**National product, measured net of a deduction for depletion of natural resources, can in certain conditions provide some indication of whether current consumption is sustainable. But the conditions are stringent, and even when they are met, other indicators may perform better.**

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 due to wear and tear, is in some ways a more meaningful measure, and 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 a non-renewable natural resource. 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 a paper published by Weitzman in 1976 (4). The paper was also 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” (5).

To understand what Weitzman did, we need some definitions. Given a long-term model including assumptions about the rates of investment in man-made capital and of extraction and use of a non-renewable resource, together with initial quantities of capital and the resource, we can infer the *time paths* of the variables, including the rate of consumption. Generally the rate of consumption will vary over time. Given also a discount rate, we can find the *present value* of the implied stream of consumption. We can also find the *shadow price* of an input by finding how much that present value increases if the initial quantity of the input is increased by one unit.

Whatever the present value of the 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*). Finally, by *properly calculated NNP* we mean consumption plus or minus adjustments for any change in man-made capital and any depletion of the resource, valued at their respective shadow prices.

We can now state Weitzman’s main conclusion as follows: if the present value of consumption is *optimised* (by suitable choice of rates of investment in capital and of use of the resource), then (subject to some technical assumptions) properly calculated NNP will equal the stationary equivalent of future consumption (6). I shall refer to this as Weitzman’s equality (Nature’s Numbers calls it the output-sustainability correspondence principle).

How exactly does this relate to sustainability, taken here to mean the possibility of maintaining consumption indefinitely at a given rate? Constant-equivalent consumption, after all, is merely a mathematical construct: it cannot be assumed (and Weitzman did not claim) that constant consumption at that rate is feasible within the parameters of the model. Moreover, it is a construct that depends on the discount rate, whereas the feasibility of constant consumption at a given rate will depend on the production function and initial quantities of inputs, but should have nothing to do with the discount rate.

The link can be made as follows. For a given model and given initial values, let OC(r) be the feasible consumption stream with optimal present value PVOC(r) at discount rate r. Let CE(r) be constant-equivalent consumption with present value PVCE(r) given r. Let NNP(r) be properly calculated initial NNP for the optimal scenario, using shadow prices consistent with r. Lastly, let CC* be the maximum feasible rate of constant consumption, and PVCC*(r) its present value given r. Then, from the definition of constant-equivalent consumption, we have 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 are both constant rates, we can infer that CC* ≤ CE(r). Assuming Weitzman’s equality, this implies CC* ≤ NNP(r).

Importantly, the argument does not depend on the value of r. If correct, therefore, it implies the following *partial* sustainability indicator (to be understood in the context of a model as outlined above):

__Sustainability Indicator 1__

Take a selection of discount rates and find properly calculated NNP consistent with the optimal consumption time path at each rate. If a putative rate of constant consumption CC* exceeds NNP at *any one *of these discount rates, then it is not sustainable forever. But if CC* is less than NNP at *all* of the discount rates, then it *may* be sustainable.

It is a merit of this indicator that it does not rely on a single discount rate. Thus it avoids the need to address the vexed question of what is the appropriate discount rate, if any, to apply to the welfare of future generations.

An important limitation however is that its application requires identification of *optimal* time paths, not just of consumption but also of capital and the resource, in order to obtain the correct shadow prices and properly calculate NNP. There is no basis here for the tempting thought that sustainability might be assessed from conventional NNP less a deduction for actual depletion of non-renewable resources valued at their market prices.

To assess the reliability of this indicator, and (consistently with my interest in the replicability of scientific research as discussed here) to explore the conditions within which Weitzman’s equality is valid, I set up a long-term model in spreadsheet form with one row per year. This implies a discrete approach, with some *ad hoc* devices to avoid circular dependencies, and therefore with results only approximating to those of a continuous time model. It has the potential however to highlight ‘awkward’ cases which may not fit the assumptions (eg of smoothly differentiable curves) on which continuous models sometimes rely.

The assumptions of my model were:

- Output of a single good which can be either consumed or invested as man-made capital.
- A Cobb-Douglas production function Y = K
^{0.3}R^{0.1}, where K is man-made capital and R is use of a non-renewable resource S, extracted at nil cost (reasons for these particular parameters are given in this post). - Constant population, labour and technology.
- No depreciation of man-made capital.
- An exogenous discount rate, unrelated (given no assumption of a competitive economy) to the marginal product of capital.
- Initial stocks: 100 units of K and 100 units of S (the respective units need not be the same).

The model is admittedly a gross simplification of any real economy: the point is that if the indicator should not work well under what might be considered ideal conditions, then it would hardly be likely to work well in application to a real economy.

Optimal scenarios were identified for six different discount rates, the largest being 4% and the smallest 0.5%. Although in principle the time horizon was infinity, the time paths of the variables were calculated for 5,000 years, the present value of consumption beyond that date even at 0.5% being insignificant. Optimal time paths were found by judicious trial and error in respect of use of the resource in the first period and allocation of output between consumption and investment, together with application of the Hotelling rule (an intertemporal efficiency condition) for use of the resource after the first period.

To find the initial shadow price of capital, the optimal time paths were also found on the assumption of one extra unit of initial capital (ie 101 units of K and 100 units of S). The shadow price (in terms of the present value of consumption as numeraire) was then calculated as the difference between the optimal present value of consumption given 101 units of K and that given 100 units. The initial shadow price of the resource was found similarly.

The maximum feasible rate of constant consumption was calculated using a formula (for the Cobb-Douglas case) found by Solow (7) and restated in a slightly simpler form by Buchholz, Dasgupta & Mitra (8).

The results are set out in Table 1 below.

From now on I take the words “properly calculated” as read. It can be seen from Table 1 that NNP at each discount rate exceeds maximum constant consumption. Thus the results are consistent with Sustainability Indicator 1. However, comparison of NNP with constant-equivalent consumption shows Weitzman’s equality holding only at 1.1% and higher rates.

Why does Weitzman’s equality not hold at all discount rates? The reason, in simple terms, is that the proof in his paper assumes that the time paths of the variables are smooth (differentiable) curves (9). This is a valid assumption when the discount rate is sufficiently high, in which case there is nothing to be gained by investment of any part of output. The optimal scenario then has constant capital and consumption of all output throughout, resulting in smooth time paths of all variables. At lower discount rates, however, investment of the *whole* of output is found to be worthwhile for a finite initial period, and then the optimal time path of consumption switches abruptly to zero investment, with consumption of the whole of output. In the jargon of dynamic optimisation, this is known as a *bang-bang solution*, and what makes it possible is that the problem of maximising the present value of consumption subject to the constraints of the model leads to a *Hamiltonian* which is linear in consumption (10). In my discrete approach, this takes the form (as the allocation of output row in Table 1 shows) of a number of years with all output invested, then one transitional year with part of output invested, and then all subsequent years with all output consumed. At low discount rates, therefore, there is a time at which the path of consumption and consequently of some other variables is not smooth.

The optimal consumption path takes the bang-bang form when the initial shadow price of capital exceeds one, implying that, at the margin, investment of output will contribute more than immediate consumption to the present value of the consumption stream. As Table 1 shows, that point is reached when the discount rate is between 1% and 1.1% (with different assumptions it might be reached at some other rate).

One other feature of the bang-bang solution should be noted. It was stated above that use of the resource after the first period was obtained via the Hotelling rule. When no investment is taking place, so that capital is constant, the effect of spreading use of the resource between years is to spread output and hence consumption between years, so the required version of the rule is that the marginal product of the resource should grow *at the discount rate*. When however the whole of output is being invested, the effect of spreading use of the resource is to spread investment between years, the requirement then being that the marginal product of the resource should grow *at a rate equal to the marginal product of capital*. My spreadsheet was designed to use, in each year, the appropriate one of these two versions of the Hotelling rule.

Although the above results are consistent with Sustainability Indicator 1, they suggest that it could be improved by making use of the apparent implication that NNP consistent with an optimal consumption time path will be minimised when the discount rate is such that the shadow price of capital is one. But we can do better than that. Since the lowest constant-equivalent consumption (3.61 at 0.5%) is less than the lowest NNP (3.66 at 1.1%), it would be better still to ignore NNP and refer directly to constant-equivalent consumption (which is also easier to find as it does not require the shadow prices). A possible formulation is:

__Sustainability Indicator 2__

Select a low discount rate, eg 0.5%, and find constant-equivalent consumption (CE) for the optimal consumption time path at that rate. If a putative rate of constant consumption CC* exceeds CE, then it is not sustainable forever. But if CC* is less than CE, then it *may* be sustainable.

For my model this works quite well, in that the difference between constant-equivalent consumption (3.61) and maximum constant consumption (3.49) is fairly small. But further work would be needed to explore whether it would work well in a wide range of circumstances. And importantly, it does not avoid the need to identify the optimal consumption time path for the discount rate.

*The spreadsheet used to obtain the above results may be downloaded here:*

NNP & Sustainability Spreadsheet Adam Bailey

**Notes and References**

- Coyle, D (revised and expanded edition 2014)
*GDP: A Brief but Affectionate History*Princeton University Press pp 12-16 - OECD (second edition 2009)
*Measuring Capital: OECD Manual*Ch 5 pp 43-51 https://www.oecd.org/std/productivity-stats/43734711.pdf - See for example Meadows D H
*et al*(1972)*The Limits to Growth*Universe Books https://www.clubofrome.org/report/the-limits-to-growth/ - 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 - Nordhaus W D & Kokkelenberg E C (eds) (1999)
*Nature’s Numbers: Expanding the National Economic Accounts to Include the Environment*p 188 https://www.nap.edu/catalog/6374/natures-numbers-expanding-the-national-economic-accounts-to-include-the - Weitzman, as 4 above, p 160.
- Solow R M (1974) Intergenerational Equity and Exhaustible Resources
*The Review of Economic Studies*Vol 41 p 39 - Buchholz W, Dasgupta S & Mitra T (2005) Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model
*Scandinavian Journal of Economics*107(3) p 553 - Weitzman, as 4 above, p 157, which states assumptions about the existence of certain partial differentials.
- Wikipedia – Bang-bang control https://en.wikipedia.org/wiki/Bang%E2%80%93bang_control