12.1.1 Definitions and Formulas

Multiplicative Model

Basically this is a power function model for multivariable data. Also referred to as a ”constant elasticity” model. (Elasticity is described in the next section.) A multiplicative model with two independent variables takes the form

where A, B, and C are all constants (parameters).

Cobb-Douglas

This is a model for total production based on the levels of labor investment, capital investment, and other investments that influence productivity. If K = capital investment, L = labor investment and P = production, the Cobb-Douglas model look like

Notice that it is a multiplicative model as discussed above. There are some important cases in the Cobb-Douglas model depending on the values of the two powers, B and C. In general, these constants are both less than 1. The model reflects the idea that if you have a lot of labor investment (lots of workers) but not enough capital (equipment for the workers to use) then productivity is hampered. If you have a lot of capital (equipment for production) but not the labor to use it, then production also suffers.

Non-constant Variance

This is a problem that often occurs in real data. The basic issue is that the residuals seem to ”fan out”. Thus, as the independent variable increases, the variability of the data around the proposed model increases systematically. (It is also possible for the variation to decrease systematically; this is less common, however.) Although the underlying pattern may be linear, non-constant variance is also ”fixed” by an appropriate transformation of the variables.