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:<sup>4</sup>

> 4 This 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)