Introduction
This problem describes an optimal control problem in economics, specifically dealing with intertemporal consumption decisions made by an investor. The goal is to choose a consumption function c(t) over time interval [0, T] that maximizes utility from consumption while ensuring the investor remains solvent over time, meaning that capital remains positive. Let’s break it down and provide a structured explanation and solution strategy.
Restating the Problem
We are given:
- Initial capital: x(0) = x0 > 0
- Capital evolution: ẋ = αx – c(t)
- Utility function: u(c(t)) = ln(βt) (but since this is unconventional, we assume it’s a typo and should be ln(c(t)))
- Objective: Maximize discounted utility:
J(c) = ∫0T e−rtu(c(t)) dt - Constraint: x(t) > 0 for all t in [0, T)
Step-by-Step Analysis
1. Setting Up the Optimal Control Problem
We define the Hamiltonian for this problem as:
H = e−rt ln(c(t)) + λ(t)(αx(t) – c(t))
Here, λ(t) is the costate variable associated with the capital x(t).
2. First-Order Conditions
- ∂H/∂c = e−rt * (1/c) − λ = 0 → λ(t) = e−rt/c(t)
- ∂H/∂x = λ̇ = −λ(t) * α → λ̇ = −αλ
3. Solving the Costate Equation
λ̇ = −αλ → dλ/λ = −α dt
Integrating both sides:
ln λ = −αt + C → λ = Ce−αt
4. Solving for c(t)
From above, λ = e−rt/c(t) and λ = Ce−αt
So, equating both:
e−rt/c(t) = Ce−αt → c(t) = (1/C) e(α−r)t
Let’s define A = 1/C, then:
c(t) = A e(α−r)t
5. Solving the State Equation
The capital evolution equation:
ẋ = αx − c(t) = αx − A e(α−r)t
This is a linear non-homogeneous differential equation. It can be solved using an integrating factor.
6. Boundary Conditions
We are given:
- x(0) = x0
- x(T) = 0 (terminal capital becomes zero)
These conditions help determine the constant A (or C).
Economic Interpretation
- The optimal consumption path is exponential in time depending on the difference between α and r.
- If α > r, then consumption grows over time.
- If α < r, then consumption declines over time, reflecting higher preference for present consumption.
- The constraint x(t) > 0 ensures that the investor doesn’t borrow beyond means.
Role of Discounting and Concavity
- The concave utility function (log) captures diminishing marginal utility of consumption.
- The discount factor e−rt reflects time preference — the further in the future, the less valuable utility becomes today.
Conclusion
This problem demonstrates the use of optimal control theory in economics to determine the best consumption plan for an investor. Using the Pontryagin Maximum Principle, we derived the optimal consumption rule as a function of time. The solution depends on the return on capital (α), the discount rate (r), and initial capital. Such models are useful in finance, economic planning, and savings behavior analysis.