Marginal rate of substitution

Summary

• \(x_2=f\left( x_1 \right)\) is the equation for a given indifference curve
• The slope of the indifference curve at \(\left( x_1,x_2 \right)\) is called the marginal rate of substitution , \(MRS\) , of good 2 for good 1.

\[MRS= \frac{dx_2}{dx_1}=f'\left( x_1 \right)\]

• Example:
• Indifference curve for \(\left( x_1=4,x_2=7 \right)\) has equation \(x_2=f\left( x_1 \right)=28/x_1\) .
• \(f'\left( x_1 \right)=-28/x_1^2\)
• \(MRS=-28/4^2=-7/4 \)
• Approximating \(MRS\)

\[MRS≈ \frac{Δx_2}{Δx_1}\]

• \(MRS\) measures the approximate increase in \(x_2\) , \(Δx_2\) , required to stay on the indifference curve for a small increase in \(x_1\) by \(Δx_1\) .
• \(MRS\) is typically negative since \(Δx_2<0\) if \(Δx_1>0\)