WPS3803
How Substitutable Is Natural Capital?1
Anil Markandya
FEEM, The World Bank and University of Bath
Suzette Pedroso-Galinato
The World Bank
World Bank Policy Research Working Paper 3803, December 2005
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange
of ideas about development issues. An objective of the series is to get the findings out quickly, even if the
presentations are less than fully polished. The papers carry the names of the authors and should be cited
accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely those of the authors.
They do not necessarily represent the view of the World Bank, its Executive Directors, or the countries they
represent. Policy Research Working Papers are available online at http://econ.worldbank.org.
1This paper is drawn from a larger study on the `Where is the Wealth of Nations?' (World Bank, 2005a). We wish
to thank Giovanni Ruta, Kirk Hamilton, Giles Atkinson and a number of others who have provided useful comments
on earlier drafts.
2
Abstract
One of the recurring themes in the sustainability literature has been the legitimacy of using an
economic framework to account for natural resources. This paper examines the potential for
substituting between different inputs in the generation of income, where the inputs include
natural resources such as land and energy resources. A nested constant elasticity of substitution
(CES) production function is used to allow flexibility in the estimated elasticities of substitution.
Also, with this specification, natural resources and other inputs are combined in different levels
of the function, thus allowing for different levels of substitutability. Institutional and economic
indicators are also incorporated in the production function estimated. Results show that the
elasticities derived from functions involving land resources were generally around one or greater,
implying a fairly high degree of substitutability. Furthermore, changes in trade openness and
private sector investment have a statistically significant and direct relationship with income
generation. No statistically significant relationship between income and any of the institutional
indicators was found.
I. Background
One of the recurring themes in the sustainability literature has been the legitimacy of
using an economic framework to account for natural resources. Those critical of such an
approach contend that wealth accounting assumes natural resource assets can be substituted by
produced assets, such as human and physical capital, on a dollar for dollar basis. This, they
argue, does not capture the limited degree to which such substitution is possible. A loss of some
natural capital, such as an entire ecosystem, surely cannot be made up with an increase in
physical capital if the very basis of social existence and well-being are destroyed in the areas
affected by that system. This makes them skeptical of the kind of wealth accounts we are
constructing here.
While we cannot hope to disentangle the full set of issues embedded in this line of
reasoning, we can at least start by focusing on the degree of substitutability between the different
assets. Underlying any wealth accounts is an implicit `production function' which is a blueprint
of the combinations of different assets with which we can achieve a given level of output. These
blueprints are usually written as a mathematical function, which describes the precise
relationship between the availability of different amounts of `inputs', such as physical and
human capital services, and the maximum output they could produce. The substitutability
between inputs is then measured as an `elasticity of substitution'. In general terms, this captures
the ease with which a decline in one input can be compensated by an increase in another, while
holding output constant. More precisely, it measures how much the ratio of two inputs (e.g.
physical capital and land) changes when their relative price changes (e.g. the price of land goes
up relative to the price of capital)2. The greater the elasticity, the easier it is to make up for the
loss of one resource by using another. Generally, an elasticity of less than one indicates limited
substitution possibilities.
A commonly used production function, which implies elasticities of one between the
inputs, is the `Cobb-Douglas' form, written as:
Yt = AtK L (1)
Income or output (Y) is expressed as a function of the levels of capital input (K), labor
input (L), an exogenous technological factor (A) and the parameters and , which give the
returns to capital and labor respectively. If the national production options could be captured by
such a function, with natural capital services included, it would have considerable implications
for sustainability. First, it would imply a degree of substitutability between natural and produced
capital that would give some comfort to those who argue we can lose some natural capital
without seriously compromising our well-being. Related to that it would validate the `Hartwick
Rule', which states that when exploiting natural resources you should ensure you save an amount
equal to the rent from those resources if you are to sustain the highest possible level of
consumption (Hamilton, 1995). This so-called `Hartwick Rule' is a useful sustainability policy
since it is open to monitoring we can check whether or not it has been adhered to.
2Where prices are not defined, we measure the change in the ratio of the inputs resulting from a change in the
marginal rate at which one factor can be substituted for another (Chiang, 1984). The discussion is complicated by
the fact that there are other definitions of substitution in the literature. We discuss this further below.
4
Economists have devoted a considerable amount of effort to estimating these elasticities,
for inputs such as capital, labor and energy but not natural resources. Although, starting in the
1970s, there were theoretical studies that modeled neoclassical economic growth with non-
produced capital such as natural resources as factors in production (e.g., Stiglitz, 1974a,b; Mitra,
1978)3; the empirical estimation of the underlying production functions was never carried out,
largely because of a lack of data.
This paper is a preliminary attempt in that direction. It is part of a larger study undertaken
by the Bank on the Wealth of Nations (World Bank, 2005a). In that study, a database of new
wealth estimates has been developed which includes both produced and non-produced capital
renewable and non-renewable resources, and human resources. This allows us to estimate a
production function that includes the services from these different resources as inputs. This paper
examines therefore the economic relationship between total wealth and income generation and
takes advantage of the new wealth estimates to estimate a production function based on a larger
set of assets. Section II briefly discusses the definitions of elasticities and describes related
studies on substitution between different inputs, including natural resources. Section III presents
the estimation of the production function and Section IV concludes.
II. Definitions of elasticities and results of earlier studies
The definition of an elasticity of substitution given at the beginning of this paper applies
unambiguously when there are only two inputs. With more than two inputs, however, the
generalization depends on what is assumed constant when the changes in the inputs of interest
are being calculated. The issues are familiar to economists a summary can be found in Kang
and Brown (1981). Based on work by Morishima, they define a `full elasticity of substitution
between inputs `i' and `j' as Fij where:
Fij = d ln(Xi / X ) j (2)
d ln( f j / fi) Yconst.
( fk / fi )const.
k i, j
Xi and Xj are levels on inputs `i' and `j' respectively. They are two of the `n' inputs that go into
producing output Y, as represented by the production function:
Y = f (X1, X2....Xn) (3)
and
fi = Y
Xi (4)
3A bibliographical compilation of studies can be found in Wagner (2004). One exception to the observation that
there is little empirical work is Berndt and Field (1981), who did look at limited natural resource substitution
between capital, labor, energy and materials. The studies generally found low elasticities between capital and
materials. They did not, however, look at land as an input in the way we do here. Nor did they work with national
level data.
5
Unfortunately, not all studies report this full elasticity of substitution (Equation 2), which
corresponds more closely to the concept of substitutability that we are interested in. More
commonly the `Allen' partial elasticity is reported, which, for inputs `i' and `j' is given by Aij:
Aij = .1 ln Xi (5)
sj ln pj
where sj is the share of total cost of production represented by input `j' and pj is the price of input
j. Readers will recognize Aij as the weighted cross price elasticity of demand for input `i' with
respect to the price of input `j'. Kang and Brown (op. cit.) cite the result from Morishima which
relates the full elasticities of substitution to the Allen elasticities as follows:
Fij = sj(Aij - Ajj) = Eij - Ejj (6)
Eij is the simple unweighted cross price elasticity of demand for input `i' it gives the
proportional change in the use of input `i' for a proportional change in the price of input `j'.
Given that in almost all conditions the own price elasticity (Ejj) is negative, the full elasticity of
substitution will be greater than the corresponding direct price elasticity. The relationship
between the Allen and the Full elasticity is less clear but, as we will see below, the latter are
typically smaller than the former.
In the simple case of two inputs, we note that the full elasticity of substitution cannot be
negative. A negative elasticity of substitution is economically nonsensical it implies that a
decline in the availability of one input can be `made up' by a decline in the availability of other
factors. With more than two inputs it is theoretically possible for this elasticity to be negative
but such a case is highly unlikely. It would require the cross price elasticity between two inputs
to be negative (indicating they are complements) and to be greater in absolute value than the
own price elasticity. Taking capital and natural resource inputs as an example, a negative full
elasticity would imply that a one percent increase in the price of the natural resource would
reduce inputs of that natural resources by a smaller percent than it decreased the inputs of capital.
Note also that by the relation defining Fij, the full price elasticity between `i' and `j' is not the
same as that between `j' and `i' i.e. Fij Fji.
Estimates of the elasticities of substitution are either reported as Allen elasticities or Full
elasticities but rarely both. Kang and Brown (op. cit.) have calculated the Full elasticities for
some studies where the Allen elasticities are reported. Table 1 gives what estimates are available
and indicates which elasticity has been reported.
Previous studies demonstrate Full elasticities that are considerably lower than the Allen
elasticities. Moreover, they are almost all positive and less than one. The one exception is the
Hudson and Jorgensen (1974) study which suggests that the capital-energy elasticity could be
negative. As that study estimates the own price elasticity for energy to be positive, we can
probably discount it.
6
The other result that has attracted a lot of attention is the difference between those studies
that find capital and energy as complements (Allen elasticity is negative) and those that find the
two factors as substitutes (Allen elasticity is positive). The differences between these studies
have been attributed to a number of factors: use of time series versus cross-section data (Griffin,
1981), response of output to changes in relative prices (Solow, 1987), and different methods of
aggregating capital (Garofalo and Malhotra, 1988).
We also note that there is little information in existing studies on the substitutability
between natural resources and other inputs. The Parks (1971) study looks at different inputs
used in manufacturing, including materials from the agricultural sector, capital and labor. In
terms of Allen elasticity, the author finds a complementarity between capital and agricultural
inputs, and a substitutability between agricultural inputs and labor. We could not recover the full
elasticity but it is likely to be low. The Moroney and Trapani (1981) study finds substitutability
between the inputs, labor and exhaustible mineral resources, in mineral-intensive production
processes. The two studies are not comparable; nor do they really throw much light on the
substitutability issues we are discussing here.
More recent studies have focused on the substitution between energy and another input,
such as labor or capital, using the same definition of elasticity that we employ. Manne and
Richels (1992) and Chang (1994) estimated the substitution possibilities between the `capital and
labor nest' and energy to be about 0.4; while Kemfert (1998) estimated the same to be about 0.5.
On the other hand, Prywes (1986) found the substitution elasticity between the `capital and
energy nest' and labor to be less than 0.5. These studies use the variables capital, labor and
energy as relating to stock of fixed assets, skilled and unskilled labor, and final energy
consumption, respectively.
III. Estimation of Nested CES production function
A. The Nested CES production function and variables
The estimation carried out here uses national level data on Gross National Income (GNI)
or economic output and sees the extent to which variations in GNI across countries at any point
in time can be explained in terms of the national availability of produced capital, human
resources and natural resources (energy and land resources). A Cobb-Douglas production
function of the form shown above is not appropriate for this estimation because it restricts the
elasticity between factors to be one. In fact, one of our objectives is to estimate the elasticity of
substitution between factors or groups of factors. A form that holds the elasticity constant but
allows it to take values different from one is the `constant elasticity of substitution' (CES)
production function. In particular, this paper uses a nested CES production function. For
example, a two level nested CES with three inputs takes the form:4
X = F[X AB(A,B),C] (7)
4This model makes the further assumption of `homothetic weak separability' for groups of inputs. Homothetic
weak separability means that the marginal rate of substitution between inputs in a certain group is independent of
output and of the level of inputs outside that group (Chiang, 1984)
7
where X is the gross output; A, B and C are inputs; and XAB represents the `joint contribution' of
A and B to production. The first level of the estimation involves A and B; while the second level
models the production of output by XAB and C. A special feature of the nested CES function is
that the elasticity of substitution between the first level inputs A and B can be different from the
elasticity of substitution between the second level inputs XAB and C. In other words, by placing
natural resources and other inputs in different levels of the function, we effectively allow for
different levels of substitutability. So, for example, natural assets may be critical (low
substitutability) while other inputs are allowed to be more substitutable among themselves.
In this paper we use related variables to estimate aggregate national level production
functions. The variables used are5:
a. Produced capital (K) is an aggregate of equipments, buildings and urban land;
b. Human capital (H) has two alternative measures: human capital, which relates
educational attainment with labor productivity (HE); or intangible capital residual (HR),
which is obtained as the difference between a country's total wealth and the sum of
produced and natural assets. Part of the intangible capital residual captures human capital
in the form of raw labor and stock of skills. For further discussion of this variable and its
rationale see World Bank (2005a; particularly, Chapter 3 and Chapter 10);
c. Production and net imports of non-renewable energy resources (E) includes oil, natural
gas, hard coal and lignite6.
d. Land resources (L), which refer to the aggregated value of crop land, pasture land and
protected areas. Land is valued in terms of the present value of the income it generates
rather than its market value.
The gross national income (GNI) and all inputs mentioned above are measured in per
capita values at 2000 prices and are taken at the national level for 208 countries. GNI data are
obtained from the World Development Indicators (World Bank, 2005). HE is derived based on
the work by Barro and Lee (2000); while the remaining variables, K, HR, E and L are the
components of wealth as described in World Bank, 2005a (Chapter 3).
The relationships of the production inputs to income are expressed in nested CES
production functions described in the Annex. Three different nested CES approaches are
examined (a) one-level function, with two inputs; (b) two-level function, with three inputs; and
(c) three-level function with four inputs. The combinations of the variables in the different CES
approaches were varied to further investigate any possible differences among substitution
elasticities for pairs of inputs.
The production function approach taken so far neglects an important set of factors that
influence differences in national income. These relate to the efficiency with which productive
5Per capita dollar values at nominal 2000 prices.
6 For energy it would be inappropriate to take the stock value of the asset, as what is relevant for production is the
flow of energy available to the economy. This is given by production plus net imports. With the other assets (K, H
and L) it is also the flow that matters but it is more reasonable to assume that the flow is proportional to the stock.
We do note, however, in the conclusions that even this assumption needs to be changed in future work.
8
assets are utilized and combined and include both institutional as well as economic factors. In
this study, we consider the following institutional indicators, which capture the efficiency with
which production can take place; as well as economic indicators, which also capture the
efficiency of economic organization:
a. institutional development indicators - indices on: voice and accountability (VA); political
instability and violence (PIV); government effectiveness (GE); regulatory burden (RB);
rule of law (RL) and control of corruption (CC). An increase in a given index measures
an improvement in the relevant indicator. Hence, they are expected to have a positive
impact on income and possibly growth. These indicators were estimated by Kaufmann,
et al, 2005.7
b. economic indicators - trade openness (TOPEN) is calculated as the ratio of exports and
imports to GDP (World Bank, 2005b); and the country's domestic credit to the private
sector as proportion of GDP (PCREDIT), which represents private sector investments
(Beck, et al., 1999)8.
Two methods of incorporating the impact of these institutional and economic indicators
were investigated. The first method involved the derivation of residuals from the regression of a
nested CES production function. The residuals are the part of income not explained by the wealth
components physical capital, human capital, land resources and energy resources, and are
regressed on the identified institutional and economic indicators. By using this method, however,
a statistically significant correlation between the residuals and any indicator would imply that
relevant variables have been omitted in the estimation of the nested CES production function.
Thus, the estimated coefficients of the nested CES production function earlier derived will be
biased and inefficient (Greene, 2000). Hence, another method is considered to be more
appropriate. The influences of the institutional and economic indicators on income will be
incorporated into the efficiency parameter of the production function, A (see Annex).
Depending on the available data for the variables of the nested CES production function,
the number of countries drops in the range of 67 to 93 countries. For a given nested CES
approach, the reduction is caused by considering only those countries that have non-missing
observations for their corresponding dependent and explanatory variables (i.e., complete case
method).9
7Data can be obtained from the website: http://www.worldbank.org/wbi/governance/pubs/govmatters4.html.
8Hnatkovska and Loayza (2004) use openness and credit as a measure of financial depth, which they find to have a
positive impact on growth. Data for this indicator can be obtained from the following website:
http://www.worldbank.org/research/projects/finstructure/database.htm.
9An "imputation method" was tried to fill the missing values for some of the countries to keep all 208 countries in
the estimation. Most of the results, however, were not found to be reasonable. For example, the imputed value of
physical capital for a low income country turned out to be too high compared to the average value of physical capital
of its income group. Hence, the imputation method was not used since it poses more problems in the estimates than
using the complete case method.
9
B. Regression results
The nested CES production functions are estimated using a non-linear estimation
method.10 The sample size in each CES approach differs because countries with missing
observations in any of the variables had to be dropped. Table 2 shows the estimated substitution
elasticities corresponding to the case where human capital is part of the measured intangible
capital residual (HR). All the statistically significant substitution elasticity estimates have a
positive sign, which is encouraging. The lowest is that between K and E at 0.37 in the three-
level production function. It is also interesting to note that most of the significant elasticities of
substitution are close to one.
A second round of regressions was carried out using the other measure of human capital
that is related to schooling and labor productivity, HE. Table 3 shows the statistically significant
elasticities of substitution, which also have a positive sign. A substitution elasticity
approximately equal to 1 is likewise found for most of the nested functions.
The results provide some interesting findings. First and foremost, there is no sign that the
elasticity of substitution between the natural resource (land) and other inputs is particularly low.
Wherever land emerges as a significant input, it has an elasticity of substitution approximately
equal to or greater than one. Second, by and large, the HE variable performs better in the
estimation equations than the HR variable. Third, the best determined forms, with all
parameters significant are those using HE, involving four factors and containing the following
combinations: (a) K, HE and L are nested together and then combine with E; or (b) K, HE and E
are nested together and then combine with L.11 It is hard to distinguish between these two
versions and so they are both used in the further analysis reported below.
From the nested CES production function estimations, the elasticity estimates of the
institutional and economic indicators can be derived. Table 4 and Table 5 show the results for
the four-factor production functions: [(K,HE,L)/E] and [(K,E,HE)/L] of Table 3, respectively. In
both Tables, the variables on trade openness and private sector investment are found to be
statistically significant. The elasticity estimates of these two variables are not very different
from each other. The results imply that for every percent increase in trade openness, gross
national income per capita increases by approximately 0.5 percent. None of the institutional
indicators, on the other hand, has a statistically significant elasticity estimate.12
10See Annex for more details.
11"Inputs nested together" refers to the joint contribution of these inputs to the production of output, as indicated in
Section III of this paper.
12In the regression where the `residuals' are expressed as a function of the institutional variables, we did find
significant values for a few institutional variables, especially the rule of law, which was encouraging as that variable
also emerges as important in other evaluations of inter-country differences in the World Bank (2005a) study.
Unfortunately, the result did not hold when the more appropriate method was used.
10
C. Simulation
The predicted value of the dependent variable can be calculated by using the estimated
coefficient estimates of the production function and the mean values of the explanatory variables.
Through this method, we try to predict what will happen to the economic output (per capita GNI
or GNIPC) if there is significant natural resource depletion. The natural resource considered in
this exercise is "land resources (L)" and the four-factor nested CES production functions used
are: [(K,HE,L)/E] and [(K,E,HE)/L] of Table 3. Table 6 presents the predicted average GNIPC
as well as the change in GNIPC given a reduction in the amount of land resources, ceteris
paribus. Based on the production function [(K,HE,L)/E], economic output is reduced by 50
percent when the amount of L declines by about 92 percent, while holding other variables
constant. For the production function [(K,E,HE)/L], on the other hand, it takes a reduction in the
amount of L by about the same percentage, ceteris paribus, to halve the economic output relative
to the baseline.
IV. Conclusions
In this paper we looked at the potential for substituting between different inputs in the
generation of GNI. Among these are land resources, one of the most important natural resources.
The estimation of a well-known production function form, which allows the elasticities of
substitution to be different from one, was carried out. The resulting elasticities involving land
resources (between L and other inputs such as physical capital, human capital and energy
resources) were generally around one or greater, which implies a fairly high degree of
substitutability. Moreover, it validates the use of a `Hartwick Rule' of saving the rents from the
exploitation of natural resources if we are to follow a maximum constant sustainable
consumption path.
There are, of course, many caveats to this result. Land resources, as measured here
include crop land, pasture land and protected areas. Each has been valued in terms of the present
value of the flow of income that it generates. Such flows, however, under-represent the
importance of protected areas for example, which provide significant non-monetary services,
including ecosystem maintenance services that are not included. Further work is needed to
include these values, and if this were done, and if the GNI measure were adjusted to allow for
these flows of `income', the resulting estimates of substitution elasticities might well change.
We intend to continue to work along these lines and to improve the estimates made here.
Another shortcoming of the method applied here is the limited number of factors
included in the original estimation. Generating national income depends not on the stock of
assets but the amounts of the stocks that are used in production and the way in which they are
used. For physical and human capital and land, we assume the rate of use is proportional to the
stock. That assumption should be improved on, to allow for different utilization rates.
The treatment of institutional factors can also be improved. In this version, they are
assumed to affect the overall efficiency of production rather than the efficiency of specific
inputs, such as capital and labor. A modified estimation equation in which K, L and HE were
11
differently affected by different institutional factors would probably find greater significance for
these factors than we have.
Finally, the paper also examined how the institutional and economic indicators will affect
the generation of GNI. Estimation results show that income generation is significantly
influenced by changes in trade openness and private sector investment. The institutional
indicators, however, have no statistically significant impact on income generation.
12
Table 1: Estimates of elasticities of substitution in previous studies
Inputs Estimates Study
Allen Elasticity Full Elasticity
Capital and natural -0.82 - Parks (1971)
resource inputs (0.47; 1.08)a - Moroney and Trapani (1981)
Labor and natural 0.90 - Parks (1971)
resource inputs (0.63 to 1.33)b - Moroney and Trapani (1981)
Capital and Labor 0.12 - Parks (1971)
1.09 (0.56 : 0.74)* Hudson and Jorgenson (1974)
1.01 - Brendt and Wood (1979)
(.06 : 0.39) (0.17 : 0.19)* Griffin and Gregory (1976)
(0.60 : 0.95) - Moroney and Trapani (1981)
0.88 - Prywes (1986)
- 0.82** Kemfert (1998)
Capital and Energy -1.39 (-0.09 : 0.24) * Hudson and Jorgenson (1974)
-3.22 0.26** Brendt and Wood (1979)
1.03: 1.07 (0.33 : 0.92)* Griffin and Gregory (1976)
-1.35 - Prywes (1986)
2.17 0.87** Chang (1994)
- 0.65** Kemfert (1998)
Labor and Energy 2.16 - Hudson and Jorgenson (1974)
0.65 - Brendt and Wood (1975)
(0.84 : 0.87) - Griffin and Gregory (1976)
0.88 - Prywes (1986)
0.35 - Chang (1994)
0.42** Kemfert (1998)
Labor and Capital - 0.40** Manne and Richels (1992)
`nest' and Energy - 0.42** Chang (1994)
0.50** Kemfert (1998)
The figures are rounded off to the nearest hundredth.
Unless stated otherwise, the elasticities above are calculated for the aggregate of the industry.
Energy the variable refers to total energy consumption
aThe given Allen elasticities are statistically significant only for two industries: primary aluminum and blast
furnaces/basic steel, respectively.
bA range is given to cover the estimated elasticities for 4 of the 6 industries studied: primary aluminum, blast
furnaces/basic steel, storage batteries and hydraulic cement.
(*) The full elasticity was calculated by Kang and Brown (1981)
(**) The full elasticity was calculated by the authors
Study focus and estimation method:
Parks (1971): Swedish manufacturing industry; Generalized Leontief Function
Moroney and Trapani (1981): Mineral-intensive manufacturing industries; Translog cost model
Hudson and Jorgenson (1974): U.S. manufacturing sector; Translog cost function
Berndt and Wood (1979): U.S. manufacturing sector; Translog cost function
Griffin and Gregory (1976): U.S. and U.K. manufacturing sectors, respectively; Translog cost function
Prywes (1986): Manufacturing sector; Nested CES estimation.
Manne and Richels (1992): Manufacturing sector; Nested CES
Chang (1994): Taiwan manufacturing industry (aggregate); Nested CES and Allen elasticities
of substitution
Kemfert (1998): German manufacturing industry (aggregate); Nested CES
13
Table 2: Substitution elasticities ^i , using intangible capital residual (HR)
( )
Inputs Substitution elasticity R-squared Adj. R- Sample
^i Standard squared size
error
A. Two factors (One level CES production function)
(1) K/HR 1.00* 3.88E-10 0.9216 0.9131 93
(2) K/E -0.48 2.02 0.9958 0.9951 78
B. Three factors (Two level CES production function)
(1) (K,HR)/L 0.9375 0.9290 93
K/HR 6.79 13.92
(K,HR)/L 1 1.00* 4.33E-10
(2) (K,HR)/E 0.9089 0.8916 70
K/HR -0.78 1.31
(K,HR)/E 1 1.00* 5.37E-10
(3) (K,E)/HR 0.87667 0.8533 70
K/E 0.65 0.69
(K,E)/HR 1 1.00* 3.96E-09
C. Four factors (Three level CES production function)
(1) (K,HR,L)/E 0.3435 0.1911 70
K/HR -0.90 0.70
(K,HR)/L 1 0.97* 0.01
(K,HR,L)/E 2 1.00* 5.46E-12
(2) (K,HR,E)/L 0.9958 0.9951 78
K/HR -0.13 0.17
(K,HR)/E 1 0.93* 0.18
(K,HR,E)/L 2 1.00* 6.52E-09
(3) (K,E,HR)/L 0.9350 0.9200 70
K/E 0.37* 0.20
(K,E)/HR 1 -0.64 0.55
(K,E,HR)/L 2 1.00* 1.27E-09
Legend: K physical capital; HR intangible capital residual (captures raw labor and stock of skills); L land
resources; E energy resources
Notes: Inputs in parenthesis imply that they are nested.
1two inputs in a nested function
2three inputs in a nested function
(*) denotes statistical significance at 5% level
The substitution elasticities and their corresponding standard errors are rounded off to the nearest hundredth.
14
Table 3: Substitution elasticities ^i , using human capital related to schooling (HE)
( )
Inputs Substitution elasticity R-squared Adj. R- Sample
^i Standard squared size
error
A. Two factors (One level CES production function)
(1) K/HE 1.00* 2.50E-08 0.9061 0.8942 81
B. Three factors (Two level CES production function)
(1) (K,HE)/L 0.9203 0.9076 81
K/HE 1.01* 0.01
(K,HE)/L 1 1.00* 2.23E-10
(2) (K,HE)/E 0.8952 0.8742 67
K/HE 1.65* 0.12
(K,HE)/E 1 1.00* 6.76E-11
(3) (K,E)/HE 0.7674 0.7209 67
K/E 0.17 0.19
(K,E)/HE 1 1.00* 8.22E-08
C. Four factors (Three level CES production function)
(1) (K,HE,L)/E 0.9037 0.8081 67
K/HE 1.78* 0.11
(K,HE)/L 1 1.14* 0.02
(K,HE,L)/E 2 1.00* 2.52E-12
(2) (K,HE,E)/L 0.9059 0.8828 67
K/HE -8.55 12.61
(K,HE)/E 1 0.48* 0.17
(K,HE,E)/L 2 1.00* 4.60E-11
(3) (K,E,HE)/L 0.9062 0.8831 67
K/E 1.57* 0.37
(K,E)/HE 1 0.92* 0.02
(K,E,HE)/L 2 1.00* 6.41E-11
Legend: K physical capital; HE human capital related to educational attainment and labor productivity; L land
resources; E energy resources
Notes: Inputs in parenthesis imply that they are nested.
1two inputs in a nested function
2three inputs in a nested function
(*) denotes statistical significance at 5% level; (**) at 10% level
The substitution elasticities and their corresponding standard errors are rounded off to the nearest hundredth.
15
Table 4: Elasticity estimates of the economic and institutional indicators
using the [(K, HE, L)/E] production function
Variable Elasticity Standard error t-statistic
TOPEN 0.47 0.10 4.53
PCREDIT 0.51 0.12 4.25
VA 0.01 0.04 0.28
PIV -0.01 0.02 -0.28
GE 0.04 0.10 0.40
RB 0.03 0.07 0.39
RL -0.07 0.10 -0.73
CC 0.01 0.09 0.17
Legends: TOPEN-trade openness; PCREDIT- variable for private sector
investment; VA - voice and accountability; PIV- political instability and violence;
GE - government effectiveness; RB - regulatory burden; RL - rule of law; and
CC - control of corruption.
Table 5: Elasticity estimates of the economic and institutional indicators
using the [(K, E, HE)/L] production function
Variable Elasticity Standard error t-statistic
TOPEN 0.50 0.09 5.27
PCREDIT 0.51 0.11 4.83
VA 0.02 0.03 0.45
PIV -0.01 0.02 -0.44
GE 0.06 0.09 0.62
RB 0.03 0.07 0.37
RL -0.08 0.09 -0.86
CC -0.02 0.08 -0.24
Legends: TOPEN-trade openness; PCREDIT- variable for private sector
investment; VA - voice and accountability; PIV- political instability and violence;
GE - government effectiveness; RB - regulatory burden; RL - rule of law; and
CC - control of corruption.
Table 6: Level of Gross National Income per capita given a reduction in the amount of land
Prod. function Baseline* Reduction in the amount of land by
20% 50% 75% 92%
(K,HE,L)/E $8,638.10 $8,068.84 $7,019.27 $5,774.25 $4,297.16
Difference from baseline** (-7%) (-19%) (-33%) (-50%)
(K,E,HE)/L $9,096.20 $8,540.27 $7,477.97 $6,147.62 $4,455.06
Difference from baseline** (-6%) (-18%) (-32%) (-51%)
*Predicted per capita GNI at the mean values of the explanatory variables.
**Rounded off to the nearest whole number
Sample size of each production function = 67
16
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18
Annex
Three different CES approaches
1. A one-level nested CES is a traditional CES production function with two inputs and written
as:
(i) physical capital (K) and human capital (H)
Y = A aK - + bH
( - )
-1 (A.1)
(ii) physical capital (K) and energy resources (E)
Y = A aK - + bE-
( )-1 (A.2)
where Y is the per capita gross national income. A is an efficiency parameter. a and b are
distribution parameters that lie between zero and one; and represents the substitution
parameter. The substitution elasticity () is calculated as: = (1 1+ ). Values of must
be greater than -1 (a value less than -1 is economically nonsensical, although it has been
observed in a number of studies see for example, Prywes, 1986). If > -1 the substitution
elasticity must of course be positive.
A, the efficiency parameter, is assumed to be a function of the economic (TOPEN and
PCREDIT) and institutional indicators described in the text. Two functional forms of A have
been tried:
(a) A = e1TOPEN +2PCREDIT+3VA+4PIV+5GE+6RB+7RL+8CC
(b) A = 1TOPEN + 2PCREDIT + 3VA + 4PIV + 5GE + 6RB + 7RL + 8CC
and the second functional form of A was found to be more appropriate.
TOPEN means trade openness; PCREDIT is a variable for private sector investment; VA,
voice and accountability; PIV, political instability and violence; GE, government
effectiveness; RB, regulatory burden; RL, rule of law; and CC, control of corruption. The
scores for each institutional indicator lie between -2.5 and 2.5, with higher scores
corresponding to better outcomes.
2. A two-level nested CES production function with three inputs is investigated for three cases:
(i) K and H in the nested function, XKH is a substitute to land resources (L):
Y1 = A1a1 b1K -1 + (1-b1)H -1
( )
1 1 +(1-a1)L-1 -1 1
(A.3)
(ii) K and H in the nested function, XKH is a substitute to energy resources (E);
Y2 = A2 a2 b2K -2 + (1-b2 )H -2
( )
2 2 +(1-a2)E-2 -1 2
(A.4)
19
(iii) K and E in the nested function, XKE is a substitute to human capital (H);
Y3 = A3 a3 b3K -3 + (1- b3 )E-3
( )3 3 + (1- a3)H -3 -1 3
(A.5)
where i and i are substitution parameters.
3. A three-level nested CES production function with four inputs is studied for these three
cases:
(i) K, H and L in the nested function, and E as a substitute to XKHL:
Y4 = A4 a4 b4 c4K - + (1- c4)H
[ ( 4 -4)4 4 + (1-b4)L- 4 ]
4 4 + (1- a4)E-4 -1 4
(A.6)
(ii) P, H and E in the nested function, and L as a substitute to XKHE ;
Y5 = A5 a5 b5 c5K - + (1- c5)H
[ ( 5 -5 )5 5 + (1-b5)E- 5 ]
5 5 + (1- a5)L-5 -1 5
(A.7)
(iii) K, E and H in the nested function, and L as a substitute to XKEH .
Y6 = A6a6 b6 c6K- + (1- c6)E-
[ ( 6 6 )
6 6 + (1-b6)H- 6 ]
6 6 + (1- a6)L6-6 -1 6
(A.8)
where i,i,i are substitution parameters; and 0 < ai,bi,ci <1.
The substitution elasticities for these CES Approaches can be described as follows:
i = 11 Gives the elasticity of substitution between K and H when `i' = 1,2,4,5
+i Gives the elasticity of substitution between K and E when `i'= 1,6
i = 11 Gives the elasticity of substitution between K/H and L when `i' = 4
+ i Gives the elasticity of substitution between K/H and E when `i' = 5
Gives the elasticity of substitution between K/E and H when `i' = 6
i = 11 Gives the elasticity of substitution between K/H and L when `i' = 1
+ i Gives the elasticity of substitution between K/H and E when `i' = 2
Gives the elasticity of substitution between K/E and H when `i' = 3
Gives the elasticity of substitution between K/H/L and E when `i' = 4
Gives the elasticity of substitution between K/H/E and L when `i' = 5
Gives the elasticity of substitution between K/E/H and L when `i' = 6
The nested CES production functions are estimated using the non-linear estimation
method via the STATA program. The non-linear estimation program uses an iterative procedure
to find the parameter values in the relationship that cause the sum of squared residuals (SSR) to
be minimized. It starts with approximate guesses of the parameter values (also called, "starting
values"), and computes the residuals and then the SSR. The starting values are a combination of
20
arbitrary values and coefficient estimates of a nested CES production function. For example, the
starting values of Equation (A.1) are arbitrary. A set of numbers is tried until convergence is
achieved. On the other hand, the starting values of Equation (A.3) are based on the coefficient
estimates of Equation (A.1). Next, it changes one of the parameter values slightly, computes
again the residuals to see if the SSR becomes smaller or larger. The iteration process goes on
until there is convergence until it finds parameter values that, when changed slightly in any
direction, causes the SSR to rise. Hence, these parameter values are the least squares estimate in
the nonlinear context.