Models with Expenses

\[[{_tV^g}+G_t(1-\%G_t)-e_t](1+i)=(b_{t+1}+E_{t+1})q_{x+t}+{_{t+1}V^g}\ p_{x+t}\]

Thiele’s Differential Equation

\[\frac{d}{dt}{_tV^g}=G_t(1-\%G_t)-e_t+(\delta_t+\mu_{x+t}){_tV^g}-(b_t+E_t)\mu_{x+t}\]