Marginal Rate of Substitution - Simple Mathematical Analysis

Simple Mathematical Analysis

Further information: Implicit differentiation

Assume the consumer utility function is defined by, 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:

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

By taking the total differential of the utility function equation, we obtain the following results:

, or substituting from above,
, or, without loss of generality, the total derivative of the utility function with respect to good x,
, that is,
.

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:

, or rearranging

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:

.

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