Source: Chapter 3, Consumer Behavior (Section 3.3)
Tags: consumer choice, utility maximisation, optimal basket, tangency condition, corner solution, interior solution, MRS equals price ratio, perfect substitutes optimum, perfect complements optimum, ECON 323
Consumer choice combines preferences (indifference curves) with the budget constraint to find the basket that maximises utility. For smooth, convex preferences the optimum is an interior solution where MRS equals the price ratio. For perfect substitutes the consumer typically spends everything on one good (corner solution). For perfect complements the optimum sits at the kink where the two goods are consumed in their fixed proportion.
Utility maximisation
Choosing the market basket on (or within) the budget constraint that reaches the highest possible indifference curve.
Interior solution
An optimum where the consumer buys positive amounts of both goods. Occurs with smooth, convex indifference curves.
Corner solution
An optimum where the consumer spends all income on just one good. Typical with perfect substitutes when MRS ≠ P_X / P_Y.
Tangency condition
At an interior optimum, the indifference curve is tangent to the budget line: MRS_XY = P_X / P_Y.
Equal marginal utility per dollar
An equivalent way to state the tangency condition: MU_X / P_X = MU_Y / P_Y. Each good delivers the same additional satisfaction per pound spent.
When indifference curves are smooth and convex (the standard case), the utility-maximising basket has two features:
It sits on the budget line. The consumer spends all income: P_X × x + P_Y × y = I.
The indifference curve is tangent to the budget line. MRS_XY = P_X / P_Y.
Because MRS_XY = MU_X / MU_Y, the tangency condition can also be written as:
MU_X / MU_Y = P_X / P_Y
Or equivalently, MU_X / P_X = MU_Y / P_Y (marginal utility per dollar is equalised across goods).
Intuition for the equal-marginal-utility-per-dollar rule:
If MU_X / P_X > MU_Y / P_Y, the consumer gets more bang for the buck from X. They should buy more X and less Y.
If MU_X / P_X < MU_Y / P_Y, do the opposite.
At the optimum, neither reallocation helps.
Worked example:
U(x, y) = x^0.5 × y^0.5
MU_X = 0.5x^(−0.5) y^(0.5), MU_Y = 0.5x^(0.5) y^(−0.5)
MRS_XY = y/x
Prices: P_X = 10, P_Y = 2, Income = 100.
Solve the two-equation system:
Tangency: y/x = 10/2 = 5, so y = 5x.
Budget: 10x + 2y = 100.
Substituting y = 5x into the budget line: 10x + 2(5x) = 100, giving 20x = 100, so x = 5 and y = 25. The optimal basket is (5, 25).
With U(x, y) = ax + by, the MRS is constant and equal to a/b. Three sub-cases arise:
MRS > P_X / P_Y: the consumer values X more (per dollar) than the market charges. Spend everything on X. Optimal basket: (I / P_X, 0).
MRS < P_X / P_Y: the consumer values Y more per dollar. Spend everything on Y. Optimal basket: (0, I / P_Y).
MRS = P_X / P_Y: the indifference curves and the budget line have the same slope. Every basket on the budget line is optimal.
Example: U(x, y) = 5x + 4y, so MRS = 5/4.
If P_X / P_Y < 5/4, buy only X.
If P_X / P_Y > 5/4, buy only Y.
If P_X / P_Y = 5/4, any point on the budget line works.
With U(x, y) = min{ax, by}, goods are consumed in fixed proportions. The kinks of the L-shaped indifference curves lie along the line ax = by (equivalently y = (a/b)x).
To find the optimum, solve two equations simultaneously:
The kink condition: y = (a/b)x (or whatever proportion is given).
The budget line: P_X × x + P_Y × y = I.
Example: kinks lie on y = 2x, budget constraint is 2x + 4y = 60.
Substitute y = 2x into the budget line: 2x + 4(2x) = 60, so 10x = 60, x = 6, y = 12. Optimal basket: (6, 12).
The exam may present an indifference map graphically (without a utility function) and ask you to identify the point on the budget line that touches the highest indifference curve. The approach is visual: trace the budget line and find where it just touches (is tangent to, or sits at the kink of) the highest reachable curve. Corner solutions are possible here too, where the budget line endpoint sits on a higher curve than any interior tangency would.
Optimality conditions (interior solution):
P_X × x + P_Y × y = I
MRS_XY = P_X / P_Y (equivalently MU_X / P_X = MU_Y / P_Y)
Perfect substitutes (U = ax + by):
MRS = a/b
If a/b > P_X/P_Y → corner at (I/P_X, 0)
If a/b < P_X/P_Y → corner at (0, I/P_Y)
If a/b = P_X/P_Y → entire budget line is optimal
Perfect complements (U = min{ax, by}):
Kink line: ax = by → y = (a/b)x
Solve kink line + budget line simultaneously
⚠️ The tangency condition MRS = P_X / P_Y only identifies an interior optimum. Always check whether a corner solution might dominate, especially with linear or unusual utility functions.
⚠️ MRS = MU_X / MU_Y is a mathematical identity that holds everywhere on the map. MRS = P_X / P_Y is the optimality condition that holds only at the best affordable basket.
⚠️ For perfect substitutes, know all three sub-cases. Exams love asking what happens when the price ratio changes relative to the constant MRS.
⚠️ For perfect complements, the optimum is always at the kink. Tangency does not apply because the indifference curve is not smooth there.
⚠️ "Marginal utility per dollar should be equal for both goods" is the intuitive version of the tangency condition. Be comfortable using either form.
⚠️ If an exam problem gives you an indifference map without a formula, find the optimum graphically: the highest curve the budget line can reach.
Q: State the two conditions that characterise the utility-maximising basket for smooth, convex preferences.
A: (1) The basket is on the budget line: P_X x + P_Y y = I. (2) The indifference curve is tangent to the budget line: MRS_XY = P_X / P_Y.
Q: A consumer has U(x, y) = xy, P_X = 4, P_Y = 1, I = 40. Find the optimal basket.
A: MRS_XY = y/x. Tangency: y/x = 4/1, so y = 4x. Budget: 4x + y = 40. Substituting: 4x + 4x = 40, x = 5, y = 20. Optimal basket: (5, 20).
Q: U(x, y) = 3x + 2y, P_X = 6, P_Y = 5. What is the optimal basket?
A: MRS = 3/2 = 1.5. Price ratio P_X/P_Y = 6/5 = 1.2. Since MRS > P_X/P_Y, the consumer values X more per dollar than the market charges. Spend everything on X: (I/6, 0). (The exact quantity depends on income.)
Q: U(x, y) = min{2x, y}. Budget constraint: 3x + 6y = 90. Find the optimum.
A: Kinks lie where 2x = y. Substitute y = 2x into the budget line: 3x + 6(2x) = 90, so 15x = 90, x = 6, y = 12. Optimal basket: (6, 12).
Q: Why does the tangency condition not work for perfect complements?
A: The indifference curve has a kink (it is not differentiable) at the optimal point, so there is no well-defined slope to set equal to the price ratio. Instead, you use the kink condition alongside the budget line.
Q: If MU_X / P_X > MU_Y / P_Y at the current basket, what should the consumer do?
A: Buy more of X and less of Y. The last dollar spent on X delivers more utility than the last dollar spent on Y, so reallocating towards X raises total utility.
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