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.
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Notes
Roughly speaking, an allocation mechanism is an algorithm or set of rules that maps initial endowments of assets into a unique future path for the economy. The allocation need not be optimal.
In what follows, all variables are assumed to be functions of time, unless otherwise stated.
In addition, the elasticity of substitution between produced capital and the exhaustible resource must be equal to 1 in the Hartwick (1977) model, and the elasticity of output with respect to produced capital must be greater than the elasticity for the resource.
To be precise, (W3) implies that \({\dot{K}}> 0\) since \({\dot{N}}<0\) (expression 16), and so \({\dot{F}}_{K}=F_{KK}{\dot{K}}< 0\).
Hamilton and Hartwick (2005) derive a generalized Hartwick rule, showing that \({\dot{C}}=F_{K}G-{\dot{G}}\) for genuine saving G. The standard Hartwick rule is a special case for \(G={\dot{G}}=0\); a path where genuine saving is identically 0 at each point in time will exhibit constant consumption.
This simplification is not necessary, but it streamlines the derivation of the main results.
Recall that the Hotelling rule, expression (26), applies in the optimal economy. As a result the growth in unit rents is completely offset by the discount rate.
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Wei T (2015) Accounting price of an exhaustible resource: a comment. Environ Resour Econ 60:579–581
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Annex: Optimal Resource Extraction with a Health Externality
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,
for constant pure rate of time preference \(\rho \). The Hamiltonian function is given by,
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,
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,
The dynamic first order condition for \({\dot{\gamma }}_{3}\) therefore gives the Hotelling rule for this economy,
Marginal rents on extraction therefore deduct the marginal damage to health \(zd^{\prime }\). Next, the dyamic first order condition on \({\dot{\gamma }}_2\) gives,
and substituting (26) yields,
This differential equation has solution,
Genuine saving G is therefore derived from the Hamiltonian function as,
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Hamilton, K., Ruta, G. Accounting Price of an Exhaustible Resource: Response and Extensions. Environ Resource Econ 68, 527–536 (2017). https://doi.org/10.1007/s10640-016-0030-6
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DOI: https://doi.org/10.1007/s10640-016-0030-6