A consumer has an income of Rs. 20 to spend on two goods, X (price Rs. 4 per unit) and Y (price Rs. 2 per unit). The marginal utility schedules are given below. Find the combination of X and Y that maximises the consumer's total utility, and verify that the equi-marginal condition is satisfied.
| Units | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| MU of X (utils) | 40 | 32 | 24 | 16 | 8 |
| MU of Y (utils) | 22 | 18 | 14 | 10 | 6 |
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Start your 14-day free trial to unlock the full solution →Step 1 — Compute marginal utility per rupee for every unit.
| Unit | MU of X | MU of X / Price (Px=4) | MU of Y | MU of Y / Price (Py=2) |
|---|---|---|---|---|
| 1 | 40 | 10 | 22 | 11 |
| 2 | 32 | 8 | 18 | 9 |
| 3 | 24 | 6 | 14 | 7 |
| 4 | 16 | 4 | 10 | 5 |
| 5 | 8 | 2 | 6 | 3 |
Step 2 — Allocate the income unit by unit, always buying whichever next unit (of X or Y) offers the higher MU per rupee, until the Rs. 20 income is exhausted.
Order of purchase by MU per rupee: Y1 (11, Rs.2, cumulative Rs.2) then X1 (10, Rs.4, cumulative Rs.6) then Y2 (9, Rs.2, cumulative Rs.8) then X2 (8, Rs.4, cumulative Rs.12) then Y3 (7, Rs.2, cumulative Rs.14) then X3 (6, Rs.4, cumulative Rs.18) then Y4 (5, Rs.2, cumulative Rs.20 — income exhausted).
This allocation buys 3 units of X (Rs. 12) and 4 units of Y (Rs. 8), spending the full Rs. 20.
Step 3 — Check the equi-marginal condition. At this allocation, the last unit of X bought has MU/Price = 6, and the last unit of Y bought has MU/Price = 5 — close to equal, and the next affordable unit (a 4th unit of X, MU/Price = 4) is worth less per rupee than the 4th unit of Y already bought (5), while a 5th unit of Y (MU/Price = 3) is worth less still, so no reallocation within the exhausted budget can raise total utility further. The condition MUx/Px = MUy/Py is met as closely as the lumpy, whole-unit data allows. …
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