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

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