Capital Accumulation and Resource Depletion: A Hartwick Rule Counterfactual

Transcription

1 Environmental & Resource Economics (2006) 34: Ó Springer 2006 DOI /s Capital Accumulation and Resource Depletion: A Hartwick Rule Counterfactual KIRK HAMILTON 1, *, GIOVANNI RUTA 1 and LIAILA TAJIBAEVA 2 1 Environment Department, The World Bank, 1818 H St NW, 20433, Washington DC, USA; 2 Department of Applied Economics, University of Minnesota, Minneapolis, MN, USA; *Author for correspondence ( khamilton@worldbank.org) 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 prescription 2 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

2 518 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 genuine 3 ) 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 K a R b ; a þ b ¼ 1. Output is divided between consumption and investment, so that FðK; RÞ ¼C þ _K. Resource extraction is assumed to be efficient, implying that S 0 ¼ R 1 0 RðsÞds the initial resource stock S 0 is exhausted over the infinite time horizon. The initial endowment of produced capital is K 0. 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 F R, that the interest rate is F K, and that the Hotelling Rule is satisfied, _F R =F R ¼ F K : (1) Genuine saving is given by, G _K F R R: (2) Hamilton and Hartwick (2005) show that consumption in this economy follows a path defined by, _C ¼ F K G _G: (3)

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, W ¼ K þ F R S ¼ Z 1 t R CðsÞe s FKðsÞds t ds: (4) 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 consumption is given by C ¼ 1)a a h i ln ð1 aþðk 0=Rð0ÞÞ a 1 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 of the initial resource stock S 0.If G=0 then consumption follows the constant 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. Corollary 1. If G ¼ G<0 8t for constant G, then _C<0 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.

4 520 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 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-rich 5 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 Source: Barro and Lee (2000). Table II. Annual growth rate of external debt (constant prices) Resource-rich developing countries 10% 4% )1% 5% Resource-poor developing countries 11% 5% 5% 7% Source: World Bank (2004).

5 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 (F R ). 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 Lo fgren (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 : (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.

6 522 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, K t ¼ XT 1 I t s ð1 cþ s : (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 Since net investment is N s K s ) K s-1, we can write the following expression for our year 2000 capital stock estimate, K 2000 ¼ K 1970 þ X2000 s¼1971 N s : (6) Here K 1970 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 N est s, then we could use expression (6) in the est obvious manner to calculate K 2000 this is how we construct our counterfactual capital stock estimates for year 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: I G s I s D s R s N s I s D s ¼ I G s þ R s:

7 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 I G, we estimate the counterfactual estimates of produced capital for each country as: K 2000 ¼ K 1970 þ X2000 ði G þ R s Þ: (7) s¼1971 K 2000 ¼ K 1970 þ X2000 s¼1971 maxðn s ; I G þ R s Þ: (8) We calculate two versions of K* in what follows one with I G ¼ 0 (the standard Hartwick rule), and a second with I G 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 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

8 524 KIRK HAMILTON ET AL. Table III. Change in produced capital under varying rules for genuine investment (I G ) Produced I G =0 (% I G =5% of I G 5% of Rent/GDP capital difference) 1987 GDP 1987 GDP average in 2000, $bn (% difference) (% difference) ( ) (%) (1995 dollars) Nigeria Venezuela, RB Congo, Rep Mauritania Gabon Trinidad and Tobago Algeria Bolivia Indonesia ) Ecuador Zambia Guyana China )62.1 ) Egypt, Arab ) Rep. Chile ) Malaysia )52.7 ) Mexico ) Peru Cameroon 24.1 ) South Africa Jamaica Colombia ) Norway ) India )52.9 ) Zimbabwe United States ) Argentina ) Togo 3.6 ) Pakistan )50.7 ) Hungary ) Morocco 93.8 )59.1 ) Brazil )59.0 ) United Kingdom ) Dominican 33.8 )73.0 ) Republic Philippines )58.4 ) Honduras 12.3 )66.9 ) Ghana

9 CAPITAL ACCUMULATION AND RESOURCE DEPLETION 525 Table III. Continued Produced capital in 2000, $bn (1995 dollars) I G =0 (% difference) I G =5% of 1987 GDP (% difference) I G 5% of 1987 GDP (% difference) Fiji 3.6 ) Benin 4.6 )72.7 ) Senegal 10.0 ) Thailand )86.3 ) Haiti 2.8 ) Korea, Rep )93.5 ) Israel )72.8 ) Cote d Ivoire 16.1 ) Bangladesh 89.7 )59.0 ) Rwanda 3.9 )83.2 ) Sweden ) Nicaragua 6.9 ) Spain )58.9 ) Denmark ) France )55.0 ) Italy ) Finland ) Belgium ) Niger Burundi 1.6 ) Portugal )71.0 ) Costa Rica 24.1 )80.0 ) El Salvador 17.1 )59.7 ) Hong Kong, )88.6 ) China Kenya 20.1 ) Madagascar 4.9 ) Sri Lanka 41.2 )88.1 ) Malawi 4.6 ) Uruguay 29.9 ) Luxembourg 43.3 )63.2 ) Paraguay 23.7 )88.6 ) Lesotho 5.7 )95.7 ) Singapore )92.7 ) Rent/GDP average ( ) (%) 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.

10 526 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 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).

11 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 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

12 528 KIRK HAMILTON ET AL. Figure 2. Actual and counterfactual produced capital (per capita) 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.

13 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 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-

14 530 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(K 0, R(0)), we do not have an analytic solution for R(0) in expression (A9). References Asheim, G. and W. Buchholz (2004), A General Approach to Welfare Measurement Through National Income Accounting, Scandinavian Journal of Economics 106, Asheim, G. B., W. Buchholz and C. Withagen (2003), The Hartwick Rule: Myths and Facts, Environmental and Resource Economics 25, Auty, R., ed. (2001), Resource Abundance and Economic Development. WIDER Studies in Development Economics. Oxford: Oxford University Press. Bohm, B., A. Gleiss, M. Wagner and D. Ziegler (2002), Dissagregated capital stock estimation for Austria methods, concepts and results, Applied Economics 34, Collins, S. M. and B. P. Bosworth (1996), Economic Growth in East Asia: Accumulation versus Assimilation, Brookings Papers on Economic Activity 2, Dasgupta, P. and G. Heal (1979), Economic Theory and Exhaustible Resources. Welwyn: Cambridge University Press. Dixit, A., P. Hammond and M. Hoel (1980), On Hartwick s Rule for Regular Maximin Paths of Capital Accumulation and Resource Depletion, Review of Economic Studies XLVII, Ferreira, S. and J. Vincent (2005), Genuine Savings: Leading Indicator of Sustainable Development?, Economic Development and Cultural Change 53, Hamilton, K. (2001), ÔThe Sustainability of Extractive EconomiesÕ, in R. Auty, ed., Resource Abundance and Economic Development. Oxford: OUP. Hamilton, K. and K. Bolt (2004), Resource Price Trends and Development Prospects, Portuguese Economic Journal 3, Hamilton, K. and M. Clemens (1999), Genuine Savings Rates in Developing Countries, World Bank Economic Review 13(2), Hamilton, K. and J. M. Hartwick (2005), Investing Exhaustible Resource Rents and the Path of Consumption, Canadian Journal of Economics 38(2),

15 CAPITAL ACCUMULATION AND RESOURCE DEPLETION 531 Hamilton, K. and C. Withagen (forthcoming), Savings Growth and the Path of Utility, Canadian Journal of Economics. Hartwick, J. M. (1977), Intergenerational Equity and the Investing of Rents from Exhaustible Resources, American Economic Review 66, Hartwick, J. M. (1993), ÔNotes on Economic Depreciation of Natural Resource Stocks and National AccountingÕ, in A. Franz and C. Stahmer, eds., Approaches to Environmental Accounting. Heidelberg: Physica-Verlag. Lange, G.-M. and M. Wright (2004), Sustainable Development in Mineral Economies: The Example of Botswana, Environment and Development Economics 9(4), Mas, Matilde, Francisco Perez and Ezequiel Uriel (2000), Estimation of the stock of capital in Spain, Review of Income and Wealth 46:1, Pezzey, J. (1989), Economic Analysis of Sustainable Growth and Sustainable Development, Environment Dept. Working Paper No. 15. Washington: The World Bank. Pezzey, J. (2004), Exact Measures of Income in a Hyperbolic Economy, Environment and Development Economics 9(4), Pritchett, L. (2000), The Tyranny of Concepts: CUDIE (Cumulated, Depreciated, Investment Effort) is not Capital, Journal of Economic Growth 5(December), Sachs, J. and A. Warner (1995), Natural Resource Abundance and Economic Growth, Development Discussion Paper No. 517a, Harvard Institute for International Development. Sarraf, M. and M. Jiwanji (2001), Beating the Resource Curse: The Case of Botswana, Environment Department Working Papers, Environmental Economics Series No. 83. Washington: The World Bank. Solow, R. (1974), Intergenerational Equity and Exhaustible Resources, Review of Economic Studies 41, Symposium on the Economics of Exhaustible Resources. Solow, R. (1986), On the Intergenerational Allocation of Natural Resources, Scandinavian Journal of Economics 88(1), Vincent, J., T. Panayotou and J. Hartwick (1997), Resource Depletion and Sustainability in Small Open Economies, Journal of Environmental Economics and Management 33, Ward, M. (1976), The Measurement of Capital. The Methodology of Capital Stock Estimates in OECD Countries, OECD, Paris. Weitzman, M. and K.-G. Lo fgren (1997), On the Welfare Significance of Green Accounting as Taught by Parable, Journal of Environmental Economics and Management 32, World Bank (2004), World Development Indicators Washington: The World Bank. Annex I: Proof of Proposition 3 Expressions (A1) (A3) establish some basic properties of the path defined by the constant net saving rule: _K F R R ¼ G) K ¼ _F R R þ F R _R so that _C ¼ F K _K þ F R _R K ¼ F K _K _F R R ¼ F K G: Constant returns to scale implies that, C ¼ F _K ¼ F K K þ F R R _K ¼ F K K G: (A1) (A2) (A3) The Hotelling rule is used to derive the following expression for the path of R:

16 532 KIRK HAMILTON ET AL. _F R ¼ F RR _R þ F RK _K ¼ F R F K ) _ R R ¼ F K þ G K : (A4) The growth rates of K and F are derived as follows: _K ¼ F R R þ G ¼ bf þ G ) _ K K ¼ b a F K þ G K (A5) _F F ¼ a _ K K þ b _ R R ¼ G K : (A6) Subtracting (A5) from (A4) we have, d dt K R ¼ 1 a F K K R ¼ K a ; which has solution; R K R ¼ h i ð1 aþt þ ð K 0=Rð0ÞÞ 1 a 1 1 a : (A7) It will be useful in what follows to derive the integral of the discount factor R R 1 0 e s FKds 0 ds. We begin by subtracting (A5) from (A6), R t F Kds d dt ðf=kþ F=K ¼ b a F K ) e 0 ¼ F a b a K 0 b: K Fð0Þ F Since a b K ¼ R K ða 1Þ a b R, (A7) implies that e t FKds a 0 ¼ K0 b h i Fð0Þ ð1 aþt þðk 0 =Rð0ÞÞ 1 a a b.we can therefore derive, Z 1 R e s FKds 0 ds ¼ a b K b 0 aða bþ 1 0 b 2 ¼ Rð0Þ b 2 F K ð0þ : (A8) R Expression (A4) implies that R ¼ Rð0Þe t R FK G ds t G 0 K, while (A6) implies that F ¼ Fð0Þe 0 K ds. 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: Wð0Þ ¼K 0 þ F R ð0þs 0 ¼ ¼ Z 1 0 ¼ afð0þ Rð0Þ Z 1 R F K Ke s FKds 0 ds Z Ce R s 0 FKds ds Z 1 0 Ge R s 0 FKds ds R Rð0Þe s FK G ds 0 K aða bþ 1 ds b 2 G F K ð0þ ¼ afð0þs 0 aða bþ 1 Rð0Þ b 2 G F K ð0þ Since F R (0)S 0 =b K a 0 R(0) b ) 1 S 0, this expression can be solved for R(0) to yield,

17 CAPITAL ACCUMULATION AND RESOURCE DEPLETION 533 R G ð0þ ¼ða bþ 1 1 a Sa0 aða bþ 1 1 K 0 K 0 þ b 2 G F K ð0þ a ¼ R H ða bþ 1 1 a: ð0þ 1 þ b 2 G Fð0Þ (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 extraction) requires that a >b. Since C G ð0þ ¼F G ð0þ F R Gð0ÞR G ð0þ G ¼ af G ð0þ G, it follows (i) that C G ð0þ<c H ð0þ, and (ii) that G<aF G ð0þ is necessary for feasibility (positive initial period consumption). This implies that R H aða bþ 1 a<r ð0þ 1 þ G b 2 ð0þ<r H ð0þ: (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 W G ð0þ ¼ a K 0 þ 1 K b 0 G: (A11) ða bþ b Rð0Þ 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 S 0 under the Hartwick Rule. Feasibility ( G<aF G ð0þ) implies that, a ða bþ K 0<W G ð0þ< a a b þ a K 0 or; W H ð0þ<w G ð0þ<w H ð0þþ a b b K 0: 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, _C ¼ F K G ¼ a K a 1 h i 1 G ¼ a ð1 aþt þðk 0 =Rð0ÞÞ 1 a G (A12) R so that, by integrating and applying expression (A3), C ¼ a h i 1 a ln ð1 aþðk 0=Rð0ÞÞ a 1 t þ 1 G þ afð0þ G (A13) Expression (Al3) implies that consumption increases without bound under the constant savings rule. QED.