Demand functions and their dependence on income

Summary

Demand functions

• If the optimal choice of \(x_1\) and \(x_2\) given \(p_1,p_2\) and \(m\) are unique then the functions

\[x_1=x_1\left( p_1,p_2,m \right)\]

• and

\[x_2=x_2\left( p_1,p_2,m \right)\]

• are called demand functions .

Demand functions and their dependence on income

• Good 1 is said to be normal if \(x_1\left( p_1,p_2,m \right)\) is increasing in \(m\)
• Good 1 is said to be inferior if \(x_1\left( p_1,p_2,m \right)\) is strictly decreasing in \(m\)

• Example:
• \(x_1\left( p_1,p_2,m \right)=m/p_1\)
• \(∂x_1/∂m=1/p_1>0\)
• Good 1 is normal.
• Example:
• \(x_1\left( p_1,p_2,m \right)=\left( 8m-m^2 \right)/10p_1\) for \(0≤m≤8\) .
• \(∂x_1/∂m=\left( 8-2m \right)/10p_1\)
• Good 1 is normal if \(m≤4\) and inferior if \(m>4\)