..... I need assistance with the following questions:
9. Intertemporal allocation of a depletable resource: suppose the demand for a resource in yeart is 1-"t = a - bqt, where t indeXes the year, P is price, q is quantity demanded, and a and b are positive constants. Suppose also that the total cost of producing the resource in year t is TCILqJ = cqt, where c is a positive constant. Suppose also that the total amount of the resource is Q. The real interest rate is r. Assume that the objective is to maximize present value of net benets to consumers consuming this resource (i.e., not a monopoly problem]. a. Set up the Lagrangian for this problem, assuming t= {I}, l, 2, ..., T) b. Show what the Kuhn-Tucker FDC are for this problem. c. Why are the Kuhn-Tucker conditions relevant, rather than equality constraints? d. Solve this problem for a = 8, b = [1.4, c = 2, Q = ID, r = 0115, and T = 2 (Le, three period model. e. The problem in (d) is a dynamic problem. 1What is meant by "dynamic?" How much would be consumed each period ifQ = l? What would be the value of relaxing the constraint Q = 21} in each period? Problem 6. Logistic population growth: suppose the growth rate of a population of wild fish is given by F(x) = 10x - 0.01x- if the population is left undisturbed, were x is population. a. What is the equilibrium natural population? b. What is the maximum sustained yield, and the corresponding population? c. Graph the population growth function. Which portions of this function correspond to biological overfishing?Problem 7. Schaefer model: suppose that the yield (harvest h) is given by h = 2Ex , where E is the amount of fishing effort and x is population (here q, the productivity parameter, equals 2). a. Find sustainable population x as a function of effort E. b. Find sustainable yield as a function of E. c. Graph the resulting Yield-Effort curve. Which portions of this curve correspond to biological overfishing?8. Gordon model: Suppose price of fish P = 10, and total cost of fishing effort is TC(E) = 1000E. a. Find the Total Revenue Product Curve. b. Suppose the fishery is managed by an owner with exclusive property rights. Find the level of effort E that maximizes the rent from the fishery, and calculate the rent. c. Find the level of effort that would occur under open access. d. Under open access, is this fishery suffering from economic overfishing? e. Under open access, is this fishery suffering from biological overfishing? Why or why not?Problem 12-1. Ludwig hires Frederic to work on a project that will yield $700 in revenue if it succeeds and $100 in revenue if it fails. Frederic's opportunity cost of working without substantial effort is $160. His additional cost of working hard is $50. If Frederic works without substantial effort, the probability that the project will succeed is 0.3. If he works hard, the probability that the project will succeed is 0.6. Design a contract whereby Ludwig will pay Frederic so that Frederic has an incentive to work hard
