To derive the Euler equation for this optimization problem, we proceed as follows:

Problem Overview

We are solving the planner's optimization problem:

$\max \int e^{-(p-n)t} \frac{C(t)^{1-\theta} - 1}{1 - \theta} \, dt$ subject to:

1. $\dot{x}(t) = A x(t) - (\delta + n)x(t) - C(t)$,
2. $x(0) = x_0 > 0$.

Here:

• $C(t)$ is consumption,
• $x(t)$ is capital,
• $p$, $n$, $\theta$, $A$, and $\delta$ are constants.

Step 1: Hamiltonian

The current-value Hamiltonian is: $H = \frac{C(t)^{1-\theta} - 1}{1 - \theta} + \mu(t) \left[ A x(t) - (\delta + n)x(t) - C(t) \right],$ where $\mu(t)$ is the costate variable (shadow price of capital).

Step 2: Necessary Conditions

The first-order conditions (FOCs) for optimality are:

1. $\frac{\partial H}{\partial C(t)} = 0$: $C(t)^{-\theta} = \mu(t).$

2. $\frac{\partial H}{\partial x(t)} = \dot{\mu}(t) - (p-n)\mu(t)$: $\dot{\mu}(t) = \mu(t) \left[ p - n - A + (\delta + n) \right].$

3. The state equation: $\dot{x}(t) = A x(t) - (\delta + n)x(t) - C(t).$

Step 3: Eliminate $\mu(t)$

From the first FOC, $\mu(t) = C(t)^{-\theta}$. Substituting this into the costate equation gives: [ \frac{d}{dt} \left[ C(t)^{-\theta} \right] = C(t)^{-\theta} \left[ p - n - A + (\delta + n) \right]. ]

Using the chain rule: $-\theta C(t)^{-\theta-1} \dot{C}(t) = C(t)^{-\theta} \left[ p - n - A + (\delta + n) \right].$

Simplify to obtain the Euler equation: $\frac{\dot{C}(t)}{C(t)} = \frac{1}{\theta} \left[ A - \delta - p \right].$

Step 4: Interpretation

The Euler equation relates the growth rate of consumption $\frac{\dot{C}(t)}{C(t)}$ to parameters of the economy, such as productivity ($A$), depreciation ($\delta$), and the discount rate ($p$).

Let me know if you need further clarifications or step-by-step verification.

Questions to Expand Understanding:

1. How does the discount rate ($p$) affect the consumption growth rate?
2. What happens to the Euler equation if $\theta = 1$ (log utility)?
3. How do changes in population growth ($n$) affect the solution?
4. Can this problem be reformulated using discrete time?
5. What is the role of the transversality condition in solving this problem?

Tip:

When solving optimization problems in continuous time, always verify the consistency of the boundary conditions (e.g., $x(0)$ and transversality conditions).