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Differential demand systems

[Home] [Up] [Formal Derivation] [Rotterdam Model] [Differential Input] [Other models] [Differential Demand]

IV.I.3 Differential demand systems

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

Differential demand systems


Define the following


which is the budget share and the marginal budget share.

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


and define


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


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


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


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


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)


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


which is indeed a differential consumer demand system.

Furthermore note that


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




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


This system does not have restrictions on the coefficients.

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Formal Derivation
Rotterdam Model
Differential Input
Other models
Differential Demand
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