This is about individual and aggregate production. Consider \( J \) firms, each with individual production possibility set \( Y_j \subseteq \mathbb{R}^L \) with \( j = 1, \ldots, J \). Each production set is non empty and closed. Given a price vector \( p \in \mathbb{R}_+^L \), firm \( j \)'s profit \( \pi_j \) is defined as \[ \pi_j(p) = \sup_{y_j \in Y_j} p \cdot y_j. \] and firm \( j \) set of optimal production choices is denoted \( y_j(p) \) (this set can be empty). The aggregate firm production set is \( Y = \sum_{j=1}^J Y_j \subseteq \mathbb{R}^L \) (the aggregate firm has access to all the \( J \) individual production possibility sets), and denote the aggregate firm profit and supply as \( \pi(p) \) and \( y(p) \) respectively. Show that \[ \pi(p) = \sum_{j=1}^J \pi_j(p). \]