Building Efficient Institutions: A Two-Stage Differential Game

Abstract

We consider a two-stage dynamic game with a corrupt government and civil society as its players. We characterize open-loop Nash equilibria and an interior switching time from a regime with high government corruption which persists in the first stage (bad regime) to a free-corruption regime and greater institutional quality (good regime, second stage). We found that an increase of optimism (pessimism) in the society will lead the civil society to invest less (more) efforts to fight corruption whereas a corrupt government will invest more (less) efforts in repression policy. Overall, the numerical results show that the higher the efficiency of the civil monitoring effort, the efficiency of institutions and the lower the discount rate; the higher the inertia which will lead the economy to a much earlier switch to good regime with low corruption as the jump occurs early.

Appendix

Proof of Proposition 1

Regime 1

The current value Hamiltonian associated with problems P _C and P _G in regime 1 is given by:

$$\displaystyle \begin{aligned} {H}^1_C(q(t),w(t),x^*(t), {\lambda}^1_C(t))&= {\lambda}^1_C(t)\left(\left(bw(t) -\beta x^*(t)\right)-a\right)\\&\quad + \left[\left(\alpha q(t)+\theta (1-w\left( t\right))\right) (1-\kappa x^*(t))-\frac{w^2(t)}{2}\right] \end{aligned} $$

$$\displaystyle \begin{aligned} {H}^1_G(q(t),w^*(t),x(t), {\lambda}^1_G(t))&= {\lambda}^1_G(t)\left(\left(bw^*(t) -\beta x(t)\right)-a\right)\\&\quad + \left[\left(\alpha q(t)+\theta (1-w^*(t)) \right) (\kappa x(t))-\frac{x^2(t)}{2}\right] \end{aligned} $$

The first order conditions (assuming interior solutions) for the open-loop Nash equilibrium are given by:

$$\displaystyle \begin{aligned} \left. \begin{aligned} &(H_C^1)_w = {b\lambda_C^1- \theta(1 -\kappa x^*) - w^*} = 0,\\ &(H_C^1)_q=\rho \lambda_C^1- \dot{\lambda}_C^1 =\alpha(1-\kappa x^*)\\ &(H_C^1)_{\lambda_C^1} = (H_G^1)_{\lambda_G^1}= \dot{q}^* = bw^*-\beta x^*-a\\ &(H_G^1)_x = {-x^* - \beta \lambda_G^1 + \kappa(\alpha q^* + \theta (1 - w^*))} = 0\\ &(H_G^1)_q=\rho \lambda_C^1- \dot{\lambda}_C^1 ={\alpha\kappa x^*} \end{aligned}\quad \right\} {} \end{aligned} $$

(8)

The open-loop equilibrium dynamics, in $(q,\lambda _C^1,\lambda _G^1)$ coordinates, are given by:

(9)

The steady-state value for the control policies and the state variable in regime 1 are

$$\displaystyle \begin{aligned} &x^{ss}_1=\frac{b(b \alpha-\theta \rho) -a\rho}{b\kappa(b \alpha-\theta \rho)+\beta \rho},\quad w^{ss}_1=\frac{(b \alpha-\theta \rho) (\beta+a\kappa)}{b\kappa(b \alpha-\theta \rho)+\beta \rho}, \text{ and}\\ & q^{ss}_1= \frac{ b {\kappa}^2 \rho {\theta}^2 - \alpha b \beta \kappa \theta+\alpha {\beta}^2 - b \rho \theta - \beta \kappa \rho \theta - \beta \kappa \rho {\theta}^2 - a {\kappa}^2 \rho {\theta}^2-a \rho }{\alpha \kappa \rho \left(\beta - b \kappa \theta\right)}\\ &\qquad \;\;+\frac{\left(\beta + a \kappa\right) \left(b^2 + 2 \kappa \theta b \beta - {\beta}^2\right)}{\kappa \left(\beta - b \kappa \theta \right) \left(b\kappa(b \alpha-\theta \rho)+\beta \rho\right)}. \end{aligned} $$

Regime 2

The current value Hamiltonian associated with problems P _C and P _G in regime 2 is given by:

$$\displaystyle \begin{aligned} {H}^2_C(q(t),w(t),x^*(t), {\lambda}^2_C(t))&= {\lambda}^2_C(t)\left(\left(bw(t) -\beta x^*(t)\right)+a\right)\\&\quad + \left[\left(\alpha q(t)+\theta (1-w\left( t\right))\right) (1-\kappa x^*(t))-\frac{w^2(t)}{2}\right] \end{aligned} $$

$$\displaystyle \begin{aligned} {H}^2_G(q(t),w^*(t),x(t), {\lambda}^2_G(t))&= {\lambda}^2_G(t)\left(\left(bw^*(t) -\beta x(t)\right)+a\right)\\&\quad + \left[\left(\alpha q(t)+\theta (1-w^*(t)) \right) (\kappa x(t))-\frac{x^2(t)}{2}\right] \end{aligned} $$

The first order conditions (assuming interior solutions) for the open-loop Nash equilibrium are given by:

$$\displaystyle \begin{aligned} \left. \begin{aligned} &(H_C^2)_w ={b\lambda_C^1- \theta(1 -\kappa x^*) - w^*} = 0,\\ &(H_C^2)_q=\rho \lambda_C^2- \dot{\lambda}_C^2 =\alpha(1-\kappa x^*)\\ &(H_C^2)_{\lambda_C^2} = (H_G^2)_{\lambda_G^2}= \dot{q}^* = bw^*-\beta x^*+a\\ &(H_G^2)_x = {-x^* - \beta \lambda_G^2 + \kappa(\alpha q^* + \theta (1 - w^*))} = 0\\ &(H_G^2)_q=\rho \lambda_C^2- \dot{\lambda}_C^2 ={\alpha \kappa x^*} \end{aligned} \quad \right\} {} \end{aligned} $$

(10)

The open-loop equilibrium dynamics, in $(q,\lambda _C^2,\lambda _G^2)$ coordinates, are given by:

(11)

The steady-state value for the control policies and state variable in regime 2 are

$$\displaystyle \begin{aligned} x^{ss}_2&=\frac{b(b \alpha-\theta \rho) +a\rho}{b\kappa(b \alpha-\theta \rho)+\beta \rho},\quad w^{ss}_2=\frac{(b \alpha-\theta \rho) (\beta-a\kappa)}{b\kappa(b \alpha-\theta \rho)+\beta \rho}, \text{ and}\\ q^{ss}_2&= \frac{ b {\kappa}^2 \rho {\theta}^2 - \alpha b \beta \kappa \theta+\alpha {\beta}^2 - b \rho \theta - \beta \kappa \rho \theta - \beta \kappa \rho {\theta}^2 + a {\kappa}^2 \rho {\theta}^2+a \rho }{\alpha \kappa \rho \left(\beta - b \kappa \theta\right)}\\ &\quad +\frac{\left(\beta - a \kappa\right) \left(b^2 + 2 \kappa \theta b \beta - {\beta}^2\right)}{\kappa \left(\beta - b \kappa \theta \right) \left(b\kappa(b \alpha-\theta \rho)+\beta \rho\right)}. \end{aligned} $$

$\blacksquare$

Proof of Proposition 2

Using the (8) and (10) in (5) and (6) we obtain the following equations:

$$\displaystyle \begin{aligned} &A_c{\lambda_C^j}^2(\tau)+B_c{\lambda_C^j}(\tau){\lambda_G^j}(\tau)+C_c{\lambda_G^j}^2(\tau)+(D_c+a){\lambda_C^j}(\tau)+ E_c{\lambda_G^j}(\tau)+F_c \\ &=A_c{\lambda_C^i}^2(\tau)+B_c{\lambda_C^i}(\tau){\lambda_G^i}(\tau)+C_c{\lambda_G^i}^2(\tau)+(D_c-a){\lambda_C^i}(\tau)+ E_c{\lambda_G^i}(\tau)+F_c, {} \end{aligned} $$

(12)

$$\displaystyle \begin{aligned} &A_g{\lambda_C^j}^2(\tau)+B_g{\lambda_C^j}(\tau){\lambda_G^j}(\tau)+C_g{\lambda_G^j}^2(\tau)+D_g{\lambda_C^j}(\tau)+ (E_g+a){\lambda_G^j}(\tau)+F_g \\ &=A_g{\lambda_C^i}^2(\tau)+B_g{\lambda_C^i}(\tau){\lambda_G^i}(\tau)+C_g{\lambda_G^i}^2(\tau)+D_g{\lambda_C^i}(\tau)+ (E_g-a){\lambda_G^i}(\tau)+F_g. {} \end{aligned} $$

(13)

where

$A_c=\frac {(b(2\beta \kappa ^3\theta ^3 + 2\beta \kappa \theta + b))}{2(\kappa ^2t^2 + 1)^2}$, $B_c=\frac {(\beta (\beta \kappa ^2\theta ^2 - b \kappa \theta + \beta ))}{(\kappa ^2\theta ^2 + 1)^2}$, $C_c=\frac {(\beta ^2 \kappa ^2\theta ^2)}{2(\kappa ^2\theta ^2 + 1)^2}$,$D_c=\frac {b(\theta +\alpha q_{th})}{\theta }-\frac {b(\theta ^2+\theta +\alpha q_{th})}{\theta (\kappa ^2\theta ^2+1)^2} -\frac {\beta \kappa (\theta ^2+\theta +\alpha q_{th})}{(\kappa ^2\theta ^2+1)}$, $E_c=\frac {(\beta \kappa (\theta ^2 + \theta + \alpha q_{th}))}{(\kappa ^2\theta ^2 + 1)^2}$, $F_c=\frac {(\theta ^2+\theta + \alpha q_{th})^2}{2\theta ^2(\kappa ^2\theta ^2+1)^2}-\frac {(\theta + \alpha q_{th})^2}{(2\theta ^2)}$,

$A_g=\frac {(b^2 \kappa ^2\theta ^2)}{2(\kappa ^2\theta ^2 + 1)^2}$, $B_g=\frac {(b(b\kappa ^2\theta ^2 + \beta \kappa \theta + b))}{(\kappa ^2\theta ^2 + 1)^2}$, $C_g=-\frac {(\beta (2b\kappa ^3\theta ^3 + 2b\kappa \theta - \beta ))}{2(\kappa ^2t^2 + 1)^2}$, $D_g=-\frac {(\beta \kappa ^2 \theta (\theta ^2 + \theta + \alpha q_{th}))}{(\kappa ^2\theta ^2 + 1)^2}$,$E_g=\frac {b(\theta +\alpha q_{th})}{\theta }-\frac {b(\theta ^2+\theta +\alpha q_{th})}{\theta (\kappa ^2\theta ^2+1)} -\frac {\beta \kappa (\theta ^2+\theta +\alpha q_{th})}{(\kappa ^2\theta ^2+1)^2}$ and $F_g=\frac {\kappa ^2(\theta ^2+\theta + \alpha q_{th})^2}{2(\kappa ^2\theta ^2+1)^2}$. From (12) and (13), the switching time τ as well as the jumps $\lambda ^j_C(\tau )-\lambda ^i_C(\tau )$ and $\lambda ^j_G(\tau )-\lambda ^i_G(\tau )$ can be calculated when the equilibrium state trajectory undergoes a transition from regime i to regime j. $\blacksquare$