How much of one good would a person give up to get a little more of the other? That trade-off rate is the most important number in consumer theory. It comes from Marginal Utility as a Partial Derivative and is the slope of an indifference curve, which you can find with Implicit Differentiation.
Key idea
The marginal rate of substitution is $$MRS = \frac{MU_x}{MU_y} = -\frac{dy}{dx}\Big|_{U \text{ fixed}}.$$ It is the amount of $y$ a person would give up for one more (small) unit of $x$ and stay equally well off: the size of the indifference curve's slope at that bundle.
Why it works
Along an indifference curve $U(x, y) = k$, $y$ is a function of $x$. Differentiate both sides with respect to $x$: $$\frac{\partial U}{\partial x} + \frac{\partial U}{\partial y}\frac{dy}{dx} = 0, \quad\text{so}\quad \frac{dy}{dx} = -\frac{MU_x}{MU_y}.$$ In words: giving up $\Delta y$ costs about $MU_y\,|\Delta y|$ in utility, and getting $\Delta x$ adds about $MU_x\,\Delta x$. Staying on the curve means the two match, so $|\Delta y|/\Delta x = MU_x/MU_y$.
Worked example: Cobb-Douglas
$U(x, y) = x^2y^3$. Find the MRS as a formula, and at $(3, 1)$.
- $MU_x = 2xy^3$ and $MU_y = 3x^2y^2$.
- $MRS = \dfrac{2xy^3}{3x^2y^2} = \dfrac{2y}{3x}$.
- At $(3, 1)$: $MRS = \dfrac{2}{9}$. She would give up about $2/9$ of a unit of $y$ for one more small unit of $x$.
Answer: $MRS = \dfrac{2y}{3x}$; $\dfrac{2}{9}$ at $(3, 1)$.
Worked example: Quasilinear
$U(x, y) = 4\sqrt{x} + y$. Find the MRS at $(16, 3)$ and at $(16, 30)$.
- $MU_x = \dfrac{2}{\sqrt{x}}$ and $MU_y = 1$.
- $MRS = \dfrac{2}{\sqrt{x}} = \dfrac{2}{4} = \dfrac{1}{2}$ at $x = 16$.
- The MRS does not depend on $y$, so it is also $\tfrac{1}{2}$ at $(16, 30)$.
Answer: $\tfrac{1}{2}$ at both bundles.
Worked example: The MRS survives a transformation
Show that $U = x^2y^3$ and $V = 2\ln x + 3\ln y$ give the same MRS.
- For V: $MU_x = 2/x$ and $MU_y = 3/y$, so $MRS = \dfrac{2/x}{3/y} = \dfrac{2y}{3x}$.
- That matches U's MRS. In general $V = f(U)$ multiplies both marginal utilities by $f'(U)$, which cancels in the ratio.
Answer: Both give $\dfrac{2y}{3x}$.
Common mistake
Writing $MU_y/MU_x$ instead of $MU_x/MU_y$, or treating the MRS as one fixed number for a curved indifference curve. Keep the good on the horizontal axis on top: the MRS of $x$ for $y$ asks how much $y$ one unit of $x$ is worth. And unless the curve is a straight line, the MRS changes from bundle to bundle, so always say where you computed it. Check: when a person has a lot of $x$ and little $y$, her MRS should be small.