If diminishing returns to individual variables are believed
present, and the price elasticity is believed dependent on only
the quantity consumed, a useful transformation is to convert the
data on the independent variables such as to give

 Wi = fo + Plln zx, + p2 In z2i + + ,m Inm,,i + i. (A.6)

On the other hand, if price is believed to be dependent on both
quantity consumed and price, then the price variable should not
be expressed in log form. The statistical assumptions regarding
the error term ej in Equation (A.6) are the same as for Equa-
tion (A.5).
 Another useful form is given by
 W, = exp [I3o+ fi zi + p/32io + I + flmi + El]. (A.7)

Equation (A.7) can be estimated easily given the transformation
 In W, = g/o + z1i + 32z2i + + nzimZ, + Ei. (A.7a)

With the above structure, W increases (decreases if the coeffi
cient is negative) at an increasing rate relative to the value of
z7. This form is convenient if it is believed price elasticity is de-
pendent only on price. The elasticity with respect to each in-
dependent variable is PkZki-
 A functional form allowing for slightly more flexibility in the
type of returns and varying price elasticity (with price changes)
is a variation on the transcendental form,5 as follows

 W = PoZit, z22 .. z/,i r exp [P ,+lz,,-+1 + +
 A z,,,,. + ej]. (A.8)

This model can be estimated with OLS procedures, after trans-
formations are accomplished, to give
 In W, = Ino + + P1 In 1 + 2 In z + .. +
 f/, In z, + Iz,+i +,, + + 3P,,,z,,i + e. (A.8a)

Any variables for which elasticities are thought to be constant
should be included in the first r variables. All other variables
would be included in the set r+1, ., where the elasticity is
variable and dependent on the level of the variable.

 5See Halter, Carter, and Hocking [10] for a discussion of the trans-
cendental form.