In One-period general equilibrium model, what if the level of the public good is endogenous variable that is chosen by the government.

We know consumer's preference and budget, firm's production function and the government is imposing a labour income tax on household.

The normal case is when the public good G is and exogenous variable, but my question is what if it is an endogenous variable?

1. Is it realistic to assume the public good is an endogenous variable chosen by the government?

2. What kind of information the government need to implement the optimal policy?

SOS, please be quick