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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?
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I am the questioner. After searching for two days, I've finally found the answer.

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}

The rest is just the standard exercise of solving the Hamiltonian equations.

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