Difficulty: Intermediate | Prerequisites: Basic consumer theory, indifference curves, budget constraints (ECON 102 or equivalent)
Cobb-Douglas utility functions are the workhorse of intermediate microeconomics. They model consumers who view goods as imperfect substitutes and always spend a fixed share of their income on each good. If you can solve a Cobb-Douglas optimisation problem, you can handle roughly half the exam questions in a typical ECON 302 course. This topic builds directly on budget constraints and indifference curve analysis from introductory micro.
A consumer with Cobb-Douglas preferences (e.g. U = X^(1/2) · Y^(1/2)) always spends a constant fraction of income on each good, regardless of prices. The optimal bundle is found by setting MRS equal to the price ratio, then substituting into the budget constraint. Total utility at the optimum is just the utility function evaluated at the optimal quantities.
Cobb-Douglas utility function
A utility function of the form U(X, Y) = X^a · Y^b, where a and b are positive constants. The exponents determine how much weight the consumer places on each good. In simple terms, it describes a consumer who likes both goods and is willing to trade between them at a rate that depends on how much of each they already have.
Marginal rate of substitution (MRS)
The rate at which a consumer is willing to give up Good Y in exchange for one more unit of Good X, holding utility constant. For Cobb-Douglas U = X^a · Y^b, MRS = (a/b) · (Y/X). Think of it as the consumer's personal exchange rate between the two goods at any given bundle.
Budget constraint
The equation Px · X + Py · Y = M, where Px and Py are prices and M is income. It defines all the bundles a consumer can just afford. In simple terms, it is the line showing every combination of X and Y that exactly uses up the consumer's money.
Optimal bundle (utility-maximising bundle)
The combination of goods that gives the highest utility the consumer can reach on their budget constraint. Found where MRS = Px/Py (the indifference curve is tangent to the budget line). Think of it as the best deal the consumer can get with the money they have.
Numeraire good
A good whose price is set to 1 so that all other prices and income are measured in units of that good. In simple terms, it is the baseline good that turns the Y-axis into "money left over for everything else."
Interior solution
An optimum where the consumer buys positive amounts of both goods. Cobb-Douglas preferences always yield interior solutions because the indifference curves never touch the axes.
The consumer has utility U(X, Y) = X^(1/2) · Y^(1/2), income M, and faces prices Px and Py.
The budget constraint is Px · X + Py · Y = M.
Because Cobb-Douglas indifference curves are smooth and convex, and never touch the axes, the optimum is always interior (both goods consumed in positive amounts).
Step 1: Compute the MRS. For U = X^(1/2) · Y^(1/2), MRS = Y/X.
Step 2: Set MRS = Px/Py. This gives Y/X = Px/Py, so Y = (Px/Py) · X.
Step 3: Substitute into the budget constraint. Px · X + Py · (Px/Py) · X = M becomes 2Px · X = M, so X* = M / (2Px).
Step 4: Back-substitute to get Y* = M / (2Py).
The key insight: the consumer spends exactly half of income on each good. This constant-share property holds for any Cobb-Douglas function. For U = X^a · Y^b, the consumer spends a/(a+b) of income on X and b/(a+b) on Y.
Given: U = X^(1/2) · Y^(1/2), M = $24, Px = $1, Py = $1.
X* = 24 / (2 · 1) = 12, Y* = 24 / (2 · 1) = 12.
Optimal bundle: (12, 12).
Plug the optimal quantities back into the utility function.
U(12, 12) = 12^(1/2) · 12^(1/2) = (12 · 12)^(1/2) = 144^(1/2) = 12.
Total utility = 12.
If Px rises to $4 (with M = $24 and Py = $1 unchanged):
X** = 24 / (2 · 4) = 3
Y** = 24 / (2 · 1) = 12
New bundle: (3, 12). The consumer buys much less X but the same amount of Y.
New utility: U(3, 12) = 3^(1/2) · 12^(1/2) = 36^(1/2) = 6. Utility falls from 12 to 6.
General Cobb-Douglas optimal demands:
For U = X^a · Y^b with budget Px · X + Py · Y = M:
X* = (a / (a + b)) · (M / Px)
Y* = (b / (a + b)) · (M / Py)
Symmetric case (a = b = 1/2):
X* = M / (2Px)
Y* = M / (2Py)
MRS for U = X^(1/2) · Y^(1/2):
MRS = Y / X
Tangency condition:
MRS = Px / Py (at the optimal bundle, the slope of the indifference curve equals the slope of the budget line)
Students often think the MRS is the same as the price ratio at all points. The MRS equals the price ratio only at the optimum. Everywhere else on the indifference curve, the consumer's personal trade-off differs from the market trade-off.
Students sometimes forget that Cobb-Douglas always gives an interior solution. You will never get a corner solution (consuming zero of one good) with this utility function, so do not waste time checking corners.
A common error is dividing income by price alone (X* = M/Px) instead of applying the expenditure-share formula. That gives the maximum possible quantity of X, not the optimal quantity.
Students occasionally confuse "utility" with "income" or "expenditure." Utility is ordinal satisfaction from consuming the bundle. It is not measured in dollars unless the problem explicitly sets it up that way.
⚠️ The constant expenditure-share result is the single most tested property of Cobb-Douglas. If U = X^a · Y^b, spending on X is always a/(a+b) of income. Memorise this.
⚠️ Cobb-Douglas problems are the starting point for nearly every compensating variation, equivalent variation, and income/substitution effect question. Get the optimal bundle right here and the rest follows.
⚠️ Expect to be asked: "What happens to the optimal bundle when Px changes?" The answer for Cobb-Douglas: X changes, Y does not (when the other price and income are held constant, the expenditure-share formula shows Y* = M/(2Py), which has no Px in it).
True or False: For U = X^(1/2) · Y^(1/2), the consumer always spends exactly half of income on Good X.
Fill in the blank: If M = $40, Px = $2, Py = $1, and U = X^(1/2) · Y^(1/2), then X* = ____.
True or False: Doubling both prices and income leaves the optimal bundle unchanged for Cobb-Douglas preferences.
Fill in the blank: The MRS for U = X^(1/2) · Y^(1/2) at the bundle (4, 16) is ____.
True or False: A corner solution is possible with Cobb-Douglas utility.
Answers: 1. True. 2. X* = 40/(2·2) = 10. 3. True (the formula M/(2Px) is homogeneous of degree zero in M and prices). 4. MRS = 16/4 = 4. 5. False.
Q: A consumer has utility U(X, Y) = X^(1/2) · Y^(1/2), income M = $50, Px = $5, and Py = $2. What is the optimal bundle?
A: X* = M/(2Px) = 50/10 = 5. Y* = M/(2Py) = 50/4 = 12.5. The optimal bundle is (5, 12.5).
Q: Using the same setup, what is the consumer's total utility at the optimum?
A: U(5, 12.5) = 5^(1/2) · 12.5^(1/2) = (62.5)^(1/2) ≈ 7.91.
Q: If Px doubles from $5 to $10 while M and Py stay the same, what is the new optimal bundle?
A: X** = 50/(2·10) = 2.5. Y** = 50/(2·2) = 12.5. The bundle shifts to (2.5, 12.5). Notice that Y does not change.
Q: Why does the quantity of Y stay the same when Px changes in the Cobb-Douglas case?
A: Because the demand for Y is Y* = M/(2Py), which contains no Px term. The expenditure share on Y is fixed at 50% of income, so changes in the other good's price do not affect it. This is a special feature of Cobb-Douglas preferences and does not hold for utility functions in general.
This material connects directly to compensating variation and equivalent variation, which measure how much a price change hurts or helps the consumer in dollar terms. It also connects to the Slutsky decomposition of a price change into substitution and income effects, both of which are computed starting from the Cobb-Douglas optimal bundle. If you are studying perfect complements (Leontief preferences), note the contrast: Cobb-Douglas consumers always substitute between goods, while Leontief consumers never do.
Cobb-Douglas, utility maximisation, optimal bundle, MRS, marginal rate of substitution, tangency condition, budget constraint, expenditure share, constant budget share, interior solution, numeraire, composite good, intermediate microeconomics, ECON 302, consumer theory