Marginal rate of substitution

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In economics, the marginal rate of substitution is the rate at which a customer is ready to give up one good in exchange for another good while maintaining the same level of satisfaction.

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[edit] Marginal rate of substitution as the slope of indifference curve

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. MRS of Y for X is the amount of Y for that 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,

\ MRS_{xy}=-m_\mathrm{indif}=-(dy/dx)
\ MRS_{xy}=MU_x/MU_y

(Please note that when comparing bundles of goods X and Y that gives 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,

\ MRS_{xy} \ge 0

[edit] Simple mathematical analysis

Assume the consumer utility function is defined by U(x,y), where U is consumer utility, x and y are goods.

Also, note that:

\ MU_x=\partial U/\partial x
\ MU_y=\partial U/\partial y

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

By taking the total differential 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{dU}{dx}= MU_x\frac{dx}{dx}+ MU_y\frac{dy}{dx}, that is,
\frac{dU}{dx}= MU_x.1 + MU_y\frac{dy}{dx}.

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{dy}{dx}, or rearranging
-\frac{dy}{dx} = \frac{MU_x}{MU_y}

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:

\ MRS_{xy}=MU_x/MU_y.\, .

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

\ m_\mathrm{indif}=m_\mathrm{budget}
\ -(MRS_{xy})=-(P_x/P_y)
\ MRS_{xy}=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 to price ratio is equal for each good.

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[edit] See also