# Utility function of habit formation model

In the habit formation RBC model, the utility can be given as follows:

$U(c_{t},n_{t}) = \ln(c_{t}) - \frac{\theta}{1+\upsilon} \bigg(\frac{n_{t}}{n_{t-1}^{\phi}}\bigg)^{1+\upsilon}$

When I do FOC wrt $n_t$, I'm confused about how to treat the variable $n_{t-1}$. Is it exogenous or should I try to find $\frac{\partial n_{t-1}}{\partial n_t}$?

Many thanks.

• Could you tell us a bit more about your model? Is this infinite horizon or are there finitely many periods? What are yo trying to maxmimize with the FOC? It looks like, if you were maximizing the lifetime utility, you would have to set this up as a dynamic programming problem rather than optimising a single period utility. Dec 5, 2017 at 11:30

I believe $n_{t-1}$ is a quantity from the past, so you should be able to treat it as an exogenous variable
$n_{t-1}$ is exogenous at time $t$. Since you must be working in a dynamic setting, $n_t$ will also show up in the utility function at time $t+1$.