Marginal utility

Summary

• Given: a utility function \(u\left( x_1,x_2 \right)\) .
• The marginal utility of good 1 , \(MU_1\) , is defined as

\[MU_1= \frac{∂u}{∂x_1}\]

• The marginal utility of good 2, \(MU_2\) , is defined as

\[MU_2= \frac{∂u}{∂x_2}\]

• \(MU_1≈Δu/Δx_1\) or \(Δu≈MU_1⋅Δx_1\) keeping \(x_2\) fixed.
• \(MU_2≈Δu/Δx_2\) or \(Δu≈MU_2⋅Δx_2\) keeping \(x_1\) fixed.
• The exact value of the marginal utility will have no interpretation.
• Important result:

\[MRS= \frac{dx_2}{dx_1}=- \frac{MU_1}{MU_2}\]

• Example
• \(u=u\left( x_1,x_2 \right)=4x_1x_2\) and \(x_1=2\) , \(x_2=4\)
• \(u=32\) and \(x_2=8/x_1\) is the equation of the indifference curve through \(\left( 2,4 \right)\)

\[ \frac{dx_2}{dx_1}=- \frac{8}{x_1^2}=-2\]

\[MU_1=4x_2=16, MU_2=4x_1=8\]

\[- \frac{MU_1}{MU_2}=- \frac{16}{8}=-2=MRS\]