<p>You have the following problem:
<span class="math-container">$$
\max_{c_t, \ell_t} \sum_{t = 0}^\infty \beta^t[\ln(c_t) + \gamma \ln(\ell_t)] \text{ s.t. } \begin{cases} y_t = k_t^\alpha(1- \ell_t)^{1- \alpha}\\
k_{t+1} = y_t - c_t\end{cases}
$$</span></p>
<p>The Bellman equation is:
<span class="math-container">$$
\begin{align*}
V(k_t) &amp;= \max_{c_t, \ell_t} \left(\ln(c_t) + \gamma \ln(\ell_t) + \beta V(y_t - c_t)\right),\\
&amp;=\max_{c_t, \ell_t} \left(\ln(c_t) + \gamma \ln(\ell_t) + \beta V(k_t^\alpha (1- \ell_t)^{1- \alpha} - c_t)\right).
\end{align*}
$$</span></p>
<p>The first order conditions give:
<span class="math-container">$$
\begin{align*}
&amp;\frac{1}{c_t} = \beta V'(k_{t+1}),\\
&amp;\frac{\gamma}{\ell_t} = \beta V'(k_{t+1})(1 - \alpha)k_t^\alpha(1- \ell_t)^{-\alpha}
\end{align*}
$$</span>
The envelope theorem gives,
<span class="math-container">$$
V'(k_t) = \beta V'(k_{t+1})\alpha k_{t}^{\alpha - 1}.
$$</span></p>

CC BY-SA 4.0
2024-11-08T08:13:41.110Z
2024-11-08T08:13:41.110Z
tdm
<p>I'm currently studying an optimal growth problem involving a representative consumer, and I'm having trouble using the Bellman equation when there are two control variables involved.
Specifically, I want to avoid using the Lagrangian method because it requires proving the existence of multipliers, which can be complex.</p>
<p>The problem is as follows: The representative consumer lives infinitely and has preferences described by the utility function <span class="math-container">$\sum_{t=0}^{\infty} \beta^t\left[\ln c_t+\gamma \ln l_t\right]$</span>, where <span class="math-container">$0&lt;\beta&lt;1$</span> is the discount factor, <span class="math-container">$c_t \geq 0$</span> is consumption, <span class="math-container">$l_t \in[0,1]$</span> is leisure, and <span class="math-container">$\gamma&gt;0$</span> is a parameter that weighs the importance of leisure. Each period, the consumer has an endowment of one unit of time, which can be allocated between work and leisure, so labor is <span class="math-container">$n_t=1-l_t$</span>.</p>
<p>The production technology is given by <span class="math-container">$y_t=k_t^\alpha n_t^{1-\alpha}$</span>, where <span class="math-container">$y_t$</span> is output, <span class="math-container">$k_t$</span> is the capital input, <span class="math-container">$n_t$</span> is the labor input, and <span class="math-container">$0&lt;\alpha&lt;1$</span> represents the output elasticity of capital. Capital depreciates completely each period, meaning there is <span class="math-container">$100 \%$</span> depreciation, so the capital accumulation equation simplifies to <span class="math-container">$k_{t+1}=y_t-c_t$</span>.</p>
<p>I need to formulate the optimal growth problem in both primitive and reduced forms, identifying the state space, the transition equation, and the utility function in reduced form. Additionally, I aim to characterize the optimal paths of consumption, leisure, and capital. I also want to determine the number of steady states, analyze their local stability, and understand how the parameter <span class="math-container">$\gamma$</span> affects the steady-state levels of consumption, leisure, and capital.</p>
<p>My main question is: How can I set up and solve this problem using the Bellman equation when there are two control variables, <span class="math-container">$c_t$</span> and <span class="math-container">$l_t$</span> ? I'm unsure how to define the value function and formulate the Bellman equation in this context. I'm also seeking guidance on the steps to follow to derive the optimality conditions using dynamic programming. Are there any standard techniques or transformations that simplify handling multiple control variables within the Bellman framework?</p>
<p>Any detailed advice or references to similar problems that have been solved using dynamic programming with multiple control variables would be greatly appreciated. Thank you for your assistance!</p>

<macroeconomics><optimal-control><lagrangian><bellman-equations>
How to Apply the Bellman Equation with Two Control Variables in an Optimal Growth Model
bruno
# How to Apply the Bellman Equation with Two Control Variables in an Optimal Growth Model

(Asked by: username_0 on 2024-11-03)

I'm currently studying an optimal growth problem involving a representative consumer, and I'm having trouble using the Bellman equation when there are two control variables involved.
Specifically, I want to avoid using the Lagrangian method because it requires proving the existence of multipliers, which can be complex.

The problem is as follows: The representative consumer lives infinitely and has preferences described by the utility function $\sum\_{t=0}^{\infty} \beta^t\left[\ln c\_t+\gamma \ln l\_t\right]$, where $0<\beta<1$ is the discount factor, $c\_t \geq 0$ is consumption, $l\_t \in[0,1]$ is leisure, and $\gamma>0$ is a parameter that weighs the importance of leisure. Each period, the consumer has an endowment of one unit of time, which can be allocated between work and leisure, so labor is $n\_t=1-l\_t$.

The production technology is given by $y\_t=k\_t^\alpha n\_t^{1-\alpha}$, where $y\_t$ is output, $k\_t$ is the capital input, $n\_t$ is the labor input, and $0<\alpha<1$ represents the output elasticity of capital. Capital depreciates completely each period, meaning there is $100 \%$ depreciation, so the capital accumulation equation simplifies to $k\_{t+1}=y\_t-c\_t$.

I need to formulate the optimal growth problem in both primitive and reduced forms, identifying the state space, the transition equation, and the utility function in reduced form. Additionally, I aim to characterize the optimal paths of consumption, leisure, and capital. I also want to determine the number of steady states, analyze their local stability, and understand how the parameter $\gamma$ affects the steady-state levels of consumption, leisure, and capital.

My main question is: How can I set up and solve this problem using the Bellman equation when there are two control variables, $c\_t$ and $l\_t$ ? I'm unsure how to define the value function and formulate the Bellman equation in this context. I'm also seeking guidance on the steps to follow to derive the optimality conditions using dynamic programming. Are there any standard techniques or transformations that simplify handling multiple control variables within the Bellman framework?

Any detailed advice or references to similar problems that have been solved using dynamic programming with multiple control variables would be greatly appreciated. Thank you for your assistance!

**Tags:** macroeconomics, optimal-control, lagrangian, bellman-equations

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## 1 answers

### Answer by username_1 on 2024-11-08. Score: 1

You have the following problem:
$$
\max\_{c\_t, \ell\_t} \sum\_{t = 0}^\infty \beta^t[\ln(c\_t) + \gamma \ln(\ell\_t)] \text{ s.t. } \begin{cases} y\_t = k\_t^\alpha(1- \ell\_t)^{1- \alpha}\\
k\_{t+1} = y\_t - c\_t\end{cases}
$$

The Bellman equation is:
$$
\begin{align\*}
V(k\_t) &= \max\_{c\_t, \ell\_t} \left(\ln(c\_t) + \gamma \ln(\ell\_t) + \beta V(y\_t - c\_t)\right),\\
&=\max\_{c\_t, \ell\_t} \left(\ln(c\_t) + \gamma \ln(\ell\_t) + \beta V(k\_t^\alpha (1- \ell\_t)^{1- \alpha} - c\_t)\right).
\end{align\*}
$$

The first order conditions give:
$$
\begin{align\*}
&\frac{1}{c\_t} = \beta V'(k\_{t+1}),\\
&\frac{\gamma}{\ell\_t} = \beta V'(k\_{t+1})(1 - \alpha)k\_t^\alpha(1- \ell\_t)^{-\alpha}
\end{align\*}
$$
The envelope theorem gives,
$$
V'(k\_t) = \beta V'(k\_{t+1})\alpha k\_{t}^{\alpha - 1}.
$$


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