Consumer Choice & Utility Maximization

You’ll be able to

Apply the utility-maximizing rule equalizing marginal utility per dollar
Explain how diminishing marginal utility generates a downward-sloping demand curve
Separate the substitution effect from the income effect

Diminishing marginal utility is why demand slopes down

Marginal utility is the satisfaction from one more unit, and it falls as consumption of the same good rises. So a consumer will only buy additional units at a lower price, because those units are worth less to them. The demand curve is therefore not an assumption — it is derived from how satisfaction behaves.

The utility-maximizing rule

MU_x / P_x = MU_y / P_y, with the entire budget spent

Per dollar, not per unit. A good costing three times as much must deliver three times the utility to be equally worth buying.

Two conditions, both required

At the optimum the ratios are equal and the budget is fully spent. A bundle with equal ratios and money left over is not optimal, because the leftover dollars could still buy utility. Exam tables are built so that a clean bundle exactly exhausts the budget, so an answer that overspends or underspends is a signal to recheck the per-dollar column.

Substitution and income effects

A price cut has two consequences. The substitution effect shifts consumption toward the good that is now relatively cheaper. The income effect is the extra real purchasing power the lower price releases, which can be spent on anything. For a normal good both effects raise quantity demanded. For an inferior good the income effect works against the substitution effect, and in the rare case where it dominates you get a Giffen good — worth knowing exists, but not something the exam requires you to compute.

Worked example

A consumer with $30 chooses between sandwiches ($5, marginal utilities 50, 40, 30, 20) and drinks ($2, marginal utilities 20, 16, 12, 8). Find the utility-maximizing bundle.

1.Utility per dollar — sandwiches: 10, 8, 6, 4. Drinks: 10, 8, 6, 4.
2.Buy in descending order: sandwich 1 and drink 1 at 10; sandwich 2 and drink 2 at 8; sandwich 3 and drink 3 at 6.
3.Cost so far: 3 sandwiches at $5 = $15, plus 3 drinks at $2 = $6, total $21.
4.$9 remains. The next best is 4 utils per dollar for either. A fourth sandwich costs $5 and a fourth drink $2 — buy both, for $7, leaving $2.
5.The final $2 buys a fifth drink if one is available; with the table given, the budget cannot be exactly exhausted, so the best affordable bundle is 4 sandwiches and 4 drinks at $28.

Answer: 4 sandwiches and 4 drinks, spending $28 of the $30. The last unit of each delivers 4 utils per dollar, so the ratios are equalized; the residual $2 buys nothing listed, which is a reminder to state the remainder rather than force a false exact fit.

Watch out

Do not equalize marginal utility itself. A good with a higher marginal utility can still be the wrong purchase if its price is higher in proportion. The comparison is always per dollar.

Checkpoint

Good A yields 60 utils at $6; good B yields 36 utils at $3. A utility-maximizing consumer should:

Checkpoint

The price of a normal good falls. The substitution and income effects:

Checkpoint

A consumer at an optimum finds that the price of good X has fallen. Immediately:

On the exam

Compute the whole utility-per-dollar column first, then allocate the budget in descending order. Working good by good is slower and makes it easy to miss that the budget is exhausted.

Answer the 3 checkpoints as you read.