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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
x_{i} = f_{i}(p_{1}, p_{2}, …, p_{n}, y)
for i = 1 to n

The choices are constrained to satisfy the condition

P^{T}X = y

where P^{T} is the transpose of P.

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

p_{j} = g_{j}(x_{j})
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

P^{T}X = R^{T}L

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).