Environmental & Resource Economics (2006) 34:517–533
DOI 10.1007/s10640-006-0011-2
Springer 2006
Capital Accumulation and Resource Depletion:
A Hartwick Rule Counterfactual
KIRK HAMILTON1,*, GIOVANNI RUTA1 and LIAILA TAJIBAEVA2
1
Environment Department, The World Bank, 1818 H St NW, 20433, Washington DC, USA;
Department of Applied Economics, University of Minnesota, Minneapolis, MN, USA; *Author
for correspondence (E-mail: [email protected])
2
Accepted 11 January 2006
Abstract. How much produced capital would resource-abundant countries have today if they
had actually followed the Hartwick Rule (invest resource rents in other assets) over the last
30 years? We employ time series data on investment and rents on exhaustible resource
extraction for 70 countries to answer this question. The results are striking: Venezuela,
Trinidad and Tobago, and Gabon would all have as much produced capital as South Korea,
while Nigeria would have five times its current level. A specific rule for sustainability –
maintain positive constant genuine investment – is shown to lead to unbounded consumption.
Key words: Hartwick rule, exhaustible resources, sustainable development
JEL classifications: Q01, Q32, Q56
1. Introduction
There is by now a substantial empirical literature documenting the ‘resource
curse’ or ‘paradox of plenty.’1 Resource-rich countries should enjoy an
advantage in the development process, and yet these countries experienced
lower GDP growth rates post-1970 than less well-endowed countries. In the
most extreme examples, levels of consumption in resource-rich countries are
lower today than they were in 1970 – development has not been sustained by
Pezzey’s (1989) definition.
The Hartwick Rule (Hartwick 1977) offers what Solow (1986) termed a ‘rule
of thumb’ for sustainability in exhaustible resource economies – a maximal
constant level of consumption can be sustained if the value of (net) investment
equals the value of rents on extracted resources at each point in time. For
countries dependent on such wasting assets this rule offers a prescription2 for
sustainable development, a prescription that Botswana in particular has largely
followed with its diamond riches (Lange and Wright 2004).
Drawing on a 30-year time series of resource rent data underlying the World
Development Indicators (World Bank 2004), we construct a ‘Hartwick Rule
counterfactual’: how wealthy, in terms of accumulated produced assets, would
518
KIRK HAMILTON ET AL.
countries be in the year 2000 if they had invested resource rents as suggested by
the Hartwick Rule since 1970? We derive the properties of a constant net (or
genuine3) saving rule – maintaining constant positive genuine savings entails a
path for consumption that rises without bound. We then test a second Hartwick
Rule counterfactual: how large would the value of produced assets be in 2000 if
genuine investments equaled a specified positive constant amount since 1970?
The results of the counterfactual calculations are, in many cases, striking.
We do not provide an empirical test of whether a given saving rule leads to
the expected path for consumption – we know of no set of countries that have
followed the Hartwick Rule strictly. Instead the counterfactual capital stock
estimates indicate to policymakers just how substantial the benefits of following sustainability rules could be in terms of the accumulation of produced
capital and, by implication, economic growth.
2. Generalizing the Hartwick Rule for Sustainability
For our purposes we assume a simple economy with an exhaustible resource
that is essential for production, as in Dasgupta and Heal (1979). For capital
K and resource extraction R, the production function is Cobb–Douglas with
constant returns to scale, F ¼ Ka Rb ; a þ b ¼ 1. Output is divided between
_ Resource extraction
consumption and investment, so that FðK; RÞR¼ C þ K.
1
is assumed to be efficient, implying that S0 ¼ 0 RðsÞds – the initial resource
stock S0 is exhausted over the infinite time horizon. The initial endowment of
produced capital is K0.
For this economy Hamilton and Hartwick (2005) establish two results that
will be used in what follows. Although they report their main result
(expression (3) below) for an economy where the present value of utility is
maximized, it is clear that the result only requires that the economy be
competitive (i.e. producers maximize profits and consumers maximize utility –
for details see Dixit et al. 1980). Competitiveness in the Dasgupta–Heal
economy implies that the resource price is equal to FR, that the interest rate is
FK, and that the Hotelling Rule is satisfied,
F_ R =FR ¼ FK :
(1)
Genuine saving is given by,
G K_ FR R:
(2)
Hamilton and Hartwick (2005) show that consumption in this economy
follows a path defined by,
_
C_ ¼ FK G G:
(3)
CAPITAL ACCUMULATION AND RESOURCE DEPLETION
519
This relates the current change in consumption to the sign and rate of growth
of genuine saving. But, since optimality is not assumed, it also provides the
basis for a general rule for sustainability (non-declining utility). If the policy
rule is to hold G=0 for all time, then this is just the standard Hartwick rule,
_
yielding constant consumption. If G>0 and G=G<F
K for all time, then
consumption is everywhere increasing – we derive a specific instance of this
more general rule in Proposition 1 below.
Hamilton and Hartwick (2005) also derive a useful wealth accounting
result for the competitive Dasgupta–Heal economy with constant returns to
scale,
Z 1
Rs
F ðsÞds
CðsÞe t K
ds:
(4)
W ¼ K þ FR S ¼
t
The following proposition derives a particular instance of a generalized
sustainability rule in the Dasgupta–Heal economy, an instance we will exploit
empirically in a subsequent section.
Proposition 1. If a>b and G ¼ G>0
8t for constant G<aFð0Þ
then conh
i
a1
a
sumption is given by C ¼ 1)a ln ð1 aÞðK0 =Rð0ÞÞ t þ 1 G þ aFð0Þ G.
Initial consumption will be lower than on the Hartwick Rule (G=0)
path,
but consumption increases without bound. Wealth on this path is greater
than under the standard Hartwick Rule, and maximum wealth is independent
then consumption follows the conof the initial resource stock S0. If G=0
stant Solow (1974)–Hartwick (1977) path.
Proof. See Annex I.
This result can be compared with the constant gross saving rate rule of
Asheim and Buchholz (2004) and the constant net saving rate rules of Pezzey
(2004) and Hamilton and Withagen (forthcoming).
There is an obvious corollary to this proposition.
then C<0
_
Corollary 1. If G ¼ G<0
8t for constant G,
8t.
Proof. The sign of C_ follows from expression (A12) in Annex I. From the
expression for C in Proposition 1 it follows that consumption will fall to zero
in finite time.
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KIRK HAMILTON ET AL.
In any simple competitive economy with produced capital and an
exhaustible resource, constant negative genuine saving leads to unsustainability. We now turn to empirical tests based on Proposition 1.
3. Linking Theory to Empirical Tests
The obvious empirical test of the extended Hartwick Rule of Proposition 1
would be to identify a set of countries where, historically, genuine saving was
equal to a non-negative constant, then test whether observed consumption
actually followed the predicted path.4 However, we are not aware of any set
of countries that have followed such a rule and the World Development
Indicators data set indicates that levels of genuine saving are highly variable
across countries and across time. Our alternative test hypothesizes that if
countries had actually invested their resource rents in other assets, then the
value of these assets should have grown commensurately over time.
The most important assets in which resource rents could be invested are
produced capital, human capital, and foreign financial assets. If resource-rich
countries had been investing resource rents in human and foreign financial
assets, while financing investments in produced capital out of other savings,
then we would expect to see relatively higher levels of human capital and
lower levels of foreign debt in these countries compared with resource-poor
nations. Tables I and II compare accumulated schooling and the growth rate
of foreign debt in resource-rich5 and resource-poor countries.
We conclude there is little evidence that resource-rich countries were
investing their resource wealth in human or foreign financial assets. If they
were investing resource rents, then it must have been in produced assets. The
empirical test presented below, therefore, compares an estimate of the actual
(year 2000) value of produced assets in each country exploiting exhaustible
Table I. Average school years per capita
Resource-rich developing countries
Resource-poor developing countries
1970
1980
1990
2000
3.34
2.64
4.03
3.97
4.90
4.89
5.79
5.47
Source: Barro and Lee (2000).
Table II. Annual growth rate of external debt (constant prices)
Resource-rich developing countries
Resource-poor developing countries
Source: World Bank (2004).
1970–1980
1980–1990
1990–2000
1970–2000
10%
11%
4%
5%
)1%
5%
5%
7%
CAPITAL ACCUMULATION AND RESOURCE DEPLETION
521
resources with its hypothetical value if historically observed resource rents
had been wholly invested in produced capital. It is a test of a ‘genuine
investment’ rule. We also compare estimated year 2000 produced assets with
the hypothetical level that would be obtained under a constant positive
genuine investment rule as suggested by Proposition 1.
There is a potential divergence between our empirical methods and the
theory of the preceding section. In the autarkic Dasgupta–Heal economy
presented above, the choice of policy rule also determines the level and path
of resource rents (FR). By using historical rents in our calculations we are
clearly diverging from the theory. However, in most instances we would
expect resource exporters to be price takers facing exogenously varying
international prices, which favors using historical rents in our analysis. If
resource prices change exogenously, a further adjustment to saving to reflect
future capital gains on resource exports is required (see Vincent et al. 1997),
but Hamilton and Bolt (2004) show that the adjustment is typically small if
historical price trends are extrapolated.
Other potential divergences between theory and empirical methods should
be noted: (i) Weitzman and Löfgren (1997) show that exogenous technological change should be reflected in genuine saving as the rate of technological change times the present value of future GNI – however, rates of total
factor productivity growth in developing countries have historically been
very small or even negative (Collins and Bosworth 1996); and (ii) Hartwick
(1993) and Hamilton (2001) show that resource discoveries should be valued
at marginal discovery cost in this model – however, if marginal and average
discovery costs do not diverge too widely than the standard national
accounts measures of saving, which treat discovery costs as a element of
investment, adequately capture discoveries.
4. Hypothetical Estimates of Capital Stocks
The question we wish to test empirically is: How rich in produced assets
would countries be today if they had actually invested resource rents in
produced capital over the last 30 years? In order to examine a variety of
counterfactuals, we construct a baseline and three alternative estimates of
produced capital stock for the year 2000 using data covering 1970–2000: (i) a
baseline capital stock derived from observed historical investment series and
a Perpetual Inventory Model (PIM); (ii) a capital stock derived assuming
zero genuine investment (the standard Hartwick Rule); (iii) a capital stock
derived from the constant genuine investment rule; and (iv) a capital stock
derived from the maximum of observed net investment and the investment
required under the constant genuine investment rule. All investment and
resource rent series are measured in constant 1995 US dollars at nominal
exchange rates.
522
KIRK HAMILTON ET AL.
The PIM is a model for estimating produced capital series by accumulating
net acquisitions of capital over time. It is easily implemented since it requires
only investment data and information on the service lives of assets and their
depreciation rates. The PIM is the method adopted by most OECD countries
to construct estimates of capital stocks (Bohm et al. 2002; Mas et al. 2000;
Ward 1976).
For each country the baseline capital stock is estimated using the PIM as,
Kt ¼
T1
X
Its ð1 cÞs :
(5)
s¼0
Here I is gross investment, while the average asset service life T is assumed
to be 20 years ,and the depreciation rate (c) 5% – these are held constant
across countries and over time. This formulation of the PIM combines
exponential depreciation of capital at rate c with ‘one-hoss shay’ depreciation at the end of the service life (i.e. the value of depreciated capital drops
to zero when it reaches age T+1). The latter assumption obviates the need
for any estimate of ‘year 0’ base capital stock. The 20-year service life is
roughly the weighted average of long-lived assets such as structures and
shorter-lived assets such as machinery and equipment. Using historical
investment series from the World Development Indicators (World Bank
2004), we use expression (5) to estimate baseline capital stocks from 1970 to
2000.
Since net investment is Ns Ks ) Ks-1, we can write the following
expression for our year 2000 capital stock estimate,
K2000 ¼ K1970 þ
2000
X
Ns :
(6)
s¼1971
Here K1970 is derived from the PIM (expression 5). Expression (6) is trivially
true for our estimated baseline capital stock series. However, if there were an
alternative net investment series Nest
s , then we could use expression (6) in the
est
– this is how we construct our counterobvious manner to calculate K2000
factual capital stock estimates for year 2000.
For gross investment I, genuine investment I G, net investment N, depreciation of produced capital D and resource depletion R we have the following
basic identities at any point in time:
IG
s Is Ds Rs
Ns Is Ds ¼ IG
s þ Rs :
CAPITAL ACCUMULATION AND RESOURCE DEPLETION
523
Using the latter expression for net investment, we calculate alternative
measures of year 2000 capital stock under varying assumptions about the
level of genuine investment. For constant IG , we estimate the counterfactual
estimates of produced capital for each country as:
K2000 ¼ K1970 þ
2000
X
ðIG þ Rs Þ:
(7)
maxðNs ; IG þ Rs Þ:
(8)
s¼1971
K
2000 ¼ K1970 þ
2000
X
s¼1971
We calculate two versions of K* in what follows – one with IG ¼ 0 (the
standard Hartwick rule), and a second with IG equal to a constant 5% of
1987 GDP (the policy rule of Proposition 1). The choice of a particular level
of constant genuine investment is obviously arbitrary. We use 5% of 1987
GDP for the following reasons: (i) there is some logic to choosing the midpoint of the time series of data from 1970 to 2000, but 1987 is a slightly better
choice, falling after the early 1980s recession, after the collapse of oil prices in
1986, and before the early 1990s recession; and (ii) a 5% genuine investment
rate is roughly the average achieved by low-income countries over 1970–
2000. Since the elasticity of output with respect to produced capital a is
implicitly greater than 0.5 in the theoretical model of the preceding section,
the choice of 5% of GDP ensures that the feasibility condition G<aF(0)
of
Proposition 1 is satisfied.
Resource depletion is estimated as the sum of total rents on the extraction
of the following commodities: crude oil, natural gas, coal, bauxite, copper,
gold, iron, lead, nickel, phosphate, silver, and zinc. While the underlying
theory suggests that scarcity rents are what should be invested under the
Hartwick Rule (i.e. price minus marginal extraction cost), the World Bank
data do not include information on marginal extraction costs. Since increasing
marginal extraction costs are likely to hold, this gives an upward bias to the
hypothetical capital stock estimates under the genuine investment rules.
When comparing estimates of the stock of produced capital for different
countries, it is worth noting that the PIM underestimates the capital stock for
countries with very old infrastructure, as in most European countries. The
value of roads, bridges, and buildings constructed many decades and even
centuries ago is not captured by the PIM. Pritchett (2000) makes a different
point, that low returns on investments in developing countries imply that the
PIM overestimates the value of capital in these countries – while this is a
valid criticism, we are primarily interested in comparing hypothetical stock
524
KIRK HAMILTON ET AL.
Table III. Change in produced capital under varying rules for genuine investment (IG)
IG=0 (% IG=5% of
IG ‡ 5% of Rent/GDP
Produced
average
capital
difference) 1987 GDP
1987 GDP
in 2000, $bn
(% difference) (% difference) (1970–2000) (%)
(1995 dollars)
Nigeria
Venezuela, RB
Congo, Rep.
Mauritania
Gabon
Trinidad and
Tobago
Algeria
Bolivia
Indonesia
Ecuador
Zambia
Guyana
China
Egypt, Arab
Rep.
Chile
Malaysia
Mexico
Peru
Cameroon
South Africa
Jamaica
Colombia
Norway
India
Zimbabwe
United States
Argentina
Togo
Pakistan
Hungary
Morocco
Brazil
United Kingdom
Dominican
Republic
Philippines
Honduras
Ghana
53.5
175.9
13.9
3.0
19.7
13.7
358.9
272.1
57.0
112.3
80.3
182.1
413.6
326.1
78.0
153.7
105.5
238.3
413.6
326.1
116.9
154.0
130.4
239.1
32.6
27.7
25.2
25.0
24.1
23.6
195.4
13.7
540.6
37.7
7.5
2.1
2899.4
159.7
50.6
116.1
)26.5
95.3
312.3
149.3
)62.1
)12.9
80.9
169.8
3.8
158.0
383.4
185.6
)45.0
28.1
83.9
177.5
32.1
158.3
388.0
191.2
5.1
36.2
23.3
12.8
12.5
11.6
11.5
11.4
10.8
9.5
151.4
305.2
975.5
132.3
24.1
349.5
13.4
198.0
456.6
965.4
14.9
16926.7
569.6
3.6
125.6
149.1
93.8
1750.5
2400.1
33.8
)3.0
)52.7
)1.5
37.2
)9.3
50.7
39.9
)19.7
)14.3
)52.9
9.1
)39.8
)6.9
)26.8
)50.7
)43.5
)59.1
)59.0
)32.7
)73.0
31.6
)31.4
35.3
98.1
54.8
109.3
87.8
30.4
24.6
)18.3
64.8
12.9
49.4
22.7
)1.7
8.7
)16.3
)6.6
27.3
)27.9
54.0
6.6
42.2
103.9
67.6
115.8
99.6
39.3
33.0
8.6
89.1
26.1
53.9
55.1
11.1
22.3
7.8
9.1
32.8
1.2
9.5
8.3
8.2
7.5
6.5
6.5
5.7
5.3
4.3
3.4
3.3
2.7
2.6
2.6
2.2
2.2
2.0
1.9
1.6
1.6
195.0
12.3
16.1
)58.4
)66.9
30.6
)14.5
)29.7
73.2
10.6
8.9
76.7
1.5
1.5
1.0
525
CAPITAL ACCUMULATION AND RESOURCE DEPLETION
Table III. Continued
IG=0 (% IG=5% of
IG‡ 5% of
Rent/GDP
Produced
average
capital
difference) 1987 GDP
1987 GDP
in 2000, $bn
(% difference) (% difference) (1970–2000) (%)
(1995 dollars)
Fiji
Benin
Senegal
Thailand
Haiti
Korea, Rep.
Israel
Cote d’Ivoire
Bangladesh
Rwanda
Sweden
Nicaragua
Spain
Denmark
France
Italy
Finland
Belgium
Niger
Burundi
Portugal
Costa Rica
El Salvador
Hong Kong,
China
Kenya
Madagascar
Sri Lanka
Malawi
Uruguay
Luxembourg
Paraguay
Lesotho
Singapore
3.6
4.6
10.0
520.6
2.8
1607.6
215.8
16.1
89.7
3.9
508.0
6.9
1623.6
437.2
3724.7
2711.2
347.6
681.9
3.0
1.6
308.8
24.1
17.1
445.9
)36.5
)72.7
)44.0
)86.3
)62.7
)93.5
)72.8
)21.2
)59.0
)83.2
)31.1
)34.9
)58.9
)33.0
)55.0
)44.8
)40.9
)48.0
9.7
)87.3
)71.0
)80.0
)59.7
)88.6
26.9
)21.7
14.2
)63.6
109.2
)68.6
)31.3
71.1
)12.9
)6.9
35.6
8.1
)15.1
21.9
)1.9
7.5
11.6
2.3
95.2
10.1
)30.8
)30.6
)2.5
)56.4
59.3
10.6
27.5
3.0
109.5
0.9
4.2
108.7
15.5
24.6
36.1
44.8
6.1
28.7
6.9
10.2
23.3
10.4
136.1
30.2
5.7
3.6
24.6
0.9
0.9
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.0
0.0
0.0
0.0
20.1
4.9
41.2
4.6
29.9
43.3
23.7
5.7
314.8
)51.9
)26.9
)88.1
)26.8
)55.5
)63.2
)88.6
)95.7
)92.7
2.0
62.4
)55.4
9.4
22.1
)22.0
)46.6
)79.9
)73.2
20.8
65.5
1.0
68.2
37.2
15.7
3.0
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Notes: Negative entries indicate that hypothetical produced assets would be lower than
observed assets under the specified rule. Column two is calculated using expression (6); column
three using expression (7) with genuine investment equal to 0; column four using expression (7)
with genuine investment equal to 5% of 1987 GDP; column five using expression (8) with
genuine investment equal to 5% of 1987 GDP.
526
KIRK HAMILTON ET AL.
levels within countries under different investment rules, which makes the
point less salient.
5. Empirical Results
How rich in produced assets would countries be in the year 2000 had they
followed the Hartwick Rule since 1970? Based on the preceding methodology,
Table III presents the year 2000 estimated produced capital stock and the
changes in this stock which would result from the alternative investment rules.
The countries shown in this table are those having both exhaustible resources
and a sufficiently long time series of data on gross investment and resource
rents. For reference, the table also shows the average share of resource rents in
GDP over 1970–2000. Note that negative entries in this table imply that
countries actually invested more than the policy rule suggests.
For the standard Hartwick rule Figure 1 scatters resource dependence,
expressed as the average share of exhaustible resource rents in GDP, against
the percentage difference between actual capital accumulation and counterfactual capital accumulation. Using 5% of GDP as the threshold for high
resource dependence, Figure 1 divides countries into the four groups shown.
The top-right quadrant of the graph displays countries with high resource
dependence and a counterfactual capital stock that is higher than the actual
(baseline) capital stock. The bottom-left quadrant displays countries with low
natural resource dependence and baseline capital stock that is higher than
would be obtained under the Hartwick rule. These two quadrants include
most of the countries in our sample, indicating a high negative correlation
Figure 1. Resource abundance and capital accumulation (standard Hartwick rule).
CAPITAL ACCUMULATION AND RESOURCE DEPLETION
527
between resource abundance and the difference between baseline and
counterfactual capital accumulation – a simple regression shows that a 1%
increase in resource dependence is associated with a 9% increased difference
between counterfactual and actual capital. Clearly the countries in the topright quadrant have not been following the Hartwick rule. Economies with
very low levels of capital accumulation despite high rents include Nigeria
(oil), Venezuela (oil), Trinidad and Tobago (oil and gas), and Zambia
(copper) – with the exception of Trinidad and Tobago, all of these countries
experienced declines in real per capita income over 1970–2000. In the
opposite quadrant, economies with low exhaustible resource rent shares but
high levels of capital accumulation include Korea, Thailand, Brazil, and
India. A number of high-income countries are also in this group.
Figure 1 shows that no country with resource rents higher than 15% of
GDP has followed the Hartwick rule. In many cases the differences are huge.
Nigeria, a major oil exporter, could have had a year 2000 stock of produced
capital five times higher than the actual stock. Moreover, if these investments
had taken place, oil would play a much smaller role in the Nigerian economy
today, with likely beneficial impacts on policies affecting other sectors of the
economy.6 Venezuela could have four times as much produced capital. In per
capita terms, the economies of Venezuela, Trinidad and Tobago and Gabon,
all rich in petroleum, could today have a stock of produced capital of roughly
$30,000 per person, comparable to South Korea (see Figure 2).
Consumption rather than investment of resources rents is common in
resource-rich countries, but there are exceptions to the trend. In the bottomright quadrant of Figure 1 are high resource dependence countries which have
invested more than the level of exhaustible resource rents. Indonesia, China,
Egypt, and Malaysia stand out in this group, while Chile and Mexico have
effectively followed the Hartwick Rule in aggregate – growth in produced
capital is completely offset by resource depletion over this 30-year span.
Among the countries with relatively low natural resource dependence and
higher counterfactual capital, we find Ghana (gold, bauxite) and Zimbabwe
(gold). This is indicative of very low levels of capital accumulation in these
economies.
Figure 3 highlights countries which have invested more than their resource
rents (as shown by the negative entries on the left side of the figure), but have
failed to maintain constant genuine investment levels of at least 5% of 1987
GDP (as shown by the entries on the right). Developing countries in this
group include Cote d’Ivoire, Madagascar, Cameroon, and Argentina. A
number of high-income countries also appear in the figure. Sweden could
have a stock of capital 36% higher if it had maintained constant genuine
investment levels at the specified target. The corresponding difference for the
UK is 27%, for Norway 25%, and for Denmark 22%. The generally low
528
KIRK HAMILTON ET AL.
Figure 2. Actual and counterfactual produced capital (per capita) – 2000.
Figure 3. Capital accumulation under the Hartwick and constant net investment rules.
level of genuine investment levels in the Nordic countries is particularly
surprising. Are these countries trading off inter-generational equity against
intra-generational equity? Further research would be required to clarify this,
a question that is beyond the scope of this paper.
CAPITAL ACCUMULATION AND RESOURCE DEPLETION
529
Finally, the next-to-last column in Table III shows the change in produced
assets for countries if they had genuine investments of at least 5% of 1987
GDP. The positive figures indicate that, with the exception of Singapore, all
countries experienced at least one year over 1970–2000 where genuine
investments were less than the prescribed constant level.
6. Conclusions
We have derived a generalization of the Hartwick Rule for sustainability, a rule
that offers the possibility of unbounded consumption. This may be viewed as
more palatable to policy makers than the constant consumption path which
follows from the standard Hartwick Rule. We test whether countries dependent on exhaustible resources were in fact investing resource rents in produced
assets and conclude that, in the majority of cases, they were not.
The Hartwick rule counterfactual calculations show that even a moderate
investment effort, equivalent to the average investment effort of the poorest
countries in the world, could have substantially increased the wealth (in terms
of produced assets) of resource-dependent economies. Of course, for the most
resource-dependent countries such as Nigeria there is nothing moderate about
the implied rate of investment – based on the resource depletion rate of 31% of
GDP reported in the World Development Indicators and consumption of fixed
capital of 6%, a Nigerian gross investment rate of 42% of GDP would have
been required in 1987 under the constant genuine investment rule.
Saving effort is of course not the whole story in sustaining development. As
Sarraf and Jiwanji (2001) document, Botswana’s successful growth strategy
was built upon a whole range of sound macroeconomic and sectoral policies,
underpinned by generally positive political economy. Savings must be channeled into productive investments that can underpin future welfare, rather than
‘showcase’ projects with minimal returns. Botswana’s absorptive capacity for
investment was a real concern to policy makers, who therefore focused on the
quality of public investments financed out of diamond revenues.
The savings rules presented here are appealing in their simplicity. Maintaining a constant positive level of genuine saving will yield a development
path where consumption grows without bound, even as exhaustible resource
stocks are run down. The real world is more complex. Poor countries place a
premium on maintaining consumption levels, with negative effects on saving
– the alternative may be starvation. At the same time financial crises, social
instability, and natural disasters all have deleterious effects on saving.
Holding to a simple policy rule in such circumstances would be no small feat.
Acknowledgements
The opinions expressed are those of the authors and not necessarily those of
the World Bank. The comments of Jean-Christophe Carret, Roberto Martin-
530
KIRK HAMILTON ET AL.
Hurtado and two reviewers are gratefully acknowledged – the usual caveats
apply. Funding by the Swedish International Development Agency is
acknowledged with thanks.
Notes
1. See Auty (2001, Ch. 1) for a good overview. One of the earliest studies was Sachs and
Warner (1995).
2. Note that Asheim et al. (2003) question whether the Hartwick rule is truly prescriptive. This
is partly because a commitment to invest resource rents now cannot commit future
generations to do the same.
3. Hamilton and Clemens (1999) use this term to denote an extended measure of net saving.
4. Ferreira and Vincent (2005) do something similar to test whether current genuine saving
predicts future changes in consumption.
5. We divide the set of countries which exploit exhaustible resources into resource-rich
(exhaustible resource rents greater or equal to 5% of GNI) and resource-poor subsets.
While China would technically be resource-rich by this definition, we exclude it from both
subsets because it is highly atypical – a resource-rich country combining rapid industrial
growth with large current account surpluses.
6. We are grateful to Alan Gelb for pointing this out.
7. Note that, since F(0)=F(K0, R(0)), we do not have an analytic solution for R(0) in
expression (A9).
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Annex I: Proof of Proposition 3
Expressions (A1)–(A3) establish some basic properties of the path defined by the constant net
saving rule:
K€ ¼ F_ R R þ FR R_
K_ FR R ¼ G)
(A1)
so that
C_ ¼ FK K_ þ FR R_ K€ ¼ FK K_ F_ R R ¼ FK G:
(A2)
Constant returns to scale implies that,
C ¼ F K_ ¼ FK K þ FR R K_ ¼ FK K G:
The Hotelling rule is used to derive the following expression for the path of R:
(A3)
532
KIRK HAMILTON ET AL.
R_
G
F_ R ¼ FRR R_ þ FRK K_ ¼ FR FK ) ¼ FK þ :
R
K
(A4)
The growth rates of K and F are derived as follows:
K_ b
G
K_ ¼ FR R þ G ¼ bF þ G ) ¼ FK þ
K a
K
(A5)
F_
K_
R_ G
¼a þb ¼ :
F
K
R K
(A6)
Subtracting (A5) from (A4) we have,
a
d K
1 K
K
¼ FK ¼
; which has solution;
dt R
a R
R
1
i1a
K h
¼ ð1 aÞt þ ðK0 =Rð0ÞÞ1a :
R
(A7)
It will be useful in what follows to derive the integral of the discount factor
begin by subtracting (A5) from (A6),
d
dt ðF=KÞ
F=K
R1
0
e
Rs
0
FK ds
ds. We
Rt
ba a
FK ds
b
F
K0 b
¼
:
¼ FK ) e 0
a
K
Fð0Þ
Rt
ba h
iba
a ða1Þba
FK ds
K0
0
Since KF b ¼ K
,
(A7)
implies
that
e
¼ Fð0Þ
ð1 aÞt þ ðK0 =Rð0ÞÞ1a . We
R
can therefore derive,
a b K0 b aða bÞ
1
:
(A8)
e
ds ¼ 2
¼
2
Rð0Þ
F
b
b
K ð0Þ
0
R t G
R t G
F ds
ds
Expression (A4) implies that R ¼ Rð0Þe 0 K K , while (A6) implies that F ¼ Fð0Þe 0 K . Now
expression (4) and expression (A3) can be used along with the preceding expressions for R and
F to derive the following expression for initial wealth:
Z 1
Rs
F ds
Wð0Þ ¼ K0 þ FR ð0ÞS0 ¼
Ce 0 K ds
0
Z 1
Z 1
Rs
Rs
F ds
0 FK ds ds
FK Ke 0 K ds Ge
¼
Z
1
Rs
0
FK ds
0
¼
aFð0Þ
Rð0Þ
0
Z1
Rð0Þe
Rs
FK G
Kds
0
ds 0
aða bÞ
1 G
FK ð0Þ
b2
aFð0ÞS0 aða bÞ
1 G
¼
2
Rð0Þ
F
b
K ð0Þ
Since FR(0)S0=b Ka0R(0)b ) 1S0, this expression can be solved for R(0) to yield,
CAPITAL ACCUMULATION AND RESOURCE DEPLETION
1
1 1
aða bÞ
1 a
G
RG ð0Þ ¼ ða bÞa Sa0 K0 K0 þ
FK ð0Þ
b2
1a
ða bÞ 1 G
¼ RH ð0Þ 1 þ
:
Fð0Þ
b2
533
(A9)
Here superscript G denotes values on the path for the constant savings rule, while superscript H
denotes values on the Hartwick rule (G=0)
path.7 Feasibility (positive initial period resource
G
it
extraction) requires that a >b. Since C ð0Þ ¼ FG ð0Þ FRG ð0ÞRG ð0Þ G ¼ aFG ð0Þ G,
G
follows (i) that CG ð0Þ<CH ð0Þ, and (ii) that G<aF
ð0Þ is necessary for feasibility (positive
initial period consumption). This implies that
1
aða bÞ a
<RG ð0Þ<RH ð0Þ:
RH ð0Þ 1 þ
b2
(A10)
Initial resource extraction is lower on the constant genuine saving path than on the Hartwick
rule path, and feasibility ensures a strict lower limit for this value.
Expression (A9) implies that
WG ð0Þ ¼
a
1 K0 b K0 þ
G:
ða bÞ
b Rð0Þ
(A11)
Total wealth is therefore greater under the constant savings rule than under the Hartwick
Rule. Note that total wealth is independent of the initial resource endowment S0 under the
G
Hartwick Rule. Feasibility (G<aF
ð0Þ) implies that,
a
a
a
a
K0 <WG ð0Þ<
þ
K0 or; WH ð0Þ<WG ð0Þ<WH ð0Þ þ K0 :
ða bÞ
ab b
b
Total wealth under the constant savings rule is therefore constrained by bounds that are
independent of initial resource endowment.
Finally, (A2) and (A7) imply that,
a1
h
i1
K
C_ ¼ FK G ¼ a
G ¼ a ð1 aÞt þ ðK0 =Rð0ÞÞ1a G
R
(A12)
so that, by integrating and applying expression (A3),
C¼
h
i
a
ln ð1 aÞðK0 =Rð0ÞÞa1 t þ 1 G þ aFð0Þ G
1a
(A13)
Expression (Al3) implies that consumption increases without bound under the constant
savings rule.
QED.