Corruption and bicameral reforms

Abstract

During the last decade unicameral proposals have been put forward in fourteen US states. In this paper we analyze the effects of the proposed constitutional reforms, in a setting where decision making is subject to ‘hard time constraints’, and lawmakers face the opposing interests of a lobby and the electorate. We show that bicameralism might lead to a decline in the lawmakers’ bargaining power vis-a-vis the lobby, thus compromising their accountability to voters. Hence, bicameralism is not a panacea against the abuse of power by elected legislators and the proposed unicameral reforms could be effective in reducing corruption among elected representatives.

Appendix

1.1 Proof of Lemma 2

Note that in the second period the high cost policy is always chosen. Hence, we conclude that to show whether the voter’s expected payoff is maximized by $\sigma ^{*}$, we only need to analyze the first period payoff for all $\sigma$. Let us start by considering the following alternative strategy

$$\begin{aligned} \sigma ^{\prime }=[\sigma ^{\prime }(C^{H})=1,\sigma ^{\prime }(C^{L})=1,\sigma ^{\prime }(\varnothing )=1] \end{aligned}$$

Under $\sigma ^{\prime }$, the high cost policy is preferred by any legislator receiving $\beta _{g_{d}}\ge 0$, since he can receive lobby transfers and choose his most preferred policy in both periods. On the other hand, under the voting strategy $\sigma ^{*}$ depending on the parameters of the model, the legislator will choose either $C^{H}$ or $C^{L}$. If $C^{H}$ is chosen, then the expected payoff under the two alternative strategies is the same. On the other hand, if $C^{L}$ is chosen then the voter prefers $\sigma ^{*}$ to $\sigma ^{\prime }$. Hence, we conclude that $\sigma ^{\prime }$ is not an equilibrium strategy. Consider next the following alternative strategy

$$\begin{aligned} \sigma ^{{\prime }{\prime }}=[\sigma ^{{\prime }{\prime }}(C^{H})=0,\sigma ^{{\prime }{\prime }}(C^{L})=0,\sigma ^{{\prime }{\prime }}(\varnothing )=0] \end{aligned}$$

Under this voting strategy the incumbent is never reappointed. Therefore, since $C^{H}$ generates a higher net profit to be shared, the legislator will always choose $C^H$. Hence, $\sigma ^{{\prime }{\prime }}$ is not an equilibrium voting strategy and more generally, by the same arguments, any strategy such that either $\sigma (C^H)=1$ or $\sigma (C^L)=0$, cannot be an equilibrium voting strategy. Finally, consider the strategy

$$\begin{aligned} \sigma ^{{\prime }{\prime }{\prime }}=[\sigma ^{{\prime }{\prime }{\prime }}(C^{H})=0,\sigma ^{{\prime }{\prime }{\prime }}(C^{L})=1,\sigma ^{{\prime }{\prime }{\prime }}(\varnothing )=0] \end{aligned}$$

Note that, as $v_{g}^{1}(C^{L})>v_{g}^{1}(\varnothing )$, if the legislator does not receive transfers from the lobby, he always implements $C^{L}$, i.e. $C^{L}$ is the outside option. Since in the bargaining game the legislator chooses between $C^{H}$ and the outside option $C^{L}$, any voting strategy that punishes or rewards him for not choosing any policy does not affect his behavior and thus the policy outcome. As a result, the voter is indifferent between $\sigma ^{*}$ and $\sigma ^{{\prime }{\prime }{\prime }}$.

$\square$

1.2 Proof of Lemma 3

In $t=2$ the following holds. Under a closed rule arrangement, the second legislator can only approve or veto the policy chosen by the first. Furthermore, since $v_{g_{2}}^{2}(C^H)>v_{g_{2}}^{2}(\varnothing )\forall \beta _{g_{2}}^{2}$, vetoing is not credible. As a consequence, if in $t=2$ the lobby can induce the first legislator to choose $C^{H}$, then she does not need to pay any positive transfer to convince the second to pass $C^{H}$. Hence, $\beta _{g_{2}}^{2}=0$. We can now determine the equilibrium transfers inducing the first legislator to choose $C^{H}$. Given that the time constraint is binding with probability $\gamma$, in case of disagreement, $g_1$ outside option is $\gamma v_{g_{1}}^{2}(\varnothing )+(1-\gamma )(B+E)$, and the lobby’s outside option is $\gamma v_{l}^{2}(\varnothing )+(1-\gamma )B$. Remembering that $v_{l}^{2}(\varnothing )=v_{g_{1}}^{2}(\varnothing )=0$, then the first legislator prefers agreement to disagreement if and only if $\beta ^{2}_{g_1}\ge \frac{C^H-\gamma (B+E)}{\pi }$, whereas the lobby always prefers agreement. Hence, if the first legislator is the proposer then $\beta ^{2}_{g_1}=1$, if the lobby is the proposer then $\beta ^{2}_{g_1}=\frac{C^H-\gamma (B+E)}{\pi }$, and $C^H$ is always chosen. Moving to $t=1$, and remembering that in $t=2$ the first legislator is the proposer with probability q and the lobby is the proposer with probability $1-q$, the second period expected payoff for $g_1$ is $\left\{ (B+E-C^H)+q\pi + (1-q)[C^H-\gamma (B+E)] \right\}$. Hence, if in the first period the first legislator rejects the first lobby offer, he obtains the disagreement payoff $\gamma v_{g_{1}}^{1}(\varnothing )+(1-\gamma )(B+E)+\delta \left\{ (B+E-C^H)+q\pi + (1-q)[C^H-\gamma (B+E)] \right\}$, whereas the agreement payoff is given by $B+E-C^H+\beta ^1_{g_1}\pi +\delta (B-C^H)$. Hence, agreement is preferred to disagreement if and only if $\beta ^{1}_{g_1}\ge \frac{C^H+\delta [E + q\pi +(1-q)C^H]}{\pi }-\gamma [1+\delta (1-q)]\frac{B+E}{\pi }$.

As for the second legislator, since $\beta _{g_{2}}^{2}=0$, his disagreement payoff is $v_{g_{2}}^{1}(\varnothing )+\delta \left[ (B+E-C^H) \right]$, whereas his payoff from agreement is $B+E-C^H+\beta ^1_{g_2}\pi +\delta (B-C^H)$. Hence the second legislator can credibly threaten to veto the proposal passed by $g_{1}$, unless he receives $\overline{\beta } _{g_{2}}^{1}=\frac{C^{H}+\delta E-(B+E)}{\pi }$. On the other hand, if $\frac{\delta E}{\pi }+\frac{C^{H}-(B+E)}{\pi }<0$, $g_{2}$ cannot credibly veto any policy chosen by $g_{1}$ and therefore $\overline{\beta }_{g_{2}}^{1}=0$.

Finally, note that $\overline{\beta }^1_{g_1} > \overline{\beta }^1_{g_2}$ if and only if $(1-\gamma )(B+E) + \delta \{q\pi + (1-q)[C^H-\gamma (B+E)]\} \ge 0$, which is always true because the second term is the expected lobby transfer, which is always weakly positive. $\square$

1.3 Lemma 6

Lemma 6

Under closed rule, the voting strategy $\sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}({\mathbf {\varnothing }})=1]$ is the unique equilibrium voting strategy.

Proof

Since the voting strategy depends only on the policy outcome, the optimality of the voting strategy relies on the same arguments as in the unicameral case. In particular, $\sigma ^{*}$ is an equilibrium voting strategy since the voter is strictly better off by choosing $\sigma ^{*}$ than under any alternative strategy $\sigma ^{\prime }$ such that either $\sigma ^{\prime }(C^H)=1$ or $\sigma ^{\prime }(C^L)=0$. Moreover, under closed rule this is the unique equilibrium voting strategy because $\sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}(\varnothing )=1]$ is strictly preferred to $\sigma ^{\prime }=[\sigma ^{\prime }(C^{H})=0,\sigma ^{\prime }(C^{L})=1,\sigma ^{\prime }(\varnothing )=0]$, since punishing or rewarding the legislators for not implementing any policy is not pay-off irrelevant. In fact, under $\sigma ^{\prime }$, the second legislator cannot extract any rent since he cannot credibly veto any policy. This implies that $C^H$ is more likely to be chosen because the feasibility constraint on lobby transfers is more easily satisfied when $\beta _{g_2}=0$. Hence the voter strictly prefers $\sigma ^{*}$ to $\sigma ^{\prime }$. $\square$

Remark 1

Consider an alternative voting strategy $\sigma ^{\prime }_{g_d}$ whereby each legislator $g_d$ is re-elected based on the policy he has passed and $\sigma ^{\prime }_{g_d}=[\sigma ^{\prime }_{g_d}(C^H)=0, \sigma ^{\prime }_{g_d}(C^L)=1, \sigma ^{\prime }_{g_d}(\varnothing )=1]$. Note that if each legislator passes the same policy, $\sigma ^{\prime }_{g_d}$ delivers the same outcome as $\sigma ^{*}$. Consider now the case where the two legislators pass different policies. Since the second legislator can only veto the policy passed by the first, the only relevant scenario is the one in which the first legislator chooses $C\in \{C^L,C^H\}$ and the second vetoes it so that no policy is passed. Since if the first legislator chooses $C^L$, the second will always pass it, we only need to consider the case in which the first legislator chooses $C^H$ and the second vetoes it. In this case under the voting strategy $\sigma ^{*}$ both legislators are re-elected, whereas under the voting strategy $\sigma ^{\prime }_{g_d}$, only the second legislator is re-elected. We can easily see that, given $\sigma ^{*}$, if the first legislator chooses $C^H$, which is subsequently vetoed, then his expected payoff is $v_{g_1}(\varnothing )+\delta (B - C^H + q\pi + (1-q) C^H)$ which is strictly lower than the payoff from choosing $C^L$, that is $B+E+\delta [B+E-C^H+q\pi + (1-q) C^H]$. In other words, the first legislator will never find it optimal to choose $C^H$ when the second legislator will veto it. The same is true under the alternative voting strategy $\sigma ^{\prime }_{g_d}$ because the payoff for $g_1$ from choosing $C^H$ that is subsequently vetoed is $v_{g_1}(\varnothing )+\delta (B-C^H)$, which is again strictly smaller than the payoff from choosing $C^L$. Hence, punishing only the first legislator for choosing $C^H$ when this outcome is subsequently vetoed does not make him more or less likely to choose $C^H$ over $C^L$.

1.4 Lemma 7

Lemma 7

Assume that the first chamber retains proposal power in the second period with probability p. Then in $t=1$ the minimum shares of rent required by the two legislators to choose the high cost policy are $\hat{\beta }_{g_1}^1=\overline{\beta }^1_{g_1}-\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$ and $\hat{\beta }_{g_{2}}^{1}=\overline{\beta }^1_{g_2}+\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$.

Proof

If the first chamber loses proposal power in the second period, the share of rents required to choose the high cost policy in the first period is $\widetilde{\beta }_{g_1}=\frac{C^H+\delta E-\gamma (B+E)}{\pi }$, whereas if she retains proposal power the share required is $\widetilde{\beta }_{g_1}+\delta [q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$. Hence, when $g_1$ retains proposal power with probability p, the share of rents required to choose the high cost policy is $\hat{\beta }_{g_1}^1=\widetilde{\beta }_{g_1}+\delta p[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$ or equivalently $\hat{\beta }_{g_1}^1=\overline{\beta }^1_{g_1}-\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$. By the same argument, the second chamber requires $\widetilde{\beta }_{g_2}=\frac{C^H+\delta E- (B+E)}{\pi }$ when she does not gain proposal power and requires $\widetilde{\beta }_{g_2}+\delta [q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$ if she gains proposal power in the second period. Hence $\hat{\beta }_{g_{2}}^{1}=\widetilde{\beta }_{g_2}+\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$, which can be rewritten as $\hat{\beta }_{g_{2}}^{1}=\overline{\beta }^1_{g_2}+\delta (1-p)[q+(1-q)\frac{C^H-\gamma (B+E)}{\pi }]$. $\square$

1.5 Proof of Proposition 2

If the minimum shares required under unicameralism and bicameralism to choose $C^{H}$ are feasible (i.e. $\overline{\beta }_{g}^{1}<1$ and $\sum _{d=1}^{2} \overline{\beta }_{g_{d}}^{1}<1$), then $C^{H}$ is chosen. On the other hand, when the minimum shares are not feasible (i.e. $\overline{\beta }_{g}^{1}>1$ and $\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}>1$), then $C^{L}$ is chosen. Note that, when $\gamma =0$, then $\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\ge \overline{\beta }_{g}^{1}$. Therefore, when $\overline{\beta }_{g}^{1}<\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<1$, both shares are feasible and $C^H$ is chosen under both legislative arrangements, whereas if $\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\ge \overline{\beta }_{g}^{1}>1$ none of the shares is feasible and $C^L$ is chosen under both legislative arrangements. On the other hand, if $\overline{\beta }_{g}^{1}<1<\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}$, only the unicameral share is feasible implying that $C^H$ is chosen under unicameralism and $C^L$ under bicameralism. Consider now the scenario where the time constraint is binding with some probability ($0<\gamma \le 1$). Then two cases arise. If $\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\ge \overline{\beta }_{g}^{1}$, then we obtain the same policy choice characterized when $\gamma =0$. On the other hand, when $\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<\overline{\beta }_{g}^{1}$ the following holds. If the minimum shares under the two legislative arrangements are feasible ($\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<\overline{\beta }_{g}^{1}<1$) then $C^H$ is chosen under both arrangements, if the same shares are not feasible ($\overline{\beta }_{g}^{1}>\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}>1$, then $C^L$ is chosen under both arrangements. Finally, if $\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}<1<\overline{\beta }_{g}^{1}$, then $C^H$ is chosen under bicameralism whereas $C^L$ is chosen under a unicameralism. $\square$

1.6 Proof of Lemma 5

In $t=2$, since $v_{g_{d}}^{2}(C^{L})>v_{g_{d}}^{2}(\varnothing )$ $\forall d$, in the absence of lobby transfers the first legislator chooses $C^{L}$ and the second legislator ratifies this choice. If amendment rights are restricted, once the policy $C^{L}$ is chosen by the first legislator, it can be amended to $C^{H}$ only if all legislators, including the first, approve the change. Remembering that in $t=2$ each legislator and the lobby l make take-it-or-leave-it offers with probability q and $1-q$, then $\beta ^{2}_{g_d}=q+(1-q)\frac{C^H}{\pi }$, with $\sum _{d=1}^2 \beta ^{2}_{g_d} \le 1$. Moving backward to the first period, for each legislator $g_d$, the payoff from $C^{H}$ is $V_{g_{d}}(C^{H})=B+E+\beta _{g_{d}}^{1}\pi -C^{H}+\delta (B-C^{H})$ and the payoff from $C^L$ is $V_{g_{d}}(C^{L})=B+E+\delta (B+E+\beta _{g_d}^{2}\pi -C^{H})$. Therefore, if amendment rights are restricted, we find that each legislators prefers $C^H$ to $C^L$ if and only if $\beta _{g_d}^{1}\ge \frac{\delta E+C^H}{\pi }+\delta [q+(1-q)\frac{C^H}{\pi }]$ with $\sum _{d=1}^2 \beta ^{2}_{g_d} \le 1$.

On the other hand, in the case of unrestricted amendment rights, if in $t=2$ the policy $C^L$ is chosen by the first legislator, the lobby can still obtain $C^H$ by paying $\beta ^{2}_{g_2}=q+(1-q)\frac{C^H}{\pi }$ to the second legislator and $\beta ^{2}_{g_1}=0$ to the first because $v_{g_{1}}^{2}(C^{L})>v_{g_{1}}^{2}(\varnothing )$. In the first period, the second legislator obtains the expected payoff $V_{g_{2}}(C^H)=B+E+\beta _{g_{d}}^{1}\pi -C^{H}+\delta (B-C^{H})$ by choosing $C^H$, and the payoff $V_{g_{2}}(C^L)=B+E+\delta (B+E+\beta _{g_2}^{2}\pi -C^{H})$ from choosing $C^L$. Hence, the second legislator prefers $C^H$ to $C^L$ if and only if $\beta _{g_2}^{1}\ge \frac{\delta E+C^H}{\pi }+\delta [q+(1-q)\frac{C^H}{\pi }]$. On the other hand, remembering that if $C^L$ is passed by the first legislator, the lobby offers $\beta _{g_2}^{1}=\frac{\delta E+C^H}{\pi }+\delta [q+(1-q)\frac{C^H}{\pi }]$ to the second legislator who amends $C^L$ to $C^H$, then by choosing $C^L$, the first legislator obtains the expected payoff $V_{g_{1}}(C^L)=B+E-C^H+\delta (B-C^H)$, whereas, by not passing any policy his expected payoff is $V_{g_{1}}(\varnothing )=\delta (B+E-C^{H})$. Hence, if $\delta E+C^H-(B+E)>0$ then $V_{g_{1}}(\varnothing )>V_{g_{1}}(C^{L})$, which implies that the first legislator can credibly threaten not to pass any policy (i.e. $V_{g_{1}}(\varnothing )>V_{g_{1}}(C^H)$) unless he receives $\beta _{g_1}^{1}\ge \frac{\delta E+C^H-(B+E)}{\pi }>0$. On the other hand, if $\delta E+C^H-(B+E)\le 0$, the first legislator cannot credibly threaten not to choose any policy, and since $V_{g_{1}}(C^{H})\ge V_{g_{1}}(C^{L})$ $\forall \beta _{g_1}^{1}$, the lobby offers $\beta _{g_1}^{1}=0$ and $C^H$ is passed. $\square$

1.7 Proof of Proposition 3

Suppose that Lemma 5 holds. If the sum of the minimum shares is feasible ($\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}\le 1$) then $C^{H}$ is chosen, whereas if $\sum _{d=1}^{2}\overline{\beta }_{g_{d}}^{1}>1$, $C^{L}$ is chosen. Since the sum of the minimum shares under open rule is at least as high than under closed rule, then: (a) whenever the sum of the minimum shares is not feasible under closed rule, it will also not be feasible under open rule; (b) When the sum of the minimum share is feasible under closed rule, it may not be feasible under open rule. From (a) and (b) we conclude that whenever the low cost policy is chosen under closed rule it will also be chosen under open rule, while the reverse is not true.

Consider now the case where amendment rights are unrestricted, with $0<\overline{\beta }^1_{g_d}<1$ for all $g_d$ and $\sum _{d=1}^{2} \overline{\beta }^1_{g_d} > 1$. In this case, the first legislator could choose $C^{H}$ if he is offered the appropriate transfer. However, given that both legislators cannot be offered the transfer necessary to pass $C^{H}$, the lobby will not find it optimal to carry out the transfer necessary to obtain $C^{H}$ in the first legislative step, knowing that this proposal will be overridden by the subsequent legislator. As a consequence, the lobby offers $\beta ^{1}_{g_1}=0$ to the first legislator. Since $g_1$ anticipates that $C^{L}$ will be overridden by the last legislator who can receive the appropriate rent share for choosing $C^{H}$, then $g_{1}$ rejects the offer and does not implement any policy. As a consequence, the second legislator legislators without proposal power will not be able to amend any proposal, and the mandate terminates with no policy implemented. $\square$

1.8 Lemma 8

Lemma 8

Under open rule, the voting strategy $\sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}({\mathbf {\varnothing }})=1]$ is an equilibrium voting strategy. Moreover, under unrestricted amendment rights it is unique.

Proof

Since the voting strategy depends only on the policy outcome, the optimality of the voting strategy relies on the same arguments as in the unicameral case. In particular, $\sigma ^{*}$ is an equilibrium voting strategy since the voter is strictly better off by choosing $\sigma ^{*}$ than under any alternative strategy $\sigma ^{\prime }$ such that either $\sigma ^{\prime }(C^H)=1$ or $\sigma ^{\prime }(C^L)=0$. Moreover, under unrestricted amendment rights this is the unique equilibrium voting strategy because $\sigma ^{*}=[\sigma ^{*}(C^{H})=0,\sigma ^{*}(C^{L})=1,\sigma ^{*}(\varnothing )=1]$ is strictly preferred to $\sigma ^{\prime }=[\sigma ^{\prime }(C^{H})=0,\sigma ^{\prime }(C^{L})=1,\sigma ^{\prime }(\varnothing )=0]$, since punishing or rewarding the legislators for not implementing any policy is not pay-off irrelevant. In fact, note that with unrestricted amendment rights, $g_1$ might not find it optimal to pass any policy in order to prevent the final implementation of $C^{H}$. As a result, $\varnothing$ can be an outside option. Suppose now that the voter adopts the voting strategy $\sigma ^{\prime }=[\sigma (C^{H})=0,\sigma (C^{L})=1,\sigma (\varnothing )=0]$. In this case, because of the punishment $\sigma (\varnothing )=0$, not choosing any policy is strictly dominated by choosing either $C^L$ or $C^H$. Hence, either the first legislator chooses $C^{L}$ and the last legislator amends it passing $C^{H}$, or both legislators pass $C^H$. Since for the voter $v_{k}^{1}(C^{H})<v_{k}^{1}(\varnothing )$, then he strictly prefers $\sigma ^{*}$ to $\sigma ^{\prime }$. $\square$