Living in groups gives benefits but also brings costs. The graph below plots per capita benefits (solid line) and per capita costs (dashed line), both measured in the same units, against group size. Four group sizes are marked on the x axis: P, Q, R, and S.
Given these patterns in benefits and costs, which one of the group sizes P, Q, R, S best represents the optimal group size?
Show Hint
Optimal group size is where the vertical gap between the benefit curve and the cost curve is largest, not where the curves cross.
Step 1: Use the marginal rule instead of reading the gap by eye.
An individual should keep joining a bigger group only as long as each extra member adds more benefit than cost. The best group size is where the extra (marginal) benefit from one more member just equals the extra (marginal) cost.
Step 2: Compare the slopes of the two curves.
The cost line is straight, so its slope, the marginal cost, stays the same at every group size. The benefit curve is steep at first and then flattens, so its slope, the marginal benefit, starts high and keeps falling as the group grows.
Step 3: Find where the two slopes match.
Right after P the benefit curve's slope is much higher than the cost line's slope, so adding members still helps. As the group keeps growing the benefit slope keeps dropping. At Q, the falling marginal benefit meets the constant marginal cost, so no further group size adds more gain than pain.
Step 4: Check the ends for confirmation.
Before Q the marginal benefit is still above marginal cost, so growing the group helps. After Q, at R and S, marginal cost has caught up to and overtaken marginal benefit, so each new member does more harm than good and net benefit falls.
Step 5: State the optimum.
The group size where marginal benefit equals marginal cost, and total net benefit peaks, is
\[ \boxed{Q} \]
