Utility function when goods are perfect substitutes

Summary

• Perfect substitutes with a one-to-one substitution: \(u\left( x_1,x_2 \right)=x_1+x_2\)
• Any monotonic transformation such as \(4x_1+4x_2\) or \({\left( x_1+x_2 \right)}^2\) will do as well.
• Perfect substitutes with an arbitrary rate of substitution: \(u\left( x_1,x_2 \right)=ax_1+bx_2\)
• \(MRS=-a/b\)