Cap-and-trade systems constitute an excellent way to allocate pollution. They can also be used for water markets. As a matter of fact, we can apply this policy to a wide variety of scenarios in which a resource changes availability over time (think of an aquifer recharging or pollution dissipating and being processed by the natural environment). We will do a general example, and then we will apply it to a Tragedy-of-the-Commons-type scenario. Suppose that there exist 100 units of a resource over a single time period-say, a supply of ground water in a season, a threshold of smog particles per day, or a total amount of SO2 in an air basin. Two firms must share this finite resource. Firm A uses the resource to provide a high-value good, earning $100 per unit of the resource, with an extraction cost given by the function 21x2, where x is the quantity of the resource extracted. Firm B provides a lower-valued good, earing $50 per unit of the resource with an extraction cost of x2. 1. Calculate the marginal net benefit of the resource for each firm. You can do this by differentiating Pxkx2, where P is $100 for Firm A and $50 for Firm B, and where k is the coefficient on the x2 component of the cost function. Do not solve for x. [10 points] 2. Calculate each firm's share of the resource. This will occur when the marginal net benefit to both firms is identical. In math terms, this means "set them equal and solve for x." Keep in mind that Firm A's share is 100xB. [10 points] 3. Calculate the "price" at which the resource will be traded. [10 points] 4. Suppose that Firm B owns the resource outright and that Firm A would like to buy its share from Firm B. How many units of the resource will trade ownership? [10 points] 5. Decide whether the incentives would change if the resource were owned collectively-for example, if it were situated on public land and either firm could use as much as they cared to until it ran out (go ahead, be creative). Describe how the incentives would change; you can use words, math, graphs, or all of the above. [10 points]