## An Inconvenient Truth about the Hartwick Rule

The relevance to the Hartwick Rule of depreciation of produced capital is not widely understood.

Suppose the inputs of a closed economy consist of produced capital K, a non-renewable resource R and labour L.  It produces a single good which can be either consumed or added to the stock of produced capital.  The quantity of output Y is determined by a constant-returns Cobb-Douglas function, implying in particular that if any one of K, R or L is nil then Y is nil.  Technology is constant, as is labour input.  Produced capital depreciates at a constant positive rate δK.

Question:  In such an economy, is it possible for consumption at some constant positive level to be maintained forever?

I have the impression that many people familiar with at least some of the vast literature on sustainability would be inclined to answer in the affirmative.  And the reason would be that they have encountered the Hartwick Rule according to which, loosely, sustainability can be achieved by investing the rents from non-renewable resources in produced capital.

The correct answer, however, is that under the conditions of the question, constant consumption cannot be sustained forever.  In simple terms, this is because use of the non-renewable resource must decline towards (but never reach) zero.  To maintain consumption, produced capital must therefore increase without limit.  Consequently depreciation of that capital must also increase without limit.  So output must become large enough to offset a huge quantity of depreciation, as well as providing for consumption and an increase in the stock of capital.  That requires a certain minimum rate of use of the resource.  Continuing use of the resource at that minimum rate must eventually exhaust any finite initial stock.  A more formal proof will be given below.

I claim no originality for this result (although I have not seen elsewhere the particular proof I set out below).  It can be found in the literature.  Indeed, it can be found in the very place that those who assume that the Hartwick Rule justifies an affirmative answer might appeal to as a source.  Most of Hartwick’s 1977 paper relies on an assumption of no depreciation (1).  In the final paragraph, however, we find the following (2):

“If there is depreciation of reproducible capital at the rate δ per unit capital per unit time, then … our savings investment rule will not [my italics] provide for the maintaining of … consumption constant over time.”

I should emphasise that the reasoning in Hartwick’s paper is entirely correct: what could mislead readers is its balance and tone, with most of the paper devoted to the unrealistic case of zero depreciation, introduced with something of a rhetorical flourish, and only a few sentences on the practically important case of capital depreciating over time.

The result may also be found in a paper by Buchholz, Dasgupta & Mitra (3), which explicitly models depreciation from the outset, albeit via the more general formula δKθ (0 ≤ θ ≤ 1).  For a class of production functions of which the Cobb-Douglas is one example, it is shown that constant consumption forever (an “equitable path” in the paper’s terminology) is impossible when δ > 0 and θ = 1 (4).  That is precisely the case to which my question applies. However, the result is one among many in the paper, some of which are given much more prominence, and could easily be missed by a casual reader.

A further (and currently open access) source for the result is the PhD thesis of Hamilton.  Here it is at least given prominence early on (5), although most of the thesis reverts to the assumption of zero depreciation (6).

I am struck however by the fact that, although many discussions of the economics of sustainability refer to the Hartwick Rule, they often fail to mention that it will not enable constant consumption to be sustained forever if produced capital depreciates at a constant positive rate.  Here are a couple of examples from textbooks.

Hanley, Shogren & White’s Environmental Economics in Theory and Practice devotes most of a section on weak sustainability rules to consideration of the Hartwick Rule (7).  Having explained the Rule and in particular the consequential feasibility of non-declining consumption, it identifies four limitations: briefly, the Rule does not hold for all types of production function or for open economies, and where it does hold, it is not necessarily consistent with constant welfare or with ecological sustainability.  But the limitation in respect of depreciation of produced capital is not mentioned.

Common & Stagl’s Ecological Economics: An Introduction refers to the Hartwick Rule in the context of discussing policies that might be adopted by a benevolent dictator in a closed economy with a constant-returns Cobb-Douglas production function using produced capital, non-renewable resource and labour inputs, with constant technology and constant population  (8). A choice of savings rate reflecting the current generation’s preferences as between current and future consumption is shown to be likely to lead to an unsustainable outcome, with consumption initially rising but eventually declining asymptotically towards zero.  It is then stated that there is a savings policy which would ensure constant consumption forever, the policy being the Hartwick Rule.  The one qualification made is that the resource must be depleted efficiently – a requirement also identified in Hartwick’s paper and generally known as the Hotelling Rule (9).  The effect of depreciation of produced capital is not mentioned.

The Hartwick Rule is also commonly referred to in policy-orientated literature relating to sustainability.  A report on an EU-funded project concerning ecosystem restoration in the UK includes the following entry in its glossary (10):

“Hartwick Rule – simple rule of thumb for sustainable development for countries that depend … on non-renewable natural resources: consumption can be maintained … if rents from non-renewable resources are continuously invested rather than used for consumption.”

A paper by van der Ploeg entitled Challenges and Opportunities for Resource Rich Economies, though acknowledging that the Hartwick Rule is “hotly debated”, appears to accept that given a Cobb-Douglas production function the Rule can make possible constant consumption in the absence of technical progress (11).  The one exception it notes is for open economies, arguing that resource-exporting countries can sustain constant consumption by investing less than implied by the Hartwick Rule (12), an assertion which would surely not be made if the relevance of depreciation were understood?

The extent of the influence of the Hartwick Rule was described by Ottenhof in a piece written for the 40th anniversary of Hartwick’s 1977 paper (13).  It states:

“the Hartwick Rule has gone on to become a pillar of sustainability economics, forever changing the way we think about the concept of sustainability.”

While it also refers to opposition among the ecological community to the weak sustainability approach associated with the Hartwick Rule, this clearly relates to arguments as to whether, or to what extent, produced capital can substitute for natural resources, and not to the effect of depreciation.

Why then is the fact that depreciation of capital undermines the significance of the Hartwick Rule not more widely recognised?  One reason may be a perception that depreciation is a minor technical issue that can safely be ignored with little consequence.  In some economic contexts such a perception would be valid. If one is considering the short-term response of an economy to a change in fiscal or monetary policy, with a focus on the effects on activity and employment, then it could be entirely reasonable to ignore depreciation.  But the long-term scenario suggested by the Hartwick Rule, with ever-increasing quantities of produced capital offsetting ever-reducing use of a renewable resource, is a context in which to ignore depreciation would be seriously misleading.

Another reason may be that the Hartwick Rule, if taken to provide a basis for sustainability, suggests many avenues for further research.  For what types of production function does the Rule hold?  Can it be extended to cases of many non-renewable resources?  How much consumption can be sustained forever?  What are the implications for measurement of national income?  How large are the rents from non-renewable resources in particular countries, and how do they compare with those countries’ investments in produced capital?  By contrast, acceptance that the Rule is not of much practical importance because capital depreciates may seem, from a research perspective, as something of a dead end.

A further reason may be an assumption that the problem with depreciation can be simply overcome by working in terms of a net rather than a gross production function.  This calls for a little explanation.  A gross production function expresses gross output – output before depreciation of capital – as a function of inputs.  Similarly, a net production function expresses output net of depreciation as a function of inputs.  The relation between the two is simple: if the gross production function is G(K,R,L), the net production function is F(K,R,L), and depreciation is δK, then:

G(K,R,L) – δK  =  F(K,R,L)              (A)

There is nothing wrong in itself in using a net production function: in some contexts it can simplify matters to do so.  The potential pitfall however is to assume that standard assumptions about the functional forms of gross production functions will simply carry over to net production functions.

Surprisingly, this fallacy can be found in Hartwick and Olewiler’s The Economics of Natural Resource Use (14).  Unlike the textbooks mentioned above, it includes depreciation in its discussion of the economics of sustainability.  It introduces a production function Q = F(K,R,L) and, since this is followed by the statement that consumption equals Q – I, where I is net investment, it is clear that this is a net production function (15).  Subsequently it is stated that, by following the Hotelling Rule and (though not referred to by name) the Hartwick Rule,  consumption can be maintained indefinitely at a constant positive level if the production function F has the Cobb-Douglas form KαRβL1-α-β (and subject to certain conditions on α and β).

The problem with this lies in the assumption of a net production function with Cobb-Douglas functional form.  From (A) above this implies that the corresponding gross production function is:

G(K,R,L)  =  KαRβL1-α-β + δK              (B)

This is an implausible form for a production function.  The implication that some output can be obtained without use of a non-renewable resource or labour is not necessarily a problem.  But it implies something much stronger, namely, that the productivity of produced capital in the absence of other inputs, indicated by the coefficient δ, is precisely what we know to be the rate of depreciation – an amazing coincidence.

Note what (B) is not saying.  There is some plausibility in a production function which divides produced capital into two parts, call them K1 and K2, the former yielding output only in conjunction with non-renewable resource inputs, and the latter yielding output without them (think of coal-fired power stations and solar panels) and with a production coefficient reflecting the actual productivity of K2.  So we might write something like:

G(K1,K2,R,L)  =  K1αRβL1-α-β + θK2             (C)

But that is not what (B) does.  It treats produced capital as homogeneous, yet capable of producing so much output on its own – precisely enough to offset depreciation -, and more in conjunction with other inputs.  I know of no reason why a production function might take such a form.

I conclude with a more formal statement and proof, in discrete time, of the above result.

Theorem The output of a closed economy in any period consists of a quantity of a single good, any part of which is either consumed within the period or added to the stock of produced capital for the next period.  Once added to the stock of produced capital it cannot subsequently be consumed. The production function is: $Y_t = AK_t^{\alpha}R_t^{\beta}\quad (\alpha,\beta > 0;\,\alpha + \beta < 1)$

where: $Y_t =$ output in period $t$; $K_t =$ stock of produced capital in period $t$; $R_t =$ quantity of a non-renewable resource used in period $t$; $A,\alpha,\beta$ are fixed parameters, the value of $A\,$ reflecting both the constant technology and the constant labour input.

The stock of produced capital in period $t\,$ is subject to depreciation of $\delta K_t\,(\delta > 0)$.  Given finite initial stocks of produced capital and the resource, no positive quantity of consumption per period can be sustained forever.

Proof:  We proceed by reductio ad absurdum.  Suppose consumption of $C\,$ per period $(C > 0)$ can be sustained forever from finite initial stocks $K_0$ of capital and $S_0$ of the resource.  Then for some $S\,$ such that $0 < S \leq S_0$: $\sum_{t=1}^{\infty} R_t = S\qquad(P1)$

From this we can infer (16): $\lim_{t\rightarrow \infty}R_t=0\qquad(P2)$

Hence given any $\epsilon > 0$, there exists a positive integer $N$ such that $R_t < \epsilon$ for all $t > N$.  For such $t$: $AK_t^{\alpha}\epsilon^{\beta} > AK_t^{\alpha}R_t^{\beta}>C\qquad(P3)$

and therefore: $K_t > \frac{(C/A)^{1/\alpha}}{\epsilon^{\beta/\alpha}}\qquad(P4)$

Since this holds for any $\epsilon > 0$, however small, we must have: $\lim_{t\rightarrow \infty}K_t=\infty \qquad(P5)$

But growth of $K$ in any one period is finite (since the production function can only yield finite output from finite inputs).  Hence there must be an infinite number of periods in which $K_t$ is both larger than $K_0$ and growing. $K_t$ can grow in a period only if output exceeds depreciation, so for each of those infinite periods we must have: $AK_t^{\alpha}R_t^{\beta} > \delta K_t\qquad(P6)$

and therefore: $R_t > (\delta/A)^{1/\beta}K_t^{(1-\alpha)/\beta} > (\delta/A)^{1/\beta}K_0^{(1-\alpha)/\beta}\qquad(P7)$

Since P7 applies to an infinite number of periods, we have: $\sum_{t=1}^{\infty}R_t > \sum_{t=1}^{\infty}(\delta/A)^{1/\beta}K_0^{(1-\alpha)/\beta} = ((\delta /A)^{1/\beta}K_0^{(1-\alpha)/\beta} \infty\,\,\,(P8)$ $\sum_{t=1}^{\infty}R_t = \infty > S_0\qquad(P9)$

Thus our supposition leads to a contradiction.  QED.

Notes and references

1. Hartwick J M (1977)  Intergenerational Equity and the Investing of Rents from Exhaustible Resources  The American Economic Review 67(5) pp 972-4.  The assumption of no depreciation is in the middle of the first paragraph on p 972.
2. Hartwick, as 1 above, p 974.
3. Buchholz W, Dasgupta S & Mitra T (2005)  Intertemporal Equity and Hartwick’s Rule in an Exhaustible Resource Model  Scandinavian Journal of Economics 107(3) pp 547-61.
4. Buchholz et al, as 3 above.  The depreciation formula is introduced on p 551 and this result is on p 553.
5. Hamilton K (1995)  Sustainable Development and Green National Accounts  PhD thesis accessible at https://core.ac.uk/download/pdf/16221331.pdf  pp 2 & 7-8.
6. Hamilton K, as 5 above, see final sentence p 9.
7. Hanley N, Shogren J F & White B (2nd edn 2007)  Environmental Economics in Theory and Practice  Palgrave Macmillan  pp 19-21
8. Common M & Stagl S (2005)  Ecological Economics: An Introduction  Cambridge University Press  pp 350-1
9. Common & Stagl, as 8 above, pp 351-2
10. Bright G  Natural Capital Restoration Project Report  https://circabc.europa.eu/sd/a/d4510f50-76ec-4332-9598-3cd762f21c64/UK-2015-Natural-capital.pdf  p 191
11. Van der Ploeg F (2006)  Challenges and Opportunities for Resource Rich Economies  EUI Working Papers RSCAS No. 2006/23  https://cadmus.eui.eu/bitstream/handle/1814/6254/RSCAS_2006_23.pdf?isAllowed=y&sequence=3  p 17
12. Van der Ploeg, as 11 above, p 18
13. Ottenhof N (2017)  Hartwick’s Rule continues to influence sustainable development after 40 years  https://economicsandpolicy.ca/2017/06/19/hartwicks-rule-continues-to-influence-sustainable-development-after-40-years/
14. Hartwick J M & Olewiler N D (2nd edn 1998)  The Economics of Natural Resource Use  Addison-Wesley
15. Hartwick & Olewiler, as 14 above, p 399
16. I am grateful to Thomas and GEdgar, participants in Mathematics Stack Exchange, for confirming the validity of this step  https://math.stackexchange.com/questions/4139757/if-sum-limits-t-1-inftyr-t-is-finite-with-r-t-geq-0-does-lim-t-rig
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