Why do we model that higher government involvement increases output? I have just read about the "real intertemporal" model of a macroeconomy in Macroeconomics by Stephen Williamson. I am concerned about how it says any increase in government spending, even if it is equal to an increase in taxes, increases the output of the economy. This is because taxes don't reduce private consumption spending as much as government spending increases demand. But, if I extend this idea to the extreme, where government involvement is exceedingly high, I don't understand why people will carry on spending more than they earn, with high taxes and no way to pay off their debt. It seems like the autonomous component of private spending is problematic when thinking about tax (as opposed to an equilibrium), because this theoretically causes people to overspend in comparison to their income. 

The deduction is that, with any positive multiplier (not just one greater than 1 as mentioned in the textbook), the more the government intervenes in the economy, the higher the GDP will be, ad infinitum, so that this model encourages higher and higher government involvement. In reality, socialist countries such as Russia have had lower output than free-market countries such as the United States, and it is well known that government bureaucracies are more inefficient than the private sector. What I'm checking here is where the theory leads. I assume this must be a Keynesian-type model, because it encourages government involvement, and that other models give a different story (neoclassical perhaps?).

As a public administration student as well, I don't like it how government spending is treated so differently to private spending, and this separation here seems to be problematic (as assuming that government departments are efficient in comparison to the private sector).
The book is talking about the short term effect. And yes, such a policy would be thought as Keynesian.

In the long run, excessive government spending will cause inflation (if unfunded) or reduce both private production and demand (due to excessive taxation).