Open Access Versus Restricted Access in a General Equilibrium with Mobile Capital

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.

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

$$\begin{aligned} \pi _j=p\phi (z)z_j - w_k x^R_{kj}, \end{aligned}$$

(24)

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.¹⁷ The first-order condition for j’s choice of capital is thus:

$$\begin{aligned} \frac{\partial \pi _i}{\partial x^R_{kj}} = p\phi (z)F_k^R\left( 1,x^{R*}_{kj}\right) - w_k=0,\;\forall j. \end{aligned}$$

(25)

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:

$$\begin{aligned} p\phi (z)F_k^R\left( n,x^{R*}_{k}\right) = w_k. \end{aligned}$$

(26)

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 $pf^{\prime }(\hat{z})$:

$$\begin{aligned} c^{M}(w_{i}, w_{j})-1= 0, \end{aligned}$$

(27)

$$\begin{aligned} c^{R}(w_{i}, w_{j})-\alpha= 0. \end{aligned}$$

(28)

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

$$\begin{aligned} \frac{\partial w_{i}}{\partial \alpha }= \frac{-c_{j}^{M}}{c_{i}^{M}c_{j} ^{R}-c_{i}^{R}c_{j}^{M}}, \end{aligned}$$

(29)

$$\begin{aligned} \frac{\partial w_{j}}{\partial \alpha }= \frac{c_{i}^{M}}{c_{i}^{M}c_{j} ^{R}-c_{i}^{R}c_{j}^{M}}. \end{aligned}$$

(30)

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:

$$\begin{aligned} \frac{\partial w_{i}}{\partial \alpha }= \frac{-y^{R}x_{j}^{M}}{x_{i}^{M} x_{j}^{R}-x_{i}^{R}x_{j}^{M}}, \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial w_{j}}{\partial \alpha }= \frac{y^{R}x_{i}^{M}}{x_{i}^{M}x_{j} ^{R}-x_{i}^{R}x_{j}^{M}}. \end{aligned}$$

(32)

We consequently have:

$$\begin{aligned} \frac{\partial w_{i}}{\partial \alpha }< 0\quad \text{ iff }\quad \xi _{i}(\mathbf {x}^{M})>\xi _{i}(\mathbf {x}^{R}), \end{aligned}$$

(33)

$$\begin{aligned} \frac{\partial w_{j}}{\partial \alpha }> 0\quad \text{ iff }\quad \xi _{i}(\mathbf {x}^{M})>\xi _{i}(\mathbf {x}^{R}). \end{aligned}$$

(34)

Without loss of generality, we posit that $\xi _{i}(\mathbf {x}^{M})>\xi _{i}(\mathbf {x}^{R})$.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. $\square$