Analytical Studies Branch Research Paper Series
Accounting for Natural Capital in Productivity of the Mining and Oil and Gas Sector

by Pat Adams and Weimin Wang
Enterprise Statistics Division, Statistics Canada
Economic Analysis Division, Statistics Canada

Release date: December 14th, 2015 Correction date: (if required)

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Abstract

This paper presents a growth accounting framework in which subsoil mineral and energy resources are recognized as natural capital input into the production process. It is the first study of its kind in Canada. Firstly, the income attributable to subsoil resources, or resource rent, is estimated as a surplus value after all extraction costs and normal returns on produced capital have been accounted for. The value of a resource reserve is then estimated as the present value of the future resource rents generated from the efficient extraction of the reserve. Lastly, with extraction as the observed service flows of natural capital, multifactor productivity (MFP) growth and the other sources of economic growth can be reassessed by updating the income shares of all inputs, and then, by estimating the contribution to growth coming from changes in the value of natural capital input.

This framework is then applied to the Canadian oil and gas extraction sector. The empirical results show that, in Canada, adding subsoil resources into production as natural capital reduces the negative MFP growth over the study period. Overall, by including subsoil resources, MFP declines by 1.5% per year over the 1981-to-2009 period, compared to a 2.2% decline without including these resources. During the same period, the real value-added growth in this industry was 2.3% per year, of which about 0.3 percentage points or 15% comes from natural capital.

Keywords: Natural resource, natural capital, resource rent, productivity

Executive summary

This paper presents a growth accounting framework in which subsoil mineral and energy resources are recognized as natural capital input into production; as such, income attributable to natural capital and value of subsoil resource reserves are estimated, and multifactor productivity (MFP) growth and the sources of economic growth are reassessed. It is the first study of its kind in Canada.

In the paper, income attributable to natural capital, or the resource rent, is first estimated. The resource rent is defined as a surplus value after all extraction costs and normal returns on produced capital have been accounted for. For the calculation of the resource rent, a rate of return on produced capital needs to be used to estimate the value of services derived from natural capital in the production process. This paper uses the long-term average of the internal rate of return on produced capital in the non-mining business industries as a whole to calculate the normal returns on produced capital in a mining industry. This is derived from the internal rate of return taken from the Canadian Productivity Accounts. By doing so, the growth accounting framework used herein remains consistent with the remainder of the MFP estimates in other industries, in that it makes use of the internal rates of return throughout to assess the cost of capital and uses what might be called an endogenous approach based on available data on rates of return. In this system, the surplus profits are zero in all business industries.

The measured resource rents can then be used for the estimation of resource reserve values using an income approach. Specifically, the value of a resource reserve is calculated as the sum of the present value of expected future resource rents generated from extracting the reserve. A discount rate needs to be chosen for this purpose. This paper adopts Hotelling’s rule in this regard. Hotelling’s rule predicts that, along the efficient (optimal) extraction path, the shadow price of a resource reserve grows at the rate of nominal interest rate on a numeraire asset. Using Hotelling’s rule, the value of a resource reserve is simply the product of the present resource rent, and the reserve life calculated using the present extraction amount.

The physical extraction of a resource reserve is used as the natural capital input in the extraction of this resource. The asset-level natural capital inputs are then aggregated into an industry-level measure. Given the resource rent and natural capital input, the industry-level MFP growth and the sources of real value-added growth can then be estimated. The impact of adding natural capital into the production process on the MFP growth would be positive (negative) if the natural capital input grows at a slower (faster) pace than produced capital. Also, the impact of these changes becomes larger (smaller) when the income share of resource rent is higher (lower).

This growth accounting framework is applied to the Canadian oil and gas extraction industry. The empirical results show that, in Canada, adding subsoil resources into production as natural capital reduces the negative MFP growth over the study period. Overall, by including subsoil resources, MFP declines by 1.5% per year over the 1981 to 2009 period, compared to a 2.2% decline without including these resources. During the same period, the real value-added growth in this industry was 2.3% per year, of which about 0.3 percentage points or 15% comes from natural capital.

1 Introduction

This paper has two objectives. The first is to estimate the resource rent generated through the extraction of subsoil mineral and energy resources, as well as the associated monetary value of resource reservesNote 1 in Canadian mining industries MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ what is referred to here as the value of natural capital. The second is to treat subsoil resources themselves as a factor input in resource extractions. This is done by estimating the flow of services derived from natural capital input, and adding it to the value of labour and produced capital inputs in the standard multifactor productivity (MFP) estimating equation. This produces a measure of MFP growth that is more complete, and provides an estimate of the significance of subsoil resources as a source of economic and productivity growth in the Canadian mineral and energy resource sector.

Subsoil mineral and energy resources are treated as non-produced and non-financial assets in the System of National Accounts (SNA). To be consistent, exploration and development expenditures are capitalized as produced capital assets in SNA. Therefore, the value of subsoil resources as non-produced assets reflects only the value of resource scarcity.

The present Canadian Productivity Accounts (CPA) calculate MFP growth as the difference between the growth in output and a weighted average growth of all inputs MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ one of which is the capital derived from investments in fixed assets. The fixed assets included in the accounts for the mining, oil and gas industries include investments in machinery and equipment, structures, and engineering assets such as mine shafts, as well as exploration and development expenditures. Natural capital MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ the value of the resources MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ is not included.

This paper offers a way in which this can be done and provides estimates of MFP growth when the cost of using natural capital is included. Specifically, the resource rent of subsoil assets is calculated as a surplus value after all extraction costs and normal returns on produced capital have been accounted for. The value of a resource reserve is then set equal to the sum of the present value of expected future resource rent flows generated from extracting the resource over its reserve life.

This treatment is akin to recognizing that the value of all the produced capital employed in the mineral industries is not equal to the cost of investments. Normally, it is assumed that well-functioning markets will bring the cost of capital and its value into equilibrium MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ the present value of the stream of earnings that are produced by it. But, on occasion, this will not occur because of the scarcity of assets or imperfections in markets. When that occurs, capital in excess of that derived from the costs of investment is employed in the industry. And that is regarded as the case, particularly in the resource sector where endowments cannot be changed by human activity MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ or at least not in the short run.

Two important parameters are required for valuing subsoil resources. One is the rate of return on produced capital that will be used for calculating the resource rent, and the other is the nominal discount rate that will be used for the net present value (NPV) of a resource reserve. The System of Environmental-Economic Accounting (SEEA) (United Nations et al. 2014, page 145) recommends that the rate of return on produced capital and the discount rate should be equal and suggests using an economy-wide interest rate, derived from returns on government bonds, as the rate of return that should be used on produced capital, as well as the nominal discount rate. This is akin to choosing an arbitrary exogenous rate of return for estimating the value of produced capital services in the MFP estimation process MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ a practice that Statistics Canada does not follow in its productivity accounts for two reasons. The rate of return that is required is the rate that the capital markets would require to cover the cost of capital. Using a government bond rate involves understating the cost of business-sector capital, since it involves greater risk. Secondly, its use generates estimates of surplus that are earned above requirements of capital markets that are difficult to interpret. This method leaves values of surplus across non-resource industries that, to be consistent with the approach adopted here, should also be incorporated into the Multifactor Productivity Program.

This paper uses an assumption that is in accord with the practice used in the CPA. The CPA calculate the internal rate of return on produced capital from the estimates of surplus and produced capital stock at an industry level. This paper assumes that, over a long term, on average, produced capital earns the same rate of return in both the mining industry and non-mining business industries as a whole.Note 2 The internal rate of return on produced capital for the non-mining business industries as a whole can then be used in calculating the cost of capital services for produced capital in the mining industry. In turn, the resource rent in a mining industry can be calculated as the residual of the surplus, estimated from the SNA, minus the produced capital services used in this industry. This approach is consistent with that followed in the CPA, and profit remains zero for all industries except those using natural capital.

Once the resource rent as the surplus is derived, an estimate of the value of natural capital that is the source of this surplus is derived from calculating the NPV of these surpluses. This is calculated using the estimates of resource reserves to estimate the years of remaining life at present extraction rates, and then calculating the NPV of the surplus. The crucial parameter that is required for this analysis is the discount rate.

This paper adopts Hotelling’s rule as the principle in the calculation of the NPV of subsoil resource reserves. Hotelling’s rule defines the optimal extraction path of non-renewable natural resources, and predicts that the net price (unit resource rent) of a non-renewable natural resource is expected to increase at the rate of nominal interest that would be earned by an appropriate asset.Note 3 Under Hotelling’s rule, the real discount rate becomes zero and the corresponding NPV of a subsoil resource reserve would reflect its value to a society, if the source reserve is efficiently extracted.

Alternate choices for the discount rate have been suggested. For example, the SEEA (United Nations et al. 2014) assumes that the unit resource rent is expected to increase at the rate of general inflation. Under this assumption, the real discount rate used would equal the real rate of interest. In this case, the value of a resource reserve would be much smaller than that calculated using Hotelling’s rule. This paper also provides an estimate of the value of natural capital that makes use of this assumption for the purposes of comparison.

The rest of the paper is organized as follows. Section 2 develops a framework for accounting for subsoil resources in production and wealth accumulation. Section 3 presents the empirical results for the Canadian mining industries, and Section 4 concludes.

2 Framework for accounting for subsoil resources

To isolate their contribution in production, subsoil resources are treated as a distinct factor of production in the same manner as labour and produced capital. Kendrick (1976) recommended that capital measures include machinery and equipment, structures, land, inventories and natural-resource capital. Following the recommendation, a Hicksian neutral production function of subsoil resource extraction can be written as

Y = A f ( L , Z K , Z N ) ,            ( 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaiabg2 da9iaadgeacaWGMbGaaiikaiaadYeacaGGSaGaamOwamaaCaaaleqa baGaam4saaaakiaacYcacaWGAbWaaWbaaSqabeaacaWGobaaaOGaai ykaiaacYcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGGaGaaiikaiaaigdacaGGPaaaaa@4A06@

where the output ( Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywaaaa@36D5@  ) is value-added based and a function of labour input ( L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36C8@  ), produced capital input ( Z K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaCa aaleqabaGaam4saaaaaaa@37D3@  ), and natural capital input ( Z N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOwamaaCa aaleqabaGaamOtaaaaaaa@37D6@  ), are augmented by productivity ( A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqaaaa@36BD@  ). For the production function to be well-defined, it is assumed that the marginal products of each factor are increasing ( f / L 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITcaWGMbaabaGaeyOaIyRaamitaiabgwMiZkaaicdaaaaaaa@3D15@ , f / Z K 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITcaWGMbaabaGaeyOaIyRaamOwamaaCaaaleqabaGaam4saaaa kiabgwMiZkaaicdaaaaaaa@3E2A@ , f / Z N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITcaWGMbaabaGaeyOaIyRaamOwamaaCaaaleqabaGaamOtaaaa kiabgwMiZkaaicdaaaaaaa@3E2D@  ) at a decreasing rate ( 2 f / L 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamit amaaCaaaleqabaGaaGOmaaaakiabgsMiJkaaicdaaaaaaa@3EEA@ , 2 f / ( Z K ) 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIy7aaeWa aeaacaWGAbWaaWbaaSqabeaacaWGlbaaaaGccaGLOaGaayzkaaWaaW baaSqabeaacaaIYaaaaOGaeyizImQaaGimaaaaaaa@4187@ , 2 f / ( Z N ) 2 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIy7aaeWa aeaacaWGAbWaaWbaaSqabeaacaWGobaaaaGccaGLOaGaayzkaaWaaW baaSqabeaacaaIYaaaaOGaeyizImQaaGimaaaaaaa@418A@  ), and that all cross-marginal products are increasing ( 2 f / L Z K 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamit aiabgkGi2kaadQfadaahaaWcbeqaaiaadUeaaaGccqGHLjYScaaIWa aaaaaa@4154@ , 2 f / L Z N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamit aiabgkGi2kaadQfadaahaaWcbeqaaiaad6eaaaGccqGHLjYScaaIWa aaaaaa@4157@ , 2 f / Z K Z N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccaWGMbaabaGaeyOaIyRaamOw amaaCaaaleqabaGaam4saaaakiabgkGi2kaadQfadaahaaWcbeqaai aad6eaaaGccqGHLjYScaaIWaaaaaaa@426C@ , 3 f / L Z K Z N 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSGbaeaacq GHciITdaahaaWcbeqaaiaaiodaaaGccaWGMbaabaGaeyOaIyRaamit aiabgkGi2kaadQfadaahaaWcbeqaaiaadUeaaaGccqGHciITcaWGAb WaaWbaaSqabeaacaWGobaaaOGaeyyzImRaaGimaaaaaaa@44A4@  ).

Equation (1) can be applied for the extraction of single or multiple subsoil resources. Logarithmically differentiating (1) yields

Y ˙ Y = α L L ˙ L + α K Z ˙ K Z K + α N Z ˙ N Z N + A ˙ A ,           (2) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaace WGzbGbaiaaaeaacaWGzbaaaiabg2da9iabeg7aHnaaBaaaleaacaWG mbaabeaakmaalaaabaGabmitayaacaaabaGaamitaaaacqGHRaWkcq aHXoqydaWgaaWcbaGaam4saaqabaGcdaWcaaqaaiqadQfagaGaamaa CaaaleqabaGaam4saaaaaOqaaiaadQfadaahaaWcbeqaaiaadUeaaa aaaOGaey4kaSIaeqySde2aaSbaaSqaaiaad6eaaeqaaOWaaSaaaeaa ceWGAbGbaiaadaahaaWcbeqaaiaad6eaaaaakeaacaWGAbWaaWbaaS qabeaacaWGobaaaaaakiabgUcaRmaalaaabaGabmyqayaacaaabaGa amyqaaaacaaMi8UaaGjcVlaayIW7caGGSaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabIca caqGYaGaaeykaaaa@5C67@

where   α L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeiiaiabeg 7aHnaaBaaaleaacaWGmbaabeaaaaa@3936@ , α K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaadUeaaeqaaaaa@3892@  and α N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaS baaSqaaiaad6eaaeqaaaaa@3895@  denote the elasticities of output with respect to labour, produced capital and natural capital, respectively. These elasticities are not observable, but can be derived by imposing the optimization conditions such that, for each factor of input, the value of its marginal products and its user costs are the same. Under the assumption of perfect competition, and given output price ( P Y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGqb GcdaahaaWcbeqaaiaadMfaaaaaaa@3890@  ) and factor input prices ( C J MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4qamaaCa aaleqabaGaamOsaaaaaaa@37BB@  ), the output elasticities can be measured as

P Y Y J = C J       α J ln ( Y ) ln ( J ) = J Y Y J = C J J P Y Y s J ,    for   J = L , Z K , Z N .           (3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa aaleqabaGaamywaaaakmaalaaabaGaeyOaIyRaamywaaqaaiabgkGi 2kaadQeaaaGaeyypa0Jaam4qamaaCaaaleqabaGaamOsaaaakiaabc cacaqGGaGaeyO0H4TaaeiiaiaabccacqaHXoqydaWgaaWcbaGaamOs aaqabaGccqGHHjIUdaWcaaqaaiabgkGi2kGacYgacaGGUbGaaiikai aadMfacaGGPaaabaGaeyOaIyRaciiBaiaac6gacaGGOaGaamOsaiaa cMcaaaGaeyypa0ZaaSaaaeaacaWGkbaabaGaamywaaaadaWcaaqaai abgkGi2kaadMfaaeaacqGHciITcaWGkbaaaiabg2da9maalaaabaGa am4qamaaCaaaleqabaGaamOsaaaakiaadQeaaeaacaWGqbWaaWbaaS qabeaacaWGzbaaaOGaamywaaaacqGHHjIUcaWGZbWaaSbaaSqaaiaa dQeaaeqaaOGaaiilaiaabccacaqGGaGaaeiiaiaabAgacaqGVbGaae OCaiaabccacaqGGaGaamOsaiabg2da9iaadYeacaGGSaGaamOwamaa CaaaleqabaGaam4saaaakiaacYcacaWGAbWaaWbaaSqabeaacaWGob aaaOGaaiOlaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGOaGaae4maiaabMcaaaa@7CC4@

Income and expenditure in extraction can be equated under the assumption of constant returns to scale, i.e.,

P Y Y = J C J J = w H + c P K K + θ P N N ,           (4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa aaleqabaGaamywaaaakiaadMfacqGH9aqpdaaeqaqaaiaadoeadaah aaWcbeqaaiaadQeaaaGccaWGkbGaeyypa0Jaam4DaiaadIeacqGHRa WkcaWGJbGaamiuamaaCaaaleqabaGaam4saaaakiaadUeacqGHRaWk cqaH4oqCcaWGqbWaaWbaaSqabeaacaWGobaaaOGaamOtaaWcbaGaam Osaaqab0GaeyyeIuoakiaacYcacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabsdaca qGPaaaaa@54C8@

where the labour cost is equal to the hours worked ( H MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaaaa@36C4@  ) multiplied by the nominal wage rate ( w MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Daaaa@36F3@  ); the cost of produced capital is equal to its nominal stock value ( P K K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa aaleqabaGaam4saaaakiaadUeaaaa@38A3@  ) multiplied by the unit user cost of produced capital ( c MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yaaaa@36DF@  ); and the user cost of natural capital is equal to its nominal stock value ( P N N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa aaleqabaGaamOtaaaakiaad6eaaaa@38A9@  ) multiplied by the resource rent parameter ( θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUdehaaa@37AD@  ). Equations (3) and (4) show that the output elasticities in (2) can be replaced with the corresponding factor shares ( s L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGmbaabeaaaaa@37EC@ , s K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGlbaabeaaaaa@37EB@ , and s N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGobaabeaaaaa@37EE@  ) in the total value-added, i.e.,

Y ˙ Y = s L L ˙ L + s K Z ˙ K Z K + s N Z ˙ N Z N + A ˙ A .           (5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaace WGzbGbaiaaaeaacaWGzbaaaiabg2da9iaadohadaWgaaWcbaGaamit aaqabaGcdaWcaaqaaiqadYeagaGaaaqaaiaadYeaaaGaey4kaSIaam 4CamaaBaaaleaacaWGlbaabeaakmaalaaabaGabmOwayaacaWaaWba aSqabeaacaWGlbaaaaGcbaGaamOwamaaCaaaleqabaGaam4saaaaaa GccqGHRaWkcaWGZbWaaSbaaSqaaiaad6eaaeqaaOWaaSaaaeaaceWG AbGbaiaadaahaaWcbeqaaiaad6eaaaaakeaacaWGAbWaaWbaaSqabe aacaWGobaaaaaakiabgUcaRmaalaaabaGabmyqayaacaaabaGaamyq aaaacaaMi8UaaGjcVlaac6cacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabwdacaqG Paaaaa@58E6@

To use Equation (5) for growth accounting, the growth of natural capital input and the resource rent associated with the use of natural capital need to be estimated.

2.1 Measuring resource rent

In this paper, the resource rent of natural capital is derived by using a residual value method.Note 4 From Equation (4), the resource rent ( R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaaaa@36CE@  ) generated from extracting a subsoil resource is calculated residually as

R = θ P N N = P Y Y w H c P K K = O S c P K K .           (6) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2 da9iabeI7aXjaadcfadaahaaWcbeqaaiaad6eaaaGccaWGobGaeyyp a0JaamiuamaaCaaaleqabaGaamywaaaakiaadMfacqGHsislcaWG3b GaamisaiabgkHiTiaadogacaWGqbWaaWbaaSqabeaacaWGlbaaaOGa am4saiabg2da9iaad+eacaWGtbGaeyOeI0Iaam4yaiaadcfadaahaa WcbeqaaiaadUeaaaGccaWGlbGaaGjcVlaac6cacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae ikaiaabAdacaqGPaaaaa@5924@

The data required for calculating the resource rent, generated from single subsoil resource extraction, comprise the corresponding gross operating surplus ( O S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGpb Gaam4uaaaa@3852@  ) calculated as nominal value-added net of labour cost, nominal value of produced capital stock, and the unit user cost of produced capital.

The unit user cost of produced capital, which is equal to the sum of a rate of return on and a rate of depreciation of produced capital, needs to be exogenous to the mining industries in order to calculate the resource rent residually. There is no consensus in the literature on the choice of the exogenous rate of return on produced capital.Note 5 The borrowing cost is one proposal. The borrowing cost in financial markets generally reflects the compensation to lenders for the provision of funds and the risk of loans not being returned. For example, a risk-free rate (the internal reference rate between banks), plus a risk premium of 1.5%, is used as the exogenous rate of return on produced capital, in the Dutch national accounts, for the calculation of resource rent in mining (Veldhuizen et al. 2012). Another example is the approach proposed in a cross-country study by Brandt, Schreyer and Zipperer (2013) for the Organisation for Economic Co-operation and Development, in which average extraction costs across countries are used to derive exogenously the resource rent of natural capital. Baldwin and Gu (2007) used a weighted average of the actual long-term debt costs, and the equity rate of return earned in Canada, for the purpose of examining how this approach compares to the endogenous estimate, when deriving capital services and MFP growth in Canada. They find that the two are relatively similar for Canada.

There are several issues related to the use of an exogenous rate of return on produced capital based on financial market information. First, using a flat exogenous rate of return will lead to high volatility in the measured resource rent, and, sometimes, negative resource rent that may not accord with long-run expectations, which are relevant for the derivation of the concept of the user cost of capital. Second, deriving a variable rate from financial market data that corresponds with longer-run expectations is difficult, because short-run financial market fluctuations may not necessarily reflect long-run expectations. Third, a rate of return obtained from financial markets is usually an after-tax measure, and needs to be converted into a before-tax measure; otherwise the resource rent would be overstated. Finally, it should be noted that, for our purposes, consistency is required between the estimates of the mining sector and other industries. Industry revenues and costs may not be equal elsewhere when an exogenous rate of return is used, and a “profit residual” may be generated across industries other than mining. While the “profit residual” is interpreted as the resource rent in a mining industry, it is more difficult to classify the reason or reasons for the residual elsewhere, other than short-run deviations from market clearing, and, therefore, leads to unnecessary white noise in interpreting the estimates for users.

To overcome these issues, this paper describes an alternate way of splitting the operating surpluses into returns on produced capital and returns on natural capital (resource rent) than those suggested by the SEEA. Specifically, the internal rates of return on produced capital are adjusted such that produced capital in a mining industry earns the same rate of return as in the non-mining business sector on average over a long period. 

2.1.1 Resource rent at the commodity level

The industry level at which MFP growth is estimated is more aggregated than the level of commodity data produced in the Environment Accounts at Statistics Canada, and each mining industry at this level involves multiple resources that are estimated separately in the Canadian System of Environmental and Resource Accounts. While the latter involve more detailed data at the commodity level, they are not at the moment fully reconciled to the industry accounts that make up the basis for the MFP estimates. To calculate the resource rent at the industry level used in the CPA, the gross operating surplus and the nominal value of produced capital stock at the commodity level are benchmarked to those at the industry level.

After the benchmarking, the internal rates of return on produced capital for the mining industries at the commodity level and the corresponding adjusted rates are then calculated. At the commodity level, data for produced capital by asset type and associated tax parameters are not readily available. Therefore, the internal rates of return on produced capital are calculated before tax and depreciation and have no asset details. Specifically, the gross internal rate of return on produced capital for commodity i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@  and industry j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@  is defined as

c i j t = O S i j t / ( P i j t K K i j t ) .           (7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGPbGaamOAaiaadshaaeqaaOGaeyypa0ZaaSGbaeaacaWG pbGaam4uamaaBaaaleaacaWGPbGaamOAaiaadshaaeqaaaGcbaWaae WaaeaacaWGqbWaa0baaSqaaiaadMgacaWGQbGaamiDaaqaaiaadUea aaGccaWGlbWaaSbaaSqaaiaadMgacaWGQbGaamiDaaqabaaakiaawI cacaGLPaaaaaGaaiOlaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae4naiaabMcaaa a@52F6@

Resource rent at the commodity level in a mining industry is calculated asNote 6

R i j t = O S i j t c ˜ i j t P i j t K K i j t ,   with   c ˜ i j t = c i j t ( c ¯ B / c ¯ i j ) ,           (8) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGPbGaamOAaiaadshaaeqaaOGaeyypa0Jaam4taiaadofa daWgaaWcbaGaamyAaiaadQgacaWG0baabeaakiabgkHiTiqadogaga acamaaBaaaleaacaWGPbGaamOAaiaadshaaeqaaOGaamiuamaaDaaa leaacaWGPbGaamOAaiaadshaaeaacaWGlbaaaOGaam4samaaBaaale aacaWGPbGaamOAaiaadshaaeqaaOGaaiilaiaabccacaqGGaGaae4D aiaabMgacaqG0bGaaeiAaiaabccacaqGGaGabm4yayaaiaWaaSbaaS qaaiaadMgacaWGQbGaamiDaaqabaGccqGH9aqpcaWGJbWaaSbaaSqa aiaadMgacaWGQbGaamiDaaqabaGcdaqadaqaamaalyaabaGabm4yay aaraWaaSbaaSqaaiaadkeaaeqaaaGcbaGabm4yayaaraWaaSbaaSqa aiaadMgacaWGQbaabeaaaaaakiaawIcacaGLPaaacaGGSaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabIcacaqG4aGaaeykaaaa@6CE5@

where c ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4yayaara WaaSbaaSqaaiaadkeaaeqaaaaa@37E9@  is the sample average of the gross internal rate of return on produced capital for the non-mining business sector, and c ¯ i j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4yayaara WaaSbaaSqaaiaadMgacaWGQbaabeaaaaa@38FF@  is that for the extraction of commodity i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36E5@  in industry j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@ .

2.1.2 Resource rent at the industry level

At the industry level, more data are available; therefore the internal rate of return on produced capital after tax can be estimated. According to the user cost formula for produced capital developed in Christensen and Jorgenson (1969), the internal rate of return on produced capital in an industry ( r i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbGaamiDaaqabaaaaa@3901@  ) can be estimated as

r i t = k ( c i k t K i k t + p k t 1 T i k t K i k t π k t p k t T i k t K i k t δ k p k t 1 K i k t ϕ i t ) k p k t 1 T i k t K i k t ,     T i k t = 1 u t z i k t I T C i k t 1 u t .            (9) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGPbGaamiDaaqabaGccqGH9aqpdaWcaaqaamaaqababaWa aeWaaeaacaWGJbWaaSbaaSqaaiaadMgacaWGRbGaamiDaaqabaGcca WGlbWaaSbaaSqaaiaadMgacaWGRbGaamiDaaqabaGccqGHRaWkcaWG WbWaaSbaaSqaaiaadUgacaWG0bGaeyOeI0IaaGymaaqabaGccaWGub WaaSbaaSqaaiaadMgacaWGRbGaamiDaaqabaGccaWGlbWaaSbaaSqa aiaadMgacaWGRbGaamiDaaqabaGccqaHapaCdaWgaaWcbaGaam4Aai aadshaaeqaaOGaeyOeI0IaamiCamaaBaaaleaacaWGRbGaamiDaaqa baGccaWGubWaaSbaaSqaaiaadMgacaWGRbGaamiDaaqabaGccaWGlb WaaSbaaSqaaiaadMgacaWGRbGaamiDaaqabaGccqaH0oazdaWgaaWc baGaam4AaaqabaGccqGHsislcaWGWbWaaSbaaSqaaiaadUgacaWG0b GaeyOeI0IaaGymaaqabaGccaWGlbWaaSbaaSqaaiaadMgacaWGRbGa amiDaaqabaGccqaHvpGzdaWgaaWcbaGaamyAaiaadshaaeqaaaGcca GLOaGaayzkaaaaleaacaWGRbaabeqdcqGHris5aaGcbaWaaabeaeaa caWGWbWaaSbaaSqaaiaadUgacaWG0bGaeyOeI0IaaGymaaqabaGcca WGubWaaSbaaSqaaiaadMgacaWGRbGaamiDaaqabaGccaWGlbWaaSba aSqaaiaadMgacaWGRbGaamiDaaqabaaabaGaam4Aaaqab0GaeyyeIu oaaaGccaGGSaGaaeiiaiaabccacaqGGaGaamivamaaBaaaleaacaWG PbGaam4AaiaadshaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaGaeyOeI0 IaamyDamaaBaaaleaacaWG0baabeaakiaadQhadaWgaaWcbaGaamyA aiaadUgacaWG0baabeaakiabgkHiTiaadMeacaWGubGaam4qamaaBa aaleaacaWGPbGaam4AaiaadshaaeqaaaGcbaGaaGymaiabgkHiTiaa dwhadaWgaaWcbaGaamiDaaqabaaaaOGaaGjcVlaayIW7caGGUaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGOaGaaeyoaiaabMcaaaa@A96A@

The asset-specific variables used in (9) include the user cost of produced capital ( c k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa aaleaacaWGRbaabeaaaaa@37FB@  ), produced capital stock ( K k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4samaaBa aaleaacaWGRbaabeaaaaa@37E3@  ), asset price ( p K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaaBa aaleaacaWGlbaabeaaaaa@37E8@  ), depreciation rate ( δ k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadUgaaeqaaaaa@38B8@  ), capital gains ( π k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaaeqaaaaa@38D0@  ), the present value of depreciation deductions for tax purposes on a dollar’s investment ( z k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaBa aaleaacaWGRbaabeaaaaa@3812@  ), and the rate of the investment tax credit ( I T C k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamysaiaads facaWGdbWaaSbaaSqaaiaadUgaaeqaaaaa@3982@  ). Other variables are the effective rate of property taxes ( ϕ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqy1dygaaa@37BF@  ) and the corporate income tax rate ( u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyDaaaa@36F1@  ). We then use Equation (9) to calculate the sample averages of the internal rate of return on produced capital for the non-mining business sector ( B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOqaaaa@36BE@  ) and a mining industry ( j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@36E6@  ) as

r ¯ B = t = 1 n r B t / n ,     r ¯ j = t = 1 n r j t / n . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadkeaaeqaaOGaeyypa0ZaaSGbaeaadaaeWaqaaiaa dkhadaWgaaWcbaGaamOqaiaadshaaeqaaaqaaiaadshacqGH9aqpca aIXaaabaGaamOBaaqdcqGHris5aaGcbaGaamOBaaaacaGGSaGaaeii aiaabccacaqGGaGabmOCayaaraWaaSbaaSqaaiaadQgaaeqaaOGaey ypa0ZaaSGbaeaadaaeWaqaaiaadkhadaWgaaWcbaGaamOAaiaadsha aeqaaaqaaiaadshacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aa GcbaGaamOBaaaacaaMi8UaaGjcVlaac6caaaa@560B@

These sample averages can sensibly be related to expectations over the same period. It is usually expected that r ¯ j > r ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadQgaaeqaaOGaeyOpa4JabmOCayaaraWaaSbaaSqa aiaadkeaaeqaaaaa@3B34@  because r ¯ j MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadQgaaeqaaaaa@3820@  includes returns on both produced and natural capital. If this is the case, one can assume that produced capital earns the same rate of return on average over the sample period in these mining industries as in the non-mining business sector.Note 7 The internal rates of return on produced capital in the mining industries with r ¯ j > r ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadQgaaeqaaOGaeyOpa4JabmOCayaaraWaaSbaaSqa aiaadkeaaeqaaaaa@3B34@  are then adjusted by the ratio of the two sample averages. However, it can be the case that the actual data gives r ¯ j r ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadQgaaeqaaOGaeyizImQabmOCayaaraWaaSbaaSqa aiaadkeaaeqaaaaa@3BE1@  in the extraction of some subsoil resources. When this happens, the resource rent in these industries will be zero. For the industries with r ¯ j > r ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadQgaaeqaaOaeaaaaaaaaa8qacqGH+aGppaGabmOC ayaaraWaaSbaaSqaaiaadkeaaeqaaaaa@3B63@ , the adjustment is made asNote 8

r ˜ j t = r j t × r ¯ B r ¯ j   if    r ¯ j > r ¯ B .           (10) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaaia WaaSbaaSqaaiaadQgacaWG0baabeaakiabg2da9iaadkhadaWgaaWc baGaamOAaiaadshaaeqaaOGaey41aq7aaSaaaeaaceWGYbGbaebada WgaaWcbaGaamOqaaqabaaakeaaceWGYbGbaebadaWgaaWcbaGaamOA aaqabaaaaOGaaeiiaiaabccacaqGPbGaaeOzaiaabccacaqGGaGaae iiaiqadkhagaqeamaaBaaaleaacaWGQbaabeaakiabg6da+iqadkha gaqeamaaBaaaleaacaWGcbaabeaakiaayIW7caaMi8UaaiOlaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGOaGaaeymaiaabcdacaqGPaaaaa@5AD8@

For a mining industry with r ¯ j > r ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadQgaaeqaaOGaeyOpa4JabmOCayaaraWaaSbaaSqa aiaadkeaaeqaaaaa@3B34@  , the adjustment made by Equation (10) does not change the pattern over time of the internal rate of return on produced capital ( r j t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa aaleaacaWGQbGaamiDaaqabaaaaa@3902@  ), but ensures that the sample averages of the adjusted rate of return on produced capital in the mining industry is the same as in the non-mining business sector, i.e.,

Average  ( r ˜ i t ) = t = 1 n ( r i t × r ¯ B r ¯ i ) / n = r ¯ B r ¯ i t = 1 n r i t / n = r ¯ B . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyqaiaabA hacaqGLbGaaeOCaiaabggacaqGNbGaaeyzaiaabccacaGGOaGabmOC ayaaiaWaaSbaaSqaaiaadMgacaWG0baabeaakiaacMcacqGH9aqpda WcgaqaamaaqadabaWaaeWaaeaacaWGYbWaaSbaaSqaaiaadMgacaWG 0baabeaakiabgEna0oaalaaabaGabmOCayaaraWaaSbaaSqaaiaadk eaaeqaaaGcbaGabmOCayaaraWaaSbaaSqaaiaadMgaaeqaaaaaaOGa ayjkaiaawMcaaaWcbaGaamiDaiabg2da9iaaigdaaeaacaWGUbaani abggHiLdaakeaacaWGUbaaaiabg2da9maalaaabaGabmOCayaaraWa aSbaaSqaaiaadkeaaeqaaaGcbaGabmOCayaaraWaaSbaaSqaaiaadM gaaeqaaaaakmaalyaabaWaaabmaeaacaWGYbWaaSbaaSqaaiaadMga caWG0baabeaaaeaacaWG0bGaeyypa0JaaGymaaqaaiaad6gaa0Gaey yeIuoaaOqaaiaad6gaaaGaeyypa0JabmOCayaaraWaaSbaaSqaaiaa dkeaaeqaaOGaaGjcVlaayIW7caGGUaaaaa@6A2C@

In addition, the internal rates of return of produced capital derived from Equation (10) are external to the mining industry of interest since it uses information of other industries. However, it uses information from the national accounts only.

The resource rents in a mining industry with r ¯ j > r ¯ B MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOCayaara WaaSbaaSqaaiaadQgaaeqaaOGaeyOpa4JabmOCayaaraWaaSbaaSqa aiaadkeaaeqaaaaa@3B34@  are then residually calculated by subtracting the returns on produced capital calculated using the adjusted rates of return on produced capital, i.e.,

R j t = O S j t k [ T j k t K j k t ( p k t 1 r ˜ j t + p k t δ k t p k t 1 π j k t ) + p k t 1 K j k t ϕ j t ]    if    r ¯ j > r ¯ .           (11) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGQbGaamiDaaqabaGccqGH9aqpcaWGpbGaam4uamaaBaaa leaacaWGQbGaamiDaaqabaGccqGHsisldaaeqaqaamaadmaabaGaam ivamaaBaaaleaacaWGQbGaam4AaiaadshaaeqaaOGaam4samaaBaaa leaacaWGQbGaam4AaiaadshaaeqaaOWaaeWaaeaacaWGWbWaaSbaaS qaaiaadUgacaWG0bGaeyOeI0IaaGymaaqabaGcceWGYbGbaGaadaWg aaWcbaGaamOAaiaadshaaeqaaOGaey4kaSIaamiCamaaBaaaleaaca WGRbGaamiDaaqabaGccqaH0oazdaWgaaWcbaGaam4Aaiaadshaaeqa aOGaeyOeI0IaamiCamaaBaaaleaacaWGRbGaamiDaiabgkHiTiaaig daaeqaaOGaeqiWda3aaSbaaSqaaiaadQgacaWGRbGaamiDaaqabaaa kiaawIcacaGLPaaacqGHRaWkcaWGWbWaaSbaaSqaaiaadUgacaWG0b GaeyOeI0IaaGymaaqabaGccaWGlbWaaSbaaSqaaiaadQgacaWGRbGa amiDaaqabaGccqaHvpGzdaWgaaWcbaGaamOAaiaadshaaeqaaaGcca GLBbGaayzxaaaaleaacaWGRbaabeqdcqGHris5aOGaaeiiaiaabcca caqGGaGaaeyAaiaabAgacaqGGaGaaeiiaiaabccaceWGYbGbaebada WgaaWcbaGaamOAaaqabaGccqGH+aGpceWGYbGbaebacaaMi8UaaGjc Vlaac6cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGXaGaaeykaaaa@8B18@

2.1.3 Resource rent benchmarking

Because of data limitations at the commodity level, the resource rent estimate at the industry level is in general more reliable when the commodity-level resource rents are all positive. In this case, the commodity-level resource rent is benchmarked using the industry-level resource rent as the control total, i.e.,

R ˜ i j t = R i j t i R i j t R j t ,    for  commodity  i  industry  j .            (12) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOuayaaia WaaSbaaSqaaiaadMgacaWGQbGaamiDaaqabaGccqGH9aqpdaWcaaqa aiaadkfadaWgaaWcbaGaamyAaiaadQgacaWG0baabeaaaOqaamaaqa babaGaamOuamaaBaaaleaacaWGPbGaamOAaiaadshaaeqaaaqaaiaa dMgaaeqaniabggHiLdaaaOGaamOuamaaBaaaleaacaWGQbGaamiDaa qabaGccaGGSaGaaeiiaiaabccacaqGGaGaaeOzaiaab+gacaqGYbGa aeiiaiaabccacaqGJbGaae4Baiaab2gacaqGTbGaae4Baiaabsgaca qGPbGaaeiDaiaabMhacaqGGaGaamyAaiabgIGiolaabccacaqGPbGa aeOBaiaabsgacaqG1bGaae4CaiaabshacaqGYbGaaeyEaiaabccaca WGQbGaaiOlaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabgdacaqGYaGaae ykaaaa@6EF5@

However, when the industry-level resource rent is zero or very small, it is recalculated as the sum of the resource rents at commodity-level,Note 9 i.e.,  

R j t = i j R i j t .           (13) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa aaleaacaWGQbGaamiDaaqabaGccqGH9aqpdaaeqaqaaiaadkfadaWg aaWcbaGaamyAaiaadQgacaWG0baabeaaaeaacaWGPbGaeyicI4Saam OAaaqab0GaeyyeIuoakiaayIW7caaMi8UaaiOlaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGOaGaaeymaiaabodacaqGPaaaaa@5000@

The resource rent generated from extracting a subsoil resource is taken here as the user cost or capital service of this natural capital asset. It is what the rental market for the assets would have to extract for the use of the natural capital if its use was rented out over the course of the year.Note 10

2.1.4 Resource rent decomposition

Let D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGeb aaaa@376F@  be the physical extraction of a subsoil asset, and P D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGqb GcdaahaaWcbeqaaiaadseaaaaaaa@387B@  be the unit user cost of the natural capital or the net price of the resource extracted at a point of time; we then have

R = P D D .            (14) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2 da9iaadcfadaahaaWcbeqaaiaadseaaaGccaWGebGaaGjcVlaayIW7 caGGUaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabsdacaqGPaaa aa@4808@

Exploration and development expenditures have been capitalized as produced capital in the SNA, implying that their returns have then been deducted in the calculation of the resource rent. As a result, the unit resource rent ( P D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGqb GcdaahaaWcbeqaaiaadseaaaaaaa@387B@  ) reflects purely the value of a subsoil resource arising from its scarcity and the quality of deposit.Note 11

Similarly to the user cost of produced capital, the resource rent can also be split into the depletion cost and returns on natural capital. Let P N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGqb GcdaahaaWcbeqaaiaad6eaaaaaaa@3885@ , δ N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaW baaSqabeaacaWGobaaaaaa@389C@  and r N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaCa aaleqabaGaamOtaaaaaaa@37EE@  denote the shadow price of, the depletion rate of, and the rate of returns on natural capital, respectively. The resource rent or the user cost of natural capital can then be written as

P D D = ( δ N + r N ) P N N = ( P N D ) depletion cost + ( P D P N ) D returns on natural capital   with  δ N = D / N .           (15) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaCa aaleqabaGaamiraaaakiaadseacqGH9aqpcaGGOaGaeqiTdq2aaWba aSqabeaacaWGobaaaOGaey4kaSIaamOCamaaCaaaleqabaGaamOtaa aakiaacMcacaWGqbWaaWbaaSqabeaacaWGobaaaOGaamOtaiabg2da 9maayaaabaGaaiikaiaadcfadaahaaWcbeqaaiaad6eaaaGccaWGeb GaaiykaaWcbaGaaeizaiaabwgacaqGWbGaaeiBaiaabwgacaqG0bGa aeyAaiaab+gacaqGUbGaaeiiaiaabogacaqGVbGaae4CaiaabshaaO Gaayjo+dGaey4kaSYaaGbaaeaacaGGOaGaamiuamaaCaaaleqabaGa amiraaaakiabgkHiTiaadcfadaahaaWcbeqaaiaad6eaaaGccaGGPa GaamiraaWcbaGaaeOCaiaabwgacaqG0bGaaeyDaiaabkhacaqGUbGa ae4CaiaabccacaqGVbGaaeOBaiaabccacaqGUbGaaeyyaiaabshaca qG1bGaaeOCaiaabggacaqGSbGaaeiiaiaabogacaqGHbGaaeiCaiaa bMgacaqG0bGaaeyyaiaabYgaaOGaayjo+dGaaeiiaiaabccacaqG3b GaaeyAaiaabshacaqGObGaaeiiaiabes7aKnaaCaaaleqabaGaamOt aaaakiabg2da9maalyaabaGaamiraaqaaiaad6eaaaGaaGjcVlaayI W7caGGUaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeii aiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaaeynaiaabMcaaaa@902A@

2.2 Valuing subsoil resource reserves

As there are often no readily available market prices for subsoil resource reserves,Note 12 the NPV of the flow of natural resource rents is used here.Note 13 The NPV method values a resource reserve from an ex-ante perspective. It converts the expected future streams of resource rents into the present value of a resource reserve. Let E t ( d t + τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGfb GcdaWgaaWcbaGaamiDaaqabaGccaGGOaqcLbuacaWGKbGcdaWgaaWc baGaamiDaiabgUcaRiabes8a0bqabaGccaGGPaaaaa@3F7A@  be the expected future nominal rate of return on a numeraire asset that is used for discounting future income flows, E t ( ρ t + τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGfb GcdaWgaaWcbaGaamiDaaqabaGccaGGOaqcLbuacqaHbpGCkmaaBaaa leaacaWG0bGaey4kaSIaeqiXdqhabeaakiaacMcaaaa@4051@  be the expected future growth rate of the unit resource rent, and T t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGub GcdaWgaaWcbaGaamiDaaqabaaaaa@38AE@  be the reserve life of a subsoil resource at a point of time. The N P V i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaadc facaWGwbWaaSbaaSqaaiaadMgaaeqaaaaa@3994@  of the reserve of a subsoil resource becomes

N P V i t = P i t N N i t = τ = 1 T i t E t ( P i t + τ D ) D i t + τ s = 1 τ ( 1 + E t ( d t + s ) ) = τ = 1 T i t s = 1 τ ( 1 + E t ( ρ t + s ) ) P i t D D i t + τ s = 1 τ ( 1 + E t ( d t + s ) ) .           (16) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaadc facaWGwbWaaSbaaSqaaiaadMgacaWG0baabeaakiabg2da9iaadcfa daqhaaWcbaGaamyAaiaadshaaeaacaWGobaaaOGaamOtamaaBaaale aacaWGPbGaamiDaaqabaGccqGH9aqpdaaeWbqaamaalaaabaGaamyr amaaBaaaleaacaWG0baabeaakmaabmaabaGaamiuamaaDaaaleaaca WGPbGaamiDaiabgUcaRiabes8a0bqaaiaadseaaaaakiaawIcacaGL PaaacaWGebWaaSbaaSqaaiaadMgacaWG0bGaey4kaSIaeqiXdqhabe aaaOqaamaaradabaWaaeWaaeaacaaIXaGaey4kaSIaamyramaaBaaa leaacaWG0baabeaakmaabmaabaGaamizamaaBaaaleaacaWG0bGaey 4kaSIaam4CaaqabaaakiaawIcacaGLPaaaaiaawIcacaGLPaaaaSqa aiaadohacqGH9aqpcaaIXaaabaGaeqiXdqhaniabg+GivdaaaaWcba GaeqiXdqNaeyypa0JaaGymaaqaaiaadsfadaWgaaadbaGaamyAaiaa dshaaeqaaaqdcqGHris5aOGaeyypa0ZaaabCaeaadaWcaaqaamaara dabaWaaeWaaeaacaaIXaGaey4kaSIaamyramaaBaaaleaacaWG0baa beaakmaabmaabaGaeqyWdi3aaSbaaSqaaiaadshacqGHRaWkcaWGZb aabeaaaOGaayjkaiaawMcaaaGaayjkaiaawMcaaaWcbaGaam4Caiab g2da9iaaigdaaeaacqaHepaDa0Gaey4dIunakiaadcfadaqhaaWcba GaamyAaiaadshaaeaacaWGebaaaOGaamiramaaBaaaleaacaWGPbGa amiDaiabgUcaRiabes8a0bqabaaakeaadaqeWaqaamaabmaabaGaaG ymaiabgUcaRiaadweadaWgaaWcbaGaamiDaaqabaGcdaqadaqaaiaa dsgadaWgaaWcbaGaamiDaiabgUcaRiaadohaaeqaaaGccaGLOaGaay zkaaaacaGLOaGaayzkaaaaleaacaWGZbGaeyypa0JaaGymaaqaaiab es8a0bqdcqGHpis1aaaaaSqaaiabes8a0jabg2da9iaaigdaaeaaca WGubWaaSbaaWqaaiaadMgacaWG0baabeaaa0GaeyyeIuoakiaayIW7 caaMi8UaaiOlaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiai aabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaabAdacaqGPaaa aa@AFA9@

For notational simplicity, we replace the period-specific discount rates and growth rates of the unit resource rent in (16) with their annual averages over the reserve life, which yields

N P V i t = P i t N N i t = P i t D τ = 1 T i t D i t + τ ( 1 + ρ t 1 + d t ) τ ,            (17) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiaadc facaWGwbWaaSbaaSqaaiaadMgacaWG0baabeaakiabg2da9iaadcfa daqhaaWcbaGaamyAaiaadshaaeaacaWGobaaaOGaamOtamaaBaaale aacaWGPbGaamiDaaqabaGccqGH9aqpcaWGqbWaa0baaSqaaiaadMga caWG0baabaGaamiraaaakmaaqahabaGaamiramaaBaaaleaacaWGPb GaamiDaiabgUcaRiabes8a0bqabaGcdaqadaqaamaalaaabaGaaGym aiabgUcaRiabeg8aYnaaBaaaleaacaWG0baabeaaaOqaaiaaigdacq GHRaWkcaWGKbWaaSbaaSqaaiaadshaaeqaaaaaaOGaayjkaiaawMca amaaCaaaleqabaGaeqiXdqhaaaqaaiabes8a0jabg2da9iaaigdaae aacaWGubWaaSbaaWqaaiaadMgacaWG0baabeaaa0GaeyyeIuoakiaa cYcacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabIcacaqGXaGaae4naiaabMcaaaa@6B97@

where

ρ t = Average τ = 1 T i t   ( E t ρ t + τ ) ,   and   d t = Average τ = 1 T i t  ( E t d t + τ ) .           (18) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadshaaeqaaOGaeyypa0JaaeyqaiaabAhacaqGLbGaaeOC aiaabggacaqGNbGaaeyzamaaDaaaleaacqaHepaDcqGH9aqpcaaIXa aabaGaamivamaaBaaameaacaWGPbGaamiDaaqabaaaaOGaaeiiaiaa cIcacaWGfbWaaSbaaSqaaiaadshaaeqaaOGaeqyWdi3aaSbaaSqaai aadshacqGHRaWkcqaHepaDaeqaaOGaaiykaiaacYcacaqGGaGaaeii aiaabggacaqGUbGaaeizaiaabccacaqGGaGaamizamaaBaaaleaaca WG0baabeaakiabg2da9iaabgeacaqG2bGaaeyzaiaabkhacaqGHbGa ae4zaiaabwgadaqhaaWcbaGaeqiXdqNaeyypa0JaaGymaaqaaiaads fadaWgaaadbaGaamyAaiaadshaaeqaaaaakiaabccacaqGOaGaamyr amaaBaaaleaacaWG0baabeaakiaadsgadaWgaaWcbaGaamiDaiabgU caRiabes8a0bqabaGccaGGPaGaaiOlaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae ymaiaabIdacaqGPaaaaa@7909@

Hotelling’s ruleNote 14 suggests that the socially and economically optimal time path of a non-renewable resource extraction is one along which the resource price, net of all extraction costs (unit resource rent), is expected to grow at the rate of return on investment (discount rate). That is

ρ t = d t .           (19) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadshaaeqaaOGaeyypa0JaamizamaaBaaaleaacaWG0baa beaakiaayIW7caaMi8UaaiOlaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeymaiaa bMdacaqGPaaaaa@48FC@

To understand the proposition, we assume that the representative agent chooses an extraction path to maximize the NPV of a resource reserve. The optimization can be written as

M a x { D i t + τ } τ = 1 T i t ( N P V i t = P i t D τ = 1 T i t D i t + τ ( 1 + ρ t 1 + d t ) τ )           (20)              s . t .       τ = 1 T i t D i t + τ = N i t . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaGfqb qabSqaamaacmaabaGaamiramaaBaaameaacaWGPbGaamiDaiabgUca Riabes8a0bqabaaaliaawUhacaGL9baadaqhaaadbaGaeqiXdqNaey ypa0JaaGymaaqaaiaadsfadaWgaaqaaiaadMgacaWG0baabeaaaaaa leqaneaacaWGnbGaamyyaiaadIhaaaGcdaqadaqaaiaad6eacaWGqb GaamOvamaaBaaaleaacaWGPbGaamiDaaqabaGccqGH9aqpcaWGqbWa a0baaSqaaiaadMgacaWG0baabaGaamiraaaakmaaqahabaGaamiram aaBaaaleaacaWGPbGaamiDaiabgUcaRiabes8a0bqabaGcdaqadaqa amaalaaabaGaaGymaiabgUcaRiabeg8aYnaaBaaaleaacaWG0baabe aaaOqaaiaaigdacqGHRaWkcaWGKbWaaSbaaSqaaiaadshaaeqaaaaa aOGaayjkaiaawMcaamaaCaaaleqabaGaeqiXdqhaaaqaaiabes8a0j abg2da9iaaigdaaeaacaWGubWaaSbaaWqaaiaadMgacaWG0baabeaa a0GaeyyeIuoaaOGaayjkaiaawMcaaiaabccacaqGGaGaaeiiaiaabc cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaaeOm aiaabcdacaqGPaaabaGaaeiiaiaabccacaqGGaGaaeiiaiaabccaca qGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaam4Caiaa c6cacaWG0bGaaiOlaiaabccacaqGGaGaaeiiaiaabccacaqGGaWaaa bCaeaacaWGebWaaSbaaSqaaiaadMgacaWG0bGaey4kaSIaeqiXdqha beaakiabg2da9iaad6eadaWgaaWcbaGaamyAaiaadshaaeqaaaqaai abes8a0jabg2da9iaaigdaaeaacaWGubWaaSbaaWqaaiaadMgacaWG 0baabeaaa0GaeyyeIuoakiaayIW7caaMi8UaaiOlaaaaaa@99E1@

The Lagrangian function for this problem can be written as

Λ i t = P i t D τ = 1 T i t D i t + τ ( 1 + ρ t 1 + d t ) τ λ i t ( N i t τ = 1 T i t D i t + τ ) .             (21) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiaadMgacaWG0baabeaakiabg2da9iaadcfadaqhaaWcbaGa amyAaiaadshaaeaacaWGebaaaOWaaabCaeaacaWGebWaaSbaaSqaai aadMgacaWG0bGaey4kaSIaeqiXdqhabeaakmaabmaabaWaaSaaaeaa caaIXaGaey4kaSIaeqyWdi3aaSbaaSqaaiaadshaaeqaaaGcbaGaaG ymaiabgUcaRiaadsgadaWgaaWcbaGaamiDaaqabaaaaaGccaGLOaGa ayzkaaWaaWbaaSqabeaacqaHepaDaaaabaGaeqiXdqNaeyypa0JaaG ymaaqaaiaadsfadaWgaaadbaGaamyAaiaadshaaeqaaaqdcqGHris5 aOGaeyOeI0Iaeq4UdW2aaSbaaSqaaiaadMgacaWG0baabeaakmaabm aabaGaamOtamaaBaaaleaacaWGPbGaamiDaaqabaGccqGHsisldaae WbqaaiaadseadaWgaaWcbaGaamyAaiaadshacqGHRaWkcqaHepaDae qaaaqaaiabes8a0jabg2da9iaaigdaaeaacaWGubWaaSbaaWqaaiaa dMgacaWG0baabeaaa0GaeyyeIuoaaOGaayjkaiaawMcaaiaayIW7ca aMi8UaaGjcVlaac6cacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGOaGaae OmaiaabgdacaqGPaaaaa@807B@

The first order conditions can be derived by taking the derivative of Equation (21), with respect to the physical extraction in each time, i.e.,

Λ i t D i t + τ = P i t D ( 1 + ρ t 1 + d t ) τ λ i t = 0 ,    for    τ = 1 , ... , T i t .           (22) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcqqHBoatdaWgaaWcbaGaamyAaiaadshaaeqaaaGcbaGaeyOa IyRaamiramaaBaaaleaacaWGPbGaamiDaiabgUcaRiabes8a0bqaba aaaOGaeyypa0JaamiuamaaDaaaleaacaWGPbGaamiDaaqaaiaadsea aaGcdaqadaqaamaalaaabaGaaGymaiabgUcaRiabeg8aYnaaBaaale aacaWG0baabeaaaOqaaiaaigdacqGHRaWkcaWGKbWaaSbaaSqaaiaa dshaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeqiXdqhaaO GaeyOeI0Iaeq4UdW2aaSbaaSqaaiaadMgacaWG0baabeaakiabg2da 9iaaicdacaGGSaGaaeiiaiaabccacaqGGaGaaeOzaiaab+gacaqGYb GaaeiiaiaabccacaqGGaGaeqiXdqNaeyypa0JaaGymaiaacYcacaGG UaGaaiOlaiaac6cacaGGSaGaamivamaaBaaaleaacaWGPbGaamiDaa qabaGccaaMi8UaaGjcVlaac6cacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdaca qGYaGaaeykaaaa@773E@

It is required that ρ t = d t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadshaaeqaaOGaeyypa0JaamizamaaBaaaleaacaWG0baa beaaaaa@3BF9@  for Equation (22) to hold. Otherwise, the current extraction is not optimal because the marginal profit of extraction ( P D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGqb GcdaahaaWcbeqaaiaadseaaaaaaa@387B@  ) and the marginal value of holding ( λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcaaMaeq4UdW gaaa@3813@  ) are not equal to each other. This is Hotelling’s rule. Substituting ρ t = d t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyWdi3aaS baaSqaaiaadshaaeqaaOGaeyypa0JaamizamaaBaaaleaacaWG0baa beaaaaa@3BF9@  into Equations (22), (21) and (17) gives

λ i t P i t N = P i t D N P V i t * = P i t N N i t = P i t D τ = 1 T i t D i t + τ * = P j i t D N i t = R i t T ^ i t ,    with   T ^ i t N i t D i t .           (23) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaH7o aBdaWgaaWcbaGaamyAaiaadshaaeqaaOGaeyyyIORaamiuamaaDaaa leaacaWGPbGaamiDaaqaaiaad6eaaaGccqGH9aqpcaWGqbWaa0baaS qaaiaadMgacaWG0baabaGaamiraaaaaOqaaiaad6eacaWGqbGaamOv amaaDaaaleaacaWGPbGaamiDaaqaaiaacQcaaaGccqGH9aqpcaWGqb Waa0baaSqaaiaadMgacaWG0baabaGaamOtaaaakiaad6eadaWgaaWc baGaamyAaiaadshaaeqaaOGaeyypa0JaamiuamaaDaaaleaacaWGPb GaamiDaaqaaiaadseaaaGcdaaeWbqaaiaadseadaqhaaWcbaGaamyA aiaadshacqGHRaWkcqaHepaDaeaacaGGQaaaaOGaeyypa0Jaamiuai aadQgadaqhaaWcbaGaamyAaiaadshaaeaacaWGebaaaOGaamOtamaa BaaaleaacaWGPbGaamiDaaqabaGccqGH9aqpcaWGsbWaaSbaaSqaai aadMgacaWG0baabeaaaeaacqaHepaDcqGH9aqpcaaIXaaabaGaamiv amaaBaaameaacaWGPbGaamiDaaqabaaaniabggHiLdGcceWGubGbaK aadaWgaaWcbaGaamyAaiaadshaaeqaaOGaaiilaiaabccacaqGGaGa aeiiaiaabEhacaqGPbGaaeiDaiaabIgacaqGGaGaaeiiaiqadsfaga qcamaaBaaaleaacaWGPbGaamiDaaqabaGccqGHHjIUdaWcaaqaaiaa d6eadaWgaaWcbaGaamyAaiaadshaaeqaaaGcbaGaamiramaaBaaale aacaWGPbGaamiDaaqabaaaaOGaaGjcVlaayIW7caGGUaGaaeiiaiaa bccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabIcacaqGYaGaae4maiaabMcaaaaa@93C3@

Therefore, along the optimal extraction path, the shadow price of a resource reserve is equal to the unit resource rent, and both are expected to grow at the rate of nominal interest rate of a numeraire asset. The NPV of a resource reserve can then be calculated as the current resource rent, multiplied by the number of periods of extraction at current level.

Hotelling’s rule also implies that the rate of return on natural capital is zero. This can be seen using Equation (15), when the shadow price of a resource reserve ( P N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGqb GcdaahaaWcbeqaaiaad6eaaaaaaa@3885@  ) is equal to the unit resource rent ( P D MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcLbuacaWGqb GcdaahaaWcbeqaaiaadseaaaaaaa@387B@  ). So the benefits today (resource rents) fully reflect the cost of future loss (depletion costs).

In the above formulation, Hotelling’s rule was used to define the optimal extraction path of non-renewable natural resources to give the conceptual and theoretical framework for understanding and analyzing the depletion of non-renewable natural resources.

In support of the use to which the rule is being used here, Miller and Upton (1985) found that, for a sample of U.S. oil and gas extraction companies, estimates of reserve values, when calculated using Hotelling’s rule, account for a significant portion of their market values. Miller and Upton (1985) also compared the accuracy of using Hotelling’s rule, as opposed to two widely cited and publicly available alternatives MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG qbaKqzGfaeaaaaaaaaa8qacaWFuacaaa@39A7@ the Securities and Exchange Commission and Herold appraisals reported that Hotelling’s rule performed better in the valuation of the resource reserves values. This supports the use to which Hotelling’s rule is being used here. It suggests that expectations are being formed to determine the values being estimated here, using something approximating Hotelling’s rule.

It is, however, the case that Livernois (2009) reports that empirical studies that examine the actual price trajectory find imperfect evidence that the actual trajectory of resource prices follows Hotelling’s rule. But the question is not whether the trajectory follows Hotelling’s rule exactly, but whether the expected values using an approximation to this rule accord with values being created in markets, which is the criterion that accords with the spirit of measurement within the SNA and the Multifactor Productivity Program.

Kronenberg (2008) discussed factors that may lead to deviations of outcomes, in the real world, from those obtained applying Hotelling’s rule. One category of these factors relates to the assumptions made for deriving Hotelling’s rule, such as perfect competition, zero extraction cost, no technical progress, fixed stock of reserves, and constant market conditions. These assumptions can be relaxed. And in this paper, we do so by calculating the value of a resource by updating information continuously on the extraction cost, reserve stock, and market conditions, implying that the corresponding optimal extraction path of a resource reserve changes over time. The other category of these factors is institutional, such as uncertain property rights and the strategic interaction between suppliers and consumers. Although these institutional factors may lead to a market failure, such that the actual extraction path is not socially optimal, valuing a resource reserve along its optimal path of extraction gives the value that can be achieved from efficient extraction of a resource reserve.Note 15

2.3 Industry-level measures  

To this point, measures on the quantity and price for each natural capital asset have been derived. The industry-level quantity and price measures are then aggregated from those for each asset using the Fisher formula. For the natural capital stock in a mining industry, its quantity and price indexes are calculated as

F Q I t N N t N t 1 = i P i t 1 N N i t i P i t 1 N N i t 1 i P i t N N i t i P i t N N i t 1 ,    F P I t N P t N P t 1 N = i P i t N N i t 1 i P i t 1 N N i t 1 i P i t N N i t i P i t 1 N N i t .           (24) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaadg facaWGjbWaa0baaSqaaiaadshaaeaacaWGobaaaOGaeyyyIO7aaSaa aeaacaWGobWaaSbaaSqaaiaadshaaeqaaaGcbaGaamOtamaaBaaale aacaWG0bGaeyOeI0IaaGymaaqabaaaaOGaeyypa0ZaaOaaaeaadaWc aaqaamaaqababaGaamiuamaaDaaaleaacaWGPbGaamiDaiabgkHiTi aaigdaaeaacaWGobaaaOGaamOtamaaBaaaleaacaWGPbGaamiDaaqa baaabaGaamyAaaqab0GaeyyeIuoaaOqaamaaqababaGaamiuamaaDa aaleaacaWGPbGaamiDaiabgkHiTiaaigdaaeaacaWGobaaaOGaamOt amaaBaaaleaacaWGPbGaamiDaiabgkHiTiaaigdaaeqaaaqaaiaadM gaaeqaniabggHiLdaaaOWaaSaaaeaadaaeqaqaaiaadcfadaqhaaWc baGaamyAaiaadshaaeaacaWGobaaaOGaamOtamaaBaaaleaacaWGPb GaamiDaaqabaaabaGaamyAaaqab0GaeyyeIuoaaOqaamaaqababaGa amiuamaaDaaaleaacaWGPbGaamiDaaqaaiaad6eaaaGccaWGobWaaS baaSqaaiaadMgacaWG0bGaeyOeI0IaaGymaaqabaaabaGaamyAaaqa b0GaeyyeIuoaaaaaleqaaOGaaiilaiaabccacaqGGaGaamOraiaadc facaWGjbWaa0baaSqaaiaadshaaeaacaWGobaaaOGaeyyyIO7aaSaa aeaacaWGqbWaa0baaSqaaiaadshaaeaacaWGobaaaaGcbaGaamiuam aaDaaaleaacaWG0bGaeyOeI0IaaGymaaqaaiaad6eaaaaaaOGaeyyp a0ZaaOaaaeaadaWcaaqaamaaqababaGaamiuamaaDaaaleaacaWGPb GaamiDaaqaaiaad6eaaaGccaWGobWaaSbaaSqaaiaadMgacaWG0bGa eyOeI0IaaGymaaqabaaabaGaamyAaaqab0GaeyyeIuoaaOqaamaaqa babaGaamiuamaaDaaaleaacaWGPbGaamiDaiabgkHiTiaaigdaaeaa caWGobaaaOGaamOtamaaBaaaleaacaWGPbGaamiDaiabgkHiTiaaig daaeqaaaqaaiaadMgaaeqaniabggHiLdaaaOWaaSaaaeaadaaeqaqa aiaadcfadaqhaaWcbaGaamyAaiaadshaaeaacaWGobaaaOGaamOtam aaBaaaleaacaWGPbGaamiDaaqabaaabaGaamyAaaqab0GaeyyeIuoa aOqaamaaqababaGaamiuamaaDaaaleaacaWGPbGaamiDaiabgkHiTi aaigdaaeaacaWGobaaaOGaamOtamaaBaaaleaacaWGPbGaamiDaaqa baaabaGaamyAaaqab0GaeyyeIuoaaaaaleqaaOGaaGjcVlaayIW7ca GGUaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaa bccacaqGGaGaaeiiaiaabIcacaqGYaGaaeinaiaabMcaaaa@BA1C@

In the case of mining, the physical extractions are the service flows provided by the natural capital. The industry-level quantity and price indexes of natural capital service (input) can then be estimated as

F Q I t Z N Z t N Z t 1 N = i P i t 1 D D i t i P i t 1 D D i t 1 i P i t D D i t i P i t D D i t 1 ,    F P I t Z N P t Z N P t 1 Z N = i P i t D D i t 1 i P i t 1 D D i t 1 i P i t D D i t i P i t 1 D D i t .           (25) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiaadg facaWGjbWaa0baaSqaaiaadshaaeaacaWGAbWaaWbaaWqabeaacaWG obaaaaaakiabggMi6oaalaaabaGaamOwamaaDaaaleaacaWG0baaba GaamOtaaaaaOqaaiaadQfadaqhaaWcbaGaamiDaiabgkHiTiaaigda aeaacaWGobaaaaaakiabg2da9maakaaabaWaaSaaaeaadaaeqaqaai aadcfadaqhaaWcbaGaamyAaiaadshacqGHsislcaaIXaaabaGaamir aaaakiaadseadaWgaaWcbaGaamyAaiaadshaaeqaaaqaaiaadMgaae qaniabggHiLdaakeaadaaeqaqaaiaadcfadaqhaaWcbaGaamyAaiaa dshacqGHsislcaaIXaaabaGaamiraaaakiaadseadaWgaaWcbaGaam yAaiaadshacqGHsislcaaIXaaabeaaaeaacaWGPbaabeqdcqGHris5 aaaakmaalaaabaWaaabeaeaacaWGqbWaa0baaSqaaiaadMgacaWG0b aabaGaamiraaaakiaadseadaWgaaWcbaGaamyAaiaadshaaeqaaaqa aiaadMgaaeqaniabggHiLdaakeaadaaeqaqaaiaadcfadaqhaaWcba GaamyAaiaadshaaeaacaWGebaaaOGaamiramaaBaaaleaacaWGPbGa amiDaiabgkHiTiaaigdaaeqaaaqaaiaadMgaaeqaniabggHiLdaaaa WcbeaakiaacYcacaqGGaGaaeiiaiaadAeacaWGqbGaamysamaaDaaa leaacaWG0baabaGaamOwamaaCaaameqabaGaamOtaaaaaaGccqGHHj IUdaWcaaqaaiaadcfadaqhaaWcbaGaamiDaaqaaiaadQfadaahaaad beqaaiaad6eaaaaaaaGcbaGaamiuamaaDaaaleaacaWG0bGaeyOeI0 IaaGymaaqaaiaadQfadaahaaadbeqaaiaad6eaaaaaaaaakiabg2da 9maakaaabaWaaSaaaeaadaaeqaqaaiaadcfadaqhaaWcbaGaamyAai aadshaaeaacaWGebaaaOGaamiramaaBaaaleaacaWGPbGaamiDaiab gkHiTiaaigdaaeqaaaqaaiaadMgaaeqaniabggHiLdaakeaadaaeqa qaaiaadcfadaqhaaWcbaGaamyAaiaadshacqGHsislcaaIXaaabaGa amiraaaakiaadseadaWgaaWcbaGaamyAaiaadshacqGHsislcaaIXa aabeaaaeaacaWGPbaabeqdcqGHris5aaaakmaalaaabaWaaabeaeaa caWGqbWaa0baaSqaaiaadMgacaWG0baabaGaamiraaaakiaadseada WgaaWcbaGaamyAaiaadshaaeqaaaqaaiaadMgaaeqaniabggHiLdaa keaadaaeqaqaaiaadcfadaqhaaWcbaGaamyAaiaadshacqGHsislca aIXaaabaGaamiraaaakiaadseadaWgaaWcbaGaamyAaiaadshaaeqa aaqaaiaadMgaaeqaniabggHiLdaaaaWcbeaakiaayIW7caaMi8Uaai OlaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqG GaGaaeiiaiaabccacaqGOaGaaeOmaiaabwdacaqGPaaaaa@BF71@

The discrete approximation of the growth accounting formula can be derived from (2) as

Δ ln ( Y t ) = s ¯ t L Δ ln ( L t ) + s ¯ t K Δ ln ( Z t K ) + s ¯ t N Δ ln ( Z t N ) + Δ ln ( M F P t ) with   s ¯ t L = ( w t 1 L t 1 / Y t 1 + w t L t / Y t ) / 2 ,   s t N = ( R t 1 / Y t 1 + R t / Y t ) / 2 ,    s ¯ t K = 1 s ¯ t L s t N .           (26) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHuo arciGGSbGaaiOBaiaacIcacaWGzbWaaSbaaSqaaiaadshaaeqaaOGa aiykaiabg2da9iqadohagaqeamaaDaaaleaacaWG0baabaGaamitaa aakiabfs5aejGacYgacaGGUbGaaiikaiaadYeadaWgaaWcbaGaamiD aaqabaGccaGGPaGaey4kaSIabm4CayaaraWaa0baaSqaaiaadshaae aacaWGlbaaaOGaeuiLdqKaciiBaiaac6gacaGGOaGaamOwamaaDaaa leaacaWG0baabaGaam4saaaakiaacMcacqGHRaWkceWGZbGbaebada qhaaWcbaGaamiDaaqaaiaad6eaaaGccqqHuoarciGGSbGaaiOBaiaa cIcacaWGAbWaa0baaSqaaiaadshaaeaacaWGobaaaOGaaiykaiabgU caRiabfs5aejGacYgacaGGUbGaaiikaiaad2eacaWGgbGaamiuamaa BaaaleaacaWG0baabeaakiaacMcaaeaacaqG3bGaaeyAaiaabshaca qGObGaaeiiaiaabccaceWGZbGbaebadaqhaaWcbaGaamiDaaqaaiaa dYeaaaGccqGH9aqpdaWcgaqaaiaacIcacaWG3bWaaSbaaSqaaiaads hacqGHsislcaaIXaaabeaakiaadYeadaWgaaWcbaGaamiDaiabgkHi TiaaigdaaeqaaaGcbaGaamywamaaBaaaleaacaWG0bGaeyOeI0IaaG ymaaqabaGccqGHRaWkdaWcgaqaaiaadEhadaWgaaWcbaGaamiDaaqa baGccaWGmbWaaSbaaSqaaiaadshaaeqaaaGcbaGaamywamaaBaaale aacaWG0baabeaakiaacMcacaGGVaGaaGOmaiaacYcacaqGGaaaaaaa caWGZbWaa0baaSqaaiaadshaaeaacaWGobaaaOGaeyypa0ZaaSGbae aacaGGOaGaamOuamaaBaaaleaacaWG0bGaeyOeI0IaaGymaaqabaaa keaacaWGzbWaaSbaaSqaaiaadshacqGHsislcaaIXaaabeaakiabgU caRmaalyaabaGaamOuamaaBaaaleaacaWG0baabeaaaOqaaiaadMfa daWgaaWcbaGaamiDaaqabaGccaGGPaGaai4laiaaikdacaGGSaaaaa aacaqGGaGaaeiiaiqadohagaqeamaaDaaaleaacaWG0baabaGaam4s aaaakiabg2da9iaaigdacqGHsislceWGZbGbaebadaqhaaWcbaGaam iDaaqaaiaadYeaaaGccqGHsislcaWGZbWaa0baaSqaaiaadshaaeaa caWGobaaaOGaaGjcVlaac6cacaqGGaGaaeiiaiaabccacaqGGaGaae iiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeikaiaabkdacaqG 2aGaaeykaaaaaa@B491@

MFP growth can then be estimated residually. It is noteworthy that the growth accounting of (26) does not take into account the impact of changes in natural capital quality, so the derived MFP growth, at this point, refers only to the (natural capital) quality-unadjusted measure.Note 16 Also, the impact of adding natural capital, as an input into production on MFP growth, relies on the relative growth of produced and natural capital. It raises MFP growth when the natural capital growth is lower than that for produced capital and vice versa.

3 Empirical results for Canadian oil and gas extraction

In this section, the growth accounting framework developed in the previous section is applied for the Canadian oil and gas mining industry as an experimental analysis. The commodity-level (asset-level) data on the gross operating surplus and nominal produced capital stock for the mineral sector is compiled by the Environment Accounts and Statistics Division of Statistics Canada based on various data sources.Note 17 These data are benchmarked to the industry-level data first and then the benchmarked data are used for the calculation of the resource rents at the commodity level. The quantity measures of the stock, depletion and addition of each subsoil resource reserve are obtained from CANSIM tables 153-0012 to 153-0015. Combined with the estimates of resource rents, these data are used for the calculation of reserve value at the commodity level and the quantity and price indexes of natural capital stock and natural capital input at the industry level. The industry-level data of value-added, labour compensation, labour and produced capital inputs come from the KLEMS (capital, labour, energy, materials and services) database used in the CPA, and the industry-level geometric-based nominal produced capital stock data come from CANSIM table 031-0002.Note 18 The gross operating surplus and the nominal capital stock data at both industry and commodity levels are used for estimating the resource rents at both commodity and industry levels. A zero real discount rate is used throughout our experimental assessment. Given that the natural capital input is measured by the amount of physical extraction, the choice of the discount rate has no impact on the measurement of MFP growth. However, the measured value of natural capital stock is much larger under Hotelling’s rule (zero real discount rate) than that with the discount rate being at 4%.Note 19 This discount rate is currently used in the Canadian System of Environmental and Resource Accounts (CSERA) and as well as in many other national statistical agencies.

Oil and gas extraction involves the extraction of natural gas, crude oil and crude bitumen. Natural gas liquids are included in the asset category of natural gas.Note 20 The estimates on the volume of reserve and extraction for each type of resources are presented first. The estimates of the nominal value of reserve and the resource rent of the extraction are presented next. The volume estimates of reserves are then aggregated across different types of resources to derive total natural capital stock, while the extractions are aggregated to derive the flow of services for the natural capital (or natural capital input), using weights based on resource rents. Finally, the contribution of the natural capital to output and its effect on MFP estimates are presented.

3.1 Resource reserve and extraction

The established reserve of oil and gas in Canada has experienced a large compositional shift towards crude bitumen. As shown in Chart 1, from 1981 to 2009, the established reserve trended down slightly for both natural gas and crude oil. It dropped by about 25% for both natural gas and crude oil over the whole sample period. At the same time, the established reserve of crude bitumen increased dramatically, especially during the periods from 1997 to 1999 and after 2005. It increased by more than 12 times, or about 9.6% per year on average.

Chart 1 Trend in established oil and gas reserves, 1981 to 2009

Description for chart 1

The title of the graph is "Chart 1 Trend in established oil and gas reserves, 1981 to 2009."
This is a line chart.
There are in total 29 categories in the horizontal axis. The primary vertical axis starts at 0 and ends at 120 with ticks every 20 points. The secondary vertical axis starts at 0 and ends at 1,400 with ticks every 200 points.
There are 3 series in this graph.
The units of the horizontal axis are years from 1981 to 2009.
The title of series 1 is "Natural gas (left scale)."
The vertical axis is "index (1981 = 100)."
The minimum value is 67.26 occurring in 2003.
The maximum value is 102.83 occurring in 1982.
The title of series 2 is "Crude oil (left scale)."
The vertical axis is "index (1981 = 100)."
The minimum value is 63.63 occurring in 1996.
The maximum value is 100.00 occurring in 1981.
The title of series 3 is "Crude bitumen (right scale)."
The vertical axis is "index (1981 = 100)."
The minimum value is 95.51 occurring in 1983.
The maximum value is 1,323.08 occurring in 2008.

Data table for chart 1
Table Summary
This table displays the results of Chart 1 Trend in established oil and gas reserves Natural gas (left scale), Crude oil (left scale) and Crude bitumen (right scale), calculated using index (1981 = 100) units of measure (appearing as column headers).
  Natural gas (left scale) Crude oil (left scale) Crude bitumen (right scale)
index (1981 = 100) index (1981 = 100) index (1981 = 100)
1981 100.00 100.00 100.00
1982 102.83 94.30 97.11
1983 101.65 95.72 95.51
1984 101.21 93.78 101.17
1985 100.01 95.49 105.66
1986 97.91 93.57 176.74
1987 95.08 91.04 176.15
1988 94.45 89.30 174.31
1989 95.59 85.50 166.83
1990 95.99 79.40 161.23
1991 95.21 74.28 154.37
1992 93.66 71.32 148.37
1993 90.49 70.33 140.80
1994 88.70 65.78 173.85
1995 89.19 66.80 176.62
1996 83.24 63.63 203.32
1997 77.90 64.29 188.92
1998 75.17 81.36 411.08
1999 73.73 77.62 581.88
2000 75.59 80.61 572.31
2001 72.55 77.88 563.08
2002 71.16 73.22 566.15
2003 67.26 71.27 529.23
2004 68.33 72.94 510.77
2005 70.52 90.88 498.46
2006 71.79 86.08 1027.69
2007 69.46 87.19 1076.92
2008 75.47 83.21 1323.08
2009 76.44 75.20 1297.23

Unlike the pattern of the established reserve over time, the extraction of all three oil and gas resources has increased, although at quite different paces (Chart 2). From 1981 to 2009, extraction grew by about 2.4% per year for natural gas, by 0.1% per year for crude oil, and by 8.4% per year for crude bitumen.

Chart 2 Trend in extraction of oil and gas reserves, 1981 to 2009

Description for chart 2

The title of the graph is "Chart 2 Trend in extraction of oil and gas reserves, 1981 to 2009."
This is a line chart.
There are in total 29 categories in the horizontal axis. The primary vertical axis starts at 0 and ends at 250 with ticks every 50 points. The secondary vertical axis starts at 0 and ends at 1,200 with ticks every 200 points.
There are 3 series in this graph.
The units of the horizontal axis are years from 1981 to 2009.
The title of series 1 is "Natural gas (left scale)."
The vertical axis is "index (1981 = 100)."
The minimum value is 90.45 occurring in 1982.
The maximum value is 228.54 occurring in 2004.
The title of series 2 is "Crude oil (left scale)."
The vertical axis is "index (1981 = 100)."
The minimum value is 97.31 occurring in 1982.
The maximum value is 124.48 occurring in 2003.
The title of series 3 is "Crude bitumen (right scale)."
The vertical axis is "index (1981 = 100)."
The minimum value is 100.00 occurring in 1981.
The maximum value is 966.29 occurring in 2009.

Data table for chart 2
Table Summary
This table displays the results of Chart 2 Trend in extraction of oil and gas reserves Natural gas (left scale), Crude oil (left scale) and Crude bitumen (right scale), calculated using index (1981 = 100) units of measure (appearing as column headers).
  Natural gas (left scale) Crude oil (left scale) Crude bitumen (right scale)
index (1981 = 100) index (1981 = 100) index (1981 = 100)
1981 100.00 100.00 100.00
1982 90.45 97.31 105.62
1983 95.86 101.19 194.38
1984 99.93 110.00 130.34
1985 108.10 106.57 173.03
1986 102.63 101.04 212.36
1987 104.43 104.78 225.84
1988 130.85 108.51 240.45
1989 132.34 103.28 260.67
1990 140.19 101.64 255.06
1991 128.50 100.75 253.93
1992 158.97 103.73 267.42
1993 180.65 108.06 276.40
1994 180.23 112.39 269.66
1995 190.19 115.07 316.85
1996 198.45 117.61 315.73
1997 201.70 118.96 369.66
1998 207.54 119.85 426.97
1999 216.97 113.73 404.49
2000 221.82 118.66 438.20
2001 228.26 118.21 471.91
2002 224.06 123.13 539.33
2003 221.39 124.48 629.21
2004 228.54 120.15 707.87
2005 220.12 116.12 643.35
2006 220.59 114.93 738.28
2007 218.68 118.81 865.17
2008 210.27 114.18 853.93
2009 194.50 104.03 966.29

3.2 Resource rent and reserve value

Chart 3 presents the estimated value of oil and gas reserves and the resource rent from the extraction of oil and gas from 1981 to 2009. As shown, the patterns of the reserve value and resource rent, over time, are quite close to each other. Both stayed low and stagnant before 1999, and then grew rapidly thereafter. The annual resource rent declined by 2.5% per year over the 1981-to-1999 period and by 17.7% per year over the 1999-to-2009 period. The corresponding growth rates for the reserve value were -4.2% and 22.9% per year for the two periods, respectively.

Chart 3 Oil and gas resource rent and reserve value, 1981 to 2009

Description for chart 3

The title of the graph is "Chart 3 Oil and gas resource rent and reserve value, 1981 to 2009."
This is a line chart.
There are in total 29 categories in the horizontal axis. The primary vertical axis starts at 0 and ends at 35 with ticks every 5 points. The secondary vertical axis starts at 0 and ends at 700 with ticks every 100 points.
There are 2 series in this graph.
The units of the horizontal axis are years from 1981 to 2009.
The title of series 1 is "Resource rent (left scale)."
The vertical axis is "billions of dollars."
The minimum value is 1.02 occurring in 1992.
The maximum value is 31.27 occurring in 2007.
The title of series 2 is "Reserve value (right scale)."
The vertical axis is "billions of dollars."
The minimum value is 13.85 occurring in 1992.
The maximum value is 614.45 occurring in 2008.

Data table for chart 3
Table Summary
This table displays the results of Chart 3 Oil and gas resource rent and reserve value Resource rent (left scale) and Reserve value (right scale), calculated using billions of dollars units of measure (appearing as column headers).
  Resource rent (left scale) Reserve value (right scale)
billions of dollars billions of dollars
1981 6.34 139.53
1982 6.87 157.05
1983 7.06 138.72
1984 6.80 130.18
1985 5.15 93.35
1986 3.38 70.79
1987 1.69 31.89
1988 1.73 28.03
1989 1.96 31.83
1990 1.79 27.71
1991 1.46 23.54
1992 1.02 13.85
1993 1.37 16.91
1994 1.81 22.54
1995 2.51 29.78
1996 3.00 34.12
1997 2.81 29.90
1998 2.48 30.90
1999 4.06 63.91
2000 6.80 99.16
2001 8.04 99.39
2002 9.02 126.70
2003 12.33 143.16
2004 18.80 216.37
2005 24.78 300.74
2006 29.00 547.31
2007 31.27 545.45
2008 26.10 614.45
2009 20.61 504.26

3.3 Natural capital stock and natural capital input

The natural capital input in this industry trended up steadily without major interruptions (Chart 4). It grew by 2.4% per year on average from 1981 to 2009. At the same time, the pattern of the natural capital stock, over time, is quite different from that of the natural capital input. The natural capital stock trended down gradually and dropped by about 17% before 1997, reflecting the down-trending movements in natural gas and crude oil reserves. After 1997, the natural capital stock exhibited a pattern, over time, similar to that of crude bitumen. It increased largely from 1997 to 1999 and after 2005, and decreased moderately from 2000 to 2005.

Chart 4 Trend in oil and gas natural capital stock and natural capital input, 1981 to 2009

Description for chart 4

The title of the graph is "Chart 4 Trend in oil and gas natural capital stock and natural capital input, 1981 to 2009."
This is a line chart.
There are in total 29 categories in the horizontal axis. The vertical axis starts at 60 and ends at 220 with ticks every 20 points.
There are 2 series in this graph.
The vertical axis is "index (1981 = 100)."
The units of the horizontal axis are years from 1981 to 2009.
The title of series 1 is "Natural capital stock."
The minimum value is 82.83 occurring in 1997.
The maximum value is 190.29 occurring in 2008.
The title of series 2 is "Natural capital input."
The minimum value is 94.18 occurring in 1982.
The maximum value is 205.22 occurring in 2007.

Data table for chart 4
Table Summary
This table displays the results of Chart 4 Trend in oil and gas natural capital stock and natural capital input Natural capital stock and Natural capital input (appearing as column headers).
  Natural capital stock Natural capital input
1981 100.00 100.00
1982 100.26 94.18
1983 99.76 103.27
1984 99.38 106.50
1985 99.54 110.35
1986 102.15 106.46
1987 99.55 109.59
1988 98.53 124.74
1989 97.71 123.54
1990 95.47 125.67
1991 92.77 120.19
1992 90.42 135.82
1993 87.57 148.20
1994 87.75 149.58
1995 88.58 157.96
1996 87.33 162.64
1997 82.83 168.17
1998 104.94 174.60
1999 119.05 176.02
2000 119.84 182.37
2001 116.30 187.50
2002 114.74 190.52
2003 108.29 194.71
2004 107.79 201.25
2005 111.78 192.05
2006 160.31 197.08
2007 163.87 205.22
2008 190.29 198.67
2009 186.14 195.71

3.4 Multifactor productivity growth

In the growth accounting framework, adding natural capital has no impact on either output (value-added) growth or the contribution of labour input. However, the income share and, hence, the contribution of produced capital input will be reduced; as a result, MFP growth would be impacted if the produced capital input and the natural capital input grew at different paces.

As shown in Chart 5, MFP growth in oil and gas extraction was positive before 1993, and became largely negative after 1993. Note that adding natural capital in the growth accounting framework has little impact on the pattern of MFP growth over time. After adjusting for natural capital, annual MFP growth increases from 1.8% to 2.0% before 1993, and from -5.1% to -4.0%, after 1993.

Overall, by including subsoil resources, MFP declines by 1.5% per year over the 1981-to-2009 period, compared to a 2.2% decline without including these resources.

Chart 5 Alternative measures of multifactor productivity, oil and gas extraction industry, 1981 to 2009

Description for chart 5

The title of the graph is "Chart 5 Alternative measures of multifactor productivity, oil and gas extraction industry, 1981 to 2009."
This is a line chart.
There are in total 29 categories in the horizontal axis. The vertical axis starts at 40 and ends at 140 with ticks every 10 points.
There are 2 series in this graph.
The vertical axis is "index (1981 = 100)."
The units of the horizontal axis are years from 1981 to 2009.
The title of series 1 is "Standard."
The minimum value is 53.02 occurring in 2009.
The maximum value is 123.46 occurring in 1993.
The title of series 2 is "Adjusted for natural capital."
The minimum value is 65.49 occurring in 2009.
The maximum value is 126.54 occurring in 1993.

Data table for chart 5
Table Summary
This table displays the results of Chart 5 Alternative measures of multifactor productivity Standard and Adjusted for natural capital (appearing as column headers).
  Standard Adjusted for natural capital
1981 100.00 100.00
1982 93.98 99.26
1983 96.80 100.79
1984 96.35 100.92
1985 97.02 102.25
1986 90.27 95.91
1987 94.48 99.79
1988 100.84 105.00
1989 97.17 101.12
1990 98.94 102.62
1991 103.14 107.53
1992 115.95 119.49
1993 123.46 126.54
1994 120.86 124.63
1995 117.75 121.56
1996 112.24 116.17
1997 108.10 112.70
1998 108.48 113.37
1999 105.66 110.94
2000 99.23 105.02
2001 89.15 95.42
2002 89.84 96.98
2003 85.40 93.17
2004 78.66 86.93
2005 69.44 80.38
2006 65.06 77.09
2007 61.49 73.49
2008 55.48 68.22
2009 53.02 65.49

3.5 Natural capital contribution to value-added growth

The contribution of the natural capital input to the industry value-added growth is moderate in oil and gas extraction. From 1981 to 2009, the log growth of value-added in oil and gas extraction was about 2.3% per year, of which about 0.3 percentage points per year or 15% came from the growth in the natural capital input (Table 1).

Table 1
Source of value-added growth, and multifactor productivity growth, oil and gas extraction industry, selected periods, 1981 to 2009
Table summary
This table displays the results of Source of value-added growth Period, 1981 to 2000, 2000 to 2008 and 1981 to 2009, calculated using percent and percentage points units of measure (appearing as column headers).
  Period
1981 to 2000 2000 to 2008 1981 to 2009
percent
Value-added growth (log), annual average 3.22 0.39 2.31
  percentage points
Contribution  
Labour input 0.08 0.84 0.32
Produced capital input 2.45 4.64 3.16
Natural capital input 0.43 0.16 0.34
Multifactor productiivty 0.26 -5.25 -1.51
  percent
Multifactor productivity growth (log), annual average before adding natural capital -0.04 -6.96 -2.27

4 Conclusion

To recognize subsoil energy and mineral resources as a capital input into the production process, this paper presents a growth accounting framework that allows the derivation of measures on natural capital stock and natural capital input in the mining industries and provides a better understanding of contribution of natural capital to economic growth and the impact of adding natural capital on productivity measurement.

The empirical results suggest a significant contribution of natural capital to the real value-added economic growth in the Canadian oil and gas extraction. However, the impact of adding natural capital in the growth accounting on the measured MFP growth changes over time. It is small before 1993 and becomes larger thereafter.

5 Appendix

Appendix Table 1
Sensitivity of natural capital value to real discount rate, oil and gas extraction industry, average, 1981 to 2009
Table summary
This table displays the results of Sensitivity of natural capital value to real discount rate Value at 0% discount divided by
value at 4% discount, calculated using ratio units of measure (appearing as column headers).
  Value at 0% discount divided by
value at 4% discount
  ratio
Total 1.49
Natural gas 1.38
Crude oil 1.21
Crude bitumen 1.80
Appendix Table 2
Input cost shares and input growth, oil and gas extraction industry, selected periods, 1981 to 2009
Table summary
This table displays the results of Input cost shares and input growth Period, 1981 to 2000, 2000 to 2008 and 1981 to 2009, calculated using percent units of measure (appearing as column headers).
  Period
1981 to 2000 2000 to 2008 1981 to 2009
percent
Annual average cost share  
Labour 12.80 9.60 11.80
Produced capital 70.80 64.90 68.50
Natural capital 16.50 25.50 19.70
Average annual input growth (log)  
Labour 1.87 9.17 4.22
Produced capital 3.57 7.17 4.73
Natural capital 3.16 0.78 2.40

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Acknowledgements

The authors would like to thank John Baldwin, Wulong Gu, Michael Wright of Statistics Canada; Pierre-Alain Pionnier of the OECD; Michael Smedes of the Australian Bureau of Statistics; Vernon Topp of the Australian Productivity Commission; Erik Veldhuizen of Statistics Netherlands; and Carl Obst of the London Group for their valuable comments and suggestions. Thanks also to  participants of the 2013 CANSEE (Canadian Society for Ecological Economics) conference at York University, Toronto; and 2014 NAPW (North American Productivity Workshop) VIII Conference at Ottawa/Gatineau, for helpful discussions. Any errors are those of the authors.

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