Corruption, conflict and the management of natural resources

Abstract

The documented link between natural resources and civil conflict is not well understood. This paper uses a political economy framework to explore the prevalence of resource-based civil conflict driven by group-level discontent. The theoretical model proposed here offers a policy-based explanation: under conditions related to the quality of governance, discontent about resource management can affect the likelihood of an insurgency. Resource policy arises endogenously as the corrupt government trades-off industry contributions and the cost induced by manifestations of resource-related discontent. The conservation effects of civil unrest are analyzed and government corruption emerges as an important determinant of conflict. The paper also presents some empirical support for the model’s predictions.

Appendix

• 1. The first order conditions for the honest government’s optimal control problem are the following:

  $$\begin{aligned} \frac{d\mathcal {H}}{dQ} =\pi _{Q}-\mu . \end{aligned}$$

  Notice that for specific profit functions that are multiplicatively separable in Q and take the form $\Pi (p,S)Q$ this condition does not depend on the control variable, and so the solution is reached by taking the most rapid approach path (MRAP) as follows: if $\frac{d\mathcal {H}}{dQ}>0$ by setting harvesting at the maximum level $Q=Q_{max}$, where $Q_{max}$ is the exploitation resulting from employing the maximum effort available (bounded perhaps by the available industry labour force and/or technological parameters), while if $\frac{d\mathcal {H}}{dQ}<0$ by setting harvesting at the minimum level $Q=Q_{min}$ (which can be zero or the break-even point for the firms). This is allowed until the shadow price of the resource $\mu$ respectively increases or decreases to equal the unit profit. The other optimality relations are derived from:

  $$\begin{aligned} \frac{d\mathcal {H}}{dS}&= \delta \mu -\dot{\mu }=\pi _{S}+U_{S}+\mu (F_{S}-\bar{H}_S)\\ \frac{d\mathcal {H}}{d\mu }&= \dot{S}=F(S)-H-\bar{H}(S). \end{aligned}$$

  while the transversality condition has to hold:

  $$\begin{aligned} \mathop {lim}\limits _{t\rightarrow \infty }\quad e^{-\delta t}\mu (t)S(t)=0. \end{aligned}$$

  Imposing the conditions of zero-growth of the state and co-state variables in a steady-state equilibrium and assuming the transversality condition holds, we obtain:

  $$\begin{aligned} (F_{S}-\bar{H}_S)+\frac{\pi _{S}+U_{S}}{\pi _{Q}}=\delta . \end{aligned}$$

  To determine the transitional path dynamics, take time-derivative of the first condition to obtain:

  $$\begin{aligned} \dot{\mu }=\pi _{QQ}\dot{Q}+\pi _{QS}\dot{S} \end{aligned}$$

  which can be substituted into the second condition as follows:

  $$\begin{aligned} (\delta -F_{S}-\bar{H}_S)\pi _{Q}-\pi _{QQ}\dot{Q}-\pi _{QS}[F(S)-Q-\bar{H}(S)]=\pi _{S}+U_{S}. \end{aligned}$$

  The $\dot{H}=0$ locus can then be determined as:

  $$\begin{aligned} H=\underbrace{\frac{\pi _{S}+U_{S}-(\delta -F_{S}-\bar{H}_S)\pi _{Q}}{\pi _{QS}}}+\left( F(S)-\bar{H}(S)\right) , \end{aligned}$$

  where the sign of its slope $\left( \frac{dH}{dS}|_{\dot{H}=0}\right)$ is ambiguous. Notice that setting the numerator of the fraction to zero, which is our golden rule above, is equivalent to imposing the steady-state harvesting condition. Transitional harvesting is decreasing towards the steady-state value when the fraction (its numerator) is positive, and increasing when it is negative.

• 2. According to the maximum principle, the first order conditions for an interior solution to this corrupt government optimal control problem are:

  $$\begin{aligned} \frac{d\mathcal {H}}{dQ}&= 0=\pi _{Q}+(\beta -1)B_{Q}-\mu \\ \frac{d\mathcal {H}}{dS}&= \delta \mu -\dot{\mu }=\pi _{S}+U_{S}+\mu (F_{S}-\bar{H}_S)\\ \frac{d\mathcal {H}}{d\mu }&= \dot{S}=F(S)-Q-\bar{H}(S), \end{aligned}$$

  and we assume the transversality condition holds. Taking the time derivative in the first condition above yields: $\dot{\mu }=\pi _{QQ}\dot{Q}+(\beta -1)B_{QQ}\dot{Q}+\pi _{QS}\dot{S}$ and plugging it into the second condition: $(\delta \!-\!F_{S}\!+\!\bar{H}_S)[\pi _{Q}\!+\!(\beta \!-\!1)B_{Q}]\!-\!\pi _{QQ}\dot{Q}\!-\!(\beta -1)B_{QQ}\dot{Q}\!-\!\pi _{QS}\dot{S}\!=\!\pi _{S}\!+\!U_{S}.$

• 3. The problem of maximizing the joint industry-government payoffs is solved in a very similar fashion: Use current value Hamiltonian

  $$\begin{aligned} \mathcal {H}=2\pi (p,S,Q)+(\beta -2)B(Q)+U(S)+\lambda [F(S)-H-\bar{H}(S)] \end{aligned}$$

  to get first order conditions:

  $$\begin{aligned} \frac{d\mathcal {H}}{dQ}&= 0=2\pi _{Q}+(\beta -2)B_{Q}-\lambda \\ \frac{d\mathcal {H}}{dS}&= \delta \lambda -\dot{\lambda }=\pi _{S}+U_{S}+\lambda (F_{S}-\bar{H}_S)\\ \frac{d\mathcal {H}}{d\lambda }&= \dot{S}=F(S)-Q-\bar{H}(S). \end{aligned}$$

  Together these conditions imply:

  $$\begin{aligned} (\delta \!-\!F_{S}\!+\!\bar{H}_S)[2\pi _{Q}\!+\!(\beta \!-\!2)B_{Q}]\!-\!2\pi _{QQ}\dot{Q}\!-\!(\beta \!-\!2)B_{QQ}\dot{Q}\!-\!2\pi _{QS}\dot{S}\!=\!2\pi _{S}\!+\!U_{S}. \end{aligned}$$

• 4. (i) A higher discount rate $\delta$ means that in order for Eq. (4) to hold, the equilibrium stock of the resource will adjust to a lower level, corresponding to a higher net growth rate $F_{S}-\bar{H}_s$ and a lower marginal stock effect (MSE), both evaluated at the optimal stock level. A more impatient corrupt government will regulate harvesting so that the resource stock settles at a lower long-run level. (ii) To show Lemma 3, assume we have two functions of the stock of the same resource $MSE_{1}(S)$ and $MSE_{2}(S)$, where $MSE_{2}(S)>MSE_{1}(S),$ $\forall S\in [0,K]$. This is then true, in particular, for $S=S_{2}^{*}:$ $MSE_{2}(S_{2})>MSE_{1}(S_{2}) (*)$. We also know that the following two relations governing optimal exploitation are verified at the optimum levels of stock $S_{1}^{*}$ and $S_{2}^{*}$:

  $$\begin{aligned} F_{S}(S_{1}^{*})+MSE_{1}(S_{1}^{*})=\delta \hbox {and}\\ F_{S}(S_{2}^{*})+MSE_{2}(S_{2}^{*})=\delta , \end{aligned}$$

  where the resource dynamics and the discount rate are common across the two problems. Assume, by contradiction, that $S_{2}^{*}<S_{1}^{*}$. It follows that $F_{S}(S_{1}^{*}<F_{S}(S_{2}^{*})$ and then for the above equations to hold concomitantly, we need $MSE_{1}(S_{1}^{*})>MSE_{2}(S_{2}^{*}) (**)$. Put together, the two starred relations imply that $MSE_{1}(S_{1}^{*})>MSE_{1}(S_{2}^{*})$, which further implies—since the functions representing the marginal stock effects under the different policy regimes are decreasing in the level of stock—that $S_{2}^{*}>S_{1}^{*}$, which is a contradiction.

• 5. It is easy to see that $MSE^{c}_{n/c}$ $<$ $MSE^{h}_{n/c}$ (where the subscripts refer to ‘no-conflict’ and the superscripts to government corruption) iff $\frac{\beta \pi _{S}+U_{S}}{\beta \pi _{H}}< \frac{\pi _{S}+U_{S}}{\pi _{H}}$. This is true in all cases, given the assumption that $\beta >1$. Alternatively, differentiate (4) with respect to the corruption coefficient to obtain:

  $$\begin{aligned}&\beta ^{2}\pi _{Q}^{2}(F_{SS}-\bar{H}_{SS})S_{\beta }+S_{\beta }[(\beta \pi _{SS}+U_{SS})\beta \pi _{Q}-(\beta \pi _S+u_S)\beta \pi _{QS}]\\&+\beta \pi _{Q} \pi _{S}-(\beta \pi _{S}+U_{S})\pi _{Q}=0, \end{aligned}$$

  which implies:

  $$\begin{aligned} S_{\beta }=\frac{U_{S}\pi _{Q}}{\beta ^{2}\pi _{Q}^{2}(F_{SS}-\bar{H}_{SS})+(\beta \pi _{SS}+U_{SS})\beta \pi _{Q}-(\beta \pi _S+u_S)\beta \pi _{QS}}<0, \end{aligned}$$

  where the inequality derives from the concavity of functions F, $\pi$ and u.

• 6. The current value Hamiltonian for the government utility maximization problem when conflict is possible is:

  $$\begin{aligned} \mathcal {H}\!=\!\pi (p,S,Q)\!+\!(\beta \!-\!1)B(Q)\!+\!U(S)\!-\!\rho (S)C(L(S,w))\!+\!\mu [F(S)\!-\!H\!-\!\bar{H}(S)] \end{aligned}$$

  and the first order conditions are:

  $$\begin{aligned} \frac{d\mathcal {H}}{dQ}&= 0 \rightarrow \pi _{Q}+(\beta -1)B_Q=\mu \\ \frac{d\mathcal {H}}{dS}&= \delta \mu -\dot{\mu }=\pi _{S} +U_{S}-\rho _{S}C-\rho C_LL_S+\mu (F_{S}-\bar{H}_S)\\ \frac{d\mathcal {H}}{d\mu }&= \dot{S}=F(S)-H-\overline{h}. \end{aligned}$$

  Take derivative with respect to time in the first condition above and plug into the second to obtain:

  $$\begin{aligned} (\delta \!-\!F_{S}\!+\!\bar{H}_S)[\pi _{Q}\!+\!(\beta \!\!-\!\!1)B_{Q}]\!\!-\![\pi _{QQ}\!+\!(\beta \!-\!1)B_{QQ}]\dot{Q}\!\!=\!\pi _{S}\!+\!U_{S}\!-\!\rho _{S}C\!-\!\rho C_LL_S \end{aligned}$$

  In the steady state:

  $$\begin{aligned} (\delta -F_{S}+\bar{H}_S)[\pi _{Q}+(\beta -1)B_{Q}]=\pi _{S}+U_{S}-\rho _{S}C-\rho C_LL_S \end{aligned}$$

  and this implies:

  $$\begin{aligned} B_{Q}=\frac{1}{\beta -1}\left[ \frac{\pi _{S}+U_{S}-\rho _{S}C-\rho C_LL_S}{\delta -F_{S}+\bar{H}_S}-\pi _{Q}\right] . \end{aligned}$$

  Similarly, the joint industry-government surplus maximization yields in the steady-state:

  $$\begin{aligned} (\delta -F_{S}+\bar{H}_S)[2\pi _{Q}+(\beta -2)B_{Q}]=2\pi _{S}+U_{S}-\rho _{S}C-\rho C_LL_S. \end{aligned}$$

  Plug in $B_{Q}$ from above to get the steady state equilibrium policy rule for renewable resource exploitation when conflict is possible:

  $$\begin{aligned} F_{S}-\bar{H}_S+\frac{\beta \pi _{S}+U_{S}-\rho _{S}C-\rho C_LL_S}{\beta \pi _{H}}=\delta . \end{aligned}$$

• 7. Compare this to the case without conflict and with a corrupt government in (4). Notice that the marginal stock effect with conflict is larger than the marginal stock effect without conflict if $-\rho _{S}C-\rho C_LL_S>0$ or if $\frac{-\rho _S}{\rho }>\frac{C_LL_S}{C}$. Given the expressions for $\rho$, C and their derivatives with respect to stock provided in the text, this becomes equivalent to

  $$\begin{aligned} \ln {\frac{\bar{q}S}{w}}\left( \bar{q}-\frac{w}{S}\right) >\frac{w}{S}-\frac{w}{S^*}+\frac{w\ln {\frac{\bar{q}S}{w}}}{S}-\frac{w\ln {\frac{\bar{q}S^*}{w}}}{S^*}, \end{aligned}$$

  which is further equivalent to

  $$\begin{aligned} \frac{(\frac{\bar{q}S}{w}-2)\ln {\frac{\bar{q}S}{w}}-1}{S}>-\frac{\ln {\frac{\bar{q}S^*}{w}}+1}{S^*}=-\varepsilon \Leftrightarrow \left( \frac{\bar{q}}{w}\ln {\frac{\bar{q}}{w}}+\varepsilon \right) S\\ +\left( \frac{\bar{q}S}{w}-2\right) \ln S-2\ln {\frac{\bar{q}}{w}}-1>0. \end{aligned}$$

  Denoting by $\theta$ the coefficient of S and with $\phi$ the constant, the previous inequality can be written: $G(S)=\theta S+(\frac{\bar{q}S}{w}-2)ln S-\phi >0$. Function $M(S)$ is defined in the text based on Eq. (4) as being equal to $(\delta -F_S+\bar{H}_S)\beta \pi _Q-\beta \pi _S+U_S$. Substituting the given functional forms, it becomes:

  $$\begin{aligned} M(S)=\frac{4pr}{K}S^4+\left( \delta p-2pr-\frac{5cr}{K}\right) S^3+c(3r-\delta )S^2\\ +\bar{h}\left( 2pw-c\bar{q}-\frac{1}{\beta }\ln {\frac{\bar{q}}{w}}\right) S-\frac{\bar{h}}{\beta }S\ln {S}-c\bar{h}w. \end{aligned}$$

  The following derives the shape of the curves in Fig. 1. To obtain the shape of the discontent effect $D=-\rho _{S}C$, calculate its first derivative with respect to $S$ as

  $$\begin{aligned} D_{S}=\frac{\bar{h} \gamma }{U^{*}S^{3}}\left[ \left( \bar{q}-\frac{w}{S}\right) -ln{\frac{\bar{q} S}{w}}\left( 2\bar{q}-3\frac{w}{S}\right) \right] . \end{aligned}$$

  Then

  $$\begin{aligned} D_{S}>0 \iff \frac{\bar{q}-\frac{w}{S}}{2\bar{q}-3\frac{w}{S}}>ln{\frac{\bar{q} S}{w}}. \end{aligned}$$

  Given that the left-hand side of this inequality is between $\left( \frac{1}{3},\frac{1}{2}\right)$, a sufficient condition for the ‘discontent’ function $D$ to slope downwards is that

  $$\begin{aligned} ln{\bar{q}}+ln{\frac{w}{S}}>\frac{1}{2} \iff S>\frac{\sqrt{e}w}{\bar{q}}. \end{aligned}$$

  The ‘(e)migration effect’ curve

  $$\begin{aligned} E=\frac{\bar{h}}{U^{*}}\left[ \left( \frac{1}{S}-\frac{1}{S^{*}}+\right) +\frac{ln{\frac{\bar{q} S}{w}}}{S}-\frac{ln{\frac{\bar{q} S^{*}}{w}}}{S*}\right] \gamma \frac{w}{S^{2}} \end{aligned}$$

  monotonically decreases in $S$. The curves illustrating the two effects are reproduced in Fig. 1a in the text. Moreover, one can show a single crossing property for the two curves in Fig. 1a, or that $E(S)-D(S)=0$ has a unique solution:

  $$\begin{aligned}&D(S)-E(S)=0 \Leftrightarrow ln{\frac{\bar{q}S}{w}}\left( \bar{q}-\frac{2w}{S}\right) -\frac{w}{S}+\frac{w}{S^{*}}\left( 1+ln{\frac{\bar{q}S^{*}}{w}}\right) \\&=0 \Leftrightarrow y(1+2ln{\bar{q}})+ln{y}(\bar{q}+2y)-k=0. \end{aligned}$$

  This last equation is monotonically increasing in $y$ and has a unique solution, where we denoted variable $y\equiv \frac{w}{S}$ and constant

  $$\begin{aligned} k\equiv \bar{q}ln{\bar{q}}+\frac{w}{S^{*}}\left( 1+ln{\frac{\bar{q}S^{*}}{w}}\right) . \end{aligned}$$

Fig. 2

Equilibrium stock levels

• 8. We have $P(S)=\rho (S)C(L(S,w))$ and it was shown above that $P_{S}=0$ has a unique solution, $P_S=\rho _SC+\rho C_LL_S<0$ if and only if $S>S'$ and $P_{S}>0$ for $S>S'$. The following determines the curvature of $P(S)$. Given the derivation above, the slopes of the two curves representing the discontent and the migration effects are:

  $$\begin{aligned} D_{S}=\frac{\bar{h}\gamma }{U^{*}S^{3}}\left[ \bar{q}-\frac{w}{S}-ln{\frac{\bar{q}S}{w}}\left( 2\bar{q}-3\frac{w}{S}\right) \right] \end{aligned}$$

  and

  $$\begin{aligned} E_{S}=-\frac{\gamma w\bar{h}}{U^{*}S^{3}}\left[ \frac{1}{S}\left( 1+2ln{\frac{\bar{q}S}{w}}\right) -\frac{1}{S^{*}}\left( 1+ln{\frac{\bar{q}S^{*}}{w}}\right) \right] , \end{aligned}$$

  respectively. Then $P_{SS}>0 \Leftrightarrow E_{S}>D_{S}$. After substituting in the functional forms and simplifying, this is equivalent to

  $$\begin{aligned} \frac{w}{S^{*}}\left( 1+ln{\frac{\bar{q}S^{*}}{w}}-\bar{q}\right) >ln{\frac{\bar{q}S}{w}}\left( \frac{5w}{S}-2\bar{q}\right) , \end{aligned}$$

  where the left hand side is a constant and the right hand side is monotonically decreasing in S. Thus, even though there is no closed form solution, $\exists S''$ such that $P_{SS}>0$ $\forall$ $S>S''$. This inflection point occurs for stock values above $S'$, like in the Fig. 1b. Again, for illustration purposes, one can get more concrete by employing specific functional forms. We calculate the condition $P_{SS}=\rho _{SS}C+2\rho _SC_LL_S+\rho C_LL_{SS}>0$ by using the functional forms provided above as follows:

  $$\begin{aligned}&\left( 2\ln {\frac{\bar{q}S}{w}}-1\right) \left( \bar{q}-\frac{w}{S}\right) -\frac{2w}{S}\ln {\frac{\bar{q}S}{w}}-2w\left( \frac{1}{S}-\frac{1}{S^*}\right) -2w\left( \frac{\ln {\frac{\bar{q}S}{w}}}{S}\right. \\&\left. -\frac{\ln {\frac{\bar{q}S^*}{w}}}{S^*}\right) >0 \Leftrightarrow 2(\frac{\bar{q}S}{w}-3)\left( \ln {\frac{\bar{q}}{w}}+\ln S\right) -1>-aS, \end{aligned}$$

  where

  $$\begin{aligned} a=\frac{2}{S^*}\left( 1+\ln {\frac{\bar{q}S^*}{w}}\right) -\frac{\bar{q}}{w} \end{aligned}$$

  which further yields

  $$\begin{aligned} \left( \frac{2\bar{q}}{w}\ln {\frac{\bar{q}}{w}}+a\right) S+2\left( \frac{\bar{q}S}{w}-3\right) \ln S-\left( 6\ln {\frac{\bar{q}}{w}}+1\right) >0. \end{aligned}$$

  Denoting with b the coefficient of S and with c the constant, this becomes: $J(S)=bS+2(\frac{\bar{q}S}{w}-3)\ln S-c>0$. The analysis is now similar to the one in section (7) above, and the following graphical solution illustrated in Fig. 2 points to the fact that $P_{SS}>0$ if and only if $S>S''$, with the latter being lower for higher $\bar{q}$ and lower $w$ and $S^*$. Differentiating (7) implicitly with respect to $\beta$ yields:

  $$\begin{aligned}&(\pi _S+\beta \pi _{SS}S_\beta +U_{SS}S_\beta -\rho _{SS}CS_\beta -\rho _{S}C_LL_SS_\beta -\rho _SC_LL_SS_\beta \\&\qquad -\rho C_LL_{SS}S_\beta )\beta \pi _H-(\beta \pi _S+u_S-\rho _SC-\rho C_LL_S)(\pi _H+\beta \pi _{HS}S_\beta )\\&\quad =-F_{SS}S_\beta \beta ^2\pi _H^2. \end{aligned}$$

  This implies:

  $$\begin{aligned} S_\beta = \frac{\pi _H(u_S-\rho _SC-\rho C_LL_S)}{(\beta \pi _{SS}+U_{SS}-\rho _{SS}C-2\rho _SC_LL_S-\rho C_LL_{SS})\beta \pi _H+F_{SS}\beta ^2\pi _H^2-(\beta \pi _S+u_S-\rho _SC-\rho C_LL_S)\beta \pi _{HS}} \end{aligned}$$

  is less than zero if $S>S'$ and $S>S''$, since under the first condition the numerator is positive as $P_{S}<0$, while under the second, the denominator is negative, as $P_{SS}>0$.