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* the "Hartwick income" defined by the gNNP when [Pi] [multiplied by] dX/dt = 0, is not (generally) the SNI(i);
* the equality between gNNP and SNI(i) holds only for a PVU-max path where the Hartwick rule is respected at every point in time;
* the gNNP for a PVU-max timepath therefore will not, in general, coincide with SNI(i), nor for that matter with SNI(ii).
Second, the measurements of dK/dt, or of the gNNP and the related Hartwickian savings ([Pi] [multiplied by] dX/dt), must be specified in terms of prices (or, as the case may be) shadow prices for the particular moment (or period) in time along the particular equilibrium path being considered.
The basic idea of the "savings" indicator - not generally valid, as we have seen - was that if Hartwick's rule is respected, then the economy is "operating within the bounds of sustainability" in the sense that current consumption [p.sub.1]C is less than the sustainable national income (presumed, somewhat erroneously, to be indicated by gNNP).
So the mathematical passage from a value of gNNP obtained empirically or for a point on a model timepath, to either version of SNI, is model specific and depends on scenario parameters.
* The transformation from GNP to gNNP for the current period can be made on the basis of current prices, neglecting for the moment whether or not these prices are PVU-optimal (but see below).
* The transformation from current gNNP to SNI(i) or SNI(ii) is, by contrast, strongly model-specific.
* the gNNP obtained from the formula gNNP = [p.sub.1]C + [Pi] [multiplied by] dX/dt is higher than the SNI(i), which means that an estimate for gNNP obtained by deducting net capital depreciation from GNP will not correctly indicate the extent to which current consumption overshoots sustainability.
Knowing the "correct" shadow relative prices and the associated quantities for calculating the gNNP depends on knowing the intertemporal production possibility frontier for economic production and environmental function (ipso facto the elasticities of substitution and the technological changes for the timespan of analysis), and knowing - or, at least, placing bounds around - the pattern of "demand" for economic goods and environmental functions on the part of future generations.
Estimates for SNI(i), presumed equal to gNNP, and for Hartwick "net savings" as sustainability indicators obtained by deductions of natural capital "depreciation" from conventional GNP using current prices, are both logically invalid and empirically suspect for the situations in which they are most urgently needed.
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