Editing Marginal rate of substitution (section)

==Simple mathematical analysis==
{{Further|Implicit differentiation}}

Assume the consumer [[utility function]] is defined by <math>U(x,y)</math>, where ''U'' is consumer utility, ''x'' and ''y'' are goods. Then the marginal rate of substitution can be computed via [[partial differentiation]], as follows.

Also, note that:

:<math>\ MU_x=\partial U/\partial x </math>
:<math>\ MU_y=\partial U/\partial y </math>

where <math>\ MU_x </math> is the [[marginal utility]] with respect to good ''x'' and <math>\ MU_y </math> is the marginal utility with respect to good ''y''.

By taking the [[total differential]] of the utility function equation, we obtain the following results:
:<math>\ dU=(\partial U/\partial x)dx + (\partial U/\partial y)dy </math>, or substituting from above,

:<math>\ dU= MU_xdx + MU_ydy </math>, or, without loss of generality, the total derivative of the utility function with respect to good ''x'',
:<math>\frac{dU}{dx}= MU_x\frac{dx}{dx}+ MU_y\frac{dy}{dx}</math>, that is,
:<math>\frac{dU}{dx}= MU_x + MU_y\frac{dy}{dx}</math>.
Through any point on the indifference curve, ''dU/dx'' = 0,    because ''U''&nbsp;=&nbsp;''c'', where ''c'' is a constant. It follows from the above equation that:
::<math> 0 = MU_x + MU_y\frac{dy}{dx}</math>, or rearranging
:<math>-\frac{dy}{dx} = \frac{MU_x}{MU_y}</math>

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:
:<math>\ MRS_{xy}= MU_x/MU_y\, </math>.

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

:<math>\ m_\mathrm{indif}=m_\mathrm{budget} </math>
:<math>\ -(MRS_{xy})=-(P_x/P_y) </math>
:<math>\ MRS_{xy}=P_x/P_y </math>

Therefore, when the consumer is choosing his utility maximized market basket on his budget line,

:<math>\ MU_x/MU_y=P_x/P_y </math>
:<math>\ MU_x/P_x=MU_y/P_y </math>

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).