Is it necessary to introduce population growth and economic growth in DSGE models with human capital?

There is not a general answer. It depends on what you are interested in.

My main interest is measuring the level of human capital in an economy in different scenarios. Typically, I have seen that human capital models divide each variable by the effective population so that a human capital LOM is originally as follows:

H_{t+1} = (1 - \delta) H_{t} + (e_{t} H_{t})^\psi \bar{H_{t}}^{1 - \psi}

becomes

n \gamma_{t} = 1 - \delta + e_{t}^{\psi}

where n is the rate of population growth and \gamma_{t} is the economic growth rate. But instead, I would like to work with the level equation without assuming population growth.

There are usually two ways to proceed:

- Write down the non-stationary model and do an explicit detrending to get a stationary model.
- Directly write down a stationary version of the model, abstracting from growth.

You want option 2. That’s typically fine (unless you are interested in substitution effects related to trend growth).