Why Substitutability Matters
When a consumer can choose between two goods, the optimal bundle depends not only on prices and income but also on how readily one good can replace the other. The CES utility function makes that substitutability explicit. Its parameter determines how strongly the consumer adjusts the ratio of the two goods when their relative price changes.
This tutorial develops the interior choice for a two-good CES model. It focuses on the optimal ratio and its response to relative prices, not on corner solutions. The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
The term “constant elasticity” refers to the fact that \(\sigma\) does not change with the bundle. A larger \(\sigma\) means the consumer’s optimal ratio responds more strongly to a change in relative prices. The weights influence the preferred ratio at a given price ratio; \(\sigma\) controls how sensitive that ratio is to prices.
The consumer faces positive prices \(p_x,p_y\), income \(m>0\), and the budget constraint \(p_xx+p_yy\le m\). The calculations below assume an interior optimum with \(x>0\) and \(y>0\), so the budget binds and the tangency condition applies. CES utility is increasing in both goods under the stated conditions, and it is concave for \(\rho<1\); therefore an interior tangency is a global maximum. If the optimum is on a boundary, the interior formulas do not by themselves establish the choice.
Deriving the Interior Choice
At an interior optimum, the marginal rate of substitution of \(x\) for \(y\) equals the price ratio \(p_x/p_y\). For nonzero \(\rho\), differentiating CES utility gives marginal utilities proportional to \(\alpha x^{\rho-1}\) and \((1-\alpha)y^{\rho-1}\). The common positive factor in the derivatives cancels in their ratio.
Solving this equality for \(x/y\), and using \(\sigma=1/(1-\rho)\), yields the key ratio formula. For \(\rho=0\), the same formula follows by differentiating the limiting Cobb-Douglas utility.
To see why the quantity formulas follow, set \(x=ry\) in the binding budget. Then \(p_xry+p_yy=m\), so \(y=m/(p_xr+p_y)\); multiplying by \(r\) gives \(x\). These expressions ensure that both quantities are positive when \(r,p_x,p_y,m\) are positive and that all income is spent.
How the Substitution Parameter Changes the Response
The ratio formula separates the effects of preferences and prices. Taking logarithms gives \(\ln(x/y)=\sigma[\ln(\alpha/(1-\alpha))+\ln(p_y/p_x)]\). Differentiating with respect to the log relative price gives \(\frac{d\ln(x/y)}{d\ln(p_y/p_x)}=\sigma\). This is a direct interpretation of the elasticity: a one-percent increase in \(p_y/p_x\), holding the preference weights fixed, produces a \(\sigma\)-percent increase in \(x/y\) locally.
More generally, if \(p_y/p_x\) is multiplied by a factor \(k>0\), the optimal \(x/y\) is multiplied by \(k^\sigma\). The percentage response is therefore not necessarily equal to the percentage price change; it depends on \(\sigma\). A high value of \(\sigma\) produces a stronger ratio response, while a low value produces a weaker response.
Worked Example: Solving a CES Choice With a High Substitution Elasticity
A consumer has utility \(U(x,y)=[0.4x^{0.5}+0.6y^{0.5}]^2\), income \(m=100\), and prices \(p_x=4\), \(p_y=1\). Assuming an interior choice, find the optimal quantities and verify the tangency and budget.
Method. Match the utility to CES form: \(\alpha=0.4\), \(1-\alpha=0.6\), and \(\rho=0.5\). Thus \(\sigma=1/(1-0.5)=2\).
First calculate the optimal ratio: \(r=[(0.4/0.6)(1/4)]^2=[(2/3)(1/4)]^2=(1/6)^2=1/36\).
Use the budget-based quantity formulas. The denominator is \(p_xr+p_y=4(1/36)+1=1/9+1=10/9\). Therefore \(y=100/(10/9)=90\), and \(x=ry=(1/36)(90)=2.5\).
Check. Spending is \(4(2.5)+1(90)=10+90=100\). The marginal-rate ratio is \((0.4/0.6)(x/y)^{-0.5}=(2/3)(1/36)^{-0.5}=(2/3)(6)=4=p_x/p_y\). Both the budget and tangency conditions hold.
Common mistake avoided. The ratio is \(x/y=1/36\), not \(y/x=1/36\). Reversing the ratio changes the quantities and fails the MRS check.
Worked Example: A Less Responsive Ratio
Let \(U(x,y)=[0.5x^{-1}+0.5y^{-1}]^{-1}\), with \(m=120\). Initially \(p_x=2\) and \(p_y=8\); then the prices change to \(p_x=8\) and \(p_y=2\). Assuming an interior optimum at both price pairs, find both bundles and compare their consumption ratios.
Method. Here \(\alpha=0.5\), \(\rho=-1\), and \(\sigma=1/(1-(-1))=0.5\). For the initial prices, \(r=[(0.5/0.5)(8/2)]^{0.5}=\sqrt{4}=2\). With \(x=2y\), the budget is \(2(2y)+8y=12y=120\), so \(y=10\) and \(x=20\).
For the changed prices, the relative price \(p_y/p_x=2/8=1/4\), so \(r=[1(1/4)]^{0.5}=1/2\). With \(x=y/2\), the budget is \(8(y/2)+2y=6y=120\), giving \(y=20\) and \(x=10\).
Check. Initial spending is \(2(20)+8(10)=120\); changed-price spending is \(8(10)+2(20)=120\). The corresponding MRS values are \(2^{-2}=1/4=p_x/p_y\) initially and \((1/2)^{-2}=4=p_x/p_y\) after the change.
The relative price \(p_y/p_x\) falls from \(4\) to \(1/4\), a factor of \(1/16\). With \(\sigma=0.5\), the optimal \(x/y\) ratio changes by \((1/16)^{0.5}=1/4\), from \(2\) to \(1/2\). The numbers confirm the predicted response.
Common mistake avoided. A sixteen-fold fall in the relative price does not imply a sixteen-fold fall in the consumption ratio. The price factor must be raised to the elasticity \(\sigma\).
Worked Example: The Cobb-Douglas Limit and a Fractional Quantity
A consumer has \(U(x,y)=x^{0.6}y^{0.4}\), income \(m=140\), and prices \(p_x=5\), \(p_y=2\). Find the interior optimum and confirm the ratio, budget, and substitution elasticity.
Method. This is the \(\rho=0\) CES limit with \(\alpha=0.6\) and \(\sigma=1\). The ratio is \(r=[(0.6/0.4)(2/5)]^1=(3/2)(2/5)=3/5\).
Substitute \(x=(3/5)y\) into the budget: \(5(3y/5)+2y=3y+2y=5y=140\). Hence \(y=28\), and \(x=(3/5)(28)=16.8\).
Check. Spending is \(5(16.8)+2(28)=84+56=140\). The MRS is \((0.6/0.4)(x/y)^{-1}=(3/2)(3/5)^{-1}=(3/2)(5/3)=5/2=p_x/p_y\).
Common mistake avoided. Do not round \(x\) to a whole number before checking the budget. The optimal quantity is \(16.8\), and rounding it to \(16\) would leave part of the budget unaccounted for and break the tangency.
Interpreting CES Parameters and Avoiding Traps
| Parameter case | Elasticity | Interpretation for the interior ratio |
|---|---|---|
| \(\rho<0\) | \(0<\sigma<1\) | The ratio responds less than proportionally to a relative-price change. |
| \(\rho=0\), Cobb-Douglas limit | \(\sigma=1\) | The ratio responds proportionally to the relative-price factor. |
| \(0<\rho<1\) | \(\sigma>1\) | The ratio responds more than proportionally to the relative-price factor. |
These comparisons apply to the interior CES ratio. They do not guarantee that an interior solution exists for every possible model or price configuration. At a boundary, the tangency equality need not hold; use the boundary and Kuhn-Tucker conditions rather than applying the ratio formula mechanically.
Several errors are especially easy to make under time pressure:
- Reversing the relative price. The formula for \(x/y\) uses \(p_y/p_x\), while the MRS condition equals \(p_x/p_y\). Check the convention by returning to the tangency equation.
- Confusing \(\rho\) and \(\sigma\). The response elasticity is \(1/(1-\rho)\), not \(\rho\). In particular, \(\rho=0\) gives \(\sigma=1\), not zero substitution.
- Stopping at the ratio. The ratio alone does not give quantities. Use the binding budget to determine their levels.
- Ignoring units of the budget. Verify that \(p_xx+p_yy=m\) using the stated prices and income, including any scale such as dollars or thousands of dollars.
Check Your Understanding
For each problem, assume an interior optimum unless otherwise stated. Check both the ratio and the budget when quantities are requested.
- A consumer has \(\alpha=0.25\), \(\rho=0.5\), \(m=90\), \(p_x=3\), and \(p_y=12\). Find \(\sigma\), the optimal ratio \(x/y\), and both quantities. (Answer: \(\sigma=2,\ x/y=16/9,\ x=120/13\approx9.2308,\ y=135/26\approx5.1923\).)
- For \(\alpha=0.5\), \(\rho=-1\), \(m=100\), \(p_x=1\), and \(p_y=9\), find the optimal bundle. (Answer: \(\sigma=0.5,\ x/y=3,\ x=25,\ y=25/3\).)
- For a CES consumer with \(\rho=0.5\), by what factor does \(x/y\) change if \(p_y/p_x\) quadruples and the weights remain fixed? (Answer: It is multiplied by \(4^2=16\).)
- A Cobb-Douglas consumer has \(\alpha=0.6\), \(m=140\), \(p_x=5\), and \(p_y=2\). Find the ratio and bundle, then verify expenditure. (Answer: \(x/y=3/5,\ x=16.8,\ y=28\); expenditure is \(84+56=140\).)
- Two CES consumers face the same change: \(p_y/p_x\) rises by a factor of \(9\). Consumer A has \(\rho=-1\), and consumer B has \(\rho=0.5\). Find the factor change in each consumer’s \(x/y\) ratio. (Answer: A has \(\sigma=0.5\), so the factor is \(9^{0.5}=3\); B has \(\sigma=2\), so the factor is \(9^2=81\).)