Limits of Contractual Democracy – Competition for Wages and Office

Abstract

Our research on wage competition dates back to 2000: We started from the observation that office-holders’ wages do not depend on their performance. As the improvement of the office-holders’ efficiency and the selection of competent office-holders are main goals of our research, we wanted to try to use salaries as an incentive for good performance. We developed a model in which an office-holder who performs badly incurs a salary reduction, as developed in Chap. 2 in the context of long-term projects. As a special feature of our model, this reduction was not meant to be imposed on the office-holders, but to be offered by the candidates during their campaign.

Appendix

Proof of Proposition 6.1

Note that if candidate 1 decides to run for office, he will be elected independently of whether candidate 2 decides to run for office or not. Therefore, candidate 2 should run for office if and only if his utility from serving as a politician is greater than zero, which is his utility from the default outcome when no candidate runs for office. Thus “run for office” is weakly dominant for candidate 2 if $b_2 +W - c_2\ge 0$. If $b_2+W-c_2<0$, “do not run” is weakly dominant. Hence, if and only if $W\ge c_2 - b_2$, candidate 2 will run for office. If $W\ge c_2 - b_2$, then candidate 1 will also run for office if

$$\begin{aligned} b_1+W-c_1\ge b_2-\frac{W}{N-1}(1+\lambda ), \end{aligned}$$

i.e. if his utility from holding office is higher than the utility obtained when candidate 2 is in office, based on the utilities in Eqs. (6.1) and (6.2). This condition can be transformed into

$$\begin{aligned} W\ge \frac{N-1}{N+\lambda }\big (c_1-(b_1-b_2)\big ). \end{aligned}$$

If $W<c_2-b_2$, candidate 1 will run for office if $b_1+W-c_1\ge 0$, i.e. if $W\ge c_1-b_1$. $\square$

Proof of Proposition 6.2

First note that in order for candidate 1 to be elected, $W_1$ must satisfy

$$\begin{aligned} W_1 \le (b_1-b_2)\frac{N-1}{1+\lambda }+W_2, \end{aligned}$$

because otherwise the public is better off electing candidate 2. This follows from Eq. (6.5). Therefore, when candidate 1 wants to be elected, he offers the wage

$$\begin{aligned} W_1 = (b_1-b_2)\frac{N-1}{1+\lambda }+W_2. \end{aligned}$$

(6.6)

A downward deviation can be excluded, because in that case candidate 1 could raise his utility by offering a higher wage, and he would still be elected. Deviation to a higher wage leads to the election of candidate 2.

Candidate 1 will not deviate to a higher wage than in (6.6) and will not leave the office to candidate 2 if

$$\begin{aligned} b_1+W_1-c_1 \ge b_2 - \frac{W_2}{N-1}(1+\lambda ). \end{aligned}$$

Inserting the equilibrium value of $W_1$ from Eq. (6.6) as a function of $W_2$, this condition becomes

$$\begin{aligned} b_1+(b_1-b_2)\frac{N-1}{1+\lambda }+ W_2-c_1 \ge b_2 - \frac{W_2}{N-1}(1+\lambda ), \end{aligned}$$

which can be transformed into

$$\begin{aligned} (b_1-b_2)\left( 1+\frac{N-1}{1+\lambda } \right) + W_2 \left( 1 + \frac{1+\lambda }{N-1}\right) \ge c_1, \end{aligned}$$

which then yields

$$\begin{aligned} W_2 \ge \frac{N-1}{N+\lambda }\,c_1-(b_1-b_2)\frac{N-1}{1+\lambda } =: W_2^{min}. \end{aligned}$$

(6.7)

Thus, candidate 1 will want to run for office if condition (6.7) is fulfilled, i.e. if the proposed remuneration $W_2$ exceeds a certain threshold. By substitution, the corresponding threshold for $W_1$ is then given by $W_1^{min}=\frac{N-1}{N+\lambda } c_1$.

We next examine the optimal choice of $W_2$ by candidate 2. A possible deviation from the proposed equilibrium in the proposition for candidate 2 would be to offer a wage $W_2^\prime = W_2-\epsilon$ for some small $\epsilon > 0$, which would lead to his election. Candidate 2 will not choose this option if

$$\begin{aligned} b_1- \frac{W_1}{N-1}(1+\lambda ) \ge b_2 + W_2^\prime - c_2, \end{aligned}$$

i.e. if his utility from being a citizen under candidate 1 is higher than his utility from holding office himself. By inserting the equilibrium value of $W_1$, as given by (6.6), we obtain the condition

$$\begin{aligned} b_1- \frac{W_2}{N-1}(1+\lambda )- (b_1-b_2)\ge b_2+W_2-\epsilon - c_2, \end{aligned}$$

which can be transformed into

$$\begin{aligned} W_2 \le \frac{N-1}{N+\lambda } (c_2 + \epsilon ). \end{aligned}$$

(6.8)

Therefore, if wage $W_2$ is small enough, candidate 2 would prefer to be a citizen under candidate 1 rather than running for office for a lower wage.

Concluding, there only exist equilibrium values for wage offers $W_2$ that satisfy both conditions (6.8) and (6.7) if

$$\begin{aligned} W_2^{max}:= \frac{N-1}{N+\lambda }\,c_2 \ge \frac{N-1}{N+\lambda }\,c_1 -(b_1-b_2)\frac{N-1}{1+\lambda } \end{aligned}$$

and hence we obtain the assumption of the proposition given by

$$\begin{aligned} (1+\lambda )(c_1-c_2) \le (N+\lambda )(b_1-b_2). \end{aligned}$$

From condition (6.6), we also obtain threshold wage

$$W_1^{max}:=(b_1-b_2)\frac{N-1}{1+\lambda }+\frac{N-1}{N+\lambda }c_2.$$

$\square$

Proof of Proposition 6.4

We first prove statement (i). In principle, six different cases can occur.

Case 1::

$\tilde{W}\le 0$, $(1+\lambda )(c_1-c_2)\le (N+\lambda )(b_1-b_2)$.

Case 2::

$\tilde{W}\le 0$, $(1+\lambda )(c_1-c_2)> (N+\lambda )(b_1-b_2)$.

Case 3::

$\tilde{W}>0$, $\tilde{W}>c_2-b_2$ and $(1+\lambda )(c_1-c_2) \le (N+\lambda )(b_1-b_2)$.

Case 4::

$\tilde{W}>0$, $\tilde{W}<c_2-b_2$ and $(1+\lambda )(c_1-c_2) \le (N+\lambda )(b_1-b_2)$.

Case 5::

$\tilde{W}>0$, $\tilde{W}>c_2-b_2$ and $(1+\lambda )(c_1-c_2) > (N+\lambda )(b_1-b_2)$.

Case 6::

$\tilde{W}>0$, $\tilde{W}<c_2-b_2$ and $(1+\lambda )(c_1-c_2) > (N+\lambda )(b_1-b_2)$.

If N is sufficiently large, we obtain $(1+\lambda )(c_1-c_2)<(N+\lambda )(b_1-b_2)$. This implies that we can drop the cases 2, 5, and 6. Now we examine the three remaining cases.

Case 1::

As candidate 1 is elected under competitive wages by Proposition 6.2, welfare is given by

$$U^{var}=Nb_1-c_1-\lambda W_1$$

As discussed in Sect. 6.6.1, under a fixed wage, the wage is set at zero, candidate 1 runs for office and is elected. We obtain

$$U^{fix}=Nb_1-c_1.$$

Thus welfare under the fixed wage scenario is no smaller than under competitive wages. Note that in both scenarios candidate 1 is elected.

Case 3::

To derive our results in case 3, we proceed in four steps.

Step 1::

Due to Proposition 6.2, candidate 1 is again elected under competitive wages. From the same proposition and the assumption of non-negative wages, we obtain

$$W_2^{min}=\max \left\{ 0,\frac{N-1}{N+\lambda }c_1-(b_1-b_2)\frac{N-1}{1+\lambda }\right\} , $$

which yields

$$W_1^{min}=\max \left\{ \frac{N-1}{1+\lambda }(b_1-b_2),\frac{N-1}{N+\lambda }c_1\right\} . $$

For sufficiently large N we obtain

$$W_1^{min}=\frac{N-1}{1+\lambda }(b_1-b_2) .$$

Therefore maximal welfare under competition for wages is given by

$$U_{max}^{var}=Nb_1-c_1-\lambda \frac{N-1}{1+\lambda }(b_1-b_2) .$$

Step 2::

Under a fixed wage, welfare depends on which candidate is elected. Following the logic in Sect. 6.6.1 and given the assumptions of case 3 and the non-negativity of wages, we have

$$U^{fix}=\max \left\{ Nb_2-c_2-\lambda \max \left\{ 0,c_2-b_2\right\} ,Nb_1-c_1-\lambda \tilde{W}\right\} . $$

We now show that for sufficiently large N, the public will always set the wage at $\tilde{W}$, so that candidate 1 runs for office and is elected. As

$$Nb_2-c_2-\lambda \max \left\{ 0,c_2-b_2\right\} \le Nb_2-c_2,$$

it suffices to show that

$$Nb_2-c_2<Nb_1-c_1-\lambda \tilde{W}.$$

Step 3::

To prove the assertion, we insert $\tilde{W}$ and obtain

$$Nb_2-c_2<Nb_1-c_1-\lambda \frac{N-1}{N+\lambda }(c_1-(b_1-b_2)) . $$

This inequality can be transformed into

$$c_1(1+\lambda \frac{N-1}{N+\lambda })-c_2<(N+\lambda \frac{N-1}{N+\lambda })(b_1-b_2) , $$

which holds for sufficiently large N (note that $\frac{N-1}{N+\lambda }\rightarrow 1$ for $N\rightarrow \infty$).

Step 4::

We can state now that welfare under fixed wages is given by

$$U^{fix}=Nb_1-c_1-\lambda \tilde{W}.$$

Welfare is higher under a fixed wage scenario if

$$Nb_1-c_1-\lambda \tilde{W}>Nb_1-c_1-\lambda \frac{N-1}{1+\lambda }(b_1-b_2).$$

Inserting $\tilde{W}$ yields

$$(1+\lambda )(c_1-(b_1-b_2))<(N+\lambda )(b_1-b_2),$$

which holds for sufficiently large N. Again, candidate 1 is elected in both scenarios.

Case 4::

Case 4 is analogue to case 3.

Under competitive wages, candidate 1 is elected and the maximal welfare is given by

$$U_{max}^{var}=Nb_1-c_1-\lambda \frac{N-1}{1+\lambda }(b_1-b_2), $$

according to the same considerations as in case 3.

Under fixed wages, welfare is given by

$$U^{fix}=Nb_1-c_1-\lambda \tilde{W}.$$

Thus, as in case 3, welfare is higher under a fixed wage if N is sufficiently large, and candidate 1 is elected in both scenarios.

Statement (ii) of the proposition follows immediately from the above considerations. If we insert $\lambda =0$, welfare is given under both wage-setting regimes by $Nb_1-c_1$, as candidate 1 is always elected. $\square$

Proof of Proposition 6.5

In principle, three different cases can occur:

Case 1::

$\tilde{W}\le 0$.

Case 2::

$\tilde{W}>0$ and $(1+\lambda )\,c_1 \le (N+\lambda )(b_1-b_2)$.

Case 3::

$\tilde{W}>0$ and $(1+\lambda )\,c_1 > (N+\lambda )(b_1-b_2)$.

We prove the statement by showing the assertions for each case.

Case 1::

Suppose $\tilde{W}\le 0$. This implies $c_1<b_1-b_2$, which can be easily verified by checking the definition of $\tilde{W}$. Then $(1+\lambda )c_1 \le (N+\lambda )(b_1-b_2)$ holds. Therefore candidate 1 is elected under competition for wages with $W_2=0$, since $W_2^{max}=0$. Accordingly, $W_1$ is given by $\frac{N-1}{1+\lambda }(b_1-b_2)$ and welfare is given by

$$\begin{aligned} U^{var} = Nb_1 - c_1 - \lambda \frac{N-1}{1+\lambda }\,(b_1-b_2). \end{aligned}$$

Since $\tilde{W}<0$, the wage is set at zero in case of a fixed wage, and candidate 1 runs for office and is elected. We obtain

$$\begin{aligned} U^{fix} = Nb_1 - c_1, \end{aligned}$$

and thus welfare is higher under the fixed wage scenario. In both scenarios candidate 1 is elected.

Case 2::

Suppose $\tilde{W}>0$ and $(1+\lambda )\,c_1 \le (N+\lambda )(b_1-b_2)$. Then candidate 1 is elected under competition for wages. Since $W_2^{max}=0$, in this case welfare is given by

$$\begin{aligned} U^{var} = Nb_1 - c_1 - \lambda \frac{N-1}{1+\lambda }\,(b_1-b_2). \end{aligned}$$

(6.9)

Under a fixed wage, the public sets the wage at $\tilde{W}$ so that candidate 1 runs for office and is elected if $N b_1 - c_1 - \lambda \tilde{W} \ge N b_2$.

As

$$\begin{aligned} \tilde{W} = \frac{N-1}{N+\lambda }\,\big (c_1-(b_1-b_2)\big ), \end{aligned}$$

this inequality can be transformed into

$$\begin{aligned} N b_1 - c_1 - \lambda \frac{N-1}{N+\lambda }\,\big (c_1-(b_1-b_2)\big ) \ge N b_2. \end{aligned}$$

This implies

$$\begin{aligned} (b_1-b_2)\ge c_1\frac{1+\lambda }{N+\lambda (2-\frac{1}{N})}, \end{aligned}$$

which always holds for $(1+\lambda )\,c_1\le (N+\lambda )(b_1-b_2)$ because

$$\begin{aligned} (b_1-b_2)\ge c_1\frac{1+\lambda }{N+\lambda }\ge c_1\frac{1+\lambda }{N+\lambda (2-\frac{1}{N})}. \end{aligned}$$

This implies that under a fixed wage scenario, candidate 1 will run and will be elected with certainty. We have welfare

$$\begin{aligned} U^{fix} = Nb_1 - c_1 - \lambda \tilde{W}. \end{aligned}$$

(6.10)

Comparing (6.9) and (6.10), welfare is higher under a fixed wage scenario if

$$\begin{aligned} \tilde{W} < \frac{N-1}{1+\lambda }\,(b_1-b_2). \end{aligned}$$

We insert $\tilde{W}$ and rearrange the terms, and obtain

$$(1+\lambda )(c_1-(b_1-b_2))<(N+\lambda )(b_1-b_2).$$

According to the assumptions in case 2, this inequality holds. Again, in both scenarios candidate 1 is elected.

Case 3::

Suppose $\tilde{W}>0$ and $(1+\lambda )\,c_1 > (N+\lambda )(b_1-b_2)$. In this case, candidate 2 is elected under competitive wages. The welfare under competition for wages is given by

$$\begin{aligned} U^{var} = Nb_2 - \lambda W_2. \end{aligned}$$

Under the fixed wage framework, welfare is

$$\begin{aligned} U^{fix} = \max \left\{ Nb_1 - c_1 - \lambda \tilde{W}, N b_2 \right\} . \end{aligned}$$

Hence welfare with wages set by the public is higher than, or equal to, what it would be under competitive wages.

While it is unambiguously clear that welfare is higher under fixed wages, it is not clear which wage the public will set in this scenario. The wage is set at $\tilde{W}$ such that candidate 1 runs for office and is elected if and only if

$$\begin{aligned} N b_1 - c_1 - \lambda \tilde{W} \ge N b_2 , \end{aligned}$$

which can be transformed into

$$\begin{aligned} (b_1-b_2)\ge c_1\frac{1+\lambda }{N+\lambda (2-\frac{1}{N})}. \end{aligned}$$

According to the assumption made in case 3, the upper inequality can either hold or not. This implies that candidate 1 may be elected under fixed wages, while under competition for wages candidate 2 will be elected for sure.

All in all, welfare is always higher under fixed wages, while candidate 1 is elected equally or more often under fixed wages than under competitive wages. $\square$

Proof of Proposition 6.6

We explore the same cases as in Proposition 6.5, but now with $\lambda =0$.

Case 1::

Suppose $\tilde{W}\le 0$. This implies $c_1<b_1-b_2$, hence $c_1<N(b_1-b_2)$ holds as well. By Proposition 6.2 and using $\lambda =0$ and $c_2=0$, we conclude that candidate 1 is elected under competition for wages. Welfare is given by

$$\begin{aligned} U^{var} = Nb_1 - c_1 . \end{aligned}$$

Under a fixed wage, the wage is set at zero, candidate 1 runs for office and is elected. We obtain

$$\begin{aligned} U^{fix} = Nb_1 - c_1. \end{aligned}$$

In both scenarios, candidate 1 is elected, and welfare is given by $Nb_1-c_1$ both under fixed wages and under competition for wages.

Case 2::

Suppose $\tilde{W}>0$ and $c_1 \le N(b_1-b_2)$. This implies that candidate 1 is elected under competition for wages. Welfare is given by

$$\begin{aligned} U^{var} = Nb_1 - c_1 . \end{aligned}$$

Under fixed wages, the public sets a wage no smaller than $\tilde{W}$, so that candidate 1 runs for office and is elected if and only if $Nb_1-c_1\ge Nb_2$. But this inequality holds by the assumptions made in case 2. Therefore welfare is given by

$$\begin{aligned} U^{fix} = Nb_1 - c_1 . \end{aligned}$$

As in case 1, candidate 1 is elected in both scenarios, and welfare is given by $Nb_1-c_1$.

Case 3::

Suppose $\tilde{W}>0$ and $c_1 > N(b_1-b_2)$. Under competition for wages, candidate 2 is elected and welfare is given by

$$\begin{aligned} U^{var} = Nb_2 . \end{aligned}$$

The public sets a wage strictly smaller than $\tilde{W}$ so that only candidate 2 will run for office and be elected if and only if $Nb_2>Nb_1-c_1$. But this inequality must hold in case 3 by assumption, thus welfare is given by

$$\begin{aligned} U^{fix} = N b_2. \end{aligned}$$

Therefore, fixed wages and competitive wages yield the same welfare, and in both scenarios candidate 2 is elected. $\square$