Problem with production function

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.

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.