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ECON 433: NATURAL RESOURCE ECONOMICS (Fall 2022)

Midterm 2 Exam

November 11, 2022

Total time: 1 hour; Maximum points = 35

The problems below require you to use Microsoft’s Excel. Please set up your spreadsheet so that I can understand what you have done and giveyou partial credit if necessary. For example, please type up theformulae you use in your spreadsheet and also any constraints that you use when setting up Solver. If you create any charts, please be sure that the x-axis and y-axis are clearly labeled.

Problem I (15 points)

Assume a fishery with a skewed logistic net growth function, F(X) = rX'1 − X/K,p , a catch-  per-unit-effort (CPUE) production function, Y = qXE, and a profit function that is given by n = pY − cE.  Assume that r = 0.1, K = 1, β = 2, p = 200, c = 1, q = 0.01.

(i)        Set up a spreadsheet and use Solver to find the steady state open access equilibrium for    this fishery.  As an initial guess for stock, use X= 0.25.  Your spreadsheet should show    the values you get when X = 0.25 and Solver’s solution. (5 points)

(ii)       Starting from the initial values of X0 = 0.25 and E0 = 1, create spreadsheet to show how

the stock and effort values converge to the open access equilibrium values.  Your spreadsheet should have columns for time, stock, net growth, effort, catch (or harvest) and profit.  Allow time to go from t = 0 to t = 500.  Assume that eta = 0.25. (5 points)

(iii)      Using the spreadsheet created in part (ii) above, create the following three charts:

a.   Time path of fish stock,

b.   Time path of effort, and

c.   Effort stock dynamics.  This chart should have effort on the vertical axis and stock on the horizontal axis.

The charts should cover the full time period, with time going up to t = 500. (3 points)

(iv)      Based on the charts created above, can you say whether the open access equilibrium is

stable or not?  Justify your answer (one sentence). (2 points)

Problem II (20 points)

Consider a fishery with a net growth function, F(Xt ) = rXt (ç 1 - Xt ö÷ , where Xt refers to the stock of fish in period t, and Yt refers to the extraction of fish in period t.

The net benefit function is, pt = aYt - bYt 2 , where a > b > 0.  Suppose that the intrinsic growth rate in this fishery is 0.5, the environmental carrying capacity is 1, a = 10, and b = 1.   Suppose  that the interest rate is 5%.

1.   Suppose that you value the stock remaining at the end of your horizon at $5 per unit.    Carefully set up a spreadsheet to show the sum of present value of net benefits from this fishery over a 10 year horizon (t = 0, 1, 2,…, 9).   In your set up,

a.   Assume an initial guess for extraction of 0.05 in every period.

b.   Assume an initial stock of 0.2.

c.   Include separate columns for net growth, net benefits, and the present value of net benefits.

2.   Your task is to determine the optimal extraction rate for this fishery.  Use SOLVER to calculate the optimal harvest strategy over this 10-year horizon.

a.   Use the initial values assumed in Part 1 above.

b.   Please incorporate the appropriate constraints when setting up Solver.

c.   Please type your constraints on the spreadsheet so I know what you have used.

3.   Now assume that you want to find the optimal extraction rate for this fishery so that the stock remaining at the end of your horizon is equal to 0.45 units, i.e., XT+1 = 0.45.  Starting with an initial guess for extraction of 0.05 in every period, use SOLVER to determine the optimal extraction trajectory over your ten-year horizon.

a.   Please show your initial set up (i.e., with an initial extraction of 0.05 in every period within the horizon).

b.   Please incorporate the appropriate constraints when setting up Solver.

c.   Please type your constraints on the spreadsheet so I know what you have used.

4.   Prepare a chart comparing the optimal extraction and stock trajectories obtained in problems 2 and 3.  Please be sure to label your chart clearly.