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Production Function

Cobb-Douglas production functions involving sector-specific capital (KS) and labor (LABS) provide potential value added (VADDP) in each sector, taking into account the level of capacity utilization (CAPUT), initially set exogenously (caputtar). In a multi-sector model the functions require sectoral exponents for capital (CDALFS) and labor that, assuming constant returns to scale, sum to one within sectors.

Solow (1956) long ago recognized that the standard Cobb-Douglas production function with a constant scaling coefficient in front of the capital and labor terms was inadequate because the expansion of capital stock and labor supply cannot account for most economic growth. It became standard practice to represent an exogenously specified growth of technology term in front of the capital and labor terms as "disembodied" technological progress (Allen, 1968: Chapter 13). Romer (1994) began to show the value of unpacking such a term and specifying its elements in the model, thereby endogenously representing this otherwise very large residual, which we can understand to represent the growth of productivity.

In IFs that total endogenous productivity growth factor (TEF) is the accumulation over time of annual values of growth in multifactor productivity (MFPGRO). There are many components contributing to growth of productivity, and there is a vast literature around them. See, for example, Barro and Sala-i-Martin (1999) for theoretical and empirical treatment of productivity drivers; also see Barro (1997) for empirical analysis (or McMahon 1999) for a focus on education.

In the development of IFs there was a fundamental philosophic choice to make. One option was to keep the multi-factor productivity function very simple, perhaps to restrict it to one or two key drivers, and to estimate the function as carefully as possible. Suggestions included focusing on the availability/price of energy and the growth in electronic networking and the knowledge society.

The second option was to develop a function that included many more factors known or strongly suspected to influence productivity and to attempt a more stylistic representation of the function, using empirical research to aid the effort when possible. The advantages of the second approach include creating a model that is much more responsive to a wide range of policy levers over the long term. The disadvantages include some inevitable complications with respect to overlap and redundancy of factor representation, as well as some considerable complexity of presentation.

Because IFs is a thinking tool, the second approach was adopted, and the production function has become an element of the model that will be subject to regular revision and, hopefully, enhancement. IFs groups the many drivers of multifactor productivity into five categories, recognizing that even the categories overlap somewhat. The base category is one that represents the elements of a convergence theory, with less developed countries gradually catching up with more developed ones. The four other categories incorporate factors that can either retard or accelerate such convergence, transforming the overall formulation into conditional convergence.

0. The convergence base. The base rate of multifactor productivity growth (MFPRATE) is the sum of the growth rate for technological advance or knowledge creation of a technological leader in the global system (mfpleadr) and a convergence premium (MFPPrem) that is specific to each country/region. The basic concept is that it can be easier for less developed countries to adopt existing technology than for leading countries to develop it (assuming some basic threshold of development has been crossed). The base rate for the leader remains an unexplained residual in the otherwise endogenous representation of MFP, but that has the value of making it available to model users to represent, if desired, technological cycles over time (e.g. Kondratief waves). The base also includes a correction term (MFPCor) that is initially set of the difference between empirical growth of MFP (calculated the first year as a residual between factor growth and output growth) and the sum of the technological leader and convergence premium terms. Over time, the correction term is phased out, but the four other terms, each of which is computed relative to the initial year, become key drivers of country-specific productivity. In fact, significant change in the other terms can either undercut the foundational convergence process or greatly augment it.

1. Knowledge creation and diffusion. On top of the foundation, changes in the R&D spending (CHGRANDD), computed from government spending (GDS) on R&D as a portion of total government spending (GOVCON) contribute to knowledge creation, notably in the more developed countries (Globerman 2000 reviewed empirical work on the private and social returns to R&D spending and found them to be in the 30-40% range; see also Griffith, Redding, and Van Reenen 2000). Many factors undoubtedly contribute to knowledge diffusion. For instance, growth in electronic and related networking should contribute to diffusion (NumNWPBoost) in spite of the fact that empirical basis for estimating that contribution is very scant. This factor is dependent in IFs on the extent of networking or the "number of networked persons" in a society (NUMNWP), relative to the potential level of networking; when the full potential is achieved, the full incremental impact of networking (numnwpgrinc) passes through to multifactor productivity growth.

2. Human capital quality. This term has two components, one from changes (CngEduc) in educational spending (GDS) and the other from changes in health expenditure (CngHlth), both relative to GDP. Barro and Sala-i-Martin (1999: 433) estimate that a 1.5% increase in government expenditures on education translates into approximately a 0.3% increase in annual economic growth (elmfped).

3. Social capital quality. There is also an addition to growth (EconFreeGF) that can come from change in the level of economic freedom (ECONFREE); the value of the parameter (elgref) was estimated in a cross-sectional relationship of change in GDP level from 1985 to 1995 with the level of economic freedom. Barro places great emphasis in his estimation work on the "rule of law" and it may be desirable to substitute such a concept in the future.

4. Physical capital quality. Robert Ayres has correctly emphasized the close relationship between energy supply availability and economic growth. For instance, a rapid increase in world energy prices (WEP) essentially makes much capital stock less valuable. IFs uses world energy price relative to world energy prices in the previous year to compute an energy price term (EnPriceTerm).

The user can in scenarios add a further exogenous factor, by region (MFPADD). Finally, a correction factor (MFPCor) adjusts the entire growth mechanism to initial empirical rates. That adjustment is phased out over time. The computation of the scaling parameter (CDA) assures that gross production is consistent with data the first year.

All of the adjustment terms (for R&D expenditures, human capital quality, and so on) are computed on an additive basis–that is, they are computed as adjustments to underlying patterns and can be added to compute the overall productivity growth rate. They are all applied to the potential value added (VADDP) in each sector. Of course, GDP is the sum across sectors of valued added. Although the production function can serve all sectors of IFs, the parameters agon and enon act as switches; when their values are one, production in the agricultural and primary energy sectors, respectively, are determined in the larger, partial equilibrium submodels and the values then override this computation.



Moving to the computation of the annual growth in multifactor productivity, we turn to the collection of terms that drive it. As discussed above, there is a base rate linked to systemic technology advance and convergence plus four terms that affect its growth over time. In addition, there is a parameter for users to intervene for any country/region (mfpadd).


The base rate includes the rate of advance in the leader (mfpleadr), the premium computed for convergence of each country/region (MFPPrem), and a correction term computed in the first year and the then dropping out over time (MFPCor).


Finally, we have the four clusters of drivers discussed above, beginning with the knowledge term.


Driver Cluster 1: Knowledge Accumulation and Diffusion



Driver Cluster 2: Human Capital



Driver Cluster 3: Social Capital



Driver Cluster 4: Physical Capital



IFs normally does not use the above equations for the first two sectors because the agriculture and energy models provide gross production for them (unless those sectors are disconnected from economics using the agon and/or enon parameters).

Physical shortages may constrain actual value added in each sector (VADD) relative to potential production. Specifically, IFs assumes that energy shortages (ENSHO), as a portion of domestic energy demand (ENDEM) and export commitments (ENX) lower actual production through a physical shortage multiplier factor (SHOMF). A parameter/switch (squeeze) controls this linkage and can turn it off.


In addition, the translation of potential into actual production depends on the imports of manufactured goods (MKAV), which serve as a proxy for both availability of intermediate goods and for technological imports. A parameter (PRODME) also controls this relationship.