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ECON 433: Natural Resource Economics

MIDTERM EXAM #1

October 11, 2021

Total time = 1 hour and 25 minutes

Total points = 50

Please answer all questions. Good luck!

I.         Briefly explain the concept of Maximum Sustainable Yield (MSY).  Suppose that the fishery you are studying has the following net growth function: (10 points)

F(Xt ) = Xt {er ( 1 −) − 1}

where Xt > 0 refers to the fish stock, r > 0 is the intrinsic growth rate, and K > 0 is the environmental carrying capacity. Algebraically calculate the MSY for this fishery.

II.        Distinguish between the descriptive and prescriptive approach to discounting.  In the context of climate change policy, what are likely consequences of using the descriptive versus the      prescriptive approach to discounting? (5 points)

III.       Explain the economic concept of a steady state? What is the underlying presumption that       allows economists to focus on the steady state rather than the interim period prior to the         steady state? (5 points)

IV.      Farmer Al has a patch of land and he waters his crops by pumping water from a well.  Over time, the constant pumping changes the height of the water level in the well according to the following equation:

Ht+1  = Ht  +  Rt − wt

where Ht represents the height of water in the well in year t, Wt is the amount of water taken out of the ground in year t to irrigate the land, Rt is the amount of water added back to the    well each time period due to rainfall.

In each period t the total agricultural output, qt, depends on the amount of water used in irrigation and is given by:

qt  = ln(awt )        where a > 0 is a constant.

Each unit of output can be sold for $p.

The average cost per gallon of water pumped from the well depends on the height of water in the well, Ht, and is

ACt = c( − Ht) where c > 0 ¸ > 0.

is the total depth of the well; you can interpret this equation as meaning that it costs more to extract a gallon of water as the height of water in the well is drawn down.

Al’s objective is to maximize the present value of net benefits from irrigating his land over (T+1) years (t = 0, 1, …, T).

1. Set up Farmer Al’s problem clearly identifying the (5 points)

(i)        objective function (ii)       constraints

(iii)      control variables

(iv)      state variables.

2.   Set up the Lagrangian and derive the first order conditions that maximize the present        value of net benefits from irrigating the land. (10 points)

3.   Provide an economic interpretation for ?冗 = p入t+1 .  Explain why this first order condition makes economic sense.  What would you suggest the farmer to do if

?冗 < p入t+1 . (6 points)

4.   Suppose now that the farmer’s planning horizon becomes infinitely long.  Write out the   first order conditions for this problem in the steady state. (4 points)

5.   Using the steady state first order conditions, derive the Fundamental Equation for this     problem.   Please show all your steps or you will not get any credit. (5 points)