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:\ 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 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. 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


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 MU''x'' is the marginal utility with respect to good ''x'' and MU''y'' 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'' (MU''x''), and ''F''(''y''), or ''dU''/''dy'', represents the marginal utility of good ''y'' (MU''y''). Also, −''dy''/''dx'' = MRS''xy'', 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.


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