Exercises: Marginal Utility and Consumer Choice

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.

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?

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.

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.

(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.

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.

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.

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.

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.