Marginal Utility and Consumer Choice

Utility and Diminishing Marginal Utility

Utility is the satisfaction you get from consuming a good. Economists measure it in made-up units called utils. The numbers are not "real"; they just let us compare how much we like things.

Here is Emma's lunch: each slice of pizza adds less than the one before.

Slices	TU (utils)	MU (utils)
0	0	—
1	30	30
2	52	22
3	66	14
4	72	6

Notice that TU keeps rising, but it rises more slowly each time. That pattern is the law of diminishing marginal utility: as you consume more of a good in a given period, the marginal utility of each additional unit eventually falls.

Why does this happen? The first slice satisfies a very hungry stomach. By the fourth, you are mostly full.

Practice★☆☆

Emma keeps track of her total utility (in "utils") from eating slices of pizza at lunch: 1 slice gives 20 utils, 2 slices give 34 utils, 3 slices give 44 utils, and 4…

Emma keeps track of her total utility (in "utils") from eating slices of pizza at lunch: 1 slice gives 20 utils, 2 slices give 34 utils, 3 slices give 44 utils, and 4 slices give 50 utils. Calculate the marginal utility of each slice. Then state whether her utility pattern shows diminishing marginal utility, and explain how you know.

Solve

The Budget Constraint

Wanting more is easy. Paying for it is the problem. A consumer's budget constraint shows every bundle of goods that can be bought with a given income at given prices.

With income $I$, and goods X and Y with prices $P_X$ and $P_Y$:

$$P_X \cdot x + P_Y \cdot y \le I$$

A bundle is affordable if its total cost is less than or equal to income. It exhausts the budget if the cost equals income exactly.

The line hits the horizontal axis at $I/P_X = 20/4 = 5$ notebooks and the vertical axis at $I/P_Y = 20/2 = 10$ pens. Its slope is $-P_X/P_Y = -2$: giving up 1 notebook frees up money for 2 pens.

Points on the line spend all the income. Points inside the shaded area are affordable but leave money unspent. Points above the line are unattainable.

Practice★☆☆

Aaliyah has an income of $\$20$ to spend on notebooks, which cost $\$4$ each, and gel pens, which cost $\$2…

Aaliyah has an income of $\$20$ to spend on notebooks, which cost $\$4$ each, and gel pens, which cost $\$2$ each. Is the bundle of 3 notebooks and 4 gel pens affordable? Does it spend her entire budget? Then write the utility-maximizing rule in terms of marginal utilities and prices that she should follow when choosing a bundle.

Solve

The Utility-Maximizing Rule

Given a budget, which bundle is best? A fair comparison must account for price. A burrito with MU of 18 utils sounds better than a drink with MU of 6 utils, but the burrito costs three times as much.

So compare marginal utility per dollar: $MU/P$. It tells you how many utils you get from the last dollar spent on that good.

Why? If $MU_X/P_X > MU_Y/P_Y$, a dollar moved from Y to X gains more utils than it loses. So you should keep shifting until the two ratios are equal. When they are equal, no switch can help.

Practice★☆☆

Malik spends his weekly allowance on burritos and drinks. The last burrito gives him 18 utils of marginal utility and costs $\$6…

Malik spends his weekly allowance on burritos and drinks. The last burrito gives him 18 utils of marginal utility and costs $\$6$. The last drink gives him 6 utils of marginal utility and costs $\$2$. Calculate the marginal utility per dollar for each good. Is Malik maximizing his utility? If not, which good should he buy more of?

Solve

Practice★★☆

Marcus spends his money on two kinds of video game rentals, Good A and Good B. At his current bundle, the marginal utility of A is 40 utils and its price is $\$8…

Marcus spends his money on two kinds of video game rentals, Good A and Good B. At his current bundle, the marginal utility of A is 40 utils and its price is $\$8$. The marginal utility of B is 30 utils and its price is $\$5$. (a) Compare the marginal utility per dollar of the two goods. (b) Explain in which direction Marcus should shift his spending. (c) Explain what will happen to the marginal utility of each good as he makes the shift, and why this eventually stops the reallocation.

Solve

Finding the Best Bundle from a Table

With a table of marginal utilities, you can build the best bundle one dollar at a time: keep choosing the unit with the highest $MU/P$ until the budget runs out.

Jayden at the fundraiser

Jayden has $\$9$. Smoothies cost $\$2$, cookies cost $\$1$.

Unit	MU smoothies	MU/P smoothies	MU cookies	MU/P cookies
1st	20	10	9	9
2nd	16	8	7	7
3rd	12	6	6	6
4th	8	4	4	4
5th	4	2	2	2

Rank all units by MU/P: 10 (smoothie 1), 9 (cookie 1), 8 (smoothie 2), 7 (cookie 2), 6 (smoothie 3), 6 (cookie 3), and so on.

Buy in that order and track spending: smoothie 1 ($\$2$), cookie 1 ($\$1$), smoothie 2 ($\$2$), cookie 2 ($\$1$), smoothie 3 ($\$2$), cookie 3 ($\$1$). Total: $\$9$.

The bundle is 3 smoothies and 3 cookies. Cost: $3(\$2) + 3(\$1) = \$9$, so the whole budget is spent.

Check the rule: what is $MU/P$ for the 3rd smoothie and the 3rd cookie?

Both equal 6, so $MU_S/P_S = MU_C/P_C$. The rule holds.

Practice★★☆

Jayden has $\$9$ to spend at a school fundraiser on smoothies (price $\$2$ each) and cookies (price $\$1…

Jayden has $\$9$ to spend at a school fundraiser on smoothies (price $\$2$ each) and cookies (price $\$1$ each). His marginal utility from each unit is shown below.

Unit	MU of smoothies	MU of cookies
1st	20	9
2nd	16	7
3rd	12	6
4th	8	4
5th	4	2

(a) Calculate the marginal utility per dollar for each unit of each good. (b) Find the bundle that maximizes Jayden's utility and show that it uses his whole budget. (c) Verify that your bundle satisfies the utility-maximizing rule.

Solve

Price Changes and the Law of Demand

The rule also explains why demand curves slope downward. When a price rises, $MU/P$ for that good falls, so the consumer shifts spending to other goods.

Cookies get more expensive

Use the same marginal utilities as before, but now cookies cost $\$2$, smoothies stay at $\$2$, and Jayden's income rises to $\$10$ (so that it buys exactly 5 units at $\$2$ each).

Both goods cost $\$2$, so $MU/P$ is half of MU. Smoothies: 10, 8, 6, 4, 2. Cookies: 4.5, 3.5, 3, 2, 1.

Rank by $MU/P$: smoothie 1 (10), smoothie 2 (8), smoothie 3 (6), cookie 1 (4.5), smoothie 4 (4), cookie 2 (3.5), and so on. Every unit costs $\$2$, so $\$10$ buys 5 units.

Ranked, the top 5 units are smoothies 1, 2, 3, cookie 1 and smoothie 4 (MU/P = 10, 8, 6, 4.5, 4). Total: $5 \times \$2 = \$10$. The bundle is 4 smoothies and 1 cookie.

Check the rule: the 4th smoothie has $MU/P = 4$ and the 1st cookie has $4.5$. The next best units are the 2nd cookie (3.5) and the 5th smoothie (2), and both are lower than 4, so they are not worth buying. The ratios are close and the budget is spent, so this is the best bundle.

How did the cookies he buys change, and what does that show?

Cookies fell from 3 to 1 when their price rose from $\$1$ to $\$2$ (and income rose slightly). Higher price, smaller quantity demanded: the law of demand.

Practice★★☆

Use the same marginal utility table as Jayden's (smoothies: 20, 16, 12, 8, 4 utils for units 1 to 5; cookies: 9, 7, 6, 4, 2 utils for units 1 to 5). Originally, with an…

Use the same marginal utility table as Jayden's (smoothies: 20, 16, 12, 8, 4 utils for units 1 to 5; cookies: 9, 7, 6, 4, 2 utils for units 1 to 5). Originally, with an income of $\$9$, smoothies cost $\$2$ and cookies cost $\$1$, and his best bundle was 3 smoothies and 3 cookies. Now the price of a cookie rises to $\$2$, the price of a smoothie stays at $\$2$, and his income is $\$10$. (a) Find his new utility-maximizing bundle using marginal utility per dollar. (b) Compare the number of cookies he buys now with before, and explain how this result illustrates the law of demand.

Solve

Practice★★★

Sofia has an income of $\$100$ per month to spend on two goods, X and Y. The price of X is $\$4$ and the price of Y is $\$6…

Sofia has an income of $\$100$ per month to spend on two goods, X and Y. The price of X is $\$4$ and the price of Y is $\$6$. Her marginal utilities depend on how many units she consumes: $MU_X = 80 - 4x$ and $MU_Y = 90 - 3y$, where $x$ and $y$ are the quantities of each good.

(a) Find the quantities of X and Y that maximize her utility. Show that the bundle exhausts her income and that the marginal utility per dollar is equal across goods.

(b) Sofia's friend suggests the bundle $(x, y) = (12, 8)$. Show that it is affordable, and then explain, using marginal utility per dollar, why it is not the best choice for her.

(c) Explain why the optimum has to satisfy both the budget constraint and the equal-ratio condition, and what would happen to her utility if she satisfied only one of them.

Solve

Practice★★★

Camila is at her utility-maximizing bundle of concert tickets (good X) and streaming credits (good Y). At that bundle, $MU_X = 20$ utils with $P_X = \$5$, and $MU_Y =…

Camila is at her utility-maximizing bundle of concert tickets (good X) and streaming credits (good Y). At that bundle, $MU_X = 20$ utils with $P_X = \$5$, and $MU_Y = 12$ utils with $P_Y = \$3$. The price of a concert ticket then falls to $\$4$, while her income and $P_Y$ stay the same.

(a) Show that she was at an optimum before the price change.

(b) After the price falls, compare the marginal utility per dollar of the two goods at her old bundle. Which good should she buy more of, and why?

(c) Suppose that, as she buys more concert tickets, the marginal utility of tickets falls, while $MU_Y$ stays at 12 for the purpose of this problem. At what value of $MU_X$ does she stop adding tickets? Explain how diminishing marginal utility connects this result to the downward-sloping demand curve.

Solve

Back to the food truck: you stop buying tacos when the last dollar on tacos gives no more satisfaction than the last dollar on a drink.

Let's practice! 3 medium exercises in this section

★★☆ Jayden has $\$9$ to spend at a school fundraiser on smoothies (price $\$2$ each) and…★★☆ Marcus spends his money on two kinds of video game rentals, Good A and Good B. At his…★★☆ Use the same marginal utility table as Jayden's (smoothies: 20, 16, 12, 8, 4 utils for…

Utility and Diminishing Marginal Utility

The Budget Constraint

The Utility-Maximizing Rule

Finding the Best Bundle from a Table

Price Changes and the Law of Demand

Summary