Why MRS Curvature Requires a Two-Good Test
A utility function can have diminishing marginal utility for each good without having diminishing MRS. The reason is that MRS compares the marginal utilities of both goods. Its change depends not only on each good’s own second derivative, but also on how the marginal utilities change together.
This tutorial develops a local second-derivative test for diminishing MRS. The test applies to a smooth two-good utility function at an interior bundle where both marginal utilities are positive. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The derivative is taken along the indifference curve, not while holding \(y\) fixed. As \(x\) increases along that curve, \(y\) typically falls. The condition therefore combines changes in both coordinates. If either marginal utility is zero, or the utility function is not twice differentiable at the bundle, the test below may not apply in this form.
Deriving the Curvature Test
Write \(M=U_x/U_y\). On a smooth indifference curve, \(U(x,y)\) is constant. Differentiating that constant-utility condition gives \(U_x+U_y(dy/dx)=0\), so \(dy/dx=-M\). The chain rule then gives the change in MRS along the curve:
Differentiating the ratio \(M=U_x/U_y\) with respect to each coordinate and substituting \(dy/dx=-M\) yields a useful Hessian expression. The Hessian is the matrix of second partial derivatives of utility.
For a geometric interpretation, let \(H\) be the Hessian of \(U\), and let \(t=(1,-M)\) be a tangent direction to the indifference curve. Since \(M=U_x/U_y\), the gradient of utility dotted with this tangent is \(U_x-MU_y=0\). The numerator in the first fraction is the Hessian’s second derivative in the tangent direction.
The same criterion can be written using a bordered Hessian. Its determinant equals the negative of the numerator in the second fraction:
Thus a positive bordered-Hessian determinant corresponds to strictly diminishing MRS under these assumptions. This criterion concerns curvature in the tangent direction; it is not a claim that every direction in the utility function’s Hessian must curve downward.
Worked Examples
Worked Example: Cobb-Douglas Utility at a Bundle
Let \(U(x,y)=\sqrt{xy}\) for \(x>0,y>0\). Find the MRS and its rate of change along the indifference curve at \((x,y)=(4,9)\), then check the result using the Hessian test.
Method. Compute the marginal utilities, form their ratio, and differentiate along the curve. The Hessian check verifies the sign using the local curvature condition.
Marginal utilities and MRS. The derivatives are \(U_x=\tfrac12\sqrt{y/x}\) and \(U_y=\tfrac12\sqrt{x/y}\). At \((4,9)\), these are \(U_x=\tfrac34\) and \(U_y=\tfrac13\), so \(M=U_x/U_y=9/4\). The indifference-curve slope is \(dy/dx=-9/4\).
In fact \(M=y/x\). Its rate of change along the curve at the given bundle is therefore \(dM/dx=(y'/x)-(y/x^2)\). Substituting \(y'=-9/4\), \(x=4\), and \(y=9\) gives \(-9/16-9/16=-9/8\). MRS is strictly diminishing there.
Hessian check. At this bundle, \(U_{xx}=-3/32\), \(U_{xy}=1/24\), and \(U_{yy}=-1/54\). The tangent-direction expression is \(U_{xx}-2M U_{xy}+M^2U_{yy}\). Substituting the values gives \(-3/32-(9/2)(1/24)+(81/16)(-1/54)=-3/32-3/16-3/32=-3/8\). Dividing by \(U_y=1/3\) gives \((-3/8)/(1/3)=-9/8\), which matches the direct derivative.
Check and mistake avoided. Both methods give \(-9/8\), so the sign and magnitude agree. The method avoids treating the MRS as if \(y\) were fixed while \(x\) changes.
Worked Example: Quasilinear Utility and Weak Curvature
Let \(U(x,y)=\ln x+y\), where \(x>0\). At \((2,3)\), calculate the rate of change of MRS along the indifference curve and verify it with the Hessian test.
Method. Use the ratio of marginal utilities, then compare its along-curve derivative with the directional second derivative. This also shows that the test can apply when one own second derivative is zero.
The marginal utilities are \(U_x=1/x\) and \(U_y=1\). Thus \(M=1/x\), and at \(x=2\), \(M=1/2\); the indifference-curve slope is \(-1/2\). Since \(M\) depends only on \(x\), its along-curve derivative is \(dM/dx=-1/x^2=-1/4\).
The second derivatives are \(U_{xx}=-1/x^2=-1/4\), \(U_{xy}=0\), and \(U_{yy}=0\). Therefore \(U_{xx}-2M U_{xy}+M^2U_{yy}=-1/4\). Since \(U_y=1\), the formula gives \(dM/dx=-1/4\), as before.
Check and mistake avoided. Both calculations give \(-1/4\). A zero value of \(U_{yy}\) does not mean that MRS is constant: the ratio still changes as \(x\) changes.
Worked Example: A Utility Transformation With an Indefinite Hessian
Consider \(U(x,y)=e^{\sqrt{xy}}\) for \(x>0,y>0\). At \((1,1)\), determine the MRS derivative and use the Hessian test to explain why a negative-semidefinite Hessian is not necessary for diminishing MRS.
Method. Calculate MRS directly and then evaluate the Hessian at the bundle. This compares curvature in the indifference-curve direction with curvature in all directions.
Let \(V=\sqrt{xy}\), so \(U=e^V\). The marginal utilities are \(U_x=e^V V_x\) and \(U_y=e^V V_y\). Since \(V_x=\tfrac12\sqrt{y/x}\) and \(V_y=\tfrac12\sqrt{x/y}\), the ratio is \(M=y/x\). Along an indifference curve, \(xy\) is constant, so \(y'=-y/x\). Differentiating \(M=y/x\) along that curve gives \(M'=y'/x-y/x^2=-2y/x^2\). At \((1,1)\), \(M=1\) and \(M'=-2\).
At \((1,1)\), \(U_x=U_y=e/2\), \(U_{xx}=U_{yy}=0\), and \(U_{xy}=e/2\). The Hessian is therefore \(\begin{pmatrix}0&e/2\\e/2&0\end{pmatrix}\), which is indefinite: its quadratic form is positive in the direction \((1,1)\) and negative in \((1,-1)\). But the indifference-curve tangent direction is \((1,-1)\), for which the Hessian quadratic form is \(-e\). Dividing by \(U_y=e/2\) gives \(-2\), matching the direct derivative.
Check and mistake avoided. The MRS is strictly diminishing even though the full Hessian is not negative semidefinite. Checking only the Hessian’s overall sign would incorrectly reject this case; the relevant direction is tangent to the indifference curve.
What the Hessian Test Does—and Does Not—Say
A negative-semidefinite Hessian guarantees \(t^{\mathsf T}Ht\leq 0\) for every direction \(t\). It therefore guarantees weakly diminishing MRS wherever the regularity conditions hold. But it is a stronger requirement than needed: the MRS test examines only the direction tangent to the indifference curve. As the exponential example showed, curvature may be positive in another direction while remaining negative along that tangent.
A common trap is to use the signs of the own second derivatives alone. For example, \(U_{xx}<0\) and \(U_{yy}<0\) do not settle the sign of \(U_{xx}-2M U_{xy}+M^2U_{yy}\). The cross-partial term can change the result. The following example shows this explicitly.
Worked Example: Diminishing Marginal Utilities but Increasing MRS
On a neighborhood of \((0.1,0.1)\), consider \(U(x,y)=x+y-\tfrac12(x^2+y^2)-2xy\). Determine whether MRS is diminishing at this bundle, even though each own second derivative is negative.
Method. Evaluate the full tangent-direction expression. The common mistake is to infer diminishing MRS just from the signs of \(U_{xx}\) and \(U_{yy}\).
The marginal utilities are \(U_x=1-x-2y\) and \(U_y=1-y-2x\). At the stated bundle, each equals \(1-0.1-0.2=0.7\), so both are positive and \(M=1\). The own second derivatives are \(U_{xx}=U_{yy}=-1\), while \(U_{xy}=-2\).
The curvature expression is \(U_{xx}-2M U_{xy}+M^2U_{yy}=-1-2(1)(-2)+(1)^2(-1)=2\). Since \(U_y=0.7\), the rate of change is \(dM/dx=2/0.7=20/7\), which is positive. MRS is increasing, not diminishing, at this bundle.
Check and mistake avoided. The result is consistent with the formula and the positive marginal utilities. The negative own second derivatives show diminishing marginal utility in each coordinate at this point, but the cross-partial term is large enough to make the tangent-direction curvature positive.
For a smooth utility function with positive marginal utilities, the practical test is therefore to calculate the entire expression, including the cross-partial term. A result at one bundle is local: it establishes curvature there, not necessarily at every point on every indifference curve. To establish diminishing MRS throughout a region, the inequality must hold at each relevant bundle in that region.
Confirm the utility function is twice differentiable at the bundle and both marginal utilities are positive.
Calculate \(M=U_x/U_y\); the indifference-curve slope is \(-M\).
Evaluate \(U_{xx}-2M U_{xy}+M^2U_{yy}\). A negative value means strictly diminishing MRS at that bundle; zero means locally constant to first order.
Check Your Understanding
For each problem, check the regularity conditions and use the along-curve derivative or the Hessian test as appropriate.
- For \(U(x,y)=x^{0.3}y^{0.7}\), derive \(M\) and \(dM/dx\) along an indifference curve. (Answer: \(M=\tfrac{3}{7}\tfrac{y}{x}\), and \(dM/dx=-\tfrac{30}{49}\tfrac{y}{x^2}<0\).)
- For \(U(x,y)=\ln x+2\ln y\), calculate \(dM/dx\) at \((2,4)\). (Answer: \(-3/4\), so MRS is strictly diminishing there.)
- For \(U(x,y)=x+y-\tfrac12(x^2+y^2)-2xy\) at \((0.1,0.1)\), state the sign of the MRS derivative and explain which Hessian term changes the result suggested by the own second derivatives. (Answer: \(dM/dx=20/7>0\); the cross-partial term \(-2M U_{xy}\) is positive and outweighs the two negative own-curvature terms.)
- For \(U(x,y)=e^{\sqrt{xy}}\) at \((1,4)\), use \(M=y/x\) and the constant-product indifference curve to find \(dM/dx\). (Answer: \(-8\).)
- At a regular bundle with \(U_y>0\), what sign of the bordered-Hessian determinant corresponds to strictly diminishing MRS? (Answer: A positive determinant, because it equals the negative of the curvature numerator.)