Here is a simple example (with a spreadsheet) of the properties of a straight line downward sloping demand curve (monopoly)
Here are two articles on Pigouvian Taxes and The Coase Theorem - I have shown you a picture of the former and will discuss the latter next week. The first article provides various examples of the use of these taxes to deal with externalities. The Coase theorem has applications beyond externalities.
It is the foundation of the transactions cost literature and the literature on contracts. This literature also provides us another way to look at markets and suggests that firms are substitutes for markets. They are both 'governance mechanisms' - they govern the conduct of transactions and the costs of using these two governance mechanisms determines which one is more likely to be used.
It also helps us to understand contracts and vertical integration. For example, we should expect to observe long term contracts or vertical integration in the aluminum value chain. This is because an alumina refinery can only accept a particular grade of bauxite and and aluminum smelter can only accept a particular grade of alumina. This type of exchange cannot be carried out using markets.
We made a distinction between the statutory incidence of taxes and the economic incidence. The economic burden of taxation is shared between consumers and producers. The respective shares depend on the elasticities of demand and supply.
Taxes distort prices (and lead to an output contraction) - so they lead to welfare losses (deadweight losses). In other words, some transactions that were taking place prior to the imposition of the tax were not taking place after the incidence of the the tax. These transactions were mutually beneficial (to buyers and sellers).
Subsidies are negative taxes - so the analysis of subsidies is similar to that of taxes. We shifted the supply curve up due to the imposition of a tax. For a subsidy we would just shift it down.