Lecture 4 · Week 5 · Besanko & Braeutigam ch. 4 · about 50 min

Consumer Choice

Putting preferences and prices together: the budget line, the tangency condition that pins down the utility-maximising bundle, and how to solve for it algebraically for Cobb-Douglas preferences.

By the end you can

Write the budget line equation and identify its slope and intercepts
State and interpret the tangency condition for an interior optimum
Solve for the utility-maximising bundle given prices, income and a Cobb-Douglas utility function
Recognise a corner solution and explain why the tangency condition can fail to apply

The budget line

A consumer’s budget line shows every bundle of two goods that exactly exhausts a fixed income at given prices.

Budget lineKey formulaFormula sheet →

P_X X + P_Y Y = M \quad\Longrightarrow\quad Y = \frac{M}{P_Y} - \frac{P_X}{P_Y}X

$M$: income
$P_X, P_Y$: prices of the two goods
$-P_X/P_Y$: the slope of the budget line

Use it when you need the set of affordable bundles, its intercepts (M/P_X on the X-axis, M/P_Y on the Y-axis), or its slope.

The tangency condition

A rational consumer with an interior optimum (some of both goods) chooses the bundle where the budget line is tangent to the highest attainable indifference curve.

Tangency (interior optimum) conditionKey formulaFormula sheet →

MRS_{XY} = \frac{P_X}{P_Y} \quad\Longleftrightarrow\quad \frac{MU_X}{MU_Y} = \frac{P_X}{P_Y} \quad\Longleftrightarrow\quad \frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}

$MU_X/P_X$: MU_Y/P_Y = utility per lira spent, equalised across both goods at the optimum

Use it when you are solving for a utility-maximising bundle with an interior solution (positive amounts of both goods).

Solving for the optimal bundle: Cobb-Douglas

Cobb-Douglas demand shortcutShortcutFormula sheet →

X^* = \frac{aM}{(a+b)P_X} \qquad Y^* = \frac{bM}{(a+b)P_Y}

$a, b$: the exponents in U = X^aY^b

Use it when you recognise Cobb-Douglas utility and want to skip the tangency derivation. With a=0.4, b=0.6 above: X^* = 0.4(300)/5 = 24, matching the long derivation exactly.

Try it: watch the tangency point move

budget lineindifference curve at the optimum

Horizontal: good X. Vertical: good Y. The optimum is where the indifference curve is tangent to the budget line.

Price of X2.00Price of Y2.00Income M100

X* (= 0.5M/Px): 25.00
Y* (= 0.5M/Py): 25.00
Spending on X: 50.0
Spending on Y: 50.0

For Cobb-Douglas utility with equal exponents, the consumer always spends exactly half of income M on each good: $P_X X^* = P_Y Y^* = 0.5M = 50.0$ , regardless of the prices.

For Cobb-Douglas preferences with equal exponents, the optimal spending share on each good never changes when income or prices change — only the quantities do. This “constant expenditure share” property is a distinctive feature of Cobb-Douglas utility, not a general law of consumer choice.

Corner solutions

Sometimes the tangency condition simply has no solution with positive amounts of both goods — the consumer optimally buys only one of the two goods. This happens with perfect substitutes whenever the goods’ price ratio does not match the consumer’s fixed MRS.

Exam practice

Summary and review

Budget line: $P_XX + P_YY = M$ ; slope $-P_X/P_Y$ ; intercepts $M/P_X$ and $M/P_Y$ .
Interior optimum (tangency): $MRS_{XY} = P_X/P_Y$ , equivalently $MU_X/P_X = MU_Y/P_Y$ .
For Cobb-Douglas $U = X^aY^b$ : $X^* = aM/[(a+b)P_X]$ , $Y^* = bM/[(a+b)P_Y]$ .
The Cobb-Douglas shortcut only applies to Cobb-Douglas utility; other forms need the full tangency derivation.
A corner solution (only one good purchased) arises when the price ratio never matches the MRS, most commonly with perfect substitutes.