# How to Calculate Marginal Utility

by Sean MannCalculating marginal utility is an important skill in economics. Marginal utility is the utility a person gains or loses by increasing or decreasing the amount of a specific good or service. A utility function can be used to describe the relationship between total utility and different amounts of the good or service. If you are given measurements of total utility for different amounts of the good or service, you can use subtraction to calculate the marginal utilities. If you are only given a utility function, you must use calculus to solve its marginal utility.

## Marginal Utility Without a Utility Function

Make a chart with three columns. The first column is the amount of the good or service you are calculating for. The second column is the total utility and the third column is the marginal utility.

Enter in available data to fill out the first and second columns. For example, you may be analyzing apples and know that zero apples gives zero utility, one apple gives 10 utility, two apples gives 15 utility, three apples gives 17 utility and so on.

Calculate the marginal utility for each row on the chart. The marginal utility of the first row is simply that row's total utility. The second row's marginal utility is the second row's total utility minus the first row's total utility. Repeat this process for all the rows on the chart. Using our previous apple example, the marginal utilities of the first four rows are 0, 10-0=10, 15-10=5 and 17-15=2.

Use the marginal utility values to answer questions about how the person's utility changes with various amounts of the good or service. The utility added from the first good to the second good is equal to the marginal utility of the second row. Using our previous apple example, the person gains 2 utility from acquiring a third apple. The tendency for each additional good to add a lower amount of utility is known as the law of diminishing marginal utility.

## Marginal Utility with a Utility Function

Use calculus to solve marginal utility problems when you are given a utility function. A utility function gives the relationship between utility and each unit of a good or service. The general form of a utility function is U(x1,x2,...xn), where the items x1, x2, etc.. up to xn all affect the person's utility. An example of a utility function is U(x,y) = 4x/y + 2, where "x" is wealth and "y" is children.

Use the marginal utility equation, which is MU(x) = dU/dx, where "x" is your variable. This equation describes the rate of change for utility given different amounts of the good. If there are multiple goods in your utility function then the marginal utility equation is a partial derivative of the utility function with respect to a specific good. Using the above example, the partial derivative of 4x/y + 2 in respect to "x" is 4/y and the partial derivative in respect to "y" is 4x.

Use the partial derivatives to solve questions about marginal utility for different amounts of the two goods. You can plug the amounts into the appropriate partial derivative to solve the marginal utility for that good. Using the above example, if "x" is equal to 5 and "y" is equal to 2, the marginal utility of "x" is 4/2=2 and the marginal utility of "y" is 4*5=20.