﻿ Inverse Demand Functions and Consumer Income
San José State University

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Thayer Watkins
Silicon Valley
USA

Inverse Demand Functions
and Consumer Income

Demand functions gives the quantities purchased as functions of prices. Let X be the column vector of consumer goods and P the column vector of prices. Consumer income will be denoted as y. The demand functions are

#### X = F(P, y) which is equivalent to xi = fi(p1, p2, …, pn, y) for i = 1 to n

The choices are constrained to satisfy the condition

#### PTX = y

where PT is the transpose of P.

The inverse demand function for the j-th good or service is

#### pj = gj(xj) with income and all of the other prices specified.

Very little can be said about inverse demand functions except that they are downward sloping. However what else can be said is that if income and all other prices are increaed by a factor (1+γ) then the inverse demand curve for the j-th good or service will also be shifted upward by a factor (1+γ). This is because

#### F((1+γ)P, (1+γ)y) = X = F(P, y)

Functions like F(P, y) are said to homogeneous to degree zero.

If incomes are determined by the supply of labor services and R is the vector of labor wage rates then

#### X = F(P, R)

If L is the vector of labor services supplied and incomes are given by

#### PTX = RTL

then all wage rates and all prices for all goods and services except the j-th one are increased by a factor of (1+γ) then the inverse demand for the j-th good or services also increases by a factor of (1+γ).