Lecture Note 1: Introduction to Exhaustible Resources (And to Optimal Control)

Updated 19 August 2008

ECON4925 Resource economics, Autumn 2008

Olav Bjerkholt:

Lecture note 1: Introduction to exhaustible resources (and to optimal control)

Perman et al. (2003, Ch. 14, esp. App.14.1; Ch 15.1-15.3); Hotelling (1931)

Note: As these notes are pieced together by reshuffling earlier notes, there may be inconsistencies and mistakes that I will appreciate to be informed about. olavbj

Introduction

Crude oil and natural gas are exhaustible natural resources, in the sense that they are available in limited quantities. The essential implication of exhaustibility is that extraction of the resource in one period directly affects production and consumption in ensuing periods. This implies that market behaviour for such goods has to be analysed within a dynamic context.

The economics for exhaustible resources have been given considerable attention in the literature, stemming back to the early work Harold Hotelling from 1931. Also a paper by L.C. Gray in 1914 ought to be mentioned, Gray deserves the credit for the discovery of optimal exhaustion and resource rent (we shall later look at his contribution in an exercise). In particular in the 1970s, after the first oil price shock, articles in the economic journals on this topic were numerous.

Characteristic features of resource markets are related to the exhaustibility aspect:

Firstly, it is rather intuitive that when resource scarcity prevails, the market price of the resource will typically exceed marginal extraction costs, in order to reflect the opportunity cost of using a resource unit today rather than conserving it for the future. This gives rise to a resource rent in the extraction activity, also called Hotelling rent, as distinct from Ricardian (differential) rent.We discuss in this lecture the Hotelling Rule, setting out how the resource rent is reflected in the market.

Secondly, due to aspects of property rights and limited access, oligopolistic behaviour typically prevails in resource markets. This leads to a monopoly rent, ie. an additional mark-up on marginal costs. It is worth noting that it is difficult to separate empirically between the various kinds of “rents” in resource markets. In particular with reference to the international oil market, there is little doubt that various kinds of coalitions and cartel behaviour historically have had significant impacts on market development and prices. Whether monopoly rent actually has dominated the potential resource rent in the market for oil, is an open question. We discuss imperfect competition in coming lectures.

A third feature of natural resource markets is that aspects of uncertainty may become very important. For instance, individual agents have to make assessments of remaining reserves of the resource. In addition, the intertemporal adjustments that occur in such markets complicate market equilibrium and induce uncertainty regarding the actual behaviour and functioning of the market. We shall look at some aspects of uncertainty and intertemporal adjustments.

We shall later discuss also taxation of non-renewable resources, mainly from the viewpoint of whether tax rules interfere with optimal depletion. Also for taxes we shall go a little beyonfd the textbook’s treatment. Then, before we leave exhaustible resources we shall look at some approaches to the understanding of natural gas markets.

The objective is not to give very detailed presentation of more complicated theory, and we avoid rigorous proofs. Rather, we intend to provide a brief overview and, hopefully, some intuition of the basic mechanisms and conclusions, to complement the even briefer presentation Perman et al. (2003).[1]

Harold Hotelling is, as mentioned, usually regarded as the founder of resource economics due to Hotelling (1931). Actually this paper did not generate much literature until it was rediscovered in the 1970s. But already in 1914 L.C. Gray made the point the non-renewable resource can be regarded as an asset. The problem of extraction and utilization of the resource can accordingly be regarded as a portfolio choice problem. A general result in this theory is that optimal allocation of assets requires marginal returns to be equalized. As we shall see, this is also the essence of the optimal path of a resource extracting firm.

When to extract a natural resource

Let us start with the very simplest situation. We assume that the resource owner has just one unit of the resource and that the cost of extracting it is . When is it optimal to extract the resource?

Clearly, if the price is given as p, then it must either hold that or . In the first case the answer is to extract now, in the latter case never!

Let us then consider a resource price that increases with a constant rate , i.e. . We must assume that , otherwise we would never extract. Why? It is straightforward to show that the resource should be extracted at a time when the price path reaches the "trigger price" given by

(1.1)

Why? We find this result simply by maximizing wrt. t. Solving for the optimal t=t*, we find that this is when the rate of increase of the net price is equal to the return on financial investments, , ie

(1.2)

(1.1) is equivalent to . As this again states that the net return on holding back on extraction should equal the interest rate at the time of depletion.

Until the “trigger price”, is reached, the return on postponing extraction exceeds the interest rate. After t*the return on keeping the resource unexploited is less than the interest rate.

The Hotelling rule for prices of exhaustible resources

Above we found that the resource rent increases by rate r, cf. relation (1.2). This equilibrium law is in the literature commonly referred to as the Hotelling rule. In his frequently cited work from 1931, discussing the developments of prices in resource markets, Harold Hotelling arrived at a condition for the market price that is formally identical to (1.3).

Thus, a main conclusion in Hotelling's study was that in equilibrium the resource rent (the net price), defined as the difference between the market price of the resource and marginal extraction costs, must increase at a rate equal to the rate of interest. The underlying assumption is again that the producers in the market possess exclusive rights to non-renewable natural resources. The only way of having a return on preserving the resource stock, is that the net price of the resource increases over time. In order for the asset market to be in equilibrium, the growth rate for the resource rent must equal the opportunity cost, i.e. the interest rate or the return on investments. Hotelling (1931) also showed that the competitive equilibrium path for the net price coincide with the conditions for optimal allocation of the total stock of the resource, to be shown below.

The Hotelling rule derived from profit maximization when unit cost is constant

The Hotelling rule in its simplest version can be set out as follows. Let S0 denote the total resource stock in the economy and Rt the total extraction at time t. We assume that unit costs are constant and equal to and that the market price, taken as given by each producer, is . We can think of this resource market as consisting of many small agents each owning a separate small part of the resource stock. We shall try to determine the depletion to maximize total profit as that will imply the maximization of the profit of each agent.

The optimal extraction path for the society as a whole is found by maximizing

(2.1)

with respect to Rt, over a possibly infinite time horizon . The maximization is constrained by the condition

(2.2)

The solution to (2.1) and (2.2) can be found by using optimal control theory. The Hamiltonian is

(2.3)

According to the necessary conditions, the optimal extraction path must fulfil the following relation:

(2.4)

The shadow price of the resource stock at time t, λt, is in the present case, with extraction costs not depending on the accumulated production, constant. We have

(2.5).

For positive extraction, (2.4) can be written, using (2.5), as

(2.6)

(2.6) expresses the Hotelling rule: along the optimal path the marginal net price, which is identical to the resource rent, should increase at the rate of discount.

What have we actually shown here? We have not determined the depletion profile, only that positive production can take place only when the resource rent increases at the rate of discount. We cannot determine the depletion until we know something about the demand side, which we shall introduce shortly. But even in the absence of information about the demand side we find from his solution that the price has to fulfil the condition that it makes the resource rent increase by .

The interpretation of the Hotelling condition as a market equilibrium condition is similar to the case of the resource extracting firm: along the socially optimal extraction path the owners of the resource stocks are indifferent between extracting and leaving the resource in the ground. If this arbitrage condition at some point is not fulfilled, some agents will be able to increase their profits by changing the speed of extraction or by auctioning the complete resource stock on the market.

All the resource owners are indifferent wrt. time of depletion. The price profile is given but its absolute level must still be determined, say by determining . The length of the horizon is still undetermined. By looking at the optimality conditions we see that the transversality condition in this case does not give us any additional information.

Let assume that the demand for the resource is given by a demand function, and d(R) be the inverse of the demand function, with d(0)=pmax..

(2.7)

It follows that demand falls to zero when price at T reaches . We call the choke price. The transversality condition in this case yields

(2.8)

If we now solve (2.7) wrt. and use the expression

(2.9)

the resource constraint (2.2) gives us one equation in one unknown, namely T.

…and from social optimization

We can complement this market solution with a corresponding one using the same assumptions but instead of revenue we use the utility of the consumption of the resource at time t (in money terms), U(Rt).

The optimal extraction path for the society as a whole is found by maximizing

(2.10)

with respect to Rt, over a possibly infinite time horizon . Again, the maximization is constrained by the condition

(2.11)

The Hamiltonian is

(2.12)

The optimal extraction path must fulfil the following relation:

(2.13)

The shadow price of the resource stock at time t, λt, is in the present case, with extraction costs not depending on the accumulated production, constant. We have . For positive extraction, (2.13) can then be written as

(2.14)

Using that in a market economy, , it is seen that (2.14) expresses the Hotelling rule: along the optimal path the marginal net price, which is identical to the resource rent, should increase at the rate of discount.The Hotelling rule for prices of exhaustible resources

Above we found that the resource rent increases by rate r, cf. relation (1.3). This equilibrium law is in the literature commonly referred to as the Hotelling rule. In his frequently cited work from 1931, discussing the developments of prices in resource markets, Harold Hotelling arrived at a condition for the market price that is formally identical to (1.3).

Consistency between socially optimal depletion and a competitive solution

Hotelling (1931) showed not only that there is a socially optimal depletion profile, but also that there is a competitive solution consistent with the socially optimal depletion profile. We approach this by solving the social planning problem for the optimal depletion profile for n identical natural resource firms and then showing that the optimality problem for one of them (essentially Gray’s problem) gives optimality conditions consistent with those of the social planning problem.

We thus assume that there are n identical natural resource firms. The socially optimal rate of depletion is conceived as the rate that maximizes the gross surplus (consumers’ surplus plus producers’ surplus) derived from the demand function given as p(.). The amount depleted (per unit of time) from each firm at time t is Rt while the amount of unextracted resource in each firm at time t is St. The cost of extraction is given for each firm by the function b(Rt, St) with bR’ > 0, bR’’ > 0 and bS’ < 0. The rate of discount is, r, the same for the social planning problem and in the competitive solution.

(3.1)

The state variable in this problem is the amount of remaining resource nSt, while the control variable is the rate of depletion nRt. The Hamiltonian of this problem is – with adjoined price (shadow price) t - as follows:

(3.2)

Assume that St* and Rt* solves the problem. Then it follows from the maximum principle that Rt* maximizes the Hamiltonian, which implies that when continuity, differentiability and concavity of the Hamiltonian hold, we have

(3.3)

Furthermore, the rate of change of the shadow price is given by

(3.4)

Alternatively, for the current value problem the Hamiltonian is

(3.5)

The first order condition is now

(3.6)

While the rate of change of the shadow price in current values is given by

(3.7)

Then, the remainder of Hotelling’s proof is to consider the profit maximizing problem of one of the n firms facing a given price path (in fact the price path resulting from the given demand curve and the depletion of the n firms). We thus have the problem:

(3.8)

The current value Hamiltonian of this problem is

(3.9)

Assume that St* and Rt* solves the problem. Then it follows from the maximum principle that Rt* maximizes the Hamiltonian, i.e.

(3.10)

Furthermore, the rate of change of the adjoined price in current values is given by

(3.11)

As the price ptis assumed to be p(nRt*), i.e. n identical firmsproducing the same amount, these conditions are exactly the same as in the social planning problem and in equilibrium t is equal to t.

Optimal control

The optimal control problem and its solution[2]

Consider the problem

with one of the terminal conditions imposed:

(i) (ii) (iii) free

In this problem is called the state variable, while is the control variable.

Suppose is an optimal pair for this problem. Then there exists a continuous function p(t) such that for all t in [t0, t1]:

The control function subject to maximizes the Hamiltonian H( t, x*(t), u, p(t)), which is defined as

Thus

for all

Furthermore,

For each of the three possible terminal conditions (i), (ii) and (iii) there is a corresponding transversality condition:

(i’)p(t1) no condition

(ii’) (with p(t1)=0 if x*(t1)>x1)

(iii’)p(t1)=0

If the problem is redefined with t1 free, then all the conditions given above are satisfied on [t0, t1*], and in addition

The optimal control problem for a nonrenewable resource

(cf. App. 14)

St = Stock of nonrenewable resource

Rt = Depletion of nonrenewable resource

Objective function /
System /
Terminal state / /
Terminal point / / / /
Present-value Hamiltonian /
Current-value Hamiltonian /
Equations of motions /
Max /
Transversality conditions /
No condition on /

[1] For a more comprehensive exposition of the theory of exhaustible resources we refer to Dasgupta and Heal (1979), also Fisher (1981). Aslaksen and Roland (1983), based on lectures by M. Hoel and S. Strøm is a good introduction in Norwegian.

[2] Cf. Sydsæter, Hammond, Strøm and Seierstad.