Marginal rate of substitution

The marginal rate of substitution is that rate at which consumer gives up or sacrifices some units of commodities y for additional unit of commodity x. in other words the MRS is that rate where consumer substitute some units of y by an additional unit of x so that consumer derives the level of satisfaction as much as he could derive previously. The marginal rate of substitution is also defined as the ratio of change in unit of commodity y to change in commodity x. therefore MRSxy=∆y/∆x.

The MRS at any point on IC is the slope of IC at that point.

Therefore MRSxy=∆y/∆x=slope of IC

Likewise it can be also expressed as MRSxy=∆y/∆x=-MUx/MUy

Where

MUx=marginal utility of x

MUY=marginal utility of y

MRSxy=(-MUx/MUy)<0

Which means that marginal rate of substitution diminishes. The diminishing MRs represents the convexity of IC. In other words as MRSxy diminishes the IC is convex to the origin.

In the figure we have a downward sloping IC consisting three points of combination A, B, C. when consumer moves from A to B the unit of x increases by MB and that of Y reduces by AM ie ∆y=AM. As unit of x increases by ∆x with the marginal utility MUx_, there is gain in satisfaction equal to +∆x.MUx. similarly if unit of y reduces by ∆y with its marginal utility the loss in total satisfaction=-∆y.MUy.

The consumer derives equal level of satisfaction from both combination A and B only when total gain in satisfaction on= total loss in satisfaction ie +∆x.MUx=-∆y.MUy

Or ∆x/∆y=-MUy/MUx

Or ∆y/∆x=-MUx/MUy

Or ∆y/∆x=MRs=-MUx/MUy

Geometrical proof of diminishing MRs

From the above figure, initially when there is no MRs, the MRs at point B=∆y/∆x=AM/BM

The MRS at point c=∆y/∆x=BN/NC

BN=NC and BN<AM=>BN<NC<AM/BM

This clearly implies that the MRs diminishes along the ic as we move from left to right. Ultimately it insures convexity of IC.