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 are<sup>5</sup> :

- 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 lignite<sup>6</sup> .

- 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

> 5 Per 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.