EC224 Week 2 — Indifference Curves, Budget Constraints & Consumer Optimum

Learning Objectives

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Each objective states what you will be able to do after completing this chapter.

LO1

Identify and sketch the four IC types and explain the logic behind each shape.

§2.1 · §2.2 · §2.3 · §2.4 · §2.5 · §2.6

LO2

Derive the budget constraint; identify slope, intercepts, and the affordable set.

§2.7 · §2.8

LO3

Predict and diagram how income changes (shift) and price changes (pivot) move the budget line.

§2.9

LO4

State and apply the two conditions for an interior optimum: budget exhaustion and tangency.

§2.10

LO5

Use the bang-for-the-buck condition to assess and correct a consumer's allocation.

§2.11

LO6

Solve algebraically for the optimal bundle under Cobb-Douglas and perfect-complement preferences.

§2.12

LO7

Select the correct optimisation method (tangency, corner, or kink) for any preference type.

§2.10 · §2.12

🗺 Chapter Concept Map

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The map below shows how all topics in this chapter connect. Each node links to the section where it is developed in detail.

Notation Used in This Chapter

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Symbol	Meaning	Symbol	Meaning
X, Y	Quantities of goods X and Y	U(X,Y)	Utility function
P_x, P_y	Prices of goods X and Y	MU_x	Marginal utility of X
I	Consumer income	MU_y	Marginal utility of Y
X, Y	Optimal quantities	MRS	Marginal rate of substitution
IC	Indifference curve	BC	Budget constraint

§ 2.1Recap: What Is an Indifference Curve?

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§ 2.1 — Recap: What Is an Indifference Curve?

In the previous week, we introduced the concept of consumer preferences and indifference curves. To briefly recap: a consumer who faces two goods, X and Y, can be described by an indifference curve — a set of all bundles (X,Y) that provide the consumer with exactly the same level of utility (satisfaction). The consumer is, by definition, indifferent among all bundles on the same curve.

Several key properties hold for well-behaved preferences:

Downward sloping: since both goods are desirable, giving up some Y must be compensated by more X to keep utility constant.
Convex: the consumer prefers variety, so they are willing to give up less and less of Y as they consume more X.
Non-crossing: two indifference curves representing different utility levels can never intersect.
Higher is better: moving outward from the origin (upper right) reaches higher utility levels.

Why This Matters

Indifference curves are the economist's way of representing preferences without asking people to assign dollar values to their satisfaction. Businesses use similar logic — through market research and conjoint analysis — to understand how customers trade off product features such as price, quality, and convenience. The same framework that describes a student choosing between food and clothing underlies how airlines design fare classes and how phone manufacturers bundle features.

✏️ Check Your Understanding

Name two properties that all well-behaved indifference curves share, and explain in one sentence why each property makes economic sense.
True or false: a consumer can be indifferent between a bundle with more of both goods and one with less of both goods. Justify your answer.

§ 2.2Types of Goods and Their Indifference Curves

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§ 2.2 — Types of Goods and Their Indifference Curves

The standard convex indifference curve is the right shape for a typical consumer who values both goods and prefers variety. But economics deals with many kinds of goods, and the shape of the indifference curve changes depending on the relationship between them. We examine four important cases.

Mini Case Study: Maya's Weekly Budget

Consider Maya, a college student with $60 per week to spend on coffee and textbooks. She values both, but they are neither perfect substitutes (she cannot read a cup of coffee) nor perfect complements (she can enjoy coffee without a textbook in hand). Her preferences are well-behaved: she prefers more of each, she prefers variety, and her willingness to give up coffee for an extra textbook declines as she already has many textbooks. Her indifference curves are the standard convex shape — downward sloping and bowing toward the origin. The concepts in this section explain how Maya's optimal choice changes if her budget grows or if textbook prices change.

Summary of IC Shapes by Good Type

Good Type	IC Shape	MRS
Standard (both goods desired)	Downward-sloping, convex	Diminishing
Perfect substitutes	Straight line, slope −a/b	Constant = a/b
Perfect complements	L-shaped (kink on ratio ray)	0, ∞, or undefined
Neutral good (Y useless)	Vertical line	∞
Economic bad (Y bad)	Upward sloping	Negative

§ 2.3Perfect Substitutes

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§ 2.3 — Perfect Substitutes

Definition: Perfect Substitutes

Perfect Substitutes: Two goods are perfect substitutes if the consumer regards them as identical — they serve the same function perfectly and the consumer is always willing to trade one for the other at a constant ratio.

The classic example is gasoline from two neighboring stations: Exxon gas and Mobil gas. For most consumers, a gallon of gasoline is a gallon of gasoline, regardless of brand.

Consider a consumer starting at bundle A = (2 Mobil, 2 Exxon) — two gallons from each station, four gallons total.

Bundle B = (3 Mobil, 1 Exxon): still four gallons total — same utility as A.
Bundle C = (4 Mobil, 0 Exxon): still four gallons total — same utility as A.

Because the consumer only cares about total quantity, all these bundles lie on the same IC. The willingness to trade is constant — always one-for-one. This constant trade-off rate means the IC is a straight line with slope −1. More generally, if the consumer trades a units of Y for b units of X, the IC has slope −a/b.

The utility function for perfect substitutes takes the linear form:

Utility Function — Perfect Substitutes

U(X,Y) = aX + bY MRS = a/b (constant)(1)

Fig. 1

Figure 1. Perfect substitutes: Exxon vs.

Key Idea

Because the consumer treats the two goods as identical, no amount of X or Y already consumed changes how willing they are to swap. The MRS is fixed. This is why the IC is a straight line rather than the bowed shape we see for ordinary goods.

Key feature: The ICs are straight, parallel, downward-sloping lines. The MRS is constant — the consumer is always willing to trade the two goods at the same fixed rate, regardless of how much of each they already have.

Application 2.1: Store Brands and the Perfect-Substitute Threat

Retailers face perfect-substitute logic every time they launch a private-label product. If consumers genuinely view a store-brand paper towel as identical to the name-brand version, the IC between the two is a straight line with slope −1. The consumer will always choose whichever is cheaper — the budget line will be tangent to the IC at a corner, buying only the cheaper brand.

This is why Procter & Gamble and Unilever invest heavily in advertising: the goal is not just awareness but preference differentiation — shifting consumers away from a linear IC toward a bowed one, so that they prefer a mix rather than the cheapest-only corner solution.

Try It

A consumer has utility U(X,Y) = 2X + 3Y. What is the MRS? Draw one indifference curve for U = 12.
If two goods are perfect substitutes, will the consumer ever buy both simultaneously? Under what special condition might they?

✏️ Check Your Understanding

A consumer has utility U(X,Y) = 2X + 3Y. What is the MRS? Draw one indifference curve for U = 12.
If two goods are perfect substitutes, will the consumer ever buy both simultaneously? Under what special condition might they?

§ 2.4Perfect Complements

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§ 2.4 — Perfect Complements

Definition: Perfect Complements

Perfect Complements: Two goods are perfect complements if they must be consumed together in fixed proportions in order to generate utility. Extra units of one good, without the corresponding extra units of the other, provide no additional utility.

The canonical example is left shoes and right shoes. A consumer with 2 left shoes and 2 right shoes has 2 functional pairs.

Bundle A = (2 left, 2 right): 2 functional pairs.
Bundle B = (3 left, 2 right): still only 2 functional pairs — the extra left shoe is useless without a matching right shoe. Same utility as A.
Bundle C = (4 left, 2 right): still 2 functional pairs. Same utility.

The IC through A = (2,2) is L-shaped: a vertical arm at X = 2 and a horizontal arm at Y = 2. The kink is at (2,2). More generally, for goods consumed in fixed proportions a:b:

Utility Function — Perfect Complements

U(X,Y) = min(X/a, Y/b)(2)

For equal proportions (a = b = 1): U = min(X,Y).

Fig. 2

Figure 2. Perfect complements: left shoes and right shoes.

Key Idea

The kink of the L is the only point on each IC where both goods are being "used." Anywhere else on the L, one good is in excess. Extra units of the surplus good add no utility.

Key feature: L-shaped ICs with the kink on the ray X/a = Y/b. The MRS is not defined at the kink, and is either 0 (horizontal arm) or ∞ (vertical arm).

Application 2.2: The Razor-and-Blade Business Model

Perfect-complement preferences underlie one of the most successful pricing strategies in business history. Gillette sells razors cheaply (near cost) and charges a premium on replacement blades. Printer companies do the same with ink cartridges. Apple's ecosystem — iPhone and App Store — exhibits the same logic. When two goods are perfect complements, the consumer must buy them in a fixed ratio. A firm that controls supply of one complement can therefore extract value through the other. This is the foundation for understanding why firms in technology, gaming (consoles and games), and healthcare (devices and consumables) adopt two-part pricing structures.

✏️ Check Your Understanding

A consumer has U(X,Y) = min(X, 2Y). Sketch two indifference curves and identify the kink points. Where do all kinks lie?
Why is the MRS undefined at the kink of an L-shaped IC? What does this mean for applying the tangency condition?

§ 2.5Neutral (Useless) Good

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§ 2.5 — Neutral (Useless) Good

Definition: Neutral Good

Neutral Good: A good is neutral if it provides zero marginal utility to the consumer — consuming more or less of it leaves utility completely unchanged. Examples include class notes for a course you have never taken, or a book in a language you do not speak.

Suppose good Y is neutral and good X is a normal good. Starting at bundle A = (2,2):

Moving to (2,3): added one unit of Y — utility unchanged.
Moving to (2,10): Y is still neutral — utility unchanged.
Any bundle of the form (2, Y) for any Y ≥ 0 has the same utility as A.

The IC through A is therefore a vertical line at X = 2. Utility only changes when X changes, so utility increases as you move horizontally to the right.

Fig. 3

Figure 3. Neutral (useless) good: vertical indifference curves.

✏️ Check Your Understanding

For the utility function U(X,Y) = X, verify that Y is neutral by computing ∂U/∂Y.
A consumer has U(X,Y) = X. If their income doubles, does their demand for Y change? Explain without algebra.

§ 2.6Economic Bads

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§ 2.6 — Economic Bads

Definition: Economic Bad

Economic Bad: A good is an economic bad if consuming more of it reduces the consumer's utility. It confers negative marginal utility — also called disutility. Classic examples: pollution, noise, commute time.

Suppose good Y is an economic bad (e.g., air pollution) and good X is a normal good (e.g., food). Starting at bundle A = (2,2):

If we add more Y (more pollution), the consumer is worse off.
To compensate and keep utility constant, we must give them more X.
Moving along the IC requires moving to the upper right: more X to compensate for more Y.

The IC through A is therefore upward sloping. Higher utility is reached by moving right (more X) or downward (less of the bad Y).

Fig. 4

Figure 4. Economic bad: upward-sloping indifference curves.

Why This Matters — Policy Application

Economic bads appear constantly in policy debates. When a government evaluates a highway expansion, residents near the road experience noise and pollution as economic bads. To keep utility constant, they need compensation — more of some other good (income, parks, soundproofing). Understanding ICs for bads helps economists design sensible compensation policies and value environmental amenities.

✏️ Check Your Understanding

Commute time is an economic bad for most workers. If a firm offers a pay rise to compensate employees for a longer commute, what does this say about the shape of their ICs over income and commute time?
For U(X,Y) = X − Y² (where Y ≥ 0), show that Y is a bad by computing ∂U/∂Y.

§ 2.7The Budget Constraint

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§ 2.7 — The Budget Constraint

Indifference curves describe what the consumer wants. But wanting something is not enough — the consumer also faces a resource constraint. This is where the budget constraint (BC) comes in.

Suppose a consumer has income I and faces prices P_x for good X and P_y for good Y. If the consumer spends all of their income:

The Budget Constraint

I = P_x · X + P_y · Y(3)

Rearranging so that Y is on the left gives us the slope-intercept form:

Slope-Intercept Form

Y = I/P_y − (P_x/P_y) · X(4)

From this form, we can read off the key properties:

Y-intercept (X = 0): Y_max = I/P_y — the maximum amount of Y if all income is spent on Y.
X-intercept (Y = 0): X_max = I/P_x — the maximum amount of X if all income is spent on X.
Slope = −P_x/P_y: the opportunity cost of X in terms of Y — for every extra unit of X, the consumer gives up P_x/P_y units of Y.

Fig. 5

Figure 5. The budget constraint.

Key Idea

The budget constraint separates the feasible from the infeasible. Any bundle on or below the line is affordable; any bundle above the line requires more money than the consumer has. The line itself represents all the ways the consumer can spend every last dollar.

Worked Example: Sheela's Budget Constraint

Sheela's Budget Constraint — I = $100, P_x = $5, P_y = $2 ▾

Step 1 — Maximum quantities

X_max = I/P_x = 100/5 = 20 Y_max = I/P_y = 100/2 = 50

Step 2 — Budget constraint equation

100 = 5X + 2Y

Step 3 — Slope

Slope = −P_x/P_y = −5/2 = −2.5

In words: For every additional unit of food Sheela buys, she must give up 2.5 units of clothing.

Step 4 — Draw the line

Plot (0, 50) on the Y-axis and (20, 0) on the X-axis; connect them with a straight line.

Try It

Suppose Sheela's income rises to $150 but prices stay the same. Without calculating, predict what happens to the budget line. Then verify by computing the new intercepts.

Answer: both intercepts scale by 1.5; slope unchanged.

✏️ Check Your Understanding

A consumer has I = $120, P_x = $4, P_y = $6. Write the budget constraint, find both intercepts, and state the slope. What is the opportunity cost of one unit of X?
True or false: if P_x and P_y both double while income stays constant, the budget line shifts inward in parallel. Justify carefully.

§ 2.8Changes in the Budget Constraint

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§ 2.8 — Changes in the Budget Constraint

Income Increases: Parallel Outward Shift

When income increases from $100 to $150 while P_x = 5 and P_y = 2 remain the same:

Y-intercept = I/P_y: increases from 50 to 75.
X-intercept = I/P_x: increases from 20 to 30.
Slope = −P_x/P_y = −5/2: unchanged (prices did not change).

Both intercepts increase proportionally and the slope is unchanged. The budget line shifts outward in parallel.

Key Idea

Income change ⇒ parallel shift of budget line. The slope is set by prices alone, so as long as prices do not change, a higher income simply slides the entire line outward. Income up: BC shifts outward. Income down: BC shifts inward.

Price Change: Pivoting the Budget Line

When P_x decreases from $5 to $4, while income = $100 and P_y = $2 remain the same:

Y-intercept = I/P_y = 100/2 = 50: unchanged (neither I nor P_y changed).
X-intercept = I/P_x: increases from 100/5 = 20 to 100/4 = 25.
Slope = −P_x/P_y: changes from −2.5 to −2 (becomes flatter).

The line pivots outward around the fixed Y-intercept. The X-intercept moves outward; the slope changes.

Key Idea

Own-price change ⇒ pivot of budget line. The intercept on the other good's axis is fixed; only the intercept on the good whose price changed moves. P_x falls: pivot outward (flatter). P_x rises: pivot inward (steeper).

Common Mistake: Confusing Shifts with Pivots

Students often draw a parallel shift when a price changes, or a pivot when income changes. The rule is simple: income changes both intercepts proportionally and leaves the slope unchanged (parallel shift); a price change fixes one intercept and moves the other, changing the slope (pivot). Quick check: if the fixed intercept has moved, it is not a pivot.

Quick-Reference: What Moves on the Budget Line?

Change	Y-intercept	X-intercept	Slope	Type
Income ↑	Moves out	Moves out	Fixed	Parallel shift out
Income ↓	Moves in	Moves in	Fixed	Parallel shift in
P_x ↓	Fixed	Moves out	Flatter	Pivot out
P_x ↑	Fixed	Moves in	Steeper	Pivot in
P_y ↓	Moves out	Fixed	Steeper	Pivot out
P_y ↑	Moves in	Fixed	Flatter	Pivot in
All prices ×k	Moves in	Moves in	Fixed	≡ income ÷k

Application 2.3: Fuel Tax vs. Income Tax — Same Revenue, Different Behaviour

Suppose the government needs to raise $500 per household in revenue. It can do so either by levying a tax on petrol (pivoting the budget line inward around the non-petrol intercept) or by reducing all households' income by $500 (a parallel shift inward). Both policies raise the same revenue. But they affect consumer choices very differently: the fuel tax changes relative prices and discourages petrol consumption specifically; the income tax leaves relative prices intact and reduces consumption of all goods proportionally.

This is why environmental economists prefer fuel taxes to income taxes when they want to reduce emissions — the pivot, not the shift, does the work.

✏️ Check Your Understanding

Sheela has I = $100, P_x = $5, P_y = $2. Draw the original budget line. Now P_y rises to $4. Which intercept is fixed? Redraw and state the new slope.
A student claims: "doubling all prices is the same as halving my income." Is she correct? Use the budget constraint equation to verify.

§ 2.9Determining the Optimal Consumption Bundle

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§ 2.9 — Determining the Optimal Consumption Bundle

We now have two pieces: (1) indifference curves, which describe what the consumer wants, and (2) the budget constraint, which describes what they can afford. The consumer's goal is:

Choose the bundle that provides the highest utility while remaining within the budget.

Why interior points are not optimal. Consider a bundle A that lies strictly inside the budget constraint. The consumer is spending less than their income — some income is unspent. But since both goods are desirable, spending that remaining income on more X or Y would push the consumer to a higher IC. Therefore, the optimal bundle must lie on the budget line.

Why crossing is not optimal. Consider bundle B on the budget line, where an IC crosses (rather than is tangent to) the budget line. At a crossing, between the two intersection points there are affordable bundles that lie above the IC through B. The consumer can do better than B by moving along the budget line. Therefore, B is not optimal.

The tangency point is optimal. Point C is where the budget line is tangent to an IC. At this point: (1) the consumer is spending all income, and (2) there is no affordable bundle on a higher IC. Point C is the optimal consumption bundle.

The Two Conditions for Optimality

From the geometric analysis, two conditions must hold at the optimal bundle:

Condition 1 — Budget Exhaustion

P_x · X* + P_y · Y* = I(5)

Condition 2 — Tangency Condition

P_x/P_y = MU_x/MU_y(6)

In words: the market's trade-off between the two goods (the price ratio) must equal the consumer's personal trade-off (the MRS). If they differ, the consumer can always rearrange spending to reach a higher IC.

Key Idea

The tangency condition says: at the optimum, the market's exchange rate between X and Y (the price ratio) exactly equals the consumer's personal exchange rate (the MRS). If they differed, the consumer could rearrange spending to get more utility without spending more money.

Common Mistake: Using Tangency for All Preference Types

The tangency condition holds only for smooth, strictly convex ICs (like Cobb-Douglas). For perfect complements, the optimum is at the kink of the L, where the MRS is undefined. For perfect substitutes, the optimum is typically at a corner. Always check the type of preferences before applying the tangency condition mechanically.

Which Method Do I Use? A Decision Guide

Selecting the Correct Optimisation Method

Smooth, strictly convex IC
(e.g. Cobb-Douglas)

Use tangency condition: P_x/P_y = MU_x/MU_y, plus budget exhaustion.

Perfect complements
U = min(X/a, Y/b)

Use kink condition: X/a = Y/b, plus budget exhaustion. Tangency does not apply.

Perfect substitutes
U = aX + bY

Compare MU_x/P_x vs. MU_y/P_y. Buy only the better-value good (corner solution).

✏️ Check Your Understanding

A consumer is at a bundle where MU_x/P_x > MU_y/P_y and the budget is exhausted. Are they at the optimum? If not, how should they adjust?
Explain in one sentence why a rational consumer with positive marginal utilities for both goods will never choose a bundle strictly inside the budget constraint.

§ 2.10Bang for the Buck

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§ 2.10 — Bang for the Buck

The optimality condition can be rearranged to yield a very intuitive interpretation. Rearranging P_x/P_y = MU_x/MU_y:

Bang for the Buck Condition

MU_x/P_x = MU_y/P_y(7)

What does each side mean?

MU_x is the extra utility from consuming one more unit of X — the benefit of X.
P_x is the cost of one unit of X — the cost of X.
MU_x/P_x is the "utility per dollar" spent on X — the bang for the buck from X.

The optimality condition says: at the optimal bundle, the utility per dollar spent is the same for every good. If this were not so — say, MU_x/P_x > MU_y/P_y — the consumer could increase total utility by spending one less dollar on Y and one more dollar on X.

Why This Matters

The bang-for-the-buck condition applies far beyond individual consumers. A firm deciding how to allocate a marketing budget across TV and digital advertising should equalize the return per dollar across channels. A hospital system allocating resources across departments, or a government allocating spending across programs, faces the same logic. Wherever resources are scarce and alternatives exist, equating marginal return per dollar is the path to efficiency.

Application 2.4: Marketing Budget Allocation

A coffee shop owner has $10,000 to split between espresso machine upgrades and barista training. The last $1,000 spent on machines increases daily throughput worth $80. The last $1,000 spent on training improves customer satisfaction, increasing repeat visits worth $130 per $1,000 over a quarter. Since the return per dollar is higher for training ($130) than for machines ($80), the owner should shift spending toward training — and keep reallocating until marginal returns per dollar are equalized. This is the bang-for-the-buck condition applied directly to a capital budgeting decision.

§ 2.11Finding the Optimal Bundle: Worked Examples

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§ 2.11 — Finding the Optimal Bundle: Worked Examples

Amanda's Optimal Consumption Bundle ▾

Amanda has income I = $200. P_x = $4 (food), P_y = $5 (clothing). Utility: U(X,Y) = √(XY) = X^1/2Y^1/2.

Part (a) — Budget line and slope

200 = 4X + 5Y Slope = −P_x/P_y = −4/5

Part (b) — Marginal utilities

MU_x = ½ X^−1/2Y^1/2 = ½√(Y/X) MU_y = ½ X^1/2Y^−1/2 = ½√(X/Y)

Part (c) — MRS

MRS = MU_x/MU_y = Y/X (the slope of the IC at any bundle)

Part (d) — Optimal bundle (P_x = 4, P_y = 5)

Condition 1 (budget): 200 = 4X + 5Y (i)

Condition 2 (tangency): 4/5 = Y/X ⇒ Y = 4X/5 (ii)

Substitute (ii) into (i): 200 = 4X + 5·(4X/5) = 4X + 4X = 8X ⇒ X* = 25

Y* = 4(25)/5 = 20

Verification: 4(25) + 5(20) = 100 + 100 = 200 = I ✓

Part (e) — New optimal bundle when P_x = $2

New budget: 200 = 2X + 5Y. New tangency: 2/5 = Y/X ⇒ Y = 2X/5

200 = 2X + 5·(2X/5) = 2X + 2X = 4X ⇒ X* = 50, Y* = 20

Observation: When the price of food fell from $4 to $2, Amanda doubled her food consumption (25 → 50). Her clothing consumption stayed at 20. This reflects the equal-expenditure-share property of √(XY): exactly half of income is always spent on each good.

Economic Insight

When two goods are consumed under Cobb-Douglas preferences, a fall in the price of one good raises its consumption but leaves consumption of the other good completely unchanged. The freed-up spending goes entirely into the cheaper good — a result that turns out to be useful for understanding demand elasticities in Chapter 3.

Cobb-Douglas Demand Functions (General Case) ▾

Generalising: U(X,Y) = X^αY^β with budget P_xX + P_yY = I.

Derivation

MU_x = αX^α−1Y^β MU_y = βX^αY^β−1

Tangency: MU_x/MU_y = P_x/P_y ⇒ αY/(βX) = P_x/P_y ⇒ Y = (βP_x)/(αP_y) · X

Substituting into budget constraint and solving:

General Cobb-Douglas Demand Functions

X* = [α/(α+β)] · I/P_x Y* = [β/(α+β)] · I/P_y

Interpretation: The consumer spends fraction α/(α+β) of income on X and fraction β/(α+β) on Y. For Amanda (α = β = 1/2): X* = I/(2P_x), Y* = I/(2P_y) — exactly half of income on each.

Economic Insight: Fixed Expenditure Shares

Chapter at a Glance

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Your one-page revision guide — everything you need before an exam or tutorial.

Key Equations

Name	Formula
Budget constraint	P_xX + P_yY = I
Slope of BC	−P_x/P_y
Tangency condition	P_x/P_y = MU_x/MU_y
Bang for the buck	MU_x/P_x = MU_y/P_y
Cobb-Douglas demand	X* = [α/(α+β)] · I/P_x

IC Shapes at a Glance

Preference	IC Shape	MRS
Standard	Convex, downward	Diminishing
Substitutes	Straight line	Constant
Complements	L-shaped	0/∞/undef.
Neutral Y	Vertical	∞
Bad Y	Upward sloping	Negative

Which Method to Use?

Smooth convex IC (Cobb-Douglas): use tangency P_x/P_y = MU_x/MU_y
Perfect complements: use kink condition X/a = Y/b
Perfect substitutes: compare MU/P; buy only the better-value good

Five Key Terms

IC — bundles giving equal utility
Budget constraint — all affordable bundles
MRS — consumer's personal trade-off rate
Tangency condition — market rate = MRS at optimum
Bang for the buck — MU/P equalised at optimum

Key Terms

Perfect Substitutes

Two goods that serve identical functions and can be traded at a constant ratio. ICs are straight, parallel, downward-sloping lines.

Perfect Complements

Two goods consumed in fixed proportions. Extra units of one without the other provide no utility. ICs are L-shaped.

Neutral Good

A good providing zero marginal utility. Extra units leave utility unchanged. ICs are vertical lines.

Economic Bad

A good conferring negative marginal utility. More of it reduces utility. ICs are upward sloping.

Budget Constraint

The equation P_xX + P_yY = I describing all bundles the consumer can afford when spending exactly all income.

Slope of the Budget Line

−P_x/P_y: the opportunity cost of good X in terms of Y. For every extra unit of X, P_x/P_y units of Y must be given up.

Parallel Shift

What happens when income changes (prices fixed): both intercepts move proportionally, slope unchanged.

Pivot (Rotation)

What happens when a price changes: one intercept is fixed, the other moves, and the slope changes.

Tangency Condition

At the optimum: slope of BC = slope of IC, i.e. P_x/P_y = MU_x/MU_y.

Bang for the Buck

The utility per dollar spent: MU_x/P_x for good X. At the optimum, MU_x/P_x = MU_y/P_y.

MRS

Marginal Rate of Substitution: the rate at which the consumer willingly trades Y for X at constant utility. Equals MU_x/MU_y in magnitude.

Cobb-Douglas Utility

U(X,Y) = X^αY^β. Generates smooth, convex ICs. Consumer spends fixed fraction α/(α+β) of income on X.

Supplementary Video Resources

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Watch in order for the best results. Each video is directly aligned to the sections listed.

Video 1 · ~9 min

Indifference Curves and the MRS

Jacob Clifford (ACDC Economics). Covers IC definition, shape, downward slope, convexity, and the MRS. Directly aligned to §2.1–§2.6. Ideal as a visual complement before attempting practice questions.

▶ Watch on YouTube

Video 2 · ~5 min

Budget Constraints

Marginal Revolution University (Tyler Cowen & Alex Tabarrok). Concise explanation of the BC equation, intercepts, slope, and affordable set. Directly reinforces §2.7–§2.8.

▶ Watch on YouTube

Video 3 · ~10 min

Consumer Optimum: Putting It Together

Khan Academy. Combines ICs and the budget constraint to locate the optimal bundle graphically and algebraically. Walks through the tangency condition intuitively. Reinforces §2.9–§2.10.

▶ Watch on YouTube

Video 4 · ~12 min

Cobb-Douglas Utility: Deriving Demand

Economics in Many Lessons. Derives Cobb-Douglas demand functions step by step using the tangency condition, with numerical examples that closely mirror Worked Example 2 in §2.11. Recommended for students wanting extra algebra practice.

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📝 Practice Questions (Q1–Q27) →

All 27 questions with full solutions — Level 1, 2 & 3