Marginal Rate Of Substitution
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In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of
utility As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
. At equilibrium consumption levels (assuming no externalities), marginal rates of substitution are identical. The marginal rate of substitution is one of the three factors from marginal productivity, the others being marginal rates of transformation and marginal productivity of a factor.


As the slope of indifference curve

Under the standard assumption of
neoclassical economics Neoclassical economics is an approach to economics in which the production, consumption and valuation (pricing) of goods and services are observed as driven by the supply and demand model. According to this line of thought, the value of a good ...
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 In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
(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 which a consumer can exchange for one unit of 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 X for good or service Y (MRSxy) is also equivalent to the
marginal utility In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a Goods (economics), good or Service (economics), service describes how much pleasure or satisfaction is gained by consumers as a result o ...
of X over the marginal utility of Y. Formally, :MRS_=-m_\mathrm=-(dy/dx) \, :MRS_=MU_x/MU_y \, It is important to note that when comparing bundles of goods X and Y that give a constant utility (points along an
indifference curve In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
), the
marginal utility In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a Goods (economics), good or Service (economics), service describes how much pleasure or satisfaction is gained by consumers as a result o ...
of X is measured in terms of units of Y that is being given up. For example, if the MRS''xy'' = 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, :\ MRS_ \geq 0


Simple mathematical analysis

Assume the consumer
utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
is defined by U(x,y), where ''U'' is consumer utility, ''x'' and ''y'' are goods. Then the marginal rate of substitution can be computed via
partial differentiation In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Par ...
, as follows. Also, note that: :\ MU_x=\partial U/\partial x :\ MU_y=\partial U/\partial y where \ MU_x is the
marginal utility In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a Goods (economics), good or Service (economics), service describes how much pleasure or satisfaction is gained by consumers as a result o ...
with respect to good ''x'' and \ MU_y is the marginal utility with respect to good ''y''. By taking the
total differential In calculus, the differential represents the principal part of the change in a function ''y'' = ''f''(''x'') with respect to changes in the independent variable. The differential ''dy'' is defined by :dy = f'(x)\,dx, where f'(x) is the ...
of the utility function equation, we obtain the following results: :\ dU=(\partial U/\partial x)dx + (\partial U/\partial y)dy , or substituting from above, :\ dU= MU_xdx + MU_ydy , or, without loss of generality, the total derivative of the utility function with respect to good ''x'', :\frac= MU_x\frac+ MU_y\frac, that is, :\frac= MU_x + MU_y\frac. 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: :: 0 = MU_x + MU_y\frac, or rearranging :-\frac = \frac 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. That turns out to equal the ratio of the marginal utilities: :\ MRS_= MU_x/MU_y.\, . ---- When consumers maximize utility with respect to a budget constraint, the indifference curve is tangent to the
budget line In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Consumer theory uses the concepts of a budget constraint and a preference ...
, therefore, with ''m'' representing slope: :\ m_\mathrm=m_\mathrm :\ -(MRS_)=-(P_x/P_y) :\ MRS_=P_x/P_y Therefore, when the consumer is choosing his utility maximized market basket on his budget line, :\ MU_x/MU_y=P_x/P_y :\ MU_x/P_x=MU_y/P_y 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. To decrease the marginal rate of substitution, the consumer must buy more of the good for which he/she wishes the marginal utility to fall for (due to the law of diminishing marginal utility).


Diminishing Marginal rate of Substitution

An important principle of economic theory is that marginal rate of substitution of X for Y diminishes as more and more of good X is substituted for good Y. In other words, as the consumer has more and more of good X, he is prepared to forego less and less of good Y. It means that as the consumer’s stock of X increases and his stock of Y decreases, he is willing to forego less and less of Y for a given increment in X. In other words, the marginal rate of substitution of X for Y falls as the consumer has more of X and less of Y. 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.


Using MRS to determine Convexity

When analyzing the utility function of consumer's in terms of determining if they are convex or not. For the horizon of two goods we can apply a quick derivative test (take the derivative of MRS) to determine if our consumer's preferences are convex. If the derivative of MRS is negative the utility curve would be concave down meaning that it has a maximum and then decreases on either side of the maximum. This utility curve may have an appearance similar to that of a lower case n. If the derivative of MRS is equal to 0 the utility curve would be linear, the slope would stay constant throughout the utility curve. If the derivative of MRS is positive the utility curve would be convex up meaning that it has a minimum and then increases on either side of the minimum. This utility curve may have an appearance similar to that of a u. These statements are shown mathematically below. :\ \frac<0 \text :\ \frac=0 \text :\ \frac>0 \text For more than two variables, the use of the Hessian matrix is required.


See also

*
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.{{citati ...
*
Marginal rate of technical substitution In microeconomic theory, the marginal rate of technical substitution (MRTS)—or technical rate of substitution (TRS)—is the amount by which the quantity of one input has to be reduced (-\Delta x_2) when one extra unit of another input is used ( ...
(the same concept on production side) *
Indifference curves In economics, an indifference curve connects points on a graph representing different quantities of two goods, points between which a consumer is ''indifferent''. That is, any combinations of two products indicated by the curve will provide the c ...
*
Consumer theory The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their pref ...
*
Convex preferences In economics, convex preferences are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly ...
*
Implicit differentiation In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...
* Labour economics


References

Adam Hayes. (2021, March 31). Inside the marginal rate of substitution. Investopedia. Jerelin, R. (2017, May 30). Diminishing marginal rate of substitution , Indifference curve , Economics. Economics Discussion * * * {{Microeconomics Consumer theory Marginal concepts