CES Utility and Substitution Patterns

Why CES Utility Is Useful

Consumers often treat two goods as substitutes without treating them as perfect substitutes. A small change in the relative amount of one good may then affect how much of the other good the consumer is willing to give up. The constant-elasticity-of-substitution, or CES, utility function represents this range of substitution patterns with a single curvature parameter.

The word “constant” refers to the elasticity of substitution: for a CES utility function, that elasticity is the same at every bundle in its domain. The curvature parameter also governs the shape of indifference curves. This tutorial is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: A two-good CES utility function is \(U(x,y)=[\alpha x^\rho+(1-\alpha)y^\rho]^{1/\rho}\), where \(x>0\), \(y>0\), \(0<\alpha<1\), and \(\rho<1\), with \(\rho\ne0\). The parameter \(\alpha\) weights the two goods, while \(\rho\) controls curvature and substitutability. The case \(\rho=0\) is defined by a limit and is not evaluated by directly substituting zero into this formula.

The positive-goods restriction keeps powers such as \(x^\rho\) well-defined for all real values of \(\rho\). The strict upper bound \(\rho<1\) covers the usual finite-curvature CES cases; the limiting case \(\rho=1\) is the linear, perfect-substitutes case. The standard CES form is increasing in each good, so a larger quantity of either good raises utility, holding the other fixed.

Marginal Utilities and the MRS

Marginal utility is the partial derivative of utility with respect to one good, holding the other good fixed. Let \(Q=\alpha x^\rho+(1-\alpha)y^\rho\), so \(U=Q^{1/\rho}\). Applying the chain rule gives:

MU_x=\frac{\partial U}{\partial x} =\alpha x^{\rho-1}Q^{1/\rho-1}, \qquad MU_y=\frac{\partial U}{\partial y} =(1-\alpha)y^{\rho-1}Q^{1/\rho-1}.

Both marginal utilities are positive for positive quantities and weights. Their ratio is the marginal rate of substitution of \(x\) for \(y\), using the convention that MRS measures the quantity of \(y\) the consumer is willing to give up for a small increase in \(x\) while staying on the same indifference curve.

MRS_{xy}=\frac{MU_x}{MU_y} =\frac{\alpha}{1-\alpha}\left(\frac{x}{y}\right)^{\rho-1}.

The common factor \(Q^{1/\rho-1}\) cancels. With this positive MRS convention, the slope of an indifference curve is \(-MRS_{xy}\): moving to more \(x\) along the same curve requires giving up some \(y\).

Derivation: Along an indifference curve, utility is constant, so its total differential is zero: \(dU=MU_x\,dx+MU_y\,dy=0\). Rearranging gives \(dy/dx=-MU_x/MU_y\). Thus the curve’s slope is negative, and the positive amount of \(y\) given up per unit increase in \(x\) is \(MRS_{xy}=MU_x/MU_y\). Substituting the marginal-utility formulas cancels their common factor and yields the CES MRS expression above.

Worked Example: Marginal Utilities and MRS with a Square-Root CES Form

Suppose \(U(x,y)=[0.4x^{0.5}+0.6y^{0.5}]^2\), and consider the bundle \(x=16\), \(y=9\). Find utility, both marginal utilities, and the MRS of \(x\) for \(y\). The parameters are \(\alpha=0.4\) and \(\rho=0.5\), and both goods are measured in their stated quantity units.

Method. Use \(Q=0.4x^{0.5}+0.6y^{0.5}\), then evaluate the CES utility and marginal-utility formulas. The MRS is \(MU_x/MU_y\).

First, \(\sqrt{16}=4\) and \(\sqrt{9}=3\). Therefore \(Q=0.4(4)+0.6(3)=1.6+1.8=3.4\), and utility is \(U=Q^2=3.4^2=11.56\).

For \(\rho=0.5\), the common marginal-utility factor is \(Q^{1/\rho-1}=Q^{2-1}=3.4\). Hence \(MU_x=0.4(16^{-0.5})(3.4)=0.4(1/4)(3.4)=0.1(3.4)=0.34\).

Similarly, \(MU_y=0.6(9^{-0.5})(3.4)=0.6(1/3)(3.4)=0.2(3.4)=0.68\). Thus \(MRS_{xy}=0.34/0.68=0.5\). At this bundle, the consumer is willing to give up 0.5 units of \(y\) for a small increase of one unit in \(x\), locally, while maintaining utility.

Check. Directly, the MRS formula gives \((0.4/0.6)(16/9)^{-0.5}=(2/3)(3/4)=1/2\), matching the ratio of marginal utilities. The common mistake to avoid is reversing that ratio: \(MU_y/MU_x=2\) is the reciprocal trade-off, not the amount of \(y\) given up per unit of \(x\).

How the Curvature Parameter Changes Substitution

The MRS expression shows that the curvature parameter affects how trade-offs respond to the quantity ratio \(x/y\). When \(\rho<1\), the exponent \(\rho-1\) is negative. As \(x/y\) rises along an indifference curve, \(MRS_{xy}\) falls: the consumer is willing to give up less \(y\) for another unit of \(x\). This is the diminishing-MRS pattern associated with convex indifference curves in the positive-goods region.

The conventional elasticity of substitution, denoted here by \(\sigma\), measures the percentage response of the quantity ratio \(x/y\) to a percentage change in the MRS, with a negative sign because the two move in opposite directions. Taking logarithms of the CES MRS gives:

\ln(MRS_{xy})= \ln\left(\frac{\alpha}{1-\alpha}\right) +(\rho-1)\ln\left(\frac{x}{y}\right).

The weights contribute a constant to this relationship. Therefore the slope of \(\ln(MRS_{xy})\) with respect to \(\ln(x/y)\) is \(\rho-1\), and the elasticity of substitution is:

\sigma =-\frac{d\ln(x/y)}{d\ln(MRS_{xy})} =-\frac{1}{\rho-1} =\frac{1}{1-\rho}.

A larger \(\sigma\) means the quantity ratio changes more in response to a given percentage change in MRS: the goods are more readily substituted. As \(\rho\) approaches 1 from below, \(\sigma\) becomes large and the indifference curves approach the straight lines of perfect substitutes. Lower values of \(\rho\) mean lower substitution elasticity and more bowed indifference curves. At \(\rho=0\), the formula for \(U\) is interpreted by its limit, not by direct substitution; that limiting CES case has \(\sigma=1\). As \(\rho\) becomes very negative, the pattern approaches that of strongly complementary goods.

Curvature parameter	Elasticity \(\sigma\)	Qualitative substitution pattern
\(\rho\) close to 1 from below	Large	Goods are relatively easy to substitute; curves approach straight lines
\(\rho=0\), interpreted as a limit	1	Unit elasticity of substitution
More negative \(\rho\)	Below 1	Goods are harder to substitute; curves are more bowed

Worked Example: A Negative Curvature Parameter

Let \(U(x,y)=[0.25x^{-1}+0.75y^{-1}]^{-1}\), with \(\alpha=0.25\), \(\rho=-1\), and bundle \(x=4\), \(y=9\). Calculate utility, both marginal utilities, and \(MRS_{xy}\). Also find the elasticity of substitution.

Method. Evaluate the bracket \(Q\), then use \(1/\rho-1=-2\) in the marginal-utility formulas. Finally, take the ratio of marginal utilities and use \(\sigma=1/(1-\rho)\).

The bracket is \(Q=0.25(1/4)+0.75(1/9)=1/16+1/12=3/48+4/48=7/48\). Since \(1/\rho=-1\), utility is \(U=Q^{-1}=48/7\), approximately \(6.8571\).

The common marginal-utility factor is \(Q^{-2}=(48/7)^2=2304/49\). Thus \(MU_x=0.25(4^{-2})(2304/49)=(1/4)(1/16)(2304/49)=36/49\).

For the other good, \(MU_y=0.75(9^{-2})(2304/49)=(3/4)(1/81)(2304/49)=64/147\). Consequently, \(MRS_{xy}=(36/49)/(64/147)=(36/49)(147/64)=108/64=27/16=1.6875\).

The elasticity is \(\sigma=1/(1-(-1))=1/2=0.5\). This is lower than one, indicating less substitution than in the unit-elastic limiting case. Check. The direct MRS formula gives \((0.25/0.75)(4/9)^{-2}=(1/3)(81/16)=27/16\), the same result. Do not mistake the negative exponent in the utility function for a negative MRS; both marginal utilities and the MRS are positive at this positive bundle.

Comparing Curvature at the Same Bundle

The role of \(\rho\) is especially clear when the weights and bundle are held fixed. Consider a bundle with \(x/y=4\) and \(\alpha=0.3\). For \(\rho=0.5\), the MRS is \((0.3/0.7)4^{-0.5}=(3/7)(1/2)=3/14\). For \(\rho=-1\), it is \((3/7)4^{-2}=3/112\). The MRS is smaller in the second case: with relatively abundant \(x\), the consumer is willing to give up less \(y\) for another unit of \(x\).

These values also illustrate the different substitution elasticities. At \(\rho=0.5\), \(\sigma=1/(1-0.5)=2\); at \(\rho=-1\), \(\sigma=1/(1+1)=0.5\). The second utility function has a lower elasticity, so its substitution pattern is less responsive to a change in relative trade-offs. MRS at one bundle is not itself the elasticity: the MRS is a local trade-off, while elasticity describes how the quantity ratio responds to a percentage change in that trade-off.

Worked Example: Comparing Two CES Substitution Patterns

Two hypothetical consumers have CES utilities with the same weights, \(\alpha=0.3\), and the same bundle, \(x=16\), \(y=4\). Consumer A has \(\rho=0.5\), and Consumer B has \(\rho=-1\). For each, calculate utility, marginal utilities, MRS, and elasticity of substitution. State which consumer has the lower substitution elasticity.

Method. Apply the CES formulas separately for each curvature parameter. The weights and bundle stay fixed, but the bracket, marginal-utility factor, and MRS exponent depend on \(\rho\).

For Consumer A, \(Q=0.3(16^{0.5})+0.7(4^{0.5})=0.3(4)+0.7(2)=1.2+1.4=2.6=13/5\). Thus \(U=Q^2=6.76\). The marginal-utility factor is \(Q^{1}=2.6\), so \(MU_x=0.3(16^{-0.5})(2.6)=0.3(1/4)(2.6)=0.195=39/200\), and \(MU_y=0.7(4^{-0.5})(2.6)=0.7(1/2)(2.6)=0.91=91/100\). Therefore \(MRS_{xy}=0.195/0.91=3/14\), and \(\sigma=1/(1-0.5)=2\).

For Consumer B, \(Q=0.3(16^{-1})+0.7(4^{-1})=0.3/16+0.7/4=3/160+28/160=31/160\). Since \(1/\rho=-1\), \(U=Q^{-1}=160/31\). The marginal-utility factor is \(Q^{-2}=(160/31)^2=25600/961\). Hence \(MU_x=0.3(16^{-2})(25600/961)=(3/10)(1/256)(25600/961)=30/961\), and \(MU_y=0.7(4^{-2})(25600/961)=(7/10)(1/16)(25600/961)=1120/961\). Thus \(MRS_{xy}=(30/961)/(1120/961)=3/112\), and \(\sigma=1/(1-(-1))=0.5\).

Conclusion and check. Consumer B has the lower substitution elasticity: \(0.5<2\). The direct MRS formula verifies both results: for A it is \((3/7)4^{-0.5}=3/14\), and for B it is \((3/7)4^{-2}=3/112\). Utility levels differ across the two functions, so comparing their numerical utility values is not the right way to compare substitution patterns; use MRS and elasticity instead.

Common Traps and Interpretation

Reversing the MRS. Under the convention used here, \(MRS_{xy}=MU_x/MU_y\), the amount of \(y\) given up per additional unit of \(x\). The indifference-curve slope is its negative.
Dropping the exponent in the derivative. The marginal-utility factor is \(Q^{1/\rho-1}\), not \(Q^{1/\rho}\). It follows from the chain rule.
Substituting \(\rho=0\) directly. The written CES formula has \(1/\rho\), so the zero case must be treated as a limit.
Calling MRS the elasticity. MRS is a trade-off at one bundle. Elasticity describes the percentage response of the quantity ratio to a percentage change in MRS.
Interpreting utility units literally. The main use here is to represent preferences and compute marginal trade-offs. A utility value such as 6.76 is not a physical quantity of satisfaction.

Key takeaway: For CES utility, marginal utilities share a common factor, so their ratio simplifies to \(MRS_{xy}=\frac{\alpha}{1-\alpha}(x/y)^{\rho-1}\). The elasticity of substitution is \(\sigma=1/(1-\rho)\): higher \(\rho\) means easier substitution, while lower \(\rho\) means more bowed indifference curves and harder substitution.

Check Your Understanding

For each problem, use the positive-goods CES model and the stated MRS convention. Answers are provided for checking.

Let \(U=[0.5x^{0.5}+0.5y^{0.5}]^2\) at \(x=9,y=4\). Find \(Q\), utility, both marginal utilities, and \(MRS_{xy}\). (Answer: \(Q=2.5\), \(U=6.25\), \(MU_x=5/12\), \(MU_y=5/8\), \(MRS_{xy}=2/3\).)
For \(U=[0.2x^{-1}+0.8y^{-1}]^{-1}\) at \(x=5,y=10\), calculate utility and \(MRS_{xy}\). (Answer: \(U=25/3\), \(MRS_{xy}=4/5\).)
A CES utility function has \(\rho=0.25\), \(\alpha=0.4\), and \(x/y=9\). Find the elasticity and the MRS. (Answer: \(\sigma=4/3\), \(MRS_{xy}=2/3\).)
At a bundle with \(x/y=16\), weights \(\alpha=0.2\), and parameter \(\rho=-1\), calculate the MRS and elasticity. (Answer: \(MRS_{xy}=1/320\), \(\sigma=1/2\).)
Two CES forms have \(\rho_A=0.8\) and \(\rho_B=-0.5\), with identical weights and positive-goods domains. Calculate each elasticity and identify which has more readily substitutable goods. (Answer: \(\sigma_A=5\), \(\sigma_B=2/3\); form A has more readily substitutable goods.)

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