Marginal rate of substitution
In economics, the marginal rate of substitution (MRS) is the rate at which a consumer is ready to give up one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no externalities), marginal rates of substitution are identical.
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. MRS of X for Y is the amount of Y for which a consumer is willing to exchange X locally. The MRS is different at each point along the indifference curve thus it is important to keep locus in the definition. Further on this assumption, or otherwise on the assumption that utility is quantified, the marginal rate of substitution of good or service Y for good or service X (MRSxy) is also equivalent to the marginal utility of X over the marginal utility of Y. Formally,
It is important to note that when comparing bundles of goods X and Y that give a constant utility (points along an indifference curve), the marginal utility of X is measured in terms of units of Y that is being given up.
For example, if the MRSxy = 2, the consumer will give up 2 units of Y to obtain 1 additional unit of X.
As one moves down a (standardly convex) indifference curve, the marginal rate of substitution decreases (as measured by the absolute value of the slope of the indifference curve, which decreases). This is known as the law of diminishing marginal rate of substitution.
Since the indifference curve is convex with respect to the origin and we have defined the MRS as the negative slope of the indifference curve,
Assume the consumer utility function is defined by , where U is consumer utility, x and y are goods. Then the marginal rate of substitution can be computed via partial differentiation, as follows.
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