Questions are about Optimal taxes and deadweight loss. Any idea about this hw?

Part III. 11. 12. 13. 14. 15. 16. Optimal taxes and deadweight loss. The Pigouvian tax does not directly depend on the slope of supply and demand, which means that the tax does not depend on how big the quantity response will be to the imposition of a tax. But, does this mean that the quantity response is irrelevant? No! The quantity response will determine the welfare gain induced by the tax. This problem provides an illustrative example. Consider a market where total private benets are equal to TB = 88062 0.5622. Total private costs are T0 = 8862+ 0.562\". Total external damages (costs) are equal to TED = 8862. What is the optimal tax? (2 points) What is the change in quantity that results from introducing the tax? (2 points) Calculate the welfare gain from introducing the optimal tax. (2 points) Now, suppose that the M B curve was much more vertical (less elastic). Specically, suppose that TB = 2,134Q 562\". What is the optimal tax now? (Think to yourself: is this different than your answer in question 11? Why or why not?) (2 points) What is the change in quantity that results from introducing the tax? (Think to yourself: is this different than your answer in question 12? Why or why not?) (2 points) Calculate the welfare gain from introducing the optimal tax. (Think to yourself: is this different than your answer in question 13? Why or why not?) (2 points) Think to yourself: what does this example suggest about the relationship between the welfare gain (elimination of deadweight loss) and the slopes of demand and supply