a model of bureaucracy and corruptionhomes.chass.utoronto.ca/~shouyong/shi1/ier04.pdf · a model of...

INTERNATIONAL ECONOMIC REVIEWVol. 45, No. 3, August 2004

A MODEL OF BUREAUCRACY AND CORRUPTION∗

BY SHOUYONG SHI AND TED TEMZELIDES1

Department of Economics, University of Toronto,Toronto, Ontario, Canada; Department of Economics,

University of Pittsburgh, Pittsburgh, PA, USA

We analyze bureaucracy and corruption in a market with decentralized ex-change and “lemons.” Exchange is modeled as a sequence of bilateral, randommatches. Agents have private information about the quality of goods they pro-duce and can supplement trade with socially inefficient bribes. Bureaucracy ismodeled as a group of agents who enjoy centralized production and consump-tion. Transaction patterns between the bureaucracy and the private sector arefully endogenous. Centralized production and consumption in the bureaucracygive rise to low power incentives for the individual bureaucrats. As a result, pri-vate agents might bribe bureaucrats, whereas they do not bribe each other. Anequilibrium with corruption and an equilibrium without corruption can coexist.We discuss some welfare implications of the model.

1. INTRODUCTION

The seminal work of Becker (1968) has stimulated a vast literature that analyzesbureaucracy and corruption as economic phenomena (see Rose-Ackerman, 1999,for references). This literature models a bureaucracy as a provider of public goodsand/or a rent-seeking organization, which controls some aspects of production,distribution, or inspection of goods and services. In this article, we deviate fromthis specification of a bureaucracy as we do not assume that it is the sole provider ofsome goods and services.2 Instead, we model a bureaucracy as a group of agentswho are subject to centralized production and consumption of goods, and who

∗ Manuscript received August 2003; revised January 2004.1 We thank Joydeep Bhattacharya for suggesting some of the references. Yi Jin provided research

assistance. We thank the participants of the BEPP workshop at Indiana University, the Universityof Pittsburgh, and the Society for Economic Dynamics meeting (Costa Rica, 2000). We also thankSteve Williamson, an Editor and two anonymous referees for some very constructive comments. Shiwould like to acknowledge the financial support from the Bank of Canada Fellowship and from theSocial Sciences and Humanities Research Council of Canada. The opinion expressed herein is the au-thors’ own and does not represent the view of the Bank of Canada. Please address correspondence to:Shouyong Shi, Department of Economics, University of Toronto, 150 St. George Street, Toronto, On-tario, Canada, M5S 3G7. Phone: 416-978-4978. Fax 416-978-6713. E-mail: [email protected].

2 An exogenously imposed interaction between a bureaucracy and the private sector seems to usto be conducive to corruption. In addition, an exogenous role of a bureaucracy is not always realistic.For example, a bureaucracy may provide public housing, rationed goods, or product safety certificates,but close substitutes for these goods and services can also be obtained in private markets. In orderto understand why corruption may arise within a benevolent bureaucracy, we need a model in whichboth the role of the bureaucracy and the trading patterns between a (possibly corrupt) bureaucracyand the private sector are endogenous.

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874 SHI AND TEMZELIDES

have access to the same technologies as the private sector in the production andmonitoring of product quality. We then ask: (i) Can bribery and corruption arise?(ii) Given the possibility of corruption, can the bureaucracy improve social welfaredespite its lack of any exogenous advantage?

Our analysis starts with an economy without a bureaucracy. Decentralized ex-changes are modeled as a sequence of bilateral, random matches. There are alarge number of infinitely-lived private agents. Each agent can produce either ahigh-quality storable good that yields no utility to himself but positive utility toother agents, or a low-quality storable good that yields no utility to anyone. Asin Akerlof (1971), a lemons problem arises since agents do not always know thequality of other agents’ goods. Upon meeting a trading partner, an agent recog-nizes the quality of his partner’s good with some probability. Agents can chooseto supplement the trade of goods with services that cost more to produce than theutility of consuming them. We call such services bribes. A trade occurs only if thetwo agents both agree to trade their goods, and if they ask for a bribe that is nohigher than what the partner offers. Under restrictions on the size and divisibilityof the bribe, we find that private agents in the economy do not bribe each other.

We then introduce a bureaucracy that produces the same set of goods andservices as the private sector. In contrast to private agents, however, bureaucratsproduce the goods collectively, and every bureaucrat consumes the same amountof goods in each period. To focus on the bureaucrats’ trading (as opposed toproduction) decisions, we assume in the benchmark model that the bureaucracyalways produces high-quality goods, and then discuss the implications of droppingthis assumption in an extension. The bureaucracy instructs its members to obeythe following trading rules: (i) not to accept or give bribes, and (ii) to trade with aprivate agent if and only if the bureaucrat can recognize the private agent’s good asa high quality one. However, a bureaucrat can choose to deviate from these tradingrules. Corruption refers to an exchange between a private agent and a bureaucratin which the private agent offers, and the bureaucrat accepts, a bribe. Individualbureaucrats’ decisions on whether to accept bribes affect social welfare, bothbecause bribes involve exchange of socially inefficient services and because theyaffect private agents’ decisions whether to produce high-quality goods. However,individual bureaucrats ignore such effects. In this sense the bureaucracy createsan externality.

The monitoring technology is imperfect. The bureaucracy can detect, only withsome probability, whether a bureaucrat has accepted bribes. When corruption isdetected, the bureaucrat is expelled from the bureaucracy forever. We are in-terested in whether a bureaucracy will be corrupt and if so, whether a corruptbureaucracy can be welfare improving.

The following results emerge from our framework. First, there is an equilibriumin which bureaucrats are corrupt. Since bribes are not exchanged between privateagents, this demonstrates that bureaucrats are more susceptible to bribery thanprivate agents. It should be emphasized that this is not so because the bureaucracyhas a monopoly in the goods or services that it offers but, rather, because pro-duction and consumption for bureaucrats are centralized. Since each bureaucratbears a small fraction of the production cost and consumption depends little on

BUREAUCRACY AND CORRUPTION 875

the bureaucrat’s own trading outcome, it is tempting for a bureaucrat to acceptbribes. Thus, our main finding is that while the bureaucracy may create a positiveexternality, which cannot be easily reproduced by the private sector, this samefeature makes bureaucrats susceptible to bribery since the externality gives riseto a low power incentive scheme.

Second, an equilibrium without corruption can coexist with an equilibrium withcorruption. We find this result interesting for the following reason. It is common toattribute the different levels of corruption among countries to different monitoringtechnologies and punishment schemes. Our model suggests that two economieswith identical fundamentals may have very different levels of corruption sincecorruption deterrence depends on beliefs as well as on fundamentals. If agentsbelieve that bureaucrats accept bribes, the bureaucracy is inefficient, and so the lossfrom being expelled from the bureaucracy is small. In this case, bureaucrats indeedaccept bribes. However, if agents believe that bureaucrats do not accept bribes,the bureaucracy is efficient, and the benefit from belonging to the bureaucracyis large enough to deter corruption. The two equilibria differ in welfare and thesize of the bureaucracy. The equilibrium without corruption Pareto dominates theequilibrium with corruption; thus, corruption reduces welfare.3

Third, when the bureaucracy is small and agents are patient, an increase in thesize of the bureaucracy increases the welfare of both private agents and bureau-crats. Thus, if a bureaucracy can commit to producing only high-quality goods, aneconomy with a corrupt bureaucracy can still be better off than an economy with-out a bureaucracy. As the size of the bureaucracy increases, the public provisionof high-quality goods crowds out private provision, but not one for one. The totalsupply of high-quality goods in the economy increases, leading to higher welfare.In addition, centralized production and consumption can make bureaucrats moreinflexible in trade; for example, they find it easier to refuse to trade when a privateagent’s good is of unknown quality. This inflexibility improves welfare by reducingthe number of lemons in the economy.

Finally, in the presence of corruption, an improvement in the ability to recognizethe quality of goods does not necessarily improve welfare. As this ability improves,the size of the bureaucracy falls, partly because the improvement enables thebureaucracy to catch and expel corrupt bureaucrats with a higher probability. Asa result, the total supply of high-quality goods in the market, as well as welfare, mayfall. For an improvement in the monitoring technology to increase welfare, eitherthe economy must be in an equilibrium without corruption, or the improvementmust be sufficiently large.

The literature on corruption is too voluminous to be surveyed here (see Rose-Ackerman, 1999, for references). Some features or results of our model exist inthis literature. For example, the role of market failure is emphasized by Rose-Ackerman (1978) and Lui (1985) in a queuing model and by Cadot (1987) ina model with asymmetric information. Also, multiple equilibria exist when thelevel of punishment on corruption is endogenous (Lui, 1986; Cadot, 1987), or

3 Interestingly, although corruption does speed up exchange in our model, it speeds up only sociallyinefficient exchange, thus encouraging the production of lemons. This is in contrast to Lui (1985).


when the returns to rent-seeking activities do not dissipate as quickly as those toregular production activities (Murphy et al., 1993). In contrast to these models,our article develops a theoretical foundation for corrupt exchange by determin-ing the exchange patterns endogenously. This allows us to demonstrate clearlyhow corruption arises from the same frictions that make the Walrasian marketunattainable, without resorting to the assumption that the bureaucracy controlsthe production, distribution, or inspection of certain goods/services. Another in-novation of our model is that it permits a coherent welfare analysis because ittakes into account how the exchange patterns may respond to policies.4

As an illustration of these differences, we contrast the welfare role of a bu-reaucracy in our article with that in Acemoglu and Verdier (2000). Assuming thatthe bureaucracy controls the inspection of project quality, Acemoglu and Verdiershow that a bureaucracy can increase welfare by forcing private agents to adopthigh-quality projects. In our model, the bureaucracy does not have such capacity;instead, it affects private agents’ behavior only indirectly through its nonnegligi-ble size in the market. To put it differently, the bureaucrats in our model improvewelfare by cleaning up their own act rather than forcing private agents to cleanup theirs as in Acemoglu and Verdier’s model.

To model decentralized exchange, we draw from the search monetary theorypioneered by Kiyotaki and Wright (1989). In this sense, the economy without abureaucracy generalizes the work of Williamson and Wright (1994) and Wang(2000), who introduce a lemons problem into search monetary models but do notallow for bribery. Whereas these authors proceed to study how money can reducethe severity of the lemons problem, we retain the nonmonetary economy andstudy the role of a bureaucracy. Our approach of modeling bribery as an exchangeof socially inefficient services follows from Engineer and Shi (1998). Finally, wemodel a bureaucracy as a nonnegligible group of agents who have centralizedconsumption and production as in Li and Wright (1998). Li and Wright assumethat the government agents always follow the government’s trading rules. Weabandon this assumption in order to analyze corruption.

Before describing the model, we wish to emphasize the following point. In whatfollows, the bureaucracy should be interpreted as a benevolent government orga-nization and not as a large private coalition of agents. Given that the bureaucracyin our model does not control any aspect of the economy, and since it producesthe same goods/services as the private sector, one may be tempted to interpretit as a large private firm (possibly a monopoly) and argue that such a firm canimprove social welfare by internalizing the externality that we associate with thebureaucracy. Our model does not lead to this interpretation. In particular, the pro-duction and the trading rules that our bureaucracy follows are aimed to improvesocial welfare, rather than to maximize private profits. Although a benevolentgovernment will find it optimal to adopt these rules, the rules are in general notthe best responses of a private firm that maximizes its own present value. Thus,

4 The welfare analyses in existing models are subject to the following version of the Lucas critique.The bureaucracy’s policies may change transaction patterns and/or the extent that the bureaucracycontrols certain elements of the economy. In this sense, assuming that these elements are fixed makespolicy evaluation unreliable. A model in which transaction patterns are endogenous provides a firststep toward addressing this criticism.


a bureaucracy that adopts such rules could not easily be interpreted as a privatefirm. We do not take a stand on the issue of whether any bureaucracy is trulybenevolent in reality. We believe that it is interesting to study why corruption canexist in a benevolent bureaucracy and whether such a bureaucracy can improvewelfare. This focus also explains why in most of the article we assume that thebureaucracy always produces high-quality goods.

2. AN ECONOMY WITHOUT BUREAUCRACY

In this section, we describe an economy without a bureaucracy and show that,under certain conditions, no trade involves the use of inefficient services (bribes).We will analyze an economy with a bureaucracy under the same conditions inSections 3 and 4.

2.1. The Environment. Time is discrete. Agents live forever and discount thefuture with a discount factor β ∈ (0, 1). Each agent can produce a perfectly storableand indivisible good, which has no consumption value for the agent himself, butcan yield utility to every other agent. The good can be either of high quality (H)or low quality (L, lemons). Producing a high-quality good costs c > 0 in utilityand yields utility u > c to the consumer. Producing a low-quality good involves nocost and yields no utility. Each agent chooses the probability according to whichhe produces a high-quality good.

Matching is random and bilateral. There is no public record-keeping, so thereis no credit or any other contractual arrangement. As in Williamson and Wright(1994), a lack of double coincidence of wants problem arises since two agents in ameeting might hold goods of differential quality. The quality of a good is privateinformation. With probability θ ∈ (0, 1), nature determines whether an agent candiscern the quality of his trading partner’s good.

Each agent can also produce nonstorable services, which can be consumed byall agents. Unlike the goods, services are socially inefficient to produce: The utilityto a consumer from each unit of service is 1 but the unit cost of providing suchservices is b> 1. Thus, an agent will never produce services for himself. For reasonsthat will become clear later, we refer to an exchange of services as “bribery.” Tosimplify the analysis, we restrict attention to the case where the amount of bribesis either 0 or B > 0.5

ASSUMPTION 1. The parameters satisfy the following conditions:

ub

> B >u − βcb − 1

(1)

b >uβc

(>1)(2)

5 If services are divisible, the set of equilibria is much larger, but this does not add additional insightsfor the main issues studied here. For the same reason, we assume away lotteries in what follows. As inEngineer and Shi (1998), who study a monetary model without private information regarding goods’quality, the cost function of service production can be extended to b(q), with b′(0) > 1 and b′′ > 0.


The first part of (1) bounds B from above, ensuring that a producer of a low-quality good is willing to offer a bribe, and the second part bounds B from belowso that a producer of a high-quality good is unwilling to offer a bribe even whenhe is offered a bribe. Condition (2) requires that there be a nonempty interval ofB that satisfies (1).

Next, we describe the exchange process. At the beginning of a period, each agentwho does not hold a good chooses the quality of good to produce. After production,agents are randomly matched in pairs and nature determines, with probability θ ,whether an agent observes the quality of the partner’s good. The realization ofthis random event is private information, so an agent does not know whether hispartner recognizes the quality of his good. The two agents then simultaneouslyannounce three decisions: whether to trade the good, the quantity of services tooffer, and the quantity of services to ask from his partner. A trade occurs if andonly if both agents choose to trade the good, and the quantity of services askedby each agent does not exceed the quantity offered by his partner. If a tradetakes place, the two agents immediately produce, swap the durable goods, andprovide the quantity of services asked for by the partner. They then depart andconsume. If a trade does not take place, the two agents depart immediately. Thistrading mechanism keeps the analysis tractable and is imposed on the basis of itssimplicity.6

An agent’s trading decision depends on what he holds, indicated by a subscripti ∈ {H, L}, and what he knows about the partner’s good, indicated by a subscriptj ∈ {H, L, U}. The meanings of j = H and j = L are clear, while the case j = U(“uninformed”) refers to the situation where nature does not reveal the qualityof his partner’s good. We refer to an agent with information (i , j) as agent (i , j).Agent (i , j) makes three trading decisions. The first is the probability with whichthe agent is willing to trade the good, which is denoted xij. The other two decisionsare the quantity of services that he requests, denoted by aij, and the quantity ofservices that the agent offers, denoted by gij. Let tij ≡ (xij, gij, aij). The tradingdecisions of an arbitrary agent are denoted by Tij = (Xij, Gij, Aij).7

An agent (i , j) in a match with an agent (i ′, j ′) chooses to trade if and only ifboth decide to trade the good, gi j ≥ Ai ′ j ′ and aij ≤ Gi′j′ . Denote

mi ′ j ′i j =

{1, if gi j ≥ Ai ′j ′ and ai j ≤ Gi ′j ′

0, otherwise

The trade takes place with probability xijmi′j′ij Xi′j′ . The notation corresponding to

m for an arbitrary agent is M. As mentioned earlier, if trade takes place, thequantities of services traded are the asked quantities, aij and Ai′j′ .

Each agent chooses his production and trading decisions taking as givenother agents’ production and trading decisions. We will focus on stationary and

6 Note that we do not allow agents to rebid. Albeit realistic, rebidding would generate a large setof equilibria, while it is not clear how it would contribute to the main issues studied here. Sequentialbargaining causes similar complexity because of the two-sided asymmetric information.

7 Throughout the article, most capital-case variables denote per capita (aggregate) variables thatindividual agents take as given.


symmetric equilibria, those in which the choices are stationary and agents of thesame type make the same choices.

2.2. Value Functions. Let Vi be the value function of an agent holding goodi prior to a match, where i ∈ {H, L}. The value function before the productiondecision is denoted by V0. Clearly, an agent always produces high-quality goodsif VH − c > VL, he always produces low-quality goods if VH − c < VL, and he isindifferent between producing high- and low-quality goods if VH − c = VL.

At the aggregate level, the production decision is in a one-to-one correspon-dence with the fraction of agents in the market who hold high-quality goods.8 Letthis fraction be P. Then, P = 1 iff all agents produce high-quality goods, P = 0iff all agents produce low-quality goods, and P ∈ (0, 1) iff agents are indifferentbetween producing high-quality and low-quality goods. Thus,

P

=1, if VH − c > VL

=0, if VH − c < VL

∈ [0, 1], if VH − c = VL

(3)

This result allows us to express V0 as follows:

V0 = VL + P(VH − VL − c)(4)

To compute the value function Vi, where i ∈ {H, L}, consider an agent whoholds a good of quality i. There are three cases regarding his information abouthis partner’s good, j ∈ {H, L, U}. For each j, the trading decisions are tij =(xij, gij, aij). Let sij denote the agent’s expected surplus from the trade in this case.Then, for i ∈ {H, L}, Vi and tij obey:

(1 − β)Vi = θ P maxtiH

xiHsiH + θ(1 − P) maxti L

xiLsiL + (1 − θ) maxtiU

xiUsiU(5)

The left-hand side of the above equation is the gain from participating in thecurrent exchange relative to holding onto the good until the next period, and theright-hand side is the expected surplus from the trade. When j ∈ H, the agent’sexpected surplus is given as follows:

siH = θ XHi mHiiH [u + aiH − bAHi + β(V0 − Vi )]

+ (1 − θ)XHUmHUiH [u + aiH − bAHU + β(V0 − Vi )]

The first group of terms is the surplus in the case where the partner also recognizesthe quality of the agent’s good, in which case the probability of trade is XHimHi

iH .

8 One can obtain this correspondence by solving the equation that requires the fraction of agentsholding high-quality goods in the market to be stationary. We omit this procedure since it does notcontribute to the analysis.


Trading in this case gives utility u from good consumption, utility aiH from servicesconsumption, and an opportunity to produce in the next period (which has apresent value βV0). The cost of trading comes from the disutility of producingservices, bAHi, as well as the foregone value of the object, βVi. Similarly, the secondgroup of terms is the surplus in the case where the partner does not recognize thequality of the agent’s good. Which of these two cases will occur is unknown at thetime when the agent makes his trading decisions tiH . Thus, these decisions mustbe the same in the two cases.

Similarly, when the agent knows the partner’s good is j = L, his expected surplusis

siL = θ XLi mLii L [ai L − bALi + β(V0 − Vi )]

+ (1 − θ)XLUmLUi L [ai L − bALU + β(V0 − Vi )]

When j = U, the agent does not know the quality of the partner’s good. Inthis case, the agent rationally believes that his partner holds a high-quality goodwith probability P, and a low-quality good with probability 1 − P. Let siUk bethe agent’s expected surplus from the trade in the case where the quality of thepartner’s good is k ∈ {H, L}. Then,

siU = PsiUH + (1 − P)siUL

where

siUH = θ XHi mHiiU [u + aiU − bAHi + β(V0 − Vi )]

+ (1 − θ)XHUmHUiU [u + aiU − bAHU + β(V0 − Vi )]

siUL = θ XLi mLiiU [aiU − bALi + β(V0 − Vi )]

+ (1 − θ)XLUmLUiU [aiU − bALU + β(V0 − Vi )]

2.3. Equilibria. Agents’ beliefs must be consistent with their strategies. If anagent believes that everyone else in the economy produces only high(low)-qualitygoods, then it is rational for the agent to treat a good of unknown quality as ahigh(low)-quality good. Thus, for all i ∈ {H, L},

P = 1 =⇒ tiU = tiH, P = 0 =⇒ tiU = ti L(6)

A symmetric equilibrium consists of the value functions (VH , VL, V0), individu-als’ production decisions, individuals’ trading strategies tij, and the representa-tive agent’s production decision and trading strategies Tij, where i ∈ {H, L} andj ∈ {H, L, U}, such that (i) for given value functions and strategies for the rep-resentative agent, all agents’ production strategies are optimal, and their tradingstrategies solve their maximization problems; (ii) agents’ trading strategies sat-isfy the consistency requirements in (6); and (iii) strategies are symmetric; i.e.,individuals’ strategies are the same as the representative agent’s strategies.


We focus on equilibria that have the following properties. First, there is a positivemeasure of high-quality goods in the market, i.e., P > 0 (if P = 0, autarky is atleast as good as any equilibrium with exchange). Second, some low-quality goodsare exchanged for high-quality goods (i.e., XHU > 0), since the informationalasymmetry in this case generates a clear welfare cost. Finally, when the quality ofthe partner’s good is unknown or observed to be low, an agent trades only if heobtains a strictly positive surplus. This restriction, which can be rationalized bythe existence of an arbitrarily small transaction cost, rules out spurious tradingwhere two low-quality good holders swap their goods.

There are three possible types of symmetric equilibria. In every equilibrium,an agent does not trade with another agent whose quality he observes to be low.Also, two agents always trade if both observe that their partner holds a high-quality good. The three types of equilibria differ in the values of P and XHU . Thefirst type, which we term a type a equilibrium, has every agent produce high-qualitywith probability 1 and always trade even when he does not discern the quality ofhis partner’s good. That is, P = XHU = 1. In the second type of equilibrium, termeda type b equilibrium, P < 1 = XHU . In this equilibrium, even though a high-qualitygood holder does not discern the partner’s quality and knows that there is positiveprobability that the partner’s good is a lemon, he trades anyway. The third typeof equilibrium, termed a type c equilibrium, has P < 1 and XHU < 1. Clearly, in atype b or a type c equilibrium, V0 = VL = VH − c.

We characterize the type a equilibrium in Proposition 1 below (proven in theAppendix) and the other two types of equilibria in Proposition 2 (proven in theAppendix). Notice that the three types of equilibria are the same as those obtainedby Williamson and Wright (1994, Section III) in a similar model that precludesthe exchange of services. Thus, Propositions 1 and 2 below can be viewed asextending their results to allow for the possibility of bribery. Although this ex-tension may seem intuitive, a formal presentation of the results is necessary herefor two reasons. First, the proof of the results is not trivial (see the Appendix),since the possibility of exchanging services significantly expands agents’ strategysets. Second, because the result that no bribery occurs in this economy will becontrasted with the possible occurrence of bribery when a bureaucracy is intro-duced, it is important that this result rests on a formal proof rather than mereintuition.

PROPOSITION 1. There exists a type a (P = 1) equilibrium if and only if θ >

c/u. In this equilibrium, agents do not ask for or offer bribes, and they trade withprobability 1.

The intuition for Proposition 1 is as follows. When it is easy to discern the qualityof goods, in the sense that θ > c/u, agents always produce only H and always tradewith each other. Because anyone who deviates to produce a lemon will be caughtwith high probability and denied trade, a deviation is not profitable.

PROPOSITION 2. Equilibria with 0 < P < 1 and VL > 0 involve no bribe exchange.An agent holding a low-quality good always wants to trade. An agent holding a


high-quality good trades if he discerns the partner’s good as of high quality; he doesnot trade if he discerns the partner’s good as of low quality; and he trades with apositive probability XHU if he cannot discern the quality of his partner’s good. Inthe last case, XHU = 1 in a type b equilibrium, and XHU < 1 in a type c equilibrium.A type b equilibrium exists if and only if c/u < θ < θ2 ≡ u/[β (2u − βc)]. A type cequilibrium exists if and only if θ3 < θ < θ2, where

θ3 ≡ 12

(1β

− 1)

+√(

1β

− 1)2

+ 4(1 − β)cu − βc

The above propositions illustrate two main features of the equilibria in our

model. First, no bribes are exchanged in any equilibrium without a bureaucracy.The precise proof of this result is involved, as shown in the Appendix, but an infor-mal argument is straightforward. First, an agent who holds a high-quality good willnot trade if he discerns that the partner holds a low-quality good, even if his partneroffers a bribe. The surplus from trading away a high-quality good for a lemon plus abribe is B − β (VH − V0) and, because VH − V0 = c in all equilibria with P ∈ (0, 1),this surplus is negative under (1) and (2). Second, an agent who holds a high-quality good will not offer a bribe, regardless of his knowledge of his partner’sgood. Even if the partner’s good is observed to be of high quality, offering a bribeyields the agent a surplus no more than u − bB − βc, which is negative under(1). Thus, only agents who hold low-quality goods may offer bribes. Third, a high-quality good holder does not ask for a bribe if he cannot discern the quality ofhis partner’s good. In such a meeting, a high-quality good holder trades away thegood with positive probability, hoping that the partner holds a high-quality goodas well. But if the partner indeed holds a high-quality good, requesting a bribe isnot optimal as it will surely result in no trade since high-quality good holders neveroffer bribes. Thus, a high-quality good holder never asks for a bribe. Similarly, ifan agent with a low-quality good wants to trade in a match with an agent whoholds a high- or unknown-quality good, the agent should never ask for a bribe.Thus, no trade involves the exchange of bribes.

Another implication of the model is that different equilibria can coexist. Forexample, whenever a type b equilibrium exists, a type a equilibrium exists as well.Beliefs about other agents’ production decisions can be self-fulfilling. If an agentbelieves that other agents will produce high-quality goods, then it is optimal forhim to produce high quality since producing a low-quality good would significantlyreduce his trading opportunities. But if an agent believes that other agents willproduce high-quality goods with a probability strictly less than 1, then it is optimalfor him to do the same since always producing a high-quality good would incura high production cost which would be wasted in trade with partners who hold(unknown) low-quality goods.

In the remainder of this article, we restrict our attention to the parameter regionθ ∈ (0, c/u). This serves two purposes, in addition to simplifying the analysis ofa bureaucracy. First, the restriction captures the severe information asymmetry,which is the type of market failure we wish to focus on. Second, the restriction


will facilitate the comparison between an economy with a bureaucracy and aneconomy without. Under this restriction, the type c equilibrium is the only possibleequilibrium in the economy without a bureaucracy, but there can be multipleequilibria in an economy with a bureaucracy. Thus, we can attribute the multiplicityexclusively to the distinct features of the bureaucracy.

For future use, we list the value functions and XHU in the type c equilibrium;

(1 − β)VH = P[θ + (1 − θ)XHU]2(u − βc) − (1 − θ)βcXHU(1 − P)

(1 − β)VL = (1 − β)(VH − c) = P(1 − θ)uXHU

}(7)

XHU = 11 − θ

[(1 − P)βcP(u − βc)

− θ

](8)

The last equation comes from the condition that VH − VL = c, which must holdfor 0 < P < 1.

3. A BUREAUCRACY OF EXOGENOUS SIZE

In this section, we introduce a bureaucracy but keep the size of the bureau-cracy and the level of punishment on corruption exogenous. We demonstrate thatbribery can exist in this economy and study how corruption depends on the levelof the punishment. This is a preliminary step toward Section 4, in which we endo-genize the size of the bureaucracy and the level of punishment.

The population now consists of two groups, private agents and bureaucrats. Themass of bureaucrats is γ ∈ (0, 1) and that of private agents is 1 − γ . Private agentsare as described in the previous section, and each behaves according to his ownincentive. In contrast, the bureaucracy functions as follows:

� The bureaucrats’ technology is such that they are only able to producegoods as a group and not individually. The bureaucracy centralizes pro-duction and consumption of goods. All bureaucrats jointly produce high-quality goods at the beginning of each period and allocate one unit to eachmember who has traded the good away in the previous period. Individualbureaucrats alone can produce only services. For now, we assume that thebureaucracy can commit to producing only high-quality goods. With re-gards to consumption, all bureaucrats pool their holdings from exchangeat the end of the period, and every bureaucrat consumes the same amountregardless of whether he succeeded in trade.9

� The bureaucracy prescribes the following trading instructions to its mem-bers: (i) not to accept bribes, and (ii) to trade if and only if their tradingpartner is known to hold a high-quality good. A bureaucrat can deviatefrom these trading instructions. If he does so and is caught, he is punished

9 More precisely, the bureaucracy uses a lottery to allocate the goods received from exchange. Everybureaucrat has the same probability of winning one unit of the good.


with an exogenous loss of utility R > 0 but is allowed to stay in the bureau-cracy (in Section 4, such a defector is expelled from the bureaucracy). Likeprivate agents, bureaucrats recognize the quality of a private agent’s goodwith probability θ .

� With probability δ∈ (0, 1), the bureaucracy can detect whether a bureaucrataccepted bribes.

Thus, individual bureaucrats in our model are best thought of not as autonomouseconomic units but, rather, as agents who represent the bureaucracy, with theadded possibility that they might accept bribes. We also remark that although thebureaucracy has the same production technology as the private agents, for nowwe assume that it does have the ability to commit to producing only high-qualitygoods, an ability which the private agents lack.

Centralized production and consumption is meant to capture an important as-pect of actual bureaucracies. That is, the main benefit of belonging to the bureau-cracy comes from all bureaucrats’ collective actions rather than from individualbureaucrats’ actions alone. This feature cuts both ways in a bureaucrat’s deci-sion on whether to deviate from the bureaucracy’s trading instructions. On theone hand, centralized consumption generates the inflexibility typically associatedwith a bureaucracy and, in this case, increases efficiency. Because a bureaucratreceives the same level of consumption of goods regardless of whether he suc-ceeds in trade, he may refuse to trade for an unknown quality good. On the otherhand, bribes may entice a bureaucrat to trade the good away since an individualbureaucrat bears little of the production cost. This is a novel implication of ourmodel that we believe also captures a main aspect of some bureaucracies. Thebureaucracy creates a positive externality, which cannot be easily reproduced bythe private sector, but which, at the same time, makes bureaucrats susceptible tobribery due to a low power incentive scheme.10

The assumption that the bureaucracy commits to producing only high-qualitygoods may not be realistic. We discuss this further in Section 5. Similarly, anactual bureaucracy may not adopt the trading instructions specified above. Wemake these assumptions because we try to analyze the behavior of a benevolentbureaucracy. These production and trading rules maximize social welfare and,hence, can be thought of as the optimal policies of a benevolent bureaucracy. Asemphasized in the introduction, these rules are typically not the best responses ofa private firm that maximizes its present value: Under the maintained restrictionthat θ < c/u, a private firm will choose P < 1 instead and will trade with positiveprobability with a trading partner who holds a good of unknown quality. Therefore,the bureaucracy in our model fits the description of a government bureaucracyand not that of a private firm.

10 One issue is whether the bureaucracy could help achieve “better” allocations, by following adifferent set of rules. For example, we could assume that a collective punishment is triggered, saythe bureaucracy shuts down, if a single bureaucrat is caught cheating. This could potentially eliminatecorruption in our model. Such punishments, however, are not robust to the introduction of an arbitrarilysmall probability of mistakes. In any case, the study of optimal mechanisms in the context of our modelis an interesting issue that is beyond the scope of this article.


To focus on the inefficiency that arises from the bureaucrats’ trading decisions,we make the following three additional assumptions to simplify other aspects ofthe bureaucrats’ behavior. First, bureaucrats must pool the goods received fromthe exchange before they consume. This rules out embezzlement. Second, eachbureaucrat faces the same matching rate as a private agent does. Third, whentwo bureaucrats meet, they simply swap their inventories and, after swapping, thegoods become consumption goods (this assumption will be relaxed in Section 5).The last two assumptions ensure that a bureaucrat is not at a disadvantage in thenumber of possible trades relative to a private agent.

To analyze agents’ decisions, let us use the subscript G to indicate variablesrelated to a bureaucrat. Consider a match between a bureaucrat and a privateagent, where the private agent holds quality i ∈ {H, L} and the bureaucrat hasinformation j ∈ {H, L, U} regarding the private agent’s good. In such a match,denote the bureaucrat’s trading decisions as tGj = (yj, aGj, gGj) and the privateagent’s decisions as tiG = (zi, aiG, giG), where y denotes the probability with whichthe bureaucrat decides to exchange the good, and zi denotes the similar probabilityfor the private agent. As before, (a, g) are the quantities of bribes requested andoffered, respectively. Let miG

Gj indicate whether these quantities are consistent witheach other (m = 1) or not (m = 0). A representative bureaucrat’s decisions in sucha match are denoted as TGj = (Yj, GGj, AGj) and the corresponding indicatorfunction is MiG

Gj. Similarly, a representative private agent’s decisions in such amatch are TiG = (Zi, GiG, AiG) and the corresponding indicator function is MGj

iG.As discussed before, we assume θ < c/u so as to focus on equilibria with 0 <

P < 1 and 0 < XHU < 1. Then V0 = VL = VH − c. The exchanges between twoprivate agents are the same as described in Proposition 2. Adapting (7) to take intoaccount the exchanges with bureaucrats, we have the following value functionsfor private agents:

(1 − β)VL = (1 − γ )(1 − θ)uPXHU + γ maxtLG

zLGsLG

(1 − β)VH = γ maxtHG zHGsHG

+ (1 − γ ){[θ + (1 − θ)XHU]2 P(u − βc) − (1 − P)

× βc(1 − θ)XHU}

Here, siG is the expected surplus of a private agent holding a good i ∈ {H, L},which is given by

siG = θYi mGiiG[u + aiG − bAGi − β(Vi − V0)] + (1 − θ)YUmGU

iG

× [u + aiG − bAGU − β(Vi − V0)]

Notice that we have imposed the consistency requirement that if a private agentmeets a bureaucrat and cannot recognize the quality of the bureaucrat’s good, hemust rationally believe that the quality is high (since bureaucrats can only producehigh-quality goods).


Every bureaucrat obtains net expected utility uG from consuming goods ineach period. This is equal to the total utility from consuming high-quality goodsminus the total production cost in the bureaucracy, divided by the size of thebureaucracy. To compute this utility, let ru be the average rate at which a bureaucratreceives a high-quality good from exchange, and rc the average rate at which ahigh-quality good produced by the bureaucracy is traded away and hence needsto be resupplied. Then,

ru = γ + (1 − γ )[θ PYH ZH MHG

GH + (1 − θ)PYU ZH MHGGU

](9)

rc = 1 − (1 − γ )

[1 − θ PYH ZH MHG

GH − θ(1 − P)YLZLMLGGL

−(1 − θ)YU(PZH MHG

GU + (1 − P)ZLMLGGU

)](10)

uG = ruu − rcc(11)

The expression in brackets in the equation for rc is the probability that the bu-reaucrat does not trade with a private agent. In a match with a private agent wherethe bureaucrat’s information about the private agent’s good is j ∈ {H, L, U}, thebureaucrat’s expected surplus is

sGj = Zj mjGGj (aGj − bAjG − δR), for j = H, L,

sGU = PZHmHGGU (aGU − bAHG − δR) + (1 − P)ZLmLG

GU(aGU − bALG − δR)

The value function for a bureaucrat, VG, is as follows:

(1 − β)VG = uG + (1 − γ )θ P maxtGH

yHsGH + (1 − γ )θ(1 − P) maxtGL

yLsGL

+ (1 − γ )(1 − θ) maxyGU

yUsGU

A notable feature of this value function is that a bureaucrat’s trading decision isdriven by bribes and the value of R. For any given R, the utility from consuming thegoods (uG) does not affect directly a bureaucrat’s decision on whether to accepta bribe. We impose a tie-breaking restriction that the bureaucrat exchanges withprobability 1 in the case where his partner holds a good of known high quality andthe surplus from bribes is 0.

Having specified the general forms of the value functions, we now turn to thestudy of equilibria. Again, we examine only symmetric equilibria where agentsof the same type and with the same information choose the same strategies. Notsurprisingly, the bureaucracy’s trading instructions are consistent with equilibriumif expected punishment to corruption exceeds the benefit of bribery, i.e., if δR> B. The following proposition verifies this intuition (see the Appendix for aproof):


PROPOSITION 3. If δR > B, no bribes are exchanged in equilibrium. Tradesbetween private agents are as described in Proposition 2. A private agent and abureaucrat trade if and only if the private agent holds a high-quality good and thebureaucrat recognizes it.

If the punishment on corruption is not severe enough, however, bribery willoccur in certain matches. We assert this result in the following proposition, whichis proved in the Appendix.

PROPOSITION 4. If R is sufficiently close to zero, then the following trading strate-gies form an equilibrium. Trades between private agents are as described in Pro-position 2. In a match between two bureaucrats, the two swap goods. In a matchbetween a bureaucrat and a private agent, trade occurs except when the privateagent’s good is of high quality and is not recognized by the bureaucrat. Moreover,bribery occurs if and only if the private agent holds a low-quality good. More pre-cisely, the trading decisions are as follows:

AGL = AGU = B, AGH = 0, GGL = GGU = GGH = 0(12)

ALG = AHG = 0, GLG = B, GHG = 0(13)

YL = YU = YH = 1, ZH = ZL = 1(14)

MGUHG = MHG

GU = 0, MHGGH = MGH

HG = 1, MGjLG = MLG

Gj = 1 ( j = L, U)(15)

Equation (12) states that the bureaucrat requests a bribe whenever he is not surewhether his partner’s good is of high quality, but a bureaucrat never offers a bribe.Equation (13) states that a private agent never requests a bribe from a bureaucrat,and that he offers a bribe only when he holds a low-quality good. Equation (14)states that the two agents always decide to trade their goods. However, a trade doesnot always occur because the quantities of services proposed by the two agentsmay be inconsistent. This occurs when the private agent holds a high-quality good,and the bureaucrat does not observe it, as indicated by (15).

In light of Proposition 4, we can reexamine two views expressed in the literature.First, bribes are sometimes deemed “natural” because they are similar to gifts be-tween private agents (Rose-Ackerman, 1999). Under our parameter restrictions,there is a clear distinction between bribes and gifts: Private agents offer inefficientservices as bribes to bureaucrats whereas they do not offer them to other privateagents nor do they receive them from bureaucrats. Second, it is sometimes saidthat bribery improves social welfare because it can speed up exchange. Althoughin our model the equilibrium with corruption involves a higher frequency of tradethan the equilibrium without corruption, it is the socially inefficient exchange thatbribery speeds up. This is because only the producers of lemons are willing tooffer bribes, so corruption encourages the production and exchange of lemons


and socially inefficient services. As a result, corruption unambiguously reduceswelfare.

4. ENDOGENOUS SIZE OF BUREAUCRACY AND PUNISHMENT ON CORRUPTION

In this section, we endogenize the amount of punishment on corruption byassuming that the bureaucracy expels a bureaucrat upon discovering that he iscorrupt. An expelled bureaucrat loses the net utility from current consumption,uG, plus the future benefit from belonging to the bureaucracy. Thus, the amountof punishment on corruption is R = uG + β(VG − V0).11 Because only a fraction(γ − n)/γ of the bureaucrats enjoy consumption goods in the bureaucracy at theend of each period and produce to replenish the goods that were traded awayduring the period, the formula of uG should be modified as follows:

uG = γ

γ − n(ruu − rcc)

The size of the bureaucracy is also endogenous in any equilibrium with corrup-tion because corrupt bureaucrats exit from the bureaucracy once they are caught.In order to obtain a stationary size of the bureaucracy, we introduce exogenousdeaths and births. Assume that each agent dies with probability α at the end ofeach period, and a measure α of new agents are born at the beginning of eachperiod. A fraction σ of the newborns are bureaucrats, and the remaining 1 − σ

are private agents. If the size of the bureaucracy is γ in this period, the size inthe next period is γ ′ = (1 − α)(γ − n) + ασ , where n is the endogenous mass ofcorrupt bureaucrats who are expelled. In a steady state, γ ′ = γ , and so γ = σ −(1 − α)n/α, which is positive iff n < ασ/(1 − α).

Given the probability of death, the effective discount factor is β = (1 − α)β̃,where β̃ now stands for the discount factor used previously. All the Bellmanequations in the previous section are valid after reinterpreting the discount factoras β = (1 − α)β̃. Again we restrict attention to the case where 0 < XHU < 1 and0 < P < 1. In this case, V0 = VL = VH − c.

4.1. Characterization of the Equilibria. Even though the amount of punish-ment on corruption is now endogenous, the agents’ trading strategies still obeyPropositions 3 and 4. The reason is that the amount of punishment, R, is exoge-nous to each individual agent, since the net utility uG and value functions (VG,V0) are treated parametrically by each bureaucrat. As a result, for a given R, eachbureaucrat’s trading behavior is entirely determined by bribes as in the previoussection.

11 Notice that the cost rc in uG in (11) is the future cost of replacing the goods that the bureaucratstrade away in the current period, rather than the cost of producing goods at the beginning of thecurrent period. When a corrupt bureaucrat is expelled, he neither incurs this replacement cost norenjoys the utility of consumption goods. Hence, it is the net utility, uG, that appears in the amount ofpunishment.


An equilibrium may or may not involve bribery. To economize on space, we unifythe characterization of the two equilibria. Let χ = 1 indicate the equilibrium withcorruption and χ = 0 the equilibrium without corruption. Substituting the tradingdecisions described by Propositions 3 and 4 into the value functions, we have

(1 − β)VH = θ(1 − γ )(1 − P)βc + γ θ(u − βc)(16)

(1 − β)VL = (1 − γ )(1 − θ)uPXHU + χγ (u − bB)(17)

(1 − β)VG = uG + χ(1 − γ )(1 − P)(B − δR)(18)

uG = γ

γ − n{(u − c)[γ + (1 − γ )θ ] − (1 − γ )(1 − P)[θ(u − c) + χc]}(19)

Here R = uG + β (VG − VL) and XHU is given by (8). Solving VH − VL = c, weobtain

P = 1 − θ(u − γβc) − (1 − β)c − χγ (u − bB)(1 − γ ) [θ(u − βc) + uβc/(u − βc)]

(20)

Notice that the size of the bureaucracy, γ , and the level of corruption, χ , affectprivate agents’ production decisions in equilibrium. This influence comes from thenonnegligible role of the bureaucracy in the market.

An equilibrium imposes three restrictions. First, a bureaucrat must have noincentive to deviate from the equilibrium strategy. In a match with a private agent,the gain to a bureaucrat from accepting a bribe is (B − δR). In the equilibriumwith corruption, the condition B > δR is required for a bureaucrat to accept abribe when the private agent’s good is known to be of low or unknown quality.In the equilibrium without corruption, the condition B ≤ δR is required for thebureaucrat to refuse a bribe. Substituting R = uG + β (VG − VL) and (VG, VL, uG,P, XHu), we can write the condition B > δR as follows:

(1 − β)B/δ + χβγ (u − bB) >γ

γ − n(u − c) [γ + (1 − γ )θ ] + (1 − γ )θβu

− θ(u − γβc) − (1 − β)c − χγ (u − bB)θ(u − βc) + uβc/(u − βc)

×{

γ

γ − n[θ(u − c) + χc] + βu

(βc

u − βc+ θ

)}(21)

In the equilibrium with corruption, this condition must hold with χ = 1. In theequilibrium without corruption, this condition must be violated with χ = 0.

Second, in both equilibria we require that XHU , P ∈ (0, 1). The restrictions that0 < XHU < 1, which imply 0 < P < 1, are equivalent to

ξ1 < γ < ξ2(22)


where

ξ1 = θ(u − βc)(1 − β − θβ)c + (1 − β)βc2

θ(u − βc)[θ(u − βc) + (1 − θ)βc] − χ(u − bB)[θu + (1 − θ)βc]

ξ2 = uc − θβc(2u − βc)θ(u − βc)2 + (1 − θ)uβc − χu(u − bB)

Third, the mass of bureaucrats must be stationary. In the equilibrium withoutcorruption, no bureaucrat is corrupt and hence no one is expelled from the bu-reaucracy; that is, n = 0 and so γ = σ . In the equilibrium with corruption, eachbureaucrat accepts a bribe when he meets a private agent who holds a low-qualitygood. Thus,

n = γ (1 − γ )(1 − P)δ(23)

Because γ = σ − (1 − α)n/α, the stationary size of γ in this equilibrium satisfies

γ [α + δ(1 − α)(1 − γ )(1 − P)] − ασ = 0(24)

Clearly, 0 < γ < σ . Substituting P from (20) with χ = 1, we can write the aboveequation as a quadratic equation of γ and show that there is a unique admissiblesolution for γ if

σ <θu − (1 − β)cu − bB + θβc

(25)

Moreover, γ → 0 when either σ → 0 or α → 1. Because the bureaucrats’ choice ofaccepting bribes affects the private sector’s production decision, P, it also affectsthe size of the bureaucracy.

An equilibrium with corruption exists if, for χ = 1, (21)–(24) hold together with(1) and (2). An equilibrium without corruption exists if, for χ = 0, (21) is violated,n = 0, γ = σ, and (22) hold together with (1) and (2).

4.2. Existence and Coexistence of Equilibria. In this section, we first establishthe existence and coexistence of the two equilibria analytically for some specialcases. Then we illustrate the existence in a broader parameter region numerically.

PROPOSITION 5. If β → 0, no equilibrium with trade exists. If β is sufficientlyclose to 1, and σ is sufficiently close to 0, an equilibrium with XHU , P ∈ (0, 1) existsin a nonempty parameter region that satisfies 0 < θ < u/(2u − c) and (1) and (2).Moreover, the equilibrium with corruption and the equilibrium without corruptioncoexist.

PROOF. Suppose that the conditions 0 <θ < u/(2u − c), (1) and (2) hold. If β →0, (8) implies XHU < 0, and so there is no equilibrium with trade. For the remainder


of the proposition, set (β, σ ) → (1, 0) so that γ → 0. Then, in both equilibria,

P −→ 1 − θuθ(u − c) + uc/(u − c)

∈ (0, 1)

The condition for XHU ∈ (0, 1), (22) becomes 0 < θ < u/(2u − c) in both equilibria.Set n = 0, γ = σ → 0, χ = 0, and β → 1. Then, (21) becomes B > +∞, whichis clearly violated. That is, an equilibrium without corruption exists. Now, setχ = 1 and consider the equilibrium with corruption. When β → 1 and σ → 0,we have n/γ → δ (1 − P) and γ /σ→ α/[α + (1 − α)δ (1 − P)]. With the abovegiven P, we can verify that the right-hand side of (21) is negative and so (21) isclearly satisfied. That is, the equilibrium with corruption exists. Moreover, the twoequilibria coexist. Since the two equilibria depend on (β, σ ) continuously, theyalso coexist for (β, σ ) sufficiently close to (1, 0). �

To distinguish the two equilibria in the remainder of this article, we use thesubscript n to denote variables in the equilibrium without corruption and b todenote variables in the equilibrium with corruption. The two equilibria do coexistin a broader region, as the following example illustrates.

EXAMPLE 1. Assume δ = θ . Let β = 0.9, σ = 0.4, α = 0.1, u = 1, c = 0.5, andb = 3.

With the parameter values in the above example, Figure 1 depicts the existenceregions of the two equilibria in the subspace (B, θ). The assumptions (1) and (2)are satisfied when 0.275 < B < 0.333. For any B in this range, the equilibriumwith corruption exists when θLb(B) < θ < θHb(B), and the equilibrium withoutcorruption exists when θLn (B) < θ < θHn(B). The two equilibria coexist whenθLb (B) < θ < θHb(B).

We emphasize that the coexistence is driven by the fact that the punishment oncorruption endogenously depends on the level of corruption. When bureaucratsdo not accept bribes, they exchange only when they can observe that their partners’goods are of high quality. In this case, the bureaucracy’s goods are never exchangedfor lemons, the cost of replacing these goods is low, and the resulting payoff tobelonging to the bureaucracy is high. This deters corruption since a bureaucratwould risk losing a high payoff if he accepted a bribe and were caught. On theother hand, if bureaucrats accept bribes, a large fraction of the bureaucracy’s goodsare exchanged for lemons and bribes. In that case, the bureaucracy’s frequencyof consumption of high-quality goods is low, and the cost of replacing goods ishigh. Both factors reduce the payoff to belonging to the bureaucracy and makecorruption more likely.

4.3. Effects of the Size of the Bureaucracy. The equilibria we study are af-fected by the parameter σ , the fraction of newborns who are bureaucrats. Sincean increase in σ results in a larger bureaucracy in both equilibria, we interpretan increase in σ as an exogenous increase in the size of the bureaucracy. This


1

0

θL n B( )

θL b B( )

θH n B( )

θH b B( )

0.3330.275 B0.28 0.29 0.3 0.31 0.32 0.33

0

0.2

0.4

0.6

0.8

1

FIGURE 1

COEXISTENCE OF TWO EQUILIBRIA. THE EQUILIBRIUM WITH CORRUPTION EXISTS IF θLb(B) <

θ < θHb(B), THE EQUILIBRIUM WITHOUT CORRUPTION EXISTS IF θn(B) < θ < θHn(B)

interpretation is exact in the equilibrium without corruption, because in that casethe size of the bureaucracy (γ ) is exactly equal to σ .

We discuss the welfare effects of σ first. A private agent’s ex ante welfare levelis V0, which equals VL (or VH − c) in the two equilibria under discussion. Abureaucrat’s ex ante welfare level is VG. The society’s welfare level is W ≡ σVG +(1 − σ )V0, which is the expected utility of a newborn agent before he realizeswhether he is a private agent or a bureaucrat. We have the following.12

PROPOSITION 6. When (β, σ ) are sufficiently close to (1, 0), the functions(V0, VG, W) are increasing in σ in both equilibria.

In other words, all measures of welfare increase in the size of the bureaucracywhen this size is sufficiently small. The gain to private agents comes primarilyfrom an increased frequency of transactions for private agents who hold high-quality goods. In an economy without a bureaucracy, a private agent holding ahigh-quality good sometimes exchanges it for a low-quality good that yields noutility. For every private agent that is replaced by a bureaucrat (who commits toproducing only high-quality goods), the number of agents producing high-qualitygoods increases by 1 − P. In this case, a private agent with a high-quality good hasa higher chance than before to meet another high-quality good holder, and so his

12 The proof is straightforward and omitted. To determine the signs of various derivatives, noticethat the equilibria exist in the limit case (β, σ ) → (1, 0) only if 0 < θ < u/(2u − c).


expected utility increases. This is the case even in the equilibrium with corruption,because a private agent holding a high-quality good does not offer bribes. Thus,VH increases in both equilibria as σ increases. Because a bureaucrat is a high-quality good producer by assumption, his value function increases for a similarreason. The value function of a low-quality good producer also increases, sinceVL = VH − c. Therefore, a private agent’s ex ante welfare increases.

The following example examines the welfare effects of σ in a larger parameterregion:

EXAMPLE 2. δ = θ = 0.4, β = 0.9, α = 0.1, u = 1, c = 0.5, b = 3, B = 0.322(≡BB).

Under these parameter values the two equilibria coexist for σ ∈ (0, 0.56).Figures 2(a) and 2(b) depict the effects of σ in this region. As Figure 2(a)shows, private agents’ and bureaucrats’ welfare levels increase with σ inboth equilibria. Thus, an increase in σ increases the overall welfare level.As Figure 2(b) shows, the size of the bureaucracy in the equilibrium withcorruption indeed increases with σ . (In the equilibrium without corruption, γ =σ , which obviously increases with σ .) Figure 2(b) also depicts the variable h,which denotes the probability with which a private agent holding a high-qualitygood succeeds in obtaining a high-quality good. This is given by

h = γ θ + (1 − γ )P [θ + (1 − θ)XHU]2

As discussed above (and shown in Figure 2(b)) h increases in σ , which is theprimary reason for why private agents’ welfare increases in σ . In this example,a private agent becomes more likely to exchange a high-quality good for a goodof unknown quality. That is, XHU increases in σ (not shown in the figures). Inaddition, P falls in σ , an effect that will be discussed later.

It is important to supplement the above welfare effect of σ with two remarks.First, the positive welfare result of a larger bureaucracy should not be construedas a general statement that a bureaucracy can always improve welfare. Rather,the result demonstrates that the exogenous advantages with which a bureaucracyis assumed to be endowed in much of the literature are not necessary for thebureaucracy to improve welfare. Because our assumption that a bureaucracy cancommit to producing high-quality goods is not always satisfied in actual economies,one should be careful when mapping our welfare result to actual observations. Wewill discuss this issue further in Section 5.

Second, one may be tempted to conclude that the positive welfare effect in ourmodel is a simple change in the decomposition between bureaucrats and privateagents, with some private agents who produce low-quality goods being replacedby bureaucrats who produce high-quality goods. This interpretation misses an in-teresting feature in Figure 2(b). Namely, the increase in the public provision ofhigh-quality goods, caused by the increase in σ , crowds out the private provision ofhigh-quality goods. Not only does the size of the private sector shrink but also pri-vate agents produce high-quality goods with a lower probability. This probability


3.2

0

V 0n σ BB,( )

V 0b σ BB,( )

V Gn σ BB,( )

V Gb σ BB,( )

0.560 σ0 0.1 0.2 0.3 0.4 0.5

0

1

2

3

a

0.8

0

P n σ BB,( )

P b σ BB,( )

h n σ BB,( )

h b σ BB,( )

γ b σ BB,( )

0.560 σ0 0.1 0.2 0.3 0.4 0.5

0

0.2

0.4

0.6

0.8b

NOTES: σ : fraction of newborns who are bureaucrats; BB = 0.322; V0: private agent’s value function;VG: bureaucrat’s value function; b(n): subscript for the equilibrium with (without) corruption; P: prob.for a private agent to produce high-quality goods; h: prob. a private agent holding good H gets a goodH in exchange; and γ : size of the bureaucracy.

FIGURE 2

(A) DEPENDENCE OF VALUE FUNCTIONS ON σ ; (B) DEPENDENCE OF (P, h, γ ) ON σ

falls because an increase in the size of the bureaucracy benefits a high-qualityholder more than a holder of a low-quality good. In equilibrium, agents must beindifferent between producing high-quality and low-quality goods. Therefore, inorder to restore the mixed-strategy equilibrium, the fraction of low-quality pri-vate agents must increase to offset the excess gain to agents holding high-qualitygoods. Despite such crowding-out, the increase in σ increases the overall frac-tion of high-quality goods in the market. That is, the crowding-out is less thanone-for-one.


4.4. Effects of Reducing the Informational Asymmetry. In our model, the in-formational asymmetry is captured by the probability of discerning the qualityof a good, θ , and the probability of detecting corruption, δ. Here, we investigatenumerically the benchmark case δ = θ first and then turn to the case where δ �= θ .

Consider the parameter values in Example 1. In addition, set B = 0.322 (≡BB),in which case the two equilibria coexist for θ ∈ (0.256, 0.445). We depict theeffects in Figures 3(a) and (b). In the equilibrium without corruption, both privateagents’ and bureaucrats’ welfare increases with θ , and so does the overall welfare

3

0

V 0n θ BB,( )

V 0b θ BB,( )

V Gn θ BB,( )

V Gb θ BB,( )

0.4450.256 θ0.3 0.35 0.4

0

1

2

3a

0.8

0

P n θ BB,( )

P b θ BB,( )

h n θ BB,( )

h b θ BB,( )

γ b θ BB,( )

0.4450.256 θ0.3 0.35 0.4

0

0.2

0.4

0.6

0.8b

NOTES: θ : prob of recognizing good’s quality; BB = 0.322; V0: private agent’s value function; VG:bureaucrat’s value function; b(n): subscript for the equilibrium with (without) corruption; P: prob. fora private agent to produce high-quality goods; h: prob. a private agent holding good H gets a good Hin exchange; and γ : size of the bureaucracy.

FIGURE 3

(A) DEPENDENCE OF VALUE FUNCTIONS ON θ ; (B) DEPENDENCE OF (P, h, γ ) ON θ


level. In the equilibrium with corruption, an increase in θ improves the welfare ofprivate agents but reduces that of bureaucrats. Thus, reducing the informationalasymmetry does not always improve welfare in the equilibrium with corruption.The overall welfare level (not shown in the figures) decreases in θ for θ < 0.35 andincreases in θ for θ > 0.35.

The positive welfare effect of θ on the welfare of private agents is easy tounderstand. In both equilibria, as the ability to discern the quality of a goodimproves, a private agent holding a high-quality good is more likely to succeed intrading his good for a high-quality good rather than for a low-quality good. Thatis, the probability h increases. As a result, the value function of such an agent, VH ,increases. Since VL = VH − c in the two equilibria, the expected utility of agentswho hold low-quality goods also increases. In the equilibrium without corruption,a bureaucrat’s welfare increases in θ because the number of exchanges with privateagents who hold high-quality goods increases.

In the equilibrium with corruption, however, bureaucrats’ welfare decreases asθ increases. This is because the probability of being expelled from the bureaucracyincreases, as reflected by the decreasing size of the bureaucracy in Figure 3(b). Anincrease in θ increases the probability of expulsion for two reasons. The first isdue to a direct effect that relies on the assumption δ = θ in the benchmark case.With δ = θ , a higher θ improves the bureaucracy’s ability to detect corruption andhence increases the probability of expulsion for a corrupt bureaucrat. The secondreason is an equilibrium effect arising from the response of P, which exists evenwithout the assumption δ = θ . Since a higher θ benefits a private agent who holdsa high-quality good more than one holding a low-quality good, an agent remainsindifferent between producing a high-quality and a low-quality good only whenP falls.13 A lower P increases the probability with which a bureaucrat meets aprivate agent who holds a low-quality good and hence increases the frequencyof corruption. As a result, the probability of expulsion from the bureaucracyincreases.

The loss to a bureaucrat from expulsion is positively related to the difference(VG − V0). When θ is small, the difference between a bureaucrat’s and a privateagent’s value function is large, and so an increase in θ has a dominating negativeeffect on overall welfare. When θ is large, the difference (VG − V0) is close tozero. In this case, the negative effect of θ is small and so an increase in θ increasesoverall welfare.

We now disentangle the effect of θ from the effect of the bureaucracy’s moni-toring ability, δ. Fix δ = 0.4, B = 0.322, and let other parameters have the values inExample 1. Then, the two equilibria coexist for θ ∈ (0.303, 0.445). The responsesof the two equilibria to changes in θ are the same as the patterns depicted inFigures 3(a) and (b). In particular, the equilibrium effect through P continues toreduce both the size of the bureaucracy and the welfare of bureaucrats. In contrastto the benchmark case, however, this negative effect is dominated by the positiveeffect from increasing θ on a private agent’s welfare. As a result, overall welfareincreases in θ .

13 This result is similar to the one obtained in a monetary model by Wang (2000).


5. DISCUSSION

In this section, we relax one restriction and two assumptions made in the pre-vious sections. The restriction to be relaxed is that XHU < 1. This will be done bystudying the equilibria with XHU = 1 in the parameter region where the equilibriawith XHU < 1 do not exist. We continue to restrict attention to P < 1. Setting theright-hand side of (8) to be greater than or equal to 1 and substituting (20) for P,we can obtain the parameter region in which XHU = 1.

We have assumed so far that the bureaucracy commits to producing high-qualitygoods. To eliminate the possible impression that this assumption is essential forthe welfare-improving role of the bureaucracy, we now assume instead that thebureaucracy is only able to commit to producing high quality with a probabilityπ ∈ (0, 1]. Because we do not model the bureaucracy’s choices, this probability isexogenously given. The other assumption to be relaxed is that two bureaucrats ina match can trade with each other and turn their goods into consumption goods.In this section, we use an indicator κ = 1 to represent this previous case (whenthis trade is possible) and κ = 0 to represent the opposite case.

In order to implement these changes, we need to take care of two other is-sues. The first issue is how the bureaucracy deals with the low-quality goods itproduces. One way is to assume that the bureaucracy distributes the low-qualitygoods, as well as the high-quality goods, among the bureaucrats. Because a bu-reaucrat who receives a low-quality good might take the good to the market, thiscreates a conflict with the bureaucracy’s rule that no bureaucrat should trade forgoods of low or unknown quality. Instead, we assume that the bureaucracy de-stroys the low-quality goods that it produces and randomly distributes only thehigh-quality goods among the bureaucrats. Only those bureaucrats who receivehigh-quality goods go to the market. Accordingly, we use γ to denote the fractionof agents in the exchange who are bureaucrats, and γ̂ to denote the size of the bu-reaucracy. The equations that describe the equilibria in Section 4 can be modifiedaccordingly.14

The other issue is in regard to the existence of an equilibrium. When κ = 0 and π

is sufficiently low, an equilibrium with a bureaucracy may fail to exist because thevalue of being a bureaucrat (VG) may be lower than the value of being a privateagent (V0). To solve this problem, we introduce a transfer from the private sectorto the bureaucracy. To do so in a way that is consistent with the environment,we assume that each private agent can produce a “transfer” good (in addition tothe good and the service), that yields no utility to other private agents but yieldspositive utility to the bureaucrats. Moreover, a bureaucrat must meet a privateagent in order to collect this transfer good and the transfer is randomly distributedamong the bureaucrats. Let τ be the amount that a bureaucrat is able to collectfrom a private agent whom he meets. The disutility from this amount to the privateagent as well as the utility to a bureaucrat are both given by τ . With this transfer,a term γ τ should be subtracted from the right-hand sides of (16) and (17), and aterm (1 − γ )τ should be added inside the brackets in (19).

14 These modifications are available upon request.


In the following discussion, we use Example 2 as the benchmark. This givesπ = 1, κ = 1, and τ = 0.

First, we allow for XHU = 1. For σ > 0.56, the equilibria with XHU < 1 nolonger exist but the equilibria with XHU = 1 do exist. For such values of σ , thefraction (P) of private agents in the market who hold high-quality goods increasesin σ (rather than the decrease in σ in Figure 2(b)). All other patterns depictedin Figures 2(a) and (b) continue to exist. When σ reaches 0.76, P reaches 1. Forσ > 0.76, no equilibrium with P < 1 exists.

Next, set π = 0.6. In this case, the equilibrium without corruption (and withP < 1) exists for σ ∈ (0.46, 0.86). For small σ , the equilibrium does not existbecause the bureaucracy is not efficient enough to deter corruption. The equilib-rium with corruption exists for all σ ∈ (0, 1), in which the fraction (P) of privateagents who hold high-quality goods ranges from 0.5 to 0.66. The dependence ofthe two equilibria on σ is similar to that depicted in Figures 2(a) and (b). In par-ticular, welfare increases in σ in both equilibria, even in the region where π <

P. Thus, the bureaucracy’s welfare-improving role does not necessarily rest onits superior ability of producing high-quality goods; rather, it relies on the factthat the bureaucracy’s commitment to sending only high-quality goods to themarket and punishing corrupt bureaucrats who are caught affect private agents’behavior.

Finally, we set π = 0.6 and κ = 0. To keep VG > V0, we set the transfer to τ =0.14. The two equilibria coexist for σ ∈ (0, 0.864). Figure 4(a) depicts the valuefunctions of the two sectors of the economy, and Figure 4(b) depicts the overallwelfare level. These figures reveal a number of interesting features. First, in theequilibrium without corruption, XHU < 1 occurs only for σ < 0.685. Thus, the valuefunctions and the welfare function in this equilibrium exhibit a kink at σ = 0.685,beyond which XHU = 1. The equilibrium with corruption does not exhibit such akink. Second, in both equilibria, welfare increases with σ , for small σ . Thus, thewelfare-improving role of the bureaucracy does not depend critically on previousassumptions that κ = 1 or π = 1. Third, the welfare level has a hump-shapeddependence onσ . Therefore, increasing the size of the bureaucracy does not alwaysincrease welfare; rather, it can reduce welfare if σ is large enough. Moreover, inthe current example the hump of the welfare function in the equilibrium withoutcorruption occurs at a level of σ where XHU < 1 and where the two equilibriacoexist. Thus, in general, increasing σ alone cannot eliminate the coexistence ofequilibria.

6. CONCLUSION

We have analyzed the exchange patterns in two economies involving decen-tralized exchange and asymmetric information regarding product quality. In theeconomy without a bureaucracy, agents do not trade socially inefficient bribeswith each other. We introduced bureaucracy as a coalition of agents who en-joy centralized production and consumption and showed that the positive exter-nality created by such a bureaucracy also gives rise to low power incentives forthe individual bureaucrats, which might lead to corruption. We endogenized the


1.7

0

V 0n σ BB,( )

V 0b σ BB,( )

V Gn σ BB,( )

V Gb σ BB,( )

0.8640 σ0 0.2 0.4 0.6 0.8

0

0.5

1

1.5

a

0.7

0

W n σ BB,( )

W b σ BB,( )

0.8640 σ0 0.2 0.4 0.6 0.8

0

0.2

0.4

0.6

b

NOTES: σ : fraction of newborns who are bureaucrats; BB = 0.322; V0: private agent’s value function;VG: bureaucrat’s ex ante value function; b(n): subscript for the equilibrium with (without) corruption;and W: society’s ex ante welfare, weighting VG by σ and V0 by 1 − σ .

FIGURE 4

(A) DEPENDENCE OF VALUE FUNCTIONS ON σ ; (B) DEPENDENCE OF THE SOCIETY’S EX ANTE WELFARE ON σ

punishment on corruption as the loss of the benefit arising from being part ofthe bureaucracy. In the economy with a bureaucracy, there are two self-fulfillingequilibria. In one bureaucrats accept bribes, whereas in the other they do not. Theequilibrium without corruption generates higher welfare and is able to support alarger bureaucracy than the equilibrium with corruption. Despite the existence ofcorruption, a bureaucracy may improve welfare.

Our welfare results should be interpreted with some caution. First, actual bu-reaucracies have many functions other than alleviating market failures arisingfrom private information, although the latter is an important function, and it isthe one emphasized in our article as well as in other articles on bureaucracy (e.g.,Acemoglu and Verdier, 2000). Second, there are features that we abstract fromthat limit the desirability of a large bureaucracy. Introducing such features will


lead to a more realistic model and might overturn some of our welfare conclu-sions. We consider our welfare results to be of value to the extent that they willremain an important part of such a more realistic story.

We believe that the search model with lemons is a rudimentary but appro-priate framework that captures the link between market failure and corruption.This framework endogenously generates trading patterns that distinguish briberyfrom gifts; the former is a corrupt exchange between a private agent and a bureau-crat, while the latter is an exchange between two private agents. In addition, themodel illustrates that corruption is not merely a transfer from private agents tobureaucrats, but rather an exchange that incurs a social deadweight loss. There isa delicate trade-off between the cost of corruption and the potential social benefitfrom a bureaucracy that can commit to producing high-quality goods.

As we mentioned earlier, the bureaucracy in existing models is typically exoge-nous in the sense that, by assumption, it controls certain elements of the exchange.The bureaucracy in our model is not fully endogenous either since we do not modelexplicitly its formation or how it commits to producing high-quality goods. Nor dowe investigate alternative mechanisms that could use incentive/penalty schemesin order to reduce corruption by individual bureaucrats. These exogenous featuresnotwithstanding, we believe that our model makes a step forward by assuming thatthe bureaucracy does not control directly any element of the exchange process.We hope that our analysis will help shift the focus of future research toward afully endogenous bureaucracy. We believe that the feature of a bureaucracy thatwe emphasized in this article, namely, centralized production and consumption,will continue to be important.

Many extensions are possible; we mention two here. The first involves allowingagents to acquire technologies that improve the ability to recognize quality. Wehave already shown that a higher ability is not necessarily welfare-improving,and so there might be a socially desirable level of such ability. Wang (2000) hasanalyzed information acquisition in a search model without corruption. It wouldbe interesting to see whether the presence of corruption changes his results. Asecond extension is to examine corruption at different levels of a hierarchicalbureaucracy. More generally, one can examine how different ways of organizing abureaucracy affect corruption, a topic emphasized by Shleifer and Vishny (1993).

APPENDIX

PROOF OF PROPOSITION 1. First, we establish the following lemma.

LEMMA 1. A symmetric equilibrium with P > 0 has the following properties:(i)XLH = MHL

LH = MHULH = 1; (ii) AiH = B implies XHj = 0 for all i , j ∈ {H, L, U};

(iii) XHL = 0 and ALH = GHU .

PROOF. Part (i) of the lemma states that an L agent who observes his partner’squality as H always wants to trade. The L agent’s surplus is u + aLH − bAHL

+ β(V0 − VL), if his partner knows his quality, and u + aLH − bAHU + β (V0

− VL) if he does not. Even under the worst terms of trade, aLH = 0 and AHj =B, the surpluses in the two types of matches are no less than u − bB, which is


positive under assumption (1). Thus, the agent always wants to trade. Part (ii) ofthe lemma states that an H holder will not trade if his partner requests a bribe.If the H holder trades and offers a bribe, his surplus is u + aHj − bB + β(V0 −VH), where j ∈ {H, L, U} indicates the agent’s information about the quality ofthe partner’s good. Since aHj ≤ B and V0 − VH ≤ − c, the surplus is no more thanu − (b − 1)B − βc, which is negative under assumption (1). For part (iii), consideran agent holding H who observes that his partner holds an L good. If he trades,his surplus is aHL − bALj + β (V0 − VH), where j ∈ {H, U}. Since V0 − VH ≤− c if P > 0, the surplus is no more than B − βc < u/b − βc < 0. Thus, xHL =XHL = 0, as in part (ii). With XHL = 0, the surplus associated with the decisionxLH is

sLH = (1 − θ)XHUmHULH (u + aLH − bAHU + β(V0 − VL))

To maximize this surplus, the choices (aLH , gLH) must ensure that mHULH = 1.

For mHULH = 1, it is necessary that aLH ≤ GHU and gLH ≥ AHU . Under the latter

restrictions, the surplus is maximized by setting aLH = GHU . This completes theproof of Lemma 1. �

Next, we move to the proof of Proposition 1. Since P = 1 in a type a equilibrium,we can use (6) to write (5) for i = H as follows:

(1 − β)VH = maxtHH

xHH XHHmHHHH(u + aHH − bAHH − βc)

Agents do not ask for bribes in this equilibrium; i.e., AHH = 0. If, to the contrary,AHH = B, then xHH = 0 by Lemma 1, and so VH = 0, which violates the requirementVH − VL > c. Similarly, XHH > 0. Given AHH = 0 and GHH ≥ 0, an individual agentearns the maximum surplus, XHH(u + GHH − βc), if he chooses aHH = GHH andgHH ≥ 0. This surplus is positive, and so agents trade with probability 1; i.e., xHH =1. Also, in equilibrium, GHH = aHH = AHH = 0.

To complete the proof we must show that it is optimal for agents to alwaysproduce high quality. Suppose that an agent deviates to producing an L good.With P = 1, the deviator always meets a partner who holds an H good. ByLemma 1, he successfully trades only if the partner does not recognize the qualityof his good, and if aLH ≤ GHU = 0 and gLH ≥ AHU = 0. Thus aLH = 0. Substitutingthese conditions and (6) into (5) for i = Lyields VL = (1 − θ)(u − βc)/[(1 − βθ)(1− β)]. We have VL < VH − c if and only if θ > c/u, in which case the deviation isnot profitable. �

PROOF OF PROPOSITION 2. We establish a series of lemmas first.

LEMMA 2. XLL = 0.

PROOF. Suppose, to the contrary, that XLL > 0. We proceed through the fol-lowing steps, concentrating on a symmetric equilibrium.


Step 1. XLL > 0 only if XLU > 0, ALL = GLU = B, and ALU = GLL = 0. Inaddition, XLL > 0 iff xLL > 0. For the choice xLL > 0 to be optimal, it is necessarythat

0 < sLL = θ XLLmLLLL(aLL − bALL) + (1 − θ)XLUmLU

LL(aLL − bALU)

Since aLL − bALL = (1 − b)ALL ≤ 0, the surplus is strictly positive only if thesecond term is positive, which requires that XLU > 0, aLL = B ≤ GLU , and ALU =0. Thus, in a symmetric equilibrium ALL = GLU = B and ALU = 0. Then, aLL −bALL < 0. If GLL = B, the agent can increase his expected surplus by deviatingto gLL = 0 < ALL , which yields mLL

LL = 0 and neutralizes the potential loss. Thiscannot be an equilibrium, and so GLL = 0.

Step 2. If XLL > 0, ALL = GLU = B, and GLL = ALU = 0, as in Step 1, thenXLU > 0 only if XHU > 0, GHU = 0, and AHU = B.

Let us examine the decision on xLU and the associated surplus sLU . SinceXHL = 0, then

sLU = P(1 − θ)XHUmHULU (u + aLU − bAHU)

+ (1 − P)[θ XLLmLL

LU(aLU − bALL)

+ (1 − θ)XLUmLULU(aLU − bALU)

](A.1)

If the agent chooses (gLU , aLU) = (GLU , ALU) = (B, 0), as in the supposed equi-librium, then mHU

LU = mLLLU = mLU

LU = 1, and sLU = P(1 − θ)XHU(u − bAHU) −(1 − P)θXLLbB. For xLU > 0, it is necessary that sLU > 0, which requires thatXHU > 0. If GHU = B, the surplus in (A.1) can be increased by choosing aLU = Brather than aLU = ALU = 0: The choice aLU = B (>ALL) neutralizes the loss aLU −bALL by inducing mLL

LU = 0, while maintaining mHULU = 1. Similarly, if AHU = 0, by

choosing gLU = 0 (<ALL) rather than gLU = GLU = B, the agent can neutralizethe loss aLU − bALL without affecting other possible trades. Thus, for xLU > 0, wemust have GHU = 0 and AHU = B.

Step 3. If XLL > 0, XLU > 0, ALL = GLU = AHU = B, and GLL = ALU =GHU = 0, as in Step 2, then XHU > 0 only if XHH > 0, AHH = 0, and GHH = B.

With GHU = 0, Lemma 1 implies that ALH = 0. With ALH = AHU = B andALU = 0,

sHU = Pθ XHHmHHHU (u + aHU − bAHH − βc)

+ P(1 − θ)XHUmHUHU(u + aHU − bB− βc)

+ (1 − P)[θ XLHmLH

HU(aHU − βc) + (1 − θ)XLUmLUHU(aHU − βc)

]Under (1), the term multiplied by (1 − P) is negative, and so is the term u + aHU −bB − βc. For xHU > 0, the first term in the above surplus must be positive under the


supposed equilibrium strategy (gHU , aHU) = (0, B). For this we need that XHH >

0 and mHHHU = 1. For mHH

HU = 1, in turn, we need that AHH ≤ gHU = 0 and GHH ≥aHU = B. Thus, AHH = 0 and GHH = B.

Step 4. If XHH > 0, XHU > 0, AHU = B, and (GHH , AHH) = (B, 0), as inStep 3, there is a gain from deviating to gHH = 0.

With AHH = 0 and AHU = B, the expected surplus associated with the choiceof xHH is

θ XHHmHHHH(u + aHH − βc) + (1 − θ)XHUmHU

HH(u + aHH − bB − βc)

Note that XHU(u + aHH − bB − βc) < 0, for any aHH ≥ 0. Since AHU = B, thispotential loss can be neutralized by deviating to gHH = 0, which induces mHU

HH =0. This deviation does not change the first term of the above surplus. Thus, thedeviation is profitable.

Therefore, there cannot be any symmetric equilibrium with XLL > 0. �

LEMMA 3. (i) XHU = 0 implies XLU = 0 and VL = 0; (ii) XHU > 0 impliesXHH > 0.

PROOF. If XHU = 0, the expected surplus associated with the choice of xLU is

(1 − P)(1 − θ)XLUmLULU(aLU − bALU)

Since aLU − bALU is negative for an exchange with ALU > 0, the best the twoagents in such a match can achieve is to swap the L-quality goods and exchangeno services. This yields zero surplus. By our tie-breaking rule, there is no exchange.Thus, xLU = XLU = 0. With XLU = XHU = 0, it is easy to verify that VL = 0. Toprove that XHU > 0 implies XHH > 0, suppose XHU > 0. The expected surplusassociated with the choice xHU is

Pθ XHHmHHHU (u + aHU − bAHH − βc)

+ P(1 − θ)XHUmHUHU(u + aHU − bAHU − βc)

+ (1 − P)[θ XLHmLH

HU(aHU − bALH − βc)

+ (1 − θ)XLUmLUHU(aHU − bALU − βc)

]≤ Pθ XHHmHH

HU (u + aHU − bAHH − βc)

+ P(1 − θ)XHUmHUHU(u + aHU − bAHU − βc)

The inequality follows from the fact that the term multiplied by 1− P is nonpositiveunder (1). For xHU > 0, it is necessary that the above surplus be nonnegative forsome (aHU , gHU). Setting (aHH , gHH) to be equal to those (aHU , gHU), the surplusassociated with the decision xHH is nonnegative, and so xHH > 0. �


LEMMA 4. There is no bribe exchanged in an equilibrium with 0 < P < 1 andVL > 0. In particular, (i) AHU = GHU = AHH = ALH = 0, (ii) XHH = MHH

HH = MHUHH =

MHHHU = MHU

HU = 1, and (iii) ALU = 0, XLU = 1, MHULU = MLU

LU = 1.

PROOF. For part (i), suppose first, to the contrary, that AHH = B. As shown inthe proof of Lemma 3, the expected surplus associated with the decision xHU isless than or equal to

Pθ XHHmHHHU (u + aHU − bAHH − βc) + P(1 − θ)XHUmHU

HU(u + aHU − bAHU − βc)

Since XHH > 0 when XHU > 0, with AHH = B, the first term is nonpositive and isstrictly negative if mHH

HU = 1. For XHU > 0, the second term must be nonnegative,and this can be achieved in a symmetric equilibrium only if aHU = AHU = 0. It isalso necessary that gHU = GHU = 0. This is because u + aHU − bAHH − βc < 0when aHU = 0 and AHH = B; if GHU = B, a deviation to gHU = 0 induces mHH

HU = 0and neutralizes this potential loss without affecting the possible expected surplusfrom trading with an HU agent. However, if GHU = AHU = 0 and AHH = B, wehave mHU

HH = 0, in which case sHH = θXHHmHHHH(u + aHH − bB − βc) < 0. This

induces xHH = 0, contradicting the result XHH > 0. Thus, AHH = 0. Set AHH = 0and suppose, contrary to the lemma, that AHU = B. Then

sHH = θ XHHmHHHH(u + aHH − βc) + (1 − θ)XHUmHU

HH(u + aHH − bB − βc)

This surplus is maximized by setting aHH = GHH . Since aHH = AHH = 0 in anysymmetric equilibrium, GHH = 0. With AHU = B and XHU > 0,

sHU < Pθ XHHmHHHU (u + aHU − βc)

For xHU > 0, the terms (aHU , gHU) must ensure mHHHU = 1, which requires aHU ≤

GHH (=0). This is not possible in a symmetric equilibrium with AHU = B. There-fore, AHU = 0. With AHH = AHU = 0, an H agent who does not observe the qualityof his partner’s good will secure the trade with a partner holding an H good nomatter whether he chooses gHU = 0 or B, provided that aHU = 0. If GHU = B, how-ever, Lemma 1 implies that ALH = GHU = B, and the choice gHU = B would invitea trade with an L partner. That is, the term (1 − P)θ XLHmLH

HU(aHU − bALH − βc)in sHU is negative if gHU = B and can be reduced to zero if gHU = 0. The deviationdoes not reduce the surpluses in other types of trade that the HU agent mightengage in. This shows that GHU = 0 in all symmetric equilibria, which impliesALH = 0 by Lemma 1. Given that part (i) of the lemma holds, the correspondingquantities of trade imply that mHH

HH = mHUHH = 1. The expected surplus associ-

ated with the decision xHH is strictly positive, and so XHH = 1. Similarly, mHHHU =

mHUHU = 1.We now prove part (iii). Since XHL = XLL = 0, and AHU = 0, we have

sLU = P(1 − θ)XHUmHULU (u + aLU) + (1 − P)(1 − θ)XLUmLU

LU(aLU − bALU)


Consider the choices aLU = 0 and gLU = 0. Since (AHU , GHU) = (0, 0), thesechoices satisfy aLU ≤ GHU and gLU ≥ GHU . Thus, mHU

LU = 1, and the first term ofthe surplus is strictly positive. If ALU = B, the choice gLU = 0 induces mLU

LU = 0and neutralizes the second term of the surplus; if ALU = 0, then the choice gLU = 0generates aLU − bALU = 0. In both cases, the expected surplus is strictly positive,and so xLU = 1. This argument also shows that aLU = ALU = B cannot be anequilibrium: If ALU = B, following the strategy aLU = ALU = B would generateat most zero surplus, which can be improved by setting aLU = 0. With ALU = 0and AHU = 0, it is immediately clear that mHU

LU = mLULU = 1. �

Now we are ready to complete the proof of Proposition 2. The above lemmashave established the trading patterns described in the proposition. In particular,there is no bribe exchange in any equilibrium. With these trading patterns, wecan simplify the value functions as (7). Then the requirement 0 < P < 1, which isequivalent to VH − c = VL, becomes

(1 − β)c = P[θ + (1 − θ)XHU]2(u − βc) − (1 − θ)XHU[Pu + (1 − P)βc](A.2)

Also, the equilibrium requirement XHU > 0 becomes

P[θ + (1 − θ)XHU](u − βc) ≥ (1 − P)βc(A.3)

If this condition holds with strict inequality, then XHU = 1; if this condition holdswith equality, then XHU ∈ (0, 1). For the type b equilibrium, set XHU = 1 in (A.2)in order to solve for P. Then, we need to verify the equilibrium requirement that(A.3) holds with strict inequality, and that 0 < P < 1. These requirements areequivalent to c/u < θ < θ2. For the type c equilibrium, we assume that (A.3) holdsas an equality, and solve this equation jointly with (A.2) for (P, XHU). Imposingthe equilibrium requirements 0 < P < 1 and 0 < XHU < 1 yields θ3 < θ < θ2. (Infact, the requirement 0 < XHU < 1 implies 0 < P < 1 in this case.) �

PROOF OF PROPOSITION 3. Suppose that δR > B. We follow the following steps.

Step 1. ZL = 1 and YL = 0. If YU = 0, then no bribes are exchanged inequilibrium.

Since bB < u, the surplus to an L agent from the exchange with a bureaucratis positive, even when the bureaucrat recognizes that the agent holds an L good.Thus, ZL = 1. Since δR > B, the surplus to the bureaucrat in such a match isnegative, if he recognizes quality; thus yL = 0. Suppose that YU = 0. If eitherYH = 0 or ZH = 0, then there is no exchange between private agents and thebureaucracy, in which case the lemma is trivially true. Thus, let us suppose YH >

0 and ZH > 0. For yH > 0, we need AHG = 0; otherwise aGH − bAHG − δR < 0,and so yH = 0. Similarly, for zH > 0, we need AGH = 0. Since AHG and AGH areboth zero, no bribes are exchanged.

Step 2. YU > 0 implies AHG = 0 and YH = 1.


Suppose YU > 0. The expected surplus associated with the decision yU is

PZHmHGGU (aGU − bAHG − δR) + (1 − P)ZLmLG

GU(aGU − bALG − δR)

≤ PZHmHGGU (aGU − bAHG − δR)

The inequality follows from the fact that aGU − bALG − δR ≤ B − δR < 0. Thus,for yU > 0, it is necessary that ZH > 0 and that aGU − bAHG − δR ≥ 0 for all (aGU ,gGU) such that aGU ≤ GHG and gGU ≥ AHG. This is impossible if AHG = B. LetAHG = 0. The same quantities (aGU , gGU) are feasible for the choices (aGH , gGH),which yield a nonnegative expected surplus for the choice yH . Thus, yH = 1.

Step 3. If YU > 0, then ALG = GGU = 0, and GLG = AGU = B.

Suppose YU > 0. Then the best choices for an L agent in a match with a bu-reaucrat are aLG = GGU and gLG ≥ AGU . If aGU = 0, this demand can be met bythe private agent, yielding mLG

GU = 1. In this case, the surplus associated with thechoice yU is strictly negative, contradicting the supposition YU > 0. Thus, aGU =AGU = B in equilibrium, which implies GLG = B. If GGU = B, then, from thedecision on aLG, ALG = aLG = B. Again, this generates mLG

GU = 1 and a loss to thebureaucrat in an exchange with an L agent. By deviating to gGU = 0 (<ALG =B), the bureaucrat can induce mLG

GU = 0 and neutralize this loss. Furthermore, thedeviation does not reduce the chance for the bureaucrat to exchange with an Hagent since AHG = 0 as shown in Step 2. Therefore, in equilibrium it must be truethat GGU = 0, which induces ALG = 0.

Step 4. If YU > 0, then AGH = GHG = 0, which yields a contradiction.

With YU > 0, we have AGU = B, as shown in Step 3, and so u + aHG −bAGU − βc < 0. The expected surplus associated with the choice of zH is at mostθYHmGH

HG(u + aHG − bAGH − βc). Since YH = 1 when YU > 0, the above sur-plus is strictly negative if AGH = B. This would imply zH = 0, which has beenshown above to be inconsistent with trade between the private sector and thebureaucracy. Thus, AGH = 0. Next, notice that when AGH = 0, the best choice fora private agent holding an H good in a match with a bureaucrat is to set gHG = 0.This is because the private agent in such a match gets a positive surplus from tradewhen the government recognizes H, and a loss when the government does notrecognize H. The choice gHG = 0 < AGH keeps the possible gain and neutralizesthe possible loss. In contrast, the choice gHG = B invites trade in both cases andreduces the expected surplus. Now that GHG = 0 < aGU , the surplus associatedwith the decision yU is strictly negative, contradicting the supposition YU > 0. Thiscompletes the proof of the lemma. �

PROOF OF PROPOSITION 4. Consider a bureaucrat’s decision on yL. GivenGLG = B and ALG = 0, the surplus associated with yL is ZLmLG

GLaGL, where mLGGL =

1 iff aGL ≤ GLG = B and gGL ≥ ALG = 0. By choosing the quantity aGL =AGL = B, the bureaucrat obtains a surplus ZLB; if he deviates to aGL = 0, thesurplus is 0. Since ZL = 1, there is no incentive for the bureaucrat to deviate from


the equilibrium choices aGL = B and yL = 1. In addition, following the equilib-rium choice gGL = GGL = 0 is not dominated by other choices (although gGL = Byields the same surplus). Similarly, consider a bureaucrat’s decision on yU . SinceAHG = ALG = 0, max{AHG, ALG} = 0 ≤ gGU . In addition, aGU ≤ B = GLG. Thus,mLG

GU = 1 for any choice aGU , gGU ∈ {0, B}. Noting, as assumed, that ZL = 1, thesurplus associated with the choice yU is PZHmHG

GUaGU + (1 − P)aGU . This equals0 if the bureaucrat chooses to deviate to aGU = 0. If the bureaucrat adheres to thesupposed equilibrium strategy aGU = AGU = B, the surplus is at least (1 − P)B >

0. Thus, aGU = B and so yU = 1, as in the supposed equilibrium. Like the quantitygGL, the decision gGU = 0 is consistent with the bureaucrat’s incentive, and gGU =B yields the same surplus. Next, consider a private agent’s decision on zL. WithYL = YU = 1 and the quantities (AGL, AGU) = (B, B), the surplus associated withzL is

θmGLLG(u + aLG − bB) + (1 − θ)mGU

LG (u + aLG − bB)

Since GGL = GGU = 0 and AGL = AGU = B, the above surplus is 0, if either aLG =B or gLG = 0. In contrast, if aLG = 0 and gLG = B, as described in equilibrium,then mGL

LG = mGULG = 1, and the surplus is u − bB > 0. Thus, aLG = 0, gLG = B, and

zL = 1. Finally, consider a meeting between a bureaucrat and a private agent witha (known) H good. Given that GHG = AHG = 0, clearly the choices aGH = gGH =0 maximize the bureaucrat’s surplus associated with yH and yield yH = 1. Giventhat GGH = AGH = GGU = 0, AGU = B, and YH = YU = 1, the choices aHG =gHG = 0 yield mGH

HG = 1, mGUHG = 0, and a surplus θ(u − βc) > 0 for the private

agent associated with the choice zH . A deviation to aHG = B reduces the surplusto 0. A deviation to gHG = B induces mGU

HG = 1 and reduces the surplus to

θ(u − βc) + (1 − θ)[u − (b − 1)B − βc] < θ(u − βc)

where the inequality follows from (1). Thus, the optimal choices for the privateagent in such a meeting are aHG = gHG = 0 and zH = 1. �

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