$\underline{\text{Household Sector:}}$
The household comprises of a single agent whose objective function is to maximize the expected value of his lifetime utility which is a function of consumption of domestically produced goods and imported goods, real money balances, and leisure: $$\max_{C_t,\frac{M_t}{P_t},H_t,D_{t+1},B_{t+1}}\dfrac1{1-\rho}E_t\sum_{t=0}^\infty \beta^i\left[ a(C_{t+i}^{1-\rho}+b\left( \frac{M_{t+i}}{P_{t+i}} \right)^{1-\rho} +(1-a-b)\ln(1-H_{t+i})^{1-\rho} \right] $$ Subject to the budget constraint: $$B_{t+1}+C_t+\frac{M_t}{P_t}+D_{t+1}=\left(\frac{W_t}{P_t}\right)H_t-T_t+\frac{M_{t-1}}{P_t}+(1+i_t^D)D_t+\Pi_t+(1+i^B_t)B_t$$
Hi I'm trying to solve an economic problem but I haven' learned dynamic programming before. Need to derive the first order conditions. I'd really appreciate if you could lend me your brain for a sec. Many thanks in advance!