Indifference Curves and MRS

This scene draws the indifference map of Cobb-Douglas utility.

An indifference curve joins bundles that feel equally good. Move the reference bundle and the whole map redraws. The marginal rate of substitution is the tangent slope there, as in the formula below. It says how much of one good replaces one unit of the other.

For example, giving up one 🍎 apple feels fine if you get two more 🍌 bananas. Then the MRS is two. The fewer apples you have, the more each apple is worth.

u(x1,x2)=x1ax21−au(x_1, x_2) = x_1^a x_2^{1-a}
MRS(x)=∂u/∂x1∂u/∂x2=a1−ax2x1MRS(x) = \frac{\partial u / \partial x_1}{\partial u / \partial x_2} = \frac{a}{1-a}\frac{x_2}{x_1}

Part of these learning paths

Examples & Exercises

Derive by differentiation that the marginal rate of substitution of Cobb-Douglas utility u(x1,x2)=x1ax21−au(x_1, x_2) = x_1^a x_2^{1-a} is MRS=a1−ax2x1MRS = \frac{a}{1-a}\frac{x_2}{x_1}.

MRS(x)=∂u/∂x1∂u/∂x2MRS(x) = \frac{\partial u / \partial x_1}{\partial u / \partial x_2}
a=0.4a = 0.4, reference (4,4)(4, 4).
Show proof

  1. Compute the partials: ∂u/∂x1=ax1a−1x21−a\partial u/\partial x_1 = a x_1^{a-1} x_2^{1-a}, ∂u/∂x2=(1−a)x1ax2−a\partial u/\partial x_2 = (1-a) x_1^a x_2^{-a}.

  2. Dividing tidies up the powers of x1x_1 and x2x_2 to MRS=a1−ax2x1MRS = \frac{a}{1-a}\frac{x_2}{x_1}. In the preset, MRS=0.40.6⋅1=23MRS = \frac{0.4}{0.6} \cdot 1 = \frac{2}{3} — the slope of the blue dashed tangent.

Show in formulas that the indifference curve flattens as x1x_1 grows (diminishing MRS), and confirm it with the sliders.

x1x_1 from 44 to 88 — MRS halves.
Show proof

  1. In MRS=a1−ax2x1MRS = \frac{a}{1-a}\frac{x_2}{x_1}, a larger x1x_1 inflates the denominator, so MRS falls — the curve flattens.

  2. At the preset (8,4)(8, 4), MRS=23⋅48=13MRS = \frac{2}{3} \cdot \frac{4}{8} = \frac{1}{3} — exactly half the 2/32/3 at (4,4)(4,4). The geometry of the convex indifference curve IS diminishing MRS.

4. Classical Demand Theory