vaprox.split: Value Function Approximation with Inaction (Split) Management

Description

Computes Chebyshev polynomial coefficients for approximating the value function $V(s)$ in a one-dimensional natural capital asset pricing model with inaction (split) management.

The function allows for a boundary stock level at which management switches from waiting (inaction) to active use, and estimates both the value function and the associated passive value parameter.

Usage

vaprox.split(aproxspace, stock, sdot, profit,
             crit.stock, split.margp, split.time = NULL)

Value

A list containing the approximation results:

degree: Degree of the Chebyshev polynomial.
lowerB: Lower bound of the approximation interval.
upperB: Upper bound of the approximation interval.
delta: Discount rate.
coefficient: Estimated Chebyshev polynomial coefficients.
shadow.stop: Shadow price evaluated at the boundary stock $\bar{s}$.
alpha: Estimated passive value parameter at $\bar{s}$.
model.type: "Split (Economic Program)".

Arguments

aproxspace: An approximation space defined by the aproxdef function.
stock: A vector or matrix of stock states $s$.
sdot: A vector or matrix of stock dynamics $\dot{s} = ds/dt$ evaluated at stock (without split management).
profit: A vector of net benefits (profit or economic program value) evaluated at stock (without split management).
crit.stock: Critical (boundary) stock level, $\bar{s}$, representing a moratorium or maximum waiting threshold.
split.margp: Marginal use value at the boundary stock, $\left.\frac{d\pi}{ds}\right|_{s=\bar{s}}$.
split.time: (Optional) Time of action $t(\bar{s})$ at the boundary stock. If NULL, the standard boundary condition is used.

Details

Following Hashida and Fenichel (2022), management is characterized by an indicator function:

$$ x(s) = \begin{cases} 1 & \text{if } s \ge \bar{s}, \\ 0 & \text{if } s < \bar{s}. \end{cases} $$

The benefit function can be decomposed as:

$$ W(s, x(s)) = \pi(s, x(s)) + \alpha(s, x(s)), $$

where $\pi(s, x(s))$ represents realized income when harvesting occurs ($x(s)=1$), and $\alpha(s, x(s))$ represents passive (amenity) value when management remains inactive ($x(s)=0$).

At the boundary $s=\bar{s}$, smooth pasting implies:

$$ V_{s}(\bar{s}) = \pi_{s}(\bar{s}). $$

The Hamilton--Jacobi--Bellman equation becomes:

$$ \delta V(s) = \pi(s) + \alpha(s) + \dot{s}\, V_{s}(s). $$

The value function is approximated by Chebyshev polynomials:

$$ V(s) = \Phi(s)\boldsymbol{\beta}, $$

where $\Phi(s)$ is the Chebyshev basis evaluated at $s$, and $\boldsymbol{\beta}$ is the vector of unknown coefficients.

Substituting into the HJB equation yields the linear system:

$$ \left[ \delta \Phi(s) - \mathrm{diag}(\dot{s}) \Phi_{s}(s) \right] \boldsymbol{\beta} = \pi(s) + \alpha(s). $$

The passive value is approximated by a first-order form $\alpha(s) = \alpha s$, and $\boldsymbol{\beta}$ and $\alpha$ are jointly determined using the boundary condition $V_{s}(\bar{s}) = \pi_{s}(\bar{s})$.

In exactly identified cases, the system is solved directly. In over-determined cases, a least-squares solution is used:

$$ \boldsymbol{\beta} = (A^{\top}A)^{-1} A^{\top} \left( \pi(s) + \alpha(s) \right), $$

where $A = \delta \Phi(s) - \mathrm{diag}(\dot{s}) \Phi_{s}(s)$.

A complete application is provided in the forestDemo vignette. See vignette("forestDemo").

References