, and $\sigma=\frac{1}{1-\rho}$ is the elasticity of substitution. One definition of the elasticity of substitution is that this parameter is defined as the substitution of $K/L$ with respect to $MRS_{K,L}$ holding output constant. In other words: $$ \sigma=\frac{d\text{log}\left(K^{*}/L^{*}\right)}{d\text{log}\left(F_{L^{*}}/F_{K^{*}}\right)} $$ Notice, that the optimal choice on part of the firm implies that $F_{L}/F_{K}=w/r$ . As such: $$ \sigma=\frac{d\text{log}\left(K^{*}/L^{*}\right)}{d\text{log}\left(w/r\right)} $$
My question is as follows: how important is the fact that we hold output constant? I understand that if we consider the problem of the firm in two stages, i.e. in the first stage as a cost minimization problem (choosing optimal $K^{*}$ and $L^{*}$ for a given level of output), and in the second stage, choosing an output level to maximize profits, then if relative prices change, it is not necessary that the profit maximizing output stays the same. In other words, if relative prices change, we shift to a different isoquant, and this definition of $\sigma$ is not the correct one.
