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This paper uses a general equilibrium assets-markets approach with arbitrageurs for valuing mineral resource deposit ownership. The results are contrasted with those delivered by a partial equilibrium approach. We show that in a general equilibrium assets-markets approach, arbitrageurs’ valuation of resource deposit rights commands a discount factor that adjusts not only for the time depreciation but also for changes in the resource stock size over time. A general equilibrium assets-markets approach with arbitrageurs leads to a more conservative management of exhaustible natural resources than a partial equilibrium approach does.

The main idea of this paper is to compare the implications of a general equilibrium (assets markets) model approach with that of a partial equilibrium (flow markets) model approach in valuing exhaustible natural resources.

A common approach used to address exhaustible resource problems is to rely on a partial equilibrium analysis. The given economy-wide interest rate is assumed to be exogenous in partial equilibrium, which implies decisions to hold mineral rights ownerships are modeled in isolation to the other non-resource assets markets. It is a one-way relationship in the sense that decisions to hold exhaustible resource ownerships do not affect the exogenous rate of interest, which is taken as given. In the [

No asset market exists in isolation. When an asset price in the market changes, this could cause a change in other asset prices. Therefore, it is worthwhile having a broad understanding of the financial markets and how each asset can impact another in order to trade accurately^{1}. The same arbitrageurs often invest across markets, and when returns change significantly in one asset market, arbitrageurs should expect similar tendencies to be happening in other markets as well. If the economy’s capital stock is efficiently allocated, then all marginal returns on asset ownerships across sectors traded in capital markets should be equalized. The role of capital markets is to allow capital to flow freely to the activity that provides the highest net gain. A general equilibrium model with arbitrageurs allows an interaction among markets by enabling capital to be withdrawn from less profitable activities and reinvested to more profitable activities.

In a general equilibrium model with arbitrageurs, there is more flexibility and interactions across all ownership rights in all sectors due to the speculative role of arbitrageurs in financial markets^{2}. Wealth may be owned in form of mineral rights ownership (mines, oilfields etc.). For an arbitrageur operating in capital markets, mineral rights ownership represents physical assets which can be used by arbitrageurs for financial speculations^{3}. When arbitrageurs have the desired mix of assets of various types in their portfolios, they are in a situation of portfolio equilibrium.

Our work can be related to the broad literature that contrasts general equilibrium and partial equilibrium effects in analyzing economic decisions (see for instance [

This paper uses a general equilibrium assets-markets approach with arbitrageurs for valuing mineral resource deposit ownership. The results are contrasted with those delivered by a partial equilibrium approach. We show that in a general equilibrium assets-markets approach, arbitrageurs’ valuation of resource deposit rights commands a discount factor that adjusts not only for the time depreciation but also for changes in the resource stock size over time. A general equilibrium assets-markets approach with arbitrageurs leads to a more conservative management of exhaustible natural resources than a partial equilibrium approach does.

This paper proceeds as follows. In Section 2, we present the valuation of mineral resource deposit ownership in a general equilibrium model with arbitrageurs. In Section 3, we compare a general equilibrium model with the [

In this paper, the general equilibrium model refers to assets markets where representative arbitrageurs can hold wealth in the form of conventional financial assets (say bonds) or in the form of exhaustible natural resource deposits. It is assumed that arbitrageurs are fully informed on the state of evolution of the natural resource stock size while making managing their portfolio. In other words, information on fundamentals of the economy including the resource stock size is incorporated in arbitrage trading activities. There is no uncertainty. The general pricing equilibrium is endogenously determined by the absence of arbitrage opportunities in assets markets [^{4}. The stock of resource is assumed to be a homogenous resource deposit^{5}. The size of the initial endowment of the resource stock is denoted by X 0 > 0 , and the remaining resource stock size at time t is denoted by X ( t ) .

Let us denote by V ( t ) the stock market value of the resource deposit ownership in a general equilibrium of assets markets, in dollars, of a remaining known homogeneous resource stock X ( t ) at time t^{6}. Denote by B ( t ) the stock market value of the conventional asset at time t.

A general equilibrium assets-market approach with arbitrageurs allows a feedback by enabling capital to be withdrawn from less profitable activities and reinvested to more profitable activities^{7}. In a general equilibrium assets-market approach with arbitrageurs, all sectors should strike the same financial rate of return. For arbitrageurs to include a resource deposit in their portfolio, the rate of change in the value of the resource deposition must equal the general equilibrium rate of return that prevails in all markets for capital assets. In the general equilibrium rate, the rate of return on mineral rights ownership V ˙ g e ( t ) / V g e ( t ) and the rate of return on other ownership categories B ˙ ( t ) / B ( t ) should be equalized^{8}. That is:

V ˙ g e ( t ) V g e ( t ) = B ˙ ( t ) B ( t ) = r ( X ( t ) , B ( t ) ) ︸ GeneralequilibriumRateofReturn (1)

where r ( X ( t ) , B ( t ) ) is the general equilibrium rate of return on assets in assets markets with arbitrageurs. The absence of arbitrage opportunities is sufficient for the existence of a general equilibrium in well functioning and competitive financial markets [

This condition can be thought of as the law of one price in a general asset market equilibrium setting^{9}.

Integrating both sides of Equation (1) leads to

V g e ( t ) = V 0 e ∫ 0 t r ( X ( τ ) , B ( τ ) ) d τ . (2)

In capital markets, the value of a capital stock is the product of the price of installed capital and the quantity of capital [^{10}:

V p e ( t ) = λ p e ( t ) X ( t ) . (3)

Substituting Equation (3) into Equation (2), the following valuation rule is obtained

λ g e ( t ) ︸ Generalequilibriumshadowprice = λ 0 ︸ Initialvalue [ X 0 X ( t ) ] ︸ Sizechangescale [ e ∫ 0 t r ( X ( τ ) , B ( τ ) ) d τ ] ︸ Timediscounting . (4)

Equation (4) tells us that from a general equilibrium assets-market approach, the endogenous discount rate used by arbitrageurs for valuing exhaustible

resource deposits rights [ X 0 X ( t ) e ∫ 0 t r ( X ( τ ) , B ( τ ) ) d τ ] adjusts not only for the time

depreciation but also for changes in the resource size over time.

In this section, we compare the features derived from partial equilibrium (flow markets) considerations with those derived from general equilibrium in assets markets.

In the [^{11},

l i m t → ∞ λ p e ( t ) X ( t ) = 0 (5)

λ ˙ p e ( t ) λ p e ( t ) = r ˜ ( t ) . (6)

l i m t → ∞ λ p e ( t ) X ( t ) = 0 (5)

In other words, from the partial equilibrium approach, stocks of natural resources are like capital goods. The shadow price of the marginal unit of resource in situ at time t is obtained as follows:

λ p e ( t ) ︸ Partialequilibriumshadowprice = λ 0 ︸ Initialvalue [ e ∫ 0 t r ˜ ( τ ) d τ ] ︸ Timediscounting . (7)

With a partial equilibrium setting, the value of one unit of the in situ resource time t is obtained by applying the same discounted cash flow technique used for valuing conventional assets whose size is fixed over time.

It is instructive to shed light on how the [

r ( X ( t ) , B ( t ) ) = r ˜ ( t ) . (8)

It is worth noting that the only case where the valuation from a general equilibrium assets-market approach with arbitrageurs equals the valuation from a partial equilibrium approach

λ g e ( t ) = λ p e ( t ) . (9)

happens if and only if

X ( t ) = X 0 at any time t > 0 . (10)

The implication (10) suggests that from a general equilibrium assets-market approach with arbitrageurs perspective, the [

Comparing the valuation of exhaustible resources using the partial equilibrium Equation (7) and the valuation of exhaustible resources using a general equilibrium (4), it follows that

λ g e ( t ) = [ X 0 X ( t ) ] λ p e ( t ) (11)

Since the resource size can only decrease over time^{12}; that is X ( t ) X 0 < 1 , it

follows that

λ g e ( t ) ︸ Generalequilibriumshadowprice > λ p e ( t ) ︸ Partialequilibriumshadowprice (12)

An interesting implication of this result is that a general equilibrium assets-market approach with arbitrageurs leads to a more conservative utilization of exhaustible natural resources. The presence of arbitrageurs allows a more conserving resource economy. The management of exhaustible resources under a general equilibrium assets-market approach leads to a more conservative policy than that obtained from a partial equilibrium framework. In a general equilibrium assets-market approach, all markets are simultaneously modeled and interact with each other.

Furthermore, as shown by [^{13}, the value of the remaining stock at time t is given

by V p e ( X ( t ) ) = ∫ t ∞ e − r ˜ ( τ − t ) λ p e ( τ ) q ( τ ) d τ = λ p e ( t ) X ( t ) , where λ p e ( τ ) is the

shadow price of the resource at time t in partial equilibrium (which is equal the price assuming that the extraction cost is zero) and q ( τ ) is the quantity extracted at time t, and X ( t ) is the size of the remaining resource stock at time t.

The relative change in value of the stock of the resource can be derived as (see [

V ˙ p e V p e = r ˜ − λ p e ( t ) q ( t ) V p e . (13)

Therefore taking into account Equation (8) and comparing Equation (13) with Equation (1) leads to

V ˙ p e V p e < V ˙ g e V g e , (14)

which implies

V g e ( t ) > V p e ( t ) . (15)

This result tells us that the value of exhaustible resources deposits analyzed from a general equilibrium assets-market approach with arbitrageurs is greater than the value of exhaustible resources deposits analyzed from a partial equilibrium model. The resource owner in partial equilibrium does not internalize the effect of the resource size change on the rate of interest, which is taken as given. A direct implication is that in doing green accounting it is important to emphasize that the valuation depends on whether a partial equilibrium or a general equilibrium assets-market approach is used. An integrated and multi-sectorial asset market approach results in greater shadow price for mineral rights ownership and is likely to be more sustainable over the long term^{14}.

In this paper, we compare the valuation of exhaustible resources from both a general equilibrium assets-market approach with arbitrageurs and a standard partial equilibrium approach. It is shown that a correction term that explicitly accounts for the change in the resource size enters the discount factor used by arbitrageurs in a general equilibrium assets-market approach. Because of the presence of this term that adjusts for changes in the resource stock size over time, capital markets analyzed from a general equilibrium model with arbitrageurs’ standpoint are more conservative than that obtained from a partial equilibrium approach in which the rate of interest is exogenously given. In some sense, our result echoes [

Our paper features a simple general model showing that integrated assets markets with informed arbitrageurs can play a better role in conserving natural resources than a partial equilibrium model approach does. This simple model can bring some insights on the link between capital market polices and long-term conservation of exhaustible natural resources, which is an important component of sustainable development. It shows that fully integrating accurate information on resource stock size into capital markets policy practices used by market participants would contribute to the promotion of a more resource conservative economy.

From a policy perspective, the conclusions derived from this paper provide a theoretical rationale behind the idea of conservation finance, a recently developed framework whose objective is to move resource conservation issues into mainstream assets markets [

Our paper suffers from some limitations, one of which is inherent to general equilibrium approaches. As mentioned by [^{15}. Another limitation of this research is that it does not account for issues such as asymmetric information, uncertainties in resource markets, technological change, and new discoveries. Future research will incorporate these features.

Kakeu, J. (2018) Valuing Exhaustible Resource Ownership: General Equilibrium Assets-Markets versus Partial Equilibrium. Theoretical Economics Letters, 8, 844-853. https://doi.org/10.4236/tel.2018.85059