Rights and judicial independence

Abstract

This paper models endogenous judicial independence (JI) as a commitment device in a political commitment game between a ruler and citizens. In a situation where citizens can observe the effectiveness of JI with some positive probability, the model shows that the ruler in fact creates an independent judiciary and credibly commits to an announced tax rate, i.e., the ruler protects private property rights. Even when citizens have no chance to observe the effectiveness of JI, the ruler can still guarantee property rights by granting human rights as a signal of JI. Although the creation of JI achieves a Pareto improvement compared with its lack, two sources of inefficiency arise. First, the equilibrium tax is inefficiently high in the sense that the tax rate is on the inefficient side of the Laffer curve. This inefficiently high tax reflects the cost of credible commitment. Second, equilibrium JI for guaranteeing human rights is inefficiently high in the sense that the ruler does not entirely use JI for credibly committing to a low tax. This inefficiently high JI represents the cost of credible signalling.

Appendix: Proofs

1.1 Proof of Lemma 1

Notice that $e_{o,1}^J$ decreases with $r_1^J$, and $e_{o,1}^J$ equals 0 when $r_1^J=1$. Thus, a system of Eqs. (2) and (3) has a unique solution of Eqs. (4) and (5).

1.2 Proof of lemma 2

Assume that $e^C (t^J_1) < \bar{e} (F^J_1, t^J_1)$ in equilibrium. The ruler can increase his payoff by lowering $F$ since citizens do not change their efforts. However, this fact contradicts the assumption that $F^J_1$ is the equilibrium JI level.

Alternatively, assume that $e^C (t^J_1) > \bar{e}(F^J_1, t^J_1)$. The ruler’s maximised payoff is ${\fancyscript{R}}=[t^J_1 \lambda / (1-t^J_1)-\gamma ]F^J_1$, and policy $(F^J_1,t^J_1)$ meets the constraint $F^J_1<(1-t^J_1) e^C (t^J_1)/ \lambda$. In this case, the ruler’s maximised payoff must be zero. Suppose that the payoff is positive. Then, the ruler can increase his payoff by increasing $F$, which contradicts the assumption that $F^J_1$ is the equilibrium JI level. Now, consider tax rate $t'=1-\epsilon$, where $\epsilon$ is positive and sufficiently small. Furthermore, let $F'$ be the JI level such that $(F',t')$ meets the constraint that $0<F'<(1-t')e^C(t')/\lambda$. Then, the ruler obtains positive payoff $R'=[t' \lambda /(1-t') -\gamma ]F'>0$. Hence, the assumption that $e^C (t^J_1) > \bar{e} (F^J_1,t^J_1)$ is contradictory.

1.3 Proof of proposition 1

• (ii) Notice that $t^J_1 = [1+\gamma / (\lambda + \gamma )]/2 > 1/2 = t^C$. Furthermore, $\partial t^J_1 / \partial \lambda <0$ and $\partial t^J_1/ \partial \gamma >0$.

• (iii) Notice that $\partial F^J_1 / \partial \alpha <0$ and $\partial F^J_1/ \partial \gamma <0$.

1.4 Proof of Proposition 2

At first, note that since citizens under state $u$ can exactly expect the equilibrium JI level on the equilibrium path, the ruler obtains a payoff at most the amount in the special case of $p=1$.

Then, I characterise the conditions that the ruler can set policy $(F_1^J, t^J_1)$. Suppose that the ruler chooses $(F_1^J, t^J_1)$. Under this policy, citizens choose their efforts to be $e^J_{o,1}(F_1^J,t^J_1) = e^C(t^J_1)$, irrespective of the information structure. In this case, the ruler’s expected payoff is

$$\begin{aligned} {\fancyscript{R}}\left( F_1^J,t^J_1,t^J_1, e^C \left( t^J_1\right) , e^C \left( t^J_1\right) \right) = t_1^J e^C \left( t_1^J\right) - \gamma \left( 1-t^J_1\right) e^C\left( t_1^J\right) / \lambda . \end{aligned}$$

Now, consider the deviation from $F_1^J$, i.e., consider the case where the ruler tries to renege on the announced tax rate. Notice that the ruler still needs to announce tax rate $t^J_1$ due to citizens’ off-equilibrium actions. Since the ruler resets the tax rate to 1, he does not establishes JI. Therefore, citizens at state $o$ do not produce. The ruler’s expected payoff in this case is

$$\begin{aligned} {\fancyscript{R}} \left( 0,t_1^J, 1, 0, e^C\left( t^J_1\right) \right) = (1-p) e^C \left( t_1^J\right) . \end{aligned}$$

I derive the following relationship:

$$\begin{aligned} {\fancyscript{R}}\left( F_1^J,t^J_1,t^J_1, e^C\left( t^J_1\right) , e^C\left( t^J_1\right) \right) \gtreqless {\fancyscript{R}} \left( 0,t_1^J, 1, 0, e^C\left( t^J_1\right) \right) \end{aligned}$$

if and only if

$$\begin{aligned} \qquad \qquad \qquad \qquad \quad t^J_1 \gtreqless \frac{(1-p) \lambda +\gamma }{\lambda +\gamma } = \tilde{t}. \end{aligned}$$

Therefore, policy $(F_1^J,t^J_1)$ is supported by equilibrium only if $t^J_1 \ge \tilde{t}$. The condition can be rewritten as $p \ge 1/2$. I can straightforwardly prove that, if condition $p \ge 1/2$ holds, the ruler’s optimal action is $(F_1^J,t^J_1)$.

Next, consider a case where $0< p < 1/2$. In this case, the ruler cannot credibly implement policy $(F_1^J,t^J_1)$ in any PBE. Since citizens at state $u$ can anticipate the exact level of $F$ on the equilibrium path, the ruler cannot receive strictly positive gains from reneging on the announced tax rate. Hence, in the equilibrium that maximises the ruler’s payoff, the equilibrium policy is the solution of the following maximisation problem:

$$\begin{aligned} \max _{F,t} \ te^J_{o,1} (F,t) - \gamma F \ \ \ \text{ s.t. } \ \ t \le \tilde{t}. \end{aligned}$$

The solution satisfies $e^J_{o,1} (F,t) = e^C (t)= \bar{e}(F,t)$, because of a similar proof to Lemma 2. Hence, the equilibrium policy is $t=\tilde{t}$ and $F=\tilde{F}$.

Finally, consider the case where $p=0$. I have the following lemma.

Lemma 3

Suppose that $p=0$. There is no PBE such that the ruler sets the strictly positive JI level and commits to an announced tax strictly lower than 1.

Proof

Suppose that the ruler sets $F^J_0>0$ in equilibrium. However, the ruler can receive a positive gain from lowering the JI level since citizens do not change their production levels. $\square$

Hence, the ruler sets $F^J_0=0$ and announces any tax rate in equilibrium. A potential announcement is 1, which equals $\lim _{p\rightarrow 0} \tilde{t}$. Furthermore, it holds that $\lim _{p\rightarrow 0} \tilde{F} = 0$.

1.5 Proof of Proposition 3

I prove this proposition in four steps. The first three steps assume that the equilibrium JI level is strictly positive and characterise the potential equilibrium actions. The final step analyses the ruler’s non-negative payoff conditions.

Step 1: Notice that since citizens can expect the exact JI level on the equilibrium path, the ruler cannot obtain strictly positive gains by reneging on the announced tax rate. Using this fact, I show the following lemma.

Lemma 4

If the equilibrium JI level is strictly positive, it is always $\bar{F}$.

Proof

Suppose that $F^H \in (0,\bar{F})$ in equilibrium. The ruler can announce a tax rate strictly less than 1. Citizens choose their effort to be $e^H (G^H, t^H) = e^J_{o,1} (F^H,t^H)>0$ on the equilibrium path. Then, the ruler’s payoff is $t^H e^H (G^H, t^H) - \gamma F^H-G^H$. However, the ruler can increase his payoff by setting $F=0$ since $e^H (G^H,t^H) - G^H >t^H e^H (G^H,t^H) -\gamma F^H-G^H$. Hence, the assumption of $F^H \in (0,\bar{F})$ leads to a contradiction.

Next, suppose $F^H \in (\bar{F},\infty )$ in equilibrium. The following inequality holds:

$$\begin{aligned} t^H e^H \left( G^H,t^H\right) -\gamma F^H - \delta G^H&\ge e^H \left( G^H,t^H\right) - (\lambda + \gamma ) \bar{F} - \delta G^H\\&> e^H \left( G^H,t^H\right) - (\lambda +\gamma ) F^H -\delta G^H. \end{aligned}$$

The first inequality means that the ruler does not obtain positive gains by establishing lower JI and reneging on the announcement. The second inequality follows from the assumption that $F^H > \bar{F}$. Therefore, it holds that $(1-t^H ) e^H (G^H,t^H) < \lambda F^H$, and $e^H (G^H,t^H) < \bar{e} (F^H,t^H)$ from Eq. (3). Consider JI level $F'= F^H - \epsilon$, where $\epsilon$ is positive and sufficiently small. When setting $F'$ instead of $F^H$, the ruler can increase his payoff without violating the constraints for credible commitment. Hence, it contradicts the assumption. $\square$

Step 2: Consider the signal. Credible signal $G^H$ must satisfy the following inequality:

$$\begin{aligned} {\fancyscript{R}} \left( \bar{F}, G^H, t^H, t^H, e^H\right)&= t^H e^H \left( G^H,t^H \right) -\gamma \bar{F} - \delta G^H \\&\ge {\fancyscript{R}} \left( 0, G^H, t^H, 1, e^H\right) = e^H \left( G^H, t^H\right) - G^H. \end{aligned}$$

With citizens’ beliefs that maximise the ruler’s equilibrium payoffs, the credible signal is

$$\begin{aligned} G(t^H) = \frac{1}{1-\delta } \left[ \left( 1-t^H\right) e^H \left( G^H,t^H\right) + \gamma \bar{F} \right] . \end{aligned}$$

Step 3: Announced tax rate $t$ is the solution of the following payoff maximisation problem:

$$\begin{aligned} \max _t \ t e^J_{o,1} \left( \bar{F},t \right) - \gamma \bar{F} - \frac{\delta }{1-\delta } \left[ (1-t) e^J_{o,1} \left( \bar{F},t\right) + \gamma \bar{F} \right] . \end{aligned}$$

The formulation implies that the ruler provides human rights at the minimum level sufficient to serve as a credible signal of JI, and that citizens know the JI level on the equilibrium path. Accordingly, I present the following lemma.

Lemma 5

In PBE such that the ruler’s payoff is maximised and $F^H = \bar{F}$, it holds that $e^H(G^H,t^H) = e^C (t^H) \le \bar{e} (\bar{F},t^H)$.

Proof

Suppose that $\bar{e} (\bar{F},t^H) < e^C (t^H)$. In this case, citizens choose $e^H (G(t^H),t^H) = \bar{e} (\bar{F},t^H)$. The ruler’s equilibrium payoff is, therefore,

$$\begin{aligned} {\fancyscript{R}} \left( \bar{F},G\left( t^H\right) ,t^H, t^H, \bar{e}\left( \bar{F},t^H\right) \right) = \left[ \frac{\left( t^H - \delta \right) \lambda }{1-t^H} - \gamma \right] \frac{\bar{F}}{1-\delta }. \end{aligned}$$

The above payoff must be non-negative in equilibrium.

Now, consider announcement $t' = t^H + \epsilon$ where $\epsilon$ is positive and sufficiently small to satisfy $\bar{e} (\bar{F},t') < e^C (t')$. The following ruler’s strategies and citizens’ actions can be supported by another PBE: JI is $\bar{F}$, announced tax is $t'$, human rights are $G(t')$ and effort on the equilibrium path is $\bar{e}(\bar{F},t')$. This occurs because

$$\begin{aligned} {\fancyscript{R}} \left( \bar{F},G(t' ),t', t', \bar{e}\left( \bar{F},t' \right) \right)&= \left[ \frac{(t' - \delta )\lambda }{1-t'} - \gamma \right] \frac{\bar{F}}{1-\delta } \\&> {\fancyscript{R}} \left( \bar{F},G\left( t^H\right) ,t^H,t^H,\bar{e} \left( \bar{F},t^H\right) \right) \ge 0. \end{aligned}$$

However, the inequality contradicts the assumption that $t^H$ is the PBE announcement that maximises the ruler’s payoff. $\square$

As a result, the candidate for the equilibrium tax rate is $\arg \max _t \{(t-\delta ) e^C (t) \ \ \text{ s.t. } \ e^C (t) \le \bar{e} (\bar{F}, t) \}$. The interior solution of this problem denoted by $t^{Int}$ is $(1+\delta )/2$. The interior solution of this problem is the solution if $\bar{F} \ge (1-t^{Int}) e^C (t^{Int}) / \lambda = F^{Bou}$. Suppose that $\bar{F} < F^{Bou}$. Then, the solution is the corner $t^{Cor}$ such that $e^C (t^{Cor} ) = \bar{e} (\bar{F}, t^{Cor} )$, i.e., $t^{Cor} = 1 - (2\alpha \lambda \bar{F})^{1/2}$. Notice that $t^{Cor}$ decreases with $\alpha ,\,\lambda$ and $\bar{F}$. Furthermore, it holds that $t^{Cor} > t^{Int} > t^C$ in this case.

Step 4: Consider the ruler’s payoff. If the ruler plays as above and receives negative payoff, the ruler chooses $F^H=0$ and $G^H=0$ instead of the above strategy. Suppose that $\bar{F} \ge F^{Bou}$. Then, the interior solution is in equilibrium if ${\fancyscript{R}} (\bar{F}, G(t^{Int}), t^{Int}, t^{Int}, e^C (t^{Int} )) \ge 0$, which can be rewritten as $\bar{F} \le (t^{Int} - \delta ) e^C (t^{Int}) / \gamma = F^{Int}$. Suppose, instead, that $\bar{F} < F^{Bou}$. Then, the corner solution is in equilibrium if ${\fancyscript{R}} (\bar{F}, G(t^{Cor}), t^{Cor}, t^{Cor}, e^C (t^{Cor} )) \ge 0$, which equals $\bar{F} \le (t^{Cor} - \delta )e^C (t^{Cor}) / \gamma = F^{Cor}$.