Accounting Price of an Exhaustible Resource: Response and Extensions

Abstract

Wei (Environ Resour Econ 60:579–581, 2015) presents a novel derivation of the accounting price for an exhaustible resource in a non-optimal economy subject to an allocation mechanism. We show that Wei (2015) and Hamilton and Ruta (Environ Resour Econ 42:53–64, 2009) are in fact employing different and mutually exclusive allocation mechanisms for the economy, and this explains the differences between the respective accounting prices. Because accounting prices must be defined subject to the allocation mechanism for the economy, the prices derived in the two papers are equally valid within their respective allocation domains. Further analysis shows that if there is declining marginal product of factors, a ‘Hartwick investment rule’ for the model economy (set investment just equal to depletion, valued at the accounting price) will lead to declining consumption for the Wei (2015) accounting price, and increasing consumption for the Hamilton and Ruta (2009) accounting price. This result is extended to consider the accounting standards recommended in the UN SEEA (System of environmental-economic accounting 2012: central framework. United Nations, European Commission, Food and Agriculture Organization of the United Nations, International Monetary Fund, Organisation for Economic Co-operation and Development, World Bank, 2012), as well as accounting for environmental externalities from resource use.

Annex: Optimal Resource Extraction with a Health Externality

For health stock H, health damage function $d\left( R \right)$, utility $U\left( {C,H} \right)$ and accumulation equations (13), (20) and (21), the objective is to maximize,

$$\begin{aligned} V=\mathop \int \nolimits _t^T U\left( {C,H} \right) \cdot e^{-\rho \left( {s-t} \right) }ds \end{aligned}$$

(24)

for constant pure rate of time preference $\rho$. The Hamiltonian function is given by,

$$\begin{aligned} \mathcal{H}=U+\gamma _1 {\dot{K}}+\gamma _{2}{\dot{H}}+\gamma _{3}{\dot{S}} \end{aligned}$$

where the $\gamma _i$ are the corresponding shadow prices. From the first order condition on consumption $(\frac{\partial U}{\partial C}=0)$ it follows that $\gamma _1 =U_C$, while the dynamic first order condition $\left( {\dot{\gamma }}_{1}=\rho \gamma _{1}-\frac{\partial H}{\partial K}\right)$ on ${\dot{U}}_{C}$ yields the standard Ramsey equation,

$$\begin{aligned} F_{K} =\rho -\left( \frac{{\dot{U}}_{C}}{U_{C}}\right) \end{aligned}$$

(25)

Defining $\gamma _2 \equiv U_C z$, where z is the value of a unit of the health stock H, the first order condition on extraction yields,

$$\begin{aligned} \gamma _3 =U_C \left( {F_R -{f}^{\prime }-zd^{\prime }}\right) \end{aligned}$$

The dynamic first order condition for ${\dot{\gamma }}_{3}$ therefore gives the Hotelling rule for this economy,

$$\begin{aligned} \frac{\frac{d}{dt}\left( {F_{R}} -f^{\prime }-zd^{\prime }\right) }{F_{R}-f^{\prime }-zd^{\prime }}=F_{K} \end{aligned}$$

(26)

Marginal rents on extraction therefore deduct the marginal damage to health $zd^{\prime }$. Next, the dyamic first order condition on ${\dot{\gamma }}_2$ gives,

$$\begin{aligned} \frac{d}{dt}\left( {U_C z} \right) =\rho U_C z-U_H \end{aligned}$$

and substituting (26) yields,

$$\begin{aligned} {\dot{z}}=F_Kz-\frac{U_H}{U_C} \end{aligned}$$

This differential equation has solution,

$$\begin{aligned} z=\mathop \int \nolimits _t^\infty \frac{U_H }{U_C }\left( s \right) \cdot e^{-\mathop \int \nolimits _t^s F_K \left( \tau \right) d\tau }ds \end{aligned}$$

(27)

Genuine saving G is therefore derived from the Hamiltonian function as,

$$\begin{aligned} G={\dot{K}}-zd\left( R\right) -\left( F_R-f^{\prime }-zd^{\prime }\right) R \end{aligned}$$

(28)