Groundwater extraction among overlapping generations: a differential game approach

Abstract

Groundwater is a common resource that has been wasted for years. Today, we pay the consequences of such inappropriate exploitation and we are aware that it is necessary to realize policies in order to guarantee the use of this resource for future generations. In fact, the irrational exploitation of water by agents, nevertheless it is a renewable resource, may cause its exhaustion. In our paper, we develop a differential game to determine the efficient extraction of groundwater resource among overlapping generations. We consider intragenerational as well as intergenerational competition between extractors that exploit the resource in different time intervals, and so the horizons of the players in the game are asynchronous. Feedback equilibria have been computed in order to determine the optimal extraction rate of “young” and “old” agents that coexist in the economy. The effects of the withdrawal by several generations are numerically and graphically analyzed in order to obtain results on the efficiency of the groundwater resource.

Appendix A

Performing the maximization problem in Eqs. (11) and (12) with respect to $w_i^j$ and $w_i^{j-1}$, for $t\,\in \,[t_j,t_j+T/2]$, we obtain:

$$\begin{aligned}&\frac{2\phi _i^{j*}(H,t)+\sum \limits _{h=1}^n\phi _h^{j-1**}(H,t)+ \sum \limits _{h=1,h\ne i}^n \phi _h^{j*}(H,t)-g}{k} \nonumber \\&\quad =[c_0-c_1 H(t)]-\frac{\displaystyle \partial V_i^{j*}(H,t)}{\partial H} \cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)} \end{aligned}$$

(25)

and

$$\begin{aligned}&\frac{2\phi _i^{j-1**}(H,t)+\sum \limits _{h=1,h\ne i}^n \phi _h^{j-1**}(H,t)+ \sum \limits _{h=1}^n\phi _{h}^{j*}(H,t)-g}{k}\nonumber \\&\quad =[c_0-c_1H(t)]-\frac{\partial V_i^{j-1**}(H,t)}{\partial H}\cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)} \end{aligned}$$

(26)

Adding in Eqs. (25) and (26) over all extractors living in the time interval $[t_j, t_j + T/2]$, that is, all the n “young” agents for the generation j and the n “old” agents for the generation $j-1$, we obtain:

$$\begin{aligned}&\displaystyle \frac{2n\phi _i^{j*}(H,t)+n^2\phi _h^{j-1**}(H,t)+n(n-1)\phi _h^{j*}(H,t)}{k}\nonumber \\&\quad =\frac{ng}{k}+n [c_0-c_1H(t)]-\sum _{i=1}^n\frac{\partial V_i^{j*}(H,t)}{\partial H}\cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)} \end{aligned}$$

(27)

for all ith extractor in the jth generation and

$$\begin{aligned}&\frac{2n\phi _i^{j-1**}(H,t)+n^2\phi _h^{j*}(H,t)+n(n-1)\phi _h^{j-1**}(H,t)}{k}\nonumber \\&\quad =\frac{ng}{k}+n [c_0-c_1H(t)]-\sum _{i=1}^n\frac{\partial V_i^{j-1**}(H,t)}{\partial H}\cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)} \end{aligned}$$

(28)

for all ith extractor in the $j-1$th generation. Equations (27) and (28) represent a system of linear equations in $\phi _i^{j*}(H,t)$ and $\phi _i^{j-1**}(H,t)$. The system admits a unique solution:

$$\begin{aligned} \phi _i^{j-1**}(H,t)= & {} \frac{g+k[c_0-c_1H(t)]-k(n+1) \frac{\partial V_i^{j-1**}(H,t)}{\partial H}\cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)}}{(1+2n)}\nonumber \\&\displaystyle +\frac{nk \frac{\partial V_i^{j*}(H,t)}{\partial H}\cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)}}{(1+2n)}\end{aligned}$$

(29)

$$\begin{aligned} \phi _i^{j*}(H,t)= & {} \frac{g+k[c_0-c_1H(t)]-k(n+1) \frac{\partial V_i^{j*}(H,t)}{\partial H}\cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)}}{(1+2n)}\nonumber \\&\displaystyle +\frac{nk\frac{\partial V_i^{j-1**}(H,t)}{\partial H}\cdot \frac{(\gamma -1)}{\Delta S}e^{r(t-t_j)}}{(1+2n)} \end{aligned}$$

(30)

Substituting Eqs. (29) and (30) into HJB given by Eqs. (11) and (12), we obtain the following system of partial differential equations (dropping for notational convenience the (H, t) arguments of the value functions):

$$\begin{aligned} -\frac{\partial V_i^{j*}}{\partial t}= & {} \left[ \frac{-(\gamma -1)^2 n^2e^{r(t-t_j)} k}{(2n+1)^2(\Delta S)^2}\right] \left( \frac{\partial V_i^{j*}}{\partial t}\right) ^2\nonumber \\&+\left\{ \frac{4[(R+g(\gamma -1))n^2+n(\frac{1}{2}g(\gamma -1)+R)+\frac{1}{4}(R+g(\gamma -1))]}{\Delta S(2n+1)^2}\right. \nonumber \\&+\left. \frac{[4(\gamma -1)\Delta S(c_0-c_1H)(n^2+\frac{1}{2}n+\frac{1}{4})-2(\gamma -1)^2n^2e^{r(t-t_j)} \frac{\partial V_i^{j-1**}}{\partial H} ]}{(\Delta S)^2(2n+1)^2}\right\} \nonumber \\&\times \left( \frac{\partial V_i^{j*}}{\partial t}\right) - \frac{\left\{ \left[ (c_0-c_1H)\Delta S+(\gamma -1)e^{r(t-t_j)} n\frac{\partial V_i^{j-1**}}{\partial H}\right] k+\Delta S g\right\} ^2}{(\Delta S)^2(2n+1)^2 k e^{r(t-t_j)}}\end{aligned}$$

(31)

$$\begin{aligned} -\frac{\partial V_i^{j-1**}}{\partial t}= & {} \left[ \frac{-(\gamma -1)^2 n^2e^{r(t-t_j)} k}{(2n+1)^2(\Delta S)^2}\right] \left( \frac{\partial V_i^{j-1**}}{\partial t}\right) ^2\nonumber \\&+\left\{ \frac{4[(R+g(\gamma -1))n^2+n(\frac{1}{2}g(\gamma -1)+R)+\frac{1}{4}(R+g(\gamma -1))]}{\Delta S(2n+1)^2}\right. \nonumber \\&+\left. \frac{[4(\gamma -1)\Delta S(c_0-c_1H)(n^2+\frac{1}{2}n+\frac{1}{4})-2(\gamma -1)^2n^2e^{r(t-t_j)} \frac{\partial V_i^{j-1**}}{\partial H}]}{(\Delta S)^2(2n+1)^2}\right\} \nonumber \\&\times \left( \frac{\partial V_i^{j-1**}}{\partial t}\right) -\frac{\left\{ \left[ (c_0-c_1H)\Delta S+(\gamma -1)e^{r(t-t_j)}n\frac{\partial V_i^{j*}}{\partial H}\right] k+\Delta S g\right\} ^2}{(\Delta S)^2(2n+1)^2 ke^{r(t-t_j)}} \end{aligned}$$

(32)

Rearranging Eqs. (31) and (32), we obtain that:

$$\begin{aligned} -\frac{\partial V_i^{j*}}{\partial t}= & {} e^{-r(t-t_j)} \left( \alpha _1 +\alpha _2 H+\alpha _3H^2\right) +\alpha _4\left( \frac{\partial V_i^{j*}}{\partial H}\right) +\alpha _5 H \left( \frac{\partial V_i^{j*}}{\partial H}\right) \nonumber \\&+\, \alpha _7 \left( \frac{\partial V_i^{j-1**}}{\partial H}\right) +\alpha _{8}H\left( \frac{\partial V_i^{j-1**}}{\partial H}\right) \nonumber \\&\,+\alpha _6e^{r(t-t_j)}\left[ \left( \frac{\partial V_i^{j*}}{\partial H}\right) +\left( \frac{\partial V_i^{j-1**}}{\partial H}\right) \right] ^2 \end{aligned}$$

(33)

and

$$\begin{aligned} -\frac{\partial V_i^{j-1**}}{\partial t}= & {} e^{-r(t-t_j)} \left( \alpha _1 +\alpha _2 H+\alpha _3H^2\right) +\alpha _4\left( \frac{\partial V_i^{j-1**}}{\partial H}\right) +\alpha _5 H \left( \frac{\partial V_i^{j-1**}}{\partial H}\right) \nonumber \\&+ \,\alpha _7 \left( \frac{\partial V_i^{j*}}{\partial H}\right) +\alpha _{8}H\left( \frac{\partial V_i^{j*}}{\partial H}\right) \nonumber \\&+\,\alpha _6e^{r(t-t_j)}\left[ \left( \frac{\partial V_i^{j-1**}}{\partial H}\right) +\left( \frac{\partial V_i^{j*}}{\partial H}\right) \right] ^2 \end{aligned}$$

(34)

where

$$\begin{aligned} \displaystyle \alpha _1= & {} -\frac{(kc_0+g)^2}{k(2n+1)^2}; \quad \displaystyle \alpha _2=\frac{2c_1(c_0k+g)}{(2n+1)^2}; \quad \displaystyle \alpha _3=\frac{-c_1^2k}{(2n+1)^2}; \\ \displaystyle \alpha _4= & {} \frac{[(\gamma -1)(kc_0+g)+R](4n^2+1)+2n[(\gamma -1)(kc_0+g)+2R]}{\Delta S(2n+1)^2};\\ \displaystyle \alpha _5= & {} \frac{-(\gamma -1)(4n^2+2n+1)kc_1}{\Delta S (2n+1)^2};\quad \displaystyle \alpha _6= \frac{-(\gamma -1)^2 n^2 k}{(2n+1)^2(\Delta S)^2};\\ \displaystyle \alpha _7= & {} -\frac{2(\gamma -1)gn}{\Delta S (2n+1)^2};\quad \displaystyle \alpha _{8}=\frac{2(\gamma -1)c_1nk}{\Delta S(2n+1)^2} . \end{aligned}$$