Households & Utility
Spend so that the marginal utility of the last dollar is equal across every good. Every question is: which side of mux/Px = muy/Py did the shock hit, and which quantity moves to restore it.
Budget constraint
- Income rises
- The line shifts outward in parallel — relative prices, so the slope, are unchanged.
- One price rises
- The line pivots around the other good's intercept.
Utility
- Total utility
- Rises with quantity, at a decreasing rate.
- Marginal utility
- The slope of total utility, and it declines. Different goods have differently-shaped mu curves.
The comparative static they ask for
Pc rises 10% and you change nothing. The rule breaks:
| Step | Numbers |
|---|---|
| Both ratios equal | garment 10 utils / $50 = 0.2 · food 2 utils / $10 = 0.2 |
| Clothing goes to $55 | 10/55 ≈ 0.18 — less utility per dollar |
| Buy less clothing, raising muc | muc reaches 11 → 11/55 = 0.2 ✓ |
Or split it: buy more food too, which lowers muf.
The two forces underneath
- Substitution effect
- The good is now relatively expensive, so buy something else. "If the price goes up, I'll buy something else."
- Income effect
- Real purchasing power fell, so you can afford less of everything. "If the price goes up, I can't buy as much."
They can push the same way or opposite ways. Indifference curves are ECON 100A.
Rules that earn points
- Write the rule before anything else: mux/Px = muy/Py.
- Say which side the shock hits and which way it moves.
- State the quantity change and the marginal utility it triggers — "buy less clothing, which raises muc."
- On the graph: income shifts parallel, price pivots.
- Equating marginal utilities instead of marginal utilities per dollar.
- Shifting the whole budget line when only one price changed. Pivot it.
- Saying "buy less clothing" and stopping. Buying less raises its marginal utility — that is what restores equality.
A household currently satisfies the Rational Spending Rule. The price of clothing then falls. To restore the rule the household should:
- ABuy more clothing, which lowers muc
- BBuy less clothing, which raises muc
- CBuy more clothing, which raises muc
- DChange nothing, since the rule only concerns marginal utilities
Answer
A — buy more clothing, which lowers muc.
A smaller denominator makes muc/Pc too large. Buying more clothing drives muc down through diminishing marginal utility until the ratio matches muf/Pf again.
B is the trap: the direction flip. Students memorize "price change → buy less" from the price-rise case and apply it to a fall.
Weekly income is $100, food is $20, clothing is $50, clothing on the horizontal axis. The price of clothing rises. The budget line:
- AShifts inward in parallel, since the household is poorer
- BPivots clockwise around the food intercept of 5 meals
- CPivots around the clothing intercept of 2 garments
- DIs unchanged, since income did not change
Answer
B — pivots around the food intercept.
Income/Pf = 100/20 = 5 meals is untouched; Income/Pc shrinks below 2 garments, so the line rotates clockwise and the slope steepens from −2.5.
A is the trap: a price rise does make you poorer in real terms, but only an income change moves the line in parallel. One price ⇒ pivot.
Coffee at Strada costs $4; frozen yogurt at Yogurt Park costs $6. muc(1) = 10, muc(2) = 8, muf(1) = 12, muf(2) = 8. The optimal daily bundle is:
- A1 coffee and 1 yogurt
- B2 coffees and 1 yogurt
- C1 coffee and 2 yogurts
- D2 coffees and 2 yogurts
Answer
B — 2 coffees and 1 yogurt.
Per dollar: coffee #1 = 10/4 = 2.5, coffee #2 = 8/4 = 2.0, yogurt #1 = 12/6 = 2.0, yogurt #2 = 8/6 ≈ 1.33. At 2 coffees and 1 yogurt, 8/$4 = 12/$6 = 2.0 — both margins deliver the same utility per dollar.
C is the trap: yogurt #1 has the highest raw marginal utility (12), so students who rank by mu instead of mu/P load up on yogurt.
"Gas got expensive, so I can't afford as much of anything this month." This statement describes:
- AThe substitution effect
- BThe income effect
- CDiminishing marginal utility
- DA parallel outward shift of the budget constraint
Answer
B — the income effect.
Real purchasing power fell, so less of everything is affordable. The substitution effect is the other sentence: "gas is relatively expensive now, so I'll take the bus."
A is the trap: both effects follow from the same price rise, so students pick whichever they saw first. Test it by asking whether the sentence is about relative price (substitution) or total purchasing power (income).
Weekly income is $100, food is $20 and clothing is $50. (a) Draw the budget constraint with both intercepts and the slope. (b) Show a rise in the price of clothing. (c) Show a rise in income.
Model answer
(a) Food intercept: 100/20 = 5 meals. Clothing intercept: 100/50 = 2 garments. Slope = −Pc/Pf = −50/20 = −2.5 — one garment costs 2.5 meals.
(b) The line pivots clockwise around the unchanged food intercept, pulling the clothing intercept inward.
(c) The line shifts outward in parallel, since relative prices and therefore the slope are unchanged.
- Both intercepts computed as Income/P, with numbers.
- Slope given as −2.5 and interpreted as a trade-off in meals per garment.
- Pivot for the price change, around the correct intercept.
- Parallel shift for income, stated as "slope unchanged".
A household consumes food and clothing and satisfies the Rational Spending Rule. The price of clothing rises 10%. Explain, using the rule, how the household restores the equality.
Model answer
The rule is muc/Pc = muf/Pf. If purchases do not change, the left side becomes muc/(1.1·Pc) < muf/Pf — clothing now yields less utility per dollar.
To restore equality the household buys less clothing, which raises muc by diminishing marginal utility running backwards, and possibly more food, which lowers muf.
With numbers: a garment gives 10 utils at $50 and food gives 2 utils at $10, so both ratios are 0.2. At $55 clothing falls to 10/55 ≈ 0.18. Cut clothing until muc reaches 11, and 11/55 = 0.2 again.
- Writing the rule with prices in the denominators before analyzing anything.
- Identifying which side falls and why.
- Naming the quantity change and the marginal utility response — not just "buy less clothing".
- Noting the adjustment can come from either good, or both.
- A worked number that lands back on equal ratios.