What drives resource prices? A qualitative review with recommendations for further development of the Hotelling model

Abstract

This paper examines which factors have an impact on the price of finite resources. Firstly, empirical and theoretical approaches used to study the development of resource prices are considered. These drivers are then studied, individually as well as altogether, starting with the Hotelling model. Implications for the future price of finite resources are subsequently reviewed, and theoretical hypotheses derived. These are then tested by means of a qualitative review based on empirical findings. This paper demonstrates how current approaches have limited applicability in determining which drivers impact the price of finite resources. One of the main reasons is identified as the Hotelling’s model’s lack of accounting for growth in demand for finite resources.

Appendices

Appendix 1

Derivation of the price path of a finite resource: Basic Hotelling model

$$ G\left({y}_t\right)={\displaystyle \sum_{t=0}^T}\frac{p_t{y}_t}{{\left(1+\gamma \right)}^t} $$

In order to solve the equation – with the constraint of the finiteness of the resource – Lagrange’s approach can be applied. Therefore, the Lagrange function

$$ L\left(y,t\right)={\displaystyle \sum_{t=0}^T}\frac{p_t{y}_t}{{\left(1+\gamma \right)}^t}+\lambda \left(\overline{y}-{\displaystyle \sum_{t=0}^T}{y}_t\right) $$

and the corresponding system of equations

$$ \begin{array}{c}\hfill \frac{\partial L}{\partial {y}_0}={p}_0-\lambda =0\hfill \\ {}\hfill \frac{\partial L}{\partial {y}_1}=\frac{p_1}{1+\gamma }-\lambda =0\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill \frac{\partial L}{\partial {y}_T}=\frac{p_T}{{\left(1+\gamma \right)}^T}-\lambda =0\hfill \end{array} $$

including the constraint

$$ \overline{y}={\displaystyle \sum_{t=0}^T}{y}_t $$

can be constructed. Solving this set of equations, it can be easily derived that maximum profit is reached if discounted prices of each period of time are equal:

$$ {p}_0=\frac{p_1}{1+\gamma }=\frac{p_2}{{\left(1+\gamma \right)}^2}=\dots =\frac{p_T}{{\left(1+\gamma \right)}^T}=\lambda $$

Transforming this equation, the price path of a finite resource can be derived as follows:

$$ {p}_t={p}_0{\left(1+\gamma \right)}^t $$

Appendix 2

Derivation of the price path of a finite resource: Simultaneous integration of supply-side drivers

If extraction costs are assumed to exist in a monopolistic market structure which also includes the risk determinant, backstop technology and recycling, the following Lagrangian function can be derived:

$$ \begin{array}{c}\hfill L\left(y,t\right)={\displaystyle \sum_{t=0}^T}\frac{p\left(y,t\right)y(t)}{{\left(1+{\gamma}_{ra}\right)}^t}-{\displaystyle \sum_{t=0}^T}\frac{c(t)y(t)}{{\left(1+{\gamma}_{ra}\right)}^t}+{\displaystyle \sum_{t=0}^T}\frac{\varDelta {c}_{tp}(t)y(t)}{{\left(1+{\gamma}_{ra}\right)}^t}\hfill \\ {}\hfill -{\displaystyle \sum_{t=0}^T}\frac{\varDelta {c}_{dg}(t)y(t)}{{\left(1+{\gamma}_{ra}\right)}^t}+{\displaystyle \sum_{t=0}^T}\frac{\varDelta {c}_{re}(t)y(t)}{{\left(1+{\gamma}_{ra}\right)}^t}-\lambda \left(\overline{y}-{\displaystyle \sum_{t=0}^T}y(t)\right)\hfill \end{array} $$

The corresponding system of equations can be set as

$$ \begin{array}{c}\hfill \frac{\partial L}{\partial {y}_0}=\frac{\partial {p}_0\left({y}_0\right)}{\partial {y}_0}{y}_0+{p}_0\left({y}_0\right)-{c}_0+\Delta {c}_{tp,0}-\Delta {c}_{dg,0}+\Delta {c}_{re,0}-\lambda =0\hfill \\ {}\hfill \frac{\partial L}{\partial {y}_1}=\frac{\frac{\partial {p}_1\left({y}_1\right)}{\partial {y}_1}{y}_1}{1+{r}_{ra}}+\frac{p_1\left({y}_1\right)}{1+{r}_{ra}}-\frac{c_1}{1+{r}_{ra}}+\frac{\Delta {c}_{tp,1}}{1+{r}_{ra}}-\frac{\varDelta {c}_{dg,1}}{1+{r}_{ra}}+\frac{\Delta {c}_{re,1}}{1+{r}_{ra}}-\lambda =0\hfill \\ {}\hfill \vdots \hfill \\ {}\hfill \frac{\partial L}{\partial {y}_T}=\frac{\frac{\partial {p}_T\left({y}_T\right)}{\partial {y}_T}{y}_T}{{\left(1+{r}_{ra}\right)}^T}+\frac{p_T\left({y}_T\right)}{{\left(1+{r}_{ra}\right)}^T}-\frac{c_T}{{\left(1+{r}_{ra}\right)}^T}+\frac{\varDelta {c}_{tp,T}}{{\left(1+{r}_{ra}\right)}^T}-\frac{\varDelta {c}_{dg,T}}{{\left(1+{r}_{ra}\right)}^T}+\frac{\varDelta {c}_{re,T}}{{\left(1+{r}_{ra}\right)}^T}-\lambda =0\hfill \end{array} $$

including the constraint

$$ \overline{y}={\displaystyle \sum_{t=0}^T}{y}_t. $$

Solving this system of equations analogously to the above-mentioned basic Hotelling model, the following optimal price path can be derived, representing the simultaneous appearance of extraction costs – including adjustments caused both by technical progress and degradation – as well as monopolistic market power, risk, backstop technology and recycling:

$$ \begin{array}{c}\hfill p\left(y,t\right)= min\left(\right(\frac{\partial {p}_0\left({y}_0\right)}{\partial \left({y}_0\right)}{y}_0{\left(1+{\gamma}_{ra}\right)}^t-\frac{\partial p\left(y,t\right)}{\partial y(t)}y(t)+{p}_0\left({y}_0\right){\left(1+{\gamma}_{ra}\right)}^t-{c}_0{\left(1+{\gamma}_{ra}\right)}^t+c(t)\hfill \\ {}\hfill +\varDelta {c}_{tp,0}{\left(1+{\gamma}_{ra}\right)}^t-\varDelta {c}_{tp,t}-\varDelta {c}_{dg,0}{\left(1+{\gamma}_{ra}\right)}^t+\varDelta {c}_{dg,t}+\varDelta {c}_{re,0}{\left(1+{\gamma}_{ra}\right)}^t\hfill \\ {}\hfill -\varDelta {c}_{re,t}\left){p}_{bs,t}\right)\hfill \end{array} $$