Unraveling short- and farsightedness in politics

Abstract

The absence of a deselection threat in incumbents’ last term in office can be negative or positive for the polity. Some politicians may reduce their efforts, while others may pursue beneficial long-term policies that may be unpopular in the short term. We propose a novel pension system that solves the effort problem while preserving the willingness to implement long-term policies. The idea is to give politicians the option to choose between a flexible and a fixed pension scheme. While in the fixed scheme, performance has no impact on the pension, the pension increases with short-term performance in the flexible scheme, using the vote share of the officeholder’s party in the next election as a performance indicator. Such a pension choice improves the well-being of citizens since officeholders are encouraged to invest in those activities that are beneficial for society. We analyze the properties and consequences of such a system. Finally, we extend the pension system with choice to non-last-term situations and derive a general welfare result.

Appendices

Appendix 1: Proofs

Proof of Proposition 3

By Proposition 1 and 2 we know that the populist chooses $e=\frac{k}{2c}$ under a fixed pension scheme and $e=\frac{k+\delta \mu \phi _0}{2c}$ under a flexible pension scheme. In both cases, he does not implement a long-term policy as he would suffer loss $d$. Hence, if given the choice, P opts for a flexible scheme if and only if

$$ \begin{aligned} {\mathbb {E}}(U^{\max }(P|{\text {flex}}\ \& \ I=0)) > {\mathbb {E}}(U^{\max }(P|{\text {fix}}\ \& \ I=0)). \end{aligned}$$

Using Eqs. (10) and (11) and inserting optimally chosen effort levels yields

$$\begin{aligned} k\left( \frac{k+\delta \mu \phi _0}{2c} \right) - c\left( \frac{k+\delta \mu \phi _0}{2c} \right) ^2 + \delta m_0+ \delta \mu \phi _0 \left( \frac{k+\delta \mu \phi _0}{2c} \right)> & {} k\left( \frac{k}{2c} \right) - c\left( \frac{k^2}{4c^2} \right) + \delta m_{\text {fix}}\\ \Leftrightarrow \left( \frac{k^2+ k\delta \mu \phi _0}{2c} \right) - \frac{(k+\delta \mu \phi _0)^2}{4c} + \delta m_0+ \frac{\delta k\mu \phi _0 +\delta ^2 (\mu \phi _0)^2}{2c}> & {} \frac{k^2}{2c} - \frac{k^2}{4c} + \delta m_{\text {fix}}\\ \Leftrightarrow m_0> & {} \frac{4m_{\text {fix}}c-2k\mu \phi _0-\delta (\mu \phi _0)^2}{4c}. \end{aligned}$$

$\square$

Proof of Proposition 4

If the statesman decides not to implement a long-term policy, his utility function is identical to that of the populist, and he chooses the same effort level. Hence, by Proposition 3, if

$$\begin{aligned} m_0>\frac{4m_{\text {fix}}c-2k\mu \phi _0-\delta (\mu \phi _0)^2}{4c}, \end{aligned}$$

the statesman chooses $I=1$ if and only if one of the following inequalities holds:

$$ \begin{aligned} {\mathbb {E}}(U^{\max }(S|{\text {flex}}\ \& \ I=0))= {\mathbb {E}}(U^{\max }(P|{\text {flex}}\ \& \ I=0)) < {\mathbb {E}}(U^{\max }(S|{\text {flex}}\ \& \ I=1)) \nonumber \\ \Leftrightarrow \dfrac{(k+\delta \mu \phi _0)^2}{4c} + \delta m_0 < \dfrac{(k+\delta \mu \phi _1)^2}{4c} + \delta (m_0+\beta ) \end{aligned}$$

(28)

or

$$ \begin{aligned} {\mathbb {E}}(U^{\max }(S|{\text {flex}}\ \& \ I=0))= {\mathbb {E}}(U^{\max }(P|{\text {flex}}\ \& \ I=0)) < & {} {\mathbb {E}}(U^{\max }(S|{\text {fix}}\ \& \ I=1)) \nonumber \\ \Leftrightarrow \dfrac{(k+\delta \mu \phi _0)^2}{4c} + \delta m_0 < & {} \dfrac{k^2}{4c} + \delta (m_{\text {fix}}+\beta ) . \end{aligned}$$

(29)

Inequality (28) is satisfied if

$$\begin{aligned} \beta > \dfrac{(k+\delta \mu \phi _0)^2-(k+\delta \mu \phi _1)^2}{4\delta c}. \end{aligned}$$

(30)

Inequality (29) is satisfied if

$$\begin{aligned} \beta > \dfrac{(k+\delta \mu \phi _0)^2-k^2}{4\delta c} + (m_0-m_{\text {fix}})=\dfrac{\delta ^2(\mu \phi _0)^2+2\delta k\mu \phi _0}{4\delta c} + (m_0-m_{\text {fix}}). \end{aligned}$$

(31)

The lower bound for $\beta$ in (30) depends on the difference $\phi _0-\phi _1$ and is zero if and only if $\mu$ is zero, which would mean that the flexible scheme reduces to a fixed scheme. We see here that a flexible scheme can never motivate every statesman to implement a long-term policy. On the contrary, as outlined in Corollary 1, the lower bound for $\beta$ in (31) can be brought down to zero if we replace $m_0$ by its lower bound $m_{\text {fix}}-\frac{2k\mu \phi _0+\delta (\mu \phi _0)^2}{4c}$ in Proposition 3(i), which we denote here by $m_0^{\text {low}}$.

Let $\beta$ satisfy Eq. (31). Then the statesman chooses the fixed pension scheme if

$$\begin{aligned} m_0<\dfrac{4m_{\text {fix}}c-2k\mu \phi _1-\delta (\mu \phi _1)^2}{4c}:=m_0^{\text {high}}. \end{aligned}$$

This results from comparing the right-hand sides of Inequalities (28) and (29) and proceeding as in Proposition 3. Since $\phi _0 > \phi _1$, it holds that

$$\begin{aligned} m_0^{\text {low}} < m_0^{\text {high}}. \end{aligned}$$

Hence, the interval

$$\begin{aligned} \left( m_0^{\text {low}}, m_0^{\text {high}}\right) \end{aligned}$$

(32)

is not empty, and each value of $m_0$ contained in this interval incentivizes the populist to choose a flexible scheme and the statesman to choose a fixed scheme, provided $\beta$ fulfills Eq. (31). It remains to be shown that interval (32) contains at least one feasible, i.e., positive value to be assigned to $m_0$. This follows by noting that $m_0^{\text {low}}(\mu =0)=m_{\text {fix}}$ and $\frac{dm_0^{\text {low}}}{d\mu } < 0$. Hence, we can choose the parameter $\mu$ in such a way that the lower bound of interval (32) is positive and each value contained in interval (32) is feasible.$\square$

Proof of Theorem 1

Part (i) and (ii)

Let $m_0$ be equal to its lower bound $m_0^{\text {low}}$ given in Proposition 4, Inequality (16). At this level of $m_0$, the populist is indifferent between the flexible and the fixed scheme, so we can assume that the populist chooses the flexible scheme and exerts higher effort. On the other hand, $m_0=m_0^{\text {low}}$ gives the statesman an incentive to choose the fixed scheme and implement the long-term policy for every $\beta >0$. This results from substituting $m_0=m_0^{\text {low}}$ in the lower bound for $\beta$ given in Proposition 4, as stated in Corollary 1.

Part (iii)

We want the pension system with choice to be budget-neutral with respect to the fixed pension scheme, which is the one currently implemented in practice. Hence, it must hold that

$$\begin{aligned} \widehat{m} = wm_{\text {fix}}+ (1-w) m_{\text {flex}}, \end{aligned}$$

where $w$ is the probability that the officeholder is a statesman and $\widehat{m}$ is the government budget for pensions under the current scheme. Substituting for the separating equilibrium value

$$\begin{aligned} m_{\text {flex}}^{\text {EQ}}=m_0^{\text {low}} + \mu \phi _0 e^{\text {opt}} = m_{\text {fix}}- \dfrac{2k\mu \phi _0+\delta (\mu \phi _0)^2}{4c} + \mu \phi _0\frac{k+\delta \mu \phi _0}{2c} \end{aligned}$$

yields

$$\begin{aligned} \widehat{m} = wm_{\text {fix}}+ (1-w) \left( m_{\text {fix}}- \dfrac{2k\mu \phi _0+\delta (\mu \phi _0)^2}{4c} + \mu \phi _0\frac{k+\delta \mu \phi _0}{2c}\right) . \end{aligned}$$

Solving for $m_{\text {fix}}$ yields

$$\begin{aligned} m_{\text {fix}}^{\text {EQ}}=\widehat{m}-\dfrac{(1-w)\delta (\mu \phi _0)^2}{4c}. \end{aligned}$$

(33)

As both $\frac{dm_0^{\text {low}}}{d\mu }$ and $\frac{dm_{\text {fix}}^{\text {EQ}}}{d\mu }$ are negative and $m_0^{\text {low}}(\mu =0)=m_{\text {fix}}$ and $m_{\text {fix}}^{\text {EQ}}(\mu =0)=\widehat{m},$ we deduce that for each feasible parameter combination $(k,c,\delta ,\phi _0,\phi _1,\widehat{m})$ we can find a $PSC(m_{\text {fix}},m_0,\mu )$ that fulfills the budget constraint.

Maximizing welfare means maximizing $\mu$, as the increase in effort for the populists is expressed by $\frac{\delta \mu \phi _0}{2c}$ and does not depend on $m_0$. In a separating equilibrium, a high value of $\mu$ requires a low value of $m_0$. If $m_0\ge 0$, feasible values for $\mu$ are

$$\begin{aligned} 0 < \mu \le \dfrac{-k+\sqrt{k^2+4\delta m_{\text {fix}}c}}{\delta \phi _0}. \end{aligned}$$

Hence we can choose a value of $\mu$ that is as close as possible to its upper bound, provided that the right-hand side of Eq. (33) is positive and the vote share $s \le 1$.$\square$

Proof of Proposition 7

As in the proof of Theorem 1, the budget neutrality requirement is expressed as

$$\begin{aligned} \widehat{m} = wm_{\text {fix}}+ (1-w) m_{\text {flex}}, \end{aligned}$$

where $w$ is the probability that the officeholder is a statesman and $\widehat{m}$ is the government budget for pensions under the current scheme. Substituting for the equilibrium value

$$\begin{aligned} m_{\text {flex}}^{\text {EQ}}=m_0^{\text {low}} + \mu \phi _0 e^{\text {opt}} = m_{\text {fix}}- \dfrac{2k\mu \phi _0+\delta (\mu \phi _0)^2}{4c} + \mu \phi _0\frac{k+\delta \mu \phi _0}{2c} \end{aligned}$$

(as determined by the value of $m_0=m_0^{\text {low}}$ in Theorem 1) in the budget neutrality equation yields

$$\begin{aligned} \widehat{m} = wm_{\text {fix}}+ (1-w) \left( m_{\text {fix}}- \dfrac{2k\mu \phi _0+\delta (\mu \phi _0)^2}{4c} + \mu \phi _0\frac{k+\delta \mu \phi _0}{2c}\right) . \end{aligned}$$

Solving for $m_{\text {fix}}$ yields

$$\begin{aligned} m_{\text {fix}}^{\text {EQ}}=\widehat{m}-\dfrac{(1-w)\delta (\mu \phi _0)^2}{4c}. \end{aligned}$$

Thus, it holds that $m_{\text {fix}}^{\text {EQ}}<\widehat{m}$. The effort that the statesman exerts under the current fixed scheme and under the fixed scheme within the pension system with choice is equal. Hence, it follows that for the statesman the utility is lower under the pension system with choice. As in the equilibrium values $m_{\text {fix}}^{\text {EQ}}$ and $m_{\text {flex}}^{\text {EQ}}$, the populist is indifferent between the two schemes, i.e., he achieves the same utility. The populist is also worse off under the pension system with choice than under the current pension scheme. Note that $m_{\text {flex}}^{\text {EQ}}$ is larger than $\widehat{m}$, but the resulting utility under the flexible scheme within the pension system with choice is lower than the utility under the fixed scheme. This is because $m_{\text {flex}}^{\text {EQ}}$ has to compensate for the loss of utility brought about by the cost of higher effort.$\square$

Appendix 2: reelections and pensions with choice

In this section we generalize the model described in Sect. 2 and assume that at the end of period 1 the officeholder can run for reelection. We start by characterizing the reelection probability.

1.1 The set-up

1.1.1 Reelection probability

The officeholder is reelected if his vote share is larger than, or equal to, $\frac{1}{2}$. As in the basic version of the model the vote share is modeled by $s=\phi _I e+\varepsilon$, where $\varepsilon$ is a random variable uniformly distributed with support $\left[ -\bar{\varepsilon },\bar{\varepsilon }\right]$ and mean 0. We use $r_{I}$ to denote the probability that the officeholder will be reelected (conditional on a specific level of effort), which depends on whether the incumbent chooses $I=1$ or $I=0$. We thus obtain

$$\begin{aligned} r_I&= \mathbb {P}\left[ s\ge \frac{1}{2} \left| \phantom {\frac{1}{1}}\right. e \right] = \mathbb {P}\left[ \phi _I e+\varepsilon \ge \frac{1}{2}\left| \phantom {\frac{1}{1}}\right. e \right] = \mathbb {P}\left[ \varepsilon \ge \frac{1}{2} - \phi _I e \left| \phantom {\frac{1}{1}}\right. e \right] \nonumber \\&= \int _{\frac{1}{2}-\phi _I e}^{\bar{\varepsilon }} \frac{1}{\bar{\varepsilon } - (-\bar{\varepsilon })} \ d\varepsilon = \frac{\bar{\varepsilon }}{2 \bar{\varepsilon }} - \frac{\frac{1}{2}-\phi _I e}{2 \bar{\varepsilon }} = \frac{\bar{\varepsilon } - \frac{1}{2}}{2 \bar{\varepsilon }} + \frac{\phi _I e}{2\bar{\varepsilon }} \nonumber \\&= v + a_I e, \end{aligned}$$

(34)

for $v=\frac{\bar{\varepsilon } - \frac{1}{2}}{2 \bar{\varepsilon }}$ and $a_I=\frac{\phi _I}{2 \bar{\varepsilon }}$. We focus on constellations where interior solutions can be used and formula (34) can be applied, which requires

$$\begin{aligned} \bar{\varepsilon }> \frac{1}{2}-\phi _I e > -\bar{\varepsilon }. \end{aligned}$$

(35)

This condition can be expressed in exogenous parameters and holds in particular if the ratio of $k$ to the effort cost parameter c is sufficiently small. Moreover, to simplify the analysis we set $\bar{\varepsilon }=\frac{1}{2}$, which yields $v=0$ and $a_I=\phi _I$.

Hence, under these assumptions and parameter choices, it holds that $r_I=\phi _I e$.

1.1.2 Sequence of events

We study the following sequence of events:

• At the beginning of the term, the incumbent decides on his pension scheme, his effort level $e$, and whether or not to undertake a long-term policy, i.e., he chooses $I\in \{0,1\}$.

• With probability $q$ the incumbent observes that his benefit from having another term is high and equal to $\widehat{W}_2$.¹⁷ With probability $1-q$ he observes that the benefit from being in office in the next term is negative and thus will not run for reelection.

We assume that $\widehat{W}_2$ is sufficiently high for the incumbent to always prefer to run for reelection in all circumstances we will consider. Hence, at the beginning of his term, the incumbent expects to run for reelection with probability $q$ ($0<q<1$).

1.1.3 Expected pensions

Under a pension system with choice, politicians simultaneously select their preferred pension scheme, their effort level e and whether or not to implement a long-term policy at the beginning of their term in period 1. In the model with reelection presented here politicians make these choices under the uncertainty of running for office and under the uncertainty of reelection. The pension scheme politicians choose in period 1 will be applied to them in period 2 if they do not run for office or if they lose elections. In the latter case, their pension with a flexible scheme will be based on the vote share they themselves received in the election and not on the vote share of their party after they have stepped down. This entails that—if $q=1$—the expected pension level with a flexible scheme is conditional on the vote share being less than $\frac{1}{2}$:

$$\begin{aligned} {\mathbb {E}}\left[ m_{\text {flex}}| q=1\right]&= {\mathbb {E}}\left[ m_0+ \mu s \left| \phantom {\frac{1}{1}}\right. s< \frac{1}{2}\right] \nonumber \\&= {\mathbb {E}}\left[ m_0+ \mu \left( \phi _I e + \varepsilon \right) \left| \phantom {\frac{1}{1}}\right. s< \frac{1}{2}\right] \nonumber \\&= m_0+ {\mathbb {E}}\left[ \mu \phi _I e \left| \phantom {\frac{1}{1}}\right. s< \frac{1}{2}\right] + {\mathbb {E}}\left[ \mu \varepsilon \left| \phantom {\frac{1}{1}}\right. s< \frac{1}{2}\right] \nonumber \\&= m_0+ \mu \phi _I e + \mu {\mathbb {E}}\left[ \varepsilon \left| \phantom {\frac{1}{1}}\right. \varepsilon < \frac{1}{2} - \phi _I e\right] . \end{aligned}$$

(36)

For the indicator function $\Phi$, it holds that¹⁸

$$\begin{aligned} {\mathbb {E}}\left[ \varepsilon \left| \phantom {\frac{1}{1}}\right. \varepsilon< \frac{1}{2} - \phi _I e\right]&= \frac{{\mathbb {E}}\left[ \Phi _{{ \varepsilon< \frac{1}{2} - \phi _I e}} \cdot \varepsilon \right] }{\mathbb {P}\left[ \varepsilon < \frac{1}{2} - \phi _I e\right] } \nonumber \\&=\frac{\int _{-\bar{\varepsilon }}^{\frac{1}{2} - \phi _I e} \varepsilon \cdot \frac{1}{2\bar{\varepsilon }} \ d \varepsilon }{1-\phi _I e} \nonumber \\&= \frac{\frac{1}{2\bar{\varepsilon }} \left[ \frac{1}{2}\varepsilon ^2\right] _{-\bar{\varepsilon }}^{\frac{1}{2} - \phi _I e}}{1 - \phi _I e} \nonumber \\&= \frac{\frac{1}{2\bar{\varepsilon }}\left( \frac{1}{2} \left( \frac{1}{2}-\phi _I e \right) ^2 - \frac{1}{2}\bar{\varepsilon }^2 \right) }{1-\phi _I e} \nonumber \\&= \frac{\frac{1}{4\bar{\varepsilon }}\left( \frac{1}{2}-\phi _I e \right) ^2 - \frac{1}{4}\bar{\varepsilon } }{1-\phi _I e} := A \end{aligned}$$

(37)

The probability of not being reelected, i.e., $\mathbb {P}\left[ s< \frac{1}{2}\right] =\mathbb {P}\left[ \varepsilon < \frac{1}{2} - \phi _I e\right] =1-\phi _I e$, follows from the result on reelection probability given at the beginning of the appendix. We assume that $1-\phi _I e$ is strictly larger than zero (i.e., there is always a chance of not being reelected). We note that

$$\begin{aligned} A <\; 0 \ \Leftrightarrow \ \ \frac{1}{4}\bar{\varepsilon }>\; \frac{1}{4\bar{\varepsilon }}\left( \frac{1}{2}-\phi _I e \right) ^2 \nonumber \\&\Leftrightarrow \ \ \bar{\varepsilon }^2>\; \left( \frac{1}{2}-\phi _I e \right) ^2 \nonumber \\&\Leftarrow \ \ \bar{\varepsilon } >\; \left| \phantom {\frac{1}{1}}\right. \frac{1}{2}-\phi _I e \left| \phantom {\frac{1}{1}}\right., \end{aligned}$$

(38)

which holds by definition, as set out at the beginning of the appendix. For $\bar{\varepsilon }=\frac{1}{2}$ (as chosen at the beginning of the appendix) it follows that:

$$\begin{aligned} A&= \frac{ \frac{1}{2} \left( \frac{1}{4}+\phi _I^2 e^2 - \phi _I e\right) - \frac{1}{8}}{1-\phi _I e} \nonumber \\&= \frac{ \frac{1}{8} + \frac{1}{2}\phi _I^2 e^2 - \frac{1}{2} \phi _I e - \frac{1}{8}}{1-\phi _I e} \nonumber \\&= \frac{\frac{1}{2}\phi _I^2 e^2 - \frac{1}{2} \phi _I e }{1-\phi _I e} \nonumber \\&= \frac{- \frac{1}{2}\phi _I e \left( 1-\phi _I e\right) }{1-\phi _I e} \nonumber \\&= - \frac{1}{2}\phi _I e \end{aligned}$$

(39)

Summarizing,

$$\begin{aligned} {\mathbb {E}}\left[ m_{\text {flex}}| q=1\right] =m_0+ \frac{1}{2} \mu \phi _I e < m_0+ \mu \phi _I e = {\mathbb {E}}\left[ m_{\text {flex}}| q=0\right] , \end{aligned}$$

(40)

as ${\mathbb {E}}\left[ \varepsilon | q=0\right] ={\mathbb {E}}[\varepsilon ]=0$.

1.1.4 Utilities of politicians

In the following we list the modified expected utility functions of the politicians, taking into account the possibility of reelection. To simplify the subsequent analysis, we set both the discount factor $\delta$ and the effort cost parameter c equal to 1.

$$ \begin{aligned} {\mathbb {E}}(U(P|{\text {fix}} \ \& \ I=0)) \nonumber \\ \quad=(1-q)(ke - e^2 + m_{\text {fix}})+q(ke - e^2 + \phi _0 e \widehat{W}_2+ (1- \phi _0 e)m_{\text {fix}}) \nonumber \\ \quad= ke - e^2 + m_{\text {fix}}+ q\phi _0 e( \widehat{W}_2-m_{\text {fix}}) \nonumber \\ \quad= - e^2 + (k+q\phi _0( \widehat{W}_2-m_{\text {fix}}))e +m_{\text {fix}}\end{aligned}$$

(41)

$$ \begin{aligned}{\mathbb {E}}\left( U(P|{\text {flex}}\ \& \ I=0\right)) \nonumber \\\quad= \left( 1-q\right) \left( ke - e^2 + m_0+ \mu \phi _0 e\right) \nonumber \\\qquad +\,q\left( ke - e^2 + \phi _0 e \widehat{W}_2+ \left( 1- \phi _0 e\right) {\mathbb {E}}\left[ m_{\text {flex}}| q=1\right] \right) \nonumber \\\quad= \left( 1-q\right) \left( ke - e^2 + m_0+ \mu \phi _0 e\right) \nonumber \\\qquad +\,q\left( ke - e^2 + \phi _0 e \widehat{W}_2+ \left( 1- \phi _0 e\right) \left( m_0+ \frac{1}{2} \mu \phi _0 e \right) \right) \nonumber \\\quad= ke - e^2 + m_0+ \mu \phi _0 e - q\mu \phi _0 e + q\phi _0 e\left( \widehat{W}_2-m_0\right) + \frac{1}{2} q \mu \phi _0 e - \frac{1}{2} q \mu \phi _0^2 e^2 \nonumber \\\quad= -\left( 1 + \frac{1}{2}q\mu \phi _0^2\right) e^2 + \left( k+\mu \phi _0+q\phi _0\left( \widehat{W}_2-m_0\right) - \frac{1}{2} q \mu \phi _0 \right) e +m_0\end{aligned}$$

(42)

$$\begin{aligned}&{\mathbb {E}}(U(S\vert{\text{fix}} \, {\&} \, I\,=\,1)) \nonumber \\&= (1-q)(ke - e^2 + m_{\text {fix}}+ \beta I)+q(ke - e^2 + \phi _I e (\widehat{W}_2+ \beta I) + (1- \phi _I e)(m_{\text {fix}}+ \beta I)) \nonumber \\&= ke - e^2 + m_{\text {fix}}+ \beta I+ q\phi _I e( \widehat{W}_2-m_{\text {fix}}) \nonumber \\&= - e^2 + (k+q\phi _I( \widehat{W}_2-m_{\text {fix}}))e +m_{\text {fix}}+ \beta I\end{aligned}$$

(43)

$$\begin{aligned}&{\mathbb {E}}(U(S\vert \text{flex}\, {\&} \,{I}\,=\,1)) \nonumber \\&=(1-q)(ke - e^2 + m_0+ \mu \phi _I e + \beta I) \nonumber \\&\quad\, +q(ke - e^2 + \phi _I e (\widehat{W}_2+ \beta I) + (1- \phi _I e)({\mathbb {E}}\left[ m_{\text {flex}}| q=1\right] +\beta I)) \nonumber \\&=(1-q)(ke - e^2 + m_0+ \mu \phi _I e + \beta I) \nonumber \\&\quad \,+q\left( ke - e^2 + \phi _I e (\widehat{W}_2+ \beta I) + (1- \phi _I e) \left( m_0+ \frac{1}{2} \mu \phi _I e + \beta I\right) \right) \nonumber \\&= ke - e^2 + m_0+ \mu \phi _I e - q\mu \phi _Ie + \beta I+ q\phi _I e( \widehat{W}_2-m_0)+ \frac{1}{2} q \mu \phi _I e - \frac{1}{2} q \mu \phi _I^2 e^2 \nonumber \\&= -\left( 1 + \frac{1}{2}q\mu \phi _I^2 \right) e^2 + \left( k+\mu \phi _I+q\phi _I(\widehat{W}_2-m_0) - \frac{1}{2} q \mu \phi _I \right) e +m_0+ \beta I\end{aligned}$$

(44)

Maximizing utility with respect to effort leads to:

$$\begin{aligned} \left( e^{\text {opt},P}_{\text {fix}}, {\mathbb {E}}(U^{\max }(P|{\text {fix}}\ \& \ I\,=\,0))\right)=\left( \dfrac{k+q\phi _0( \widehat{W}_2-m_{\text {fix}})}{2 }, \dfrac{(k+q\phi _0( \widehat{W}_2-m_{\text {fix}}))^2}{4 }+m_{\text {fix}}\right) \\ \left( e^{\text {opt},P}_{\text {flex}}, {\mathbb {E}}(U^{\max }(P|{\text {flex}}\ \&\ I\,=\,0))\right)= \left( \dfrac{k+\mu \phi _0 (1 - \frac{1}{2}q) + q\phi _0( \widehat{W}_2-m_0)}{2 + q\mu \phi _0^2}, \dfrac{(k+\mu \phi _0 (1 - \frac{1}{2}q)+q\phi _0( \widehat{W}_2-m_0))^2}{2 ( 2+ q\mu \phi _0^2)}+m_0\right) \\ \left( e^{\text {opt},S}_{\text {fix},I}, {\mathbb {E}}(U^{\max }(S|{\text {fix}}\ \&\ I\,=\,1)) \right)= \left( \dfrac{k+q\phi _I( \widehat{W}_2-m_{\text {fix}})}{2 }, \dfrac{(k+q\phi _I( \widehat{W}_2-m_{\text {fix}}))^2}{4}+m_{\text {fix}}+ \beta I\right) \\ \left( e^{\text {opt},S}_{\text {flex},I}, {\mathbb {E}}(U^{\max }(S|{\text {flex}}\ \&\ I \,=\,1))\right)= \left( \dfrac{k+\mu \phi _I (1 - \frac{1}{2}q) +q\phi _I( \widehat{W}_2-m_0)}{2 + q\mu \phi _I^2}, \dfrac{(k+\mu \phi _I(1 - \frac{1}{2}q)+q\phi _I( \widehat{W}_2-m_0))^2}{2 (2 + q\mu \phi _I^2 )}+m_0+ \beta I\right) \end{aligned}$$

1.2 Pension system with choice

In the model with reelection, it is no longer trivial that the populist exerts higher effort under a flexible pension scheme. The critical condition is given in the following Proposition.

Proposition 8

If the incumbent is a populist, effort is higher under a flexible scheme if and only if

$$\begin{aligned} m_0< m_{0}^{\text {critical}}:= \dfrac{2\mu - q\mu + 2 qm_{\text {fix}}- qk\mu \phi _0 - q^2 \mu \phi _0^2 (\widehat{W}_2-m_{\text {fix}})}{2q}. \end{aligned}$$

(45)

Proof of Proposition 8

The effort exerted by the populist under a flexible pension scheme is higher than the effort exerted under a fixed scheme if and only if

$$\begin{aligned} e^{\text {opt},P}_{\text {flex}}&> e^{\text {opt},P}_{\text {fix}} \\ \Leftrightarrow \dfrac{k+\mu \phi _0 - \frac{1}{2}q\mu \phi _0 + q\phi _0( \widehat{W}_2-m_0)}{2 + q\mu \phi _0^2}&> \dfrac{k+q\phi _0( \widehat{W}_2-m_{\text {fix}})}{2 } \\ \Leftrightarrow m_0&< \dfrac{2\mu - q\mu + 2 qm_{\text {fix}}- qk\mu \phi _0 - q^2 \mu \phi _0^2 (\widehat{W}_2-m_{\text {fix}})}{2q} := m_{0}^{\text {critical}}. \end{aligned}$$

$\square$

Next we look for a welfare improving PSC for which the populist is indifferent between the flexible and fixed scheme, as this generates the weakest condition on $\beta$ under which the statesman implements a long-term policy. The next proposition shows that such a PSC does not always exist.

Proposition 9

A $PSC(m_{\text {fix}},m_0,\mu )$ with the following properties:

• (i) $PSC(m_{\text {fix}},m_0,\mu )$ is feasible;

• (ii) the populist is indifferent between the flexible and fixed scheme;

• (iii) $PSC(m_{\text {fix}},m_0,\mu )$ is welfare-enhancing with respect to the fixed scheme if the incumbent is a populist;

can be constructed in a neighborhood of $q=0$ but does not always exist in a neighborhood of $q=1$.

Proof of Proposition 9

We sketch the main steps of the proof.

2.1 Step 1

W.l.o.g. we assume $0<q<1$. The populist is indifferent between the fixed and flexible schemes if and only if

$$\begin{aligned}{\mathbb {E}}(U^{\max }(P|{\text{flex}}\ \& \ I\ = \ 0)) - {\mathbb {E}}(U^{\max }(P|{\text{fix}}\ \&\ I\ = \ 0)) = 0 \\\Leftrightarrow \dfrac{(k+\mu \phi _0-\frac{1}{2}q\mu \phi _0+q\phi _0( \widehat{W}_2-m_0))^2}{2 ( 2+ q\mu \phi _0^2)}+m_0- \left( \dfrac{(k+q\phi _0( \widehat{W}_2-m_{\text {fix}}))^2}{4}+m_{\text {fix}}\right) = 0 \end{aligned}$$

Solving the above equality w.r.t. $m_0$ yields two solutions $m^{\text {low}}_{0}< m^{\text {high}}_{0}$:

$$\begin{aligned} &m^{\text {low}}_{0}= \dfrac{ q^2 \phi _0^2 (2 \widehat{W}_2- \mu ) +2 qk\phi _0 - 4 - \sqrt{2 (2+q\mu \phi _0^2)}\sqrt{(q^2 \phi _0^2 (\widehat{W}_2- m_{\text {fix}}) + qk\phi _0 - 2)^2 - 2 q\mu \phi _0^2(1 - q)}}{2 q^2 \phi _0^2}, \\ &\text {and} \\ &m^{\text {high}}_{0}= \dfrac{q^2 \phi _0^2 (2\widehat{W}_2- \mu ) +2 qk\phi _0 - 4 + \sqrt{2 (2+q\mu \phi _0^2)}\sqrt{(q^2 \phi _0^2 (\widehat{W}_2- m_{\text {fix}}) + qk\phi _0 - 2)^2 - 2 q\mu \phi _0^2(1 - q)}}{2 q^2 \phi _0^2}. \end{aligned}$$

For a given $m_{\text {fix}}$ and $\mu$, the populist only chooses the flexible scheme if $m_0$ is either lower than or equal to $m_0^{\text {low}}$ or if $m_0$ is larger than or equal to $m_0^{\text {high}}$. This property can be explained as follows: As the effort exerted by the politician decreases if $m_0$ increases, there are small values of $m_0$ that induce high effort resulting in higher utility under the flexible scheme than under a fixed scheme, as reelection chances are high. On the other hand, low effort is connected with high values of $m_0$ (when the indifference requirement holds for a fixed $m_{\text {fix}}$). This flexible scheme is attractive for the populist as the fixed part is high. In the intermediate range of values for $m_0$, the optimal effort choice of the populist does not provide sufficient benefits for the populist either in terms of higher reelection chance or higher pension benefits.

2.2 Step 2

The above values are well defined if

$$\begin{aligned} -\frac{2}{q\phi _0^2}<\mu < \frac{(q^2 \phi _0^2 (\widehat{W}_2- m_{\text {fix}}) + qk\phi _0 - 2))^2}{2 q\phi _0^2(1 - q)} := \mu ^* \end{aligned}$$

(46)

The functions defined by $m^{\text {low}}_{0}$ and $m^{\text {high}}_{0}$ are continuous for $q \in (0,1]$. It holds that

$$\begin{aligned} \lim _{q\rightarrow 0^+} m^{\text {low}}_{0}= - \infty , \end{aligned}$$

which indicates that $m^{\text {low}}_{0}$ is not a feasible choice for small q, as in such cases $m^{\text {low}}_{0}$ will be negative.

2.3 Step 3

Properties (ii) and (iii) hold together if and only if

$$\begin{aligned} &m_{0}^{\text {critical}}- m^{\text {high}}_{0}< 0 \\ \Leftrightarrow \ \dfrac{2 q\phi _0^2 \mu - q^2 \phi _0^3 k\mu - 2 q\phi _0 k+ 4 - q^3 \phi _0^4 \mu ( \widehat{W}_2- m_{\text {fix}}) - 2 q^2 \phi _0^2 (\widehat{W}_2-m_{\text {fix}})}{2 q^2 \phi _0^2} \\&\quad- \dfrac{\sqrt{2 (2+q\mu \phi _0^2)}\sqrt{(q^2 \phi _0^2 (\widehat{W}_2- m_{\text {fix}}) qk\phi _0 - 2)^2 - 2 q\mu \phi _0^2(1 - q)}}{2 q^2 \phi _0^2} <0, \end{aligned}$$

where $m_{0}^{\text {critical}}$ is as defined in Proposition 8. We study the function

$$\begin{aligned} f(\mu ):=m_{0}^{\text {critical}}(\mu )- m^{\text {high}}_{0}(\mu ) \end{aligned}$$

w.r.t. $\mu$. The function f is continuous in $\mu$ if (46) holds. Consider the interval $C:=\left( 0, \mu ^* \right)$ for some $\mu ^*$. As the function f is continuous for $-\dfrac{ 2}{q\phi _0^2}< \mu < \mu ^*$, for a given parameter combination $\left( k, \phi _0, q, \widehat{W}_2, m_{\text {fix}}\right)$ f will be either positive or negative on C.

2.4 Step 4

We examine the extreme cases $q=0$ and $q=1$. As f is continuous in $q \in (0,1]$, one can show that in a neighborhood of $q=0$ we can find parameters that fulfill $m_{0}^{\text {critical}}- m^{\text {high}}_{0}> 0$.

We now turn to the case $q=1$. Consider the derivative of f with respect to $\mu$ evaluated in $\mu =0$. If for a given parameterization of the problem this value is positive, then f will be positive on C. This would mean that $m_{0}^{\text {critical}}- m^{\text {high}}_{0}> 0$ can be satisfied for a feasible value of $\mu$. It holds that

$$\begin{aligned} \frac{d f}{d \mu }\left| \phantom {\frac{1}{1}}\right. _{\mu =0, q=1}=\frac{3}{2}-\frac{3}{4}k\phi _0 - \frac{3}{4} \phi _0^2 \left( \widehat{W}_2- m_{\text {fix}}\right) > 0 \end{aligned}$$

if

$$\begin{aligned} \widehat{W}_2- m_{\text {fix}}< \frac{2 - k\phi _0}{ \phi _0^2}. \end{aligned}$$

(47)

Only in this case is it possible to fulfill $m_{0}^{\text {critical}}- m^{\text {high}}_{0}> 0$, otherwise not. If

$$\begin{aligned} \frac{2 - k\phi _0}{ \phi _0^2} \le 0, \end{aligned}$$

then requirement (47) contradicts the assumption $\widehat{W}_2>m_{\text {fix}}$.$\square$

The proof of Proposition 9 reveals that if the reelection mechanism is taken into account it is not always possible to design a welfare-increasing pension system with choice where the populist is indifferent between the schemes. Imposing the indifference requirement entails more than technical simplification. A PSC satisfying this condition enables statesmen with relatively low $\beta$ to implement long-term policies, while ensuring that populists increase effort by choosing the flexible scheme. Hence the indifference condition offers the best opportunity for the PSC to increase welfare.

2.5 Extended system with choice

Proposition 9 gives a formal account of the complication with the pension system with choice. In Sect. 8 we introduced the extended pension system with choice. We proceed here by formalizing the observations listed there. The sequence of events is as described at the begining of this appendix.

Theorem 2

If

$$\begin{aligned} \beta > \beta ^{\text {crit2}} = \dfrac{q^2 }{4}(\phi ^2_0-\phi ^2_1)(\widehat{W}_2-m_{\text {fix}})^2+ \dfrac{q k}{2}(\phi _0- \phi _1) (\widehat{W}_2-m_{\text {fix}}), \end{aligned}$$

then there exists a $PSC^{ext}(m_{\text {fix}},m_0,\mu )$ for every feasible problem parameterization $\left( k, c=1, \delta =1, \phi _0, \phi _1, q, \widehat{W}_2\right)$ such that

• (i) $S$ chooses the fixed scheme, $I=1$ and $e=\dfrac{k+q\phi _1( \widehat{W}_2-m_{\text {fix}})}{2 }:=e^{\text {opt},\text {ext}}_{\text {fix}};$

• (ii) $P$ chooses the flexible scheme, $I=0$ and

  $$\begin{aligned} e=\dfrac{k+\mu \phi _0 (1-q)+q\phi _0( \widehat{W}_2-m_{\text {fix}})}{2}:=e^{\text {opt},\text {ext}}_{\text {flex}}; \end{aligned}$$

• (iii) effort exerted under a flexible scheme is higher than under a fixed scheme for all $0 \le q <1$;

• (iv) expected expenditures under the extended pension system with choice and under the current standard fixed pension system are equal.

Proof of Theorem 2

Parts (i), (ii), and (iii)

W.l.o.g. we assume $q\ne 0$. The effort levels exerted by P and S solve the maximization problems of the respective utility functions w.r.t. e given the pension schemes within the extended pension system with choice. The expected utilities for the populist are given as

$$ \begin{aligned}&{\mathbb {E}}(U(P|{\text {fix}}^{\text {ext}} \ \& \ I=0)) \\&\quad= (1-q)(ke - e^2 + m_{\text {fix}})+q(ke - e^2 + \phi _0 e \widehat{W}_2+ (1- \phi _0 e)m_{\text {fix}}), \\&\text {and} \\&{\mathbb {E}}(U(P|{\text {flex}}^{\text {ext}} \ \& \ I=0)) \\&\quad=(1-q)(ke - e^2 + m_0+ \mu \phi _0 e)+q(ke - e^2 + \phi _0 e \widehat{W}_2+ (1- \phi _0 e)m_{\text {fix}}). \end{aligned}$$

The expected utilities for the statesman are given analogously as

$$\begin{aligned}&{\mathbb {E}}(U(S\vert{\text{fix}^{\text{ext}}} \, \& \, I=1))= (1-q)(ke - e^2 + m_{\text {fix}}+ \beta I)+q(ke - e^2 + \phi _I e (\widehat{W}_2+ \beta I) + (1- \phi _I e)(m_{\text {fix}}+ \beta I))\\& \text {and} \\&{\mathbb {E}}(U(S\vert{\text{flex}^{\text{ext}}} \, \& \, I=1)) \\&\quad= (1-q)(ke - e^2 + m_0+ \mu \phi _I e + \beta I)+q(ke - e^2 + \phi _I e (\widehat{W}_2+ \beta I) + (1- \phi _I e)(m_{\text {fix}}+ \beta I)) \end{aligned}$$

Note that the populist always exerts higher effort under the flexible scheme than under the fixed scheme. Effort levels are equal between the schemes only when $q=1$, i.e., when the officeholder will stand for reelection with certainty.

The populist chooses the flexible scheme only if the resulting expected utility is higher than with the fixed scheme. This holds when

$$\begin{aligned} m_0> m_{\text {fix}}-\frac{2q\mu \phi _0^2(\widehat{W}_2-m_{\text {fix}}) + (1-q)\mu ^2\phi _0^2+2k\mu \phi _0}{4} , \end{aligned}$$

(48)

which follows from comparing the expected utilities in both cases. Analogously, the statesman chooses the fixed scheme if

$$\begin{aligned} m_0< m_{\text {fix}}-\frac{2q\mu \phi _1^2(\widehat{W}_2-m_{\text {fix}}) + (1-q)\mu ^2\phi _1^2+2k\mu \phi _1}{4}, \end{aligned}$$

(49)

where we have assumed that $\beta$ is so large that he chooses $I=1$. As $\phi _0>\phi _1$ there exists a non-empty interval of $m_0$ values such that the two types of officeholders choose different schemes, provided $\beta$ is sufficiently high. By setting $m_0$ equal to its lower bound in Inequality (48), we make the populist indifferent between the two pension schemes. In this setting, the lower bound on $\beta$ ensuring that the statesman implements a long-term policy has to satisfy

$$\begin{aligned}{\mathbb {E}}(U^{\max }(S|{\text {flex}}^{\text {ext}}\& \ {I\,=\,0})) = {\mathbb {E}}(U^{\max }(P|{\text {flex}}^{\text {ext}}\& \ {I\,=\,0})) = {\mathbb {E}}(U^{\max }(P|{\text {fix}}^{\text {ext}}\& \ {I\,=\,0})) < {\mathbb {E}}(U^{\max }(S|{\text {fix}}^{\text {ext}}\& \ {I\,=\,1})) \\\quad\Leftrightarrow \dfrac{(k+q\phi _0( \widehat{W}_2-m_{\text {fix}}))^2}{4 }+m_{\text {fix}} < \dfrac{(k+q\phi _1( \widehat{W}_2-m_{\text {fix}}))^2}{4 }+m_{\text {fix}}+ \beta \\& \quad\Leftrightarrow \beta > \dfrac{q^2 }{4}(\phi ^2_0-\phi ^2_1)(\widehat{W}_2-m_{\text {fix}})^2+ \dfrac{q k}{2}(\phi _0- \phi _1) (\widehat{W}_2-m_{\text {fix}}) := \beta ^{\text {crit2}}. \end{aligned}$$

Part (iv)

Budget neutrality can be shown in the same way as in Theorem 1.$\square$

We note that if a reelection mechanism is taken into account, it is no longer possible in this setting to motivate every statesman to implement a long-term policy, but only those that have a sufficiently high value of $\beta$ or a sufficiently low value of q, meaning that they do not wish to stand for reelection. This occurs because choice $I=1$ impairs their reelection chances and this loss can only be compensated by $\beta$. Once again, the indifference requirement for the populist ensures that condition $\beta \geq \beta ^{\text {crit2}}$ is the weakest possible condition. It arises under a pure fixed scheme (current standard scheme) as well. Hence the extended pension system with choice does not deter any more statesmen from choosing $I=1$ than the pure fixed scheme and gives populists an incentive for higher effort.

The characterization in Theorem 2 and the budget requirements enable us to make welfare comparisons.

Corollary 3

The extended pension system with choice is welfare-enhancing

• with respect to the fixed pension scheme, as populists work harder in their last term,

• with respect to the flexible pension scheme , as all statesmen implement a long-term policy if $q=0$,

• with respect to the pension system with choice, as the system can be applied to all problem parameterizations.

Restricted to last-term situations, the extended pension system with choice is equivalent to the pension system with choice, which is welfare-increasing by Corollary 2. If $q=1$, the impact of the extended system with choice is equivalent to that of a fixed scheme. The effort exerted in this case is

$$\begin{aligned} e = \dfrac{k+ \phi _I( \widehat{W}_2-m_{\text {fix}})}{2 }, \end{aligned}$$

(50)

which is larger than the effort

$$\begin{aligned} e = \dfrac{k+ \mu \phi _I }{2 } \end{aligned}$$

exerted in a last term under the flexible scheme within the pension system with choice if and only if $\widehat{W}_2-m_{\text {fix}}> \mu$.

Note that if the incumbent is rejected in the elections, his pension level is equal to $m_{\text {fix}}$. Even in the case of $q=1$, an extended pension system with choice creates higher effort incentives, as the fixed pension level under the system with choice is lower than the pension amount in the current fixed scheme because of budget neutrality as in Proposition 7. We note that for $q=1$ the incumbent is indifferent between the fixed and flexible scheme, as he will never be subject to the flexible scheme.

Appendix 3: List of symbols

e :

politician’s level of effort

$\bar{e}$ :

maximum level of effort

$\widehat{e}$ :

level of effort under the fixed pension system

b :

utility of a representative voter

$k$ :

constant coefficient in the per-capita benefit equation $b=ke$

$c$ :

constant coefficient defining the cost of exerting effort

m :

pension level

$I$ :

indicator variable, $I=1$ stands for the implementation of the long-term policy

$\beta$ :

future benefit for the statesman if he implements the long-term policy

fix:

fixed scheme under the pension system with choice

flex:

flexible scheme under the pension system with choice

$\widehat{m}$ :

pension amount under current scheme

$m_{\text {fix}}$ :

pension level under the fixed pension scheme

$m_{\text {flex}}$ :

pension level under the flexible pension scheme

W :

welfare function

$\alpha$ :

weight of the level of effort in the welfare function

$m_0$ :

fixed pension payment under the flexible pension scheme

s :

vote share: $s=\phi _I e + \varepsilon$

$\mu$ :

coefficient determining the level of flexible payment within the flexible scheme

$\phi _I$ :

coefficient in the vote share depending on $I$, it holds $\phi _0>\phi _1$

$\varepsilon$ :

random factor in the vote share

$\bar{\varepsilon }$ :

upper boundary of the support interval for the random variable $\varepsilon$

$m_0^{\text {low}}$ :

value of $m_0$ for which P is indifferent between fix and flex

$m_0^{\text {high}}$ :

value of $m_0$ for which S is indifferent between fix and flex

g :

coefficient giving the benefit deriving from future career opportunities

w :

probability that the incumbent is a statesman

$q$ :

probability that the politician wishes to stand for reelection

$\widehat{W}_2$ :

benefit for the politician of holding office in period 2

$r_I$ :

probability of reelection in period 2 in dependence of $I$

T :

type of officeholder (S or P)

$\Phi$ :

indicator function