Political cronyism

Abstract

This paper analyzes incentives for cronyism in politics within a political agency model with moral hazard. The analysis focuses on the institutional features, which define contractual and appointment procedures within political organizations. The institutional framework does not allow explicit contracting with politicians. They are motivated by reelection incentives and just need to guarantee that their team performance exceeds the minimum threshold required for reelection. This lowers the returns to bringing in efficient individuals in the politician’s team. Moreover, the nature of political promotions (such that a crony’s career is tied to that of his patron) leads to the alignment between political objectives of the politician and his cronies. This further increases the politician’s incentives to appoint less efficient friends.

Appendices

Appendices

Throughout the Appendix, $F$ will be used to denote the normal distribution function, and $f$ for the corresponding density.

Appendix 1: Subordinates’ maximization problem

The first-order conditions (FOCs) with respect to subordinate $i$’s support $s_{i}\ge 0$ for the politician are

$$\begin{aligned} \begin{array}{lcc} \frac{1}{\sqrt{s_{i}+\sum _{j\ne i}s_{j}}}-s_{i}=0 &{} \quad \text {if} &{} i\in C, \\ -s_{i}=0 &{}\quad \text {if} &{} i\in E. \end{array} \end{aligned}$$

The second-order conditions (SOCs) with respect to $s_{i}$ hold. The FOCs yield equilibrium levels of support $s_{i}^{p}$, which depend on a number of cronies $m$ appointed by the politician:

$$\begin{aligned} s_{i}^{p}=\left\{ \begin{array}{lcc} m^{-\frac{1}{3}} &{} \quad \text {if} &{} i\in C, \\ 0 &{} \quad \text {if} &{} i\in E. \end{array} \right. \end{aligned}$$

(6.1)

The FOCs with respect to subordinate $i$’s effort $a_{i}\in \left[ 0, \overline{a}\right]$, taking reelection threshold $p\in \mathbb {R}$ as given, are

$$\begin{aligned} \begin{array}{lcc} f_{\widehat{\varepsilon }}\left( p-a_{i}-\sum _{j\ne i}a_{j}\right) -a_{i}=0 &{} \quad \text {if} &{} i\in C, \\ f_{\widehat{\varepsilon }}\left( p-a_{i}-\sum _{j\ne i}a_{j}\right) -\frac{ a_{i}}{\gamma }=0 &{} \quad \text {if} &{} i\in E. \end{array} \end{aligned}$$

(6.2)

The second derivatives with respect to $a_{i}$ are

$$\begin{aligned} \begin{array}{lcc} -f_{\widehat{\varepsilon }}^{\prime }\left( p-a_{i}-\sum _{j\ne i}a_{j}\right) -1 &{} \text {if} &{} i\in C, \\ -f_{\widehat{\varepsilon }}^{\prime }\left( p-a_{i}-\sum _{j\ne i}a_{j}\right) -\frac{1}{\gamma } &{} \text {if} &{} i\in E. \end{array} \end{aligned}$$

(6.3)

If (6.3) evaluated at critical points characterized by the FOCs (6.2) are strictly negative then the SOCs of the subordinates’ maximization problem are satisfied. In this case, the assumption that the upper bound of the set of efforts, $\overline{a}$, is large ensures that the best response functions are characterized by the FOCs (6.2). The subordinates’ best response functions are continuous functions from a nonempty, compact, convex set, $\left[ 0,\overline{a}\right]$, into itself. Thus, the standard existence results are applied to establish the existence of equilibrium efforts $a_{i}^{p}$, $i=1,...,n$, characterized by the FOCs ( 6.2).

However, if (6.3) evaluated at critical points characterized by the FOCs (6.2) are non-negative then corner solutions are to be considered as the FOCs (6.2) do not yield maximum. If a subordinate exerts maximal level of effort $\overline{a}$ then his net payoff from holding office becomes

$$\begin{aligned} 1-F_{\widehat{\varepsilon }}\left( p-\overline{a}-\sum _{j\ne i}a_{j}\right) -\frac{\overline{a}^{2}}{2}. \end{aligned}$$

If he makes no effort then his net payoff is

$$\begin{aligned} 1-F_{\widehat{\varepsilon }}\left( p-\sum _{j\ne i}a_{j}\right) . \end{aligned}$$

The assumption that the upper bound of the set of efforts, $\overline{a}$, is large guarantees that the latter is greater than the former. Thus, $a_{i}^{p}=0$ if the corresponding SOC does not hold.

Appendix 2: Politician’s maximization problem

The FOCs (6.2) and the second derivatives (6.3) of the subordinates’ maximization problem imply that both in interior optima and in corner solutions, all cronies exert the same level of effort, denoted by $a_{C}^{p}$, and all experts exert the same level of effort, denoted by $a_{E}^{p}$. Then substituting $s_{i}^{p}$ defined in (6.1) into the politician’s objective function $\Pi \left( m\right)$ yields

$$\begin{aligned} \Pi \left( m\right) =1-F_{\widehat{\varepsilon }}\left( p-ma_{C}^{p}-\left( n-m\right) a_{E}^{p}\right) +2m^{\frac{1}{3}}. \end{aligned}$$

The FOC with respect to $m$ is

$$\begin{aligned} -f_{\widehat{\varepsilon }}\left( p-ma_{C}^{p}-\left( n-m\right) a_{E}^{p}\right) \cdot \left( a_{E}^{p}-a_{C}^{p}\right) +\frac{2}{3}m^{- \frac{2}{3}}=0. \end{aligned}$$

(6.4)

The second derivative with respect to $m$ is

$$\begin{aligned} -f_{\widehat{\varepsilon }}^{\prime }\left( p-ma_{C}^{p}-\left( n-m\right) a_{E}^{p}\right) \cdot \left( a_{E}^{p}-a_{C}^{p}\right) ^{2}-\frac{4}{9}m^{- \frac{5}{3}}. \end{aligned}$$

(6.5)

If (6.5) evaluated at a critical point characterized by the FOC (6.4) is strictly negative then the SOC of the politician’s maximization problem is satisfied and therefore the FOC (6.4) yields a number of cronies which maximizes the politician’s objective function $\Pi \left( \cdot \right)$ for a given reelection threshold $p$. Then an equilibrium number of cronies $m^{p}$ is given by

$$\begin{aligned} m^{p}=\left\{ \begin{array}{ccc} \left\lfloor m\right\rfloor &{} \quad \text {if} &{} \Pi \left( \left\lfloor m\right\rfloor \right) \ge \Pi \left( \left\lceil m\right\rceil \right) , \\ \left\lceil m\right\rceil &{} \quad \text {if} &{} \Pi \left( \left\lfloor m\right\rfloor \right) <\Pi \left( \left\lceil m\right\rceil \right) , \end{array} \right. \end{aligned}$$

(6.6)

where $m$ is characterized by the FOC (6.4), $\left\lfloor \cdot \right\rfloor$ denotes the floor function and $\left\lceil \cdot \right\rceil$ the ceiling function.

However, if (6.5) evaluated at a critical point characterized by the FOC (6.4) is non-negative then corner solutions are to be considered as the FOC (6.4) does not yield maximum. If the politician delegates tasks only to cronies then his utility is

$$\begin{aligned} \Pi \left( n\right) =1-F_{\widehat{\varepsilon }}\left( p-na_{C}^{p}\right) +2n^{\frac{1}{3}}. \end{aligned}$$

If he appoints only experts then his utility becomes

$$\begin{aligned} \Pi \left( 0\right) =1-F_{\widehat{\varepsilon }}\left( p-na_{E}^{p}\right) . \end{aligned}$$

Note that $\Pi \left( n\right) >\Pi \left( 0\right)$ for $n\ge 1$. Thus, $m^{p}=n$ if the SOC does not hold.

Appendix 3: Proof of proposition 1

The voter chooses $p$ to maximize

$$\begin{aligned} m^{p}a_{C}^{p}+\left( n-m^{p}\right) a_{E}^{p}=na_{E}^{p}-m^{p}\left( a_{E}^{p}-a_{C}^{p}\right) , \end{aligned}$$

(6.7)

where $a_{C}^{p}$ and $a_{E}^{p}$ are either characterized by the FOCs (6.2) or equal to $0$ (either both or one of them). $m^{p}$ is either given by (6.6) or equal to $n$.

First, I consider the case in which both $a_{C}^{p}$ and $a_{E}^{p}$ are characterized by the FOCs (6.2) and $m^{p}$ is given by (6.6). The FOCs (6.2) yield

$$\begin{aligned} \begin{array}{l} a_{C}^{p}=f_{\widehat{\varepsilon }}\left( p-m^{p}a_{C}^{p}-\left( n-m^{p}\right) a_{E}^{p}\right) , \\ a_{E}^{p}=\gamma f_{\widehat{\varepsilon }}\left( p-m^{p}a_{C}^{p}-\left( n-m^{p}\right) a_{E}^{p}\right) , \end{array} \end{aligned}$$

(6.8)

while the FOC (6.4) yields

$$\begin{aligned} m=\left( \frac{3}{2}f_{\widehat{\varepsilon }}\left( p-m^{p}a_{C}^{p}-\left( n-m^{p}\right) a_{E}^{p}\right) \left( a_{E}^{p}-a_{C}^{p}\right) \right) ^{- \frac{3}{2}}. \end{aligned}$$

(6.9)

Substituting (6.8) and (6.9) into (6.7) and rearranging yields

$$\begin{aligned}&m^{p}a_{C}^{p}+\left( n-m^{p}\right) a_{E}^{p}\\&= \left\{ \begin{array}{lll} \overline{f}_{\widehat{\varepsilon }}\left( \left\lfloor \left( \frac{3}{2} \left( \gamma -1\right) \overline{f}_{\widehat{\varepsilon }}^{2}\right) ^{- \frac{3}{2}}\right\rfloor \left( 1-\gamma \right) +n\gamma \right) &{} \text {if } &{} \Pi \left( \left\lfloor \left( \frac{3}{2}\left( \gamma -1\right) \overline{f}_{\widehat{\varepsilon }}^{2}\right) ^{-\frac{3}{2} }\right\rfloor \right) \ge \Pi \left( \left\lceil \left( \frac{3}{2}\left( \gamma -1\right) \overline{f}_{\widehat{\varepsilon }}^{2}\right) ^{-\frac{3 }{2}}\right\rceil \right) , \\ \overline{f}_{\widehat{\varepsilon }}\left( \left\lceil \left( \frac{3}{2} \left( \gamma -1\right) \overline{f}_{\widehat{\varepsilon }}^{2}\right) ^{- \frac{3}{2}}\right\rceil \left( 1-\gamma \right) +n\gamma \right) &{} \text {if} &{} \Pi \left( \left\lfloor \left( \frac{3}{2}\left( \gamma -1\right) \overline{f}_{\widehat{\varepsilon }}^{2}\right) ^{-\frac{3}{2} }\right\rfloor \right) <\Pi \left( \left\lceil \left( \frac{3}{2}\left( \gamma -1\right) \overline{f}_{\widehat{\varepsilon }}^{2}\right) ^{-\frac{3 }{2}}\right\rceil \right) , \end{array} \right. \end{aligned}$$

which increases with $\overline{f}_{\widehat{\varepsilon }}\equiv f_{ \widehat{\varepsilon }}\left( p-m^{p}a_{C}^{p}-\left( n-m^{p}\right) a_{E}^{p}\right)$. The voter therefore chooses $p$ to maximize the density function $f_{\widehat{\varepsilon }}\left( p-m^{p}a_{C}^{p}-\left( n-m^{p}\right) a_{E}^{p}\right)$, which takes the maximum value of $f_{ \widehat{\varepsilon }}\left( 0\right) =\frac{1}{\sqrt{2\pi n}\sigma }$ at $p=m^{p}a_{C}^{p}+\left( n-m^{p}\right) a_{E}^{p}$. Substituting $p=m^{p}a_{C}^{p}+\left( n-m^{p}\right) a_{E}^{p}$ into (6.8) and (6.9) yields the equilibrium levels of subordinates’ efforts $a_{C}^{*}$ and $a_{E}^{*}$

$$\begin{aligned} \begin{array}{l} a_{C}^{*}=\frac{1}{\sqrt{2\pi n}\sigma }, \\ a_{E}^{*}=\frac{\gamma }{\sqrt{2\pi n}\sigma }, \end{array} \end{aligned}$$

and the number of cronies $m$ which maximizes the politician’s objective function

$$\begin{aligned} m=\left( \tfrac{4\pi n\sigma ^{2}}{3\left( \gamma -1\right) }\right) ^{\frac{ 3}{2}}. \end{aligned}$$

Since $\Pi \left( \left\lfloor m\right\rfloor \right) <\Pi \left( \left\lceil m\right\rceil \right)$ then the equilibrium number of cronies $m^{*}$ (bounded by $n$) is equal to

$$\begin{aligned} m^{*}=\min \left[ \left\lceil \left( \tfrac{4\pi n\sigma ^{2}}{3\left( \gamma -1\right) }\right) ^{\frac{3}{2}}\right\rceil ,n\right] . \end{aligned}$$

The equilibrium reelection threshold (as well as the expected performance of the politician’s team) is given by

$$\begin{aligned} p^{*}=\max \left[ \tfrac{1}{\sqrt{2\pi n}\sigma }\left( \gamma n-\left( \gamma -1\right) \left\lceil \left( \tfrac{4\pi n\sigma ^{2}}{3\left( \gamma -1\right) }\right) ^{\frac{3}{2}}\right\rceil \right) ,\tfrac{\sqrt{n}}{ \sqrt{2\pi }\sigma }\right] . \end{aligned}$$

Substituting the equilibrium efforts, number of cronies and reelection threshold into the second derivatives (6.3) and (6.5) yields

$$\begin{aligned} -f_{\widehat{\varepsilon }}^{\prime }\left( 0\right) -1&= -1, \\ -f_{\widehat{\varepsilon }}^{\prime }\left( 0\right) -\frac{1}{\gamma }&= - \frac{1}{\gamma }, \\ -f_{\widehat{\varepsilon }}^{\prime }\left( 0\right) \cdot \left( \tfrac{ \gamma -1}{\sqrt{2\pi n}\sigma }\right) ^{2}-\frac{4}{9}\left( m^{*}\right) ^{-\frac{5}{3}}&= -\frac{4}{9}\left( m^{*}\right) ^{-\frac{5}{3 }}, \end{aligned}$$

which are strictly negative. It follows that the SOCs of the subordinates’ and politician’s maximization problems are satisfied.

I turn now to the case in which both $a_{C}^{p}$ and $a_{E}^{p}$ are characterized by the FOCs (6.2) and $m^{p}=n$. The voter chooses $p$ to maximize

$$\begin{aligned} na_{C}^{p}=nf_{\widehat{\varepsilon }}\left( p-na_{C}^{p}\right) , \end{aligned}$$

which increases with $f_{\widehat{\varepsilon }}\left( p-na_{C}^{p}\right)$ . The voter thus chooses $p$ to maximize the density function $f_{\widehat{ \varepsilon }}\left( p-na_{C}^{p}\right)$. To ensure that the corner solution $m^{p}=n$ is an equilibrium of the politician’s maximization problem, the second derivative of the politician’s maximization problem (6.5) evaluated at a critical point characterized by FOC (6.4) has to be non-negative. This implies that $-f_{\widehat{\varepsilon } }^{\prime }\left( p-ma_{C}^{p}-\left( n-m\right) a_{E}^{p}\right)$ with $m$ characterized by FOC (6.4) has to be strictly positive implying $p-ma_{C}^{p}-\left( n-m\right) a_{E}^{p}>0$. After tedious algebra, one might show that $\max _{p>ma_{C}^{p}+\left( n-m\right) a_{E}^{p}}\left[ f_{ \widehat{\varepsilon }}\left( p-na_{C}^{p}\right) \right] <f_{\widehat{ \varepsilon }}\left( 0\right) =\frac{1}{\sqrt{2\pi n}\sigma }$. It follows therefore that the expected performance of the political organization in this case is strictly lower than in the case in which $m^{p}$ is characterized by the FOC (6.4). Thus, this is not an equilibrium.

Finally, I consider the case in which either $a_{C}^{p}=0$, or $a_{E}^{p}=0$ , or both $a_{C}^{p}=a_{E}^{p}=0$. To guarantee that this corner solution is an equilibrium of the subordinates’ maximization problem, the corresponding second derivative evaluated at the critical point has to be non-negative. Therefore, the voter chooses $p$ from the set in which the corresponding second derivative is non-negative. But in this case, the expected performance of the political organization is strictly lower than in the case in which the subordinates’ efforts are characterized by the FOCs (6.2 ). Thus, this is not an equilibrium.

Then the unique equilibrium of the voter’s maximization problem is given by

$$\begin{aligned} p^{*}=\max \left[ \tfrac{1}{\sqrt{2\pi n}\sigma }\left( \gamma n-\left( \gamma -1\right) \left\lceil \left( \tfrac{4\pi n\sigma ^{2}}{3\left( \gamma -1\right) }\right) ^{\frac{3}{2}}\right\rceil \right) ,\tfrac{\sqrt{n}}{ \sqrt{2\pi }\sigma }\right] . \end{aligned}$$