§ 2.1-2.4
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Indifference Curves
An indifference curve (IC) shows all bundles $(X, Y)$ that give a consumer exactly the same utility. Geometrically, ICs are level curves of the utility function $U(X,Y)$: just as a topographic map shows lines of equal elevation, each IC traces a line of equal satisfaction. The shape of the IC encodes the entire preference structure.
Key Idea
Well-behaved ICs are downward-sloping (both goods have positive marginal utility), convex toward the origin (diminishing marginal rate of substitution), and never intersect (a direct consequence of transitive preferences).
Economic Intuition - Preference Types & the Shape of ICs
The shape of the IC encodes everything about preferences. For Cobb-Douglas, a controls the spending share on X: a = 0.7 means 70% of income always goes to X regardless of prices - try it. Perfect substitutes have constant MRS = 1 everywhere, so the consumer is always willing to swap one X for one Y. Perfect complements have ICs that are L-shaped: consuming more of one good without the other adds zero utility (like left and right shoes). Moving to a higher IC always means higher utility - so ICs further from the origin are always preferred.
Fig 2.1▸ Indifference Curve Explorer▾
3D Utility Surface
Single IC at U0 (drag P to see MRS)
Preference Type
Cobb-Douglas
Utility Function
U(X,Y) = X^α · Y^(1-α)
IC Shape
Smooth & convex
MRS
Diminishing
The consumer substitutes X for Y at a decreasing rate, convex ICs reflect the preference for variety. Moving to higher ICs (upper-right) means higher utility.
Current utility level
U0 = 2.00
Economic Intuition - Why IC Shape Drives Market Strategy
The shape of indifference curves has direct business implications. When two goods are perfect substitutes (straight-line ICs), consumers buy only the cheapest option - this is exactly the threat private-label brands pose to name brands. Procter & Gamble invests in advertising to shift consumers away from linear ICs toward convex ones, creating a preference for the brand itself. Perfect complements (L-shaped ICs) underlie the razor-and-blade business model: Gillette prices handles near cost and charges a premium on blades because both must be used in fixed proportions. Apple's iPhone/App Store ecosystem follows the same logic. Economic bads (upward-sloping ICs) are central to environmental policy: a longer commute is a bad, so workers require higher wages as compensation - the wage premium reflects the upward slope of their IC over (income, commute time).
Common Mistake
Indifference curves never intersect. Suppose two ICs crossed at bundle A. Then A lies on both curves, meaning the consumer is indifferent between bundles on either curve. But bundles on the outer IC represent strictly more utility, which contradicts the indifference. The formal culprit is transitivity of preferences.
§ 2.5
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The Budget Constraint
The budget constraint $P_X X + P_Y Y = I$ defines the boundary of the affordable set. Its slope $-P_X/P_Y$ is the relative price of $X$ in terms of $Y$: it tells you how many units of $Y$ you must give up to get one more unit of $X$.
Budget Constraint, slope-intercept form
$$Y = \frac{I}{P_Y} - \frac{P_X}{P_Y}\, X$$
Y-intercept: \(\,I / P_Y\)
X-intercept: \(\,I / P_X\)
Slope: \(\,-P_X / P_Y\)
Economic Intuition - LO 5: The Budget Constraint as an Opportunity Set
The slope \(−P_X/P_Y\) is the relative price: how many units of \(Y\) you must give up to get one more unit of \(X\) at market prices. This is the market's trade-off. When income rises, both intercepts move out proportionally - the slope is unchanged because the price ratio is unchanged. When \(P_X\) rises, only the \(X\)-intercept moves inward and the line pivots: \(Y\) becomes relatively cheaper, so the consumer's opportunity set shrinks on the \(X\)-axis.
Fig 2.3▸ Interactive Budget Line▾
The shaded triangle is the affordable set. Use the sliders to change income or prices and watch the intercepts move.
Budget Equation
5X + 4Y = 100
X-intercept \(I/P_X\)
20.00
Y-intercept \(I/P_Y\)
25.00
Slope \(-P_X/P_Y\)
-1.250
Opp. cost of 1 unit of X
1.25 units of Y
Affordable set area
250.00
What moves the line?
Income I → parallel shift. Both intercepts scale; slope unchanged.
PX → pivot around Y-intercept. Slope steepens.
PY → pivot around X-intercept. Slope flattens.
Economic insight
The slope \(-P_X/P_Y\) is the relative price: how many units of Y you give up for one more unit of X. Only a price change alters this trade-off; an income change leaves the slope unchanged.
§ 2.6
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Income & Price Changes
Income changes shift the budget line in parallel (slope unchanged). Price changes pivot it around the fixed intercept.
Economic Intuition - LO 6: Shifts vs. Pivots
An income change is a parallel shift: it multiplies both intercepts by the same factor (1 + Δ%) so the slope −P_X/P_Y is unchanged. A price change pivots the line around the unchanged intercept: raising P_X rotates the line inward around the fixed Y-intercept (I/P_Y stays constant, I/P_X falls). The key test: did the slope change? If yes → price change. If no → income change.
| 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 |
Fig 2.4▸ Shift vs. Pivot Explorer▾
Legend
B₀ Original
B₁ New
Change
Income ↑ 30%
Original slope
−1.25
New slope
−1.25
Fixed intercept
Neither (parallel)
A parallel shift means only income changed. Both intercepts move proportionally; the slope is set by prices alone.
Real-world example
A fuel tax pivots the budget line inward: only the fuel intercept moves and the slope changes. A lump-sum income tax shifts it inward in parallel; both intercepts move equally, slope unchanged. Both raise equal revenue, but only the fuel tax changes relative prices and nudges consumption.
Quick test
Did the slope change? Yes → price change. Did both intercepts move proportionally? Yes → income change. Is one intercept fixed? Yes → that good's price changed.
§ 2.7
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The Consumer's Best Choice
The consumer's optimal bundle satisfies two conditions at once. First, the bundle must lie on the budget line (all income is spent). Second, the indifference curve through that bundle must be tangent to the budget line. At the tangency, the slope of the IC equals the slope of the budget line:
Two Conditions for the Best Affordable Bundle
Condition 1, Budget Exhausted
$$P_X X^* + P_Y Y^* = I$$
Every dollar is spent.
Condition 2, Tangency
$$\frac{P_X}{P_Y} = \frac{MU_X}{MU_Y} = MRS$$
Market trade-off = personal trade-off.
Economic Intuition - LO 7: Why Tangency is Optimal
At any non-tangency point on the budget line, the IC slope (MRS) ≠ budget line slope (P_X/P_Y). If MRS > P_X/P_Y, the consumer values X more than markets do - they should buy more X. They keep adjusting until MRS = P_X/P_Y. At the tangency, the personal willingness to trade (MRS) exactly matches the market's trade-off (price ratio). No further reallocation can improve utility. Watch the figure: the optimal IC is tangent to the budget line; the ICs above are unaffordable; the ICs below are sub-optimal.
Fig 2.5▸ Consumer Optimum, Find the Best Bundle▾
Live Derivation
X* (optimal X)
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Y* (optimal Y)
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U* (max utility)
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IC (below/above)
Optimal IC
Budget line
Optimum
Common Mistake
The tangency condition applies only to smooth, convex ICs. For perfect complements, use the kink condition $X/a = Y/b$. For perfect substitutes, compare $MU_X/P_X$ vs $MU_Y/P_Y$ and buy only the better-value good.
§ 2.8
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Marginal Utility per Dollar
The tangency condition $P_X/P_Y = MU_X/MU_Y$ can be rearranged to show the same optimality in a different form. Dividing both sides by prices gives the marginal utility per dollar condition:
Marginal Utility per Dollar Condition (rearranged tangency)
$$\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}$$
Utility per dollar must be equalized across all goods at the optimum
Key Idea
If $MU_X/P_X > MU_Y/P_Y$, spending one less dollar on $Y$ and one more on $X$ raises utility. The consumer keeps reallocating until both ratios are equal. This equimarginal principle extends far beyond consumer theory: it governs optimal advertising budgets, hospital resource allocation, and government program spending.
Economic Intuition - LO 8: The Equimarginal Principle in Business
The bang-for-the-buck condition $MU_X/P_X = MU_Y/P_Y$ is the foundation of rational resource allocation wherever budgets are constrained. A coffee shop owner splitting $10,000 between espresso machine upgrades and barista training should keep spending on whichever gives more revenue per dollar, stopping only when the marginal return per dollar is equal across both investments - that is the condition satisfied by the optimum in the figure. Netflix applies the same logic across content categories: drama, comedy, documentary each get funding up to the point where the last dollar spent generates equal viewer-minutes per dollar. A hospital system allocating staff across departments faces the identical problem. The great insight of consumer theory is that this is not a special business rule - it is the universal condition for optimality whenever resources are scarce and alternatives exist.
Fig 2.6▸ Tangency Explorer: Drag the Bundle▾
MU(X) / P(X), bang for buck on X
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MU(Y) / P(Y), bang for buck on Y
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Verdict
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Action
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Drag the orange Bundle anywhere, even below the budget line. When it's below the constraint, arrows show how to reach the budget line (orange) then slide to the optimum (red). At the optimum both MU/P ratios are equal.
Cobb-Douglas Demand Functions
For $U(X,Y) = X^\alpha Y^\beta$, the optimal demand functions take a clean closed form. A consumer with Cobb-Douglas preferences always spends a fixed fraction of income on each good, regardless of prices. The income shares $\alpha/(\alpha+\beta)$ and $\beta/(\alpha+\beta)$ are constant.
Cobb-Douglas Demand Functions
$$X^* = \frac{\alpha}{\alpha+\beta} \cdot \frac{I}{P_X} \qquad Y^* = \frac{\beta}{\alpha+\beta} \cdot \frac{I}{P_Y}$$
The spending share \(\alpha/(\alpha+\beta)\) on good \(X\) is constant, independent of income and prices
Application
When economists report that "households spend roughly 30% of income on housing regardless of income level," they're observing Cobb-Douglas behaviour. The spending share $\alpha/(\alpha+\beta)$ is constant, independent of prices or income.
Economic Intuition - The Income Expansion Path & Engel’s Law
For Cobb-Douglas preferences the income expansion path is always a straight ray through the origin: as income doubles, both X* and Y* exactly double and the bundle moves proportionally outward. This is what economists call homotheticity. The fixed spending-share property ($\alpha/(\alpha+\beta)$ on X, always) is its signature: no matter how rich or poor the consumer, the fraction of income allocated to each good stays constant. When a government macro-model assumes that households spend a fixed share of income on food regardless of income level, they are embedding Cobb-Douglas preferences.
§ 2.10
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MRS Intuition: Watch It Fall Along the Curve
The MRS is the slope of the indifference curve at any point. For a Cobb-Douglas consumer, MRS falls as you move right along the curve (diminishing MRS). Drag the orange point to see the tangent, the local trade-off triangle, and the live MRS value update in real time.
Fig 2.8▸ MRS Intuition Slider▾
Point P on IC
(x, y)
MRS at P = MUx/MUy
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Tangent slope (= -MRS)
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What the triangle shows
As you drag P right, the tangent flattens. The green triangle shows the local ΔX and ΔY at P: how many units of Y the consumer gives up for one more X. MRS = ΔY/ΔX at that point. Diminishing MRS means this ratio shrinks as X grows.
MRS magnitude (flat left, steep right)
§ 2.11
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Why Indifference Curves Cannot Cross
Indifference curves never intersect. If they did, the transitivity of preferences would be violated. Drag the inner IC upward until the two curves cross, then watch the logical contradiction appear.
Fig 2.9▸ Why ICs Cannot Cross▾
Inner IC lowMove up to cross
Drag the slider right to move the inner IC. When the two curves intersect, a logical contradiction will appear here.
The Contradiction
If IC1 and IC2 cross at point A: bundle B on IC2 satisfies A ~ B (same utility as A). Bundle C on IC1 satisfies A ~ C. By transitivity, B ~ C. But B has strictly more of both goods than C, which is impossible under monotone preferences.
§ 2.12
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Corner Solution Explorer: Perfect Substitutes
With perfect substitutes, the optimal bundle jumps from one axis to the other as the price ratio crosses the MRS threshold. Adjust Px and watch the optimal bundle animate from one corner to the other. The knife-edge case where the entire budget line is optimal is highlighted.
Fig 2.10▸ Corner Solution Explorer▾
MUx/Px (bang per $ on X)
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MUy/Py (bang per $ on Y)
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MRS threshold (a/b)
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Optimal bundle
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Adjust Px to see the bundle switch corners.