How to Apply the Bellman Equation with Two Control Variables in an Optimal Growth Model

Question

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!

Answer 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, $$ V'(k_t) = \beta V'(k_{t+1})\alpha k_{t}^{\alpha - 1}. $$