| vaprox {capn} | R Documentation |
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 |
sdata |
A data.frame or matrix of [stock,sdot,benefit]=[ |
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}) = \mathbf{\mu}(\mathbf{S})\mathbf{\beta}, \mathbf{\mu}(\mathbf{S})
is Chebyshev polynomials and \mathbf{\beta} is their coeffcients.
Then, p_{i} = \mathbf{\mu}_{s_{i}}(\mathbf{S})\mathbf{\beta} by the orthogonality of Chebyshev basis.
Adopting the properties above, we can get the unknown coefficient vector \beta from:
\delta \mathbf{\mu}(\mathbf{S})\mathbf{\beta} = W(\mathbf{S}) + \displaystyle \sum_{i=1}^{d} diag (\dot{s}_{i}) \mathbf{\mu}_{s_{i}}(\mathbf{S})\mathbf{\beta}, and thus,
\beta = \left( \delta \mathbf{\mu}(\mathbf{S}) - \displaystyle \sum_{i=1}^{d} diag (\dot{s}_{i}) \mathbf{\mu}_{s_{i}}(\mathbf{S}) \right)^{-1} W(\mathbf{S}) .
In a case of over-determined (more nodes than approaximation degrees),
\beta = \left( \left( \delta \mathbf{\mu}(\mathbf{S}) - \displaystyle diag (\dot{s}_{i}) \sum_{i=1}^{d} \mathbf{\mu}_{s_{i}}(\mathbf{S}) \right)^{T}
\left( \delta \mathbf{\mu}(\mathbf{S}) - \displaystyle \sum_{i=1}^{d} diag (\dot{s}_{i}) \mathbf{\mu}_{s_{i}}(\mathbf{S}) \right) \right)^{-1}
\times \left( \delta \mathbf{\mu}(\mathbf{S}) - \displaystyle \sum_{i=1}^{d} diag (\dot{s}_{i}) \mathbf{\mu}_{s_{i}}(\mathbf{S}) \right)^{T} W(\mathbf{S}) .
For more detils see Fenichel and Abbott (2014), Fenichel et al. (2016), and Yun et al. (2017).
Value
A list of approximation resuts: deg, lb, ub, delta, and coefficients. Use results$item
(or results[["item"]]) to import each result item.
degree |
degree of Chebyshev polynomial |
lowerB |
lower bound of Chebyshev nodes |
upperB |
upper bound of Chebyshev nodes |
delta |
discount rate |
coefficient |
Chebyshev polynomial coefficients |
References
Fenichel, Eli P. and Joshua K. Abbott. (2014) "Natural Capital: From Metaphor to Measurement."
Journal of the Association of Environmental Economists. 1(1/2):1-27.
Fenichel, Eli P., Joshua K. Abbott, Jude Bayham, Whitney Boone, Erin M. K. Haacker, and Lisa Pfeiffer. (2016) "Measuring the Value of Groundwater and Other Forms of Natural Capital."
Proceedings of the National Academy of Sciences.113:2382-2387.
Yun, Seong Do, Barbara Hutniczak, Joshua K. Abbott, and Eli P. Fenichel. (2017) "Ecosystem Based Management and the Welath of Ecosystems" Proceedings of the National Academy of Sciences. (forthcoming).
See Also
Examples
## 1-D Reef-fish example: see Fenichel and Abbott (2014)
data("GOM")
nodes <- chebnodegen(param$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)