2. (% de k) (a) Find the area as a function of g, A.(q), below the marginal cost curve, MC(g) = c+dg, where > 0 and d > 0. A.(q) is the shaded area shown in the figure below. The area under the marginal cost curve, A.(g), represents the total cost to the company to produce g units of oil. Hint: divide the area under the curve MC/(q) into the area of a triangle and the area of a rectangle and add them together. The width of the rectangle is and the height of the rectangle is c. MC(q) =c+dq q (b) Find the area as a function of , A,(g), below the marginal revenue curve, MR(g) = a bg, for 7 o, %]. Here a > 0 and b > 0. The area under the marginal revenue curve, A,(q), represents the total revenue the company gains by producing units of oil. Hint: again divide the area under the curve M R(q) into the area of a triangle and the area of a rectangle and add them together. The width of the rectangle is and the height is a bg. (c) Assume that @ > c and g [0, %] The total profit the oil company makes as a function of quantity produced, , is given by P(q) = A,(q) Ac(g). Find the quantity * that the company should produce in order the maximize its profit. Make sure to justify why the quantity you found is indeed the global maximum. (d) What is the total profit that the company makes when it is producing the quantity which maximizes its profit? In other words, what is P(q")? () Another method of finding the quantity * that the company should produce in order the maximize its profit is to look for where M R(q) and MC/(q) intersect. Why should this intersection point be where profit is maximized? In the scenario above, the oil company will typically produce the quantity of oil that maximizes its profit, with no consideration to the climate emissions produced in the extraction of oil. One method that governments use to reduce the climate emissions involved in the extraction of oil is to impose a carbon tax on the fuel used to extract oil. In BC, the carbon tax is currently $65/tonne. This essentially increases the marginal cost of producing some quantity g of oil by some fixed positive amount . 3. (Y Y Yy) Assume the carbon tax is applied so that the Marginal Cost is now MC;(q) = +t +dg and that a > (c +t). (a) What is the quantity produced by the oil company that maximizes profits now? For the other parts of this question, call this quantity . The profit function is now Py(q) = A,(gq) A.(q), where A(q) is the area under the new marginal cost curve, MC;(g). Assume again that g [0, %] and make sure to justify why the quantity you found is indeed the global maximum. (b) What is the total profit that the company makes when it is producing the quantity which maximizes its profit now? In other words, what is P;()? (c) Is the new quantity where profit is maximized, g, less than the quantity that maximized profit with no carbon tax, q'? (d) How big does the carbon tax, , need to be so that the new quantity produced where profit is maximized is half of the old quantity produced? In other words, how big does need to be in order . for G = %7