Consider $an$ agent whose preferences over any couple $(x_1,x_2),$ where $x_1{R}_{+}$ and $x_2{R}_{+},e.g.,$ apples and oranges, $is$ such that she prefers the bundle that $is$ closest $to$ having the same number $of$ apples and oranges. Write a utility function $u$ : $R_{+}^2{R}_{+}$ which represent these preferences

A politician remarks $Our$ recent increases $in$ the wage rates $of$ teachers has been a total success! The shortage $of$ teachers has been reduced drastically. Another, similar wage increase should eliminate this shortage entirely$$

$(b)$ Explain and illustrate $in$ a diagram what $is$ meant $by$ "income effects" and "substitution effects" $of$ a wage rate change.

$(c)$ Explain and illustrate how you would model the labour supply decision $of$ a potential teacher. $Do$ you agree that the wage increase will increase the labour supply $in$ this case? Carefully outline the assumptions underlying your argument.