Search and Matching in Political Corruption

2.1 Search and Matching Setup

In the following model, we examine the dynamic interaction between entrepreneurs, seeking bribery opportunities to boost their profits with additional rents, and politicians who may provide bribery opportunities in exchange for bribes. For simplicity, we assume that each politician can offer the entrepreneur only one opportunity. We regard both entrepreneurs and politicians as atomistic competitors seeking to maximize their respective objective functions, with no moral restraint to engage in corruption. We view them all as prospective bribe-givers and -receivers. Finding the right match is a costly process for prospective bribers, requiring a variety of efforts for “greasing the wheels” and lobbying, i.e. making the targeted politicians responsive to their requests. Likewise, politicians must possess the “proper talents” in order to persuade the bribers that they are a suitable match and provide the extra-rents they are seeking.

Consider N politicians. We assume this number as fixed and use it as the normalizing variable. The matching process is governed by a matching function that returns the number of successful one-to-one matches of bribery (μN) as a function of searching entrepreneurs (λN) and searching politicians (ωN). Thus, the number of matched politicians, which corresponds to the number of matched entrepreneurs, is (1 − ω)N, while the total number of entrepreneurs in the corruption market is (1 − ω + λ)N. The matching function maintains the standard assumption of being increasing in both arguments, concave and homogeneous of degree 1:

It is useful to work in relative terms, expressing the “tightness” of the corruption market as η = λ ω . This variable is particularly useful because it summarizes the searching externalities that the two sides of the market cause each other: when corruption tightness is high, there is a relatively higher number of entrepreneurs searching for a bribery opportunity, matching will be easier for politicians, but more time-consuming for entrepreneurs, since there are more prospective bribers “competing” for the right bribee. In contrast, when corruption tightness is low, there are more bribery opportunities for potential bribers to choose from, making it easier for them to find the ideal match.

The above mechanism can also be described in terms of transition probabilities and mean duration of search for unmatched entrepreneurs and politicians. Since unmatched entrepreneurs and politicians randomly meet and agree to a corruption agreement, the process of changing state is Poisson with rates μ(ωN, λN)/λN = q(η) and μ(ωN, λN)/ωN = ηq(η), respectively.^[3]

Bribing is subject to an exogenous probability of being detected, π. Detection disrupts the corruption deal and also acts as the corresponding transition probability of moving back to the unmatched pool for both the briber and the bribee.

The previous considerations suggest that the variation in the mean number of unmatched politicians can be calculated as the difference between two flows: (i) the flow of corrupt politicians who are caught bribing, and (ii) the flow of unmatched politicians who reach a corruption agreement with a briber:

Since the purpose of our analysis is to examine the dynamic persistence and pervasiveness of corruption, it is useful to express the above equation in terms of the mean number of corrupt politicians (γ = 1 − ω), whose variation is therefore given by the difference between the flow of unmatched politicians who match with a briber in a corruption deal and the flow of bribees who are detected:

Thus, it is straightforward to interpret γ as a measure of the level of corruption pervasiveness in political institutions.

2.2 Bellman Equations

The determination of the path for corruption market dynamics follows an asset equations approach, treating both prospective bribers and bribees as optimizing and forward-looking agents. In what follows, we are going to present the characterization of the dynamic model in the steady state, while the out-of-steady-state details are presented in Appendix B. As stated previously, entrepreneurs in the corruption market can be in one of two states: (i) unmatched, prospective bribers in search for a politician offering a bribing opportunity, whose present discounted value is denoted by E _u , or (ii) actual bribers, who have reached an agreement with a politician, whose present discounted value is denoted by E _m . The entrepreneur is expected to incur a flow cost, c, while searching for corruptible politicians. We think of c as the cost of various legal (e.g. lobbying) and illegal acts, as preparatory to the canonical bribe, intended to build a “good” reputation in the corruption market that could attract the attention of politicians and ease the interaction. These costs can include the time and effort invested in networking, research, and communication; expenses related to hosting events, conferences, or meetings to facilitate connections; or costs associated with producing materials (e.g. reports, policy briefs) to capture policymakers’ attention. Therefore, c is not a sum of money that is directly pocketed by individual politicians, but it serves the role of establishing networks and connections, and signaling available bribers in the corruption market. In mathematical terms and considering the steady state:

where ρ is the discount rate. We assume for entrepreneurs a free entry condition in the corruption market. Prospective bribers search for opportunities to bribe if there are profitable prospects. If everyone can enter, however, all profit opportunities would finally be exploited, and the expected utility of being a prospective briber will be driven to zero, i.e. E _u  = 0. The implication of this assumption is that the number of entrepreneurs seeking bribery opportunities is a jump variable; that is, entrepreneurs enter or exit the corruption market instantaneously. Eventually, equation (4) can be simplified as follows:

This condition states that the expected value of being a briber must equate the expected costs of finding the right bribery opportunity (i.e. the flow cost of searching times its mean duration).

If matching occurs, the entrepreneur becomes an actual briber and receives a rent r while paying the bribe amount b, which is determined endogenously. However, as mentioned above, a briber also faces an exogenous probability of being detected. In the event of being caught, the briber pays a fine, s, assumed proportional to the amount of the rent r and returns to the unmatched pool. The asset equation for E _m in steady state reads as follows:

The present discounted value of being a corrupt politician is denoted by P _m , which combines the flow of returns provided by the bribe with the net expected returns of changing state: if the corruption deal is discovered, the bribee pays a reputational and pecuniary fine, p, again proportional to the amount of the rent r and returns to the unmatched pool:

The assumption that the fine paid by the politician also includes a loss in political reputation implies that p > s.

The present discounted value of the expected income stream of the unmatched politicians, denoted by P _u , takes into account only the net value of the possibility of state change: with probability ηq(η) the politician may be approached by the right briber and agree to a corruption deal, thereby becoming a bribee:

Every briber-bribee match generates a surplus in terms of expected net rent and briber’s savings in search costs. We assume that this surplus is split by Nash bargaining. In formal terms, the bribe is determined as follows:

where β denotes the bargaining power of the politician.^[4]

4.1 Comparative Statics and Dynamics

We turn our attention to the analysis of the comparative statics and the relevant dynamics that are consequential to an exogenous shock affecting a parameter of the model, ceteris paribus.^[8] An exogenous shock entails a modification of the curves (10) and (11) in the top-side panel of Figure 1, which results in a new equilibrium in the (η, b) space. The adjustment of both η and b is instantaneous. The new value of η sets in motion the dynamics in equation (3), which ultimately leads to a new value of the pervasiveness of corruption in the political system, γ.

Consider first an exogenous increase in π. A higher probability of being detected reduces the briber’s expected net rent, while increasing the expected value of the search costs (see equation (10)). In the upper panel of Figure 2, this modification corresponds to an inward shift of the A(η, b) curve. Consequently, the entrepreneur will provide a lower bribe for every given level of corruption tightness. An increased detection probability simultaneously changes the B(η, b) curve.^[9] Using the implicit function theorem, we demonstrate that η will unambiguously decrease in equilibrium, while for a large spectrum of parameters the value of b diminishes. As a result, matching becomes more difficult as fewer entrepreneurs seek bribery opportunities, indirectly reducing the number of politicians who enter the corruption pool.

Figure 2:

Comparative statics and dynamics – increase in π. Note: the upper panel illustrates the comparative statics for β > 1/2. If β ≤ 1/2 then the B(η, b) curve shifts upward. The direction of the effect on η (and γ) will remain the same.

To capture the effect of a change in π on the level of corruption pervasiveness, we need to analyze the new equilibrium in the (η, γ) space, which is illustrated in the bottom-side panel of Figure 2. The η ̇ = 0 line shifts leftward reflecting the reduced new equilibrium of η. The γ ̇ = 0 curve shifts downward in response to an increase in the outflow from the corruption pool subsequent to a larger π as clearly inferred from equation (3). Therefore, the final effect of both inflows and outflows of politicians in the corruption pool results in a lower level of corruption pervasiveness. In sum, if we consider a rise in π to be the result of a more efficient judiciary, the model predicts a decline in the political corruption levels.

Differently from the case analyzed above, any modifications in any other exogenous parameters affect the level of corruption pervasiveness only indirectly, through their effect on corruption tightness and, as a consequence, the politician’s matching probability. In other words, only the inflow of politicians in the corruption pool is affected, while no outflow takes place. We can fully describe the mechanics of the comparative statics by looking at the shifts of the curves in the top-side panel of Figure 1. In the bottom-side panel of Figure 1, the equilibrium level of γ emerges from the shift of the η ̇ = 0 line, while the γ ̇ = 0 curve remains unchanged. Hence, in the subsequent analysis, we will mainly focus on the effects on corruption tightness and omit the simple graphical representation of the comparative statics.

An exogenous increase in s has the effect of reducing both b and η, since politicians can extract lower bribes (i.e. B(η, b) shifts downward), and entrepreneurs face a reduction in the expected net rents from bribing (i.e. A(η, b) shifts inward). Similarly, an exogenous increase in p will also reduce η, but unlike the penalty s, the equilibrium bribe will increase since politicians will demand a higher risk premium to engage in a corruption agreement. A larger bribe makes entrepreneurs more reluctant to enter the corruption market. Graphically, the B(η, b) curve would shift upward, whilst the A(η, b) curve would remain unaltered. In sum, for both types of sanctions, the reduction of η implies fewer matches and, consequently, a lower share of corrupt politicians.

By increasing market competition and accountability in government decision-making, it becomes more difficult for corrupt actors to establish and maintain relationships with politicians, which translates in the context of our model into an exogenous increase in search and signaling costs c. This can include policies such as mandatory public disclosure of lobbying activities, campaign finance reporting, and asset declarations by public officials. The main effect of such an increase is an inward shift of the A(η, b) curve because the entrepreneur needs to offer a lower bribe to offset the increased costs. At the same time, the savings from search costs once a match occurs will increase, thereby inducing an upward shift of the B(η, b) curve. Both b and η will decrease as a result of the new equilibrium level. Thus, if increased search (and signaling) costs reflect greater transparency and competition we can expect a reduction in corruption levels (Campos et al. 2021; Knack, Biletska, and Kacker 2019; Strîmbu and González 2018).

The effect of an exogenous increase in the discount rate is similar to the case of an increase in search costs, where both b and η decrease. This is not surprising because the discount rate applies to the time in which entrepreneurs search for corruption opportunities. Specifically, the A(η, b) curve shifts downward, while the B(η, b) curve remains unchanged. The finding of an inverse relationship between discounting and corruption levels is confirmed by Helland and Sørensen (2012) in the context of electoral dynamics with political corruption. Furthermore, if a rise in ρ is directly correlated with a higher real interest rate, for example, due to a lower inflation rate, then the model shows a direct relationship between inflation and corruption pervasiveness. This last finding is supported by empirical evidence (Al-Marhubi 2000; Braun and Di Tella 2004).

An exogenous increase in the politician’s bargaining power results in a reduction of η and an increase in b, since β has a direct positive effect in the Nash bargaining condition (11). Thus, if larger βs signal political regimes with stronger military and poorer civil rights, the model predicts larger bribes and lower corruption rates (Cheung, Rau, and Stouraitis 2012; Svensson 2003).

Finally, an exogenous increase in the rent value from corruption will clearly induce more entrepreneurs to search for corruption opportunities, resulting in an outward shift of the A(η, b) curve. As a result, η would increase. From a Nash bargaining perspective, there is a larger expected net surplus to be shared, which graphically implies an upward shift of the B(η, b) curve. The result is larger bribes with more corruption.

All the above cases are summarized in Table 1 and in the following proposition.

4.2 Comparative Policy Analysis

The results of comparative statics outlined in Table 1 do not provide enough information in terms of which policy is most effective in curbing corruption. In this section, we compare the effectiveness of different policies in reducing corruption according to its relative scale. For instance, we can obtain some insights from the comparison of the elasticity of γ with respect to each available policy instrument, i.e. the percentage change in γ after a unit percentage increase in the designated deterrence policy. Consider the impact of the following deterrence policies: (1) increasing detection efficiency; (2) raising penalties from detection; (3) increasing the briber’s search costs. For example, when assessing the relative effectiveness of detection efficiency and severity of briber’s penalty, if ɛ _γ,π  > ɛ _γ,s then improving detection efficiency is a more effective tool in curbing corruption than imposing harsher penalties to bribers.^[10]

The policy implications are summarized by the ranking of the following elasticities:

For relatively small-scale political corruption (i.e. r < r _L ), an intervention on search costs appears to be the most effective anti-corruption measure. Increasing transparency levels in lobbying operations and establishing greater competition in public procurement would raise the search/signaling costs of corruption opportunities to the point where they are no longer profitable. As the rent from corruption increases (i.e. r > r _L ), the impact of search costs diminishes, particularly for relatively large-scale corruption rents (i.e. r > r _H ). With relatively larger rents, increasing search/signaling costs does not effectively discourage entry in the corruption market. When the stakes are higher, the most effective deterrent strategy changes from inhibiting the bribers’ entry signal into the corruption market to ex-post policy instruments such as especially investing in investigative resources and judicial efficiency, but also applying harsher fines.

The intuition behind this result is straightforward: given a certain level of search costs, an increase in these costs would have a more considerable impact on the matching surplus to split on smaller than larger rents, when compared to an enhancement in judicial efficiency or the imposition of stricter sanctions.^[11] This outcome depends on the functioning of c compared to π or s or p, since c is akin to a lump sum deducted from the total rent, while π, s and p would diminish the expected value of the rent. As with any lump sum, when the rent is relatively small, an increase in the value of the lump sum could render the rent-seeking activity unsustainable due to its direct impact on the matching surplus. In contrast, an increase in π, s or p would only indirectly reduce the surplus, primarily through its impact on anticipated rents. The underlying principle aligns with the role of fixed and variable costs in firms’ entry decisions. When marginal revenues are comparatively low, an increase in fixed costs may significantly influence firms’ decision to enter the market more than an increase in variable costs. Conversely, when marginal revenues are relatively high, the impact of an increase in fixed costs is distributed over a larger revenue base, thereby exerting a lesser influence on firms’ entry decisions.

The model also predicts that reputational penalties for politicians have a greater impact on corruption than normal fines. In other words, the larger the role of reputational costs in the objective function of politicians, the more significant the impact of harsher sanctions on corrupt politicians relative to bribers.

To enhance our comprehension of these results, we should recognize that as the intensity of common deterrence tools (namely, π, s, and p) escalates, ceteris paribus, the rent thresholds decline, thereby diminishing the effectiveness of adopting c as an anti-corruption policy across a broad spectrum of corruption activities. Conversely, if the magnitude of c is significantly high, it would elevate the rent thresholds, thus rendering any policy aimed at boosting transparency and signaling costs more effective. This outcome might initially appear counterintuitive when juxtaposed with the law of decreasing marginal returns prevalent in various economic variables. Nevertheless, every policy possesses a threshold beyond which it can effectively deter criminal behavior. In other words, in a theoretical framework, if an anti-corruption policy surpasses a certain threshold, bribers no longer derive any positive net benefit from corruption.^[12]

We can now state the following proposition that summarizes the results of the comparative policy analysis.

Identifying empirical instances where more stringent policies on lobbying and other signaling costs are preferred for relatively small-scale corruption activities can be challenging. Nevertheless, Campos and Giovannoni (2017) provide cogent evidence supporting the increased likelihood of lobbying, as a precursor to corruption, being markedly more prevalent in smaller electoral districts, which serve as a proxy for relatively small-rent levels, compared to larger ones. The underlying rationale is the amplified incentives for firms to establish political connections when: (1) there are fewer officials to be elected, thereby enhancing the efficacy of the corruption signal, and (2) the cost to attract the attention of politicians and foster interaction is relatively lower. Consequently, policies explicitly aimed at mitigating reputation-building and curtailing undue interactions between entrepreneurs and local politicians warrant comprehensive consideration.

Similarly, identifying real-world examples of policies explicitly aimed at either major or minor corruption cases can also be a complex undertaking. This challenge primarily stems from the fact that high-profile cases are predominantly the ones garnering substantial media attention. Once a scandal breaks, the tendency is to implement anti-corruption policies across the board. For instance, in the aftermath of the 2014 Operation Car Wash scandal in Brazil, the 2016 Law on the Responsibility of Federal State Companies was instituted, mandating several measures (Costa, Lima, and Goldschmidt 2020): (1) Implementation of codes of conduct; (2) Establishment of statutory audit committees; (3) Overhaul of the appointment procedures for board members and senior management; (4) Enhancement of procurement process transparency; (5) Intensification of monitoring and enforcement.

In real-world settings, the adoption of anti-corruption policies necessitates a comparative assessment of the costs associated with each deterrence strategy, an element not accounted for in our analysis. For example, streamlining procedures and bolstering transparency, which demand coordinated interventions across multiple levels, often present a more politically complex challenge than the symbolic demonstration of a “firm hand”. In other words, the implementation of conventional anti-corruption policies designed to elevate the anticipated cost of crime, such as the increase of penalties, often proves to be more straightforward. From a media standpoint, such strategies may offer enhanced political benefits for politicians.