One Citizen per Period

The simplest type of interaction between citizens and the government is a two-player game that takes place at time t: a citizen j and a government official g are matched for an interaction. The citizen has an endowment w and can spend it on two goods: a private good P (which has a price normalized to 1 per unit, and provides utility normalized to 1 per unit) or a government good G. In the following, only one market for G goods is considered at a time, so the subscript g on each G good is omitted.

The utility function of the citizen for the private and government good is linear and additive: the endowment is transformed into utility at rate 1 for the private good, so w enters into the utility function without a coefficient before it. The government good enters the utility function at rate α_j, and it is constrained by assumption to be binary: G_j ∈ {0,1}. For the problem to be at all interesting, α_j has to be greater than 1, so that the government good is desirable for the citizen. Any R_j ‘rent’ (bribe, tax) payments to the government are subtracted from the utility function, as they reduce the private good consumption of the citizen. The utility function is therefore:

$${{U}_j}\, = \,w\, + \,{{\alpha }_j}{{G}_j}\,{\rm{ - }}\,{{R}_j}.$$

The government official's utility is also linear in money (which in the corrupt system comes in the form of a ‘rent’ or bribe, hence the R notation):

$${{U}_g}\, = \,{{R}_g}.$$

Multiple interactions, different citizens

The commitment problem looks different however, if multiple, repeated interactions are considered. Repeated games will have multiple equilibria, as suggested by the various folk theorem results (for example, Friedman Reference Friedman1971). The folk theorem for repeated games in which the stage game is a dynamic game is given in Wen (Reference Wen2002). The result in Wen (Reference Wen2002) is similar to the one in the folk theorems for repeated simultaneous games, with the exception of the replacement of the minmax value for repeated simultaneous games with a more general notion of “effective minmax.” The effective minmax is, roughly speaking, the lowest utility that can be imposed on a player in one period without other players also minmaxing themselves. In the present case, this value is easy to determine as w for citizens and zero for the government. Each citizen can choose to not make any rent payments and consume only private goods, thus receiving a utility of w. The government can choose to not provide any G and therefore not incur any costs, and get zero utility. The feasibility condition in the folk theorem is that the sum of payoffs cannot be greater than the maximum amount of utility available in the game each period, which in this case is $$$w\, + \,\alpha _{{t,j}}^{{MAX}} \,{\rm{ - }}\,c$$$ . Here $$$\alpha _{{t,j}}^{{MAX}} $$$ is the highest value of α at time t. Therefore the folk theorem of Wen (Reference Wen2002) states that any payoff vector that generates utilities per period higher than zero for the government, and higher than w for citizens, but that sums up to less than $$$w\, + \,\alpha _{j}^{{MAX}} \,{\rm{ - }}\,c$$$ in each period can be sustained in a subgame-perfect equilibrium for a high enough discount rate.

Among these multiple equilibria, this analysis concentrates on those that, loosely speaking, are cooperative in the sense of having some notion of Pareto optimality associated with them. More formally, the analysis will assume that players only consider equilibria Γ*, which has the following properties:

(1) Γ* is Pareto efficient in expectation.

This means that Γ* is not weakly Pareto dominated in expectation. $$$EV_{{EQ}}^{k} $$$ denotes the expected payoff of player k in equilibrium EQ. An equilibrium $$${{\rGamma }^{ \ast '}} $$$ will not be considered if there exists another Γ* such that $$$EV_{{{{\rGamma }^{ \,\ast '}} }}^{k} \leq EV_{{{{\rGamma \,}^ \ast } }}^{k} \,\forall k \in {\rm{\{ }}N,g{\rm{\} }}$$$ , with strict inequality at least once, in which N is a continuum from which the citizens are selected.

A separate requirement from the cooperative one would be that the community enforcement process only selects equilibria in which the citizens are not overpaying for the good, that is, the rent level is not higher than what is absolutely necessary for the government to provide the good to whatever citizen it provides to in equilibrium. This requirement helps remove multiple equilibria in which all that is different is the size of the rent (which is just a transfer from citizens to the government), and thus allows the analysis to concentrate on other dimensions of redistribution. Formally:

(2) Γ* is such that $$$V_{{t,{{\rGamma \,}^ \ast } }}^{g} \, = \,V_{{t,OSD}}^{g}, \forall t.$$$

This requires that the government's participation (incentive compatibility) condition in the equilibrium always holds with equality: the utility of the government under the equilibrium has to be equal to the utility of the government under the best one-stage deviation from the given equilibrium at every moment.

Consider a game in which the basic two-stage game presented above is repeated numerous times between the official and a different potential customer each period. In such an infinitely repeated game, all players discount the future with rate $$$\delta \, \in \,{\rm{(0}},{\rm{1)}}$$$ . In this section, consider the simplest matching structure between the official and possible ‘customers’: each period, the official is matched to play the game described above with a single customer. The customers are drawn from a continuum of size N, so each customer will only be drawn once. The crucial parameters that influence what sorts of equilibria can arise in this random matching game are the players’ information sets: the information set Λ_t,j of player j at time t is determined by which of the interactions that have taken place until time t − 1 are known to j, in terms of their actions/outcomes. A player who knows about (some of) the actions taken until time t − 1 at time t can condition his or her actions based on this knowledge. This leads to splitting the citizens into two groups: I for insiders, whose information sets include at least the last meetings between the government official and a member of I (the more realistic situation in which they include all previous interactions between insiders and the official is identical in terms of equilibrium outcomes), and O for outsiders, whose information sets include only their own participation in the game.

Given the set of histories H_t up to time t, the information set of player j at time t is Λ_t,j. Denote by A_t.i the set of available actions for player j at time t, by a_t,j ∈ A_t,j the action chosen by player j at time t, and by a _{(t − 1)I ,j} or a _{(t − 1)O ,j} the action of j in the previous time period in which an I player has been matched, and the corresponding notion for O players, respectively. If the history h is such that h ∈ Λ_{t ,j}, then player j does not know whether they are at history h or some other $$$h^{\prime}\, \in \,{{\rLambda }_{t,j}}$$$ . The information sets of the two kinds of players, assumed to be common knowledge, therefore are:

$$no{ \qquad\qquad\qquad\quad{{\rLambda }_{t,i}}\, = \, {\rm{\{ }}{{h}_t}|{{a}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i}} \in {{h}_t},{{A}_{\tau, o}} \in {{h}_t},{{a}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,g}} \in {{h}_t},{{A}_{\:{\rm{(}}\tau \:{\rm{)}}O,g}} \in {{h}_t}, \cr \qquad\forall \,\tau \in \overline{{0,t}}, \:\:\forall i \in I,\:\:\forall \,o \in O{\rm{\} }} $$

$${{\rLambda }_{t,o}}\, = \,{\rm{\{ }}{{h}_t}\,|\,{{A}_{\tau, i}} \in {{h}_t},\,{{A}_{\tau, o}} \in {{h}_t},{{A}_{\tau, g}} \in {{h}_t},\,\forall \,\tau \in \overline{{0,t}}, \,\forall i \in I,\forall o \in O{\rm{\} }}{\rm{.}}$$

The matching process is given by random draws from the mass of citizens, and the distribution of α parameters in the mass of citizens is given by a continuous random variable with cumulative distribution function F:

$$\alpha \, \sim \,{{F}_\alpha }{\rm{(}}\alpha {\rm{)}}{\rm{.}}$$

The repeated nature of the game opens up the possibility of cooperation sustained by community enforcement.Footnote ³ In the following, an equilibrium that fulfills Condition 1 will be derived, and shown to be the only one that satisfies it while also satisfying Condition 2. The basic intuition of community enforcement of ‘cooperative’ equilibria is that the players who are matched after a certain interaction has taken place use punishment strategies against players who have deviated from the prescribed equilibrium behavior in that interaction. The matching of I or O players is stochastic, with the simple law that the probability of an i match in every stage is p and the probability of an o match is 1 − p. The parameter p is a measure of the extent of secrecy that exists in society with respect to corrupt transactions: a higher p means less secrecy, as more people are in the group of insiders. In the extreme case in which p = 1, everyone has full information about corruption, meaning that there are no constraints on revealing corrupt transactions.

Denote by M_{t ,i} the event “an insider player has been matched with the government official at stage t,” and by M_{t ,o} the event that “an outsider player has been matched with the government official at stage t.” Also denote by i′ all I players that are not i. Given this set-up, the following statement can be shown to be true (proof in online Appendix 1):

Proposition 1: Assume $$${{\alpha }_i}\,\geq \,\frac{{c{\rm{(}}1\, - \,\delta \,{\rm{(1}}\, - \,p{\rm{))}}}}{{\delta p}}$$$ , ∀i. The following strategies generate a cooperative subgame perfect equilibrium of the repeated game, in periods when the government official is matched with players from I:

∀τ < t:

$${{\sigma }_{t,i}}\, = \,\left\{ {\matrix{ {{{R}_{t,i}}\, = \,} \hfill &amp;#x0026; {c\frac{{{\rm{1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{)}}}}{{\delta p}}\:\;if\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\,\geq \,c\frac{{{\rm{1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p)}}{{\delta p}}\:\;and\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, = \,{\rm{1}}} \right)} \hfill \cr {{{R}_{t,i}}\, = \,} \hfill &amp;#x0026; {\; \; \; \; {\rm{0}}\:\;if\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, \lt \,c\frac{{{\rm{1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{)}}}}{{\delta p}}\:\;or\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, \ne \,G} \right)} \hfill \\\end} \right$$

∀τ < t:

$${{\sigma }_{t,g}}\, = \,\left\{ {\matrix{ {{{G}_{t,i}}} \hfill { = \,{\rm{1}}\:\;if\;\:\left( {{{M}_{t,i}}\:\;and\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\,\geq \,c\frac{{{\rm{1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{)}}}}{{\delta p}}\:\;and\;\:{{R}_{t,i}}\,\geq \,c\frac{{{\rm{1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{)}}}}{{\delta p}}\:\;and\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, = \,{\rm{1}}} \right)} \right)} \hfill \cr {{{G}_{t,i}}} {\, = \,0\:\:\:\;if\;\:\left( {{{M}_{t,i}}\:\;and\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, \lt \,c\frac{{{\rm{1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{)}}}}{{\delta p}}\:\;or\;\:{{R}_{t,i}}\, \lt \,c\frac{{{\rm{1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{)}}}}{{\delta p}}\:\;or\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, \ne \,{\rm{1}}} \right)} \right)} \hfill \cr {{{G}_{t,o}}} {\, = \,0\:\:\:\;if\;\:{{M}_{t,o}}}. \hfill \\\end} \right$$

$$\forall \,\tau \, \lt \,t\,{{\sigma }_{t,o}}\, = \,\left( {{{R}_{t,o}}\, = \,0} \right).$$

This condition $$${{\alpha }_i}\,\geq \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{))}}}}{{\delta p}}$$$ states that the government good is worth enough that sacrificing some income to secure it is worth it for player i. If this condition holds for any α_i, the cooperative equilibrium always exists and features the provision of G in every matching with an insider player, the payment of a bribe R_i, and the provision of no G (and the payment of no bribes) when interacting with outsiders. The assumption of no credit constraints means that α_i (and therefore the rent paid) can be greater than w if needed (or alternatively, that α_i ≤ w, ∀i). Note that an equivalent equilibrium is generated if insider players observe all previous matches with other insider players: similar to the equilibrium presented above, they would condition the start of the punishment phase on observing one defection from cooperative behavior.

The equilibrium derived above is Pareto optimal, as the maximal level of utility is generated at each stage. The utilities in each period are w + α_j − R_j for the citizen and R_j − c for the government, so total utility is w + α_j − c, which is the maximal amount of utility available in the game, as long as c < α_j. Among the allocations that are Pareto optimal, the one generated by the equilibrium is the only one that is also subgame perfect: the citizens would not give up more income at each stage for the government, as this would not be subgame perfect for them, and the government would similarly not give up any of its utility, as this would violate subgame perfection.

Multiple Citizens per Period

The model above omits an important feature of real-life corrupt transactions: the fact that the G good might be rival at any given time t, that is, that more than one person would derive utility from getting it from the official. This situation will generate multiple equilibria, so the question that arises is how to select among the multiple possible equilibria. Applying Condition 1 will restrict the possible equilibria to a class of equilibria that shares most of the properties of the one for single matches, and further requirements will suggest a natural candidate for a focal point among this class of equilibria.

Consider a modified version of the selection process for customers for G at time t: now instead of drawing one citizen, insider or outsider, from the distribution F_α (α), draw n_i insiders, and n_o outsiders from F_α (α). The parameters n_i and n_o are themselves random:

$$no{ \qquad\qquad\qquad\qquad\qquad\qquad \ \qquad\qquad&amp;#x0026; {{n}_i}\, \sim \,{{F}_{{{N}_i}}}{\rm{(}}{{n}_i}{\rm{)}} \cr &amp;#x0026; {{n}_o}\, \sim \,{{F}_{{{N}_o}}}{\rm{(}}{{n}_o}{\rm{)}} $$

The random variable F_n is generic, but it is useful to impose the limitation that n_i > 0 or n_o > 0 for simplicity of computation. That is, stages in which no customer is matched are ignored, and there is at least one potential customer in each stage. Also maintained is the assumption that the customers are drawn from a continuum. So the government now faces a ‘demand function’ in each period made up of n_i + n_o potential customers, ranked by α (their valuation of the government good) and can distribute G to only one of them. It is useful to denote by $$$\alpha _{{t,i}}^{{{\rm{(}}r{\rm{)}}}} $$$ , r = 1, 2, … the ranking of the α parameters of the insiders drawn at time t, with $$$\alpha _{{t,i}}^{{{\rm{(1)}}}} $$$ being the highest, and so on. It is also useful to denote by q the probability that the draws from the two distributions generate a match with at least one inside player: q = Pr(n_i > 0). Further, denote by C_t,j the event “player j has been chosen by the government as partner at stage t,” and by $$${{{\rm M}}_t}$$$ the set of all citizens that have been matched at time t. Also denote by M_{t ,i} the event “at least one insider player has been matched at time t” (which has probability q) and by M_t,o the event “only outsider players have been matched at time t” (probability 1 – q).

Condition 1 eliminates all equilibria in which G is not allocated to someone who is part of the demand at each time t. Therefore, only consider equilibria in which someone is always the recipient of the good, from among those who have been matched at time t. The surviving equilibria are all similar to the one in the ‘multiple citizens per period’ section, in that the government selects an insider citizen from among those forming the demand and offers G in exchange for a rent; deviations from that behavior lead to a trigger punishment strategy for all players. One equilibrium, however, has a number of attractive properties that suggest it is the most natural focal point: an equilibrium in which the government asks for bids from the insider citizens who have been matched each period, and selects the one with the highest ability to pay the rent. This equilibrium has four attractive properties. First, it is the most attractive for the government among those that satisfy all the other conditions, as it generates the largest bribe. In as much as the government's announced strategy can be a focal point, this will be the outcome of the game. Second, it is strictly better in expectation than the non-cooperative equilibrium, for every citizen, as every citizen has a positive probability of being selected as a partner, as everyone has a positive probability of ending up in a single match with the government (though it is not the only equilibrium that has this property). Third, it maximizes total welfare in the group made up of g and I, as the citizen who wins the good is the one with the highest valuation of it. Fourth, it satisfies Condition 2, which requires that citizens do not overpay for G. The formal description of this equilibrium follows here, and the proof is provided in online Appendix 1:

∀τ < t:

$${{\sigma }_{t,i}}\, = \,\left\{ {\matrix{ {{{R}_{t,i}}\, = \,} {\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;if\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\,\geq \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;and\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, = \,G\:\;and\;\:{{C}_{t,i}}\:\;and\;\:{{n}_{t,i}}\, = \,{\rm{1}}} \right)} \hfill \cr {{{R}_{t,i}}\, = \,} {\; \; \alpha _{{t,i}}^{{{\rm{(2)}}}} \:\;if\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\,\geq \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;and\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, = \,G\:\;and\;\:{{C}_{t,i}}\:\;and\;\:{{n}_{t,i}}\, \gt \,{\rm{1}}} \right)} \hfill \cr {{{R}_{t,i}}\, = \,} {\; \; \; \; {\rm{0}}\:\;if\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, \lt \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;or\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, \ne \,G\:\;or\;\:\neg {{C}_{t,i}}} \right)} \hfill } \right$$

∀τ < t

$${{\sigma }_{t,g}}\, = \,\left\{ {\matrix {{j\:\;s.t.\;\:{{C}_{t,j}}} {\, = \,\arg \,{{{\mathop {\max }\limits_j }}^V} _{{\!\!\!\!\!\!\!\!\!\!COOP}}^{g} {\rm{(}}{{C}_{t,j}}{\rm{)}},\:\;s.t. \ \ j \in {{{\rm M}}_t}} \hfill \cr {{{G}_{t,i}}} {\, = \,G\:\;if\;\:\left( {{{n}_{t,i}}\, \gt \,0\:\;and\;\:{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\,\geq \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;and\;\:{{R}_{t,i}}\,\geq \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;and\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, = \,G} \right)} \hfill \cr {{{G}_{t,i}}} {\, = \,{\rm{0}}\:\:\:\;if\;\:\left( {{{n}_{t,i}}\, \gt \,0\:\;and\;\:\left( {{{R}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i}}\, \lt \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;or\;\: {{R}_{t,i}}\, \lt \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(}}1\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}\:\;or\;\:{{G}_{{\rm{(}}t\,{\rm{ - }}\,{\rm{1)}}I,i&#x0027;}}\, \ne \,G} \right)} \right)} \hfill \cr {{{G}_{t,o}}} {\, = \,{\rm{0}}} \hfill } \right $$

$$\forall \,\tau \, \lt \,t\:\:\:{{\sigma }_{t,o}}\, = \,\left( {{{R}_{t,o}}\, = \,0} \right)$$

The player with the highest valuation of G at time t wins the contest for G by paying (slightly more than) the maximum that the second player is willing to pay. The assumption that this rent payment is enough to keep the government in the cooperative equilibrium is similar to the one above: $$${{\alpha }^{{\rm{(2)}}}} \,\geq \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}$$$ . The major difference between this equilibrium and the one described in the previous case is that the distribution of gains between the government and citizens is different in the case of multiple matches. The presence of competition between bribe payers leads to the government capturing most of the rents from such interactions.

Spot Transactions

One type of interaction not covered by the model above is that of a spot transaction, in which there are no commitment problems on the part of the government. As has been argued, these kinds of transactions are not likely to be characteristic of corrupt interactions in most cases, but they are included here for completeness. In the spot transaction case, the insider-outsider effect disappears, and the government awards the good to the citizen who pays the highest price in each stage. That is, the government official's problem is to maximize R_j − c subject to the fact that R_j will be received only if c is also paid. This leads the government official to select the j player who can pay the highest R_j. That citizen will maximize their own utility w + α_jG_j − R_j knowing that G_j = 1 only if the payment R_j is the maximal offer received by the government official. This leads the citizen with the highest valuation of the good to pay a bribe worth $$$R_{j}^{\ast} \, = \,\alpha _{j}^{{{\rm{(2)}}}} $$$ if more than one player has been matched, and $$$R_{j}^{\ast} \, = \,c$$$ if they are the only match. Therefore, the insider or outsider who has the highest α gets the G good, and pays a rent worth the maximal rent that the player with the second-highest α is willing to pay. So the spot transaction case retains the result that the citizen with the highest private utility wins the good, but does not retain the distinction between insiders and outsiders. However, these kinds of transactions are likely to be rare (or altogether non-existent) in practice.

Repeated Interaction

The case of repeated interactions between the official and either one citizen or multiple citizens leads to the possibility of achieving cooperative (corrupt) outcomes just like in the community-enforcement case. Because the community-enforcement and repeated-interaction models are largely analogous, the presentation in this section is abbreviated. The factor that determines whether a given citizen will be able to participate in such corrupt interactions is in this case the frequency of interactions with the corrupt official, corresponding to the discount rate. In the case of a single player being repeatedly matched with a corrupt official (for example, repeated government contracts being awarded by the same official to potentially the same businessperson), the two players can achieve a cooperative equilibrium in which the optimal outcome is achieved, just like in the community-enforcement case, if the discount rate is high enough. Therefore in this case, the likelihood that each citizen will be matched with the official in the future, for any kind of future interaction, determines whether such equilibria can exist. The equilibrium derivation for the case of one citizen being matched with one government official for repeated interactions will be analogous to the one in the ‘one citizen per period’ section; the condition for equilibrium existence between an official and citizen i is $$${{\delta }_i}\,\geq \,\frac{c}{{{{\alpha }_i}}}$$$ , where δ_i is the discount rate faced by both government official and citizen i in their repeated interaction. (This is just the condition for equilibrium existence from the ‘one citizen per period’ section, $$${{\alpha }_i}\,\geq \,\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta {\rm{(1}}\,{\rm{ - }}\,p{\rm{))}}}}{{\delta p}}$$$ , where p = 1, because the next match is a player who knows the history of the game, and terms have been rearranged to express the condition as a function of the discount rate.) Therefore, citizens can now be split into two groups: those for whom the discount rate δ_i meets this condition, and those for whom it does not. The gains/losses for the two groups can be analyzed in a manner that is identical to the one for the two insider and outsider groups. Therefore the discussion in the next section, which refers to relative gains and losses from corruption, can be applied to the frequent-rare interaction distinction just as it can be applied to the insider-outsider distinction. Moreover, in practice, the group of insiders and the group of frequent interactors will probably have very significant overlap: those who have lots of information about the hidden behavior of official j are probably, to a large extent, the same as those who will potentially interact with official j frequently.

The conclusion is similar for cases in which more than one player desires the good in each stage. Consider a situation similar to the single-player one, where in each period t set of n players competes for a rival good. Assume that the n citizens are split between a group H of citizens (who are matched at each stage with the official) and a group L of citizens (who meet infrequently with the official). The H players will have a discount rate δ_H > 0, and the low-frequency players will have a low discount rate, approximated by zero for simplicity, δ_L = 0. The most efficient equilibrium for the government official is to select the H player with the highest valuation of the good, who can pay a price equal to the second-highest valuation, just like in the non-repeated version. Now all that has changed is that the punishment in case of defection will come from that same player rather than from other future matches.

The conclusion of the repeated interaction model is very similar to that of the single-interaction model: only citizens who meet government officials frequently can participate in corrupt exchanges, and therefore any gains to be had from corruption can only be shared among them. The results of the distributive effects of corruption over groups to be presented in the rest of the article can be understood as applying equally to the insider-outsider distinction and the frequent-infrequent interaction distinction, especially since in practice the two sets of categories overlap significantly.

Extortion/Forced Consumption

The case of meetings between citizens and government officials in which nothing valuable to the citizen is transferred to them is not covered by the previous cases. In this case, the interaction between the official and the citizen is characterized by the possibility that the official will impose a cost k on the citizen unless a payment R, R ≤ k, is made by the citizen, corresponding to extortion. For such transactions to happen, the official must be able to commit to leaving the citizen alone once the payment has been made. Assuming that this happens, therefore assuming a spot transaction, the equilibrium outcome is that in each unlucky match, the citizen will have to pay R = k as a rent to the official. Keeping the assumption of random matching, and denoting by p_e the probability of being matched with an official who extorts the citizen in each period, all citizens will incur a cost p_ek per period in the corrupt regime (extortion is assumed to not take place in the non-corrupt regime). The higher p_e is, the less attractive corruption is for any citizen. However, in societies in which such practices are rare, the other distributive effects of corruption would take precedence in citizens’ calculations.

Equilibrium Determination of α

Thus far, the α indicating how useful the government good is for each of the citizens has been kept exogenous. But there are many cases in which agents react to knowing that G is for sale by modifying their behavior in ways that maximize the usefulness of this fact. An obvious example is a “get out of jail” card. The knowledge that this product can be bought on the open market will make an agent commit the crime that is associated with the card, knowing that he or she can avoid the negative consequences of jail by buying the card. Assuming the equilibrium presented in the ‘multiple citizens per period’ section, if more than one customer is matched with the government, the one with the highest ability to maximize α will have an advantage over the others.

To analyze the nature of this maximization, and the effects it creates for utility distribution, consider the maximization problem if G is not for sale. Time subscripts are not included, as they would be redundant here. In this non-corrupt case, in each match between citizens and an official, G is distributed according to social utility. The major reason why utilitarian social and private utility would differ is the existence of externalities: citizen j maximizing his or her utility might induce an externality e on other citizens. So the identity that connects social and private utilities is assumed to be:

$${{\alpha }_j}\, = \,\alpha _{j}^{B} \, + \,{{e}_j}.$$

The parameters $$$\alpha _{j}^{B} $$$ and e_j arise in equilibrium, once player j knows that he or she will be the one to receive G. The externality e_j is assumed to always be positive for the citizen whose utility function it enters directly, and negative for all other citizens (it is a negative externality). The parameter α ^B indicates the ‘baseline’ utility of distributing good G for citizen j—that is, j's private utility net of any externalities. For many of the goods that are subject to corrupt transactions, this baseline utility is zero: there is no benefit to having a get-out-of-jail card, or a pass to extract rents from citizens, that is not also a cost for someone else. In these cases, all of the gains for j are captured by the externality term.

Whether citizen j is the one receiving the good is determined by the nature of the regime: non-corrupt or corrupt. In the non-corrupt system, the good is distributed to the citizen with the highest baseline utility for it, as long as he or she does not impose externalities. That is, the j* who gets the good G is determined by:

$$\forall j \, \, s.t. \, \, \, {\rm{(}}{{e}_j}\, = \,0 \, \;and\; \, \alpha _{j}^{B} \, \gt \,0{\rm{)}} \, \, \, \, {{j}^ \ast } \, = \,\arg \,\mathop {\max }\limits_j \alpha _{j}^{B} .$$

Implicit in this definition is that if the baseline utility of anyone receiving the good is not strictly positive, then the good is not distributed at all, $$${{j}^ \ast } \, = \,\varnothing $$$ . This simple social welfare function combines the two basic principles of utilitarian individual utility maximization and Pareto optimality. The citizen's problem is therefore to set the optimal level of (potential) $$$\alpha _{j}^{B} $$$ and e_j, knowing that they could be selected by the government for the allocation of G based on the results of this optimization. The maximization problem of the citizen in the non-corrupt system is:

$$no{\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ \ \qquad &amp;#x0026; \mathop {\max }\limits_{{{\alpha }_j},{{e}_j}} \alpha _{j}^{B} \, + \,{{e}_j} \cr &amp;#x0026; \, s.t. \, \, \, \, e\, = \,{\rm{0}}, $$

which yields the obvious solution (max α ^B)_j, e_j = 0, where (max α ^B)_j stands for the highest α ^B that citizen j can achieve. The externality e_j is already set to 0 in the maximization problem, since it is a given that any other level of e_j will not maximize the citizen's utility, as he or she will not receive G.

In the corrupt system, G is always distributed to the citizen with the highest α_j, that is, the one with the highest private utility for the good. The citizen who gets the good is therefore:

$${{i}^ \ast } \, = \,\arg \,\mathop {\max }\limits_i \alpha _{i}^{B} \, + \,{{e}_i}.$$

So the players who manage to make e_i higher have an advantage over those who do not. Also, if $$$\alpha _{i}^{B} $$$ is zero—that is, there is no private utility for the good that is not also an externality—all that matters is the citizen's ability to impose externalities: the citizens who are most skilled at this get the government good.

Distributive Effects

This section formally discusses what should intuitively emerge from the sections above: the presence of corruption, when compared to the non-corrupt system in which goods are not allocated based on willingness to pay and private utility, distributes income toward insiders (those with high capability to impose externalities) and toward the government, while reducing total output. The analysis first concentrates on the cases in the ‘one citizen per period’ and ‘multiple citizens per period’ sections above (single matches and multiple matches in one stage), as the spot transaction and repeated game situations are very relevant (the former) or make predictions that are analogous to the ones for non-repeated interaction (the latter). The equilibrium assumed for the case of multiple matches is the one described in the ‘multiple citizens per period’ section.

Denote by $$${{\rDelta }_j}{\rm{(}} \cdot {\rm{)}}\,{\rm{ = }}\,U_{j}^{{CORR}} {\rm{(}} \cdot {\rm{)}}\,{\rm{ - }}\,U_{j}^{{REF}} {\rm{(}} \cdot {\rm{)}}$$$ the net gain from corruption when compared to reform (REF). The relevant gains and losses are: Δ_i (α_i, e_i), the gain for insiders given fixed α and e; Δ_o (α_o, e_o), the gain for outsiders; Δ_j (e), the gain as a function of an agent's e; Δ_g, the gain for the government; and $$$\rDelta \left( {\mathop{\sum}\nolimits_{I,O,g} U} \right)$$$ , the total gain in utility for the society. It is useful to denote by $$$\bar{e}$$$ the per capita level of the externality for players who are not the ones imposing it. This term will enter their utilities with a negative sign.

The change in utility for outsiders of the corrupt system is given by:

$${{\rDelta }_o}{\rm{(}}{{\alpha }_o},{{e}_o}{\rm{)}}\, = \,\left\{ {\matrix{ {{\rm{ - }}\bar{e}\,{\rm{ - }}\,\alpha _{o}^{B} } &amp;#x0026; { \, \;if\; \, C_{o}^{{REF}} } \cr {{\rm{ - }}\bar{e}\,{\rm{ - }}\,0} &amp;#x0026; { \, \;if\; \, \neg C_{o}^{{REF}} }. \right$$

In the corrupt system, outsider players are never chosen to receive G, but have to suffer the expected value of the externality imposed by the winner. In the reformed system, the outsiders can either be allocated the G (which gives them the baseline utility $$$\alpha _{o}^{B} $$$ ) or not win it, while not having an externality imposed on them, and get 0. So whatever the case, outsider players lose from the corrupt system.

Denote by R_MIN either $$$\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,p{\rm{))}}}}{{\delta p}}$$$ or $$$\frac{{c{\rm{(1}}\,{\rm{ - }}\,\delta \,{\rm{(1}}\,{\rm{ - }}\,q{\rm{))}}}}{{\delta q}}$$$ , depending on whether the analysis is performed on the model with single matches or on the model with multiple matches. R_MIN is the minimal payment made to the government at each stage when matched alone. The change in utility for insider players is given by:

$${{\rDelta }_i}{\rm{(}}{{\alpha }_i},{{e}_i}{\rm{)}}\, = \,\left\{ {\matrix{ {\left\{ {\matrix{ {{{\alpha }_i}\,{\rm{ - }}\,\alpha _{i}^{{(2)}} } &amp;#x0026; { \, \;if\; \, {{n}_{i,t}}\, \gt \,1} \cr {{{\alpha }_i}\,{\rm{ - }}\,{{R}_{MIN}}} &amp;#x0026; { \, \;if\; \, {{n}_{i,t}}\, = \,1} \\\end} \right} \hfill &amp;#x0026; {\;and\;if\; \, {\rm{(}}C_{i}^{{CORR}} \, \;and\; \, \neg C_{i}^{{REF}} {\rm{)}} \right} \right \hfill }\hfill\cr \left \{ {\matrix{ {{{\alpha }_i}\,{\rm{ - }}\,\alpha _{i}^{{(2)}} \,{\rm{ - }}\,\alpha _{i}^{B} } &amp;#x0026; { \, \;if\; \, {{n}_{i,t}}\, \gt \,1} \hfill\cr {{{\alpha }_i}\,{\rm{ - }}\,{{R}_{MIN}}\,{\rm{ - }}\,\alpha _{i}^{B} } &amp;#x0026; { \, \;if\; \, {{n}_{i,t}}\, = \,1} \\\end} \right} \hfill &amp;#x0026; \;and\;if\; \, {\rm{(}}C_{i}^{{CORR}} \, \;and\; \, {{C}^{REF}} {\rm{)}} \right \right \hfill \cr \left \hskip -7pc \matrix{ {{\rm{ - }}{{{\bar{e}}}_i}\,{\rm{ - }}\,\alpha _{i}^{B} } \hfill \qquad{if\; \, {\rm{(}}\neg C_{i}^{{CORR}} \, \;and\; \, C_{i}^{{REF}} {\rm{)}}} \hfill \cr {{\rm{ - }}{{{\bar{e}}}_i}\,{\rm{ - }}\,0} \hfill \ \, \qquad {\;if\; \, {\rm{(}}\neg C_{i}^{{CORR}} \, \;and\; \, \neg C_{i}^{{REF}} {\rm{)}}} \hfill \\\end} \hfill \right$$

The cases in which n_i,t > 1 are obviously only relevant if allowing multiple matches. Insider players gain from the corrupt system if they are chosen as recipients of G in this system but not in the reformed system (first case). If they are chosen in both systems, the effect is ambiguous. On the one hand, they have to pay the bribe $$$\alpha _{i}^{{{\rm{(2)}}}} $$$ or R_MIN for getting the good in the corrupt system, and no bribe in the reformed system. On the other hand, the utility from the good in the corrupt system, α_i, can be greater than the utility in the reformed system, $$$\alpha _{i}^{B} $$$ , because the former might include the externality e_i. So the insider player gains from the corrupt system in this subcase if:

$$\left\{ {\matrix{ {{{\alpha }_i}\,{\rm{ - }}\,\alpha _{i}^{{{\rm{(2)}}}} \,{\rm{ - }}\,\alpha _{i}^{B} \, \gt \,{\rm{0}}\, \Leftrightarrow \,{{e}_i}\, \gt \,\alpha _{i}^{{(2)}} } &amp;#x0026; { \, \;if\; \, {{n}_{i,t}}\,{\rm{ \gt }}\,{\rm{1}}} \cr {{{\alpha }_i}\,{\rm{ - }}\,{{R}_{MIN}}\,{\rm{ - }}\,\alpha _{i}^{B} \, \gt \,{\rm{0}}\, \Leftrightarrow \,{{e}_i}\, \gt \,{{R}_{MIN}}} &amp;#x0026; { \, \;if\; \, {{n}_{i,t}}\,{\rm{ = }}\,{\rm{1}}.} \\\end} \right$$

That is, an insider player who is very skilled at imposing externalities can benefit from the corrupt system, even if he or she would have been awarded G in the bribe-free reformed system, if the externality he or she can impose on other players is higher than the effective value of the bribe.

The third subcase—when the insider player is not awarded G in the corrupt system, but would be awarded G in the reformed system—can arise when the insider player has a high baseline utility for the good, but less of an ability to impose externalities than other I players. In this case, this player loses from the corrupt system. If the insider player would not be awarded G in either the reformed or the corrupt system, he or she loses from the corrupt system because of the externality that is imposed in such a system.

The conclusion of the first two analyses of changes in utilities is that outsider players always lose from the corrupt system, while insider players can often gain from it. In any given stage, the insider player who wins G under corruption either clearly gains from the corrupt system (if they would not have received G in the reformed system) or gains conditional on having a large e_i (or low R_MIN or low $$$\alpha _{i}^{{{\rm{(2)}}}} $$$ ). As different I players will be drawn in different stages, all of them have a chance to benefit from corruption at some point. Also notable would be a realistic case in which the externalities $$$\bar{e}$$$ are imposed on a large mass of citizens that is either outside the I group, or outside the I and O group altogether. In this case, the $$$\bar{e}$$$ terms would be very small for an individual I player, so over time their negative effect in stages where that I player is not chosen to receive G would be small, while in the stages where he or she is chosen to receive G the gains would be large. In the case of an equilibrium in which the citizen who gets G pays something else than α ⁽²⁾ (in the multiple-matches case), the intuition of the results is not changed: insider players can often benefit from corruption, while outsider players never benefit.

The effect of corruption given e_j is that those with a high capacity to impose externalities have to gain from corruption if they are also insiders, as they are both more likely to win G and to benefit from it once they have gained it. (In the case of multiple matches, this assumes the equilibrium presented in the ‘multiple citizens per period’ section). The change in utility from corruption is given in the following expression, in which the underset notation indicates whether e has a positive (+), negative (−) or neutral (0) effect on the given quantity:

$${{\rDelta }_i}{\rm{(}}e{\rm{)}}\,{\rm{ = }}\,{\mathop{{E{\rm{[}}C_{i}^{{CORR}} &#x007C;{{e}_i}{\rm{]}}}}\limits_{{{\rm{( + }}/0{\rm{)}}}}} {\mathop{{{{U}^{CORR}} {\rm{(}}{{e}_i}{\rm{)}}}}\limits_{{{\rm{( + )}}}}} \,{\rm{ - }}\,{\mathop{{E{\rm{[}}C_{i}^{{REF}} {\rm{]}}}}\limits_{{{\rm{(0)}}}}} {\mathop{{{{U}^{REF}} {\rm{(}}e{\rm{)}}\,}}\limits_{{{\rm{(0)}}}}} {\rm{ - }}\,{\mathop{{E{\rm{[}}\neg C_{i}^{{CORR}} &#x007C;{{e}_i}{\rm{]}}}}\limits_{{{\rm{( - /0)}}}}} {\mathop{{{\rm{(}}{{{\bar{e}}}_i}{\rm{)}}}}\limits_{{{\rm{(0)}}}}} .$$

This quantity is affected by e_i in the following way. The second term is not affected by e at all. In the first term, the expectation of winning G is increased by e (when competing against other I players), as well as the utility of winning it (always). In the third term, the loss from externalities from others is not affected by I's own e_i, but the probability of not being chosen is negatively affected by e_i. So overall, e_i has a positive effect on an insider player's utility from corruption; that is, the corrupt system favors those with a greater ability to generate externalities. For an outsider player, e_o has no effect on their gain or loss from corruption, as they either do not get the good (in a corrupt system), or get it in an environment in which e_o does not matter (in a reformed system). If the rule for choosing partners in the case of multiple matches is different, then, while U ^(CORR) (e_i) is still positively affected by e, it may be that $$$E{\rm{[}}C_{i}^{{CORR}} &#x007C;{{e}_i}{\rm{]}}$$$ has no natural connection with e_i if the rule for choosing partners is not based on choosing the one with the highest α.

For the government, the effect of corruption is either neutral or positive in each period:

$${{\rDelta }_g}\, = \,\left\{ {\matrix{ {{{R}_{MIN}}\,{\rm{ - }}\,c\,{\rm{ - }}\,{\rm{0}}} &amp;#x0026; { \, \;if\; \, {{n}_i}\, = \,{\rm{1}}} \cr {\alpha _{i}^{{{\rm{(2)}}}} \,{\rm{ - }}\,c\,{\rm{ - }}\,{\rm{0}}} &amp;#x0026; { \, \;if\; \, {{n}_i}\, \gt \,{\rm{1}}} \cr {{\rm{0}}\,{\rm{ - }}\,{\rm{0}}} &amp;#x0026; { \, \;if\; \, {{n}_i}\, = \,{\rm{0}}} \\\end} \right$$

In the first two cases, the effect is positive, as R_MIN > c from its formula, and $$$\alpha _{i}^{{{\rm{(2)}}}} \,\geq \,{{R}_{MIN}}\, \Leftrightarrow \,\alpha _{i}^{{{\rm{(2)}}}} \,\geq \,c$$$ by assumption. So the overall effect of the corrupt system on the government is positive, even if the rent that emerges in the case of multiple matches is not $$$\alpha _{i}^{{{\rm{(2)}}}} $$$ , as the rent paid always has to be large enough to cover the cost c for the government.

To compute the overall effect on total utility of the three kinds of agents, first note that externalities and rent payments can be eliminated from the calculation, as they are just transfers from one group to another. Also notable is the fact that in a corrupt system, many transactions do not happen at all: if the only match is one or more outsider players, G is not allocated at all, and it does not generate utility for anyone even if it would have positive social utility if it were allocated. Another reason that corruption reduces total utility is that even when G is allocated in a corrupt system, it is less likely to be allocated to the individual with the highest baseline utility for it. In a corrupt system, in each stage, a citizen with baseline utility $$$\alpha _{i}^{{B \ast }} $$$ gets the good. In a reformed system, in each stage a citizen with baseline utility $$$\alpha _{j}^{{B \ast }} $$$ gets the good. There are two reasons why $$$\alpha _{i}^{{B \ast }} $$$ might be lower than $$$\alpha _{j}^{{B \ast }} $$$ , and no case in which it is higher. First, as j can be either i or o, the pool from which the highest-α ^Bj player is drawn is larger than the pool from which the highest-α i player is drawn. Second, as distribution in the corrupt system is based on α_i, not $$$\alpha _{i}^{B} $$$ , in general, the insider player with the highest $$$\alpha _{i}^{B} $$$ will not be chosen. That is, there are social losses under corruption because the government goods are awarded only to insider players and are awarded based on private utility, not social utility:

$${{\rDelta }_{i,o,g}}\, = \,\left\{ {\matrix{ {{\rm{ - }}\alpha _{{j \ast }}^{B} } \hfill &amp;#x0026; { \, \;if\; \, {{M}_{t,o}}} \hfill \cr {\alpha _{{i \ast }}^{B} \,{\rm{ - }}\,c\,{\rm{ - }}\,\alpha _{{j \ast }}^{B} } \hfill &amp;#x0026; { \, \;if\; \, {{M}_{t,i}}} \hfill \\\end} \right$$

$${ &amp;#x0026; \, where \ i \ast \, = \,{\rm{\{ }}i&#x007C;{{\alpha }_i}\,\geq \,{{\alpha }_{i&#x0027;}}\forall i&#x0027; \in I\backslash {\rm{\{ }}i{\rm{\} \} }} \cr &amp;#x0026; j \ast \, = \,{\rm{\{ }}j&#x007C;\alpha _{j}^{B} \,\geq \,\alpha _{j}^{B} \forall j&#x0027; \in {\rm{(}}I\, + \,O{\rm{)}}\backslash {\rm{\{ }}i{\rm{\} \} }}{\rm{.}} \cr}$$

In periods when only an outsider player is matched, the net social effect is negative or zero: no transaction takes place under corruption, whereas G would be allocated to the citizen with the highest baseline valuation in a non-corrupt system. If some insider players have been matched, the net effect is given by the difference between the baseline utility of the citizen receiving the good in a corrupt system (minus the cost of providing the good) and the baseline utility of the person receiving the good in a non-corrupt system. For reasonable parameter values, this will be highly negative in the case of goods that have any social utility in the first place. The major component of the difference is $$$\alpha _{{i \ast }}^{B} \,{\rm{ - }}\,\alpha _{{j \ast }}^{B} $$$ , and $$$\alpha _{{i \ast }}^{B} $$$ will generally be smaller than $$$\alpha _{{j \ast }}^{B} $$$ . If the winner of the good in the multiple matches case is not the one with the highest α, then the effect can only be more negative, as the utility of whoever gets the good can only be lower.

A corollary to Expression 22 is that the amount of secrecy regarding corruption (which is given by the size of the I group, parameterized as the probability p) affects total social welfare. Keeping everything else constant, societies in which secrecy is greater suffer a greater loss, as more agents are part of the outside group with which transactions cannot be made.

In the case of market structures that feature repeated interactions with the same player(s), the model predicts that the only ones who are able to engage in corrupt transactions are those with frequent/long-term meetings with the official, as reflected in their discount rates for these interactions. Denoting by H the group with discount rates higher than the threshold for sustaining cooperation presented in the ‘repeated interaction’ section, and by L the group of citizens that does not meet this threshold (considering first the case of one-on-one matches between government officials and citizens), the net per-period effect of the corrupt system for the L group is equal to the net effect for the outsider group O defined in the single-interaction case: $$${{\rDelta }_l}{\rm{(}}{{\alpha }_l},{{e}_l}{\rm{)}}\,{\rm{ = }}\,{{\rDelta }_o}{\rm{(}}{{\alpha }_o},{{e}_o}{\rm{)}}$$$ . This is because, just as in the case of outsiders in the single-interaction model, these citizens cannot engage in corrupt transactions. For the H group, which can engage in corrupt transactions, again the per-period effect on utility is the same compared with the non-corrupt baseline as for the I group: $$${{\rDelta }_h}{\rm{(}}{{\alpha }_h},{{e}_h}{\rm{)}}\,{\rm{ = }}\,{{\rDelta }_i}{\rm{(}}{{\alpha }_i},{{e}_i}{\rm{)}}$$$ , replacing R_MIN with $$$\frac{c}{\delta }$$$ , given that now, effectively, p = 1. The argument is similar for the rival goods, multiple-customers case: H citizens can potentially gain from corruption, whereas L players will lose by an argument analogous to the single-match case.

The possibility of interactions in which the government official simply extorts citizens, described in the ‘extortion/forced consumption’ section, qualifies the statements about redistributive effects above. In each unlucky match with a government official who extorts citizens, a simple transfer is worth k. As stated in the ‘extortion/forced consumption’ section, if citizens in any of the situations analyzed above have chance p_e of being part of such an interaction, then any net effect of corruption would take the form $$${{\rDelta }_{NET}}\, = \,{\rm{(1}}\,{\rm{ - }}\,{{p}_k}{\rm{)}}\rDelta \,{\rm{ - }}\,{{p}_k}k$$$ , where Δ stands for any of the differences presented in the previous paragraphs. The term −p_kk uniformly reduces every citizen's utility from corruption, and therefore the higher p_k is, the more muted the distinctions between citizens as a result of non-extortionary corrupt interactions are.

The magnitude of the net social effect under reasonable parameter values if there is some social utility associated with the good is worth exploring. The nature of the distribution F _α primarily determines it, and a realistic model for α will lead to high social losses from corruption. F _α is the distribution of baseline utilities for the G good in the population. G goods that generate any kind of social utility, such as permits to engage in regular economic activity, will generally require very specific complementarities between the good and the consumer in order for the good to provide the optimal utility. For example, there are very few agents in a given country (maybe even in the world) who are very competent at building roads, operating retail operations, providing military equipment, etc. That is, the distribution F _α is likely to be severely skewed toward zero, and very long-tailed. The first source of inefficiency is that limiting draws from this distribution to only insider players is likely to miss the few high-α ^B players. Moreover, even after taking the draws only from the insider group, the one insider who gets chosen is not the one with the highest baseline utility, but the one with the highest value of the sum of baseline value and externality. Since imposing externalities is itself a skill, it can also be modeled as being distributed as a long-tailed distribution, which is either independent or positively correlated with the baseline α ^B distribution. A Monte Carlo simulation is useful for illustrating this logic. Take the case in which there are multiple matches of insider players and model the distribution of α ^B and e as log-Normal:

$$no{ \qquad \qquad \qquad \qquad \qquad \qquad \ \ \qquad &amp;#x0026; {{\alpha }^B} \, : \,\ln N{\rm{(}}{{\mu }_{{{\alpha }^B} }},\sigma _{{{{\alpha }^B} }}^{2} {\rm{)}} \cr &amp;#x0026; e\, : \,\ln N{\rm{(}}{{\mu }_e},\sigma _{e}^{2} {\rm{)}}{\rm{.}} $$

Set n_i and n_o to be the expected value of their distributions, which are generic:

$$no{ \qquad \qquad \qquad \qquad \qquad \qquad \ \ \qquad \qquad &amp;#x0026; {{n}_i}\, \equiv \,E{{f}_{{{n}_i}}}\, \cr &amp;#x0026; {{n}_o}\, \equiv \,E{{f}_{{{n}_o}}}. $$

Also set c = 5. For the simulation, take N = 100,000 repeated draws from the distribution above, at different values of the relevant parameters, and compute the expected value of the social loss $$$E\left[ {\alpha _{{i \ast }}^{B} \,{\rm{ - }}\,c\,{\rm{ - }}\,\alpha _{{j \ast }}^{B} } \right]$$$ . Results are presented in Table 1. The social gain is always negative. A small I group relative to the O group, indicating greater secrecy, means a larger loss. The size of the externalities here does not matter, as they are not correlated with α ^B. High variation in the skill α ^B induces a larger loss, because getting the difference between the right agent for the job and the one chosen through the government's optimization problem will increase as the differences in skill in the population increase.

Table 1

The Net Social Effect of Corruption: Simulations