capn (version 1.0.0)

vaprox: Calculating V-approximation coefficients

Description

The function provides the V-approximation coefficients of the defined Chebyshev polynomials in aproxdef.

Usage

vaprox(aproxspace, sdata)

Arguments

aproxspace

An approximation space defined by aproxdef function

sdata

A data.frame or matrix of [stock,sdot,benefit]=[$\mathbf{S}$,$\dot{\mathbf{S}}$,$W$]

Value

A list of approximation resuts: 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

Details

The V-approximation is finding the shadow price of $i$-th stock, $p_i$ for $i=1,\cdots ,d$ from the relation:

$\delta V=W(\mathbf{S})+p_1{\dot{s}}_1+p_2{\dot{s}}_2+\cdots +p_d{\dot{s}}_d$,

where $\delta$ is the given discount rate, $V$ is the intertemporal welfare function, $\mathbf{S}=(s_1,s_2,\cdots ,s_d)$ is a vector of stocks, $W(\mathbf{S})$ is the net benefits accruing to society, and ${\dot{s}}_i$ is the growth of stock $s_i$. By the definition of the shadow price, we know:

$p_i=\frac{\partial V}{\partial s_i}$.

Consider approximation $V(\mathbf{S})=\mu (\mathbf{S})\beta$, $\mu (\mathbf{S})$ is Chebyshev polynomials and $\beta$ is their coeffcients. Then, $p_i={\mu }_{s_i}(\mathbf{S})\beta$ by the orthogonality of Chebyshev basis. Adopting the properties above, we can get the unknown coefficient vector $\beta$ from:

$\delta \mu (\mathbf{S})\beta =W(\mathbf{S})+{\displaystyle \sum _{i=1}^{d}diag({\dot{s}}_i){\mu }_{s_i}(\mathbf{S})\beta }$, and thus,

$\beta ={(\delta \mu (\mathbf{S})-{\displaystyle \sum _{i=1}^{d}diag({\dot{s}}_i){\mu }_{s_i}(\mathbf{S})})}^{-1}W(\mathbf{S})$.

In a case of over-determined (more nodes than approaximation degrees),

$\beta ={\left({(\delta \mu (\mathbf{S})-{\displaystyle diag({\dot{s}}_i)\sum _{i=1}^d{\mu }_{s_i}(\mathbf{S})})}^T(\delta \mu (\mathbf{S})-{\displaystyle \sum _{i=1}^{d}diag({\dot{s}}_i){\mu }_{s_i}(\mathbf{S})})\right)}^{-1}$ $\times {(\delta \mu (\mathbf{S})-{\displaystyle \sum _{i=1}^{d}diag({\dot{s}}_i){\mu }_{s_i}(\mathbf{S})})}^{T}W(\mathbf{S})$.

For more detils see Fenichel and Abbott (2014), Fenichel et al. (2016), and Yun et al. (2017).

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. Fenichel, Eli P., Joshua K. Abbott, Jude Bayham, Whitney Boone, Erin M. K. Haacker, and Lisa Pfeiffer. (2016) "Measuring the Value of Groundwater and Other Forms of Natural Capital." Proceedings of the National Academy of Sciences.113:2382-2387. Yun, Seong Do, Barbara Hutniczak, Joshua K. Abbott, and Eli P. Fenichel. (2017) "Ecosystem Based Management and the Welath of Ecosystems" Proceedings of the National Academy of Sciences. (forthcoming).

See Also

aproxdef, vsim

Examples

## 1-D Reef-fish example: see Fenichel and Abbott (2014)
data("GOM")
nodes <- chebnodegen(param$nodes,param$lowerK,param$upperK)
simuDataV <- cbind(nodes,sdot(nodes,param),profit(nodes,param))
Aspace <- aproxdef(param$order,param$lowerK,param$upperK,param$delta)
vC <- vaprox(Aspace,simuDataV)

## 2-D Prey-Predator example
data("lvdata")
aproxdeg <- c(20,20)
lower <- c(0.1,0.1)
upper <- c(1.5,1.5)
delta <- 0.03
lvspace <- aproxdef(aproxdeg,lower,upper,delta)
vaproxc <- vaprox(lvspace,lvaproxdata)