Constant Difference of Elasticity

Definition

Constant Difference of Elasticity (CDE):

CDE function

Consider a expenditure minimization problem as:

Where:

p: price vector: note here we do not conduct linearization, so all variables are level, no matter upper or lowercase
x: demand vector in quantity
u: utility
E: minimum expenditure
f(x): utility function
G(p, u): minimum expenditure function, homogeneous of degree 1 in price, so we can normalize the minimum expenditure with price:

Normalized expenditure function

Where:

z: normalized price. at the normalized price level, we can achieve target utility with minimized cost equals 1.

To obtain CDE expenditure function, Hanoch (1975) restricts the number of substitution effects to N by imposing additivity in normalized price. The implicitly additive expenditure function form becomes

Where

: N parameters which determines substitution possibilities of commodities in consumption, < 1. SUBPAR
: N expansion parameters. They appear owing to non-homotheticity in consumption. > 0. INCPAR.
: scale parameters to specific the function. > 0

This function represents the minimized expenditure (G) given the utility level (u) and normalized price (z)

We can obtain the demand equation from CDE function via envelope result:

And convert the demand to linearized form.

Note: I think it is based on the Shephard's lemma.

Features of CDE

Special cases of CDE function:

Leontief: if
Cobb-Douglas function, if
Non-homothetic CES function: if
Homogeneous CES: if

CDE function lies between CES and fully flexible forms.

Calibration of CDE

CDE parameters (e, b) can be easily calibrated with existing data on income and own price elasticities.

With CDE function and the N of b parameters and N of e parameters (recall they are both indexed with i), we can deduce N own-price elasticities according to the note of demand system with CDE.

Or, if we have N own-price elasticities and N expenditure elasticities, we may also deduce the value of CDE parameter e and b.

Note