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# Marginal rate of substitution

In economics, the marginal rate of substitution (MRS for short) is the rate at which consumers are willing to give up units of one good in exchange for more units of another good. Put another way, the MRS of good X for good Y is the amount of good Y that a person is willing to give up to obtain one additional unit of good X. The MRS measures the value that the consumer places on one extra unit of a good, where the opportunity cost is quantified by amount of another good sacrificed. Mathematically, the MRS is the negative slope or derivative (evaluated at a point) of the indifference curve. Therefore, the MRS at any point on an indifference curve is equal in magnitude to the slope of that indifference curve. The marginal rate of substitution of good X for good Y (MRSxy) is also equivalent to the marginal utility of X over the marginal utility of Y. Formally,

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

For example, if the MRSxy = 2, the consumer will give up 2 units of good Y to obtain 1 additional unit of good X.

As you move down a convex indifference curve, the marginal rate of substitution decreases since the magnitude of the slope of the indifference curve is decreasing.

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,

$\ MRS_{xy} \ge 0$

## 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:

$\ MU_x=dU/dx$
$\ MU_y=dU/dy$

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

By differentiating the utility function equation, we obtain the following results:

$\ dU=F(x)dx + F(y)dy$
$\ dU=(dU/dx)\Delta x + (dU/dy)\Delta y$
$\ dU=MU_y\Delta x + MU_x\Delta y$

Since dU = 0 for any indifference curve (because U = c, where c is a constant), it follows that:

$\ 0=F(x)dx + F(y)dy$
$\ -(dy/dx)=F(x)/F(y)$

and

$\ 0=MU_y\Delta x + MU_x\Delta y$
$\ -(\Delta y/\Delta x)=MU_x/MU_y$

Where F(x), or dU/dx, represents the marginal utility of good x (MUx), and F(y), or dU/dy, represents the marginal utility of good y (MUy). Also, −dy/dx = MRSxy, so MRSxy equals minus the slope of the indifference curve. Therefore:

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

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.