7. (20 points) Adam Smith claimed that the invisible hand of the market leads to common good. For quite a long time, it is understood as equilibrium allocations maximize the sum of utilities across agents. (a) Prove this understanding is dramatically wrong, by proposing a class of counter ex- amples. 2 (b) Prove this understanding is actually correct for a special class of economies, in which agents have quasi-lincar utility functions: for an economy (I, (Ui, Wi, w:")ier) with I agents and N+1 goods, a consumption bundle is denoted by