Chapter 21: The Theory of Consumer Choice

Budget constraints, indifference curves, and the geometry behind every demand curve

Section 1

The Budget Constraint — What You Can Afford

Every consumer faces a simple fact: income is limited. If you spend more on one thing, you must spend less on another. The budget constraint is the line on a graph that shows every bundle of two goods the consumer can just afford to buy with her entire income.

To keep things simple, Mankiw's example uses two goods: pizza (at $10) and Pepsi (at $2), with an income of $1,000 per month. Put pizza on the horizontal axis and Pepsi on the vertical axis.

Income equation Px·X + Py·Y = I
Horizontal intercept X = I / Px all income on pizza → 1000/10 = 100 pizzas
Vertical intercept Y = I / Py all income on Pepsi → 1000/2 = 500 liters
Slope −Px / Py −10/2 = −5 liters of Pepsi per pizza

The slope is the relative price — the opportunity cost of one good in units of the other. Giving up one pizza frees up $10, which buys 5 extra liters of Pepsi. That trade-off is exactly what the market is offering the consumer.

Income = $1,000. Px = $10, Py = $2. Every bundle on the line costs exactly $1,000; every bundle below is affordable but leaves money on the table.

Shifts vs. pivots: Income changes shift the budget line in or out in a parallel way (the slope, which depends only on relative prices, doesn't change). A price change pivots the line around the unchanged intercept — the relative price flips, so the slope flips too.

Section 2

Preferences — What You Want

The budget constraint tells us what a consumer can afford. To figure out what she will buy, we need to know her preferences. We represent preferences with indifference curves — each curve connects every bundle that gives the same level of satisfaction.

The slope of the indifference curve at any point is called the marginal rate of substitution (MRS). It measures how much Pepsi the consumer would accept in exchange for giving up one pizza and still feel equally happy. Since she likes both goods, the curve slopes downward.

MRS = |slope of indifference curve| = MUx / MUy

Four Properties of Indifference Curves

1. Higher is better I2 > I1 more of both goods = more satisfaction
2. Slope down Less of X → more of Y to stay equally happy
3. Never cross No two ICs intersect would be a logical contradiction
4. Bowed inward MRS decreases as X rises scarce goods are worth more

Property 4 is the most interesting. If you already have lots of pizza and little Pepsi, you'd gladly give up a pizza for just a little Pepsi — your MRS is small. But if you only have a tiny bit of pizza, giving up one is very costly, so you demand a lot of Pepsi in return — your MRS is large. That's what makes the curve bow toward the origin.

Points A and B are equally satisfying (both sit on I₁). Point D is strictly preferred because it sits on the higher curve I₂ — more of both goods.

Section 3

Two Extreme Cases

The shape of an indifference curve reveals how substitutable the two goods are. Two extreme cases are worth knowing — they come up on exams and in real-world applications.

Perfect Substitutes

Straight lines

Nickels and dimes. You only care about total value, so you'd always trade 2 nickels for 1 dime. MRS is constant, and the indifference curves are straight lines.

Perfect Complements

Right angles (L-shapes)

Left shoes and right shoes. A 5th left shoe is useless without a 5th right shoe. The ICs are L-shaped: extra of one good beyond a match adds no satisfaction at all.

(a) Perfect substitutes: constant trade-off, straight-line ICs. (b) Perfect complements: kinked at fixed ratios, L-shaped ICs.

Most real goods sit between these extremes. Coke and Pepsi are close to substitutes (bowed but almost straight). Skis and ski bindings are close to complements (bowed but almost L-shaped). Normal bowed curves are the default.

Section 4

The Consumer's Optimum

Put the budget constraint and indifference curves on the same graph. The consumer wants the highest possible indifference curve, but she can only reach bundles on or below the budget line. So the question becomes: which is the highest IC that still touches the budget line?

The answer is the IC that is exactly tangent to the budget constraint. At that point, the slope of the IC equals the slope of the budget line. That's the consumer's optimum.

Tangency condition MRS = Px / Py IC slope = budget line slope
Equivalent form MUx / Px = MUy / Py equal bang-per-buck on every good

The second form is intuitive: at the optimum, the last dollar spent on pizza gives the same added satisfaction as the last dollar spent on Pepsi. If one good gave more bang-per-buck than the other, the consumer would shift spending toward it — and that's not an optimum.

Optimum is where the budget line is tangent to the highest reachable IC (here, I₂). The consumer cannot reach I₃; any point on I₁ is worse than the optimum.

Why tangency? If the budget line crossed an IC, the consumer could slide along the budget line toward a higher IC — meaning the original point wasn't the best. The only way that move is impossible is when the slopes match: tangency.

Section 5

When Income or Prices Change

Once we have a tangency, we can ask: what happens when income rises, or when a price drops? The budget constraint moves, the old optimum is no longer tangent, and the consumer re-optimizes at a new point.

Income change — parallel shift

An increase in income shifts the budget line outward parallel to itself. Prices are unchanged, so the slope is unchanged. If both goods are normal goods, the consumer buys more of both. If one good is an inferior good, she buys less of it when her income rises.

Normal good: quantity rises with income (most goods).
Inferior good: quantity falls with income. Classic examples are ramen noodles, bus rides, and generic store-brand foods — when people get richer they switch to better substitutes.

Price change — pivot

A drop in the price of Pepsi pivots the budget line outward around the pizza intercept. If you were spending all $1,000 on pizza, your 100 pizzas are unchanged. But the Pepsi intercept rises: with cheaper Pepsi, $1,000 now buys more liters. The line becomes steeper and the slope flips to reflect the new relative price.

What the consumer does with this new opportunity depends on preferences. Normally she'll buy a lot more of the now-cheaper good. But the total change hides two separate forces — the substitution effect and the income effect — which we decompose in the next section.

Section 6

Income and Substitution Effects

When the price of a good falls, two things happen at once:

  • The substitution effect: the good is now cheaper relative to everything else, so the consumer shifts spending toward it.
  • The income effect: the price cut effectively makes the consumer richer — her $1,000 now buys more — so she can consume more of both goods.

To see these effects cleanly, we decompose the move from the old optimum (A) to the new optimum (C) into two steps. First imagine a hypothetical compensated budget line: a line drawn with the new relative prices but just barely letting the consumer reach her old indifference curve. The point where that line is tangent to the old IC is point B.

Substitution effect A → B move along the same IC to the new MRS
Income effect B → C parallel shift up to the new (higher) IC
Total effect A → C what you actually observe

A Pepsi price drop ($2 → $1) moves the optimum from A to C. The dashed compensated budget line isolates the substitution effect (A → B along I₁) from the income effect (B → C, shift to higher I₂).

Giffen Goods — the Weird Exception

Normally both effects push in the same direction for the good whose price fell, so demand curves slope down. But here's a weird edge case: what if the good is strongly inferior, so that the income effect pushes in the opposite direction from the substitution effect, and the income effect is so big it wins?

Then a price drop actually reduces consumption of the good — a violation of the law of demand. Economists call this a Giffen good. Mankiw's classic example is potatoes during the Irish famine: when the price of potatoes rose, the poor became so much worse off that they had to give up meat entirely and eat even more potatoes.

Giffen goods are extremely rare. The only rigorously documented case is a 2008 field experiment on rice subsidies in rural Hunan, China. For exam purposes: all Giffen goods are inferior goods, but almost no inferior goods are Giffen.

Section 7

Applications: Wages and Interest Rates

The theory of consumer choice isn't just about pizza and Pepsi. Any trade-off a person faces can be modeled as a choice between two "goods." Two applications matter for the final exam.

1. Do Higher Wages Raise Labor Supply?

Replace pizza and Pepsi with consumption (on the vertical axis) and leisure (on the horizontal axis). The wage is the price of an hour of leisure — every hour of leisure costs you one hour of wages. A wage increase pivots the budget line outward, the same way a Pepsi price cut did.

The substitution effect says: leisure is now more expensive, so substitute away from leisure and work more. But the income effect says: you're richer, and if leisure is a normal good, you'll want more of it — so work less. The net effect is ambiguous.

The backward-bending labor supply curve: at low wages, the substitution effect dominates and people work more as wages rise. But at high wages, the income effect takes over — you're rich enough that you'd rather have an extra day off than an extra day of pay. Historical data (shorter work weeks over the past century) and studies of lottery winners both show the income effect is real and large.

2. Do Higher Interest Rates Raise Saving?

Now replace the two goods with consumption when young (horizontal) and consumption when old (vertical). The interest rate is the price of today's consumption in terms of tomorrow's. At r = 10%, giving up $1 of consumption today gives you $1.10 of consumption tomorrow.

A rise in the interest rate pivots the budget line outward: savers get more old-age consumption per dollar saved. Again there are two effects.

Effect Direction
Substitution effect Future consumption is cheaper relative to present consumption → save more, consume less today.
Income effect You're richer → consume more of both today and tomorrow → save less today.
Total effect Ambiguous. If substitution dominates, higher r raises saving. If income dominates, higher r lowers saving.

This is actually a big policy question: should we cut taxes on interest income to encourage saving? Theory can't give a clean answer — you have to look at data, and the data aren't conclusive.

Section 8

Key Takeaways

  • Budget constraint: line of all bundles costing exactly income I. Slope = −Px/Py. Intercepts are I/Px and I/Py.
  • Indifference curves represent preferences. They slope down, don't cross, and bow toward the origin. Slope magnitude = MRS.
  • Substitutes vs. complements: perfect substitutes have straight-line ICs; perfect complements have L-shaped ICs. Most real goods are in between.
  • Consumer optimum: the tangency point where MRS = Px/Py. Equivalent: equal marginal utility per dollar across all goods.
  • Income shifts are parallel; price changes pivot the budget line. Normal goods rise with income; inferior goods fall.
  • Income + substitution effects: a price change is decomposed into a move along the old IC (SE) and a shift to a new IC (IE). Giffen goods are the rare exception where IE > SE and they point opposite ways.
  • Applications: wages (labor supply can bend backward if income effect dominates) and interest rates (saving may rise or fall with r).