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I. Explain the concept of Maximum Sustainable Yield.  Why does it differ from the optimal extraction of a renewable resource recommended by an economist? (5 points)

II. Describe one fishery management strategy favored by economists that can be used to achieve the bioeconomic equilibrium. Explain how it works to ensure economic efficiency in the fishery market. (5 points)

III. Why do economists consider old growth forests a non-renewable resource instead of a renewable resource?  Give two reasons. (5 points)

IV. (25 points total)

Suppose that X_t represents the amount of water (in acre-feet) in an underground aquifer and W_t the amount of water pumped (also in acre-feet) from the aquifer, both in period t. Suppose that there is a constant rate of recharge R > 0 (in acre-feet) into the aquifer in each period so that the change in the amount of water in the aquifer is given by the difference equation

X_t+1 – X_t = R – W_t.

The net benefit from water pumped from the aquifer in period t is given by the function

where a > b > 0 and c > 0 are parameters.

a) Set up the present value maximization problem assuming an infinitely long horizon.  Clearly identify the objective function, constraint and control variables. (5 points)

b) Write out the Lagrangian for this problem. (5 points)

c) Derive the first-order conditions for W_t, X_t, and λ_t. Show your steps. (5 points)

d) Evaluate the first-order conditions in steady state, and show that they imply the two-equation system: (5 points)

e) What is the analytic expression for X*? Hint: It is the positive of a quadratic equation. (5 points)

V. (30 points total)

You own an oil well and your goal is to maximize the sum of present value of profits.  The market for oil is perfectly competitive and you can sell your oil at a constant price of $p per unit (i.e., the price of oil does not change over time).

Your total cost of extract is given by , where c > 0, and q_t > 0 is the quantity extracted in period t.  Profits in period t are given by , and you know that at the end of your horizon, i.e., at t = T,

The resource dynamic equation for this problem can be represented by a single equation, , where R₀ is the initial stock of oil in your well.

1. Write out the Lagrangian for this problem and the first order conditions.  Let the discount rate be δ. (3 points)

2. Using the first order conditions, derive the Hotelling Rule for this problem. (3 points)

3. Using the Hotelling Rule above, derive an expression for the extraction path.  That is, derive an expression for q_t as a function of p, c, δ and q₀. (3 points)

4. Using the fact that and the equation derived in part 3, derive an expression for q₀.  (This expression should be a function of p, c, δ and T.) (3 points)

5. Substitute the expression for q₀ obtained in part 4 into the expression for q_t obtained in part 3. (3 points)

6. Next, use your answer from part 5 and the fact that

to derive an equation that you can use to solve for T, the year when the oil in your well is completely depleted.

Make use of the fact that . (5 points)

7. The answer for part 6 should be:

Suppose p = 1, c = 1.14, δ = 0.05, R₀ = 1.  Use Excel’s SOLVER to find the value of T that solves this equation.  Assume an initial guess for T = 5. (5 points)

Please set up your spreadsheet in a manner that allows me to understand what you have done. Include the solution when T=5 (initial guess) and after you have completed SOLVER and obtained the optimal value for T.

8. Rounding up/down the value of T obtained in part 6 to the nearest integer, use Excel to calculate (5 points)

a) the quantity extracted in each period over your horizon, and

b) the sum of the present value of profits from your well.

On your spreadsheet include separate columns for q_t, and the present value of .

VI. Suppose that the volume function for an even-aged forest is given by

where a and b are positive constants.  Suppose the market price of timber is $p/cubic foot and the total cost of cutting and planting is $c/cubic foot so that the profit from clear cutting the trees at time T is given by .

Suppose that for this industry a = 2, b = 0.0004, p = 1, and δ = 0.05 and c = 200.

1. Derive an expression for the mean annual increment (MAI).  What is the rotation length that maximizes the MAI?  Show your steps. (5 points)

2. Use Solver to find the rotation length that maximizes the MAI.  Start with an initial guess of T_MAI = 40. Please ensure your spreadsheet shows the initial guess for T_MAI = 40 as well as your SOLVER solution. (5 points)

3. Suppose you wish to maximize the present value of profits from a single cutting.  Set up the optimization problem and derive the first order condition to determine the rotation length. Show your steps. (No need to solve the first order condition.) (5 points)

4. Use Solver to find the optimal single rotation, T_s, that is the rotation length that maximizes .  Start with an initial guess of T_s = 40. Please ensure your spreadsheet shows the initial guess for T_s = 40 as well as your SOLVER solution. (5 points)

5. Use EXCEL’s SOLVER to calculate the Faustmann rotation, ie, the rotation that maximizes

Start with an initial guess of T* = 40. Please ensure your spreadsheet shows the initial guess for T* = 40 as well as your SOLVER solution. (5 points)