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Consider the Hotelling model for the depletable resources. Arm operating in a perfectly competitive market and facing a demand equation p[tl= a- b qitl maximizes its present value of prot subject to the resource constraint. The dynamic optimization problem solution yields the optimal production and price paths. a. What is the economic meaning of the first-order-condition (F.O.C) results: betat[p(tl- cm]= lambda (beta is the discount factor, p=price, cqrt=derivative of cost function with respect to production, and lambda: Lagrangian multiplier.) b. The F.O.C. also yields: pltJ= Cqr + beta "t (lambda). What are the two terms in the right-hand side? How is this result different than the condition for non-depletable resources? If the solutions to this problem are (assuming MEC=0): qltl = (albl [1-(1+rlt'T] and p(t) = a (1+rlt'T, and demand equation is pitl=500 - 0.05 qitl, c. How do q and p change over time? d. How a change in interest rate will affect production and prices? Is this consistent with the rise in oil prices in the 2000s? e. Applying this model to oil, what is the price of alternative energy when oil runs out? If the cost of alternative energy is expected to fall in the future, how will it affect the oil price path? Why? f. Given r = 0.1 and T=50, what are q and p in periods 0, 20, and 50