National product, calculated net of a deduction for depletion of natural resources, could in theory provide some indication of whether current consumption is permanently sustainable.  But the conditions that would be needed to make net national product into a practical sustainability indicator cannot be met.

When gross national product (GNP) and related economic aggregates were first developed by Kuznets and others in the 1930’s and 1940’s, there was debate as to whether the aim should be to measure activity and output, or welfare and well-being.  Against a background of mass unemployment and then World War II, the debate was won by those who wanted the former (1).  To this day, GNP as calculated in most countries remains a measure of activity and output, and (as many critics have pointed out) it is easy to find examples of activities which raise GNP but do not enhance and may even lower welfare.

It has always been recognised that net national product (NNP), which equals GNP less an allowance for depreciation of capital assets is in some ways a more meaningful measure. Most countries publish estimates of NNP as well as GNP.  Nevertheless, most economic discussion focuses on gross aggregates, including GDP (gross domestic product, which is similar to GNP but excludes certain international income flows).  This seems to be partly because of the short-term link between activity and employment, and partly because of difficulties – both conceptual and practical – in measuring depreciation (2).

In the 1970’s, growing interest in environmentalism and concerns regarding resource depletion (3) led some economists to explore long-term macroeconomic models in which the essential inputs to production include non-renewable natural resources.  This led to the idea that NNP might be adapted – by including a suitable deduction for resource depletion –  to provide an indicator of sustainability.  Much of the academic literature on this topic stems from Weitzman’s 1976 paper On the Welfare Significance of National Product in a Dynamic Economy (4). 

Weitzman explored the properties of NNP within a long-term model whose assumptions included the following (5):

• A single consumption good (or a single number representing consumption).
• Many capital goods, taken to include not only equipment and infrastructure but also human capital and non-renewable natural resources.
• A production possibility set determined as a function of the available capital goods.
• A fixed own rate of interest r on the consumption good.

NNP was taken to be measured as consumption plus the value of net investment in capital goods.  For a non-renewable resource, it would include a deduction for any depletion. 

Given initial quantities of the capital goods, there can be determined within the model an optimal consumption stream, that is, one which maximises the discounted present value of future consumption, taking the fixed own rate of interest as the discount rate.  To achieve that path requires an optimal balance at all times between consumption and net investment and, within net investment, between investment in and depletion of the various capital goods.

Whatever the present value of the optimal consumption stream may be, there must exist a rate of consumption which, if maintained constant forever, has the same present value.  Weitzman calls this the stationary equivalent of future consumption: others have called it, more conveniently, constant-equivalent consumption, and I shall use the latter term here. 

We can now state Weitzman’s main conclusion as follows: NNP equals constant-equivalent consumption (6). I shall refer to this as Weitzman’s equality.  His formulation of the equality (7) shows that NNP depends on the discount rate, so it is appropriate to write NNP(r); constant-equivalent consumption, being calculated from the optimal consumption stream for a particular discount rate, also depends on the discount rate.

The impact of Weitzman’s paper went beyond the academic literature.  It was an important influence on Nature’s Numbers (1999), a report commissioned by the US government on expanding the US national accounts to “include the environment” (8).  According to this report:

“One of the most surprising results of modern economic theory is the output-sustainability correspondence principle …. This … holds that under idealized conditions, net national product and sustainable income are identical.” (9)

Now there are two problems with this statement. One is that, even if it were technically correct, making it in a report to government carries a strong implication that it is practically important.  That requires that the necessary “idealized conditions” are a reasonable approximation to reality or, more precisely, any differences between those idealized conditions and reality are such that net national product is a reasonable approximation to sustainable income.  I shall show below that this is not the case.  The second problem is that the statement is not even technically correct. As I shall explain shortly, it is a basic misunderstanding to interpret Weitzman’s constant-equivalent consumption as sustainable income.

It is worth noting that the term “output-sustainability correspondence principle” has not found favour with economists (or anyone else).  On googling it, I obtained few hits, most of them linking to Nature’s Numbers itself.  As a name for Weitzman’s equality, the term is doubly misleading: sustainability is not constant-equivalent consumption, and “output” does not convey the subtleties of Weitzman’s concept of net national product.  

If we are concerned to use economic analysis to assess the sustainability or otherwise of an economy that uses non-renewable resources, then we will want to develop long-term models similar in some respects to that of Weitzman.  But there is one fundamental difference.  The natural way to define sustainability within such a model is in terms of the feasibility of a constant rate of consumption continuing forever.  We might ask whether a particular rate of consumption, for example the current rate, can continue indefinitely, or we might ask what is the maximum rate of consumption that can be so sustained.  In either case, we are concerned with a putative actual rate of consumption, a rate that (subject to its distribution) would determine the living standard of the people living at any time.  Weitzman’s constant-equivalent consumption, however, is merely a theoretical construct.  We cannot assume that constant consumption at that rate is feasible within the parameters of his model (indeed, Weitzman shows that it is not feasible (10)).  Even if the optimal consumption stream were achieved, actual consumption would be more than constant-equivalent consumption at some times, and less at others, and for those unfortunate enough to live at times when it was less, it would be no consolation (even if it were so) that constant-equivalent consumption would have been sufficient for their needs or aspirations.

So does Weitzman’s equality have any bearing on the question of sustainability?  In fact, a  link can be made as follows. For a given model, given initial values, and discount rate r, we make the following definitions:

• PVOC(r) is the present value of the optimal feasible consumption stream OC(r) at discount rate r.
• PVCE(r) is the present value of constant-equivalent consumption CE(r) at discount rate r.
• PVCC*(r) is the present value at discount rate r of the maximum feasible rate of constant consumption CC*. Note that we do not write CC*(r): the maximum feasible rate depends only on the initial quantities of capital goods, including natural resources, and the production technology.

From the definition of constant-equivalent consumption, PVOC(r) = PVCE(r).  Since OC(r) is optimal given r, we must have PVCC*(r) ≤ PVOC(r) and therefore PVCC*(r) ≤ PVCE(r). Because CC* and CE(r) are both constant rates, we can infer that CC* ≤ CE(r). Given Weitzman’s equality NNP(r) = CE(r), this implies CC* ≤ NNP(r).   Importantly, this argument does not depend on the value of r.

So from Weitzman’s equality we can infer what might be termed a partial sustainability indicator: if a putative rate of constant consumption CC* exceeds NNP(r), then it is not sustainable forever.  But if CC* is less than NNP(r), then it may be sustainable. 

Before proceeding, it will be useful to show that this indicator can be expressed in another way.  Writing C for consumption, I for the vector of net investments in each of the capital goods, and q for the vector of prices of those goods, we have NNP(r) = C + qI.  The above indicator implies that for sustainability we must have C ≤ CC* ≤ NNP(r), from which we can infer C ≤ C + qI and therefore qI ≥ 0, a condition sometimes termed the genuine savings indicator.  This can also be put the other way round: if qI < 0, then current consumption is definitely unsustainable.

The fact that the above indicator is only partial is no reason to belittle it.  On the contrary, assessing long-term sustainability is both an important and a complex matter, so any contribution should be welcome.

The real problem with this indicator is that it relies on the assumptions underlying Weitzman’s equality – or as Nature’s Numbers put it, the idealized conditions.  Weitzman’s paper is quite concise and, as with much of the literature on this topic, it is not entirely straightforward to discern the precise assumptions on which its results depend.  That’s not just my opinion; Asheim’s 2003 paper Green National Accounting for Welfare and Sustainability: A Taxonomy of Assumptions and Results (11) begins with a similar sentiment.

Asheim proceeds to consider five possible assumptions that long-term models might or might not make, and to identify conclusions which can be drawn from particular combinations of those assumptions. In relation to sustainability as defined above, the two most relevant assumptions are:

• Optimality: the economy is on a path which maximises the present value of the consumption stream, and prices of capital goods are such as to support that path. I take the latter to mean that the time paths of prices are such that, if producers act at all times so as to maximise their profits given those prices, then the choices of goods produced and inputs used at any time will be such that the present value of the consumption stream will be maximised.
• Stationary technology: there is no technical progress and therefore the amount and composition of output at any time depends solely on the quantities of inputs and the allocation of inputs to the production of different goods.

Asheim’s conclusions regarding sustainability are summarised in a table (12), the blanks indicating combinations of assumptions from which no conclusions can be drawn.  What is striking is that without the optimality assumption, no conclusions can be drawn, regardless of whether other assumptions hold or not.  Our indicator above, in the form that current consumption may be sustainable if qI ≥ 0 and is unsustainable if qI < 0 (I cannot see why Asheim switches between these equivalent formulations) is shown as requiring both optimality and stationary technology.

These two assumptions are in fact both implicit in Weitzman’s paper, though perhaps not stated as clearly as they might have been.  His initial definition of NNP includes C for consumption and K for capital (13), but the fact that these letters soon acquire asterisks serves as a warning sign that something is going on.  The something is that C* and K*, together with the prices of each capital good, are on paths determined within, or just as if they were within, a competitive economy with perfect foresight.  The paths are such as both to maximise income at any time and to meet the condition for intertemporal efficiency (14).  These conditions ensure that the paths optimise the present value of the consumption stream, subject to the constraints of the initial quantities of capital goods and the production possibility set.

What this means, in simple terms, is that the optimality assumption is built into Weitzman’s concept of NNP.  That has two implications.  Firstly, it means that his NNP is very different from any NNP calculated using actual prices determined in real markets that are often imperfectly competitive and lack perfect foresight.  I say “any” NNP because that includes NNP as conventionally estimated by national income statisticians, but also estimates of “Green NNP” which make allowance for natural resource depletions but value those depletions on the basis of actual prices.  Consequently, Weitzman’s equality, and the sustainability indicator derived from it, is not valid for NNP based on actual prices; only for his concept of NNP.

The second implication is that to estimate Weitzman’s NNP so as to use it in applying the above sustainability indicator would raise enormous practical difficulties.  Instead of valuing net investment in a capital good using its actual price, it would be necessary to use a price that made full allowance for the future, a price that reflects the direct and indirect contributions to output the good can make into a far distant future in which natural resources may be much scarcer and man-made capital more plentiful than at present.  More precisely, the price would need to be the shadow price as it would be calculated from a giant dynamic optimisation problem with no time horizon seeking to maximise a constant level of consumption subject to the constraints of a production function and initial quantities of each capital good and natural resource.  Without knowing what prices such a model would yield, it is fairly obvious that they would not always correspond very closely to actual prices.  Consider for example the sharp fall in the price of oil in 2008, which surely had much to do with the effect of problems originating in financial markets on short-term demand for oil, and little to do with long-term considerations? 

Solving such a dynamic optimisation problem may be possible (with suitable computing power), notwithstanding the very large number of goods that would need to be considered.  But the nature of such problems is that small changes to the terms of the problem sometimes result in large changes to the solution.  As explained in this post, we cannot estimate a production function with sufficient accuracy for future circumstances in which the ratio of man-made capital to non-renewable natural resources may be many times higher than at present.  Nor in many cases can we measure current reserves of natural resources with the necessary accuracy; moreover the issue for many resources is not so much their finite quantity as the increasing cost of extraction from progressively less accessible sources, which would further complicate the problem.  The practical conclusion is that we do not have the necessary knowledge to calculate Weitzman’s NNP.

If, however, we did have that knowledge, we could simply infer the maximum constant level of consumption from the solution to the problem.  There would be no need to follow the roundabout approach of using the prices determined by the problem to calculate Weitzman’s NNP as a basis for applying the sustainability indicator.

Turning to the assumption of stationary technology, Weitzman adopted a neat device to broaden the scope of circumstances that can be considered to satisfy that assumption.  His broad definition of capital goods enabled him to treat technical progress of a gradual, step-by-step kind as capital accumulation in the forms of increasing stocks of knowledge and enhancements to the human skills used in applying knowledge.  In so far as technical progress can be handled in this way, it implies no change to the functional relation between the the inputs to production and the production possibility set, just more inputs resulting in more production.  It does however add an extra complication to the measurement of NNP by adding to the complexity of the production relation, and to the number and type of capital goods for which net investment must be measured and valued.  Thus it renders even more impracticable what we have already identified as impracticable because of the optimality assumption.

Weitzman also recognised, however, that this device would not do for sudden and unanticipated inventions.  In such a case, he asserted, the discounted present value of the increase in the future consumption stream made possible by the invention should immediately be capitalised by increasing NNP (15).  Whatever the merits of that approach, it means that NNP will suddenly peak and then (barring another such invention) revert to something like its previous level.  Such fluctuations in NNP imply that in those circumstances it will no longer equal constant-equivalent consumption, which should increase to reflect the invention but not subsequently fall.

A further very important issue in assessing sustainability is how to bring population growth within the scope of the analysis.  Weitzman suggests, without developing the point, that one could simply calculate everything on a per capita basis (16).  Certainly, the natural interpretation of sustainability in circumstances of growing population is constant consumption per capita.  But that leads to the problem that the appropriate prices for valuing net investment will depend on the future consumption levels needed to ensure constant per capita consumption, which will depend on future population growth, the extent of which in the long term is unpredictable. 

A final feature of Weitzman’s NNP is that it is to be calculated not as a total for a period but as a rate at a point in time within a continuous time framework.  In this respect also it differs from NNP as measured by national income statisticians, who start (for flow variables such as consumption and investment) from data for a period.  In principle, this could make a significant difference where net investment in a capital good within the period is large, so that if its shadow price could be measured it would be found to have changed during the period.  It is hard to say how much difference this might make in practice, but it is one more difficulty in making the leap from practical measurement of NNP to a reliable sustainbility indicator.

The conclusion I draw from all this is that NNP does not provide a practical sustainability indicator. That also applies to the genuine savings reformulation.  I should perhaps add that this is not to say I believe any other sustainability indicator to be better.  The problem of assessing long-term sustainability just is extremely challenging.

This post was substantially amended on 15 January 2019 to focus more clearly on the key points and eliminate some errors.

Notes and References

1. Coyle, D (revised and expanded edition 2014) GDP: A Brief but Affectionate History Princeton University Press  pp 12-16
2. OECD (second edition 2009) Measuring Capital: OECD Manual Ch 5 pp 43-51  https://www.oecd.org/std/productivity-stats/43734711.pdf
3. See for example Meadows D H et al (1972) The Limits to Growth Universe Books  https://www.clubofrome.org/report/the-limits-to-growth/
4. Weitzman M L (1976) On the Welfare Significance of National Product in a Dynamic Economy  The Quarterly Journal of Economics  90(1) pp 156-162
5. Weitzman, as above, pp 156-8
6. Weitzman, as above, p 160
7. Weitzman, as above, equation (10) p 161
8. Nordhaus W D & Kokkelenberg E C (eds) (1999) Nature’s Numbers: Expanding the National Economic Accounts to Include the Environment https://www.nap.edu/catalog/6374/natures-numbers-expanding-the-national-economic-accounts-to-include-the
9. Nordhaus & Kokkelenberg, as above, p 36
10. Weitzman, as above, pp 159-160.
11. Asheim G B (2003) Green National Accounting for Welfare and Sustainability: A Taxonomy of Assumptions and Results  Scottish Journal of Political Economy  Vol 50(2) pp 113-130
12. Asheim, as above, Table 2 p 128
13. Weitzman, as above, Identity (2) p 157
14. Weitzman, as above, p 158
15. Weitzman, as above, p 162
16. Weitzman, as above, p 157