Government Spending

Recall the initial model, which is the blue part.

To add the government part, we should add the green part, which means that the government got involved into the process.

Household

$$U=lnC+\beta ln(L-l)+v(G)$$

The aim for the household is to maximize their utility:

$$Ma{x}_{C,l}U=\sum _{t=0}\frac{u_t}{(1+\rho {)}^t}\text{ s,t }\dot{K}=I-\delta K=wl+RK-T-C-\delta K$$

Take the hamiltonian.