# How to set up the Hamiltonian for this utility maximization problem?

Consider a representative household who accumulates capitals, earns labour and capital incomes, consumes part of its incomes, buys bonds and pays taxes.

The household maximizes its lifetime utility

$$\max_{c_t, l_t, k_t, b_t} \int_0^\infty e^{-\rho t} u(c_t, 1-l_t) dt$$

subjected to the budget constraint:

$$\dot{k_t} + \dot{b_t} = w_t l_t + (R_t - \delta) k_t + r_t b_t - c_t - \tau_t$$

Endogenuous variables:

1. $$c_t =$$ consumptions
2. $$l_t =$$ labours
3. $$k_t =$$ capitals
4. $$b_t =$$ bonds

Exogenuous variables:

1. $$\tau_t =$$ taxes
2. $$w_t =$$ labour wage
3. $$R_t =$$ capital rent
4. $$r_t =$$ interest rate
5. $$\rho =$$ discount rate
6. $$\delta =$$ depreciation rate

I want to solve this problem using the Hamiltonian method. The problem is, there are two state variables but only one costate variable.

Questions:

1. Did I write the budget constraint correctly?
2. How to set up the Hamiltonian for this problem?
3. How to derive $$R_t = r_t - \delta$$ in this problem?

The trick is to define a new control variable $$x_t = \dot{k_t}$$. With this we can transform the original constraint into two new constraints:
\begin{align} \dot{k_t} &= x_t \\ \dot{b_t} &= w_t l_t + (R_t - \delta) k_t + r_t b_t - c_t - x_t - \tau_t \end{align}
There are now $$3$$ control variables $$c_t, l_t, x_t$$, $$2$$ state variables $$k_t, b_t$$ and $$2$$ constraints, so we should define two costate variables $$\lambda_t, \mu_t$$. The Hamiltonian is
\begin{align} \mathcal{H} = & e^{-\rho t} u(c_t, 1-l_t) + \lambda_t x_t + \\ & \mu_t [w_t l_t + (R_t - \delta) k_t + r_t b_t - c_t - x_t - \tau_t] \end{align}