1. Consider an investor who at time \( t=0 \) is endowed with initial capital of \( x(0)=x_{0}>0 \). At an time \( t \in[0, T] \) where \( T>0 \) is given he decides about his rate of consumption \( c(t) \in[0, \bar{c}] \) wher \( \bar{c}>0 \) is a large maximum allowable rate of consumption thus his capital stock evolves according to \( \dot{x}=\alpha x-c(t) \) where \( \alpha>0 \) is a given rate of return the investor's time-t utility for consuming at rate \( \mathrm{c}(\mathrm{t}) \) is \( \mathrm{u}(\mathrm{c}(\mathrm{t})) \) where \( \mathrm{u}=\mathrm{P}+\rightarrow \mathrm{P} \) is his increasing strictly concave utility function. The investor's problem is to find a consumption plan \( \mathrm{c}(\mathrm{t}), \mathrm{t} \in[0, \mathrm{~T}] \) so as to maximize his discounted utility. \[ (c)=\int_{0}^{T} e^{-r t} u(c(t)) d t \] \[ u(c(t))=\ln (\beta t) \] \[ x(0)=x_{0} \] where \( \mathrm{r} \geq 0 \) is a given discount rate subject to the solvency constraint that the capital stock \( \mathrm{x}(\mathrm{t}) \) must be positive for all \( \mathrm{t} \in[0, \mathrm{~T}) \). [The value \( \beta \) lies in \( [0, \infty) \).]

To solve the investor's problem, we need to maximize the discounted utility function given the constraints on capital stock and consumption. Let's break down the problem step by step. ### Step 1: Understand the Problem 1. **Initial Capital**: The investor starts with an initial capital \( x(0) = x_0 > 0 \). 2. **Consumption Rate**: The consumption rate \( c(t) \) is bounded by \( 0 \leq c(t) \leq \bar{c} \). 3. **Capital Dynamics**: The capital stock evolves according to the differential equation: \[ \dot{x} = \alpha x - c(t) \] 4. **Utility Function**: The utility from consumption is given by: \[ u(c(t)) = \ln(\beta c(t)) \] 5. **Discounted Utility**: The objective is to maximize the following integral: \[ J = \int_{0}^{T} e^{-rt} u(c(t)) dt = \int_{0}^{T} e^{-rt} \ln(\beta c(t)) dt \] 6. **Solvency Constraint**: The capital stock must remain positive for all \( t \in [0, T) \). ### Step 2: Set Up the Hamiltonian To solve this optimization problem, we can use the calculus of variations or optimal control theory. We define the Hamiltonian \( H \) as follows: \[ H = e^{-rt} \ln(\beta c(t)) + \lambda(t)(\alpha x(t) - c(t)) \] where \( \lambda(t) \) is the costate variable associated with the capital stock. ### Step 3: Derive the Necessary Conditions 1. **First-Order Condition for Consumption**: To find the optimal consumption \( c(t) \), we take the derivative of the Hamiltonian with respect to \( c(t) \) and set it to zero: \[ \frac{\partial H}{\partial c} = \frac{e^{-rt}}{c(t)} - \lambda(t) = 0 \] This gives us: \[ \lambda(t) = \frac{e^{-rt}}{c(t)} \] 2. **Costate Equation**: The costate variable evolves according to: \[ \dot{\lambda}(t) = r\lambda(t) - \frac{\partial H}{\partial x} = -\lambda(t) \alpha \] This simplifies to: \[ \dot{\lambda}(t) = -\alpha \lambda(t) \] 3. **Capital Dynamics**: The capital dynamics are given by: \[ \dot{x} = \alpha x - c(t) \] ### Step 4: Solve the System of Equations We have a system of equations to solve: 1. \(\dot{x} = \alpha x - c(t)\) 2. \(\dot{\lambda} = -\alpha \lambda\) 3. The first-order condition for \( c(t) \). Let's solve the costate equation first: \[ \lambda(t) = \lambda(0)e^{-\alpha t} \] Now substituting \( \lambda(t) \) back into the first-order condition: \[ \frac{e^{-rt}}{c(t)} = \lambda(0)e^{-\alpha t} \] This leads to: \[ c(t) = \frac{e^{-rt}}{\lambda(0)e^{-\alpha t}} = \frac{e^{(\alpha - r)t}}{\lambda(0)} \] ### Step 5: Determine the Optimal Consumption Plan The optimal consumption plan is: \[ c(t) = \frac{e^{(\alpha - r)t}}{\lambda(0)} \] This consumption must satisfy the constraint \( 0 \leq c(t) \leq \bar{c} \). ### Step 6: Ensure Solvency To ensure that the capital stock remains positive, we need to analyze the dynamics: \[ \dot{x} = \alpha x - c(t) \] Substituting \( c(t) \) into this equation, we can analyze the conditions under which \( x(t) > 0 \). ### Conclusion The optimal consumption plan is given by: \[ c(t) = \frac{e^{(\alpha - r)t}}{\lambda(0)} \] subject to the constraints \( 0 \leq c(t) \leq \bar{c} \) and ensuring that \( x(t) > 0 \) for all \( t \in [0, T) \). The specific value of \( \lambda(0) \) can be determined based on the initial conditions and the requirement that the consumption does not exceed the maximum allowable rate.