Exercise 23.4. Prove Proposition 23.3. [Hint: to prove the second part of this proposition, first note that equilibrium wage will be given by whichever group has lower net (after tax) productivity. Then write the utility of workers under two scenarios, first, when the elite have lower net productivity, and second when the middle class have lower net productivity. In writing these expressions, recall that the group with the lower productivity will employ 1 − θL¯ workers, since Condition 22.1 holds. Derive the optimal tax policies for workers in these two scenarios and then compare the utility at these optimal policies].