Question on form of utility

Dear Professor Pfeifer,

I want to consider heterogenous housing in the DSGE model. I firstly set household’s utility function as {{E}_{0}}\sum\limits_{t=0}^{\infty }{{{\beta }^{t}}\left( \log {{c}_{t}}+j\log {{h}_{t}}-\frac{1}{1+\gamma }{{n}^{1+\gamma }} \right)}, in which {{h}_{t}}={{\left[ {{\omega }^{1-\eta }}{{\left( h_{t}^{H} \right)}^{\eta }}+{{\left( 1-\omega \right)}^{1-\eta }}{{\left( h_{t}^{R} \right)}^{\eta }} \right]}^{\frac{1}{\eta }}}. But then I found that it could hardly solve the steady state. So I change the utility function as {{E}_{0}}\sum\limits_{t=0}^{\infty }{{{\beta }^{t}}\left( \log {{c}_{t}}+{{j}^{H}}\log h_{t}^{H}+{{j}^{R}}\log h_{t}^{R}-\frac{1}{1+\gamma }{{n}^{1+\gamma }} \right)}. Is it proper to set the utility function like this? Is there any difference between the two types of utility function?

Obviously, there is a difference. The first specification has a parameterized elasticity of substitution via \eta. In the second case, you essentially set the elasticity of substitution to 1, i.e. the Cobb-Douglas case.