## Abstract

Open Access versus Restricted Access in a General Equilibrium with Mobile Capital We consider an economy with two sectors, resource and manufacturing, in a general-equilibrium setting. Two property regimes in the resource sector are compared, open access versus restricted access, with both labor and capital mobility. We first contribute by deriving the multi-factor demand conditions under open access. We provide necessary and sufficient conditions for a factor to gain from a property regime change. Redistributive effects depend crucially on relative factor intensities. Contrary to common wisdom, labor may gain from being “expelled” from the resource sector following privatization.

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## Notes

- 1.
Throughout the analysis, depending on the context or associated literature, terms like “privatization”, “restricted access” or “regulated access” are used interchangeably to represent a managed resource under a property regime to be defined precisely in Sect. 3. This is in opposition to a situation of an unmanaged resource, where terms like “open access”, “free access” or “uncontrolled access” are used. We refrain from using the expression “common property” as it may or may not represent a managed resource, depending on the literature.

- 2.
- 3.
The introduction of stock dynamics goes beyond the scope of this paper. It would, however, be straightforward to use our specification and introduce stocks dynamics by assuming period-by-period rent dissipation, as done in Manning et al. (2018) and Squires and Squires and Vestergaard (2013). See Brooks et al. (1999) for a game-theoretic justification of this approach. We leave that for future work.

- 4.
Finnoff and Tschirhart (2008), Finnoff and Tschirhart (2011) and Deacon et al. (2011) explicitly make use of such a “nested” representation of effective efforts in a CGE model, using a CES technology. Manning et al. (2018) implicitly do the same using a Cobb–Douglas formulation. This in fact only requires the (mild) assumption of homotheticity.

- 5.
The subscript of a function denotes a partial derivative.

- 6.
- 7.
It is straightforward to extend this proposition to any number of inputs or, more generally, to any margin that is left unrestricted.

- 8.
Note that the excessive use of capital under open access is distinct from the concept of “capital stuffing”, as the latter refers to situations of regulated fisheries (Townsend 1985).

- 9.
We are grateful to a referee for raising this point as we believe that its consideration contributes greatly to the analysis.

- 10.
For the case of fisheries, IREPA (2006) estimates annual depreciation rates of 7% for hulls, 25% for engines and 35% for “other equipment”. See also the interesting discussion by Wilen (2009) regarding processing stranded capital arguing that “Most capital involved in fisheries processing is malleable and not likely to be devalued as a result of rationalization”.

- 11.
This is a consequence of Shephard’s lemma which states that \({\tilde{\mathbf{a}}}^{S}=(c_{i} ^{S}({\tilde{\mathbf{w}}}),c_{j}^{S}({\tilde{\mathbf{w}}}))\). See, for instance, chapter 3 of Woodland (1982).

- 12.
Note also that if the cost functions were to intersect at multiple points, only one would be compatible with any specific total factor endowment.

- 13.
Even in 2017 Dean (2017), the New York Times reports on a surprisingly similar situation in Myanmar.

- 14.
Note that the use of factor income shares as estimators for the parameters of the Cobb–Douglas production function requires that all input uses respect the first-order conditions for profit maximization. This is an unrealistic assumption to make for most developing countries as the land and labor markets are often non-competitive. We understand that it is for this reason that Mundlak, Butzer and Larson (2012) chose to concentrate the discussion on elasticities.

- 15.
We are grateful to the Editor for raising this point.

- 16.
The value for \(\theta\) corresponds to an elasticity of substitution of .782.

- 17.
- 18.
Note that the problem is undefined for \(\xi _{i}(\mathbf {x}^{M})=\xi _{i}(\mathbf {x}^{R})\).

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This paper has benefited from comments by Stefan Ambec, Brian Copeland, Pierre Lasserre, Martin Weitzman and seminar participants at the University of Ottawa, Université de Rouen, the CREE Meetings, the Conference on Environment and Natural Resources Management in Developing and Transition Economies at Université d’Auvergne, the Montreal Natural Resources and Environmental Economics Workshop, Economix at Université Paris Ouest Nanterre La Défense, the School of Economics at the University of Queensland and the Congreso Sobre México at Universidad Iberoamericana.

## Appendices

### Appendix A: Self-employment and Cost-Minimization Under an Open-Access Regime

In this section, we show that cost minimization occurs under open access with self-employed users/workers, as long as they are free to move between the sectors.

Let *n* denote the total number of workers exploiting the resource. (We keep this number fixed for now. It will be made endogenous below.) \(z_j=F^R(x^R_{lj},x^R_{kj})\) denotes the effective effort expended on the resource by worker *j*, \(j\in \{1,2,\ldots ,n\}\). The total effort expended on the resource is thus given by \(z=\sum _{j=1}^{n}z_j\), which yields total output *f*(*z*) and average product of effort \(\phi (z)\). Suppose, to simplify, that a worker inelastically supplies one unit of labor and that she cannot split her time between the manufacturing and the resource sector. We have \(x^R_{lj}=1\), \(\forall j\), and individual effort is thus uniquely determined by a worker’s choice of capital input \(x^R_{kj}\) at cost \(w_k\). We now turn to this choice.

Worker *j*’s net payoff from the resource is given by

where \(z_j=F^R(1,x^R_{kj})\). Now if *n* is large enough—as must be the case under open access over a large resource site—each resource worker will take \(\phi (z)\) as given when deciding on her individual effort level.^{Footnote 17} The first-order condition for *j*’s choice of capital is thus:

This condition has the same interpretation as the one provided above for conditions (12). Now with constant returns to scale, we have \(F_k^R\left( 1,x^{R*}_{kj}\right) =F_k^R\left( n,x^{R*}_{k}\right)\), where \(x^{R*}_{k}=nx^{R*}_{kj}\). Consequently, for a given number of workers, the equilibrium total amount of capital hired on the resource site is given by:

For the case of capital, we therefore have the same condition as in (12). Given *n*, this condition determines the equilibrium total and individual efforts being supplied, respectively \(z^*\) and \(z^*/n\), along with the corresponding equilibrium individual net payoff \(\pi ^*=(p\phi (z^*)z^* - w_k x^{R*}_{k})/n\). Let us now turn to the determination of *n*.

Given that workers are free to move between the sectors, the equilibrium allocation of workers between manufacturing employment and resource exploitation must be such that they are indifferent between them. This requires that \(\pi ^*=w_l\). Given that the prevailing wage \(w_l\) in the manufacturing sector must be regarded as the true opportunity cost of workers exploiting the resource, this last condition implies total rent dissipation, i.e., \(p\phi (z^*)z^* =w_ln + w_k x^{R*}_{k}\). This leads to the following proposition:

###
**Proposition 6**

*With self-employed resource users who are free to move between the sectors, the open-access equilibrium is equivalently characterized by conditions* (12).

###
*Proof*

The case with \(i=k\) has already been derived as condition (26) above. We now derive the case for \(i=l\). Because \(F^{R}(\mathbf{x} ^{R})\) has constant returns to scale, we have \(z=x^R_lF^R_l+x^R_kF^R_k\). Substituting (26), this gives \(z=x^R_lF^R_l+w_kx^R_k/p\phi (z)\) or, equivalently, \(p\phi (z)z=p\phi (z)x^R_lF^R_l+w_kx^R_k\). Making use of the above total rent dissipation result, this implies that \(p\phi (z)x^R_lF^R_l+w_kx^R_k=w_lx^R_l + w_k x^{R}_{k}\), where \(x^R_l\) substitutes for *n*. This simplifies to \(p\phi (z)F^R_l=w_l\). \(\square\)

### Appendix B: Proof that \(c^{R}({\hat{\mathbf{w}}})\le c^{R}({\tilde{\mathbf{w}}})\)

Recall that equilibrium conditions for factor payments must respect condition (6) and either of (11) or (10). Let us express those by the following set of two equations, where parameter \(\alpha\) is either equal to \(p\phi (\tilde{z})\) or \(pf^{\prime }(\hat{z})\):

Differentiating these two expressions with respect to parameter \(\alpha\) and making use of Cramer’s rule yields:

Now according to Shephard’s lemma, \(c_{i}^{S}\) denotes the quantity of factor *i* used in sector *S* per unit of output, i.e., \(y^{S}c_{i}^{S}=x_{i}^{S}\), \(S\in \{M,R\}\), and similarly for factor *j*. Inserting this into the above two equations yields:

We consequently have:

Without loss of generality, we posit that \(\xi _{i}(\mathbf {x}^{M})>\xi _{i}(\mathbf {x}^{R})\).^{Footnote 18} The above therefore implies that an increase in \(\alpha\) leads to a decrease in \(w_{i}/w_{j}\).

Assume now that \(c^{R}(\hat{w})>c^{R}(\tilde{w})\). This implies that \(\alpha\) must take on a larger value under RA than OA and thus, according to the above result, \(\hat{w}_{i}/\hat{w}_{j}<\tilde{w}_{i}/\tilde{w}_{j}\). But \(c^{R} (\hat{w})>c^{R}(\tilde{w})\) also implies that \(\hat{z}<\tilde{z}\), as can be readily seen from Fig. 2. Now according to Corollary 2, \(\xi _{i}(\mathbf {x}^{M})\ge \xi _{i}(\mathbf {x}^{R})\) implies \(\hat{w}_{i}/\hat{w}_{j}\ge \tilde{w}_{i}/\tilde{w}_{j}\) when \(\hat{z} <\tilde{z}\). A contradiction. \(\square\)

### Appendix C: Proof that \(\hat{z}<\tilde{z}\)

Assume to the contrary that \(\hat{z}\ge \tilde{z}\). Then, it must be the case that \(c^{R}(\hat{w})<c^{R}(\tilde{w})\). In line with the analysis of Appendix 1 above, this calls for a lower value of \(\alpha\) under RA as compared to OA and therefore \(\hat{w}_{i}/\hat{w}_{j}>\tilde{w}_{i}/\tilde{w}_{j}\). Consequently, factor *i* is used less intensively under RA than OA and, as can be readily seen in Fig. 1, this requires \({\hat{\mathbf{x}}}^{R}\ll {\tilde{\mathbf{x}}}^{R}\) and thus \(\hat{z}<\tilde{z}\). A contradiction. \(\square\)

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Congar, R., Hotte, L. Open Access Versus Restricted Access in a General Equilibrium with Mobile Capital.
*Environ Resource Econ* (2021). https://doi.org/10.1007/s10640-021-00542-4

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### Keywords

- Property rights
- Natural resources
- Open access
- General equilibrium
- Mobile capital
- Factor payments
- Factor intensities
- Income distribution

### JEL Classification

- D02
- D23
- D33
- K11
- Q2
- N5
- O13