# 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

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