# Question on form of utility

**URL:** <https://forum.dynare.org/t/question-on-form-of-utility/20879>\
**Category:** General DSGE Modeling\
**Created:** [12 August 2022 02:27 UTC](https://forum.dynare.org/t/question-on-form-of-utility/20879 "2022-08-12T02:27:54Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![jamess](https://forum.dynare.org/letter_avatar_proxy/v4/letter/j/87869e/32.png) [@jamess](https://forum.dynare.org/u/jamess)\
**Post date:** [12 August 2022 02:27 UTC](https://forum.dynare.org/t/question-on-form-of-utility/20879/1 "2022-08-12T02:27:54Z")

</div>

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?

---

<div class="post-metadata">

**Author:** ![jpfeifer](https://forum.dynare.org/user_avatar/forum.dynare.org/jpfeifer/32/5044_2.png) [@jpfeifer](https://forum.dynare.org/u/jpfeifer)\
**Post date:** [12 August 2022 07:10 UTC](https://forum.dynare.org/t/question-on-form-of-utility/20879/2 "2022-08-12T07:10:38Z")

</div>

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.