Two Different Questions About Utility
When a consumer has more of a good, the additional utility from another unit may fall. That statement concerns marginal utility as one quantity changes while the other is held fixed. The marginal rate of substitution (MRS) asks a different question: as the consumer moves along an indifference curve, how much of one good would compensate for a small increase in the other?
These ideas are related through marginal utilities, but they are not interchangeable. In particular, diminishing marginal utility for each good does not by itself guarantee diminishing MRS. To see why, it is important to track which quantity changes and what is held constant.
Suppose both marginal utilities are positive. Using the convention that MRS is the amount of \(y\) the consumer is willing to give up for a small increase in \(x\), its value is \(MU_x/MU_y\). Along a smooth indifference curve, the curve's slope is negative MRS. Diminishing MRS means this positive trade-off falls as \(x\) increases along that same curve—not merely as \(x\) changes with \(y\) fixed.
Why the Indifference-Curve Condition Is Different
The distinction follows from implicit differentiation. Along an indifference curve, utility does not change. If \(u(x,y)\) is differentiable, a small change therefore satisfies \(du=MU_x\,dx+MU_y\,dy=0\). Rearranging gives the indifference-curve slope and the positive MRS:
When \(x\) increases along the curve, \(y\) generally changes too. Consequently, both the numerator and denominator of MRS can change. The own-good conditions \(u_{xx}<0\) and \(u_{yy}<0\) tell us what happens to each marginal utility when its own good changes and the other is fixed. They do not, on their own, determine the combined change in the ratio along the curve.
For diminishing marginal utility of each good, inspect \(u_{xx}\) and \(u_{yy}\), holding the other good fixed.
When both marginal utilities are positive, calculate \(MU_x/MU_y\).
Use \(dy/dx=-MRS_{xy}\) to account for the change in \(y\) as \(x\) rises. Only then assess whether the MRS falls.
Worked Examples: When Both Properties Hold
Worked Example: A Separable Utility Function
Let \(u(x,y)=\ln x+\ln y\), for \(x>0\) and \(y>0\). Compare the bundles \((1,4)\) and \((2,2)\), and determine whether both marginal utilities diminish in their own goods and whether MRS diminishes between these points along the same indifference curve.
Method. Calculate the marginal utilities and identify the utility level set. This keeps the comparison along the curve distinct from a one-good change holding the other good fixed.
Marginal utilities. Differentiating gives \(MU_x=1/x\) and \(MU_y=1/y\). The own-good derivatives are \(\partial MU_x/\partial x=-1/x^2<0\) and \(\partial MU_y/\partial y=-1/y^2<0\). Thus each good has diminishing marginal utility when the other is fixed.
Check the indifference curve. At \((1,4)\), utility is \(\ln 1+\ln 4=\ln 4\). At \((2,2)\), utility is \(\ln 2+\ln 2=\ln 4\). The bundles therefore lie on the same indifference curve.
Compare MRS. Since \(MRS_{xy}=(1/x)/(1/y)=y/x\), it is \(4/1=4\) at \((1,4)\), and \(2/2=1\) at \((2,2)\). As \(x\) rises from 1 to 2 on this curve, MRS falls from 4 to 1. The calculation confirms diminishing MRS for this example.
Check. The utility levels agree, and the computed MRS values have the stated direction. The mistake to avoid is comparing bundles off the same indifference curve and treating that comparison as a change in MRS along a curve.
Worked Example: A Cobb-Douglas Utility Function
Consider \(u(x,y)=x^{1/3}y^{2/3}\) for positive quantities. Compare \((1,2)\) with \((4,1)\). Verify that both marginal utilities diminish in their own goods and determine how MRS changes along the comparison.
Method. Differentiate to find each marginal utility, verify the utility level, and then compare the MRS values. The comparison is meaningful along an indifference curve only if utility is unchanged.
Marginal utilities. Differentiation gives \(MU_x=(1/3)x^{-2/3}y^{2/3}\) and \(MU_y=(2/3)x^{1/3}y^{-1/3}\). Their own-good derivatives are \(\partial MU_x/\partial x=-(2/9)x^{-5/3}y^{2/3}<0\) and \(\partial MU_y/\partial y=-(2/9)x^{1/3}y^{-4/3}<0\). Thus both marginal utilities diminish in their own goods for positive \(x,y\).
Check utility. At \((1,2)\), \(u=1^{1/3}2^{2/3}=2^{2/3}\). At \((4,1)\), \(u=4^{1/3}1^{2/3}=2^{2/3}\). The bundles are on the same indifference curve.
Calculate MRS. Taking the ratio of the marginal utilities gives \(MRS_{xy}=y/(2x)\). At \((1,2)\), this is \(2/(2\cdot1)=1\). At \((4,1)\), it is \(1/(2\cdot4)=1/8\). MRS falls as \(x\) rises along this curve.
Check. Both bundles have utility \(2^{2/3}\), and the MRS calculation uses the ratio \(MU_x/MU_y\), not its reciprocal. Reversing that ratio is a common error because it changes which good is being traded for which.
A Counterexample: Diminishing Marginal Utilities Without Diminishing MRS
The previous examples show that diminishing marginal utility and diminishing MRS can occur together. They do not establish that the first property guarantees the second. For that, examine a utility function with an interaction between the goods. The following calculation is local: it uses a neighborhood of \((1,1)\) where both marginal utilities are positive, so the MRS is defined.
Worked Example: Marginal Utility Falls While MRS Rises
Let \(u(x,y)=5x+5y-\tfrac12x^2-\tfrac12y^2-2xy\), in a neighborhood of \((1,1)\). Determine whether each marginal utility diminishes in its own good, then find the direction in which MRS changes as \(x\) increases along the indifference curve through \((1,1)\).
Method. Differentiate the utility function, check the own-good derivatives, and use the indifference-curve slope to differentiate the MRS along the curve.
Marginal utilities and own-good changes. We have \(MU_x=5-x-2y\) and \(MU_y=5-y-2x\). Therefore \(u_{xx}=-1<0\) and \(u_{yy}=-1<0\). At \((1,1)\), both marginal utilities equal \(2\), so both are positive there and remain positive in a sufficiently small neighborhood.
Find the curve's slope. At \((1,1)\), \(MRS_{xy}=2/2=1\). The indifference-curve slope is therefore \(dy/dx=-1\). As \(x\) increases locally along this curve, \(y\) decreases at that rate at the point.
Differentiate the marginal utilities along the curve. By the chain rule, \(d(MU_x)/dx=u_{xx}+u_{xy}(dy/dx)=-1+(-2)(-1)=1\). Similarly, \(d(MU_y)/dx=u_{yx}+u_{yy}(dy/dx)=-2+(-1)(-1)=-1\). The numerator of MRS is increasing locally while its denominator is decreasing locally.
Differentiate their ratio. At the point, \(MU_x=MU_y=2\), so
Thus MRS rises locally as \(x\) increases along the indifference curve, even though each marginal utility decreases when its own good alone increases. The cross effect, represented by \(u_{xy}=-2\), matters for how the marginal utilities change together along the curve.
Check. The own-good second derivatives are both negative, while the derivative of MRS along the curve is positive. This directly disproves the claim that diminishing marginal utility of each good is sufficient for diminishing MRS. The mistake to avoid is using the own-good derivatives as if they described movement along an indifference curve.
What the Comparison Does—and Does Not—Establish
A utility function is a representation of preference rankings, not a physical measure of satisfaction with fixed units. If \(v=g(u)\), where \(g\) is strictly increasing and differentiable, then \(v\) represents the same rankings. Its marginal utilities are \(v_x=g'(u)u_x\) and \(v_y=g'(u)u_y\). The factor \(g'(u)\) can change how marginal utility behaves as utility changes, so diminishing marginal utility is not generally invariant to a change of utility representation.
By contrast, where \(g'(u)>0\) and \(MU_y>0\), the MRS ratio is unchanged: \(v_x/v_y=[g'(u)u_x]/[g'(u)u_y]=u_x/u_y\). This is why an MRS comparison describes a preference trade-off more directly than the numerical size of marginal utility. The key takeaway is not that the properties can never occur together; the examples show that they often do. It is that one property cannot be inferred from the other without additional structure.
- Diminishing marginal utility: vary one good and hold the other fixed.
- Diminishing MRS: move along an indifference curve and track the ratio of marginal utilities.
- When checking MRS: verify both marginal utilities are positive and keep the direction of the trade-off clear.
This distinction is useful whenever a question asks what a utility function implies about preferences. A negative own second derivative answers a question about the change in one marginal utility. It does not, by itself, answer whether the consumer's willingness to exchange the goods declines along an indifference curve.
Check Your Understanding
For each problem, distinguish own-good marginal utility changes from MRS changes along a curve.
- For \(u(x,y)=\ln x+2\ln y\), calculate \(MU_x\), \(MU_y\), and \(MRS_{xy}\) at \((2,1)\). (Answer: \(MU_x=1/2\), \(MU_y=2\), and \(MRS_{xy}=1/4\).)
- For \(u(x,y)=x^{2/5}y^{3/5}\), compare \((1,1)\) with \((8,1/4)\). Verify that they have equal utility and calculate MRS at each point. (Answer: Both have utility 1; MRS falls from \(2/3\) to \(1/48\).)
- Let \(u(x,y)=6x-x^2+4y-\tfrac12y^2\). At \((1,2)\), find both marginal utilities, MRS, and the derivative of MRS with respect to \(x\) along the indifference curve. (Answer: \(MU_x=4\), \(MU_y=2\), \(MRS_{xy}=2\), and \(d(MRS_{xy})/dx=-3\).)
- Let \(u(x,y)=7x+7y-\tfrac12x^2-\tfrac12y^2-3xy\). At \((1,1)\), verify diminishing own marginal utilities and calculate the local change in MRS as \(x\) increases along the indifference curve. (Answer: \(u_{xx}=u_{yy}=-1\); \(MRS_{xy}=1\), and \(d(MRS_{xy})/dx=4/3>0\), so MRS rises locally.)
- If \(\tilde u=g(u)\) for differentiable \(g\) with \(g'(u)>0\), express the transformed MRS in terms of the original MRS. (Answer: \(\tilde{MRS}_{xy}=[g'(u)u_x]/[g'(u)u_y]=MRS_{xy}\).)