Difficulty: Intermediate | Prerequisites: Utility Theory, Indifference Curves, Budget Sets notes; basic calculus (partial derivatives)
This is where the whole consumer theory toolkit comes together. You take a utility function, calculate the MRS, set it equal to the price ratio, and combine with the budget constraint to find the bundle that maximises utility. If you can do this problem type fluently, you can handle nearly any consumer choice question on an introductory microeconomics exam.
To find the utility-maximising bundle, calculate the MRS (ratio of marginal utilities), set it equal to the price ratio, and solve simultaneously with the budget constraint. If the MRS does not equal the price ratio at a given bundle, the consumer can do better by trading one good for the other.
Marginal rate of substitution of apples for bananas, MRS(a, b)
The rate at which the consumer is willing to trade bananas for apples while staying equally satisfied. Calculated as MU_a / MU_b. In simple terms, it answers: "How many bananas is the consumer willing to give up for one more apple?"
Partial derivative
The derivative of a multivariable function with respect to one variable, holding the others constant. Used to find marginal utility: MU_a = dU/da.
Tangency condition
The optimality condition where the slope of the indifference curve (MRS) equals the slope of the budget line (price ratio): MRS = P_a / P_b. At this point, the consumer's personal trade-off matches the market trade-off.
Cobb-Douglas utility
A utility function of the form U = A * x^alpha * y^beta (plus possibly a constant). The problem's U(a,b) = 7a^(1/3)b^(1/4) + 4 is Cobb-Douglas in structure. These functions always produce interior solutions with smooth, convex indifference curves.
Optimal bundle
The bundle of goods that gives the highest utility the consumer can afford. Found by solving the tangency condition and the budget constraint simultaneously.
U(a, b) = 7a^(1/3) * b^(1/4) + 4.
MU_a = dU/da = 7 * (1/3) * a^(-2/3) * b^(1/4) = (7/3) * a^(-2/3) * b^(1/4).
MU_b = dU/db = 7 * (1/4) * a^(1/3) * b^(-3/4) = (7/4) * a^(1/3) * b^(-3/4).
MRS(a, b) = MU_a / MU_b = [(7/3) * a^(-2/3) * b^(1/4)] / [(7/4) * a^(1/3) * b^(-3/4)].
Simplify: MRS = (4/3) * (b/a).
The 7 and the +4 in the utility function cancel out and do not affect the MRS. This is a useful shortcut to remember.
(A) a = 4, b = 3: MRS = (4/3)(3/4) = 1.
(B) a = 6, b = 18: MRS = (4/3)(18/6) = (4/3)(3) = 4.
The price ratio is P_a / P_b = 3/1 = 3.
At bundle (A): MRS = 1, but the price ratio = 3. Since MRS < P_a/P_b, the consumer values an extra apple less than the market charges for it. She should trade apples for bananas (buy fewer apples, more bananas).
At bundle (B): MRS = 4, and the price ratio = 3. Since MRS > P_a/P_b, the consumer values an extra apple more than it costs. She should trade bananas for apples (buy more apples, fewer bananas).
Neither bundle is optimal.
3a + 1b = 21, or b = 21 - 3a.
Set MRS = price ratio: (4/3)(b/a) = 3.
Solve: b/a = 9/4, so b = (9/4)a.
Substitute into the budget constraint: 3a + (9/4)a = 21.
Combine: (12/4)a + (9/4)a = (21/4)a = 21.
a = 4.
b = (9/4)(4) = 9.
The utility-maximising bundle is a = 4 apples, b = 9 bananas.
Check: 3(4) + 1(9) = 12 + 9 = 21. The budget is fully spent.
MU_a = \frac{7}{3} \, a^{-2/3} \, b^{1/4}MU_b = \frac{7}{4} \, a^{1/3} \, b^{-3/4}MRS(a,b) = \frac{MU_a}{MU_b} = \frac{4b}{3a}\text{Tangency: } \frac{4b}{3a} = \frac{P_a}{P_b} = 3 \implies b = \frac{9}{4}a\text{Budget: } 3a + b = 21Substitute to get a = 4, b = 9.
The tangency condition describes how rational consumers allocate spending at the margin. If you have ever compared whether an extra hour of study is worth more than an extra hour of sleep, you are implicitly weighing marginal rates of substitution against the "price" (time cost) of each activity.
Students often forget to take the ratio of marginal utilities and instead set the marginal utilities equal to each other. The MRS is MU_a / MU_b, not MU_a = MU_b.
When MRS > price ratio, students sometimes think the consumer should buy fewer apples. The logic runs the other way: MRS > P_a/P_b means the consumer values apples more than the market does, so she should buy more apples.
Students frequently forget to check their answer against the budget constraint. Always verify that the total spend equals income.
The +4 in the utility function does not affect the MRS or the optimal bundle. Students sometimes try to include it in the derivatives. A constant drops out when you differentiate.
"Calculate the MRS" and "find the optimal bundle" are the two most common computational questions in consumer choice. You need to be able to do both quickly.
When given a bundle and asked whether the consumer is optimising, compare MRS to P_a/P_b. If they are not equal, state which direction the consumer should trade.
Always show your budget constraint check at the end. Markers look for it.
For Cobb-Douglas utility, the optimal spending share on each good is determined by the exponents. This is a powerful shortcut: the consumer spends a fraction alpha/(alpha + beta) of income on good a.
Fill in the blank: MRS(a, b) for U = 7a^(1/3)b^(1/4) + 4 simplifies to ______.
(4/3)(b/a), or equivalently 4b/(3a).
True or false: At the optimal bundle, MRS = P_b / P_a.
False. MRS = P_a / P_b (the price of the good on the horizontal axis divided by the price of the good on the vertical axis).
Fill in the blank: If MRS > P_a/P_b, the consumer should buy more ______ and fewer ______.
More apples, fewer bananas.
True or false: The constant +4 in U(a,b) = 7a^(1/3)b^(1/4) + 4 changes the optimal bundle.
False. A constant drops out in differentiation and does not affect the MRS or the optimum.
The optimal bundle is a = ______, b = ______.
a = 4, b = 9.
Q: Given U(a, b) = 7a^(1/3)b^(1/4) + 4, calculate MRS(a, b) and evaluate it at (a = 4, b = 3) and (a = 6, b = 18).
A: MU_a = (7/3)a^(-2/3)b^(1/4). MU_b = (7/4)a^(1/3)b^(-3/4). MRS = MU_a/MU_b = (4/3)(b/a). At (4, 3): MRS = (4/3)(3/4) = 1. At (6, 18): MRS = (4/3)(18/6) = 4.
Q: P_a = 3, P_b = 1. Is the consumer maximising utility at (4, 3)? At (6, 18)? What should she do in each case?
A: Price ratio = 3. At (4, 3): MRS = 1 < 3, so the consumer values apples less than the market price implies. She should trade apples for bananas. At (6, 18): MRS = 4 > 3, so she values apples more than their relative cost. She should trade bananas for apples.
Q: Write the budget constraint and find the utility-maximising bundle.
A: Budget constraint: 3a + b = 21. Tangency condition: (4/3)(b/a) = 3, giving b = (9/4)a. Substitute: 3a + (9/4)a = 21, so (21/4)a = 21, a = 4. Then b = (9/4)(4) = 9. Optimal bundle: 4 apples, 9 bananas. Check: 3(4) + 9 = 21.
Q: Why does the +4 in the utility function not matter for finding the optimal bundle?
A: Adding a constant to a utility function is a monotonic transformation that preserves the ranking of all bundles. When you differentiate to find marginal utilities, the constant drops out (its derivative is zero). The MRS, the tangency condition, and therefore the optimal bundle are all unaffected.
This topic ties together everything from the first three sets of notes. The MRS comes from the utility function (Topic 1), the shape of the indifference curve reflects preferences (Topic 2), and the budget constraint defines what is affordable (Topic 3). From here, the course typically moves to demand curves: the optimal bundle at each price traces out the consumer's demand function.
MRS, marginal rate of substitution, marginal utility, partial derivative, tangency condition, optimal bundle, utility maximisation, Cobb-Douglas, consumer choice, budget constraint, price ratio, interior solution, apples and bananas, consumer equilibrium, Lagrangian, constrained optimisation, Intro to Microeconomics, NYU Econ