Income offer curve and Engel curve

Summary

• The income offer curve is the collection of optimal choices \(x_1,x_2\) for all possible values of income \(m\) (prices fixed)
• Example
• \(u\left( x_1,x_2 \right)=x_1x_2\) , \(p_1=1,p_2=2\)
• Optimal choice: \(x_1=m/2\) , \(x_2=m/4\) or \(x_2=x_1/2\)
• Income offer curve: Straight line through origin with slope 1/2
• An Engel curve show the relationship between optimal choice of a good and income

• For a normal good, if the proportion of income spent on good 1, \(p_1x_1\left( p_1,p_2,m \right)/m\) , is strictly increas ing in \(m\) then we say that good 1 is a luxury good . Otherwise we say that it is a necessary good.
• Example:
• Cobb-Douglas utility function: \(u\left( x_1,x_2 \right)=x_1^ax_2^b\) .
• \(x_1\left( p_1,p_2,m \right)= \frac{a}{a+b} \frac{m}{p_1} \)
• Proportion of income spent on good 1: \( \frac{a}{a+b}\)
• Does not depend on \(m\) so neither luxury nor necessary
• Example:
• \(x_1\left( p_1,p_2,m \right)=\left( 8m-m^2 \right)/10p_1\) for \(0≤m<8\) .
• Proportion of income spent on good 1: \(0.8-0.1m\) so necessary
• Proportion of income spent on good 2: \(0.2+0.1m\) so luxury