Suppose that a pricetaking firm is extracting an exhaustible resource subject to increasing costs as the stock of the resource is depleted. That is: Max/y Mire it dt where I 2 RR; Subject to: {' (2R, )dt = S The firm wants to maximize its profits over 0 to T period (the interest rate is i which is identical to the social optimal discount rate) and the mine is worth nothing after the firm ceases operations. a) Set up the firm's optimization problem, F.O.C, and derive the Hotelling efficiency condition. Interpret your results. 16] b) Draw an appropriate phase diagram which will show the solution to the above depletion model. Explain how each phase can be a solution. 10 c) Show and discuss the effect of increasing two of the parameters of your choice on (b) above