At equilibrium consumption levels (assuming no externalities), marginal rates of substitution are identical.
[1] Under the standard assumption of neoclassical economics that goods and services are continuously divisible, the marginal rates of substitution will be the same regardless of the direction of exchange, and will correspond to the slope of an indifference curve (more precisely, to the slope multiplied by −1) passing through the consumption bundle in question, at that point: mathematically, it is the implicit derivative.
The MRS is different at each point along the indifference curve thus it is important to keep locus in the definition.
By taking the total differential of the utility function equation, we obtain the following results: Through any point on the indifference curve, dU/dx = 0, because U = c, where c is a constant.
It follows from the above equation that: The marginal rate of substitution is defined as the absolute value of the slope of the indifference curve at whichever commodity bundle quantities are of interest.
If this equality did not hold, the consumer could increase his/her utility by cutting spending on the good with lower marginal utility per unit of money and increase spending on the other good.
That the marginal rate of substitution of X for Y diminishes can also be known from drawing tangents at different points on an indifference curve.
When analyzing the utility function of consumer's in terms of determining if they are convex or not.
Diminishing marginal rate of substitution | Indifference curve | Economics.