pdotaprox: Calculating Pdot-approximation coefficients
In capn: Capital Asset Pricing for Nature

Description Usage Arguments Details Value References See Also Examples

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

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

`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} ≤ft( \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} ≤ft( \frac{dW}{ds} \right)

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

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

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

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

\dot{p} = \frac{ ≤ft( ≤ft(W_{ss}\dot{s} + \ddot{p} \right) ≤ft( δ - \dot{s}_{s} \right) + ≤ft( W_{s} + \dot{p} \right) ≤ft(\dot{s}_{ss} \dot{s} \right) \right) } { ≤ft( δ - \dot{s}_{s} \right)^{2} }

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

\mathbf{μ β} = diag ≤ft( δ - \dot{s}_{s} \right)^{-2} ≤ft[ ≤ft(W_{ss}\dot{s} + diag (\dot{s}) \mathbf{μ}_{s} \mathbf{β} \right)≤ft( δ - \dot{s}_{s} \right) + diag ≤ft(\dot{s}_{ss} \dot{s} \right) ≤ft( W_{s} + \mathbf{μ β} \right) \right] , and

\mathbf{β} = ≤ft[ diag ≤ft( δ - \dot{s}_{s} \right)^{2} \mathbf{μ} - diag ≤ft( \dot{s}≤ft( δ - \dot{s}_{s} \right) \right) \mathbf{μ}_{s} - diag (\dot{s}_{ss} \dot{s} ) \mathbf{μ} \right]^{-1} ≤ft( W_{ss} \dot{s} ≤ft( δ - \dot{s}_{s} \right) + W_{s} \dot{s}_{ss} \dot{s} \right) .

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

\mathbf{β} = ≤ft( A^{T}A \right)^{-1} A^{T}B .

For more detils see Fenichel and Abbott (2014).

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

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.

aproxdef, pdotsim

## 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])