Marginal Rate of Substitution Formula: How to Calculate MRS

The marginal rate of substitution (MRS) determines whether or not a consumer would replace one product or service for another one, assuming the two goods offer the same level of satisfaction and utility. By plotting the units sold on a chart and calculating the curve between the two—called an indifference curve—economists can determine consumers’ general preferences for two different goods (also called substitutes).

Economists use the marginal rate of substitution formula, or MRS formula, to complete this kind of indifference curve analysis. In doing this calculation, economists can figure out whether or not the consumer substitutes are evenly desirable, or if the two goods or services even have the same utility function.

The marginal rate of substitution formula, or MRS formula, is:

The formula’s different variables are:

• dy/dx: This is a derivative of y, with respect to x. In mathematical terms, a derivative refers to the rate of change of a function compared to a variable. In this case, y and x are the variables, and they represent the quantity of each item or product.
• MU: This represents the marginal utility—the measure of a customer’s happiness or satisfaction with a product—for two different goods, represented in the equation by x and y.
• x, y: These variables represent two different goods, both with the same level of utility.

Assuming two different goods or products have the same utility function, the slope of the indifference curve will show the trade-offs made by consumers—as Good X increases on the graph, Good Y decreases and vice versa. In other words, the curve will indicate consumers’ general preferences for one versus the other, with the curve potentially even shifting over time as economists recalculate using up-to-date figures.

For example, consider a shop with two items for sale: coffee and energy drinks. As a researcher, you might talk to a consumer to determine what mixture of coffee and energy drinks will make them happy. Continuing the hypothetical, you then plot the marginal utility figures out on a chart, with one product—coffee, for example—on the x-axis as Good X, and the other—energy drinks—on the y-axis as Good Y.

In this example, as the consumer acquires an additional unit of energy drinks (a higher number of units), they will have less coffee (a lower number of units). If you were to measure multiple consumers and connect the points on both the x-axis and the y-axis, you would form the slope called the indifference curve.

When economists are plotting the usage of goods and services to complete an indifference curve analysis, the slope the figures form often follows the law of diminishing marginal rate of substitution, or diminishing MRS. This means that the slope formed on the graph goes into a convex and negative shape. This can show how the combination of goods relate to each other and how MRS decreases. In addition, when you factor in a person’s consumption bundle—the total number of units they consume—you can see how the substitution effect relates to their overall consumption.

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