Consumer surplus with continuous demand

Summary

• Setup
• Two goods model
• Both goods infinitely divisible
• \(p_2=1\) , \(p_1x_1+x_2=m\)
• 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
• Result:
• First order condition: \(v'\left( x_1 \right)=p_1\)
• Inverse demand function: \(p_1\left( x_1 \right)=v'\left( x_1 \right)\)

\[v\left( x_1 \right)=\int_{0}^{x_1}{ p_1\left( t \right)dt }\]

• \(v\left( x_1 \right)\) is gross benefit or the gross consumer’s surplus from consuming \(x_1\) units of good 1
• \(v\left( x_1 \right)-p_1x_1\) is consumer’s surplus or the net consumer’s surplus from consuming \(x_1\) units of good 1