measures of income, saving and wealth can provide a hypothetical upper bound on what a prudent fiscal policy would look like.

First we look back at the recent resource discovery record that can be inferred from changes in proven reserves, or _S_ , over a period of time. We then apply these discovery data to alternative measures of national income using the marginalist approach suggested by the theoretical model, and the modified Hicksian approach. Next we exploit data on total resources (proven reserves plus the expected value of probable resources) to examine how expanding the national balance sheet to reflect total resources would affect measures of depletion and net saving.

_(i) Modified Hicksian approaches to discoveries compared to marginalist approaches_ To compare modified Hicksian versus marginalist approaches for the calculation of _aNNI_ , we first adjust investment (K̇) by subtracting the depreciation of produced capital, calculate alternative measures of resource depletion using expressions (27) and (28), and then we apply expressions (29) and (30) in order to arrive at our alternative income measures.

Total resource rents are estimated using data on world resource prices and (average) extraction costs from World Bank (2011a).<sup>15</sup> The values of resource stocks are estimated as present values of resource rents using a social discount rate of 4%.

We use published data on proven reserves _S_ for oil resources over the period 2000-2009 drawn from BP (2011). Following the usual definition, these proven reserves are stocks of a resource that can be exploited profitably at current prices and costs. As noted by Gelb _et al_ . (2012), these data can be used to infer imputed discoveries in any given year. That is, for example, the physical difference between the (closing) stock in 2008 and the (closing) stock in 2009 (gross of extraction over the year) can be thought of as the implied amount of resources added to proven reserves over the year. Broadly speaking, these added resources might come from two sources (for a discussion, see Mitchell, 2004): (i) known resources and, (ii) unknown resources. In the former, additions to proven resources are from discovered sources which were contingent on, for example, technological improvement which would allow extraction to be economically feasible. This might entail a technological development that now allows more of an oil resource to be extracted from known wells. In the latter, these additions are from more speculative or ‘undiscovered’ sources where, for example, geological evidence suggests that resources are present with some probability. Of course, proven reserves might also be revised downwards. More generally, evolving knowledge will determine assessments about the allocation of resources between these categories as well as leading to revisions in estimated total resources.

Table A.1 in the Annex to this paper describes some basic data needed to calculate the implications of a marginalist approach to income measurement given the apparent extent of imputed discoveries over 2000-2009. As well as the physical extent of imputed discoveries, the table also describes this calculation as a percentage of proven reserves. For example, in the case of Ecuador in 2008, imputed discoveries corresponds to almost 68% of the closing stock of proven oil reserves. This information about imputed discoveries is also conveyed in terms of the number of implied years that are added to the reserve life (calculated as the

> 15 Note that if world resource prices are distorted, as a result of monopoly or oligopoly for example, the values of resource rents will also diverge from efficient levels.

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