Optimal choice for discrete goods and quasi linear preferences

Summary

Discrete goods

• Example: Good 1 can only be consumed in integer units,
• \(u\left( x_1,x_2 \right)=x_1x_2\)
• \(p_1=2,p_2=1\) and \(m=5\)
• Possible bundles: \(\left( 0,5 \right),\left( 1,3 \right),\left( 2,1 \right)\)
• \(\left( 1,3 \right)\) is the optimal choice .

Quasilinear preferences

\[u\left( x_1,x_2 \right)=v\left( x_1 \right)+x_2\]

• Preferences will be strictly convex if and only if \(v\left( x_1 \right)\) is strictly concave

\[MRS=- \frac{MU_1}{MU_2}=-v'\left( x_1 \right)\]

• Example:
• \(v\left( x_1 \right)=\sqrt{x_1}\)
• \(MRS=-v'\left( x_1 \right)=- \frac{1}{2\sqrt{x_1}}\)
• Optimal choice for \(x_1\) (if it can be afforded) does not depend on \(m\) :

\[x_1= \frac{p_2^2}{4p_1^2}\]