Cartel Behaviour in an Exhaustible Resource Industry

Abstract

Under low-elasticity demand, a resource monopoly is supposed to be constrained either by the presence of competing participants in the market or by the existence of a substitute for the natural resource. In the presence of a competitive fringe, the cartel’s activity in an exhaustible resource industry violates the Herfindahl principle presented in Chap. 2 that an advantageous oil region depletes its resource stock before a disadvantageous region begins extraction. In the cartel-fringe models of resource industry considered in this chapter, the time sequencing of production dramatically changes and permits the case where the low-cost cartel and the high-cost fringe produce simultaneously. In the presence of a backstop technology, a perfect substitute provides a ceiling on the exhaustible resource price. In the model with such technology, the fringe fully exhausts its resource before the cartel becomes a monopoly that sells the resource at the backstop price. The cartel initially makes a strategic choice of resource allocation over time between the transition phase and the backstop phase.

Appendices

Appendices

1.1 A.1 Formula (9.11)

The solution for differential equation (9.10) is

$$ P(t)=\frac{\alpha +c}{2}+{w}_0{e}^{rt}=c+\frac{\alpha -c}{2}+{w}_0{e}^{rt}. $$

(9.51)

The terminal condition x(T_M) = 0 implies that P(T_M) = α, hence, from Eq. (9.51),

$$ \alpha =\frac{\alpha +c}{2}+{w}_0{e}^{r{T}_M} $$

and ${w}_0=\frac{1}{2}\left(\alpha -c\right){e}^{-r{T}_M}$.

1.2 A.2 Condition (9.15)

The demand elasticity is $\sigma (x)=-\mathcal{P}(x)/x{\mathcal{P}}^{\prime }(x)$ for the domain of x such that ${\mathcal{P}}^{\prime }(x)<0$. The derivative of demand elasticity is

$$ {\sigma}^{\prime }(x)=-\frac{x\mathcal{P}^{\prime }{(x)}^2-\mathcal{P}(x)\left({\mathcal{P}}^{\prime }(x)+x{\mathcal{P}}^{\prime \prime }(x)\right)}{{\left(x{\mathcal{P}}^{\prime }(x)\right)}^2}. $$

It is negative if, and only if,

$$ \frac{x{\mathcal{P}}^{\prime }(x)}{\mathcal{P}(x)}<1+\frac{x{\mathcal{P}}^{\prime \prime }(x)}{{\mathcal{P}}^{\prime }(x)}, $$

(9.52)

which implies

$$ 2+\frac{x{\mathcal{P}}^{\prime \prime }(x)}{{\mathcal{P}}^{\prime }(x)}>0, $$

(9.53)

because $-x{\mathcal{P}}^{\prime }(x)/\mathcal{P}(x)<1$ for ρ(x) > 0.

Consider the equation for the marginal revenue growth (9.8) with c = 0. On the one hand, $\dot{\rho}(x)=\dot{\mathcal{P}}(x)+{\mathcal{P}}^{\prime }(x)\dot{x}+x{\mathcal{P}}^{\prime \prime }(x)\dot{x}$. On the other hand, $\dot{x}=\dot{\mathcal{P}}(x)/{\mathcal{P}}^{\prime }(x)$. Hence, $\dot{\rho}(x)=\dot{\mathcal{P}}(x)\left(2+x{\mathcal{P}}^{\prime \prime }(x)/{\mathcal{P}}^{\prime }(x)\right)$and Eq. (9.8) is equivalent to

$$ \dot{\mathcal{P}}(x)=r\frac{\mathcal{P}(x)+x{\mathcal{P}}^{\prime }(x)}{2+x{\mathcal{P}}^{\prime \prime }(x)/{\mathcal{P}}^{\prime }(x)}. $$

Condition $\dot{\mathcal{P}}(x)<r\mathcal{P}(x)$ holds for x > 0 if, and only if, $\mathcal{P}(x)+x{\mathcal{P}}^{\prime }(x)<\mathcal{P}(x)\left(2+x{\mathcal{P}}^{\prime \prime }(x)/{\mathcal{P}}^{\prime }(x)\right)$, which is equivalent to (9.52), the necessary and sufficient condition for σ^′(x) < 0. Hence, condition σ^′(x) < 0 is necessary and sufficient for $\dot{P}< rP$. Condition (9.53) is necessary and sufficient for ${\rho}^{\prime }(x)=2{\mathcal{P}}^{\prime }(x)+x{\mathcal{P}}^{\prime \prime }(x)<0$, hence σ^′(x) < 0 is sufficient for ρ^′(x) < 0.

1.3 A.3 Equation (9.24)

The general solution of differential equation (9.20) is

$$ {x}_b(t)=D-k{e}^{rt}, $$

where D = β⁻¹(α − c_b − Δc) and k is an unknown constant. From (9.23) we have:

$$ {\displaystyle \begin{array}{l}\beta {x}_a(t)=\beta \left(q(t)-{x}_b(t)\right)=\left(\alpha -{c}_b\right)\left(1-{e}^{r\left(t-{T}_b\right)}\right)-\beta D+\beta {ke}^{rt}\\ {}\kern6em =\beta {ke}^{rt}-\left(\alpha -{c}_b\right){e}^{r\left(t-{T}_b\right)}+\varDelta c.\end{array}} $$

The termination condition for the cartel is x_a(T_a) = 0, implying that

$$ \beta k{e}^{r{T}_a}=\left(\alpha -{c}_b\right){e}^{r\left({T}_a-{T}_b\right)}-\Delta c\ \mathrm{or} $$

$$ \beta k{e}^{rt}=\left(\alpha -{c}_b\right){e}^{r\left(t-{T}_b\right)}-\Delta c{e}^{r\left(t-{T}_a\right)}. $$

Consequently, $\beta {x}_a(t)=\Delta c-\Delta c{e}^{r\left(t-{T}_a\right)}=\Delta c\left(1-{e}^{r\left(t-{T}_a\right)}\right)$.

1.4 A.4 The Objective Function (9.39)

During the transition phase, t ≤ T_b, the cartel produces x_a = y(P) − x_b = P^−σ − x_b. The initial present value of the cartel’s rent during this phase is

$$ \underset{0}{\overset{T_b}{\int }}{e}^{- rt}P(t){x}_a(t) dt={P}_0\underset{0}{\overset{T_b}{\int }}\left(P{(t)}^{-\sigma }-{x}_b(t)\right) dt={P}_0\left(Q-{S}_b\right)={P}_0\left({S}_a-B\right) $$

due to (9.36) and the resource balance B = S_a + S_b − Q. During the backstop phase, T_b ≤ t ≤ T_a, the cartel’s output is x_a(t) ≡ F^−σ, and the initial present value for this phase is

$$ \underset{T_b}{\overset{T_a}{\int }}{e}^{- rt}F{x}_a(t) dt=F\underset{T_b}{\overset{T_a}{\int }}{F}^{-\sigma }{e}^{- rt} dt={e}^{-r{T}_b}F\frac{1-{e}^{-r\Delta T}}{r{F}^{\sigma }}={e}^{-r{T}_b}F\frac{1-{e}^{- rB{F}^{\sigma }}}{r{F}^{\sigma }}, $$

since ΔT = BF^σ. Thus, the cartel’s objective function (9.30) transforms to

$$ {W}_a=\underset{P_0,B}{\max}\left[{P}_0\left({S}_a-B\right)+{e}^{-r{T}_b}F\frac{1-{e}^{- rB{F}^{\sigma }}}{r{F}^{\sigma }}\right]. $$