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).