Marginal rate of substitution

From Wikipedia, the free encyclopedia

In economics, the marginal rate of substitution (“MRS” for short) is the least-favorable rate at which an agent is willing to exchange units of one good or service for units of another. The MRS measures the value that the consumer places on one extra unit of a good or service, where the opportunity cost is quantified by amount of another sacrificed.

Under the standard assumption of neoclassical economics that goods and services are continuously divisible, the marginal rates of substition 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 endowment in question, at that endowment. 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 (MRS_xy) is also equivalent to the marginal utility of X over the marginal utility of Y. Formally,

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

$\ MRS_{xy}=MU_x/MU_y$

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,

$\ MRS_{xy} \ge 0$

[edit] Mathematical analysis of the marginal rate of substitution

Assume the consumer utility function is defined as:

$\ U=F(x,y)$

Where U is consumer utility, x and y are goods, and F is the utility function.

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,

$\frac{dU}{dx}= MU_x\frac{dx}{dx}+ MU_y\frac{dy}{dx}$ , that is,

$\frac{dU}{dx}= MU_x.1 + MU_y\frac{dy}{dx}$ . (Eq. 1)

Through any point on the indifference curve, dU/dx = 0, because U = c, where c is a constant. It follows from (Eq. 1) 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.

[edit] References

Microeconomics (2005, 6th Edition) by Pindyck, and Rubinfeld

Category: Consumer theory

Marginal rate of substitution

From Wikipedia, the free encyclopedia

[edit] Mathematical analysis of the marginal rate of substitution

[edit] References

[edit] See also

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