Marginal Analysis and Consumer Choice: Study Guide (Microeconomics 1.6)

Thinking at the margin

In Topic 1.5 you compared the total benefit of a choice with its total cost. Many everyday decisions are not “yes or no” but “how much?” How many hours should you study? How many games should you play? How many coffees should you buy this week?

For “how much” questions, economists look at the margin: the next unit.

• Marginal benefit (MB): the extra benefit from one more unit of an activity.
• Marginal cost (MC): the extra cost of one more unit of that activity.
• Marginal analysis: comparing the MB of one more unit with its MC to decide whether to increase, decrease or keep the level of an activity.

The decision rule is short:

Comparison for the next unit	What a rational person does	Why
MB > MC	do more	the next unit adds more benefit than cost, so total net benefit rises
MB < MC	do less	the last unit added more cost than benefit, so cutting it raises total net benefit
MB = MC	keep the level	this is the optimal quantity

Total net benefit = total benefit − total cost. Adding each unit whose MB is greater than its MC keeps raising total net benefit. So the quantity where MB = MC is the same quantity where total net benefit is largest. When an extra unit costs nothing (MC = 0), this means you keep going until total benefit itself is at its peak, where MB falls to zero.

The same rule works for firms. A firm compares the extra revenue from one more unit of output with its extra cost. You will use this idea again and again, from Unit 2 onward.

Sunk costs do not change the best quantity

A sunk cost is a cost you have already paid and cannot get back, whatever you do next. A fixed cost that is already committed behaves the same way. The same is true of a benefit already fixed by a past choice.

These amounts are the same whichever quantity you choose now. They change your total net benefit, but they do not change the MB or MC of the next unit. So they cannot change the optimal quantity.

Example: you have paid a non-refundable $30 to join a climbing wall for a month. Each visit also costs $2 for shoe hire. Whether you should go a sixth time depends only on the MB of the sixth visit and its $2 MC. The $30 is gone either way. Saying “I must go more to get my money’s worth” is the sunk cost fallacy.

The model of rational consumer choice

The second half of the topic applies marginal analysis to a consumer spending money. The model rests on these assumptions:

1. Consumers face constraints. Their income is limited and they must pay the market prices of goods. They cannot buy everything they want.
2. Consumers are rational. They make choices on purpose, given their constraints, to get the greatest possible total utility.
3. Utility is the satisfaction a person gets from consuming a good or service. We measure it in imaginary units called utils.
4. Preferences are known and consistent. The consumer knows how much each extra unit is worth to them.
5. Diminishing marginal utility. Each extra unit of a good gives less extra satisfaction than the one before.

In the language of Topic 1.5, a consumer’s total benefit is their total utility, so the consumer’s marginal benefit is their marginal utility.

Total utility and marginal utility

• Total utility (TU): the total satisfaction from all the units consumed.
• Marginal utility (MU): the extra satisfaction from one more unit. MU = change in TU ÷ change in quantity. With one unit at a time, MU is simply the difference between one TU value and the next.

Kai, a fictional student, is at a party where pizza is free. His utility is:

Slices of pizza	0	1	2	3	4	5	6	7
Total utility (utils)	0	20	36	48	56	60	60	56
Marginal utility (utils)	–	20	16	12	8	4	0	−4

Notice three things:

• MU falls with each slice. This is diminishing marginal utility. The first slice, when Kai is hungry, gives the most satisfaction.
• TU keeps rising while MU is positive, but by smaller amounts each time. Diminishing MU does not mean total utility falls.
• TU is highest (60 utils) when MU reaches zero, at the sixth slice. The seventh slice has negative MU and makes Kai worse off. The pizza is free, so MC = 0, and Kai’s best choice is where MU = MC = 0: 6 slices. (Kai is indifferent between 5 and 6 slices, since the sixth adds nothing. By the usual convention we choose the quantity where MB = MC.)

If each slice cost money, Kai would stop sooner. That is the next idea.

Worked example 1: marginal analysis with a table and a graph

Question. Ravi visits a fictional bowling alley. He paid a $10 entry fee at the door, which is non-refundable. Each game then costs $4. His marginal benefit from each game, valued in dollars, is:

Game	1st	2nd	3rd	4th	5th	6th
MB ($)	11	8	6	4	2	1

(a) Use marginal analysis to find Ravi’s optimal number of games. (b) Check your answer using total benefit and total cost. (c) Does the $10 entry fee change the answer? (d) The price per game rises to $6. What is the new optimal number?

(a) MC = $4 for every game. Compare MB with MC game by game:

1. Games 1, 2 and 3: MB ($11, $8, $6) > MC ($4). Each adds to net benefit, so play them.
2. Game 4: MB = $4 = MC. Playing it adds nothing and loses nothing.
3. Game 5: MB = $2 < MC = $4. It would lower net benefit, so stop before it.

The optimal quantity is where MB = MC: 4 games.

(b) Total benefit is the sum of the MBs; total cost (not counting the entry fee) is $4 × games.

Games	1	2	3	4	5	6
Total benefit ($)	11	19	25	29	31	32
Total cost of games ($)	4	8	12	16	20	24
Total net benefit ($)	7	11	13	13	11	8

Total net benefit peaks at $13, at 3 and 4 games. The 4th game adds exactly zero, which is why the two values tie. Using the convention “go to where MB = MC”, the answer is 4 games. Both methods agree.

(c) No. The $10 is sunk: Ravi pays it whether he plays 0 games or 6. It lowers his overall gain for the evening to $13 − $10 = $3, but it does not change the MB or MC of any game, so the optimum is still 4. (The entry fee matters only before he pays it, when the choice is “go or not go”. That is a total decision, from Topic 1.5.)

(d) Now MC = $6. MB > MC for games 1 and 2, MB = MC for game 3, and MB < MC from game 4. The new optimum is 3 games. At $6 per game, total net benefit is $5, $7, $7 and $5 for 1 to 4 games, so 3 games is again at the peak.

Interpretation. A higher marginal cost moves the optimum to a smaller quantity. A sunk cost does not move it at all.

Spending a budget on two goods

When a consumer buys two goods with a fixed income, comparing MU alone is not enough, because the goods have different prices. A good with high MU may also be expensive. The fair comparison is marginal utility per dollar:

MU per dollar = MU ÷ price

A consumer maximizes total utility when both of these hold:

1. All income is spent (or the budget is used up as fully as whole units allow).
2. MU_A ÷ P_A = MU_B ÷ P_B for the last unit of each good bought.

Why does the second condition matter? Suppose the last dollar spent on good A gives 10 utils and the last dollar spent on good B gives 6 utils. Moving a dollar from B to A loses about 6 utils and gains about 10. Total utility rises. Because of diminishing MU, buying more A lowers MU_A ÷ P_A and buying less B raises MU_B ÷ P_B. Keep moving spending until the two ratios are equal. Then no swap can raise total utility.

A practical method with a table: spend each next purchase on the good with the highest MU per dollar, until the money runs out.

Worked example 2: the utility-maximizing combination

Question. Tomas has $12 a week to spend on smoothies ($3 each) and muffins ($2 each). His marginal utilities are:

Unit	1st	2nd	3rd	4th	5th	6th
MU of smoothies (utils)	30	21	12	6	3	–
MU of muffins (utils)	18	16	14	8	4	2

(a) Calculate MU per dollar for each unit. (b) Find the utility-maximizing combination and show that it meets both conditions. (c) Calculate Tomas’s total utility. (d) Explain why 1 smoothie and 4 muffins is not the best choice.

(a) Divide each MU by the price.

Unit	1st	2nd	3rd	4th	5th	6th
Smoothies: MU ÷ $3	10	7	4	2	1	–
Muffins: MU ÷ $2	9	8	7	4	2	1

(b) Buy in order of highest MU per dollar, keeping track of spending:

1. 1st smoothie (10 utils per $): spent $3.
2. 1st muffin (9): spent $5.
3. 2nd muffin (8): spent $7.
4. 2nd smoothie (7) and 3rd muffin (7): spent $7 + $3 + $2 = $12. The money is used up.

Tomas buys 2 smoothies and 3 muffins. Check both conditions:

• Spending: 2 × $3 + 3 × $2 = $6 + $6 = $12, all his income.
• Last units: MU ÷ P for the 2nd smoothie = 21 ÷ 3 = 7; for the 3rd muffin = 14 ÷ 2 = 7. The ratios are equal.

(c) Total utility = (30 + 21) + (18 + 16 + 14) = 51 + 48 = 99 utils.

(d) 1 smoothie and 4 muffins costs $3 + $8 = $11. The last muffin gives 8 ÷ 2 = 4 utils per dollar, but a 2nd smoothie would give 21 ÷ 3 = 7 utils per dollar. Spending is in the wrong place, and $1 is unused. Its total utility is 30 + 56 = 86 utils, less than 99.

Check. Every other affordable combination gives less: for example, 4 smoothies give 69 utils and 6 muffins give 62 utils. Notice that buying only the good with the highest first MU (smoothies) is a poor plan, because MU diminishes and smoothies cost more.

Common misconceptions

• “Diminishing marginal utility means total utility falls.” TU keeps rising as long as MU is positive; it just rises by less each time.
• “Keep going until MB is zero.” Only if the next unit is free. When there is a price, stop where MB = MC.
• “I’ve paid already, so I should use it more.” Sunk costs do not change the MB or MC of the next unit, so they cannot change the best quantity.
• “A rational consumer buys the good with the highest marginal utility.” Compare MU per dollar, not MU alone. A high-MU good can be poor value if it is expensive.
• “Equal MU per dollar is enough.” The consumer must also spend the budget. Equal ratios at a lower spending level leave money that could raise utility.
• “The best quantity is where MB is highest.” The first unit usually has the highest MB, but stopping there throws away other units whose MB is still above MC.
• “Average benefit is the same as marginal benefit.” Average benefit is TB ÷ quantity; MB is the change in TB from one more unit. Decisions about “one more” need the marginal value.

Where this leads

This is the last topic in Unit 1. In Unit 2 (Supply and Demand), diminishing marginal utility and the MU-per-dollar rule help explain why consumers buy more of a good when its price falls, and the MB = MC rule reappears in the models of firm behaviour later in the course. Test yourself with the practice questions, then use the revision notes and the checklist to consolidate. To review total benefit and total cost first, see the cost-benefit analysis study guide, or return to the course roadmap.