Worked Derivations of Utility, MU, and MRS

From a Utility Function to a Slope

A utility function lets us represent how a consumer ranks bundles of goods. For a specified bundle, a common calculation sequence is to evaluate utility, differentiate with respect to each good to find marginal utilities, and use their ratio to determine the marginal rate of substitution (MRS). The MRS also gives the magnitude of the indifference-curve slope; the slope itself has the opposite sign under the convention used here.

This tutorial works through that sequence for linear, Cobb-Douglas, and quasilinear functions. The purpose is to interpret and calculate these quantities at a bundle, not to solve a consumer's choice problem. A utility number is a representation of a ranking: its absolute value does not by itself measure happiness or permit comparisons between different consumers.

Definition: For a differentiable utility function \(U(x,y)\), the marginal utility of good \(x\) is \(MU_x=\partial U/\partial x\), holding \(y\) fixed; the marginal utility of good \(y\) is \(MU_y=\partial U/\partial y\), holding \(x\) fixed. When \(MU_y\ne 0\), define the MRS of \(x\) for \(y\) as \(MRS_{xy}=MU_x/MU_y\). At a differentiable point on an indifference curve, its slope in an \((x,y)\) graph is \(dy/dx=-MRS_{xy}\).

These formulas describe a local comparison at a bundle. The MRS says how many units of \(y\) a consumer would just be willing to give up for a small increase in \(x\), keeping utility unchanged. The ratio definition requires \(MU_y\ne 0\); the usual finite-slope calculation also assumes the utility function is differentiable at the bundle. If a function has a kink or a relevant partial derivative is zero, the ratio or slope may be undefined or require separate treatment.

Why the MRS Is the Negative of the Slope

An indifference curve consists of bundles with the same utility. Along a differentiable portion of that curve, a small change in \(x\) and \(y\) must leave utility unchanged. The total differential of \(U\) is therefore zero:

dU=MU_x\,dx+MU_y\,dy=0.

If \(MU_y\ne 0\), divide by \(dx\) and solve for the slope:

MU_x+MU_y\frac{dy}{dx}=0 \quad\Longrightarrow\quad \frac{dy}{dx}=-\frac{MU_x}{MU_y}=-MRS_{xy}.

This derivation explains both the ratio and the sign. When both marginal utilities are positive, a small increase in \(x\) must be offset by a decrease in \(y\) to keep utility constant, so the curve slopes downward. The MRS is reported as a positive magnitude in that case, while the graph's slope is negative.

Three Utility Forms and Their Calculations

The same sequence applies across utility forms, but the derivatives and resulting MRS differ. In the examples, goods are measured in units for which the stated utility functions apply, and bundles are evaluated at the indicated quantities. For the Cobb-Douglas and quasilinear examples, quantities are positive so the derivatives and ratios used below are well-defined.

Useful formulas: For linear utility \(U(x,y)=ax+by\), where \(a,b>0\), \(MU_x=a\), \(MU_y=b\), and \(MRS_{xy}=a/b\). For Cobb-Douglas utility \(U(x,y)=A x^\alpha y^\beta\), where \(A,\alpha,\beta>0\) and \(x,y>0\), \(MU_x=\alpha U/x\), \(MU_y=\beta U/y\), and \(MRS_{xy}=(\alpha/\beta)(y/x)\). For quasilinear utility \(U(x,y)=v(x)+y\), where \(v\) is differentiable, \(MU_x=v'(x)\), \(MU_y=1\), and \(MRS_{xy}=v'(x)\).

For the Cobb-Douglas formulas, differentiation gives \(MU_x=A\alpha x^{\alpha-1}y^\beta\). Since \(U=A x^\alpha y^\beta\), this is \(\alpha U/x\); similarly, \(MU_y=\beta U/y\). Dividing the two derivatives cancels the common utility factor and gives \((\alpha/\beta)(y/x)\). These expressions apply where \(x,y>0\); the ratio formula should not be applied at a boundary point without checking the derivatives there.

Worked Example: Linear Utility at a Bundle

Let \(U(x,y)=4x+3y\). At the bundle \((x,y)=(2,5)\), calculate utility, both marginal utilities, the MRS, and the indifference-curve slope. The method is to evaluate the function first, differentiate each good separately, and then use the stated MRS convention.

Utility. Substitute \(x=2\) and \(y=5\): \(U(2,5)=4(2)+3(5)=8+15=23\).

Marginal utilities. Holding \(y\) fixed, \(\partial U/\partial x=4\), so \(MU_x=4\). Holding \(x\) fixed, \(\partial U/\partial y=3\), so \(MU_y=3\). The derivatives are constant, so these values also apply at \((2,5)\).

MRS and slope. The ratio is \(MRS_{xy}=MU_x/MU_y=4/3\). Thus the indifference-curve slope is \(dy/dx=-4/3\).

Check and trap avoided. Along an indifference curve, \(4x+3y=23\), so \(3y=23-4x\) and \(y=23/3-(4/3)x\). Its slope is indeed \(-4/3\). A common error is to report \(+4/3\) as the graph's slope. That is the positive MRS magnitude; the slope is its negative.

Worked Example: Cobb-Douglas Utility at a Bundle

Let \(U(x,y)=x^{1/3}y^{2/3}\), with \(x,y>0\). At \((x,y)=(8,27)\), calculate utility, the marginal utilities, the MRS, and the curve's slope. The method uses partial derivatives, followed by either their ratio or the simplified Cobb-Douglas formula.

Utility. Since \(8^{1/3}=2\) and \(27^{2/3}=(27^{1/3})^2=3^2=9\), \(U(8,27)=2(9)=18\).

Marginal utility of \(x\). Differentiate while holding \(y\) fixed: \(MU_x=(1/3)x^{-2/3}y^{2/3}\). At the bundle, \(8^{-2/3}=1/4\) and \(27^{2/3}=9\), so \(MU_x=(1/3)(1/4)(9)=9/12=3/4\). Equivalently, \((1/3)U/x=(1/3)(18/8)=18/24=3/4\).

Marginal utility of \(y\). Differentiate while holding \(x\) fixed: \(MU_y=(2/3)x^{1/3}y^{-1/3}\). At the bundle this is \((2/3)(2)(1/3)=4/9\). Equivalently, \((2/3)U/y=(2/3)(18/27)=36/81=4/9\).

MRS and slope. The ratio is \((3/4)/(4/9)=(3/4)(9/4)=27/16=1.6875\). Therefore \(dy/dx=-27/16=-1.6875\).

Check and trap avoided. The simplified formula gives \((\alpha/\beta)(y/x)=( (1/3)/(2/3))(27/8)=(1/2)(27/8)=27/16\), matching the ratio of derivatives. A frequent trap is reversing the ratio and calculating \(MU_y/MU_x=16/27\); that is not \(MRS_{xy}\) under the definition here.

Worked Example: Quasilinear Utility at a Bundle

Let \(U(x,y)=2\sqrt{x}+y\), for \(x>0\) and \(y\ge 0\). At \((x,y)=(9,4)\), calculate utility, marginal utilities, the MRS, and the slope. The method is to differentiate the square-root term with respect to \(x\); the additive \(y\) term makes the marginal utility of \(y\) constant.

Utility. \(U(9,4)=2\sqrt{9}+4=2(3)+4=10\).

Marginal utilities. Differentiating with respect to \(x\), \(MU_x=2(1/(2\sqrt{x}))=1/\sqrt{x}\). At \(x=9\), this is \(1/3\). Differentiating with respect to \(y\) gives \(MU_y=1\).

MRS and slope. \(MRS_{xy}=(1/3)/1=1/3\), so the indifference-curve slope is \(dy/dx=-1/3\).

Check and trap avoided. Along utility level 10, \(2\sqrt{x}+y=10\), or \(y=10-2\sqrt{x}\). Differentiating gives \(dy/dx=-1/\sqrt{x}\), which at \(x=9\) is \(-1/3\). A common error is to treat \(MU_x\) as constant because the function is quasilinear. Only \(MU_y\) is constant here; \(MU_x=1/\sqrt{x}\) varies with \(x\).

A Consistent Calculation Routine

At an interior bundle where the function is differentiable and \(MU_y\ne 0\), the calculation can be organized as follows. Keeping the steps separate helps distinguish utility levels from derivatives and the MRS magnitude from the signed slope.

Evaluate utility.
Substitute the specified quantities into \(U(x,y)\). This gives the utility level of that bundle.

Find each partial derivative.
Differentiate with respect to one good at a time, holding the other good fixed. Then evaluate both derivatives at the specified bundle.

Calculate the MRS.
Divide \(MU_x\) by \(MU_y\), following \(MRS_{xy}=MU_x/MU_y\). Keep the order of goods consistent with the notation.

Report the slope and check it.
Use \(dy/dx=-MRS_{xy}\). When practical, differentiate the equation for the indifference curve directly to confirm the sign and value.

Interpreting Results and Avoiding Errors

These calculations matter because they translate a utility representation into a local trade-off between goods. The derivative \(MU_x\) describes the change in utility from a small increase in \(x\), holding \(y\) fixed. The MRS compares that change with the marginal utility of \(y\), while the slope describes how much \(y\) must change to keep utility fixed. These quantities are related, but they are not interchangeable.

Do not confuse utility with marginal utility. \(U(x,y)\) is the function's value at the bundle; \(MU_x\) and \(MU_y\) are derivatives.
Keep the MRS order straight. With \(MRS_{xy}=MU_x/MU_y\), reversing numerator and denominator changes the result.
Distinguish magnitude from signed slope. With positive marginal utilities, \(MRS_{xy}\) is positive and the indifference-curve slope is negative.
Evaluate derivatives at the stated bundle. For nonlinear functions, marginal utilities and the MRS can vary across bundles.
Check the assumptions. A derivative ratio may not describe a finite slope at a point where the function is not differentiable or \(MU_y=0\).

The formula for a Cobb-Douglas MRS also shows why the ratio changes with the bundle: holding the exponents fixed, it is proportional to \(y/x\). For the quasilinear function \(v(x)+y\), by contrast, the MRS depends on \(x\) through \(v'(x)\), not on \(y\). The linear example has a constant MRS because both marginal utilities are constant.

Key takeaway: Evaluate \(U\), calculate \(MU_x\) and \(MU_y\), form \(MRS_{xy}=MU_x/MU_y\), and then report the indifference-curve slope as \(-MRS_{xy}\). Always state which quantity is the positive trade-off magnitude and which is the signed graph slope.

Check Your Understanding

For each bundle, find utility, both marginal utilities, the MRS of \(x\) for \(y\), and the indifference-curve slope.

For \(U(x,y)=7x+2y\), evaluate the quantities at \((3,6)\). (Answer: \(U=33\), \(MU_x=7\), \(MU_y=2\), \(MRS_{xy}=3.5\), slope \(=-3.5\).)
For \(U(x,y)=x^{1/2}y^{1/2}\), evaluate them at \((25,16)\). (Answer: \(U=20\), \(MU_x=0.4\), \(MU_y=0.625\), \(MRS_{xy}=0.64\), slope \(=-0.64\).)
For \(U(x,y)=3\sqrt{x}+2y\), evaluate them at \((9,4)\). (Answer: \(U=17\), \(MU_x=0.5\), \(MU_y=2\), \(MRS_{xy}=0.25\), slope \(=-0.25\).)
For \(U(x,y)=2x^{1/4}y^{3/4}\), evaluate them at \((16,81)\). (Answer: \(U=108\), \(MU_x=27/16\), \(MU_y=1\), \(MRS_{xy}=27/16\), slope \(=-27/16\).)
A differentiable utility function has \(MU_x=6\) and \(MU_y=4\) at a bundle. Using the convention in this tutorial, state the MRS and the indifference-curve slope, and verify the sign through the constant-utility condition. (Answer: \(MRS_{xy}=3/2\), slope \(=-3/2\); \(6\,dx+4\,dy=0\) implies \(dy/dx=-6/4=-3/2\).)

What You'll Learn