Utility Maximization and the Downward-Sloping Demand Curve

Objectives

• Students will be able to apply the utility-maximizing rule to allocate a fixed budget so that marginal utility per dollar is equal across goods (MUx/Px = MUy/Py).
• Students will be able to calculate marginal utility per dollar from a utility table and identify the optimal consumption bundle.
• Students will be able to decompose the effect of a price change into a substitution effect and an income effect.
• Students will be able to explain how the utility-maximizing rule produces a downward-sloping demand curve.

Materials (all free, no student accounts needed)

• Consumer Choice module lesson
• Supply and Demand sandbox graph
• Consumer Choice practice set
• AP Micro cram sheet

Five-minute warm-up, no prep

Open one of these on the projector. Students say what they think happens to price and quantity before anything moves, lock it in, and then the graph plays out step by step. No accounts, nothing graded or stored.

• Tea Gets Pricier
• Coffee Beans Get Expensive
• Printers Get Pricier, Ink Suffers

Warm-up (6 min)

• Do Now, using the module's third-slice-of-pizza framing: "The first slice of pizza is amazing, the fourth is meh. In one sentence, why do you eventually stop even when pizza is still tasty?"
• Cold-call two students, then name it: marginal utility falls as you consume more. Tell them today they turn that feeling into a spending rule.

Direct instruction (15 min)

• Project the Budget Constraints and Utility-Maximizing Rule sections of the module lesson. Put up a two-good utility table and compute marginal utility per dollar (MU divided by price) for each unit.
• Walk the rule: keep spending the next dollar on whichever good has the higher MU per dollar until the budget runs out, and at the optimum MUx/Px = MUy/Py.
• Stress the trap out loud: you compare MU per DOLLAR, not raw MU. A cheaper good with lower MU can still be the better buy.
• Introduce the two effects of a price drop: the substitution effect (the good is now cheaper relative to others, so buy more of it) and the income effect (your fixed budget now stretches further). Note that for a normal good they push the same way.

Guided practice (15 min)

• Work one bundle live from a utility table on the board: cold-call students for each "which good gets the next dollar?" decision until the budget is exhausted, then confirm MU per dollar is equalized.
• Now connect the rule to demand. Project /sandbox/supply-demand and drag the price line to a high level; cold-call a student to read the quantity demanded off the demand curve. Drag it lower and read the higher quantity.
• Ask the class: "The utility rule just made each buyer want more when the price fell. Which part is the substitution effect and which is the income effect?" Have a student label each for the price drop you just showed.
• Whiteboard check: give a two-good table where the higher-MU good is more expensive, and have every pair decide which good the next dollar buys. Reveal that the cheaper, lower-MU good wins on MU per dollar.

Independent practice (9 min)

• Students complete the utility-maximization items in the Consumer Choice practice set, finding the optimal bundle from a table and a budget.
• Each student writes one sentence explaining, in MU-per-dollar terms, why buying the highest-marginal-utility good first can be the wrong move.

Exit ticket

• Given this two-good utility table and a $12 budget, find the optimal bundle and show that MU per dollar is equal for both goods at your answer.
• The price of a normal good falls. In one sentence each, describe the substitution effect and the income effect on the quantity you buy.
• Fill in the rule: at the optimal bundle, ____ divided by price is equal across all goods.

Homework

• Finish the Consumer Choice practice set and note any table item you could not complete.
• In three sentences, describe a purchase you made where a cheaper option beat a more appealing one on value per dollar, and connect it to the MU-per-dollar rule.

Differentiation

• Early finishers: change one price in the utility table and ask them to re-solve the optimal bundle and explain, in substitution and income terms, why the quantity of that good changed.
• Support: provide the utility table with the MU-per-dollar column already set up as blank cells so students divide and compare without building the table.
• Extension for strong classes: ask why the income effect can run against the substitution effect for an inferior good, and what that would do to the demand curve.

Misconceptions to head off

• Belief: maximize utility by buying the good with the highest marginal utility. Correction: compare marginal utility PER DOLLAR (MU divided by price); a cheaper good with lower MU can be the better buy.
• Belief: at the optimum, marginal utility is equal across goods. Correction: it is marginal utility per dollar that is equalized, not marginal utility itself, because prices differ.
• Belief: total utility is maximized where marginal utility is highest. Correction: total utility keeps rising as long as marginal utility is positive; within a budget the optimum is set by the MU-per-dollar rule, not by the peak of marginal utility.
• Belief: the substitution and income effects always push the same direction. Correction: for a normal good they reinforce each other, but for an inferior good the income effect works against the substitution effect.