Incumbent Policy Choice

At the beginning of the game, Nature draws a state of the world $ \omega \in \{0,1\} $ , where $ Pr[\omega =1]=\frac{1}{2} $ .Footnote ⁸ When $ \omega =0, $ a reform policy is never beneficial, but when $ \omega =1, $ implementing a reform may provide societal benefits. The incumbent then receives a signal about $ \omega $ , denoted by $ s\in \{0,1\} $ . The distribution of the signal depends on the Incumbent’s type: if the Incumbent is competent, $ {\theta}_I=H $ , then she learns the state perfectly, $ s=\omega $ , but if the Incumbent is incompetent, $ {\theta}_I=L $ , then she receives an uninformative signal (i.e., $ Pr[s=1]=Pr[s=0]=\frac{1}{2} $ ). Since the Incumbent does not know her own type she treats this as a noisy, but informative signal in either case. Following the Incumbent’s observation of $ s, $ she chooses a policy $ x\in \{0,1\} $ , where $ x=1 $ is the reform policy and $ x=0 $ can be interpreted as no reform or retention of the status quo.

Citizen Coproduction and Policy Outcomes

Following the Incumbent’s policy choice each citizen simultaneously chooses whether or not to contribute to policy coproduction. We denote citizen i’s decision with $ {a}_i\in \{0,1\} $ , where $ {a}_i=1 $ means citizen i contributes to coproduction and $ {a}_i=0 $ means citizen i does not contribute.

The policy outcome depends on both the Incumbent’s policy choice and aggregate citizen coproduction. It can be a policy success, $ y=1 $ , or a policy failure, $ y=0 $ . The distribution of this policy outcome is given by

⁽¹⁾ $$ \begin{array}{rl}Pr[y=1\mid x,\omega ]=\left\{\begin{array}{ll}0,\hskip1em \qquad & \mathrm{if}\hskip0.3em \omega =0\hskip0.3em \mathrm{or}\hskip0.3em x=0,\qquad \\ {}\overline{a},\hskip1em \qquad & \mathrm{if}\hskip0.3em x=\omega =1,\qquad \end{array}\right.& \end{array} $$

where $ \overline{a}={\int}_0^1{a}_idi $ measures the proportion of coproducers in the population. If the Incumbent does not pursue reform, $ x=0 $ , then there is no chance for a policy success. If instead the Incumbent does pursue reform, $ x=1 $ , then the likelihood of success depends on how many citizens contributed to coproduction (i.e., $ \overline{a} $ ).

Elections

After observing x, $ \overline{a} $ , and y, the citizens simultaneously vote to retain the Incumbent or replace her with the Challenger. The politician who wins the most votes wins the election, which we denote with $ W\in \{I,C\} $ . Citizens want to elect a competent politician.

Payoffs

The payoffs for each citizen i are given by

(2) $$ \begin{array}{rl}{u}_i(y,{a}_i)=y(1+{\gamma}_i{a}_i)-c{a}_i-kx+{1}_H\left({\theta}_W\right).& \end{array} $$

The term $ y(1+{\gamma}_i{a}_i) $ captures two benefits of a successful reform: a common benefit of $ 1 $ to all citizens and a selective additional benefit equal to $ {\gamma}_i $ only to those citizens who contributed to coproduction. This is the sense in which we interpret $ {\gamma}_i $ as player i’s private coproduction or cooperation incentive, since it represents the value that they place on helping to achieve a successful outcome. The parameter $ c\in (0,1) $ represents the cost of contributing to policy coproduction, which we assume to be constant across citizens.Footnote ⁹ Finally, $ k\in (0,1) $ represents the common cost of the reform, which is bounded between zero and one to represent the idea that the safe policy is inferior to a successful reform but superior to an unsuccessful reform. The last term is an indicator capturing the idea that citizens want to elect the more competent candidate.Footnote ¹⁰

The payoff to the Incumbent is given by

(3) $$ \begin{array}{rl}{u}_I(y,x)={1}_{\{W=I\}}\beta +y-kx,& \end{array} $$

where $ {1}_{\{W=I\}} $ is an indicator function that takes the value of one if and only if the Incumbent wins reelection, and $ \beta \ge 0 $ is a benefit to winning the election. Outside of office benefits, the Incumbent’s payoff captures the fact that she prefers to implement a successful reform over choosing the default policy and prefers the default policy over an unsuccessful reform.Footnote ¹¹

Equilibrium

We analyze perfect Bayesian equilibria in which citizens play symmetric strategies. Strategies for each citizen consist of the following:

• • A coproduction strategy, which takes the form of a cutoff $ \overset{\sim }{\gamma } $ such that citizen i chooses $ {a}_i=1 $ and contributes to coproduction, given policy $ x=1 $ , if $ {\gamma}_i\ge \overset{\sim }{\gamma }. $

• • A voting strategy $ {\tilde{v}}_i(x,\overline{a},y) $ , which gives each citizen i’s vote choice for each combination of x and y given aggregate coproduction level $ \overline{a} $ .

A strategy for the Incumbent consists of a policy strategy $ {\sigma}_x(s) $ which gives a probability of choosing $ x=1 $ for each signal.

Players’ beliefs consist of a joint distribution over $ {\theta}_I $ and $ \omega . $ The Incumbent’s belief after receiving a signal s is denoted by $ {\mu}^x(s). $ The citizens’ beliefs after observing x are denoted by $ {\mu}_i^a(x) $ for each citizen i. Finally, the citizens’ beliefs after observing the policy outcome are $ {\mu}_i^v(x,y,\overline{a}). $ The superscripts on the beliefs remind us of the decision that must be made when the player holds this belief. For a given decision, a player may only care about $ \omega $ or only care about $ {\theta}_I $ , so it is convenient to keep track of marginal beliefs. Given some beliefs $ \mu $ at an arbitrary history h and player j (with I representing the incumbent and i an arbitrary citizen), the marginal belief that $ \omega =1 $ is $ {\widehat{\omega}}_j(h)={\sum}_{\theta^{\prime}\in \{L,H\}}\mu (\omega =1\wedge {\theta}_I={\theta}^{\prime }|h) $ and the marginal belief that $ {\theta}_I=H $ is $ {\widehat{\theta}}_j(h)={\sum}_{\omega^{\prime}\in \{0,1\}}\mu ({\theta}_I=H\wedge \omega ={\omega}^{\prime }|h). $ Equilibrium requires that each component of the player’s strategy maximizes their expected utility at each information set given beliefs and that the beliefs are consistent with Bayes’s rule at all information sets reached with positive probability.

Before moving to the analysis, we briefly reinterpret the model in light of some substantive examples, keeping in mind that we only wish to capture some of the main forces of these situations rather that explicitly capturing all of their features.

Election Stage

By the time citizens vote, they have observed the Incumbent’s policy choice, the aggregate coproduction level, and the policy outcome. These choices potentially reveal information about the Incumbent’s competence, $ {\theta}_I $ , which then dictates whether or not it is optimal to retain the Incumbent or replace her with a randomly chosen Challenger (who is competent with probability one-half). That is, if $ {\widehat{\theta}}_I(h)\ge \frac{1}{2} $ , then citizens vote to reelect and if $ {\widehat{\theta}}_I(h)<\frac{1}{2} $ , then they vote to remove the Incumbent and replace her with the Challenger.

Citizen beliefs depend on the Incumbent’s policy strategy, which can be either uninformative (i.e., $ {\sigma}_x(0)={\sigma}_x(1) $ ), in which case the Incumbent chooses to implement the reform with the same probability regardless of her signal, or informative (i.e., $ {\sigma}_x(0)=0 $ and $ {\sigma}_x(1)=1 $ ), in which case the Incumbent always follows her signal and implements the reform if and only if she receives $ s=1 $ . Table 1 collects the relevant beliefs depending on (i) the Incumbent’s policy strategy and (ii) the history $ h=(x,y,\overline{a}) $ and presents the conditions on prior beliefs about the Incumbent’s competence ( $ {\rho}_I $ ) to be retained following different histories. Lemma 1 characterizes citizen voting behavior, given a generic level of coproduction $ \overline{a} $ .

Table 1.

Citizen Beliefs Conditional on Incumbent Policy Strategy and History

Table 1. Long description

The table consists of four columns: Incumbent policy strategy, History (x, y, a-bar), Posterior belief theta-hat sub I (h), and Retention condition.

* Row 1: Uninformative strategy where sigma sub x (0) equals sigma sub x (1). For all histories, the posterior belief is rho sub I and the retention condition is rho sub I is greater than or equal to 1 over 2.

* Row 2: Informative strategy where sigma sub x (0) equals 0 and sigma sub x (1) equals 1. This is broken down into three historical outcomes:

- No reform (0, 0, a-bar): Posterior belief is rho sub I and retention condition is rho sub I is greater than or equal to 1 over 2.

- Reform failure (1, 0, a-bar): Posterior belief is the fraction with numerator 2 open parenthesis 1 minus a-bar close parenthesis rho sub I all over denominator 2 minus a-bar open parenthesis 1 plus rho sub I close parenthesis. The retention condition is rho sub I is greater than or equal to rho-bar (a-bar) which is defined as the fraction 2 minus a-bar all over 4 minus 3 a-bar.

- Reform success (1, 1, a-bar): Posterior belief is 2 rho sub I all over rho sub I plus 1. The retention condition is rho sub I is greater than or equal to 1 over 3.

A few aspects of Lemma 1 deserve further attention. First, when the Incumbent plays an uninformative policy strategy citizens learn nothing about competence and therefore reelect the Incumbent solely based on prior beliefs about her competence: if she was ahead of the Challenger (i.e., $ {\rho}_I>\frac{1}{2} $ ), then she is reelected and if she was trailing the Challenger (i.e., $ {\rho}_I<\frac{1}{2} $ ), then she is replaced.

Second, if the Incumbent plays an informative strategy, then citizens are only able to update about Incumbent competence when reform is pursued. If $ x=0, $ then citizens learn nothing about the Incumbent and therefore reelect the Incumbent if and only if she is initially more likely to be competent than the Challenger ( $ {\rho}_I\ge \frac{1}{2} $ ). It is worth noting that this characteristic of equilibrium is fragile and depends critically on the assumption that $ Pr[\omega =1]=\frac{1}{2} $ . As discussed above, however, this assumption is deliberate as it rules out the standard pandering incentives that arise in, for example, Canes-Wrone, Herron, and Shotts (Reference Canes-Wrone, Herron and Shotts2001) due to the fact that one state of the world is more likely ex ante.

Otherwise, when $ x=1 $ and reform is pursued, citizens learn that the Incumbent received a positive signal $ s=1 $ and policy outcomes y matter. Specifically, the retention threshold is more lenient following a policy success, $ y=1 $ , relative to a policy failure, $ y=0 $ : $ \frac{1}{3}<\overline{\rho}(\overline{a}) $ . This illustrates that policy success is always good news about Incumbent competence and policy failure is always bad news. However, how bad policy failure affects the chances for Incumbent reelection depend further on citizen coproduction, $ \overline{a} $ .

The derivative of the retention threshold following failed policy reform, $ \overline{\rho}(\overline{a}) $ , with respect to $ \overline{a} $ is positive, implying that higher levels of citizen coproduction lead to a more demanding retention threshold. The citizens remove more Incumbents, in terms of their prior likelihood of being competent, following a failed reform when coproduction is higher. This is illustrated graphically in the first three scenarios (from the top) of Figure 1: as coproduction levels increase the relevant retention threshold following failed reform is more demanding. The Incumbent has to initially be more likely to be competent to be able to withstand the way that citizens negatively update about her competence in the face of policy failure when more citizens participate in coproduction.

Figure 1.

Election Stage with Informative Policy Strategy, Conditional on (x, y, ā)

Note: The gray shaded regions denote the priors about Incumbent competence ( $ {\rho}_I $ ) that would lead citizens to vote to remove the Incumbent and the non-shaded regions denote the range of priors that would lead citizens to vote to retain the Incumbent. The top three rows illustrate citizen electoral strategies following a failed reform at different levels of coproduction: the top row illustrating high coproduction, $ \overline{a}=\frac{3}{4} $ ; the second row moderate coproduction, $ \overline{a}=\frac{1}{2} $ ; and the third row low coproduction, $ \overline{a}=\frac{1}{4} $ . The bottom rows denote citizen electoral strategies following no reform and a successful reform, respectively. From top to bottom, the rows are ordered by how demanding citizen retention rules are.

Figure 1. Long description

The diagram consists of five horizontal bars stacked vertically above a numerical axis ranging from 0 to 1. Each bar is divided into a gray shaded region on the left labeled Replace and a white region on the right labeled Retain.

* The top three bars are grouped under the heading Failed reform.

* The first bar, where a bar equals 3 over 4, shows the Replace: reform failure region extending to the rightmost point, approximately 3 over 4 on the axis.

* The second bar, where a bar equals 1 over 2, shows the Replace: reform failure region ending at the midpoint of the axis.

* The third bar, where a bar equals 1 over 4, shows the Replace: reform failure region ending at approximately 5 over 8 on the axis.

* The fourth bar is titled No reform (independent of a bar). The Replace: no reform region ends exactly at the 1 over 2 mark.

* The fifth bar is titled Successful reform (independent of a bar). The Replace: reform success region is the smallest, ending at the 1 over 3 mark.

The horizontal axis at the bottom provides specific inflection points. From left to right, the marked points are 0, 1 over 3, 1 over 2, rho bar (1 over 4), rho bar (1 over 2), rho bar (3 over 4), and 1. The gray Replace regions shrink as the reform outcome improves from failure to success.

Moreover, following a failed reform, the retention threshold can never drop below the threshold required to retain the Incumbent following no reform: $ \overline{\rho}(\overline{a})\ge \frac{1}{2} $ : $ {\lim}_{\overline{a}\to 0}\overline{\rho}(\overline{a})=\frac{1}{2} $ and $ {\lim}_{\overline{a}\to 1}\overline{\rho}(\overline{a})\to 1 $ . This follows from the fact that policy success depends on more than the Incumbent’s competence; it also depends on citizen coproduction. If there is no citizen coproduction at all ( $ \overline{a}=0 $ ), then a failed reform is completely uninformative. A reform, assuming that $ \omega =1 $ , succeeds proportional to the aggregate level of coproduction, which implies that, when there is no coproduction, there can be no policy success regardless of the type of Incumbent in office. On the other hand, if there is full coproduction ( $ \overline{a}=1 $ ), then the only way a reform could have failed is because the Incumbent chose $ x=1 $ when $ \omega =0 $ , which, assuming an informative policy strategy, will only happen when an incompetent Incumbent is in office. Thus, in that case, no Incumbent will be retained since the citizens learn that the Incumbent must have been incompetent. The main theoretical upshot is that citizen coproduction alters how informative policy failures are since success and failure is the joint product of both politician and citizen choices.

The fact that $ \overline{\rho}(\overline{a})\in \left[\frac{1}{2},\hskip2.77695pt 1\right] $ also means that there are essentially four relevant regions for what happens when policy choices are informative. First, the Incumbent’s initial likelihood of competence may be so poor that there is nothing she can do to win the election. This is the case when $ {\rho}_I<\frac{1}{3} $ . Even if the Incumbent pursues successful reform, she cannot overcome her poor initial standing with citizens. Second, the Incumbent may have to produce a successful reform in order to win the election. This is the case when $ \frac{1}{3}\le {\rho}_I<\frac{1}{2} $ . The Incumbent has a moderately low prior probability of competence, but this disadvantage could be overcome by successful policy reform. Third, the Incumbent may have a relatively strong prior likelihood of competence: $ \frac{1}{2}\le {\rho}_I<\overline{\rho}(\overline{a}) $ . In this case, the Incumbent will win the election unless she pursues reform and it fails. Finally, the Incumbent may have such a strong initial likelihood of competence that she could never lose, even if reform fails. Her initial standing with the citizens is so high that even a failed reform, which does downgrade citizens’ opinion of her, is not enough to propel the Challenger to a victory. This is the case when $ {\rho}_I>\overline{\rho}(\overline{a}) $ . These regions are illustrated in Figure 1.Footnote ¹³

Coproduction Stage

In the previous section, we analyzed citizen voting behavior taking aggregate coproduction as given. We now analyze the coproduction stage and characterize when citizens choose to contribute to the success of a reform. Citizens face a coordination problem where their coproduction decisions are a function of the cost of coproduction, c, and their belief in the likelihood of achieving a policy success, which depends on their beliefs about $ \omega $ and the total amount of coproduction in the population. While the coproduction stage has similarities to a global game (e.g., Carlsson and Van Damme Reference Carlsson and Van Damme1993; Morris and Shin Reference Morris and Shin1998; Reference Morris, Shin, Dewatripont, Hansen and Turnovsky2003), for example, there are strategic complementarities, it is not a global game. In particular, the model does not satisfy the limit dominance property central to global games (i.e., there is no $ {\gamma}_i $ such that coproduction is a dominant strategy).Footnote ¹⁴ Accordingly, as we show below, there is not a unique equilibrium to this stage of the game: some equilibria are characterized by zero coproduction while others have positive coproduction.

In fact, following any policy choice, there is always an equilibrium in which there is zero coproduction. To see this, suppose that all citizens choose $ {a}_i=0 $ . Then the probability of a policy success is zero regardless of the state. Each citizen i is therefore best responding by playing $ {a}_i=0 $ , since no citizen captures the benefit $ {\gamma}_i $ and would still pay the cost c in order to coproduce.

Of course, there can also be equilibria in which some citizens engage in coproduction and there is the possibility of successful policy reform. We start by observing that citizens’ optimal strategies are cutoff rules with respect to $ \gamma . $ A citizen has a best response to choose $ {a}_i=1 $ and participate in coproduction if:

$$ \begin{array}{rll}\Pr [y=1]{\gamma}_i\ge c.& & \end{array} $$

If $ x=0, $ then $ \Pr [y=1]=0 $ and no citizens will coproduce. Otherwise, if $ x=1 $ , then $ \Pr [y=1] $ can be decomposed as follows. Taking as given the proportion $ {\overline{a}}_{-i} $ of other citizens who coproduce, and conditioning beliefs on the observed choice of x, we have $ \Pr [y=1]={\widehat{\omega}}_i(x){\overline{a}}_{-i} $ and can reduce each citizen’s incentive compatibility condition to coproduce:

(4) $$ \begin{array}{rl}{\widehat{\omega}}_i(x){\overline{a}}_{-i}{\gamma}_i\ge c.& \end{array} $$

This condition is monotone in $ {\gamma}_i $ , which implies that citizens use a cutoff rule. Define $ \overline{\gamma}(x) $ as the common cutoff rule used by citizens after observing the Incumbent’s choice of policy. Condition presented in Equation 4, recalling that $ {\gamma}_i\sim U[0,1] $ and rearranging, can be rewritten as

$$ \begin{array}{rll}{\gamma}_i\ge \frac{c}{{\widehat{\omega}}_i(x)(1-\overline{\gamma}(x))}.& & \end{array} $$

The equilibrium cutoff rule is at the point of indifference, which yields an interior solutionFootnote ¹⁵

(5) $$ \begin{array}{r}{\overline{\gamma}}^{*}(x)=\frac{{\widehat{\omega}}_i(x)-\sqrt{{\widehat{\omega}}_i(x)({\widehat{\omega}}_i(x)-4c)}}{2{\widehat{\omega}}_i(x)}.\end{array} $$

This cutoff is interior only when $ {\widehat{\omega}}_i\ge 4c $ . Otherwise, if $ {\widehat{\omega}}_i<4c $ , then the only equilibrium is one in which there is zero coproduction. When citizens believe that $ \omega =1 $ is sufficiently unlikely, no individual has a strong enough incentive to cooperate and bear the costs of coproduction, as everyone is too pessimistic about the chances of successful reform. Citizens need to believe that $ \omega =1 $ is sufficiently likely, relative to coproduction costs, to justify individual decisions to engage in coproduction. We can now characterize equilibrium behavior in the coproduction phase.

When Do Incumbents Pursue Policy Reform?

The Incumbent’s beliefs about $ \omega $ following their private signal s, denoted by $ {\widehat{\omega}}_I(s) $ , are

$$ \begin{array}{rll}{\widehat{\omega}}_I(0)=\frac{1-{\rho}_I}{2},& & \\ {}{\widehat{\omega}}_I(1)=\frac{1+{\rho}_I}{2}.& & \end{array} $$

Naturally, an Incumbent who receives $ s=0 $ is more pessimistic that $ \omega =1 $ than an Incumbent who receives $ s=1 $ , and the difference is increasing in the Incumbent’s belief that they are competent, $ {\rho}_I $ . However, the Incumbent’s policy choice depends not only on their beliefs about whether the state of the world is conducive to reform, they also take into account how anticipated citizen coproduction affects both their reelection prospects and the likelihood of a policy success.

Accordingly, the Incumbent implements policy reform if and only if:

(6) $$ {\displaystyle \begin{array}{l}\underset{\mathrm{Electoral}\ \mathrm{considerations}}{\underbrace{\beta \left(\Pr [W=I|s,x=1,\overline{a}]-\Pr [W=I|x=0]\right)}}\\ {}+\underset{\mathrm{Policy}\ \mathrm{considerations}}{\underbrace{\Pr [y=1|s,x=1,\overline{a}]}}\ge \underset{\begin{array}{c}\mathrm{Reform}\\ {}\mathrm{costs}\end{array}}{\underbrace{k.}}\end{array}} $$

The direct effect of the Incumbent’s information on anticipated policy success always leads to an Incumbent who received $ s=1 $ being more confident of reform leading to success, all else equal (i.e., $ \Pr [y=1|s=1,x=1,\overline{a}]>\Pr [y=1|s=0,x=1,\overline{a}] $ ), which makes Equation 6 easier to satisfy. That is, from a pure policy perspective, an Incumbent who received favorable information ( $ s=1 $ ) has stronger incentives to pursue reform than an Incumbent who received unfavorable information ( $ s=0 $ ). In terms of electoral motivations, the likelihood of being reelected after no reform, $ x=0 $ , is deterministic: if $ {\rho}_I\ge \frac{1}{2} $ , then the Incumbent is reelected (i.e., $ \Pr [W=I|x=0,{\rho}_I\ge \frac{1}{2}]=1 $ ) and if $ {\rho}_I<\frac{1}{2} $ , then she is replaced (i.e., $ \Pr [W=I|x=0,{\rho}_I<\frac{1}{2}]=0 $ ). Thus, the net effect of pursuing reform is always either weakly negative—in a context where $ {\rho}_I\ge \frac{1}{2} $ —or weakly positive—in a context where $ {\rho}_I<\frac{1}{2} $ . When the net effect is negative, increased office motivations make it harder to satisfy the condition in Equation 6 while when it is positive, it makes the condition easier to satisfy. That is, when the Incumbent would be reelected for sure after $ x=0, $ pursuing reform could be risky if a failure would lead to her losing the election and if she would be replaced for sure after $ x=0, $ then there may be an incentive to pursue reform in the hopes of producing a success to improve voters’ evaluation of her competence and secure reelection. How these incentives manifest depend on the Incumbent’s strategy—uninformative or informative—and the associated level of citizen coproduction.

Lemma 2 characterizes citizen coproduction generally, but the Incumbent’s policy strategy—uninformative or informative—also affects equilibrium coproduction because it affects citizen beliefs about $ \omega $ . Aggregate coproduction depends on the equilibrium cutoff $ \overline{\gamma} $ , which further depends on coproduction costs and citizen beliefs about $ \omega $ . When the Incumbent plays an uninformative strategy so that x is the same regardless of s equilibrium coproduction, after plugging in citizen prior beliefs about $ \omega $ , is given by

(7) $$ \begin{array}{rlll}{\overline{a}}^{\mathrm{un}}(c)& =1-{\overline{\gamma}}^L(1),& & \\ {}& =1-\frac{\frac{1}{2}-\sqrt{\frac{1}{2}\left(\frac{1}{2}-4c\right)}}{2\left(\frac{1}{2}\right)},& & \\ {}& =\frac{1+\sqrt{1-8c}}{2}.\end{array} $$

Citizens retain their prior beliefs, $ {\widehat{\omega}}_i=\frac{1}{2} $ , since they cannot infer anything about the Incumbent’s signal. Thus, equilibrium coproduction only depends on the costs of participating c. Intuitively, aggregate coproduction is decreasing in coproduction costs. If instead the Incumbent plays an informative policy strategy so that $ x(s)=s $ equilibrium coproduction, after plugging in the Incumbent’s updated beliefs after receiving their signal for citizen beliefs, is given by

(8) $$ \begin{array}{llll}{\overline{a}}^{\mathrm{inf}}(c,{\rho}_I)& =1-{\overline{\gamma}}^L(1),& & \\ {}& =1-\frac{\frac{1+{\rho}_I}{2}-\sqrt{\frac{1+{\rho}_I}{2}\left(\frac{1+{\rho}_I}{2}-4c\right)}}{2\left(\frac{1+{\rho}_I}{2}\right)},& & \\ {}& =\frac{1+{\rho}_I+\sqrt{(1+{\rho}_I)(1-8c+{\rho}_I)}}{2(1+{\rho}_I)}.\end{array} $$

In contrast to when the Incumbent plays an uninformative policy strategy, now citizens are able to infer the Incumbent’s signal s and update their beliefs about $ \omega $ . Since the Incumbent’s strategy is informative, the citizens learn that the Incumbent received signal $ s=1 $ when they observe policy reform $ x=1 $ . Accordingly, now aggregate coproduction is increasing in the prior about the Incumbent, $ {\rho}_I $ , because the more likely it is that the Incumbent is competent the more likely it is that they accurately learned that $ \omega =1 $ . Aggregate coproduction is also decreasing in coproduction costs c, just as when the Incumbent plays an uninformative policy strategy.

Comparing across the two policy strategies, citizen coproduction is always higher when the Incumbent plays an informative strategy and only pursues reform when they received a positive signal $ s=1 $ : $ {\overline{a}}^{\mathrm{inf}}(c,{\rho}_I)>{\overline{a}}^{\mathrm{un}}(c) $ . The coproduction gap is also increasing in the Incumbent’s initial competence probability $ {\rho}_I $ . The more likely it is that the Incumbent is competent, the bigger the coproduction premium is to playing an informative policy strategy.

In the equilibria to follow,Footnote ¹⁶ we characterize the Incumbent’s incentives in different political environments to determine when she pursues policy reform.

Uninformative Equilibria

In this section, we assume that the Incumbent either never implements reform or always implements reform and characterize when these strategies constitute equilibria. In either case, the citizens do not learn anything about $ \omega $ and, therefore, the reelection rule is to retain the Incumbent if and only if $ {\rho}_I\ge \frac{1}{2} $ . The Incumbent’s reelection prospects are not responsive to policy choice and therefore only the Incumbent’s policy motivations matter. Accordingly, taking into account equilibrium coproduction, the Incumbent pursues reform if and only if:

$$ \begin{array}{rll}{\widehat{\omega}}_I(s)\cdot {\overline{a}}^{\mathrm{un}}(c)\ge k,& & \end{array} $$

which yields two reform cost thresholds that depend on the Incumbent’s private signal s:

$$ \begin{array}{lll}{k}^{*}(0):=\left(\frac{1-{\rho}_I}{2}\right)\left(\frac{1+\sqrt{1-8c}}{2}\right)=\frac{\left(1-{\rho}_I\right)\left(1+\sqrt{1-8c}\right)}{4},& & \\ {}{k}^{*}(1):=\left(\frac{1+{\rho}_I}{2}\right)\left(\frac{1+\sqrt{1-8c}}{2}\right)=\frac{\left(1+{\rho}_I\right)\left(1+\sqrt{1-8c}\right)}{4},& & \end{array} $$

where $ 0<{k}^{*}(0)<{k}^{*}(1)<1 $ . An incumbent who receives signal s chooses to implement reform if and only if $ k\le {k}^{*}(s) $ . With these thresholds in hand, we can now characterize uninformative equilibria, beginning with an equilibrium where the Incumbent never pursues reform.

Proposition 1. The Incumbent never pursues policy reform if any of the following conditions are satisfied:

1. 1. There is zero citizen coproduction: $ \overline{a}=0 $ .

2. 2. Reform costs are prohibitively large: $ k>{k}^{*}(1) $ .

If $ c>\frac{1}{8} $ or $ k>{k}^{*}(1), $ then the unique uninformative equilibrium features zero citizen coproduction and the Incumbent never pursuing reform.

There are two reasons that the Incumbent may never choose reform regardless of her information: (i) there is zero citizen coproduction, which we show in the previous section is always possible or (ii) there would be positive citizen coproduction, but the cost of reform k is too large. While an equilibrium with zero coproduction and no reform always exists, it is the unique equilibrium if either coproduction costs are too high or reform costs are too high.

Of course, it is also possible that there would be positive coproduction following a policy reform and that reform costs are so low that the Incumbent will always pursue reform.

Proposition 2. There is an equilibrium in which the Incumbent always pursues reform if and only if both of the following conditions are satisfied:

1. 1. There is positive citizen coproduction, which requires that $ c<\frac{1}{8} $ .

2. 2. Reform costs are sufficiently low: $ k\le {k}^{*}(0) $ .

In terms of uninformative equilibria in which the Incumbent always pursues reform both conditions—positive citizen coproduction and low reform costs—are necessary and jointly sufficient. If either of the conditions fails, then there are equilibria in which the Incumbent will forego reform and choose $ x=0 $ . If there is no citizen coproduction then the Incumbent, regardless of the signal s, will always set $ x=0 $ . If there is positive citizen coproduction but $ k>{k}^{*}(1), $ then even an Incumbent who received $ s=1 $ will forego reform. Having characterized equilibria in which the Incumbent either never pursues reform or always pursues reform we now turn to informative equilibria.

Informative Equilibrium

In an informative equilibrium, the Incumbent follows her signal and reveals that information to the citizens. Whether or not she pursues policy reform depends on where she stands electorally prior to her policy choice. When her initial competence probability is either very low, $ {\rho}_I<\frac{1}{3} $ , or very high, $ {\rho}_I\ge \overline{\rho}(\overline{a}) $ , then her reelection prospects do not depend on her policy choice—she either always loses or always wins. If the Incumbent’s prior competence probability is moderately low, $ {\rho}_I\in \left[\frac{1}{3},\hskip2.77695pt \frac{1}{2}\right) $ , then she can win reelection only if she pursues successful policy reform. Finally, if the Incumbent’s prior competence probability is moderately high, $ {\rho}_I\in \left[\frac{1}{2},\hskip2.77695pt \overline{\rho}(\overline{a})\right) $ , then she wins reelection unless she produces failed policy reform. The Incumbent’s incentives to pursue policy reform depend critically on whether or not the electoral environment is competitive and, if it is, whether she is behind or ahead ex ante.

Uncompetitive Election. When the election is uncompetitive the Incumbent pursues policy reform solely based on the likelihood of a policy success, which depends on their beliefs about the state of the world given their signal ( $ {\widehat{\omega}}_I(s) $ ) and the anticipated level of coproduction ( $ {\overline{a}}^{\mathrm{inf}}(c,{\rho}_I) $ ). Accordingly, the Incumbent, after receiving signal s, pursues policy reform if and only if:

$$ \begin{array}{rll}{\widehat{\omega}}_I(s)\cdot {\overline{a}}^{\mathrm{inf}}(c,{\rho}_I)\ge k.& & \end{array} $$

This incentive compatibility condition yields two thresholds depending on the Incumbent’s private signal s:

(9) $$ {\displaystyle \begin{array}{rcl}{k}^{**}(0)& :=& \left(\frac{1-{\rho}_I}{2}\right)\left(\frac{1+{\rho}_I+\sqrt{\left(1+{\rho}_I\right)\left(1-8c+{\rho}_I\right)}}{2\left(1+{\rho}_I\right)}\right)\\ {}& =& \left(\frac{1-{\rho}_I}{1+{\rho}_I}\right)\left(\frac{1+{\rho}_I+\sqrt{\left(1+{\rho}_I\right)\left(1-8c+{\rho}_I\right)}}{4}\right),\end{array}} $$

(10) $$ {\displaystyle \begin{array}{rcl}{k}^{**}(1)& :=& \left(\frac{1+{\rho}_I}{2}\right)\left(\frac{1+{\rho}_I+\sqrt{\left(1+{\rho}_I\right)\left(1-8c+{\rho}_I\right)}}{2\left(1+{\rho}_I\right)}\right)\\ {}& =& \frac{1+{\rho}_I+\sqrt{\left(1+{\rho}_I\right)\left(1-8c+{\rho}_I\right)}}{4},\end{array}} $$

where $ 0<{k}^{**}(0)<{k}^{**}(1)<1 $ . The Incumbent engages in informative policymaking where she pursues reform ( $ x=1 $ ) only when $ s=1 $ if and only if $ k\in ({k}^{**}(0),\hskip2.77695pt {k}^{**}(1)) $ . Reform costs need to be high enough that the Incumbent would not pursue reform after a pessimistic signal ( $ s=0 $ ) but low enough that the Incumbent will pursue reform after a positive signal ( $ s=1 $ ). This decision depends solely on the Incumbent’s assessment about how likely it is that reform will be successful since electoral considerations do not come into play.

Competitive Election. In a competitive election, reelection incentives are activated: either the Incumbent is slightly behind the Challenger (i.e., $ {\rho}_I\in (\frac{1}{3},\hskip2.77695pt \frac{1}{2}) $ ) or slightly ahead of the Challenger (i.e., $ {\rho}_I\in (\frac{1}{2},\hskip2.77695pt \overline{\rho}(\overline{a}) $ ) ex ante. In the former case, this pushes the Incumbent toward gambling for reelection since she will lose reelection unless she produces successful policy reform. In the latter case, the Incumbent is incentivized to play it safe and forego reform since she will be reelected unless she produces failed reform.

First, consider the case where the Incumbent can only win reelection by implementing a successful policy reform: $ {\rho}_I\in \left(\frac{1}{3},\hskip2.77695pt \frac{1}{2}\right] $ . Given signal s, the Incumbent chooses $ x=1 $ if:

(11) $$ \begin{array}{r}{\widehat{\omega}}_I(s)\cdot {\overline{a}}^{\mathrm{inf}}(c,{\rho}_I)\cdot (1+\beta )\ge k.\end{array} $$

Compared to the settings where reelection is determined based on the Incumbent’s prior likelihood of competence, in this case, there is the additional incentive to produce a policy reform in order to secure reelection and enjoy the benefit $ \beta $ . Once again, we can state these conditions in terms of $ {k}^{**}(s) $ . Specifically, in this case, an informative policy strategy is incentive compatible if

$$ \begin{array}{rll}(1+\beta ){k}^{**}(0)<k\le (1+\beta ){k}^{**}(1).& & \end{array} $$

Now that the Incumbent also has an incentive to pursue policy reform, since the only way she can secure reelection is by producing a successful reform, the cost thresholds that support pursuit of reform are higher relative to uncompetitive elections. As the benefit from winning reelection ( $ \beta $ ) increases so do the relevant thresholds on k, regardless of the Incumbent’s signal.

Now consider the case where the Incumbent always wins reelection unless she pursues reform and it fails: $ {\rho}_I\in \left(\frac{1}{2},\hskip2.77695pt \overline{\rho}({\overline{a}}^{\mathrm{inf}}(c,{\rho}_I))\right] $ . In this case, the Incumbent pursues reform if and only if:

(12) $$ \begin{array}{r}{\widehat{\omega}}_I(s)\cdot {\overline{a}}^{\mathrm{inf}}(c,{\rho}_I)\cdot (1+\beta )-\beta \ge k.\end{array} $$

Compared to the previous case, now the Incumbent has an incentive to forego reform because doing so secures reelection with certainty and the associated benefit $ \beta $ . This makes pursuing policy reform more demanding. Again, expressing this in terms of cost thresholds conditional on the Incumbent’s signal, we have that informative policy choice is supportable if:

$$ \begin{array}{rll}(1+\beta ){k}^{**}(0)-\beta <k\le (1+\beta ){k}^{**}(1)-\beta .& & \end{array} $$

Now the relevant thresholds on k shift down, proportional to office benefits $ \beta $ , relative to the previous cases. On the one hand, this means that incumbents who receive signal $ s=0 $ are less likely to set $ x=1 $ , but, on the other hand, it is also less likely that incumbents who receive signal $ s=1 $ will implement reform.Footnote ¹⁷ With analysis of both uncompetitive and competitive elections in hand, we can now characterize informative equilibria.

Proposition 3. Suppose c is small enough so that positive coproduction is feasible (i.e., $ c<\frac{1+{\rho}_I}{8} $ ). There exists an equilibrium with informative policy choices and positive coproduction under the following circumstances:

• • The election is uncompetitive (i.e., $ {\rho}_I<\frac{1}{3} $ or $ {\rho}_I>\overline{\rho}({\overline{a}}^{\mathrm{inf}}(c,{\rho}_I)) $ ) and $ {k}^{**}(0)<k\le {k}^{**}(1). $

• • The election is competitive and the Incumbent trails the Challenger so that she wins reelection only if she pursues successful policy reform (i.e., $ {\rho}_I\in \left(\frac{1}{3},\hskip2.83864pt \frac{1}{2}\right] $ ) and $ (1+\beta ){k}^{**}(0)<k\le (1+\beta ){k}^{**}(1). $

• • The election is competitive and the Incumbent leads the Challenger so that she wins reelection unless she pursues failed policy reform (i.e., $ {\rho}_I\in \left(\frac{1}{2},\hskip2.83864pt \overline{\rho}({\overline{a}}^{\mathrm{inf}}(c,{\rho}_I))\right] $ ) and $ (1+\beta ){k}^{**}(0)-\beta <k\le (1+\beta ){k}^{**}(1)-\beta . $

If instead the cost of coproduction is prohibitively large, that is, $ c>\frac{1+{\rho}_I}{8} $ , then the unique equilibrium is uninformative and features zero coproduction and the Incumbent never pursuing reform.

Proposition 3 highlights how the Incumbent’s incentives respond to the electoral environment. The Incumbent is most likely to pursue reform when the election is competitive and she is moderately unpopular ( $ {\rho}_I\in (\frac{1}{3},\hskip2.77695pt \frac{1}{2}) $ ) and least likely to pursue reform when the election is competitive and she is moderately popular ( $ {\rho}_I\in (\frac{1}{2},\hskip2.77695pt \overline{\rho}({\overline{a}}^{\mathrm{inf}}(c,{\rho}_I))) $ ). This follows directly from the way that the electoral environment impacts the Incumbent’s incentives. When the election is competitive the Incumbent considers her electoral motivations and when she is moderately unpopular these motivations push her to gamble for reelection, making policy reform more attractive, while when she is moderately popular, she can secure reelection as long as she does not produce failed reform, which pushes her away from policy reform. These incentives are amplified by increased benefits to securing reelection (i.e., by increasing $ \beta $ ). In an uncompetitive electoral environment, the Incumbent pursues reform solely based on her policy motivations. This is the moderate case where policy reform is more likely than when the Incumbent holds a small lead over the Challenger but less likely than when she sits at a slight electoral disadvantage ex ante.

These dynamics are illustrated graphically in Figure 2, which plots equilibrium regions based on the Incumbent’s prior competence probability $ {\rho}_I $ (x-axis) and reform costs k (y-axis). As noted above, there can always be an uninformative equilibrium in which the Incumbent does not pursue reform ( $ x=0 $ ) and there is zero coproduction. This is the unique equilibrium anytime coproduction costs are too high or reform costs are too high. There are also informative equilibria where the Incumbent follows her signal when reform costs are intermediate (light gray regions). Comparing panels a and b in Figure 2 also illustrates the impact of office benefits $ \beta $ on Incumbent incentives. Specifically, we can see that when the Incumbent’s prior competence probability is moderately low and she trails the Challenger in a competitive election (i.e., $ {\rho}_I\in (\frac{1}{3},\frac{1}{2} $ ), the thresholds that support informative policy choice both shift up and reform is more likely as $ \beta $ increases.Footnote ¹⁸ When instead the Incumbent is moderately popular and leads the Challenger (i.e., $ {\rho}_I\in (\frac{1}{2},\overline{\rho}(\overline{a})) $ ), the thresholds shift down and reform becomes less likely as $ \beta $ increases.

Figure 2.

Equilibrium Regions

Note: The x-axis is the Incumbent’s prior competence probability $ {\rho}_I $ and the y-axis is the cost of reform k. In all figures, $ c=\frac{1}{10} $ and $ \beta \in \left\{\frac{1}{4},\hskip2.77695pt \frac{3}{4}\right\} $ , from (a) to (b). Each region is labeled with the equilibria that exist within that region: NR denotes an uninformative equilibrium where the Incumbent never pursues reform and citizens never engage in positive coproduction; AR denotes an uninformative equilibrium where the Incumbent always pursues reform provided that and citizens engage in positive coproduction; and FS denotes informative equilibrium where the Incumbent pursues reform if and only if $ s=1 $ and citizens engage in positive coproduction.

Figure 2. Long description

A two-panel figure with panel (a) labeled Low office motivation beta equals 1 over 4 and panel (b) labeled High office motivation beta equals 3 over 4. Both panels use a horizontal x-axis for rho sub I (competence probability) from 0 to 1 and a vertical y-axis for k (cost of reform) from 0 to 1.

In panel (a):

* The top region is dark gray, labeled Never Reform (N R).

* A large central light gray region is labeled Follow Signal and Never Reform (F S and N R).

* The bottom-left region is medium gray, labeled Always Reform and Never Reform (A R and N R).

* As rho sub I increases past 1 over 2, a triangular region at the bottom is labeled A R, F S, and N R.

* Near rho sub I equals 1, small slivers are labeled N R and A R and N R.

In panel (b):

* The dark gray N R region is larger, particularly between rho sub I values of 1 over 2 and 1.

* The light gray F S and N R region is split into two distinct sections by a vertical dark gray N R block between rho sub I values of 1 over 3 and 1 over 2.

* The bottom regions for A R and N R and the combined A R, F S, and N R follow a similar downward-sloping trend as in panel (a) but with slightly different boundary intersections.

Substantive Examples

We return to the examples of education policy and disease outbreak prevention in order to interpret the implications of the model for empirical work on these topics.

Citizen Coproduction and Political Inefficiencies

Having characterized equilibria, we now turn to the policy distortions that are introduced through electoral accountability and how citizen coproduction interacts with those distortions. To do so, we focus on informative equilibria and take as our benchmark what the Incumbent would do in terms of implementing policy reform if she had no electoral considerations. This is a useful benchmark in that it illustrates how the Incumbent would pursue reform if the only considerations she had were based on her private information and how that informs the likelihood of policy success. We will compare her propensity to pursue reform when electoral considerations are introduced. When the Incumbent’s policy behavior differs when electoral considerations are introduced we will have identified distortions introduced by elections. Since citizen coproduction can impact the likelihood that the Incumbent produces a policy success, and therefore increases her chances of reelection, coproduction will also shape the nature of these distortions.

As shown in the previous analysis, when the Incumbent is in an uncompetitive election her decision to pursue policy reform does not depend on electoral considerations. Thus, there is no difference between the benchmark in which we assume away elections: the Incumbent already behaves as if there is no election since her policy choice has no bearing on whether or not she is reelected. However, when the Incumbent has an intermediate prior competence probability electoral considerations play a central role. When she trails the Challenger, but could win reelection with successful reform, these electoral considerations lead to more policy reform than would occur without electoral pressures. This follows from the incentives for the Incumbent to gamble for reelection. If instead the Incumbent would win reelection unless she produces failed reform then electoral pressures lead to distortions that imply less policy reform than the no-election benchmark. Now the Incumbent’s incentives are to play it safe precisely because that is in her best interest from an electoral perspective. These distortions are amplified as the Incumbent’s benefits from holding office, $ \beta $ , increase. This is illustrated in Figure 3 comparing the left column (low office benefit) to the right column (high office benefit). Increasing $ \beta $ expands the gray shaded regions that illustrate the policy distortions introduced through electoral pressure.

Figure 3.

Policy Reform Thresholds for Informative Policymaking as Office Benefits $ \beta $ and Coproduction Cost c Vary

Note: The solid black lines display the reform cost thresholds given $ \beta $ and c in an informative equilibrium. The dashed lines illustrate the reform cost thresholds when there are no electoral considerations. The light gray shaded regions illustrate how electoral considerations lead to more policy reform and the dark gray shaded regions illustrate how electoral considerations lead to less policy reform.

Figure 3. Long description

Four panels labeled a through d are arranged in a two by two grid. Each panel shares the same axes: the horizontal x-axis represents rho sub I ranging from 0 to 1, and the vertical y-axis represents k ranging from 0 to 1. Two solid black lines originate from k equals 1 half at rho sub I equals 0, diverging symmetrically toward the top and bottom right. Dashed lines run parallel or adjacent to these solid lines.

* Panel a: beta equals 1 fourth and c equals 1 over 20. Light gray regions appear between rho sub I equals 1 third and 1 half. Dark gray regions appear between rho sub I equals 1 half and rho bar of a super star.

* Panel b: beta equals 3 fourths and c equals 1 over 20. The light gray region above the midline is significantly taller, extending to k equals 1 between rho sub I equals 1 third and 1 half.

* Panel c: beta equals 1 fourth and c equals 1 over 10. Similar to panel a, but the shaded regions are vertically narrower, indicating a smaller range for policy reform.

* Panel d: beta equals 3 fourths and c equals 1 over 10. Combines the increased height of the light gray region from panel b with the narrower dark gray regions from panel c.

In all panels, light gray shading indicates areas where electoral considerations increase policy reform, while dark gray shading indicates areas where they decrease it.

Comparing the top row (low coproduction cost) to the bottom row (high coproduction cost) in Figure 3 illustrates how citizen coproduction impacts these distortions. Holding office benefits constant (within column) we can see that raising coproduction costs leads to a smaller range of incumbents who face competitive elections where they lead the Challenger (i.e., the retention threshold $ \overline{\rho}(\overline{a}) $ decreases as coproduction costs increase because aggregate coproduction declines). The level of citizen coproduction scales the range of the policy distortions by fundamentally altering the electoral landscape: as citizen coproduction decreases some incumbents who previously faced competitive elections with incentives to play it safe are now electorally insulated and win reelection with certainty, which eliminates any electoral pressure they previously felt. They are free to pursue policy reform as if there were no election, which eliminates policy distortions. This dynamic provides the following implication.Footnote ¹⁹

Counterintuitively, increasing the costs of coproduction can promote more policy reform even though doing so reduces coproduction and the likelihood that reform is successful. This highlights a core accountability pathology that arises in our model: reducing (increasing) coproduction makes policy failures less (more) informative about Incumbent competence, which expands (contracts) the range of incumbents, in terms of prior probability of competence, who face uncompetitive elections where they win with certainty. This allows them to disregard any electoral pressures and instead pursue policy reform purely based on their informed assessment about whether or not reform is justified. Put another way, citizen coproduction amplifies electoral pressures when they are present, but also dictates when they arise by altering the electoral environment itself.

Subsidizing Coproduction

In many cases, politicians are in a position to subsidize citizen coproduction when policies are pursued that require public buy-in. For example, new policies might introduce administrative burdens through the need for citizens to fill out complicated paperwork, to learn about their program eligibility, or to find the time to participate in the new program (e.g., Herd and Moynihan Reference Herd and Moynihan2019). As part of policy reform, then, politicians might build in ways to lower these costs—for example, online applications, information campaigns, and hiring personnel. However, given the subtle dynamics explored in the previous section it is not clear that incumbent politicians concerned about reelection, in addition to producing successful policy reforms, will always find it optimal to subsidize coproduction. Since subsidizing coproduction will induce higher coproduction, and higher coproduction will increase the informativeness of policy failures, some incumbents would benefit from relatively higher coproduction costs that drive down coproduction.Footnote ²⁰

Table 2 presents the Incumbent’s expected utility from policy reform for each electoral environment. Within each environment, the Incumbent always benefits from promoting higher coproduction through subsidizing coproduction costs. This is evident from the fact that each expected utility is strictly increasing in $ \overline{a} $ , which increases as c decreases. In any scenario, so long as changes in c do not alter the electoral environment she faces the Incumbent’s expected utility is increasing in aggregate coproduction, which implies that she benefits from lowering coproduction costs.

Table 2.

Incumbent Expected Utility for Pursuing Policy Reform by Electoral Environment

Table 2. Long description

The table consists of two columns: Electoral environment and Expected utility from x = 1.

* Row 1: Uncompetitive, losing. Defined by rho sub I < 1/3. The expected utility is omega-hat sub I (s) dot a-bar minus k.

* Row 2: Competitive, trailing. Defined by rho sub I in the interval (1/3, 1/2). The expected utility is omega-hat sub I (s) dot a-bar dot (1 plus beta) minus k.

* Row 3: Competitive, leading. Defined by rho sub I in the interval (1/2, rho-bar (a-bar)). The expected utility is omega-hat sub I (s) dot a-bar dot (1 plus beta) minus k.

* Row 4: Uncompetitive, winning. Defined by rho sub I > rho-bar (a-bar). The expected utility is omega-hat sub I (s) dot a-bar plus beta minus k.

However, when coproduction costs change and aggregate coproduction shifts so does the retention threshold $ \overline{\rho}(\overline{a}) $ —as aggregate coproduction increases (decreases), the threshold also increases (decreases). Incumbents with prior probabilities of competence around that threshold face a trade-off: subsidizing coproduction boosts aggregate coproduction, which also raises $ \overline{\rho}(\overline{a}) $ and increases the range of incumbents who face a competitive election rather than an uncompetitive election where they win with certainty. Conversely, as coproduction costs increase, the threshold $ \overline{\rho}(\overline{a}) $ decreases and some incumbents move from a competitive electoral environment to an uncompetitive one. Thus, some incumbents benefit from higher coproduction costs that suppress aggregate coproduction because it alters the electoral environment from a competitive one where incumbents are pushed to play it safe to an uncompetitive one where they win reelection with certainty.

Consider the scenario where there are two levels of aggregate coproduction, $ \overline{a} $ and $ {\overline{a}}^{\prime } $ , where $ \overline{a}>{\overline{a}}^{\prime } $ . This produces two retention thresholds $ \overline{\rho}(\overline{a}) $ and $ \overline{\rho}({\overline{a}}^{\prime }) $ , where $ \overline{\rho}({\overline{a}}^{\prime })<\overline{\rho}(\overline{a}) $ . For incumbents with prior competence probabilities at the original threshold $ \overline{\rho}(\overline{a}), $ a decrease in aggregate coproduction and shift to $ \overline{\rho}({\overline{a}}^{\prime }) $ shifts them from a competitive election to an uncompetitive election. This is beneficial whenever the expected utility from being in an uncompetitive electoral environment where the incumbent wins with certainty (but coproduction is lower) outweighs the expected utility from being in a competitive electoral environment where the incumbent wins unless she produces a failed reform (but coproduction is higher):

$$ \begin{array}{rll}{\displaystyle \underset{\begin{array}{c}\mathrm{Competitive}\ \mathrm{election};\\ {}\mathrm{higher}\ \mathrm{coproduction}\end{array}}{\underbrace{{\widehat{\omega}}_I(s)\cdot \overline{a}\cdot (1+\beta )-k}}}<{\displaystyle \underset{\begin{array}{c}\mathrm{Uncompetitive}\ \mathrm{election};\\ {}\mathrm{lower}\ \mathrm{coproduction}\end{array}}{\underbrace{{\widehat{\omega}}_I(s)\cdot {\overline{a}}^{\prime }+\beta -k}}}.& & \end{array} $$

Rearranging this inequality shows that incumbents benefit from lower coproduction in exchange for electoral safety if office benefits are sufficiently large:

$$ \begin{array}{rll}\frac{{\widehat{\omega}}_I(s)\cdot (\overline{a}-{\overline{a}}^{\prime })}{1-{\widehat{\omega}}_I(s)\cdot \overline{a}}<\beta .& & \end{array} $$

When the benefits from winning office are sufficiently large, relative to the decreased likelihood of policy success, measured by the reduction in aggregate coproduction $ \overline{a}-{\overline{a}}^{\prime } $ , an incumbent would prefer increased coproduction costs. That is, they would introduce, for example, administrative burdens to discourage coproduction even though doing so reduces the chances of producing a policy success. This follows from the fact that local changes in aggregate coproduction and the associated effects on expected policy benefits are continuous while a shift from an uncompetitive to competitive electoral environment leads to a discontinuous jump in expected electoral benefits (from $ {\widehat{\omega}}_I(s)\overline{a}\beta $ to $ \beta $ ). In these cases, the incumbent does not benefit from subsidizing coproduction even though doing so increases the likelihood of a policy success, which also increases the chances of higher policy benefits and an improved evaluation of competence from voters. Citizen coproduction structures the underlying political environment in a way that can lead to reduced electoral pressures, which is attractive to some incumbents even though it comes at the cost of potential policy success.