# Explain the definition of a primal shifter versus an input shifter parameters in the standard CES function

I have run into a CES function that seems to be very closer to standard but with a small disaggregation of the share parameter into two parameters (primal share) and (input shift). I am hoping someone can provide a reference to that introduction and the reasoning behind the disaggregation.

The standard form of the CES function is of course:

$$\begin{equation} V=A\left[\sum_{i=1}^n\alpha_ix_i^\rho\right]^{1/\rho}, \end{equation}$$

Where $$\begin{equation} a_i - share \ parameter \\ \sigma=\frac{1}{1-\rho} - elasticity \end{equation}$$

However GTAP CGE documents describe the CES (standard form as the following):

In production, the CES function is used to select an optimal combination of inputs (either goods and/or factors) subject to a CES production function. In consumer demand, the CES is used as a utility (or sub-utility) or preference function. In either case, the purpose is to minimize the cost of purchasing the ’inputs’ subject to the production or utility function. In generic terms the system takes the following form: $$\begin{equation} min \ X_i = \sum_i{P_i X_i} \end{equation}$$

subject to constraint: $$\begin{equation} V=A\left[\sum_{i=1}^n\alpha_i(\lambda_iX_i)^\rho\right]^{1/\rho}, \end{equation}$$

The objective function represents aggregate expenditure. The constraint expression will be referred to as the CES primal function. The parameter $$A$$ is an aggregate shifter that can be used to shift the overall production function (or utility function). Each input, $$X_i$$, is multiplied by an input-specific shifter, $$\lambda_i$$, that can be used to implement input specific productivity increases (for example biased technological change), or specific changes in consumer preferences.

The (primal) share coefficients, $$\alpha_i$$, are typically calibrated to some base year data and held fixed. The CES exponent, $$\rho$$, is linked to the curvature of the CES function (and will be explained further below). For given component prices, $$P_i$$, and a given level of production or utility $$V$$ , solving the optimization program above will yield optimal demand functions for the inputs, $$X_i$$.

My question surrounds the change of the term $$a_ix_i^\rho$$ to $$a_i(\lambda_ix_i)^\rho$$. Clearly there is some qualitative / model interpretation reason for creating this difference. The paper says that $$a_i$$ becomes the primal share parameter, and $$\lambda_i$$ becomes the input shift parameter which can represent specific input factor productivity increases or changes in preference.

I am hoping someone can answer the following:

1. What is the primal shift parameter and how does it differ from the original shift parameter?
2. I assume the $$A$$ factor is just the original TFP (techonology) factor, is that correct?
3. How exactly does $$\lambda_i$$ represent preference changes or techonology increases? Can that be described using some mathematical intuition?
4. Where does this formulation come from? Is there a paper that built on top of the original CES paper from Solow, Minhas et. al.?

Would greatly appreciate the help!

It is correct $$a_i$$ is the primal share parameter denoting how much of $$x_i$$ is converted to output $$V$$ the inclusion in of the $$\lambda_i$$ term becomes useful when creating production trees.

As you may be aware in GTAP and most CGE models there are nested production trees that describe the production process. Generally they are split in the following way:

                      Final Output
|
|----------------------------------|
|                                  |
Various Input Commodities           |---------------|
Capital          Labor


The above is a simplification but it follows the general pattern of a production tree or what is commonly referred to as the CES technology tree. As you will note there are composite goods and "raw" goods in this technology tree. For example Value-Added is a composite of Capital and Labor. Intermediate Inputs is a composite of all the various input commodities. And final output is a composite of Intermediate Inputs and Valued Added bundles. These can be called composites or bundles, the term is interchangeable. All composite goods in the tree are formulated using the CES equation, and the input goods will have their respective primal shares. What becomes more interesting is at the raw good level, for example Capital and Labor. These are not composites in the CES Technology Tree, they are raw inputs. Often times we want to model a productivity increase of one of these factors. This is where the $$\lambda_i$$ term comes in, it is the productivity multiplier of one of the input factors in the production tree. Often times labor productivity increases are modeled using this term.

The composite goods will not have $$\lambda_i$$ productivity terms, because conceptually, it doesn't make sense that composite goods increase in productivity. Composites are just a useful way of constructing a production tree where there is substitutability between some of the factors of production but not all of the factors of production. For example we can substitute capital with labor and vice versa. But you may not be able to subsitute land with people or capital.

So

$$\begin{equation} V=A\left[\sum_{i=1}^n\alpha_ix_i^\rho\right]^{1/\rho}, \end{equation}$$

and

$$\begin{equation} V=A\left[\sum_{i=1}^n\alpha_i(\lambda_iX_i)^\rho\right]^{1/\rho}, \end{equation}$$

are both forms of the CES Technology function. But the former is used for composite goods in the CES Production Tree, whereas the latter is used for "leaf" or "raw" input goods (often capital, labor types, land etc) in the CES Technology Tree.