Problem with production function

General DSGE Modeling
1

Dear Professor Pfeifer,

When I want to add expenditure of goverment into production function, I see two types of functions like: {{Y}_{t}}={{\left( {{K}_{t-1}} \right)}^{\alpha }}{{\left( {{L}_{t}} \right)}^{1-\alpha }}{{\left( {{G}_{t}} \right)}^{\mu }} and {{Y}_{t}}={{\left( {{K}_{t-1}} \right)}^{\alpha }}{{\left( {{L}_{t}} \right)}^{1-\alpha-\mu }}{{\left( {{G}_{t}} \right)}^{\mu }}. I can not decide which one should be used. What is the difference of economic meaning between them ?

Any reply will be appreciated. Thank you for your time.

2

The first one has constant returns in the private factors, while the second one has decreasing returns to the private factors. The first one is a lot more common.