Lecture 5 · Week 5 · Besanko & Braeutigam ch. 4 · about 60 min

Consumer Choice: the Optimal Bundle

Putting preferences and the budget line together: why the best affordable bundle is where an indifference curve is tangent to the budget line, how to solve for it step by step, the Lagrange method, and the corner solutions of perfect substitutes, perfect complements and quasilinear preferences.

By the end you can

Explain with a diagram why the optimum lies on the budget line, on the highest reachable indifference curve
State the tangency condition MRS = Px/Py and its 'utility per lira' form
Solve for the optimal bundle with the two-equation method and with the Lagrangian
Use the Cobb-Douglas shortcut and know when it does not apply
Find corner solutions for perfect substitutes and quasilinear utility, and the kink solution for perfect complements

Key idea · Exam recipe (memorise this)

Write $MU_x$ , $MU_y$ and set $MU_x/MU_y = P_x/P_y$ .
Solve it for $y$ in terms of $x$ (board: $y/x = 0.5 \Rightarrow x = 2y$ ).
Put that into $P_x x + P_y y = I$ and solve. Check the budget adds up.
Cobb-Douglas $x^ay^b$ : $x^* = \frac{a}{a+b}\frac{I}{P_x}$ . Substitutes $ax + by$ : buy only the higher $MU/P$ . Complements $\min(ax, by)$ : $ax = by$ .
Graph (board letters): budget line from $I/P_y$ to $I/P_x$ ; $U_1$ cuts it at B and C; $U_2$ touches it at the optimum A; D on $U_3$ is out of reach. Dashed lines from A to the axes, and write “at A: slope of budget line = slope of indifference curve”.

How the exam asks it: four core exam templates

In consumer choice exams, questions follow one of four canonical structures: interior tangency with Cobb-Douglas preferences, testing off-optimum bundles (like board points B and C), corner solutions with perfect substitutes, or kink solutions with Leontief complements.

Exam template 1: Interior optimum and off-optimum evaluation (the board example)

Question: A consumer has utility $U(x, y) = xy$ , income $I = 800$ dollars, and faces prices $P_x = 20$ dollars and $P_y = 40$ dollars.
a) Derive the tangency condition and express $x$ in terms of $y$ .
b) Solve for the optimal consumption bundle (point A on the board).
c) Compute the maximum utility level achieved at point A.
d) Basket B = (8, 16) lies on the budget line. Explain why basket B is not the optimal choice.

Exam template 2: Cobb-Douglas general derivation and shortcut

Question: A consumer has utility $U = x^{0.4}y^{0.6}$ , income $I = 300$ dollars, and faces prices $P_x = 5$ dollars and $P_y = 6$ dollars.
a) Find the marginal utilities $MU_x$ and $MU_y$ .
b) Derive the tangency condition and express $y$ in terms of $x$ .
c) Solve for the optimal bundle $(x^*, y^*)$ using the budget constraint.
d) Verify the result using the Cobb-Douglas expenditure share shortcut.

Exam template 3: Perfect substitutes and corner solutions

Question: A consumer views two sizes of bottled water as perfect substitutes: $U(x, y) = x + 2y$ , where $x$ represents 500 ml bottles and $y$ represents 1 L bottles. Income is $I = 96$ TL, $P_x = 3$ TL, and $P_y = 8$ TL.
a) Calculate the marginal utility per lira for each good.
b) Determine the utility-maximising bundle $(x^*, y^*)$ and state whether this is an interior or corner solution.
c) Compute the maximum utility achieved.

Exam template 4: Perfect complements (kink solution)

Question: A consumer has fixed-proportions (Leontief) utility $U(x, y) = \min(2x, y)$ . Income is $I = 800$ TL, $P_x = 20$ TL, and $P_y = 40$ TL.
a) State the condition for avoiding waste at the optimum and derive the optimal expansion ray.
b) Solve for the utility-maximising bundle $(x^*, y^*)$ .
c) Compute the maximum utility level achieved.

Cheat-sheet & the 4-step method

· Cheat-sheet: consumer optimum by preference type

Preference Type	Utility Function $U(x, y)$	Optimum Condition	When to Use	How to Write the Exam Answer
Cobb-Douglas	$x^a y^b$	$MRS = \dfrac{a}{b}\dfrac{y}{x} = \dfrac{P_x}{P_y}$ and $P_x x + P_y y = I$	Smooth interior solutions	State $MU_x, MU_y$ ; set $MRS = P_x/P_y \implies$ substitute into budget; verify expenditure shares
Cobb-Douglas Shortcut	$x^a y^b$	$x^* = \dfrac{a}{a+b}\dfrac{I}{P_x}$ , $y^* = \dfrac{b}{a+b}\dfrac{I}{P_y}$	Multiple choice or checking work	State share formulas and compute quantities in one line
Perfect Substitutes	$ax + by$	Compare $\dfrac{a}{P_x}$ vs $\dfrac{b}{P_y}$	Linear indifference curves	If $a/P_x \gt b/P_y$ buy only $x$ ( $I/P_x, 0$ ); if $a/P_x \lt b/P_y$ buy only $y$ ( $0, I/P_y$ )
Perfect Complements	$\min(ax, by)$	Ray $ax = by$ and $P_x x + P_y y = I$	L-shaped indifference curves	Set $ax = by$ ; substitute into budget line; no calculus
Quasilinear	$v(x) + y$	$v'(x) = \dfrac{P_x}{P_y}$ ; $y = \dfrac{I - P_x x^*}{P_y}$	Linear in numeraire $y$	Tangency pins down $x^$ ; if $P_x x^ \le I$ interior $(x^, y^)$ ; if $P_x x^* \gt I$ corner $(I/P_x, 0)$

Key idea · The 4-step method for any consumer choice question

Step 1: Identify the preference structure.
- Is it Cobb-Douglas ( $x^a y^b$ ), substitutes ( $ax+by$ ), complements ( $\min(ax, by)$ ), or quasilinear ( $v(x)+y$ )?
Step 2: Apply the appropriate optimality condition.
- Cobb-Douglas / Quasilinear: set $MRS = MU_x/MU_y = P_x/P_y$ .
- Perfect substitutes: compare utility per dollar $MU_x/P_x$ versus $MU_y/P_y$ .
- Perfect complements: set the arguments equal ( $ax = by$ ).
Step 3: Combine with the budget constraint.
- Substitute the relationship between $x$ and $y$ into $P_x x + P_y y = I$ .
- Solve for the numerical values of $x^*$ and $y^*$ .
Step 4: Check budget balance and conclude in bold.
- Verify that $P_x x^* + P_y y^* = I$ and neither quantity is negative.
- Calculate maximum utility $U^*$ and state the final result in a bold exam sentence.

The consumer’s problem

Lecture 3 gave us what the consumer wants (indifference curves), Lecture 4 what they can afford (the budget line). The optimal choice puts the two together. It is a constrained optimization problem, exactly as in Lecture 1:

The consumer's problemKey formulaFormula sheet →

\max_{x,\,y}\ U(x, y) \quad \text{subject to} \quad P_x x + P_y y = I

$U(x,y)$: the objective function (utility)
$P_x x + P_y y \le I$: the constraint (budget line, spent in full at the optimum)
$x, y$: endogenous choice variables
$P_x, P_y, I$: exogenous

Use it when you are asked for the utility-maximising (optimal) bundle, also called the consumer's optimum or equilibrium.

We keep the board notation: food $x$ on the horizontal axis, clothing $y$ on the vertical axis, income $I$ .

The picture: why the optimum is a tangency

Step through the diagram. It uses the board budget line ( $I = 800$ , $P_x = 20$ , $P_y = 40$ ) and the utility function $U = xy$ .

Three things are true at the optimum:

It is on the budget line, not inside it. With more-is-better preferences, leftover money could buy more of something and raise utility.
It is on the highest indifference curve the budget allows. Curves further out ( $U_3$ ) are better but unaffordable. Curves that cut the line ( $U_1$ at B and C) can be beaten by moving along the line.
At an interior optimum the indifference curve is tangent to the budget line, so their slopes are equal.

The tangency condition

The lecturer derives the condition on the board in three lines. Learn it in this order: it is a typical “show that” question.

Why the optimal choice is not point B

The lecturer’s board example: $P_x = 20$ , $P_y = 40$ , $I = 800$ , $U = xy$ , and B = (8, 16). B is on the budget line ( $20 \cdot 8 + 40 \cdot 16 = 160 + 640 = 800$ ), so it is affordable, but it lies on the lower curve ( $U = 8 \cdot 16 = 128$ ).

$U = xy$ gives $MU_x = y$ and $MU_y = x$ . At B: $MU_x = 16$ , $MU_y = 8$ .
$MRS = MU_x/MU_y = 16/8 = 2$ , but $P_x/P_y = 20/40 = 0.5$ . They are not equal: no tangency at B, so B is not an optimal choice.
Per lira: $MU_x/P_x = 16/20 = 0.8$ on food, $MU_y/P_y = 8/40 = 0.2$ on clothing. Food gives four times as much utility per lira.
Move 40 lira from clothing to food: give up 1 clothing (utility falls by about 8) and buy 2 food (utility rises by about $2 imes 16 = 32$ ). Spending is unchanged, utility rises.
So the consumer moves down the budget line from B toward A = (20, 10), where $MRS = 10/20 = 0.5 = P_x/P_y$ and $U = 200 \gt 128$ .

Point C = (32, 4) is on the same curve and the mirror image: $MRS = 4/32 = 0.125 \lt 0.5$ ( $MU_x/P_x = 0.2 \lt MU_y/P_y = 0.8$ ), so the consumer moves up the line toward A by buying more clothing and less food.

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}

$MU_x/P_x$: extra utility from the last lira spent on food
$MU_y/P_y$: extra utility from the last lira spent on clothing

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

Solving the board example

The Cobb-Douglas shortcut

Cobb-Douglas demand shortcutShortcutFormula sheet →

x^* = \frac{a}{a+b}\cdot\frac{I}{P_x} \qquad y^* = \frac{b}{a+b}\cdot\frac{I}{P_y}

$a, b$: the exponents in U = x^a y^b
$a/(a+b)$: the share of income spent on x

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.

The consumer spends a fixed share $a/(a+b)$ of income on $x$ and $b/(a+b)$ on $y$ , whatever the prices. With $U = xy$ ( $a = b = 1$ ) that is half and half: 400 dollars each in the board example, so $x^* = 400/20 = 20$ and $y^* = 400/40 = 10$ .

The Lagrange method

Some lecturers and Besanko and Braeutigam’s appendix solve the same problem with a Lagrangian. It gives exactly the same answer and is useful when the algebra gets messy.

Lagrangian for the consumerFormula sheet →

\Lambda(x, y, \lambda) = U(x, y) + \lambda\,(I - P_x x - P_y y)

$\lambda$: the Lagrange multiplier, the marginal utility of income (extra utility from one more lira)

Use it when you want a systematic way to solve the constrained problem: set the three partial derivatives equal to zero.

Try it: move the optimum

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 the spending share on each good never changes when income or prices change: only the quantities do. This “constant expenditure share” property is special to Cobb-Douglas.

Corner solutions

Sometimes there is no tangency with positive amounts of both goods, and the consumer buys only one good. This is a corner solution. The tangency condition does not hold there; instead, compare utility per lira.

Corner solution testFormula sheet →

\frac{MU_x}{P_x} \gt \frac{MU_y}{P_y} \text{ at every bundle} \ \Rightarrow\ x^* = \frac{I}{P_x},\ y^* = 0

$MU_x/P_x, MU_y/P_y$: utility per lira on each good

Use it when the MRS never equals the price ratio (perfect substitutes), or the tangency point would need a negative quantity.

Perfect substitutes

Quasilinear utility

With quasilinear utility the tangency gives a fixed amount of $x$ , whatever the income. If income is too small to afford it, the consumer ends up at a corner.

Perfect complements

With L-shaped indifference curves there is no tangency (the MRS is not defined at the corner). The optimum is at the kink, on the budget line.

Exam practice

Summary and review

The consumer maximises $U(x, y)$ subject to $P_x x + P_y y = I$ .
The optimum is on the budget line and on the highest reachable indifference curve.
Interior optimum: $MRS_{xy} = P_x/P_y$ , equivalently $MU_x/P_x = MU_y/P_y$ . Solve it together with the budget line.
If $MRS \gt P_x/P_y$ buy more $x$ ; if $MRS \lt P_x/P_y$ buy more $y$ .
Cobb-Douglas $U = x^ay^b$ : $x^* = \frac{a}{a+b}\frac{I}{P_x}$ , $y^* = \frac{b}{a+b}\frac{I}{P_y}$ (constant spending shares).
Lagrangian: $\Lambda = U + \lambda(I - P_xx - P_yy)$ ; $\lambda$ is the marginal utility of income.
Perfect substitutes: buy only the good with the higher $MU/P$ (corner).
Perfect complements $\min(ax, by)$ : $ax = by$ plus the budget line (kink).
Quasilinear: tangency fixes $x$ ; with too little income the consumer is at a corner.