Marginal Rate of Substitution

The Marginal Rate of Substitution (MRS) measures how much of good $Y$ a consumer is willing to give up to get one more unit of good $X$ , while staying on the same indifference curve. Geometrically, it’s the absolute value of the slope of the IC at any given point. It answers the question: “What’s my personal exchange rate between these two goods?”

Key Term

Marginal Rate of Substitution (MRS)

The MRS of $X$ for $Y$ is the rate at which the consumer is willing to trade $Y$ for $X$ while maintaining the same utility:

$M R S_{X Y} = - \frac{Δ Y}{Δ X} = \frac{M U _{X}}{M U _{Y}}$

where $M U_{X}$ and $M U_{Y}$ are the marginal utilities of goods $X$ and $Y$ .

MRS equals the ratio of marginal utilities (and the absolute slope of the IC)

Interactive — Marginal Rate of Substitution Along an IC

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An indifference curve with a tangent line showing the slope at the chosen point.

A crucial feature of typical ICs is diminishing MRS. As you move down along an IC (gaining $X$ , losing $Y$ ), you become less and less willing to give up additional $Y$ for each extra unit of $X$ . Why? Because $X$ becomes relatively abundant (lower marginal value) while $Y$ becomes scarce (higher marginal value). This is what makes ICs convex to the origin — the slope gets flatter as you move right.

Real-world example: suppose you have 1 pair of jeans and 10 t-shirts. You’d happily trade 3 t-shirts for another pair of jeans (MRS = 3). But if you already have 5 pairs of jeans and only 2 t-shirts, you’d demand something much bigger to part with a t-shirt — your MRS is now very low. Your willingness to trade changes as your bundle changes. That’s diminishing MRS in action.

Worked Example

A consumer has utility $U (x, y) = x^{1/2} y^{1/2}$ . At the bundle (4, 16), calculate the MRS of $x$ for $y$ .

Check yourself · no marks

If the MRS at a point on an IC is 2, and the price ratio $P_{x} / P_{y}$ is 3, should the consumer buy more x or more y?

Commit to one first. Guessing and being wrong beats reading the answer cold.

Practice · 1 / 4

No penalty for a wrong answer

Work with the MRS:

What does ‘diminishing MRS’ mean in plain English?

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