Question

Demand Curves are homogeneous of degree zero in prices and income. Please explain your answer by equations.

Answer 1

The demand curves are  said to be homogenous of degree zero, if when all its arguments (i.e price and income ) are multiplied by any number t>0, then the value of the demand curves get multiplied by t⁰.( Any number raised to the power 0 equals one).Thus the demand curves get multiplied by 1.

If the money income and the prices were to change by the same proportion, then the quantity demanded remains unchanged

Initial Budget Constraint:p1x1+p2x2=m

x1 and x2 are the 2 goods being taken into consideration and p1 and p2 are the prices of the 2 goods

m: Income

For the proof of the homogenity of zero degree, let us assume that the prices and income change by the same proportion ,say $\alpha$

Now the Budget Constraint becomes

$\alpha$p1x1+$\alpha$p2x2=$\alpha$m

$\alpha$ is the proportionality factor in the above case

Now the Lagrange function becomes as follows:

L=f(x1,x2)+$\lambda$( $\alpha$m- $\alpha$p1x1- $\alpha$p2x2)

The First order conditions for maximization

f1-$\lambda$$\alpha$ p1=0.....(1)

f2-$\lambda$$\alpha$ p2=0.....(2)

$\alpha$m- $\alpha$p1x1- $\alpha$p2x2=0

The previous equation can be expressed as follows:

$\alpha$(m-p1x1-p2x2)=0

Since $\alpha$ is not equal to 0,

m-p1x1-p2x2=0

Removing $\alpha$ from the equations 1 and 2 , rearranging and dividing equation 1 by equation 2  we get

f1/f2=p1/p2

which is the original equilibrium condition for the consumer.Therefore the demand function for the price-income set ($\alpha$p1,$\alpha$p2,$\alpha$m) is derived from the same equations as the price income set (p1,p2,m).This proves the homogeniety condition of the demand function.Such functions are homogenous of degree zero in prices and income