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 $Δx_1>0$