Economics of Non-renewable Energy Supply

Abstract

This chapter provides an introduction to the economic concepts related to fossil fuel supply. The chapter presents a typical chain of activities in the fossil fuel supply, namely exploration , development and production and discusses the relevant economic decision-making issues for each activity. It also considers the influence of government intervention on the investment decisions through fiscal instruments.

Annex 7.1: Mathematical Treatment of Depletion

A Simple Model of Extraction of Exhaustible Resources

The basic model of the extraction of non-renewable resources was initially proposed by Hotelling (1931). The problem is to find the optimal depletion path of a firm that seeks to extract such resources to maximize its profit. There is a vast body of academic literature on this subject—see Devarajan and Fisher (1981), Fisher (1981) and Krautkraemer (1998) for further details. The basic model is based on the following assumptions: (a) the size of the resource stock is known, (b) the entire reserve is exhausted during the project life, (c) interest rate is fixed,

We define the following terms:

• y_t is the quantity of resource extracted in period t;

• X_t is the resource stock at the beginning of period t = fixed at $\overline{{X_{0} }}$ at time 0.

• C = C(y_t, X_t) = total extraction cost,

• P(y_t) is the inverse demand function for the resource

• r is the discount rate

• T = time horizon

The objective is to maximize the net benefit

$$ {\text{Max}}\,\left( {{\text{y}}_{\text{t}} } \right)\sum\limits_{{{\text{t}} = 0}}^{\text{T}} {\left[\frac{1}{{(1 + {\text{r}})^{\text{t}} }}({\text{p}}_{\text{t}} {\text{y}}_{\text{t}} - {\text{c}}({\text{y}}_{\text{t}} ,{\text{X}}_{\text{t}}) )\right]} $$

(7.14)

S.t.

$$ {\text{X}}_{0} = \overline{\text{X}}_{0} ;{\text{X}}_{\text{T}} = \overline{\text{X}}_{\text{T}} $$

(7.15)

and

$$ \frac{{{\text{dX}}_{\text{t}} }}{\text{dt}} = - {\text{y}}_{\text{t}} ,{\text{or}}\,{\text{X}}_{{{\text{t}} + 1}} - {\text{X}}_{\text{t}} = - {\text{y}}_{\text{t}} $$

(7.16)

The Lagrange function is given by

$$ \begin{aligned} {\text{L}} = & \sum\limits_{{{\text{t}} = 0}}^{\text{T}} {\left[\frac{1}{{(1 + {\text{r}})^{\text{t}} }}({\text{p}}_{\text{t}} {\text{y}}_{\text{t}} - {\text{c}}({\text{y}}_{\text{t}} ,{\text{X}}_{\text{t}} ))\right]} \\ & \quad + \sum\limits_{{{\text{t}} = 0}}^{{{\text{T}} - 1}} {\mu_{\text{t}} ({\text{X}}_{\text{t}} - {\text{X}}_{{{\text{t}} + 1}} - {\text{y}}_{\text{t}} )} + \alpha (\overline{\text{X}}_{0} - {\text{X}}_{0} ) + \beta (\overline{\text{X}}_{\text{T}} - {\text{X}}_{\text{T}} ) \\ \end{aligned} $$

(7.17)

First order condition resulting from differentiation with respect to y_t is:

$$ \frac{{{\text{p}}_{\text{t}} - (\partial {\text{c}}/\partial {\text{y}}_{\text{t}} )}}{{(1 + {\text{r}})^{\text{t}} }} - \mu_{\text{t}} = 0; $$

(7.18)

which can be rewritten as

$$ {\text{p}}_{\text{t}} - \frac{{\partial {\text{c}}}}{{\partial {\text{y}}_{\text{t}} }} = \mu_{\text{t}} (1 + {\text{r}})^{\text{t}} = \lambda_{\text{t}} $$

(7.19)

The net price is equal to royalty and in the special case where cost of extraction is negligible the price should grow at the rate of interest. The term on the right hand side of Eq. 9.5 is the user cost , which is directly related to the shadow price of the resource. It suggests that for non-renewable resources, the price should contain an additional element that takes care of the effect of resource depletion. This is the opportunity cost of using the resource now instead of leaving it for the future. In the special case when the cost of extraction is insignificant or zero, the price becomes equal to the rent and hence the rate of price change is just equal to the rate of interest. This is the fundamental result in the economics of exhaustible resources.

1.1 A.7.1.1 Effect of Monopoly on Depletion

Consider the case of pure monopoly—where one single monopoly producer is functioning in the industry. The problem here is similar to the competitive market. The only difference is in the first condition of optimal depletion because the monopolist will take into account the influence of his output decision on price. The first order condition resulting from differentiation with respect to y_t is given by

$$ \frac{{{\text{p}}_{\text{t}} + {\text{y}}_{\text{t}} \frac{\text{dp}}{{{\text{dy}}_{\text{t}} }} - (\partial {\text{c}}/\partial {\text{y}}_{\text{t}} )}}{{(1 + {\text{r}})^{\text{t}} }} - \mu_{\text{t}} = 0; {\text{or}}\,{\text{MR}} - {\text{MC}} = {\text{Royalty}} $$

(7.20)

Introducing price elasticity in the above equation we get

$$ \frac{{{\text{p}}_{\text{t}} \left(1 + \frac{1}{{{\text{e}}_{\text{p}} }}\right) - (\partial {\text{c}}/\partial {\text{y}}_{\text{t}} )}}{{(1 + {\text{r}})^{\text{t}} }} - \mu_{\text{t}} = 0; $$

(7.21)

which can be re-written as

$$ {\text{p}}_{\text{t}} \left(1 + \frac{1}{{{\text{e}}_{\text{p}} }}\right) - (\partial {\text{c}}/\partial {\text{y}}_{\text{t}} ) = \mu_{\text{t}} (1 + {\text{r}})^{\text{t}} = \lambda_{\text{t}} ; $$

(7.22)

$$ {\text{p}}_{\text{t}} = \lambda_{\text{t}} + (\partial {\text{c}}/\partial {\text{y}}_{\text{t}} ) - \frac{{\lambda_{\text{t}} + (\partial {\text{c}}/\partial {\text{y}}_{\text{t}} )}}{{1 + {\text{e}}_{\text{p}} }} $$

(7.23)

This implies that the price under monopoly would have three components: marginal cost of extraction, royalty and a monopoly rent. This third component is positive for all elasticity values greater than −1.0. In those cases, price under monopoly would be greater than the price under competition.

For a linear demand function, it can be shown that the optimal price path in the case of a monopoly is two times less rapid than that of a competitive market price path. Obviously, the two prices start at different levels and the price charged by the monopolist includes the monopoly rent. This is shown graphically in Fig. 7.16. The optimal extraction path also follows a similar path—under the competitive market situation, the resource is exhausted twice as fast as that under the monopoly in the above case (see Fig. 7.17).

Fig. 7.16

Price path in competitive and monopoly cases

Fig. 7.17

Optimal extraction path

Relating the above idea to the oil market would then suggest that the price change under the OPEC era in the 1970s was an adjustment process where the competitive price path was abandoned in favour of a monopolistic price path. This is shown in Fig. 7.18. Surely, this slows down the extraction and the resource will last longer in this case.

Fig. 7.18

Change in price path under OPEC after the first oil shock

1.2 A.7.1.2 Effect of Discount Rate on Depletion Path

As the discount rate plays an important role in the net worth calculation, the discount rate influences the decision about using non-renewable resources now or in the future. A high discount rate leads to higher rate of extraction initially but the output declines fast and therefore, the resource is exploited quickly (see Fig. 7.19). On the other hand, a lower discount rate prolongs the resource availability through a lower rate of initial extraction and a slower rate of extraction.

Fig. 7.19

Effect of discount rate on the extraction path

The price path for different discount rates again follows the similar pattern (see Fig. 7.20). A high discount rate reduces the initial price but the price path is steeper compared to a low discount rate, which in turn causes to reach the backstop prices earlier.

Fig. 7.20

Price path under different discount rates

It needs to be mentioned here that although this application of the Hotelling principles to depletion has given rise to a large volume of academic literature, energy prices do not seem to follow the prescriptions of the theory. As shown in Fig. 7.21, the crude oil price did not follow the price path suggested by the theory, although prices have hardened in recent times. The theory relies on a number of restrictive assumptions and despite much theoretical interest, has not helped much in understanding the fuel price behaviour.

Fig. 7.21

Data source BP Statistical Review of World Energy, 2010

Oil price trend.

Therefore, from a practical point of view, the relevance and influence of the theory has been quite limited.