Possible Solutions for Homework 2

 

Economics of the Environment and Natural Resources/Economics of Sustainability

K Foster, CCNY, Spring 2012

 

 

1.        Write a short essay (about 200 words) on one of the geoengineering topics discussed by Ken Caldeira that you found particularly interesting (which might reflect a bit of additional research on your part, particularly if you couldn't be there).  You need not agree with him, of course!  Each person should write their own essay although you should have someone in your study group proofread.

Answers will vary.

 

  1. Consider the market for a product with an output that pollutes the air.  The industry's Supply curve (only including private internal costs) can be represented as QS = 3PS.  The demand can be approximated as QD = 100 – 5PD.  The industry's marginal external costs from pollution occur as MEC = 0.5Q – 10 whenever Q, the quantity produced, is greater than 10.
    1. What is the privately chosen equilibrium quantity and price, when neither demanders nor suppliers take account of external costs?

The private equilibrium is the quantity where private marginal costs (supply) equals demand price (marginal benefit), so 3P = 100 – 5P, or P*=12.5, Q*=37.5 [alternately solve in terms of costs not quantities, so 1/3 Q = 20 - .2Q, which gives the same answer.

    1. What is the MSC, the marginal social cost (the vertical sum of MC and MEC)?

MSC is the vertical sum so add the costs: MC = 1/3 Q and MEC = .5Q – 10, so the MSC = .833Q – 10 (wherever MEC>0 i.e. Q>20; apologies for typo).

    1. What is the social optimum level of production of this good?  What is the deadweight loss created by a lack of government action?

The social optimum is where MSC = demand, so .833Q – 10 = 20 – .2Q or Q** = 29.03.  The graph from Excel is here,

The DWL is the triangle with horizontal length as the difference between competitive output, Q*=37.5, and social optimum, Q** = 29.03 (so 8.47) and height which is the size of the MEC at 37.5, so 8.75, so the area of the triangle is .5*8.47*8.75 = 37.05.

    1. Suppose the government introduced a tax (per unit of output) to try to move closer to optimum.  (Recall that this means that PD = PS + Tax.)  What tax would reduce DWL the most?

The best tax would bring the level of output to the social optimum, 29.03.  A tax exactly the size of the MEC at that level of output (4.52) would do it.  In more generality, a tax of level T would solve PD = PS + T.  Substitute to find 20 - .2Q = 1/3 Q + T; Q = (20 – T)/.533.  What level of T would make this Q=29.03?  4.516.

    1. If the government instead restricted the level of output through regulation, what regulation would be set?

Clearly the simplest regulation would set a maximum production level of 29.03; alternately a price cap of 9.68 would leave excess demand (consumers would be willing to buy more at this price) but suppliers would not willingly make more.

    1. If demand for this product suddenly rose so QD = 12 – 2PD,  what would be the effects of the tax or regulation that was imposed above?  Is there DWL now?

Sorry, I really effed up this question; the given demand curve is nowhere near an increase!  If it were increased to Q = 120 – 2P, then

So neither the optimal tax nor the quantity restriction would give zero DWL.

At the tax of T=4.516, we'd have Pd = Ps+T, so 60 - .5Q = 1/3 Q + 4.516, so Q = 66.58.  The tax would give the grey triangle of DWL; the quantity restriction would give the blue DWL.

 

  1. Consider fracking, which drills for natural gas but pollutes water supplies.  A particular well site being considered would impact drinking water supplies over an area of 100 [assume this is in thousands of acres].  The impacted area could be reduced at a cost [measured in tens of thousands of dollars]; denote the area cleaned up as x [thousands of acres] then the cost of avoidance is 3x.  The drinking water facility could find new sources of water at cost 2y, [y in thousands of acres to be newly sourced].  So for example if the well site reduces impact by 10 then it pays 3*10 while the drinking water facility pays 2*90.
    1. Absent any regulation or coordination, how much cleaning would be done by the frackers?  What would be the costs to each side?

Originally the frackers do nothing so the cost to the drinking water facility is 200.

    1. What would be the optimal amount of impact reduction and cleanup, chosen by a social planner who weighted the costs of both parties equally?

A social planner would note that y=100 – x so the cost to the drinking water facility (DWF) is 2(100 – x).  Set 2(100 – x) = 3x and solve x*=40.

    1. What would the Coase Theorem suggest would be the outcome, if regulations demanded that the frackers pay the cost of drinking water sourcing?  If regulations gave the frackers free disposal?

The Coase Theorem would suggest that the two sides could reach an accommodation near the social optimum.

    1. If instead of a single drinking water facility, there were 1000 separate wells, what would be the likely outcome, from a Coase perspective?

Now transactions costs would be much higher so they might not come to a resolution.

    1. What if the fracker's cost of avoidance were ?

In this case, 2(100 – x) = 3x + .25x2; solve to get 0= -200 + 5x + .25x2; so x=20.

    1. The value of the natural gas has not yet been mentioned: how would softening prices affect the analysis?  

Lower prices mean the frackers would want to drill fewer than 100.