pdotaprox: Compute Pdot-Approximation Coefficients

Description

Computes Pdot-approximation coefficients for the Chebyshev polynomial representation of the shadow price derivative using the approximation space defined by aproxdef. Currently, only the one-dimensional case is supported.

Usage

pdotaprox(aproxspace, stock, sdot, dsdotds, dsdotdss, dwds, dwdss)

Value

A list containing the approximation results. Individual elements can be accessed using results$item or results[["item"]]. The list includes:

degree: Degree of the Chebyshev polynomial.
lowerB: Lower bound of the Chebyshev domain.
upperB: Upper bound of the Chebyshev domain.
delta: Discount rate.
coefficient: Estimated Chebyshev polynomial coefficients.

Arguments

aproxspace: An approximation space defined by the aproxdef function.
stock: A vector of stock values, $s$.
sdot: A vector of stock growth rates, $\dot{s} = \frac{ds}{dt}$.
dsdotds: A vector of derivatives of stock growth with respect to stock, $\frac{d \dot{s}}{d s}$.
dsdotdss: A vector of second derivatives of stock growth with respect to stock, $ \frac{d}{ds} \left( \frac{d \dot{s}}{ds} \right)$.
dwds: A vector of marginal welfare with respect to stock, $\frac{dW}{ds}$.
dwdss: A vector of second derivatives of marginal welfare with respect to stock, $\frac{d}{ds} \left( \frac{dW}{ds} \right)$.

Details

The Pdot-approximation solves for the shadow price of a stock, $p(s)$, using the relationship:

$p(s) = \frac{W_{s}(s) + \dot{p}(s)}{\delta - \dot{s}_{s}}$,

where $W_{s} = \frac{dW}{ds}$, $ \dot{p}(s) = \frac{dp}{dt}$, $\dot{s}_{s} = \frac{d\dot{s}}{ds}$, and $\delta$ is the discount rate.

Taking the time derivative of this expression yields:

$ \dot{p} = \frac{ \left( W_{ss} \dot{s} + \ddot{p} \right) \left( \delta - \dot{s}_{s} \right) + \left( W_{s} + \dot{p} \right) \left( \dot{s}_{ss} \dot{s} \right) }{ \left( \delta - \dot{s}_{s} \right)^2 } $.

Let the approximation be $ \dot{p}(s) = \mathbf{\Phi}(s)\mathbf{\beta} $, where $\mathbf{\Phi}(s)$ is a vector of Chebyshev basis functions and $\mathbf{\beta}$ is the coefficient vector. Then,

$ \ddot{p} = \frac{d \dot{p}}{ds} \frac{ds}{dt} = \mathrm{diag}(\dot{s}) \mathbf{\Phi}_{s}(s) \mathbf{\beta} $.

Substituting and rearranging yields the linear system:

$ \mathbf{\beta} = A^{-1} B, $
where

$ A = \mathrm{diag} \left( \delta - \dot{s}_{s} \right)^2 \mathbf{\mu} - \mathrm{diag} \left( \dot{s} (\delta - \dot{s}_{s}) \right) \mathbf{\mu}_{s} - \mathrm{diag} \left( \dot{s}_{ss} \dot{s} \right) \mathbf{\mu}, $

and

$ B = W_{ss} \dot{s} (\delta - \dot{s}_{s}) + W_{s} \dot{s}_{ss} \dot{s}. $

In the over-determined case, coefficients are obtained via least squares:

$ \mathbf{\beta} = \left( A^{T} A \right)^{-1} A^{T} B. $

For additional theoretical background, see Fenichel and Abbott (2014).

References

Fenichel, Eli P. and Joshua K. Abbott. (2014). Natural Capital: From Metaphor to Measurement. Journal of the Association of Environmental Economists, 1(1/2), 1--27. tools:::Rd_expr_doi("10.1086/676034")

Examples

Run this code

## 1-D Reef-fish example: see Fenichel and Abbott (2014)
data("GOM")

param <- GOM$param
simData <- GOM$simData

Aspace <- aproxdef(param$order, param$lowerK, param$upperK, param$delta)

pdotC <- pdotaprox(
  Aspace,
  simData$stock,
  simData$sdot,
  simData$dsdotds,
  simData$dsdotdss,
  simData$dwds,
  simData$dwdss
)

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