ARE/ESP 175 Natural Resource Economics Assignment 2

Problem 1 [2.5 points]

You have just inherited an opal mine. Opal is currently a hot commodity, but you know based on fashion trends it will instantaneously become valueless after two years. Therefore you need to decide how much opal to extract in time periods t = {0, 1}. Rents in period t are T(t) = pqt − cqt(2), where qt is the amount of opal extracted in period t, p > 0 is the fixed price per unit opal over the next two years, and c > 0 is a cost parameter. S0 is the initial stock of opal. Total extraction cannot exceed the initial stock of opal: q0 + q1  ≤ S0 .

1.A. [1 point]

You want to maximize discounted profits over 2 years with a discount factor of 6 = 1/(1+r). Write down the Lagrangian and solve for the first-order conditions (FOCs), assuming you want to extract all of the opal in your deposit. What are the two conditions we can derive from the FOCs and what do they tell us about the solution to this problem?

1.B. [1.5 points]

Suppose we have the following parameter values: p = 4; c = 1; r = 0.1; and S0  = 5. Solve for the optimal extraction path (q0, q1) over the two periods. Your answer should be numbers, not parameters. Is it optimal to extract all of the opal from the ground? Explain. (Hint: Recall that we can relax the constraint in a constrained optimization problem by setting the Lagrange multiplier equal to zero.)

Problem 2 [2.5 points]

Kim Kardashian just posted a picture of her opal jewelry on Instagram, and you know that this will now make opal a pop- ular fashion choice for the foreseeable future. That is, now you can choose the optimal final time period (T) for extraction.

The current period is t = 0, and you are managing the extraction activities over the interval t = 0, 1, 2, ..., T. Rents in period t are T(t) = pqt − cqt(2), where qt is the amount of opal extracted in period t, p > 0 is the fixed price per unit opal over the next T years, and c > 0 is a cost parameter. S0 is the initial stock of opal. Total extraction cannot exceed the initial stock of opal: ∑0 qt  ≤ S0 .

2.A. [1 point]

Your goal is to maximize discounted profits over T years with a discount factor of 6  =  1/(1 + r).  Write down the constrained maximization problem you are facing and the associated Lagrangian. Solve for and interpret the first-order conditions (FOCs) —i.e. what must be true for extraction rates to be optimal? Note: don’t input the parameter values from Problem 1 into Problem 2yet —we’ll do that later.

2.B.

Suppose again that we have the following parameter values: p = 4; c = 1; r = 0.1; and S0 = 5.

2.B.i.  [1 point]    Use Excel and Solver to find the optimal extraction path —i.e., find the T and qt  for t  = 0, 1, ..., T that maximize the net present value of the rents from the mine. Plot the optimal extraction path that you found above. Discuss whether your extraction path makes sense. If possible, provide a screenshot of your graph; otherwise, provide a hand-drawn graph that closely resembles your Excel graph.

2.B.ii. [0.5 points]    Should you extract the entire opal deposit if you can choose the time at which the deposit is depleted?

Problem 3: The Green Paradox [5 points]

Burning fossil fuels increases the atmospheric stock of greenhouse gasses (e.g., CO2), which in turn affects global climate. Such changes in global climate have lead to serious environmental and economic damages. Higher extraction (and burn- ing) of fossil fuels leads to higher atmospheric stocks of greenhouse gasses, worsening the climate problem. The risks associated with climate change have generated interest in the development of cleaner alternative energy sources, such as solar and wind power, which emit little or no carbon. In this problem, we will investigate whether a policy that encourages a speedier transition to clean energy sources will mitigate the climate problem.

Assume that the fossil fuel industry faces demand for aggregate output qt, represented by a linear inverse demand curve: pt  = a − bqt . Assume that there is a maximum (or “choke-off”) price that occurs at ā ≤ a, at which point consumers substitute entirely to an alternative energy technology so that demand for fossil fuels is zero if price is greater than ā. In

scheduling their production, firms are assumed to know about this backstop/substitute. Here, ā could represent the price at which consumers are willing to invest in clean energy systems, such as solar panels or wind turbines.

Let industry profits in any given period tbe denoted as Tt  = pt qt −cqt, and assume that the industry chooses an extraction path {q0 , ..., qT } to maximize the net present value of industry profits over T years, subject to the stock constraint S0  = ∑0 qt . Let 6 = 1/(1 + r) indicate the industry’s discount factor.

3.A.

Suppose that the fossil fuel industry is perfectly competitive so that all firms are the same and small enough that they take the price of minerals p as given.

3.A.i.  [1 point]    Write down the Lagrangian function that characterizes the industry’s maximization problem. Write down and interpret the first-order conditions.

3.A.ii. [1 point]    What is the terminal condition of this problem? Explain intuitively why the terminal condition must hold.

3.A.iii. [1 point]    Using the flow, stock, and terminal conditions, derive an implicit equation (or expression) for the date of exhaustion T.

Hint 1: use thefact that the demandfor minerals equals the supply of minerals in every period.

Hint 2: do not try to solvefor T. Just write down an equation that must be truefor the date of exhaustionfor a profit-maximizing competitive mineral industry. We will solvefor T in Excel in the Problem B.                                                                       3.A.iv. [0.5 points]    Suppose that we have the following parameter values: a = 1; ā = 0.8; b = 0.1; r = 0.05; c = 0.1; and S0  = 75. Use Excel and Solver to determine the extraction path for the competitive industry. Plot both the extraction and price paths in two separate graphs. Explain whether they make sense. If possible, provide a screenshot of your graphs; otherwise, provide hand-drawn graphs that closely resemble your Excel graphs.

3.B.

Now suppose that the government has decided to take action against climate change by instituting a policy that imme- diately reduces the atmospheric stock of greenhouses. In particular, it has decided to subsidize the production of clean

energy technology, such as solar panels and wind turbines, which emit no greenhouse gases. As a result of the policy, the