Marginal rate of substitution
Encyclopedia
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 utility
.
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 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
, the marginal rate of substitution of good or service X for good or service Y (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,
, 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,
, 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:
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 per unit of money spent is equal for each good. 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.
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
.
Marginal rate of substitution as the slope of indifference curve
Under the standard assumption of neoclassical economicsNeoclassical economics
Neoclassical economics is a term variously used for approaches to economics focusing on the determination of prices, outputs, and income distributions in markets through supply and demand, often mediated through a hypothesized maximization of utility by income-constrained individuals and of profits...
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 showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the 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 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 mathematical analysis, 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. In this sense, a measure is a generalization of the concepts of length, area, and volume...
, 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 service is the utility gained from an increase in the consumption of that good or service...
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
Indifference curve
In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the indifference curve...
), the marginal utility
Marginal utility
In economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...
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 utilityMarginal utility
In economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...
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,, 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 per unit of money spent is equal for each good. 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.
See also
- marginal conceptsMarginal conceptsIn 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.- Marginality :...
- indifference curves
- consumer theoryConsumer theoryConsumer choice is a theory of microeconomics that relates preferences for consumption goods and services to consumption expenditures and ultimately to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most closely studied relations in...
- convex preferencesConvex preferencesIn economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...
- microeconomicsMicroeconomicsMicroeconomics is a branch of economics that studies the behavior of how the individual modern household and firms make decisions to allocate limited resources. Typically, it applies to markets where goods or services are being bought and sold...
- implicit differentiation