How to Calculate the Optimal Bundle With Cobb-Douglas Utility
With U = X^a × Y^b, the optimal bundle is X* = (a/(a+b)) × I ÷ Px and Y* = (b/(a+b)) × I ÷ Py, the point where MRS equals Px ÷ Py.
The Cobb-Douglas Optimum formula
Calculator
Enter the exponents of U = X^a × Y^b, income and the two prices to get the utility-maximizing bundle.
The consumer spends the share a ÷ (a + b) of income on X.
- Optimal Y (Y*)
- 6
- Spending on X
- $60
- MRS at the optimum
- 0.6
- Utility at the optimum
- 2,400
The rest of income, the share b ÷ (a + b), goes to Y.
Always the same fraction of income, whatever the price of X.
Equals the price ratio Px ÷ Py = 0.6 at the tangency.
No other bundle that costs the same gives more.
How to calculate Cobb-Douglas Optimum, step by step
- 1Read the exponents, income and prices. From U = X^a × Y^b take the exponents a and b, then note income I and the two prices Px and Py.
- 2Find each good's budget share. Good X gets a ÷ (a + b) of income and good Y gets b ÷ (a + b). The two shares always add up to 1.
- 3Split income by the shares. Spending on X = (a ÷ (a + b)) × I, and spending on Y = (b ÷ (a + b)) × I.
- 4Divide each spending amount by its price. X* = spending on X ÷ Px and Y* = spending on Y ÷ Py. These are the utility-maximizing quantities.
- 5Check the tangency and the budget. MRS = (a × Y*) ÷ (b × X*) should equal Px ÷ Py, and Px × X* + Py × Y* should equal I. If either check fails, recheck the shares.
Worked example: Cobb-Douglas Optimum
Utility is U = X^2 × Y, so a = 2 and b = 1. Income is $90, X costs $3 and Y costs $5. The exponents sum to 2 + 1 = 3, so X takes 2 ÷ 3 of the budget and Y takes 1 ÷ 3. Spending on X = (2 ÷ 3) × $90 = $60, so X* = $60 ÷ $3 = 20 units. Spending on Y = (1 ÷ 3) × $90 = $30, so Y* = $30 ÷ $5 = 6 units. Budget check: (20 × $3) + (6 × $5) = $60 + $30 = $90, all of it spent. Tangency check: MRS = (a × Y) ÷ (b × X) = (2 × 6) ÷ (1 × 20) = 12 ÷ 20 = 0.6, and Px ÷ Py = 3 ÷ 5 = 0.6, so the two match. In AP terms, MUx = 2 × X × Y = 2 × 20 × 6 = 240 and MUy = X^2 = 20^2 = 400, so MUx ÷ Px = 240 ÷ 3 = 80 and MUy ÷ Py = 400 ÷ 5 = 80 utils per dollar. Utility at the optimum is 20^2 × 6 = 400 × 6 = 2,400. Two other bundles that also cost exactly $90 do worse: 10 of X and 12 of Y gives 10^2 × 12 = 100 × 12 = 1,200, and 25 of X and 3 of Y gives 25^2 × 3 = 625 × 3 = 1,875.
Where the formula comes from
For U = X^a × Y^b the marginal utilities are MUx = a × U ÷ X and MUy = b × U ÷ Y, so MRS = MUx ÷ MUy = (a × Y) ÷ (b × X). Setting that equal to the price ratio Px ÷ Py and cross-multiplying gives a × Py × Y = b × Px × X, which says spending on Y is always b ÷ a times spending on X.
Substitute that into the budget line Px × X + Py × Y = I and you get Px × X × (1 + b ÷ a) = I. Since 1 + b ÷ a = (a + b) ÷ a, this simplifies to Px × X = (a ÷ (a + b)) × I. Divide by Px to get X*, and run the same steps with the goods swapped to get Y*.
What the fixed shares mean for demand
Because spending on X is always a ÷ (a + b) of income, a change in Px changes the quantity of X but never the amount spent on it. In the example, if Px rose from $3 to $4, X* would fall to $60 ÷ $4 = 15 units, spending on X would stay at $60, and Y* would stay at 6. Demand that keeps spending constant as price changes is unit elastic, so Cobb-Douglas demand sits exactly on the dividing line of the total revenue test at every price.
Income works the other way round. A 10% rise in income raises both X* and Y* by exactly 10%, an income elasticity of 1 for each good. The price of Y never appears in the formula for X*, so the cross-price elasticity between the two goods is zero.
Cobb-Douglas Optimum questions
Why does a Cobb-Douglas consumer spend a fixed share of income on each good?
At the tangency, spending on Y always equals b ÷ a times spending on X, whatever the prices are. Combined with the budget line, that pins spending on X at a ÷ (a + b) of income and spending on Y at b ÷ (a + b).
Do the exponents in Cobb-Douglas utility have to add up to 1?
No. Dividing each exponent by their sum gives the budget shares either way. U = X^2 × Y and U = X^(2/3) × Y^(1/3) rank every bundle in the same order, because the second is the cube root of the first, so they pick the same optimal bundle.
How does the Cobb-Douglas optimum relate to the utility-maximizing rule MUx/Px = MUy/Py?
MRS = Px ÷ Py is the same condition as MUx ÷ Px = MUy ÷ Py, just rearranged. At the Cobb-Douglas optimum the marginal utility per dollar is equal across both goods and the whole budget is spent, which is exactly the AP rule.
Can the optimum be a corner solution with Cobb-Douglas utility?
Not when both exponents are positive. Utility is zero whenever either good is zero, so any affordable bundle with some of each good beats a corner, and the formula always returns positive amounts of both goods.
What happens to the optimal bundle when the price of X rises?
X* falls in inverse proportion to Px (doubling Px halves X*), and Y* does not change, because the price of X does not appear in the formula for Y*. Spending on X stays at the same share of income, so the consumer simply buys fewer units of X with it. In the example, raising Px from $3 to $4 cuts X* from 20 to 15 while spending on X stays at $60.
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