# Consumer surplus, discrete goods

Summary

Consumer’s surplus, discrete good with quasilinear preferences

- Setup
- Two goods model
- Good 1 is discrete (integer)
- \(p_2=1\) , \(p_1x_1+x_2=m\)
- Quasilinear preferences \(u\left( x_1,x_2 \right)=v\left( x_1 \right)+x_2\)
- Strictly convex preferences: \(v\left( x_1 \right)\) concave
- Optimal choice \(x_1\) :
- \(x_1=n\) if and only if \(r_{n+1}≤p_1≤r_n\)
- Deriving \(v\left( n \right)\)
- \(r_n=v\left( n \right)-v\left( n-1 \right)\)
- Normalize: \(v\left( 0 \right)=0\)

\[v\left( n \right)=r_1+…+r_n=\sum_{i=1}^{n}{ r_i }\]

- \(v\left( n \right)\) is called the gross benefit or the gross consumer’s surplus from consuming \(n\) units of good 1
- If \(x_1=n\) then \(x_2=m-p_1n\) and

\[u\left( n,m-p_1n \right)=v\left( n \right)+m-p_1n\]

- The consumer’s surplus or net c onsumer’s surplus from consuming \(n\) units of good 1 is defined as

\[CS\left( n \right)=v\left( n \right)-p_1n=\sum_{i=1}^{n}{ r_i }-p_1n\]