Extinction in common property resource models: an analytically tractable example

Abstract

We discuss an analytically tractable discrete-time dynamic game in which a finite number of players extract a renewable resource. We characterize a symmetric Markov-perfect Nash equilibrium of this game and derive a necessary and sufficient condition under which the resource does not become extinct in equilibrium. This condition requires that the intrinsic growth rate of the resource exceeds a certain threshold value that depends on the number of players and on their time-preference rates.

Appendix

In this “Appendix,” we prove that for every $x_0\ge 0$, the trajectory generated by (10) and the consumption stream defined by $c_t=\lambda f(x_t)$ form an optimal solution to problem (1)–(3).

Suppose first that $x_0=0$. Because of $f(0)=0$, it follows that every feasible state trajectory satisfies $x_t=0$ for all $t\in \mathbb {N}_0$. Since this is the only feasible state trajectory, it must be an optimal one.

Now assume that $x_0>0$ and denote by $(x_t)_{t=0}^{+\infty }$ the unique trajectory of (10) emanating from $x_0$. Let $(\tilde{x}_t)_{t=0}^{+\infty }$ be any other feasible state trajectory with the same initial state $\tilde{x}_0=x_0$. Finally, denote by $c_t=\lambda f(x_t)$ and $\tilde{c}_t=f(\tilde{x}_t)-\tilde{x}_{t+1}$ the corresponding control paths. We need to show that

$$\begin{aligned} \lim _{T\rightarrow +\infty }\sum _{t=0}^T\rho ^t\left[ u(c_t)-u(\tilde{c}_t)\right] \ge 0. \end{aligned}$$

(22)

We define

$$\begin{aligned} p_t=u'(c_t) \end{aligned}$$

(23)

for all $t\in \mathbb {N}_0$. Since $x_t>0$ and $u$ is strictly increasing and continuously differentiable on $(0,+\infty ),\,p_t$ is well defined for all $t\in \mathbb {N}_0$, and it holds that $p_t>0$. Since $c_t$ maximizes the right-hand side of the Bellman equation given by $u(c)+\rho W(f(x_t)-c)$ and since $c_t\in (0,f(x_t))$, it follows that

$$\begin{aligned} p_t=u'(c_t)=\rho W'(f(x_t)-c_t)=\rho W'(x_{t+1}). \end{aligned}$$

Furthermore, from the envelope theorem applied to the optimization problem in (6), we obtain

$$\begin{aligned} W'(x_t)=\rho W'(f(x_t)-c_t)f'(x_t)=\rho W'(x_{t+1})f'(x_t). \end{aligned}$$

Combining the last two equations, we get

$$\begin{aligned} p_t=\rho p_{t+1}f'(x_{t+1}) \end{aligned}$$

(24)

for all $t\in \mathbb {N}_0$. Let us define for $(x,c)\in \mathbb {R}^2_+$ and $t\in \mathbb {N}_0$

$$\begin{aligned} L(x,c,t)=u(c)+p_t[f(x)-c] \end{aligned}$$

and note that $L$ is strictly concave in $(x,c)$ for all $t\in \mathbb {N}_0$. Furthermore, it holds that

$$\begin{aligned} L_x(x_t,c_t,t)&= p_tf'(x_t),\end{aligned}$$

(25)

$$\begin{aligned} L_c(x_t,c_t,t)&= u'(c_t)-p_t=0, \end{aligned}$$

(26)

where we have used (23). We have

$$\begin{aligned}&\sum _{t=0}^T\rho ^t\left[ u(c_t)-u(\tilde{c}_t)\right] \\&= \sum _{t=0}^T\rho ^t\left\{ L(x_t,c_t,t)-p_t[f(x_t)-c_t]-L(\tilde{x}_t,\tilde{c}_t,t)+p_t[f(\tilde{x}_t)-\tilde{c}_t]\right\} \\&\ge \sum _{t=0}^T\rho ^t\left[ L_x(x_t,c_t,t)(x_t-\tilde{x}_t)+L_c(x_t,c_t,t)(c_t-\tilde{c}_t)-p_t(x_{t+1}-\tilde{x}_{t+1})\right] \\&= \sum _{t=0}^T\rho ^t\left[ p_tf'(x_t)(x_t-\tilde{x}_t)-p_t(x_{t+1}-\tilde{x}_{t+1})\right] \\&= p_0f'(x_0)(x_0-\tilde{x}_0)+\sum _{t=0}^T\rho ^t\left[ \rho p_{t+1}f'(x_{t+1})(x_{t+1}-\tilde{x}_{t+1})-p_t(x_{t+1}-\tilde{x}_{t+1})\right] \\&-\rho ^{T+1}p_{T+1}f'(x_{T+1})(x_{T+1}-\tilde{x}_{T+1})\\&= \sum _{t=0}^T\rho ^t\left[ p_t(x_{t+1}-\tilde{x}_{t+1})-p_t(x_{t+1}-\tilde{x}_{t+1})\right] -\rho ^{T}p_{T}(x_{T+1}-\tilde{x}_{T+1})\\&\ge -\rho ^{T}p_{T}x_{T+1}. \end{aligned}$$

In the first step, we have used the definition of the function $L$ and, in the second step, the concavity of $L$ in $(x,c)$ and the feasibility of the two solutions. In the third step, we have made use of (25)–(26); in step four, we have rearranged the terms; in step five, we have used $x_0=\tilde{x}_0$ and (24). The last step follows from $\tilde{x}_{T+1}\ge 0$ and $p_{T}\ge 0$. From the above result, it follows that (22) is implied by the transversality condition

$$\begin{aligned} \lim _{T\rightarrow +\infty }\rho ^{T}p_{T}x_{T+1}=0. \end{aligned}$$

(27)

We distinguish two cases. First, if $a\rho >1$, then we know from Lemma 2(a) that $\lim _{T\rightarrow +\infty }x_T=x^*>0$, and therefore, $\lim _{T\rightarrow +\infty }p_T=\lim _{T\rightarrow +\infty }u'(c_T)=\lim _{T\rightarrow +\infty }u'(\lambda f(x_T))=u'(\lambda f(x^*))$. Since both $x_{T+1}$ and $p_T$ have finite limits and $\rho \in (0,1)$, it is obvious that (27) holds.

Second, let us assume that $a\rho \le 1$ holds. We have

$$\begin{aligned} p_T=u'(c_T)=u'(\lambda f(x_T))=u'([\lambda /(1-\lambda )]x_{T+1})=\left( \frac{\lambda }{1-\lambda }\right) ^{-\eta }x_{T+1}^{-\eta } \end{aligned}$$

and it follows therefore that (27) holds if $\lim _{T\rightarrow +\infty }\rho ^Tx_{T+1}^{1-\eta }=0$. Since this property has been proven in Lemma 2(b), we have verified the transversality condition (27) also in this case.