Topics Today (3/21/13)  Nonrenewable Resources  Hotelling’s Rule of Resource ExtractionMajor Issue #3 Are we running out of natural resources?  Oil and coal make up most of our current energy production, but have no natural growth rate.  Fish and forests are renewable, but how do we keep them at sustainable levels?Hotelling’s Rule of Nonrenewable Resource Extraction  Motivating question: Suppose you own an oil field, and the cost of extracting oil from the field is zero. How should you extract oil from the field over time?  What does it depend on?  Suppose we have a firm trying to decide how much oil to extract this year and how much to extract next year.  Assumptions:  Profit-maximizing firm.  Competitive industry.  Costless extraction.  Perfect information on future prices.Hotelling’s Rule of Resource Extraction  The present value of a barrel of oil would have to be the same in both periods. => p0=p1 / (1+r) => (1+r)p0=p1 => (p1– p0)/p0=r : Hotelling’s Rule Example: Suppose the price of oil next year (year 1) is $110 per barrel. According to Hotelling’s Rule, what must be the price today (year 0) if the discount rate is 10%? Hotelling’s Rule of Resource ExtractionHotelling’s Rule of Resource Extraction  Hotelling’s rule: (p1 – p0) / p0 = r  The price of oil increases at a rate equal to the discount rate.  Oil prices will rise through time.  In the context of our simple model, what is the intuition behind this result:  Suppose p0 >p1 / (1+r) => p0(1+r)>p1 ; extract all the oil in the current year  Suppose p0<p1 / (1+r) => p0(1+r)<p1; leave all of the oil in the groundHotelling’s Rule of Resource Extraction  But what about the case where the costs of extraction are increasing as the resource stock declines?  If we introduce costs, then the maginal net benefit of the resource is MNB = p – MC, where MC is the marginal cost of extraction.  A modified Hotelling’s rule follows: (MNB1 – MNB0) / MNB0 = r  Marginal net benefit of the resource increases over time at a rate equal to the discount rateHotelling’s Rule of Resource Extraction  Ex/ Suppose there are 100 tons of Davite, a newly discovered non-renewable mineral. Assume the following:  Two-period model (0 and 1).  Discount rate = 10%.  The Davite industry is competitive  The marginal cost of producing Davite is $10.  Inverse demand curve for Davite: p = 80 – q, where p is price and q is quantity.  Initially, 50 tons produced in the first period and 50 tons produced in the second period.Hotelling’s Rule of Resource Extraction 80 80 30 10 $ $ 50 25 Supply Demand 30 10 25 50 Demand Supply Tons / Year Tons / Year 80 80 30 10 $ $ 50 25 Supply Demand 30 10 25 50 Demand Supply Tons / Year Tons / Year Period 0 Period 1Hotelling’s Rule of Resource Extraction  Is this equal division of Davite a market equilibrium?  Initially, at the margin the net benefit for all producers is $20 per unit in both periods (periods 0 and 1)  Marginal NPV of $20 in period 1 is 20/1.1 = 18.18 < 20 ; market incentive to extract more in period 0; therefore not a market equilibrium  Does it maximize social net benefits?  Since it’s a private good in a competitive market, the market equilibrium yields the economically efficient allocation. But as just noted, this allocation is not a market equilibrium, must not maximize social net benefits.  Alternative argument: NPV of the last unit is more valuable in the current period than in the next period, which means social net benefit can be improved by greater consumption in Period 0.  Efficient allocation: homework exerciseHotelling’s Rule of Resource Extraction  Graphical demonstration of the socially efficient allocation of the resource stock : 80 10 $ Tons / Year 70 Demand MC 80 10 $ Tons / Year Period 0 70 Demand MC 70 $ Tons / Year Period 1 70 Marginal Net Benefit 70 $ Tons / Year Period 0 70Hotelling’s Rule of Resource Extraction  Graphically with two periods, the socially efficient allocation of extraction is the following: 70 $ Marginal Net Benefit in 0 PV of Marginal Net Benefit in 1 70/1.1 = 63.6 $ 70 $ Quantity in 0 Q0= 50.95; Q1= 49.05 Marginal Net PV of Marginal 70/1.1 = 63.6 Quantity in 1 $ 100 19.05 Q: Do you understand the construction of this diagram? A=yes B=noHotelling’s Rule of Resource Extraction  Here’s another way to understand the construction of the problem  Recall the two conditions for cost effectiveness involving two firms:  E1+E2=E*  MAC1=MAC2  We showed this result with a graph where we “flipped” the MAC curve for one of the two firms with the curve for that firm starting at the maximum emissions (see, for instance, lecture 6 slide 6, and lecture 8 slide 4)  Structurally this is the exact same problem here:  E1 in the cost effectiveness problem is replaced by the extraction in period 1, Q1 and E2 is replaced by the extraction in period 2, Q2; E* is replaced by the mineral stock S;  MAC1 is replaced by MNB1 and MAC2 is replaced by discounted MNB2, MNB2/(1+r);  That is , we are solving two equations in two unknowns,  Q1+Q2=S  MNB1=MNB2/(1+r)  We show this problem graphically just like we showed the problem of cost effectiveness with two firms: we flip one of the MNB curves and start the curve at the maximum available stockHotelling’s Rule of Resource Extraction  Two observations of the model:  First, notice that the efficient solution does not require maximizing current social net benefits. Why?  Because there is an opportunity cost of consuming resources today, namely the value of consuming those resources in the future.  This is called the marginal user cost (also called “scarcity rent”.  Second, notice that efficient allocation requires consuming more in the first period than in the second.  This reflects the opportunity cost of holding Davite in the ground.  Opportunity cost of foregone investment.Hotelling’s Rule of Resource Extraction  What happens to the two-period model with higher discount rates? Graphically: 70 $ Quantity in 0 q0= 51.82; q1= 48.18 Marginal Net Benefit in 0 PV of Marginal Net Benefit in 1 70/1.1 = 63.6 Quantity in 1 $ 70/1.2 = 58.3 70 $ Marginal Net PV of Marginal 70/1.1 = 63.6 $ 70/1.2 = 58.3 Q: Do you understand why the PV MNB1 curve pivots down from the horizontal axis when the