Calculate the Marginal Rate of Substitution between two goods.
Result
How to Calculate the Marginal Rate of Substitution (MRS)
The Marginal Rate of Substitution (MRS) is a fundamental concept in microeconomics that measures the rate at which a consumer is willing to give up one good in exchange for another while maintaining the same level of utility (satisfaction). It represents the slope of the indifference curve at any given point.
Understanding MRS is crucial for analyzing consumer choice behavior and market demand. It effectively answers the question: "How much of Good Y must I sacrifice to gain one more unit of Good X without feeling better or worse off?"
The Formulas
There are two primary ways to calculate MRS, depending on the data available to you:
1. Using Marginal Utilities
If you know the utility derived from the last unit of each good, the MRS relates to the ratio of their marginal utilities.
MRSxy = MUx / MUy
Where:
MUx = Marginal Utility of Good X (the extra satisfaction from one more unit of X).
MUy = Marginal Utility of Good Y (the extra satisfaction from one more unit of Y).
2. Using Discrete Changes (Slope Formula)
If you are looking at a movement along an indifference curve between two specific bundles of goods, you calculate the MRS as the negative ratio of the changes in quantities.
MRSxy = – (ΔY / ΔX) = – (Y2 – Y1) / (X2 – X1)
Where:
ΔY = Change in quantity of Good Y (Final Y – Initial Y).
ΔX = Change in quantity of Good X (Final X – Initial X).
Note: The result is mathematically negative because you give up one good to get the other, but MRS is typically reported as a positive magnitude (absolute value) in economics textbooks.
Example Calculation
Let's assume a consumer is choosing between Coffee (Good Y) and Donuts (Good X).
Scenario 1 (Marginal Utility): Suppose the marginal utility of the last Donut consumed is 20 utils, and the marginal utility of the last Coffee is 10 utils.
MRS = 20 / 10 = 2.
This means the consumer is willing to give up 2 Coffees to get 1 more Donut.
Scenario 2 (Bundle Change): A consumer moves from Bundle A (10 Coffees, 2 Donuts) to Bundle B (6 Coffees, 4 Donuts) and remains equally happy (same indifference curve).
Change in Coffee (Y) = 6 – 10 = -4.
Change in Donuts (X) = 4 – 2 = +2.
MRS = -(-4 / 2) = -(-2) = 2.
Diminishing Marginal Rate of Substitution
The Law of Diminishing Marginal Rate of Substitution states that as a consumer acquires more of Good X (and has less of Good Y), the amount of Good Y they are willing to give up for yet another unit of Good X decreases. Graphically, this is why indifference curves are convex to the origin (bowed inward).
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