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**Non-renewable Resources: Optimal Extraction**

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**Categories of Natural Resources**

Nonrenewable vs. Renewable Nonrenewable – finite quantity, rate of generation insignificant compared with rate of use. Renewable – high rate of generation or regeneration. With nonrenewables we are concerned with determining efficient inter-temporal consumption. As we begin, a note about terminology in this course. People often categorize natural resources as either exhaustible or inexhaustible. Those resources of which we have finite stocks and which don’t regenerate themselves at all, or at very slow rates, are considered to be “exhaustible”. Examples: coal, petroleum, mineral ores Those which have stocks that can grow quickly over time are considered to be “inexhaustible”. Examples: fish, trees, water This distinction turns out to be highly problematic, because the so-called exhaustible resources (as we will see) are, in effect, very difficult to exhaust, or even impossible. The cheapest resources to exhaust are extracted first (closest to surface deposits, higher grade ores) As costs of extraction (and exploration for new deposits) increase, market prices increase as supply/MC curve shifts left – higher prices choke off demand At higher prices, substitutes become more and more attractive (renewable energy resources, desalinating seawater) Example: Stanley Jevons was a 19th century British economist who became famous for his book The Coal Question. In it he wrote that Britain's industrial vitality depended on coal and, therefore, would decline as that resource was exhausted. As coal reserves ran out, he wrote, the price of coal would rise. This would make it feasible for producers to extract coal from poorer or deeper seams. Although he was right about the incentive to mine more costly seams, he was almost surely wrong that the main factor was the cost of coal. Jevons failed to appreciate the fact that as the price of an energy source rises, entrepreneurs have a strong incentive to invent, develop, and produce alternate sources. In particular, he did not anticipate oil or natural gas. Also, he did not take account of the incentive, as the price of coal rose, to use it more efficiently. On the other hand, the so-called “inexhaustible” resources – the biological ones – can, indeed, be totally depleted. Species extinction provides many examples. Because stocks do not have to be driven to zero in order for extinction to occur (passenger pigeons, some whale species) – the natural and human-caused rate of decline can suddenly overwhelm the growth rate. Because such extinctions can also be caused by loss of habitat – not just by hunting down the last mating pair. So this first distinction is problematic. As a result, we will use different terminology in this course, and I want you to be very careful and specific about this. Non-renewable vs. Renewable Read three sub-bullets under non-renewable vs. renewable. Last bullet: why is this true? It will take us more than a few lectures to get to this point, but today we will start with non-renewables. And the typical situation for non-renewables is that they tend to be privately owned – decisions about their extraction are made by firms in reasonably competitive markets (there is a range: oil vs. coal). The typical situation for renewable resources is that they are not privately owned (with the exception of many forests and a few other important examples). In fact, they are frequently open-access resources, and we have already discussed the fact that market rates of use for open-access resources will be higher-than-efficient rates. It turns out that when resources are scarce (non-renewable or not), there is an extra economic cost to their consumption – the opportunity cost of forgone future consumption. In economics, we call this “scarcity rent” or “marginal user cost”. Today we will talk in detail about what that means in the context of a non-renewable resource. And when we get to fish, forests, etc. we will talk about scarcity rent in the context of renewable resources. But it turns out that private ownership provides powerful incentives for the owner to “internalize” this additional external cost – the cost of forgone future consumption. Because if they don’t, they will not be maximizing profits over time. So a very important distinction between non-renewables and renewables tends to be the prevalence of private ownership of non-renewables; and the open-access nature of renewables. So keep this distinction in the back of your mind today as we go through the introduction to non-renewables and the economic concept of scarcity, which has to do with market forces as much as it has to do with the actual known quantities estimated by geology.

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**Defining Dynamic Efficiency**

When a policy or program produces streams of benefits and costs over time, it is dynamic, rather than static. In a dynamic setting, the economically efficient allocation maximizes the present value of net benefits. At this allocation, PV(marginal net benefits) are equal across time periods. If this weren’t true, it would be possible to increase the present value of net benefits by re-allocating consumption across time periods. This is going to become much clearer as we go through an example. But switch from static to dynamic efficiency is important, so wanted to mention the difference up front. To deal with dynamic problems, we are going to need to tweak our efficiency rule for this new context, and introduce some new tools.

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**Parameters of Our Two-period Problem: Extraction of 20 Barrels of Oil.**

The scenario that we are going to discuss today is the same two-period model that Tietenberg goes through in Chapter 5 of the text – same demand curve, same constant MC. So you can go through this again with Tietenberg if you find that helpful. Read demand curve and MC from the slide. We have some demand curve – which is going to be the same in each period, to keep things simple here – and some constant marginal extraction cost (equal to $2/unit). We have a total stock of oil, which is equal to 20 units. The discount rate is 10%, or .10 And when I use the abbreviation PVNB, I mean “present value of net benefits”>

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**Problem with Static Efficiency and Non-renewables**

Demand for oil: MB = q $ PERIOD #1 8 MB(Q) 2 MC 15 barrels extracted (Q)

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**Problem with Static Efficiency and Non-renewables**

Demand for oil: MB = q $ $ PERIOD #1 PERIOD #2 8 8 MB(Q) MB(Q) 2 MC 2 MC OK, in our two-period model, there are two periods – one and two. And after that, time comes to an end for our purposes. (We will move on to the case of n periods/infinite horizon soon enough, but this is where we start.) What is the efficient rate at which we should extract this non-renewable resource in each period? Let’s start out by assuming that we can just figure out the efficient quantity of oil to extract from an oil well by solving the same static efficiency problem that we have been solving in each period. We will just maximize net benefits in the first period by solving for the efficient quantity. Then some time will pass and we will be in the second period, and we will do the same thing again. Because demand and MC are not going to change, the answer to the static efficiency problem will be the same in each period. If we do that, we will want to consume 15 units in the first period, and 15 units in the second. Total consumption will be 30 units. But wait, we have just exceeded our finite stock of 20 units! So where there is scarcity, we can’t just solve the static efficiency problem “myopically” in each period. If we actually had a stock of 30 units or 50 or 100 – if the resource was not scarce, given our parameters, there would be no problem with doing this. But the resource IS scarce. So we will have to figure out a new rule for economically efficient extraction. 15 barrels extracted (Q) 15 barrels extracted (Q) 15+15=30 > 20 units available

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**First Candidate for Two-period Consumption Allocation**

Candidate 1: Extract 15 in period 1, and leave whatever is left over (5) for consumption in period 2. $ $ PERIOD #1 PERIOD #2 8 8 MB(Q) MB(Q) NB2 NB1 2 MC 2 MC Explain “straw man” OK, let’s start taking the finite stock into account and propose two alternative extraction scenarios: 1. How about extracting the statically efficient quantity in period #1 (15 units), and just leaving what is left over for consumption in period 2? Let’s calculate the net benefits from that scenario. Since we are no longer “myopic” and we are taking the two-period nature of the problem into account, we are solving for the present value of net benefits. (I discussed how to do this in the lecture on economic efficiency – there is also a handout available on the web, and you have a practice problem in the current problem set. Hopefully this will be clear to everyone.) The present value of net benefits is just the sum of present values from each period. The first period is the present (t0), so we don’t have to discount those net benefits. Calculate them as the area below the demand curve and above the supply/MC curve, to the left of the extracted quantity (15), as usual. The second period is (t1), so we discount those net benefits. Calculate them as the area under the demand curve and above the marginal cost curve, to the left of the quantity extracted (5). Then discount using r=.10, or 10%. Sum them – we get PVNB=67.73. Or how about leaving 15 units for period 2 consumption, since we know that is going to be the static efficient quantity in period 2, and consuming only 5 units in the current period? If we go through the same process to figure out the present value of net benefits under this extraction scenario, we get PVNB=65.91. We could keep doing this – solving for different extraction scenarios that meet the requirement S<=20 and seeing if we can find the one that maximizes the present value of net benefits. However, the number of possible combinations of quantities in periods 1 and 2 is infinite, so we would have a difficult job ahead of us. As it turns out, both of these options (and all other options except the one for which we will solve in a moment) will fail to maximize the present discounted value of net benefits. (This is why they are “straw men.” You could have guessed this, because I chose two extremes – one weighted very heavily toward current consumption, and one weighted very heavily toward future consumption.) There is a much easier way to do this, as there was for the static efficiency problem. The dynamically efficient extraction schedule or allocation will the one for which the PVNB from the last unit consumed in period 1 is exactly equal to the PVNB from the last unit consumed in period 2. The PV of (marginal net benefits) will be equal across periods. The present value of marginal net benefits in each period must be equal in order to maximize the present value of total net benefits over time. Think about that intuitively before we go on to solve the problem by applying this dynamic efficiency rule. What if it were not true? If the net benefit of the last unit consumed in period 1 were higher than the net benefit of the last unit consumed in period 2, then we would re-allocate some consumption from the second period to the first – we could increase total present value of net benefits by doing so. Opposite would be true if the imbalance went the other way. So we are going to shift consumption from period to period so that the marginal net benefits in each period are exactly equal, in present value terms. 15 barrels extracted (Q) 5 barrels extracted (Q) NMB(shaded) = 26 NMB(shaded) = 22 PVNB = /(1+.10) = = 46

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**Second Candidate for Two-period Consumption Allocation**

Candidate 2: Extract 5 in period 1, and leave 15 for consumption in period 2. $ PERIOD #1 PERIOD #2 8 8 MB(Q) MB(Q) NB1 NB2 2 MC 2 MC 5 barrels extracted (Q) 15 barrels extracted (Q) NMB = 22 NMB = 26 PVNB = /(1+.10) = = 45

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**Algebraic Solution to Dynamically Efficient Allocation in Two Periods**

In a dynamic setting, the economically efficient allocation maximizes the present value of net benefits. At this allocation, PV(marginal net benefits) are equal across time periods.

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**Non-renewable Resource Extraction: The Two-period Model**

Marginal Net Benefit in Period #2 ($) discounted at 10% r Marginal Net Benefit in Period #1 ($) PV of MB- MC in Period #1 6 PV of MB- MC in Period #2 5.45 5 4 4 3 3 2 2 1 1 Explain – We will graph the PV of marginal NB in each period on the same graph. To do this, we are going to put quantity consumed in period 1 on the horizontal axis moving from left to right (what we are used to); and quantity consumed in period 2 on the horizontal axis moving from right to left (the opposite of what we are used to). The vertical axis is marginal net benefits in each period – remember we are going to discount benefits and costs that occur in the second period. PV(marginal NB) is just discounted (marginal benefits-marginal costs). So here, in period #1, this PV(marginal) curve is just our original MB curve (the demand curve), minus the constant MC of 2 at each unit. (See – MNB=MB-MC, and we know MB=8-0.4q, and MC=2. So MNB=8-0.4q-2=6-0.4q.) In period 2, we have to discount MNB to get their present value. So the period 2 MNB curve is just: ((6-0.4q)/1.10). IF THE CURVES ARE MARGINAL NET BENEFITS, THEN HOW DO WE GET TOTAL NET BENEFITS IN EACH PERIOD? CALCULATE THE AREA UNDER THE CURVES! For example, for our first “straw man” allocation candidate: 15 in the first period, 5 in the second, what would PV(total net benefits) look like? It would be the sum of two areas – area under whole triangle under PV(MNB1) on the left, plus the area under PV(MNB2) up to a quantity of 5 units consumed. SHADE THESE AREAS WITH THE OVERHEAD MARKER. Just add them up. So that is what it looks like graphically. And we want to maximize the present value of net benefits, or the total area under the two PV(MNB) curves, to the left of the quantity consumed in period 1, and to the right of the quantity consumed in period 2. Q in Period #1 5 10 15 20 20 15 10 5 Q in Period #2

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**Dynamically Efficient Allocation in the Two-period Model**

Marginal Net Benefit in Period #2 ($) discounted at 10% r Marginal Net Benefit in Period #1 ($) PV of MB- MC in Period #1 6 PV of MB- MC in Period #2 5.45 5 4 4 3 3 2 2 1 1 Remember, that the dynamically efficient allocation will be the one in which these two equations representing PV(marginal net benefits) in each period are equal to each other, otherwise we could re-allocate consumption and increase net benefits. So how do we figure that out? We can use algebra: set the two curves equal to each other and solve for q1, q2 , using the additional fact that q1+q2=20; then substitute those quantities into the equations for PV(MNB) and solve for p1, and p2; or we can figure it out graphically, as shown here. Show the area(s) that represent PV(total net benefits) by shading them – won’t get any bigger than this – that is the meaning of dynamic efficiency. You can go through the algebra problem on your own – the answers you should get are down at the bottom of this slide. You will have another opportunity to a two-period non-renewable resource extraction problem on problem set 3. Ok, so we just solved for the dynamically efficient allocation of consumption of a non-renewable resource, in a simple two-period context. How can we characterize the result? production falls over time (q2<q1) prices rise over time consumption that might have seemed efficient from a myopic perspective is no longer efficient when we take scarcity into account (solving the static efficiency problem, we would have consumed 15 units in the first period, but now we consume just a little bit more than 10). In solving this problem, we obtained a result that may have seemed strange to some of you, especially those of you who have had economics before. We obtained market prices for this non-renewable resource in each period, and IN NEITHER PERIOD WAS THE MARKET PRICE EQUAL TO THE MARGINAL EXTRACTION COST (which, as you will recall, was $2 per unit). Why is this strange? Well, in a competitive market, we expect price to be exactly equal to the marginal cost of producing the last unit of the good in question – this is another way of looking at economic efficiency (other than saying it maximizes net benefits) and what happens when we look for the point at which MB intersects MC. But in the case of a scarce non-renewable resource, we have just solved for market prices which are, in both periods, GREATER THAN marginal extraction cost. We know we have dynamic efficiency, but it looks like we may be violating the static efficiency criterion that we have been working with. This isn’t REALLY the case, of course. When resources are scarce, there is an extra piece of marginal cost that we don’t have to worry about when resources are not scarce. That is, if this were an unlimited resource, we would solve the static efficiency problem in each period without running into the limited stock the way we did earlier this lecture. And the market price would exactly equal that constant marginal extraction cost (2). In the case of scarcity, however, greater current consumption comes with an extra cost in terms of foregone potential future consumption. The present value (at the margin) of these FOREGONE FUTURE CONSUMPTION OPPORTUNITIES is MARGINAL USER COST, or SCARCITY RENT. WHEN RESOURCES ARE SCARCE, GREATER CURRENT USE DIMINISHES FUTURE OPPORTUNITIES. MUC IS THE PRESENT VALUE OF THESE LOST OPPORTUNITIES AT THE MARGIN. Q in Period #1 5 10 15 20 20 15 10 5 Q in Period #2 q1=10.239 q2=9.761 From demand function, p1=3.90 [MB = q] p2=4.10

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**Dynamic Efficiency with Constant Marginal Extraction Costs**

P ($) MUC is marginal user cost MEC is marginal extraction cost 8 Period #1 p1 = 3.90 MUC1 = 1.90 2 MEC demand 5 10 15 20 Q P ($) q1 = 8 Period #2 Another way of looking at this – once we use the two-period graph to determine the dynamically efficient allocation (quantities and prices in both periods), we can use that information to graph demand, price and marginal cost in each period as if we had solved the problem separately. Here again, we can see that the quantity that will be extracted in an efficient market is at the intersection of price and demand, which in the case of scarce non-renewables is different than (LESS THAN) the intersection of marginal extraction cost and demand. The fact that we have a new cost to be concerned about because of scarcity – the cost of forgone future consumption – there is a separation between the marginal extraction cost curve and the “total marginal cost” curve, which is represented by price, the sum of MEC and MUC. Looking at it this way, it is also clear why we can think of the opportunity cost of lost future consumption as a negative externality to current consumption (it imposes an extra cost – MUC). However, in the case of privately-owned non-renewables, there are very powerful market incentives for resource producers to take this extra cost into account. So in many cases, this “externality” WILL be internalized by the market acting on its own accord. Let’s get into that a bit. p2 = 4.10 MUC2 = 2.10 2 MEC demand 5 10 15 20 Q q2 = 9.761

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**Scarcity and Marginal User Cost**

Marginal user cost (or scarcity rent) of current consumption is the opportunity cost of forgone future consumption. For non-renewables, MUC=P-MEC This extra cost is a negative externality from the extraction of non-renewable resources. Must be internalized for market equilibrium allocation to be efficient. Let’s try to get a better handle on exactly what this marginal user cost represents. The price of non-renewable resources includes two components: MEC and MUC. We are not just concerned about accounting for the marginal cost of extracting these resources (MEC), but also about the marginal forgone future opportunity to consume them later (MUC). If you have had any kind of dynamic programming (engineers in the crowd), the MUC is the co-state variable in an optimal control (dynamic optimization) problem; it is the dynamic equivalent of a Lagrange multiplier in a static optimization problem. Example: If people in Las Vegas, NV (explain Las Vegas – need to apply 5-6 feet of water per year to a lawn to keep it green) use large quantities of water to keep lush, green lawns, this may be entirely appropriate from an economic perspective if the water is from large, quickly replenishable water supplies. However, if doing so denies drinking water to future generations by reducing the level of an aquifer that is not replenishable for practical purposes, then it may be very inappropriate. The economic concept of scarcity, while it is different from what you think of if you are used to thinking in terms of reserve-to-use ratios, etc. – how much is left – does provide a critical signal of the costs imposed on the future by current consumption. And this signal tells us that consumption that might be appropriate when resources are plentiful may not be appropriate when resources are scarce. You can also think of it as the in-situ value of a resource – the value of the resource as a capital asset in the ground. If markets do not take scarcity rent, MUC, into account in allocating consumption of non-renewable resources over time, market equilibrium will be inefficient – extraction will be “too fast” compared to the efficient rate, and prices will be “too low.” GO BACK TO THE EXAMPLE WE JUST WORKED ON AND FIND MUC IN BOTH PERIODS. (Put up previous slide.) OK, so we have said P=MEC+MUC, which is the same thing as saying that MUC=P-MEC. So in the first period in the problem we just solved, MUC=3.90-2=1.90; and in the second period, MUC=4.10-2=2.10. As it turns out, in this diagram, MUC is represented by the height of the intersection of the two PVMNB curves on this two-period graph. The present value of the MUC is the same in both periods (1.90), but in real terms, MUC=1.90 in the first period, and 2.10 in the second. NOW TURN TO NEXT SLIDE (10) AND LOOK AT THE ONE-PERIOD GRAPHS WE HAVE BEEN DRAWING. SAME THING, DIFFERENT REPRESENTATION.

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The Hotelling Rule At the dynamically efficient extraction allocation of a non-renewable resource with constant marginal extraction cost, the marginal user cost rises over time at the rate of interest (the opportunity cost of capital). Therefore, price also rises at the rate of interest [since MEC is constant – refer fig. on pg. 12] No-arbitrage condition: if it were possible to make more ($) by shifting consumption around, the private owner would do that. The key to understanding the market’s treatment of non-renewables lies in thinking of them like any other capital asset. Private owners of non-renewable natural resources will treat resource stocks like any other capital asset. If non-renewables in the ground are capital assets, then the rate of growth of MUC (the in-ground/in-situ value of the resource) is a capital gain. Resource stocks, because they are limited in nature, are characterized by two costs when the private owner extracts them – the marginal extraction cost (what it costs to pump the oil), and the marginal user cost (the opportunity cost of NOT extracting that unit tomorrow). The marginal user cost is thus the value of a unit of that non-renewable resource IN THE GROUND, as a capital asset that can be spent today or saved for tomorrow. When we determined that the present value of MUC is the same in both periods (1.90) is the same thing as saying that, in real terms, MUC is rising at the rate of interest (r). MUC1=1.90, and MUC2=(1.90)*(1+r) This is a critical result in natural resource economics, and it characterizes something called the Hotelling Rule, named for Harold Hotelling, an economist who we have already discussed as the proposer of the travel cost method for valuing recreational uses of natural resources and environmental amenities. Explain the math – the time derivative of MUC (or price minus MEC) is equal to the interest rate. These are all expressions of the same relationship. In the discrete, two-period case that we are solving, this is equivalent to saying that the change in MUC from period 1 to period 2, as a percentage of MUC in period 1, will be equal to the interest rate. We are going to spend another few minutes here to make sure everyone gets this, because it is a really important result. What is the intuition behind it? If the value of the resource in the ground is growing at a rate that is faster than the opportunity cost of that capital – faster than the amount that you would earn by extracting every unit, selling it all, and putting the money in the bank, how much would the private owner extract? NONE. (If dMUC/dt>r , extract nothing. It is earning you more value in the ground). On the other hand, if the value of the resource in the ground is growing at a rate that is slower than the opportunity cost of that capital – slower than the amount that you would earn by extracting every unit, selling it all, and putting the money in the bank, how much would the private owner extract? EVERYTHING. (If dMUC/dt<r , extract EVERYTHING. It is will earn you more value in the bank). This intuition is like a “no-arbitrage condition”. If it were possible for the private owner of a resource to make more money by shifting extraction toward the present or toward the future, the private owner would do that. Thus when we look across the market at equilibrium, extraction rates of privately-owned non-renewable resources should be preserving this equation (the Hotelling Rule). If they are not, there is more money to be made by shifting extraction from the present to the future, or vice-versa. (Arbitrage opportunities exist). And that is like $20 bills just sitting around on the ground. The competitive pressures of the market are extremely powerful. So if everything is working right, and there are CONSTANT MEC (this turns out to be important), private ownership will preserve the dynamically efficient balance between current and future extraction.

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**Assumptions of Hotelling Model**

Constant marginal extraction costs: only MUC changes over time. Private, competitive owners of non-renewable resources: property rights are well defined. Future price path is known (or “equilibrium in expectations”) Keep in mind the assumptions we have carried through the analysis thus far. Have repeated them a few times, but important to put them together and look at them. Read bullet 1 – will look at what happens if not today Read bullet 2 – will also look at what happens if not today Read bullet 3 – Harder to understand this one if you aren’t familiar with capital theory. But similar to the idea of how the stock market works under equilibrium assumptions. We are not saying everyone has perfect information about what is going to happen. But what is in play is called the “self-fulfilling consensus view” of dynamic expectations. Futures markets and long-term contracts for the delivery of natural resources do exist – so can see how have some idea about future price path. As long as everyone has the same basic information, the resulting “equilibrium in expectations” is enough for our theory – don’t have to have perfect information.

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**Generalizing from 2 Periods to N Periods**

Generalizes to the n-period case. Hotelling rule still holds for constant MEC. Exhaustion of the resource will occur at the point where MEC+MUC=“reservation price” or “choke price”, if such a price exists. What does the choke price or reservation price represent? We derived the two-period results graphically. Due to the limitations of two-dimensional graphs, there are some conclusions of the n-period models that I will have to present intuitively. Will use some graphs where it helps. Again, can be done rigorously, but need optimal control theory/calculus of variations. So instead will have to take my word for a lot of this intuition. Read bullet 1. Read bullet 2. It turns out that in the dynamically efficient n-period case, the resource will be exhausted precisely at the moment when MEC+MUC=the maximum price anyone would be WTP in the market for a unit of the resource. This is called the “reservation price” or the “choke price”. It is the economic meaning of the intersection of the demand curve and the vertical (price) axis. This is an important result – in the simple model, all it takes for complete exhaustion to be dynamically efficient is: Constant MEC; and Existence of a reservation/choke price. (demand not asymptotic to the price axis) In the real world, this may not always be the case. For example, something my have very low substitution possibilities – maybe it is a unique resource. (So unlike oil, natural gas, coal, solar energy, etc. – hard to imagine substitutes.) For such goods, demand may never “hit” the vertical axis. Will only get closer and closer as the price tends toward infinity. Draw a heuristic example of asymptotic demand. Imagine taking out smaller and smaller quantities of the resource with teaspoons as the price gets higher and higher… NEXT SLIDE SHOWS THE LINEAR (NON-ASYMPTOTIC) CASE GRAPHICALLY.

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**Transition to a Backstop (substitute) Technology**

$ ‘Choke’ Price MEC of backstop Market price of the non-renewable will rise until it reaches MEC of the backstop technology. Remember, the backstop is infinitely available, so it has no MUC – no scarcity rent. Again, think of seawater and solar energy. So its costs are only extraction costs. At the point in time at which the total marginal cost of the non-renewable is equal to the MEC of the backstop, the non-renewable will be completely exhausted. Recall that in the two-period example we went through in class, the market price in the first period was $3.90/unit, and the price in the second period was $4.10/unit. An example of a backstop technology in the two-period case would be as if we introduced another technology in the market that could be provided at a price of $ What would happen? Well, we would predict that in the second period, the price of the non-renewable would be capped at $4.00 – without the substitute it was $4.10. This “pushing down” of the price in the second period will also push down the price in the first period. So this means that extraction/consumption in the first period would speed up (be higher) in comparison to the no-backstop case. Because we have a limited stock of the non-renewable, extraction/consumption in the second period would be lower than the no-backstop case. So in the second period, some consumption would be satisfied through finishing up the non-renewable, and some would be provided by the new, infinitely available resource. In the n-period case, the infinitely-available resource would take over from there. Think about it intuitively. If oil producers know that solar energy is available at a marginal cost of $X/unit, then they also know that demand for energy produced from oil will be zero at anything higher than $X/unit. The market prices take these expectations about the future into account. Jumping from there to this idea that the transition from oil to solar will be smooth and will take place just as the price of oil (MEC+MUC) is equal to the price of solar takes a few more assumptions! But the basic intuition is straightforward. Time Time at which non-renewable resource is exhausted and a backstop is discovered , thus shifting to the backstop use so that price of the non-ren. res. tapers off

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**What sets P*? A Transition to Other Non-Renewables**

We can consider either: Same resource, but ores of different quality (coal with high or low energy content); or Different resources entirely (coal vs. oil) Multiple transitions, based on incremental exhaustion of “better” resources (less costly). Can think of backstop technology at end of process, as well. If there are ores of two different quality (can think of them as low MEC vs. high MEC, as well – same idea), or two different non-renewable resources, will transition from resource to resource (or coal seam to coal seam) in same way that we discussed transition to a backstop. At each step, the market price will act as a signal for the next resource to “kick in”. Until that point, by Hotelling’s reasoning, it is more valuable for the owner of the “waiting” resource to wait to enter the market (begin extracting) until the price reaches a point at which MUC is rising at exactly r. (Before then, MUC is rising faster than r – it is more profitable to keep the resource in the ground.) Then, at exactly the point at which MUC rises slower than r, that resource will be exhausted. So it is like a chain of price paths strung together, until either everything is exhausted, or we transition to a backstop technology.

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**Multiple non-renewable transitions, with backstop technology**

$ MEC of backstop So the same idea here. First we would start by exploiting the “best” ore -- maybe the ore or resource with the lowest marginal cost per unit of energy output or some other measure of quality. Imagine that we have successive coal mines – for the first, there is high-quality coal just laying around in mounds on top of the earth. So we exploit that first. Given the limited stock, marginal extraction costs, and expectations over demand, we can figure out what we expect the price path to look like. That price path also takes into account the fact that there are other coal mines around – ones for which the cost of extracting coal may be higher and/or the quality of the coal may be lower. At some price, p1, the “best” non-renewable will be exhausted. That price, p1, signals the producer of the next-best type of coal to start extracting – These “price signals” are expressions of the Hotelling Rule. Take p1, for example. P1 is the price at which, on the margin, the rate of growth of MUC of resource 2 tips from being greater than the interest rate (so the producers hang on to resource 2 in the ground) to being equal to the interest rate. That balance is kept until p2, when resource 2 is exhausted. At that point, production of resource 3 kicks in because the rate of growth of MUC3 has just tipped from being faster than the interest rate, to being equal to the interest rate. And so on, and so forth. This is an expression of what we talked about in the beginning of our look at non-renewables – at increasing price levels, the “effective” reserves will grow – it becomes economically sensible to start extracting resources that looked way too expensive when prices were low. And eventually, we could still transition to a backstop technology – something that typically has a pretty high extraction cost, but is available in infinite quantities. And the MEC of the backstop acts as a “ceiling” on the whole price path for extraction of that resource. The market price will never rise above that, because we could always just use the renewable/infinite resource, instead. (No one will pay more than that.) Time Non-renewable resource 2 is exhausted Non-renewable resource 3 is exhausted Time at which non-renewable resource 1 is exhausted

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**Extraction Today Affects Future Costs: Increasing Marginal Extraction Costs**

Cost of extracting one unit of the resource increases as the stock gets smaller. Common terminology: “Stock effect” Now there is an extra cost to extracting today – the effect on future extraction costs. Disincentive to extract – extraction rate slows. Typically, with stock effects, exhaustion is not dynamically efficient. READ FIRST BULLET – define increasing MEC. HERE WE ARE TALKING ABOUT MOVEMENT ALONG THE MEC CURVE – NOT SHIFTING THE CURVE (MAKE DISTINCTION LATER WITH EXPLORATION AND TECHNOLOGY). This is quite typical. In fact, most non-renewables that you can think of probably have increasing MEC. As we have to drill deeper and deeper into an oil well, the lift cost of bringing up one more barrel is going to increase. As the level of an aquifer gets lower and lower, the pumping cost of bringing up one more acre-foot is going to increase, etc. etc. READ BULLET 2: stock effect READ BULLET 3: Define stock effect Now there are some extra gains to be had from leaving a unit of stock in the ground above and beyond the capital gains from doing so (the MUC) – if I do, my extraction costs in the next period will be lower. The competitive extractor looks ahead to the effect on future costs of extracting today, rather than waiting. READ BULLET 4 – each producer’s extraction rate will slow, because the rate of return rises less rapidly. Waiting until the next period now allows me to gain: The rate of growth of the capital asset in the ground; The forgone increase in next-period costs that I would have experienced by extracting today. Hotelling is still relevant, but things are much more complicated – it is going to be very difficult to tell from the data what might be happening to MUC over time in the presence of rising MEC – we’ll see that next class. In fact, because MUC is a measure of economic scarcity, under increasing MEC, MUC will actually decline, rather than rise over time. As MEC gets larger and larger and you are miles deep in a coal seam chipping away at rock with a small needle or something – the price required to make such technologies economically sensible may be so high that the opportunity cost of current extraction (MUC) can drop to zero. Finally, READ LAST BULLET: -- when there are stock effects, even if there is a “choke price” for demand, it may never be efficient to completely exhaust a resource. Reasonably easy to imagine this – costs increase asymptotically as the stock dwindles to zero. Those last units may just be so expensive to mine that it will never be economic to do so.

21
**Stock of a Non-renewable Resource**

How would we define the stock of a non-renewable resource? Ore/reserves that are feasible to extract at current prices and technologies. Is this an exogenous entity? How would technological change affect stock? How does this create incentives for firms? What about the choke price?

22
**Will the Market Achieve Dynamic Efficiency?**

Yes, under certain assumptions, many of which are met in the markets for non-renewables. Private owners of resources will consider scarcity, not simply their extraction costs, or they risk missing out on a capital gain. Can we tell from market data whether the markets for non-renewables are dynamically efficient? READ BULLET 1 READ BULLET 2 Very, very tricky to tell whether dynamic efficiency is achieved – whether Hotelling Rule holds in a general sense in the real world . Will talk about this more on Thursday. Here is the basic idea, however. With constant MEC, P=MEC+MUC, and MUC rises at the rate of interest, says Hotelling. If we try to look at data on P to figure out what might be happening with MUC (which is usually what people do – MEC data is proprietary information, hard to get over time), then we will get bogged down in what might be happening with technological change, discovery and exploration, rising marginal extraction costs as a function of the stock. More later on this…

23
**Non-competitive market structure (monopolies, cartels) **

Conditions Under Which Dynamically Efficient Extraction Will Not Occur in Private Markets Non-competitive market structure (monopolies, cartels) Asymmetric information Incomplete markets Externalities in production or consumption Public goods Tragedy of the commons/open access resources Divergence between private and social discount rates As usual, I am going to draw your attention to the assumptions that do need to be met in order for market equilibrium to be efficient (here, dynamically so.) Most of these conditions will seem very familiar to you from the static efficiency analysis that we did in the beginning of the class. But now we can say something about whether they tend to hold or not in the markets for non-renewables. Here, again, we are talking about deviations from the “perfect vacuum” of any given market– deviations that can result in extraction rates that are higher or lower than economically efficient rates. In any of these cases, we can’t expect the market acting on its own to maximize the present value of the stream of net benefits from a non-renewable natural resource. READ FIRST BULLET – PROPERTY RIGHTS: THIS TURNS OUT THE BE THE ****BIG***** ONE. WILL SEE THIS WITH RENEWABLE RESOURCES, WHICH TEND TO BE OPEN ACCESS. -- would result in higher than efficient rates – lower than efficient prices -- faster than efficient exhaustion. And, as we have just discussed, with non-renewables property rights tend to be reasonably well-defined, which is why the market tends to do pretty well on its own. READ SECOND BULLET -- INFORMATION: Again, it is asymmetric information we are worried about, doesn’t have to be perfect. Equilibrium in expectations. Not a terrible worry -- active futures markets for many non-renewable natural resources, and market is a pretty efficient transmitter of information about discovery of new discoveries, etc. Effect of information asymmetry could cut either way – extraction/exhaustion/prices could be higher or lower than dynamically efficient levels. READ THIRD BULLET – MARKET POWER -- have talked a bit about market power today. Monopolist as conservationist’s friend, if conservationist is a person who would like to see extraction rates below dynamically efficient rates. Because that is what we get – longer time to exhaustion where exhaustion is optimal. With an important caveat about OPEC and the market for oil, the market for non-renewables also tends to be characterized by a reasonable amount of competition. READ FOURTH BULLET – EXTERNALITIES. Of all of these conditions, we have said that #1 tends to be taken care of. #2 is not a terrible problem in markets as they exist today #3 – with the important caveat about OPEC, often OK. But these last two bullets –- externalities and the “new” one for the dynamic context – this one that has to do with the rate at which we discount future consumption – are where dynamic efficiency tends to break down, if it breaks down, in the markets for non-renewables. Externalities – if in extracting and consuming oil or coal, all of the costs are not internalized, we will have extraction that is faster than dynamically efficient extraction. And there will be resulting efficiency losses in a dynamic context. Same idea as before. This is why many environmental economists talk about taxing fossil fuels, etc. READ LAST BULLET – DISCOUNT RATE. This is a new one – remember we didn’t have to worry about discounting in the static context. Last class, we talked about this issue of choosing a discount rate when doing a benefit-cost analysis. Well, the rate at which the market discounts future consumption/extraction of non-renewables is some amalgam of the rate at which a bunch of individual private owners of non-renewables discount future consumption/extraction. And this may well differ from the rate you would choose if you were a social planner, looking out over time and thinking about intergenerational efficiency. The article by Bob Solow that you read for today “The Economics of Resources or the Resources of Economics”, speaks about this pretty eloquently. Many cases in which we can imagine that this is true. If private r is higher than social r, extraction/exhaustion will be FASTER than dynamically efficient rates. If private r is lower than social r, extraction/exhaustion will be SLOWER than dynamically efficient rates. So we have completed the basic economic theory of non-renewable natural resources. We built the basic Hotelling model and saw what happened when we changed one or more of the assumptions. Started with a two-period model, and extended it to n periods. On Thursday, we will discuss some studies that take the theory to the data – the difficult business of figuring out whether the theory matches the real world. Then we will launch into renewable resources by starting with water – since it can be either renewable or non-renewable depending on the nature of the water resource (groundwater vs. surface water).

24
**Non-competitive Markets: Monopoly**

For monopolist, Hotelling Rule is slightly different: For most “reasonable” demand functions, monopolist extracts more slowly, exhausts resource later than competitive private owner. Monopolist increases total profits from resource by restricting output in early time periods – monopoly rents. This is because restricting output raises the price in the early time periods and more profits can be reaped early rather than later, therefore slower extraction – total PV increases by restricting output in the early time periods. Recall that in the case of a competitive owner, the Hotelling Rule told us that (P-MEC) – the given market price minus marginal extraction costs – would rise at the rate of interest. For the monopolist, the concept is similar. But the monopolist can manipulate (increase) the market price by restricting output/extraction. So as in the static case, she will do this and earn monopoly rents. Just to give you the formal statement of the change to the Hotelling Rule: it is now marginal revenue minus marginal cost that will rise at the rate of interest. The monopolist will extract at the rate that preserves this equation. The result is that, for most reasonable assumptions that we could make about the demand curve over time, the monopolist will extract more slowly, and will exhaust a resource at a later point in time than a competitive private owner. This is not done out of any kind of concern for future generations, but because she is able to increase the total present value of the resource by restricting output (thus raising price) in the early time periods. Bob Solow’s article – says the “monopolist is the conservationist’s friend” – if a conservationist can be defined as someone who prefers extraction of resources at rate slower than the dynamically efficient rate.

25
**At q1 NB = P – C = p1c1 which is greater than, At q2 NB = P – C = p2c2 **

For a monopolist, MC (S) is rising and not constant – increased supply only at higher price and thus control over supply of resource P MB (D) MC (S) p1 p2 c2 c1 q q Q At q1 NB = P – C = p1c1 which is greater than, At q2 NB = P – C = p2c2

26
**For a given price, q1 < q2 to equate PVMB1 = PVMB2**

27
**Exploration and Technological Progress**

Technological progress can shift the MEC function downward over time. Exploration and discovery can also shift the MEC function downward over time. Both technological R&D and exploration exhibit “diminishing returns” over time. While costs may fall initially, when diminishing returns set in, costs will begin to rise. READ FIRST BULLET This means that, even if we are extracting lower-grade ores than we were 30 years ago, and it costs more to get energy out of them, total marginal costs can fall over time. New technologies can still make it cheaper per unit of energy. Same thing for exploration and discovery. Higher prices stimulate exploration for new sources. Some of those exploration efforts pay off, and we discover new sources. Both of these forces, however, exhibit diminishing returns: -- in general, “easiest” innovations with most substantial effects on MEC happen first – benefits to each marginal technological improvement decline over time. -- in general, “easiest” new resource stocks to find are found first – must exert more and more exploration effort to get the same amount of discovery over time. So, while these two forces may result in falling costs initially, eventually we would expect them to turn upwards. Draw example: U-shaped marginal extraction costs – MEC and price on vertical axis, time on horizontal.

28
**solve Demand = MB = 25 – 0.8q Supply = MC = 5/unit Stock = 40 units**

Discount rate, r = 10% = 0.10 MUC = marginal user cost = P – MC t = time period Find q1 & q2 Find p1 & p2 Find MUC in t1 & t2

29
**Dynamically Efficient Allocation in the Two-period Model**

25 [NMB1] [NMB2/1.1] 25 18.18 q q2

30
**Dynamic Efficiency with Constant Marginal Extraction Costs**

P ($) MUC is marginal user cost MEC is marginal extraction cost 25 Period #1 p1 = 8.81 MUC1 = 3.81 5 MEC demand 10 20 30 40 Q P ($) q1 = 20.24 25 Period #2 Another way of looking at this – once we use the two-period graph to determine the dynamically efficient allocation (quantities and prices in both periods), we can use that information to graph demand, price and marginal cost in each period as if we had solved the problem separately. Here again, we can see that the quantity that will be extracted in an efficient market is at the intersection of price and demand, which in the case of scarce non-renewables is different than (LESS THAN) the intersection of marginal extraction cost and demand. The fact that we have a new cost to be concerned about because of scarcity – the cost of forgone future consumption – there is a separation between the marginal extraction cost curve and the “total marginal cost” curve, which is represented by price, the sum of MEC and MUC. Looking at it this way, it is also clear why we can think of the opportunity cost of lost future consumption as a negative externality to current consumption (it imposes an extra cost – MUC). However, in the case of privately-owned non-renewables, there are very powerful market incentives for resource producers to take this extra cost into account. So in many cases, this “externality” WILL be internalized by the market acting on its own accord. Let’s get into that a bit. p2 = 9.19 MUC2 = 4.19 5 MEC demand 10 20 30 40 Q q2 = 19.76

31
**Conclusions/observations**

Because q1 > q2, p1 < p2 Given MEC, higher p2 implies higher MUC in future time periods For a given p, q1 < q2 to equate PVMB1 = PVMB2 => p > MC = monopoly rents As extraction continues, stock depletes so that MUC continuously increases for all future time periods

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