Utility, total utility and marginal utility
Utility = the power of a good to satisfy a want. It is personal: a glass of water gives more utility to a thirsty person.
Total utility (TU) = total satisfaction from all units consumed. Marginal utility (MU) = extra satisfaction from one more unit: MUn = TUn − TUn−1. Also TU = sum of all MUs.
| Roti | MU | TU |
|---|---|---|
| 1 | 20 | 20 |
| 2 | 16 | 36 |
| 3 | 12 | 48 |
| 4 | 8 | 56 |
| 5 | 4 | 60 |
| 6 | 0 | 60 |
| 7 | −4 | 56 |
Cardinal approach: utility is measured in numbers called utils, or in rupees.
Law of diminishing marginal utility
As we consume more units of a good, one after another, the MU of each extra unit falls.
- While MU is positive, TU rises (at a slower rate).
- When MU = 0, TU is at its maximum (point of satiety).
- When MU is negative, TU falls.
Assumptions: units are of the same size and quality, consumed without a long gap, tastes and income do not change.
Equilibrium by utility analysis
One good
Measure MU in rupees (MU in utils ÷ MU of one rupee). The consumer buys while MU ≥ price, and stops where MUx = Px. If MU > price, buying more adds gain; if MU < price, she is paying more than the joy she gets, so she buys less. Because MU falls, this point is reached.
Two goods
Condition: MUx/Px = MUy/Py = MU of money. If MUx/Px > MUy/Py, a rupee gives more joy on X, so she shifts spending to X; MUx falls and MUy rises until the ratios are equal.
Budget set and budget line
Bundle = a combination (x1, x2) of two goods. The budget set is all bundles the consumer can buy with income M at prices P1, P2: P1x1 + P2x2 ≤ M.
The budget line has the bundles that cost exactly M: P1x1 + P2x2 = M. Intercepts: M/P1 on the x-axis, M/P2 on the y-axis. Slope = −P1/P2, the market rate of exchange.
Changes in the budget line
- Income rises: parallel shift outward (slope unchanged).
- Income falls: parallel shift inward.
- P1 falls: the line rotates out on the x-axis (M/P1 grows), y-intercept same.
- P2 rises: the line rotates in on the y-axis.
Indifference curves and the indifference map
An indifference curve (IC) joins bundles that give the consumer the same satisfaction. An indifference map is a family of ICs. This is the ordinal approach: we only rank bundles (better, worse, same), not measure utility.
Monotonic preferences: more of at least one good (and not less of the other) is better.
MRS (marginal rate of substitution) = units of good 2 the consumer will give up for one more unit of good 1 while staying equally happy = −Δx2/Δx1, the slope of the IC.
Properties
- ICs slope downward (to get more of one, give up some of the other).
- ICs are convex to the origin, because MRS diminishes.
- A higher IC gives more satisfaction (monotonic preferences).
- Two ICs never intersect.
Consumer's equilibrium: conditions
The consumer chooses the bundle on her budget line that lies on the highest possible IC. At that point:
- MRS = P1/P2 (slope of IC = slope of budget line; the line is tangent to the IC).
- MRS must be diminishing at that point (IC convex).
If MRS > P1/P2, she values good 1 more than the market does, so she buys more good 1; MRS falls until equal. If MRS < P1/P2, she buys less good 1.
Try it at home
Drink glasses of water after a run and score each glass from 0 to 10 for how good it felt. Write MU and TU in a table. Do your scores fall? At which glass does MU become 0? In the 3D, change income and watch point E slide to a higher IC.
Key formulas and definitions
- MUn = TUn − TUn−1; TU = ΣMU
- MU in ₹ = MU in utils ÷ MU of ₹1
- One good: MUx = Px
- Two goods: MUx/Px = MUy/Py = MU of money
- Budget line: P1x1 + P2x2 = M; slope = −P1/P2
- MRS = −Δx2/Δx1; equilibrium: MRS = P1/P2 with MRS falling
Worked examples
1. TU of 1, 2, 3 ice creams is 30, 50, 60. Find MU of each.
MU1 = 30; MU2 = 50 − 30 = 20; MU3 = 60 − 50 = 10. MU falls: diminishing MU.
2. MU (₹) of successive mangoes: 30, 24, 18, 12, 6. Price ₹18. How many will she buy?
She buys while MU ≥ 18. MU of the 3rd = 18 = price, so she buys 3 mangoes.
3. MUx = 20, Px = 4, MUy = 30, Py = 5. Is she in equilibrium? What should she do?
MUx/Px = 5, MUy/Py = 6. A rupee on Y gives more, so she buys more Y and less X until the ratios are equal.
4. M = ₹120, P1 = ₹6, P2 = ₹10. Write the budget line and its intercepts and slope.
6x1 + 10x2 = 120. x1-intercept = 120/6 = 20; x2-intercept = 120/10 = 12; slope = −6/10 = −0.6.
5. In the same case P1 falls to ₹4. What happens to the budget line?
x1-intercept becomes 120/4 = 30; the x2-intercept stays 12. The line rotates outward on the x-axis and becomes flatter (slope −0.4).
6. At a bundle MRS = 3 and P1/P2 = 2. What will the consumer do?
She is willing to give 3 units of good 2 for one more good 1, but the market asks only 2. So she buys more good 1; MRS falls until it equals 2.
Common mistakes
- Saying TU is maximum when MU is maximum. TU is maximum when MU = 0.
- Drawing the budget line slope as −P2/P1. With good 1 on the x-axis it is −P1/P2.
- Letting two indifference curves cross. That would mean one bundle gives two different satisfaction levels.
- Forgetting the second condition: MRS must be falling (IC convex) at equilibrium.