AP Practice 2

AP Practice 2

A sound economic decision compares the additional benefit and additional cost of an action while respecting the decision-maker’s constraints.

This original, unofficial short free-response practice combines consumer optimization, elasticity, firm behavior, perfect competition, and a simple two-player game. It is designed for approximately 12 minutes, matching the suggested time for a short Section II question. Show calculations, label every graph completely, and answer each subpart directly.

Practice prompt

A student has $30 to spend on smoothies and sandwiches. Each smoothie costs $5, and each sandwich costs $3. The student’s marginal utility schedule is:

Quantity	Marginal utility of smoothies	Marginal utility of sandwiches
1	40	24
2	30	18
3	20	12
4	10	6

(a) Determine the student’s utility-maximizing combination of smoothies and sandwiches. Show your reasoning using marginal utility per dollar.

The price of a smoothie increases from $5 to $6. The quantity of smoothies demanded falls from 10 to 8.

(b) Calculate the price elasticity of demand using the percentage-change definition. Classify demand as elastic or inelastic.

A perfectly competitive firm sells reusable bottles for a market price of $18. Its marginal-cost schedule is:

Quantity	Marginal cost
1	$10
2	$14
3	$18
4	$24

(c) Identify the firm’s profit-maximizing quantity. Explain why the firm does not produce the fourth unit. On a correctly labeled graph, show the firm’s demand, marginal revenue, and marginal-cost curves.

Two stores, North and South, independently decide whether to include a particular brand of shoes in a weekend sale. Their profits are shown below; the first number in each cell is North’s payoff and the second is South’s payoff.

	South: Include	South: Do not include
North: Include	$(8,8)$	$(15,5)$
North: Do not include	$(5,15)$	$(12,12)$

(d) Identify whether either store has a dominant strategy. Determine whether both stores choosing Do not include is a Nash equilibrium. Explain the strategic-interdependence problem represented by the game.

Worked reasoning

(a) Consumer optimization under a budget constraint

The budget constraint is the limit created by income and prices. The student should allocate spending where the marginal utility per dollar is greatest, not simply choose the item with the greatest total utility.

$$ \frac{MU_{\text{smoothie}}}{P_{\text{smoothie}}} \qquad \text{and} \qquad \frac{MU_{\text{sandwich}}}{P_{\text{sandwich}}} $$

For the first unit, a smoothie provides $40/5 = 8$ utility units per dollar, while a sandwich provides $24/3 = 8$. For the second units, the ratios are $30/5 = 6$ and $18/3 = 6$; for the third units, they are $20/5 = 4$ and $12/3 = 4$; and for the fourth units, they are $10/5 = 2$ and $6/3 = 2$.

The student can purchase six total items with the available budget if the combination costs $30. One utility-maximizing combination is 3 smoothies and 5 sandwiches, but this costs $30 only if the student can purchase five sandwiches beyond the listed schedule. Using only the listed schedule, the highest fully supported combination is 3 smoothies and 3 sandwiches, costing $24; the remaining $6 cannot purchase another smoothie but can purchase two additional sandwiches if their marginal utilities are known. Therefore, the data are insufficient to determine a unique final bundle beyond the listed schedule.

Examiner-rewarded reasoning: identify the constraint, calculate marginal utility per dollar, compare the ratios, and state when additional information is required. A conceptual statement such as “the student buys the goods with the most utility” is incomplete because optimization requires numerical comparison.

(b) Price elasticity of demand

Price elasticity of demand measures how responsive quantity demanded is to a change in price:

$$ E_d = \frac{%\Delta Q_d}{%\Delta P} $$

Using the midpoint method:

$$ %\Delta Q_d = \frac{8-10}{(8+10)/2} = \frac{-2}{9} \approx -0.222 $$

$$ %\Delta P = \frac{6-5}{(6+5)/2} = \frac{1}{5.5} \approx 0.182 $$

$$ E_d = \frac{-0.222}{0.182} \approx -1.22 $$

The absolute value is approximately $1.22$, so demand is elastic. The negative sign reflects the law of demand; classification uses the absolute value.

(c) Perfectly competitive firm

A perfectly competitive firm is a price taker, so its demand and marginal-revenue curves are horizontal at the market price:

$$ P = MR = $18 $$

The firm produces 3 units, where $MC = MR = $18$. The fourth unit is not produced because its marginal cost, $24$, exceeds the marginal revenue, $18$; producing it would reduce profit.

The graph should include a vertical axis labeled price, cost, and revenue; a horizontal axis labeled quantity; a horizontal curve labeled $d = MR = P = $18$; an upward-sloping $MC$ curve; and the profit-maximizing quantity $Q^* = 3$ where $MC$ intersects $MR$.

(d) Strategy and equilibrium

A dominant strategy is an action that produces a higher payoff regardless of the other player’s action. North should choose Include: if South includes, North earns $8 rather than $5; if South does not include, North earns $15 rather than $12. Thus, Include is North’s dominant strategy. By symmetry, Include is also South’s dominant strategy.

Both stores choosing Do not include is not a Nash equilibrium. At $(12,12)$, either store could switch alone to Include and raise its payoff from $12$ to $15. The Nash equilibrium is $(\text{Include},\text{Include})$, because neither store benefits from changing its action unilaterally once the other includes.

This is strategic interdependence: each store’s payoff depends on its own decision and the rival’s decision. Oligopolists may struggle to achieve the monopoly outcome because each firm has an incentive to pursue its individual gain, even when mutual restraint would produce higher combined profits.

Misconception check and error review

Misconception: “A firm should produce where price equals average total cost.” Correction: profit maximization occurs where $MR = MC$; in perfect competition, this becomes $P = MR = MC$, provided the firm’s short-run production condition is satisfied.

After practicing, mark each error as concept, calculation, graph, or explanation. Redo only the missed step without looking at the solution, then write one sentence explaining why the tempting incorrect answer fails.