pdotaprox {capn}

Calculating Pdot-approximation coefficients

Description

The function provides the Pdot-approximation coefficients of the defined Chebyshev polynomials in aproxdef. For now, only unidimensional case is developed.

Usage

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

Arguments

aproxspace

An approximation space defined by aproxdef function

stock

An array of stock, s

sdot

An array of ds/dt, \dot{s}=\frac{ds}{dt}

dsdotds

An array of d(sdot)/ds, \frac{d \dot{s}}{d s}

dsdotdss

An array of d/ds(d(sdot)/ds), \frac{d}{ds} \left( \frac{d \dot{s}}{ds} \right)

dwds

An array of dw/ds, \frac{dW}{ds}

dwdss

An array of d/ds(dw/ds), \frac{d}{ds} \left( \frac{dW}{ds} \right)

Details

The Pdot-approximation is finding the shadow price of a stock, p from the relation:

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

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

In order to operationalize this approach, we take the time derivative of this expression:

\dot{p} = \frac{ \left( \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) \right) } { \left( \delta - \dot{s}_{s} \right)^{2} }

Consider approximation \dot{p}(s) = \mathbf{\mu}(s)\mathbf{\beta}, \mathbf{\mu}(s) is Chebyshev polynomials and \mathbf{\beta} is their coeffcients. Then, \ddot{p} = \frac{ d \dot{p}}{ds} \frac{ds}{dt} = diag (\dot{s}) \mathbf{\mu}_{s}(s) \mathbf{\beta} by the orthogonality of Chebyshev basis. Adopting the properties above, we can get the unknown coefficient vector \beta from:

\mathbf{\mu \beta} = diag \left( \delta - \dot{s}_{s} \right)^{-2} \left[ \left(W_{ss}\dot{s} + diag (\dot{s}) \mathbf{\mu}_{s} \mathbf{\beta} \right)\left( \delta - \dot{s}_{s} \right) + diag \left(\dot{s}_{ss} \dot{s} \right) \left( W_{s} + \mathbf{\mu \beta} \right) \right] , and

\mathbf{\beta} = \left[ diag \left( \delta - \dot{s}_{s} \right)^{2} \mathbf{\mu} - diag \left( \dot{s}\left( \delta - \dot{s}_{s} \right) \right) \mathbf{\mu}_{s} - diag (\dot{s}_{ss} \dot{s} ) \mathbf{\mu} \right]^{-1} \left( W_{ss} \dot{s} \left( \delta - \dot{s}_{s} \right) + W_{s} \dot{s}_{ss} \dot{s} \right) .

If we suppose A = \left[ diag \left( \delta - \dot{s}_{s} \right)^{2} \mathbf{\mu} - diag \left( \dot{s}\left( \delta - \dot{s}_{s} \right) \right) \mathbf{\mu}_{s} - diag (\dot{s}_{ss} \dot{s} ) \mathbf{\mu} \right] and
B = \left( W_{ss} \dot{s} \left( \delta - \dot{s}_{s} \right) + W_{s} \dot{s}_{ss} \dot{s} \right) , then over-determined case can be calculated:

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

For more detils see Fenichel and Abbott (2014).

Value

A list of approximation results: deg, lb, ub, delta, and coefficients. Use results$item
(or results[["item"]]) to import each result item.

degree

degree of Chebyshev polynomial

lowerB

lower bound of Chebyshev nodes

upperB

upper bound of Chebyshev nodes

delta

discount rate

coefficient

Chebyshev polynomial coefficients

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.

See Also

aproxdef, pdotsim

Examples

## 1-D Reef-fish example: see Fenichel and Abbott (2014)
data("GOM")
nodes <- chebnodegen(param$nodes,param$lowerK,param$upperK)
simuDataPdot <- cbind(nodes,sdot(nodes,param),
                      dsdotds(nodes,param),dsdotdss(nodes,param),
                      dwds(nodes,param),dwdss(nodes,param))
Aspace <- aproxdef(param$order,param$lowerK,param$upperK,param$delta)
pdotC <- pdotaprox(Aspace,simuDataPdot[,1],simuDataPdot[,2],
                   simuDataPdot[,3],simuDataPdot[,4],
                   simuDataPdot[,5],simuDataPdot[,6])