Emily's second method in part $($b$)$ arrives at the ideal purchase policy, taking into account the long$-$term consequences of each decision. The method in part $($a$),$ however, also incorporates long$-$term considerations and can yield results just as good as the method in part $($b$).$

Emily's second method in part $($b$)$ finds the cost for the best overall purchase policy. Conversely, the method in part $($a$)$ focuses on finding the best purchase policy for the given month, but this method is actually more comprehensive and effective than the one in part $($b$).$

Emily's second method in part $($b$)$ calculates the cost of the most effective purchase policy considering the implications of each decision over time. Conversely, the method in part $($a$)$ concentrates solely on identifying the best course of action for the current month, failing to consider its influence on subsequent choices. This short$-$term perspective in $($a$)$ often results in less favorable outcomes compared to the more comprehensive approach in part $($b$).$

Emily's second method in part $($b$)$ determines the optimal purchase policy that takes into account the long$-$term consequences of each decision. On the other hand, the method in part $($a$)$ focuses solely on finding the best approach for the immediate month, neglecting its potential impact on future decisions. Consequently, the method in $($a$)$ tends to yield suboptimal results compared to the method in part $($b$)$