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In economics, the marginal rate of substitution 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 satisfaction.

r the standard assumption of neoclassical economics
Neoclassical economics

Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distribution s in markets through supply and demand, often as mediated through a hypothesized maximization of income-constrained utility by individuals and of cost-constrained profits of firms employing avai...
 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
Indifference curve

In microeconomic theory, an indifference curve is a graph of a function showing different bundles of good , each measured as to quantity, between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another....
 (more precisely, to the slope multiplied by -1) passing through the consumption bundle in question, at that point: mathematically, it is the implicit derivative.






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In economics, the marginal rate of substitution 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 satisfaction.

Marginal rate of substitution as the slope of indifference curve

Under the standard assumption of neoclassical economics
Neoclassical economics

Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distribution s in markets through supply and demand, often as mediated through a hypothesized maximization of income-constrained utility by individuals and of cost-constrained profits of firms employing avai...
 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
Indifference curve

In microeconomic theory, an indifference curve is a graph of a function showing different bundles of good , each measured as to quantity, between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another....
 (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 Y for X is the amount of Y for which a consumer is willing to exchange for X locally. The MRS is different at each point along the indifference curve thus it is important to keep locally in the definition. Further on this assumption, or otherwise on the assumption that utility is quantified
Measure (mathematics)

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset....
, the marginal rate of substitution of good or service X for good or service Y (MRSxy) is also equivalent to the marginal utility
Marginal utility

In economics, the marginal utility of a Good or of a Service is the utility of the specific use to which an agent would put a given increase in that good or service, or of the specific use that would be abandoned in response to a given decrease....
 of X over the marginal utility of Y. Formally,

(Please note that when comparing bundles of goods X and Y that gives a constant utility (points along an indifference curve
Indifference curve

In microeconomic theory, an indifference curve is a graph of a function showing different bundles of good , each measured as to quantity, between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another....
), the marginal utility
Marginal utility

In economics, the marginal utility of a Good or of a Service is the utility of the specific use to which an agent would put a given increase in that good or service, or of the specific use that would be abandoned in response to a given decrease....
 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,

Simple mathematical analysis


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 implicit differentiation, as follows.

Also, note that:

where is the marginal utility with respect to good x and is the marginal utility with respect to good y.

By taking the total differential of the utility function equation, we obtain the following results: , or substituting from above,

, or, without loss of generality, the total derivative of the utility function with respect to good x
X

X is the twenty-fourth letter in the modern Latin alphabet. Its name in English language is spelled ex , plural exes .History...
, , that is, . 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:
, or rearranging


The marginal rate of substitution is defined by minus the slope of the indifference curve at whichever commodity bundle quantities are of interest. That turns out to equal the ratio of the marginal utilities: .

When consumers maximize utility with respect to a budget constraint, the indifference curve is tangent to the budget line, therefore, with m representing slope:

Therefore, when the consumer is choosing his utility maximized market basket on his budget line,

This important result tells us that utility is maximized when the consumer's budget is allocated so that the marginal utility to price ratio is equal for each good.

See also

  • marginal concepts
    Marginal concepts

    In economics, marginal concepts are associated with a specific change in the quantity used of a Good or Service , as opposed to some notion of the over-all significance of that class of good or service, or of some total quantity thereof....
  • indifference curves
  • consumer theory
    Consumer theory

    Consumer theory is a theory of microeconomics that relates preferences to supply and demand. The link between personal preferences, consumption, and the demand curve is one of the most complex relations in economics....
  • convex preferences
    Convex preferences

    In economics, convex preferences are a property of utility functions commonly represented in an indifference curve as a bulge toward the origin for normal goods....
  • microeconomics
    Microeconomics

    Microeconomics is a branch of economics that studies how individuals, households and firms and some states make decisions to allocate limited resources, typically in markets where goods or services are being bought and sold....
  • implicit differentiation