# Differential demand systems

#### IV.I.3 Differential demand systems

Review eq. (IV.I.1-26) and substitute (IV.I.1-19) into it (dropping *)

(IV.I.3-1)

Define the following

(IV.I.3-2)

which is the budget share and the marginal budget share.

Also denote Divisia's volume index and Frisch's price index as

(IV.I.3-3)

and define

(IV.I.3-4)

In general (c.q. with or without preference independence), eq. (IV.I.3-4) can be interpreted by rewriting it as

(IV.I.3-5)

As a next step, define the inverse of the income elasticity of the marginal utility of income

(IV.I.3-6)

Computing the total differential of p'q = m yields (using (IV.I.3-2), and (IV.I.3-3))

(IV.I.3-7)

This result can be used to easily prove that (IV.I.3-1) can be written as (after premultiplying with 1/m P)

(IV.I.3-8)

Already the Divisia volume index pops up in the differential demand system. Now, using the above definitions we can replace the following terms of (IV.I.3-8)

(IV.I.3-9)

Hence we may write (IV.I.3-1) via (IV.I.3-8) and (IV.I.3-9) as

(IV.I.3-10)

which is indeed a differential consumer demand system.

Furthermore note that

(IV.I.3-11)

(which is a symmetric matrix) or on using (IV.I.1-19)

(IV.I.3-12)

Hence

(IV.I.3-13)

due to (IV.I.1-21).

(IV.I.3-14)

This system does not have restrictions on the coefficients.

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