<p>I am the questioner. After searching for two days, I've finally found the answer.</p>
<p>The trick is to define a new control variable <span class="math-container">$ x_t = \dot{k_t} $</span>. With this we can transform the original constraint into <em>two</em> new constraints:</p>
<p><span class="math-container">\begin{align}
\dot{k_t} &amp;= x_t \\
\dot{b_t} &amp;= w_t l_t + (R_t - \delta) k_t + r_t b_t - c_t - x_t - \tau_t
\end{align}</span></p>
<p>There are now <span class="math-container">$3$</span> control variables <span class="math-container">$c_t, l_t, x_t$</span>, <span class="math-container">$2$</span> state variables <span class="math-container">$k_t, b_t$</span> and <span class="math-container">$2$</span> constraints, so we should define two costate variables <span class="math-container">$\lambda_t, \mu_t$</span>. The Hamiltonian is</p>
<p><span class="math-container">\begin{align}
\mathcal{H} = &amp; e^{-\rho t} u(c_t, 1-l_t) + \lambda_t x_t + \\
&amp; \mu_t [w_t l_t + (R_t - \delta) k_t + r_t b_t - c_t - x_t - \tau_t]
\end{align}</span></p>
<p>The rest is just the standard exercise of solving the Hamiltonian equations.</p>

CC BY-SA 4.0
2021-12-01T09:38:27.587Z
2021-12-01T09:38:27.587Z
user141240
<p>Consider a representative household who accumulates capitals, earns labour and capital incomes, consumes part of its incomes, buys bonds and pays taxes.</p>
<p>The household maximizes its lifetime utility</p>
<p><span class="math-container">$$ \max_{c_t, l_t, k_t, b_t} \int_0^\infty e^{-\rho t} u(c_t, 1-l_t) dt $$</span></p>
<p>subjected to the budget constraint:</p>
<p><span class="math-container">$$ \dot{k_t} + \dot{b_t} = w_t l_t + (R_t - \delta) k_t + r_t b_t - c_t - \tau_t $$</span></p>
<p>Endogenuous variables:</p>
<ol>
<li><span class="math-container">$ c_t = $</span> consumptions</li>
<li><span class="math-container">$ l_t = $</span> labours</li>
<li><span class="math-container">$ k_t = $</span> capitals</li>
<li><span class="math-container">$ b_t = $</span> bonds</li>
</ol>
<p>Exogenuous variables:</p>
<ol>
<li><span class="math-container">$ \tau_t = $</span> taxes</li>
<li><span class="math-container">$ w_t = $</span> labour wage</li>
<li><span class="math-container">$ R_t = $</span> capital rent</li>
<li><span class="math-container">$ r_t = $</span> interest rate</li>
<li><span class="math-container">$ \rho = $</span> discount rate</li>
<li><span class="math-container">$ \delta = $</span> depreciation rate</li>
</ol>
<p>I want to solve this problem using the Hamiltonian method. The problem is, there are two state variables but only one costate variable.</p>
<p>Questions:</p>
<ol>
<li>Did I write the budget constraint correctly?</li>
<li>How to set up the Hamiltonian for this problem?</li>
<li>How to derive <span class="math-container">$ R_t = r_t - \delta $</span> in this problem?</li>
</ol>

<macroeconomics><mathematical-economics>
How to set up the Hamiltonian for this utility maximization problem?
user141240
# How to set up the Hamiltonian for this utility maximization problem?

(Asked by: username_0 on 2021-11-29)

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?

**Tags:** macroeconomics, mathematical-economics


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## 1 answers

### Answer by username_0 on 2021-12-01. Score: 2 | Accepted answer

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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