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: Identify what "optimal group size" means.
The optimal group size is the size at which an individual gets the biggest net gain from group living. Net gain is simply benefit minus cost, so we want the group size where the vertical gap between the benefit curve and the cost curve is largest.
Step 2: Read the shape of each curve.
The benefit curve rises fast at small group sizes and then flattens out, a curve of diminishing returns. The cost curve rises roughly in a straight line, so cost keeps climbing steadily as the group gets bigger. Both curves start from almost the same low point at P.
Step 3: Track the gap between the curves.
Near P the two curves sit close together, so net gain is small. Moving from P to Q, benefit shoots up much faster than cost, so the gap widens and net gain grows.
Step 4: See what happens beyond that point.
Past Q, the benefit curve keeps bending over while the cost line keeps climbing at the same steady rate, so the gap between them starts to shrink again. By R the gap is smaller than at Q, and by S the two curves meet, where net gain drops to zero.
Step 5: Pick the group size with the biggest net gain.
Since the gap between benefit and cost is largest at Q, an individual gains the most from group living at that size. Group sizes smaller than Q have not yet captured the full benefit, and group sizes larger than Q add more cost than benefit.
Final Answer:
The optimal group size is
\[ \boxed{Q} \]
