AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR GERVAN FEARON AND LUTZ-ALEXANDER BUSCH Abstrat. This paper investigates the impat of a ministry s budget size on the hoie between auditing a (Niskanen) bureau and employing ompetitive bidding in the provision of a publily funded good. The ministry s marginal expeted payoff inreases fastest with budget size for autioning as opposed to auditing. However, the ministry is shown to swith from a purely publi provision to a ompetitive provision as the budget size inreases even if expeted fixed osts favor the audit. The study ontributes to the literature by extending the Niskanen framework and by endogenizing the institutional arrangements for the provision of publi setor goods. Keywords: Niskanen bureau, publi good provision, mehanism design, audit, aution. JEL: H11, H4, D44, D73 1. Introdution Just as the private setor, in reent years governments have looked towards outsouring and privatization for the provision of goods and servies by the publi setor (De Fraja, 1993; Feenstra and Hanson, 1996). An inreasing number of prison failities in the USA are operated by private ontrators and reportedly have 10 perent lower osts than their publi setor ounterparts (Hart et al. (1997).) Signifiant ost advantages for privatization over publi provision have been found for refuse olletion servies in ities with 50,000 or more residents, while no signifiant differene was found for ities with fewer residents (Stevens (1978), Savas (1977).) For poliing, it has been found that ities with a population of 80,000 or more had more ostly and less effiient poliing servies than smaller ities, even in the presene of monitoring or auditing ativities by itizens (Davis and Hayes (1993).) The key eonomi problem faing a ministry responsible for the provision of a publi good lies in the asymmetri information about osts between the produer of the good or servie and the ministry. Date: Marh 23, 2005. We would like to thank two anonymous referees for insightful omments that helped us to make signifiant improvements to the paper. We are grateful to Al Slivinski and Ig Horstmann for ritique and valuable omments. The first author would like to thank Jim Davies for his support and enouragement and has benefited from disussions with Vitor Aguirregabiria, Kathy Mosou, Jeffrey Smith, Ron Wintrobe, and Augustine Wong. 1

2 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH This is true both if the produer is a profit maximizing private firm and if the produer is a government bureau. As a result, the ministry has to pay informational rents to eah type of provider. If the provider is a profit maximizing firm, informational rents are minimized through the use of an aution mehanism, whih introdues ompetition at the bidding stage (Osborne and Gaebler (1993).) MAfee and MMillan (1987) suggest that suh ompetitive provision tends to involve first prie sealed bid autions. Publi setor bureaus have been haraterized as Niskanen budget maximizing institutions (Niskanen (1975), Mueller (1997).) Bendor et al (1985) and Breton and Wintrobe (1975) suggest that monitoring and management ompensation pakages may lessen the informational rents of suh Niskanen bureaus. Monitoring has been defined by Mookherje and Png (1992) as the ommitment of resoures towards the detetion of an offense prior to reeiving information about an ourrene. They term the ommitment of resoures after an offense has been reported as investigation. Auditing, as analyzed by Baron and Besanko (1984) in a prinipal-agent model with asymmetri information about a firm s realized osts, involves the expenditure of resoures in order to detet misreporting after reeiving a signal about osts. This paper investigates the impat of budget size on the hoie between auditing and ompetitive bidding in the provision of a publily funded good. Auditing and ompetitive bidding raise several questions. First, what is the impat of the budget size on the funding soure s hoie between ompetitive bidding and auditing? Seond, what ost distributions will result in the ministry having a non-trivial hoie between ompetitive bidding and auditing the bureau for a given budgetary alloation? Third, what are the impliations of the number of firms apable of providing the publi setor good on the ministry s hoie of ompetitive bidding or auditing? In order to address these questions, the equilibrium outomes of two mehanisms are ompared: One is an auditing mehanism offered by the ministry, as prinipal, to a bureau, the agent. As in Baron and Besanko (1984) and Laker and Weinberg (1989), the first-order approah to the prinipal-agent problem is used. In this approah, the global inentive ompatibility onstraint for the agent is replaed by a loal inentive ompatibility onstraint, allowing the use of optimal ontrol tehniques. The seond mehanism under investigation is an aution mehanism. Due to its greater analytial simpliity, a seond prie sealed bid aution is used to haraterize the outomes. The revenue equivalene theorem assures

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 3 us that this aution will lead to the same (expeted) utility for the ministry as the first-prie sealed bid aution observed in pratie. 1 It is well known that an agent ould be indued to truthfully report private information in the presene of infinite penalties for misreporting. The ultural norm that the punishment fit the rime however implies that there is a finite maximal penalty whih the ourts will uphold. This level is imposed exogenously. Auditing, thus, must our with positive probability sine even if the prinipal detets misreporting the penalty is bounded. Hene the auditing mehanism, just like the aution, is not apable of extrating all informational rents. Autions, of ourse, are well known to be effiient mehanisms, i.e., to minimize the informational rents of the agent. Sine in pratie both auditing and autioning are ostly to implement, it is nevertheless not obvious whih mehanism is better under what irumstanes. The ministry thus faes a non-trivial hoie between providing the publily funded good either via ompetitive bidding or an audited bureau. The main findings are as follows: Most generally, the optimal way to ontrat for the provision of the publily funded good depends on the fixed osts of the mehanism, the number of potential bidders, and the distribution of osts. More speifially, we onfirm that the informational rents of the provider inrease faster with budget size under auditing than in the aution mehanism. This informational rent advantage of the aution mehanism an offset a fixed ost disadvantage relative to the audit mehanism. Hene, as the budget size inreases, swithing from auditing to the aution mehanism may our at suffiiently high budget sizes even if auditing enjoys a fixed ost advantage over the aution mehanism. In the absene of this fixed ost advantage, the aution mehanism stritly dominates the audit mehanism for all budget sizes. An inrease in the number of firms partiipating in the aution results in a deline in the budget level that indues swithing from auditing to the aution, sine the ministry s expeted payoff from the aution is inreasing in the number of bidders. We an also show that for small enough budget levels the ministry will hoose to produe the good without auditing or onduting an aution, having some able provider do so at the highest possible ost, thereby avoiding any fixed osts assoiated with either mehanism. These results are onsistent with the norm for many organizations that do not ondut 1 See MAfee and MMillan (1987), or Milgrom (1987), among others. Jehle and Reny (2001) also provides an overview of autions, the revenue equivalene theorem, and mehanism design.

4 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH audits or autions for suffiiently small prourement ontrats or ativities, as well as the examples at the beginning of the introdution. The resulting important role of the alloation of fixed osts in government is further disussed in the onlusion. The paper is organized into five setions. In setion 2, the formal model is developed. Setion 3 haraterizes the equilibrium outomes for the auditing and ompetitive bidding mehanisms. In setion 4, the ministry s equilibrium hoie is determined. Setion 5 provides onlusions. All mathematial derivations and proofs are in a series of Appendies. 2. The Model A ministry has an exogenously fixed program budget R R + and ares only about the level of a publily funded good, x R +. The ministry does not possess the tehnology to produe the publily funded good. Instead, the good is produed with a onstant marginal ost tehnology by either a bureau or a firm. In either ase, the atual level of marginal ost is private information. The ministry only knows that the marginal ost [, ], > 0, is distributed aording to a density f(). The bureau is an extended Niskanen bureau that ares about its disretionary budget in exess of its prodution osts. Letting B denote the budget transferred to the bureau, and x the level of prodution, the bureau s preferenes are represented by the utility funtion v(b, x) = B x. The bureau has a reservation level of utility, v, whih represents the payoff if its resoures are re-deployed away from the provision of x. For simpliity, the reservation utility is assumed to be zero. The firm is one of N firms apable of produing the good. Firms are assumed to be (expeted) profit maximizers with zero profits as their reservation utility. Eah firm s atual marginal ost of prodution is determined by an independent, identially distributed draw from the ost distribution. The ministry has three alternative institutional arrangements to obtain the publily funded good, denoted by a {A, C, E}. It an design an auditing mehanism for the bureau (denoted by A); it an aution off the right to produe the good to one of the firms in a ompetitive bidding proess, using a seond-prie sealed bid aution with a reserve prie (denoted by C); or the ministry an hoose to not design a mehanism at all using unsupervised prodution (denoted by E). In this last ase, the ministry hands over all of the budget to a produer and demands the output level onsistent with the highest

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 5 possible ost realization. The resulting level of utility for the ministry, R/, onstitutes a minimum utility level whih an be ahieved under either publi or private provision. 2.1. Auditing. For auditing (A), the ministry possesses an audit tehnology that generates a signal of true osts. This signal, denoted ĉ, is distributed on [, ]. The distribution of the signal is onditional on the atual value of the bureau s ost realization, and has density h(ĉ ). The signal is assumed to be unbiased, that is ĉh(ĉ )dĉ =. It is also assumed that the onditional distribution satisfies the monotone likelihood ratio property: Let H(ĉ ) denote the (umulative) distribution funtion, then it is assumed that H(ĉ ) < H(ĉ ) ĉ if >. Use of this audit tehnology osts resoures. Let q denote the probability of an audit ourring, then the assoiated audit ost is m(q). The fixed ost of auditing, assoiated with maintaining the neessary infrastruture, is m(0) = m 0. Assumption A1: Audit osts m(q) satisfy: (i) m(0) = m 0 > 0; (ii) m (q) > 0, m (q) > 0 for q (0, 1); (iii) m (0) = 0; and (iv) lim q 1 m(q) = lim q 1 m (q) =. 2 If an audit is performed and generates a ost signal below the ost reported by the bureau, a penalty is imposed. This penalty is assumed to be exogenously determined by the ourts and will be speified later. 3 Auditing is treated as a prinipal-agent problem and addressed in the usual mehanism design approah. The ministry is the prinipal and reeives a report from the bureau, denoted by w. Based on this report, the prinipal hooses an audit ontrat onsisting of a transfer from the ministry to the bureau, t(w), the level of good provided by the bureau, x(w), and an auditing intensity q(w) [0, 1]. In the ase where the audit returns a ost signal below reported ost, the bureau is penalized. Suppose that there exists an exogenous penalty funtion whih furthermore is proportional to the mis-statement of osts, but whih annot reward the bureau; then, in the event that the bureau is audited, a penalty of max[0, p(w ĉ)] is assessed, where ĉ is the ost signal, and p is an exogenous parameter. 4 The penalty is assumed to be paid into the onsolidated revenue fund of the government and is not available to a 2 A1 implies that the ost of monitoring, inluding the assoiated administrative osts, is onvex. Hettih and Winer (1988) make a similar assumption about administrative osts and it is onsistent with the empirial observations of Davis and Hayes (1993). Laffont and Tirole (1986) make a similar laim in relation to the disutility of effort in terms of money. 3 Andreoni (1991), Bose (1995), and Dikens et al (1989) provide a omprehensive overview of penalties and their enforeability. 4 Baron and Besanko (1984), in ontrast, show that the optimal penalty funtion has the bang-bang feature for linear audit osts, for whih an interior monitoring probability is a knife edge ase.

6 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH speifi ministry. As in Baron and Besanko (1984) where the penalty is refunded to onsumers and not to the regulator, there is no revision in the quantity of the good provided by the bureau after an audit. The bureau s expeted utility from stating a ost of w when the true ost is is 5 v(w; ) = (t(w) x(w)) q(w)eĉ[max[0, p(w ĉ)] ] = (t(w) x(w)) q(w)p = t(w) x(w) q(w)p w w (w ĉ)h(ĉ ) dĉ H(ĉ )dĉ. (1) It is important to note that both v(w; ) and v(w; )/ w are ontinuous at w =. By the revelation priniple, attention an be restrited to truth-telling mehanisms. The partiipation onstraint for the bureau is given by v(, ) = t() x() q()p Global inentive ompatibility requires that H(ĉ ) dĉ 0. (2) v(; ) v(w; ) w, [, ]. (3) We employ a loal inentive ompatibility approah, as shown in Appendix (A). This restrits the derivative of the bureau s value funtion v(), denoted by v(), and is given by v() = x() q()p H(ĉ ) dĉ. (4) The ministry s problem then is to maximize its expeted payoff by hoosing a ontrat (t(w), x(w), q(w)) to offer to the bureau for any reported ost w. It faes the budget onstraint R m(q(w)) t(w) 0, as well as the partiipation onstraint (2) and the inentive ompatibility onstraint (4). Optimal ontrol theory is used to solve this problem. Speifially, the bureau s utility v( ) is onsidered to be the state variable and the ontrol variables are t(w), x(w), and q(w). Truth-telling together with (2) implies: t() = v() + x() + q()p H(ĉ )dĉ, (5) 5 Note that R w wh(ĉ ) dĉ = wh(w ), while integration by parts yields R w ĉh(ĉ ) dĉ = wh(w ) R w H(ĉ ) dĉ.

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 7 so that t(w) an be eliminated as a ontrol. The ministry s problem then is: 6 s.t. (i) max q( ),x( ) (ii) { } x(s)f(s)ds v() = x() q()p H(ĉ ) dĉ R m(q()) v() x() q()p (iii) v() 0 H(ĉ )dĉ 0 (6) (iv) v() = 0 This problem is analysed fully in Appendix (B). A priori there are two ases of interest: first, when both the transition equation (i) and the budget onstraint (ii) are binding, and, seond, the ase when the budget onstraint (ii) is slak while positive output is produed, indiating that the ministry funds some prodution but has unused budget room. The latter annot be part of the solution however. The intuition behind this is as follows: Sine the ministry does not are about money but only output, the free budget does not add to utility. It ould be spent instead on an inreased audit probability, whih redues the slope of the value funtion, relaxing the inentive ompatibility onstraint, and thus allowing for higher output. Hene the ministry would inrease the audit probability (and output), ultimately exhausting the budget. It follows that the solution to (6) will involve both the transition equation and the budget onstraint binding at positive output levels. Just as in an aution with a reserve prie, the optimal ontrat may involve a region of ost reports for whih the ministry does not fund the prodution of output by the bureau. Denote this utoff value above whih no prodution ours by ˆr. It is determined by the distribution of osts aording to ˆrf(ˆr) F (ˆr) = 0 (see Appendix B.) 7 As in an aution with a reserve prie, it may be better for the ministry (in expeted utility terms) not to reeive output for some (high) realizations of ost and in turn to reeive more output if the ost realization is low. The exlusion of the high ost types relaxes the inentive ompatibility onstraint for the lower ost types and helps to redue their informational rents. In the optimal ontrat the ministry audits with q(; p) > 0 for ost reports in the interval [, ˆr) where 6 See Kamien and Shwartz (1981) for an outline of the tehniques used to solve suh programming problems. 7 As an example, the p.d.f. f() = 2/ 3, 1 < < from Hoag and Craig (1978, p. 32) an be shown to result in ˆr = 3 1/2 and F (ˆr)/ˆr = 10/27 for ˆr = {r : f(r)r F (r) = 0}.

8 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH it transfers funds and demands output. The minstry does not audit (q(; p) = 0) for ost reports ˆr, and does not transfer funds nor demand output. 2.1.1. Ministry Payoff. The ministry s expeted (over ost realizations) payoff from the audit mehanism is denoted by E [u(, R : A)]. In what follows, this payoff shedule will be haraterized. By definition, E [u(, R : A)] = x(s)f(s)ds. Let ˆr denote the highest ost for whih a positive output is required as per the disussion above. Using (6(ii)) to isolate x( ), the ministry s expeted payoff shedule an be rewritten as: ˆr x(s)f(s)ds = ( v(s) q(s)p ) H(ĉ s) dĉ f(s)ds. (7) s Using the solution to the optimal ontrol problem (6), it is shown in Appendix (C) that the ministry s expeted payoff shedule an be expressed as: ˆr ( E [u(, R : A)] = K(R) + χ(s; p) + R m 0 1 s s ζ(t; p)dt q(s)p s ) H(ĉ s)dĉ f(s)ds, (8) where the funtions χ(; p) and ζ(t; p) are as defined in equations (C3) and (B12), respetively. K(R) is a onstant of integration whih depends on the budget level R. Appendix (C) also shows that E [u(, R : A)] R = F (ˆr) > 0. (9) 1ˆr Hene the ministry s expeted payoff is linear and inreasing in the budget level, and we have an expression for the slope whih we an be used to ompare provision through auditing versus ompetitive bidding. 2.2. Competitive bidding. Under ompetitive bidding (C), the ministry onduts a seond-prie sealed bid aution with a reserve prie. The aution inurs a fixed ost b F 0. This fixed ost aptures the osts of the neessary infrastruture assoiated with operating the aution, whih may inlude the ost of hiring staff for drafting the bid, addressing questions, and evaluating the responses, as well as advertisement osts and the osts of other regulatory requirements impliit in publi setor bidding ativities. The free budget available for goods provision then is (R b F ). In the aution, the ministry aepts sealed bids on per unit ost from the N firms that are apable of produing the good. If all bids exeed the reserve prie no prodution will our and no budget is transfered. Otherwise, the lowest ost bid wins the aution and in return reeives the available budget. The winning bidder is required to produe output at a level

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 9 orresponding to the lower of the unit ost of the seond lowest bid or the reserve prie. The output level thus will be x = (R b F )/ units, where denotes that unit ost. The strategies for the N firms are their bids w( n ). As usual, the firms weakly dominant strategy in a seond-prie sealed bid aution is to bid truthfully, that is w( i ) = i. This is due to the fat that a firm s payoff, onditional on winning the aution, is independent of its bid. Let = ( 1, 2,..., N ) denote the N-vetor of submitted bids. The i th order statisti is denoted by (i), suh that (1) (2) (N). G (i) denotes the probability distribution of this i th order statisti of true osts, with the assoiated density g (i). Sine osts are distributed iid aording to F ( ) with density f( ), we know that 8 G (i) () = g (i) () = N ( N u u=i ) F () u (1 F ()) N u, (10) N! (i 1)!(N i)! f()f ()i 1 (1 F ()) N i. (11) Let r denote the reservation level of ost, above whih no prodution ours. 9 There are three relevant events in omputing the ministry s expeted utility from the aution: (1) If (1) > r then no bid is below the reserve level and the ministry obtains a zero payoff from the aution sine no private firm prodution is funded. This event ours with probability (1 G (1) (r)). (2) If (2) < r then at least two bids are below the reserve level. In this ase the level of output will orrespond to the seond lowest of these osts (2), while prodution and all of the budget net of fixed osts is awarded to the lowest bidder. Hene the output level will be (R b F )/ (2). Of ourse, (2) is a random variable, and the expetation needs to be taken over the interval (, r) using the density funtion g (2) (). (3) If (1) < r but (2) > r then the lowest bid will win at the reserve level of ost. Hene prodution ours with an output level orresponding to a prodution ost of r. The assoiated output level is (R b F )/r. Prodution is by the lowest ost firm, whih reeives all of the budget net of fixed osts. The probability of this event is (1 [1 G (1) (r)] G (2) (r)) = G (1) (r) G (2) (r), sine with probability 1 G (1) (r) event (1) ours, with probability G (2) (r) event (2) ours, and the three events are mutually exlusive and exhaustive. 10 8 See, for example, David, H.A., 1981, Order Statistis, 2nd edition, John Wiley and Sons In, New York. 9 Note that its value is, a priori, different from the value «of ˆr in the previous setion. 10 N It an be shown that G(1) (r)) G (2) (r) = F (r)[1 F (r)] 1 N 1.

10 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH We an now state the ministry s expeted utility from the aution, denoted E [u(, R, N : C)], whih it will maximize by hoosing an optimal r: E [u(, R, N : C)] = R b F g (2) ()d + [G (1) (r)) G (2) (r)] R b F r. (12) The optimal reserve prie r for the ministry is derived in Appendix (D), where it is shown that it satisfies: r = F (r ) f(r ) if f (r )(1 F (r )) 2(N 2)f(r ) 2. (13) Equation (13) is related to the usual monotoni hazard rate ondition. In partiular, the standard assumption that v (1 F (v))/f(v) is nondereasing implies that f (r)(1 F (r)) 2f(r) 2, whih learly is suffiient to allow (13) to hold for all N > 3. Hene, a ompetitive setor of more than three firms is a suffiient ondition for r to define an equilibrium reservation level of ost for the ministry. 11 It is interesting to observe that the ultural or institutional norm in many publi institutions is to require a minimum of three bids or quotes. 2.2.1. Ministry payoff. For r satisfying (13), the expeted profits of the ministry are { 1 E [u(, R, N : C)] = {R b F } 1 r (1 F (r )) N (1 F ()) N 2 d } NF ()(1 F ()) N 1 d. (14) E [u(, R, N : C) approahes {R b F }/ as N approahes infinity. For R b F, the ministry an hoose to do nothing and E [u(, R, N : C) = 0 in that ase. The expeted payoff is stritly positive otherwise. Clearly, it is linear and inreasing in R, sine the term in the large brakets above is independent of R. 2 3. Charaterization of institutional hoie The institutional hoie on the part of the ministry involves the ministry hoosing to have the good produed in the publi setor through auditing or in the private setor through ompetitive bidding, provided that one of them produes a higher expeted utility than R/. This lower bound derives from the fat that the ministry an have any firm or bureau provide the good at this ost level without supervision. The ministry hooses ompetitive bidding if E [u(, R, N : C)] max { E [u(, R : A)], R }, 11 Note that the exlusion level of osts in the auditing framework, following equation (B.13) in Appendix B2, is defined by the same ondition as here for the aution mehanism. It follows that autions and auditing both exlude the same set of ost realizations.

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 11 and audited provision by a bureau if E [u(, R : A)] max { } E [u(, R, N : C)], R. If neither inequality holds the provision is by either method, sine an unsupervised provider is engaged. Of partiular interest is the question how the institutional hoie depends on the budget size and how it varies with it. As has already been noted, both institutions expeted payoffs are linear in the budget, R. It was shown earlier that E [u(, R : A)]/ R = F (ˆr)/ˆr in the ase of auditing, while it follows from (14) that for an aution E [u( )] R = 1 (1 F (r )) N r (1 F ()) N 2 NF ()(1 F ()) N 1 d 2 d. (15) Clearly, for the aution lim N > E [u( )]/ R = 1/. Therefore the expeted payoff is inreasing more rapidly under ompetitive bidding than under auditing if there is a very large number of firms. On the other hand, for N = 1, E [u(, R, 1 : C)]/ R = F (r )/r. Note that r = ˆr, sine both values are defined by the same equation. Of ourse, N = 1 is not a valid number of firms in an aution. However, sine E [u(, R, N : C)]/ R > E [u(, R, N 1 : C)]/ R for all N (see Appendix (F)) it follows that for all valid values of N (i.e., N 3) the expeted payoff from the aution is inreasing more rapidly than that of the auditing mehanism. What is the intuition behind this result? It derives from the well known fat that autions are optimal mehanisms, that is, extrat the maximal amount of informational rents from the informed agent. The auditing sheme hosen is not able to ahieve the same degree of rent extration. It follows that the ompetitive mehanism yields a higher inrement in expeted utility to the ministry than the audit mehanism for any given inrement in budget size. More expliitly, sine the providers reservation utility has been normalized to zero, interim informational rents (that is, the rents one the provider knows true ost) an be defined as the expeted profits of a provider from partiipating in the mehanism. 12 For the aution mehanism the profits for a firm with ost realization (1) are {( min (R b F ) ) } (2) (1), (R b F ) r (1) (2) r, (16) sine the required output orresponds to either the seond lowest ost or the reserve, whihever is lower. For the audit mehanism the bureau s utility is given by v() defined in equation (B.16) in Appendix 12 Note that under full information eah provider would be held to the reservation level of utility for all ost realizations.

12 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH B. While these expressions do not lend themselves to easy omparison, it an be noted that for both the aution and the audit mehanism the informational rent of a provider at the (idential) reservation ost levels (ˆr, r ) is zero. For all lower ost realizations informational rents arue. They are higher for the audited bureau ompared to the winning bidder in an aution. Furthermore, as the budget size is inreased, informational rents also inrease. For the aution the rate of inrease is For the bureau, the rate of inrease in informational rents is v() R { min 1 (1), 1 } (1) (2) r. (17) = K(R) R 1 ds = 1. (18) s2 ˆr Clearly, then, the bureau obtains a larger inrease in informational rents unless the aution happens to have the seond lowest ost realization at or above the reservation level an event that ours with a probability stritly less than one. Informational rent onsiderations alone would thus suggest that ompetitive provision is always superior to audited provision. However, the ministry s expeted payoff not only depends on the informational rents given up, but also on the osts assoiated with operating eah of the mehanisms. These osts have the effet of reduing the budget available for atual goods provision. Sine the aution is a more effiient mehanism, it will dominate the audited provision anytime the atual budget available for goods prodution is the same for both mehanisms. Of ourse, if autions are not only more effiient in terms of informational rents, but also ost less to run in terms of their fixed osts (b F ) ompared to the (expeted) auditing fixed osts, then the aution will definitely yield a higher ministry utility. 13 An example of this ase is depited in Figure 1, where ministry expeted utility is plotted against budget size for unsupervised provision, ompetitive provision, and audited provision. As the figure demonstrates, the ministry will hoose ompetitive bidding for all suffiiently high budget levels R > R 0 (N), where: R 0 (N) = {R : E [u(, R, N : C)] R/ = 0}. If R R 0 (N), then the ministry hooses to produe the good without oversight, earning a payoff R/. Sine the expeted utility from the ompetitive mehanism depends positively on the market depth N, it is lear that R 0 (N) will be lower the larger is N. Note 13 The expeted fixed ost of auditing is denoted by E[M +P ] for simpliity. It onsists of the expeted osts of performing the audit, plus the expeted value of the funds transferred to the bureau in order to over the bureau s expeted penalties.

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 13 Expeted Payoff E [u(, R, N : C)] E [u(, R : A)] R b F E[M +P ] R R 0 (N) Figure 1. Expeted payoffs if E[M +P ] > b F also that mandatory ompetitive provision poliy would only be optimal if aution osts are not higher than any osts assoiated with ompletely unsupervised provision (here, zero.) If, however, the osts assoiated with running an aution (b F ) exeed the (expeted) osts assoiated with auditing, then the audit allows a larger budget to be alloated to goods prodution. Even though the audit is not as effiient in transforming this budget into ministry utility, it may yield a higher ministry utility due to the larger budget available for prodution. Figure 2 shows an example of this situation. There now exist two ritial budget sizes at whih provision hoie may swith. At the first, R 1, the audited bureau beomes better than unsupervised high ost provision, at the seond, R 2 (N), ompetitive provision starts to outperform the audited bureau. Here R 1 = {R : [E [u(, R, : A) R/] = 0} and R 2 (N) = {R : E [u(, R, N : C)] E [u(, R, : A)] = 0}. The ministry hooses ompetitive bidding if R > R 2 (N) > R 1 or if R > R 0 (N), R 2 (N) < R 1. If R 2 (N) R R 1, the ministry hooses auditing, and the ministry onduts no supervision (or oversight) if R R 1. As in the previous ase, sine the marginal expeted payoff of ompetitive provision depends positively on the number of firms partiipating in the aution R 2 (N) is dereasing in N. The following proposition summarizes these results.

14 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH Expeted Payoff R E[M +P ] b F R 0 (N) R 2 (N) R 1 E [u(, R, N : C)] E [u(, R : A)] R Figure 2. Expeted payoffs if E[M +P ] < b F Proposition 1. If autions have lower fixed osts than auditing, the ministry will hoose a ompetitive bidding proess for all suffiiently high budget levels. If, however, auditing has suffiiently lower (expeted) osts, the ministry may first swith from high ost provision to an audited bureau as budget size inreases, and only for even higher budget levels will ompetitive provision beome optimal. In either ase, the budget level at whih ompetitive provision beomes optimal is lower, the larger the ompetitive setor. This proposition predits that the institutional arrangement in the publi setor is dependent on the size of the ministry s budget, the size of the ompetitive market, and the relative sizes of the fixed osts of auditing and autioning. This latter point is of partiular poliy relevane, sine the alloation of fixed osts of supervision or ompetitive prourement is endogenous in multi-produt government settings. As is further disussed below, the optimality of ompetitive provision an therefore be affeted by administrative rules onerning ost alloation. 4. Disussion The paper analyses a ministry s hoie in the provision of a publi setor good in the presene of asymmetri information. The ministry opes with asymmetri information about provision osts by

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 15 either auditing a Niskanen budget maximizing bureau, or by ontrating with private firms. In the ase of auditing, the penalty whih an be imposed is exogenously limited, limiting the minstry s abilty to extrat informational rents from the bureau. In the ontrating ase, a seond prie sealed bid aution is used in order to minimize informational rents. The paper extends the Niskanen model of government bureauray, and endogenizes the hoie between auditing and ompetitive bidding in the publi setor. We show that at very low budget levels it may be optimal not to invest any resoures towards the redution of informational rents, but to provide the good without any supervision. This result is onsistent with the ommon praties of many publi institutions that permit the issuane of prourement ontrats to a single firm without tender when these ontrats are suffiiently small. It is noteworthy that the implementation of government regulations to require the use of autions independent of the budget level may therefore not imporove welfare. For higher budget levels the optimal hoie involves an attempt to limit informational rents. The approah that is best will depend on the osts assoiated with eah and the budget size. An aution is the optimal hoie independent of the size of the budget if the fixed osts for ompetitive bidding are low relative to those expeted for auditing. If, however, the fixed osts for ompetitive bidding are high relative to those for auditing, the optimal hoie an depend on the budget level. In that ase the institutional arrangement an involve swithing from publi setor to private setor prodution as budget size inreases. This is driven by the fat that autions are the optimal way to limit informational rents, and so will ultimately outperform any other mehanism (e.g., auditing and unsupervised prodution) even if it has an initial ost advantage. Both autions and audited provision may involve a reservation level of ost, above whih the ministry will not fund the provision of the good. The optimal level of this reservation ost depends on the distribution of osts, as is usual for autions and shown here for auditing. Indeed, it is the same for both prourement hoies. As is ommon pratie for publi setor prourement, it is found that the aution mehanism requires more than three firms to partiipate. The size of government is reportedly positively orrelated to overall eonomi ativity as measured in terms of gross domesti produt per apita (Alesina and Waziary, 1998). Holsey and Borherding (1997)

16 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH also suggest that government budget rises faster than inome in the short run while its long-run inome elastiity is unity (i.e., a restating of Wayne s Law). While Wayne s Law is not supported by all empirial studies, the size of government is generally found to be positively orrelated to per apita inome and the inome of the median voter. 14 It follows that the results predit that there will be growing reliane on private provision of publi setor goods during periods of eonomi boom and expanding government budgets even when fixed osts favor auditing over autions. Also, developed ountries tend to have larger government budgets in absolute and as a perentage of GDP than developing nations. Again, if fixed osts, inluding the availability of tehnial ability, favor auditing over autions, then it an be antiipated that the private setor will be more appropriately involved in the provision of publi setor goods in developed ountries as ompared to developing nations. In general, the implementation of institutional arrangements in government without regard for the fixed osts of maintaining these strutures or the size of the government budgetary resoures may not be welfare improving. This study predits that the government opes with information asymmetries by aiming to minimize informational rents. The optimal institutional arrangement of government is found to vary depending on the osts of eah arrangement. However, in a government environment with multiple produts, the alloation of fixed osts between programs is endogenous. Sine it also is a signifiant determinant of the hoie between publi or private setor provision, this study suggests that the administrative rules assoiated with the alloation of these osts are important in prediting and determining the institutional arrangements of government. Bureaurats an influene these fixed osts through the alloation methodology, and may therefore be instrumental in determining the fixed ost of onduting publi setor autions or publi setor supervision. Hene, the institutional rules surrounding the alloation of these osts are an important determinant of the institutional arrangements established for the provision of publi setor goods. 14 Alesina and Waziary (1998) report that total government expenditures, inluding transfers and interest payments, as a perentage of GDP is positively related to per apita inome based on a ross-setion analysis of over 90 ountries. This relationship also holds for publi expenditures on defense, eduation and publi investment when population has been taken into aount.

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 17 APPENDIX A. Loal Inentive Compatibility in Auditing: The global inentive ompatibility onstraint (3) implies an infinite number of onstraints. Rogerson (1985) and Jewitt (1988) outline the diffiulties involved in using this type of onstraint. The use of a loal inentive ompatibility onstraint is suggested as an alternative to the global approah, and is also used by Baron and Besanko (1984), for example. The loal inentive ompatibility onstraint approah uses the fat that the optimality of truth telling involves loally maximizing the bureau s utility or payoff. The first order ondition for the bureau s optimal hoie of reported ost is v(w;) w = 0. Together with truth telling (w = ), we therefore require v(w; ) w w= = t () x () q ()p H(ĉ ) dĉ q()ph( ) = 0. (A.1) The seond order ondition is 2 v(w; ) w w = t () x () q ()p w= H(ĉ ) dĉ 2q ()ph(ĉ ) q()ph( ) 0. Let v() = v(, ) denote the resulting value funtion for the bureau under truth telling. (A.2) By the envelope theorem v() = x() q()p H(ĉ ) dĉ. (A.3) This restrition will be imposed on the equilibrium value funtion and takes the plae of (3). B. The Optimal Control Problem: The ministry s problem is: s.t. (i) max q( ),x( ) (ii) { } x(s)f(s)ds v() = x() q()p H(ĉ ) dĉ R m(q()) v() x() q()p (iii) v() 0 H(ĉ )dĉ 0 (B.1) (iv) v() = 0

18 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH Let λ() denote the (differentiable) multiplier on the transition equation (i) and let µ() denote the (differentiable) multiplier on (ii), whih has to satisfy: µ() 0; ( µ() R m(q()) v() x() q()p The first order neessary onditions for the problem in (B.1) are: ) H(ĉ )dĉ = 0. (B.2) R m(q()) v() x() q()p v() = x() q()p v() = 0 λ () = µ() λ() = 0 f() λ() µ() = 0 λ()p H(ĉ ) dĉ H(ĉ ) dĉ µ()p H(ĉ )dĉ 0 µ() 0 H(ĉ )dĉ µ()m (q()) = 0 (B.3) (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) The Kuhn-Tuker onstraint qualifiations will hold as long as (B.1)(ii) is onave. Constraints: Transition equation and budget onstraint binding. The transition equation and the budget onstraint are binding if λ 0 and µ 0. Equations (B.6) and (B.8) then imply the linear first order differential equation and hene λ () + λ() 1 = f() λ() = 1 (F () + k) (B.11) (B.10) for some onstant k. Sine λ() = 0 by (B.7), it follows that k = 0, and so: λ() = F ()/ if µ() > 0. (B.12) Knowing λ(), equation (B.9) gives rise to: [ m F () (q(; p)) = p (f() F ()) H(ĉ ) dĉ ] H(ĉ )dĉ. (B.13) Denote the RHS of (B.13) by ζ(; p). Sine H(, )/ < 0, the first term in the square brakets is negative for low. The seond term is, of ourse, positive, so that the square brakets are guaranteed

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 19 to be negative. At =, the square braket is zero, so that q(; p) = 0. The audit intensity, q(, p), is inreasing in the penalty p. Clearly, in the absene of a penalty (i.e., p = 0) the ministry has no inentive to audit the bureau as auditing would provide no deterrent to misreporting. For p > 0, an audit does serve as a deterrent and so q(; p) > 0 as haraterized by equation (B.13). Suppose there exists an ˆr [, ] suh that ˆrf(ˆr) F (ˆr) = 0. As ˆr from below the first term in the square brakets approahes negative infinity. The impliation is that marginal monitoring ost will also have to approah infinity. Sine total monitoring ost then will also beome large (by A1) this is inonsistent with (B.1(ii)). It follows that a orner solution must our at some point prior to (at a ost below) ˆr. In that ase, auditing ours with q(; p) > 0 for the set of [, ˆr) and q(; p) = 0 for > ˆr sine a reported ost above the reservation prie (ˆr) results in the ministry not funding the prodution of the good. Given λ() = 0, it follows from (B.9) that µ() [ p ] H(ĉ ) ĉ + m (q()) = 0. Therefore q() = 0. This in turn implies that m(q()) = m 0 and so: m(q(, p)) = m 0 + ζ(t; p)dt. But from (B.4) we an solve for x() and substitute into (B.3). This yields R m 0 ζ(t; p)d t v() + v() + q(, p)p H(ĉ ) dĉ q(, p)p H(ĉ )dĉ = 0 (B.14) and therefore v() v() = R + m 0 + ( ζ(t; p)dt q(, p)p H(ĉ ) dĉ ) H(ĉ )dĉ. (B.15) It follows that ( 1 v() = K(R) s 2 R m 0 ζ(t; p)dt+ q(s, p)ps H(ĉ s) dĉ q(s, p)p s ) H(ĉ s)dĉ ds (B.16) where K(R) is a onstant of integration whih does depend on the parameter R. (B.4) now allows the haraterization of x(). This onludes the ase of a binding transition equation and budget onstraint. Budget onstraint not binding. The budget onstraint is not binding and the ministry will have money left over if µ( ) = 0. (B.6) implies a onstant λ( ) for this ase, while (B.8) implies that either f() = λ = onst in whih ase positive output may be produed, or output must be zero. This is due to the fat that it

20 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH annot be infinite sine then the budget onstraint would be binding. Suppose that output is positive: Then (B.9) indiates that the auditing probability should be inreased (sine the LHS of (B.9) is greater than zero) but as q( ) is inreased it osts more money and ultimately the budget onstraint must bind. Hene, positive output is inompatible with a slak budget. It follows that, if the budget is slak, output and auditing all are zero. Positive output and auditing therefore implies the budget onstraint is binding and the non-binding ondition is onsidered no further. C. Deriving the Ministry s Expeted Payoff under Auditing Equation (7) in the text defines the ministry s (expeted) payoff shedule E [u(, R : A)]. Differentiating (B.16) with respet to gives: v() = K(R) ( 1 R m 0 1 s 2 ( R m 0 ζ(t; p)dt + q(; p)p ζ(t; p)dt + q(s; p)ps H(ĉ ) dĉ q(; p)p H(ĉ s) dĉ q(s; p)p s ) H(ĉ )dĉ ) H(ĉ s)dĉ ds (C.1) Define χ(, p) as: ( 1 χ(, p) = s 2 R m 0 ζ(t; p)dt + q(s; p)ps H(ĉ s) dĉ q(s; p)p s ) H(ĉ s)dĉ ds. (C.2) Equation (8) follows immediately. We are interested in the derivative of E [u(, R : A)] with respet to R. χ(; p)/ R = 1/ 1/ > 0. Also, sine v(ˆr) = 0 if ˆr < or v() = 0 otherwise, we know that ˆr ( 1 0 = v(ˆr) = ˆrK(R) ˆr s 2 R m 0 ζ(t; p)dt+ q(s, p)ps H(ĉ s) dĉ q(s, p)p s Taking the total derivative yields 0 = ˆr dk ˆr ˆr s 2 dr ds, and from that dk ˆr dr = 1 s 2 ds = 1ˆr + 1. Taking the partial derivative of (8) we therefore obtain E [u(, R : A)] R = ˆr ( 1 ˆr 1 + 1 1 s + 1 ) f(s)ds = F (ˆr) > 0 s 1ˆr (C.5) ) H(ĉ s)dĉ ds. (C.3) (C.4) whih is equation (9) in the text.

AUDITING AND COMPETITIVE BIDDING IN THE PUBLIC SECTOR 21 D. The optimal reserve level of ost in an aution: The ministry s problem is max r E [u(, R, N : C)] = This has first order ondition: R b F {R b F } ( rg (1) (r) G (1) (r) + G (2) (r) ) The seond order ondition for this problem is ( [ 2 rg(1) (r) G (1) (r) + G (2) (r) ] r g (2) ()d + [G (1) (r)) G (2) (r)] R b F r. (D.1) r 2 = 0. (D.2) ( ) ) rg (1) (r) + g {R b F } (2)(r) r 2 0. (D.3) Using (D.2) in (D.3) implies that rg (1) (r) + g (2)(r) 0. Expanding g i ( ), we obtain N! ( r f (r)(1 F (r)) N 1 (N 2)f(r)f(r)(1 F (r)) N 2) + (N 1)! N! (1)!(N 2)! f(r)f (r)(1 F (r))n 2 0 (D.4) and after simplifiation rf (r)(1 F (r)) (N 2)f(r)(rf(r) + F (r)) 0. This in turn neessitates f (r)(1 F (r)) f(r) 2 (N 3)f(r) 2 + (N 2) f(r)f (r). (D.5) r In what follows, it is assumed that equation (D.5) holds. The first order ondition (D.2) being satisfied implies rg (1) (r) (G (1) (r) G (2) (r)) = 0 rnf(r)(1 F (r)) N 1 NF (r)(1 F (r)) N 1 = 0 rf(r) F (r) = 0 (D.6) Using (D.6) in (D.5), the optimal reserve prie r satisfies: r = F (r ) f(r ) when f (r )(1 F (r )) 2(N 2)f(r ) 2. (D.7) For r satisfying (D.7), the expeted profits of the ministry then are E [u(, R, N : C] = R b F N(N 1)f()F ()(1 F ()) N 2 d+ Integrating by parts twie yields equation (14) in the text. R b F r NF (r )(1 F (r )) N 1. (D.8)

22 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH F. The effet of N on the Marginal expeted payoff from the aution. E [u(, R, N : C)] E [u(, R, N 1 : C)] = (F.1) R R 1 (1 F (r )) N (1 F ()) N NF ()(1 F ()) N 1 r 2 d 2 d 1 + (1 F (r )) N 1 r + F (r ) (1 F (r )) N 1 r F (r ) (1 F (r )) N 1 r + (1 F ()) N 1 2 d + 2 (1 F ())N 2 F () 2 d + (N 1)F ()(1 F ()) N 2 (N 1)F () 2 (1 F ()) N 2 2 d = (F.2) NF () 2 (1 F ()) N 2 2 d = (F.3) 2 d > 0. (F.4) REFERENCES Alesina, A., Waziarg, R., 1998. Openness, ountry size and government. Journal of Publi Eonomis. 69: 305-321. Andreoni, J., 1991. Reasonable doubt and the optimal magnitude of fines: should the penalty fit the rime? Rand Journal of Eonomis. 22( 3), 385-395. Baron, David P., Besanko, David, 1984. Regulation, Asymmetri Information and Auditing. The Rand Journal of Eonomis, 15(4): 447-470. Bendor, J., Taylor, S., Van Gaalen, R., 1985. Bureaurati expertise versus legislative authority: A model of deeption and monitoring in budgeting. The Amerian Politial Siene Review. 79: 1041-1060. Bose, P., 1995. Regulatory errors, optimal fines and the level of ompliane. Journal of Publi Eonomis. 56: 475-484. Breton, A., Wintrobe, R., 1975. The equilibrium size of a budget-maximizing bureau: A note on Niskanen s theory of bureauray. Journal of Politial Eonomy. 83(1): 195-207. David, H.A., 1981. Order Statistis, 2nd Edition, John Wiley and Sons In., New York, USA. Davis, L.D., Hayes, K, 1993. The Demand for Good Government. The Review of Eonomis and Statistis. 75(1): 148-152. Dikens, W.T., Katz, L.F., Lang, K., Summers, L. H., 1989. Employee rime and the monitoring puzzle. Journal of Labor Eonomis. 7(3): 331-347.

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24 GERVAN FEARON AND LUTZ-ALEXANDER BUSCH Mookherje, D., Png, I.P.L., 1992. Monitoring vis-a-vis investigation in enforement of law. The Amerian Eonomi Review. 82(3): 556-565. Mueller, D.C., 1997. Perspetives on Publi Choie: A Handbook. Cambridge University Press, United Kingdom. Niskanen, W.A., 1975. Bureaurats and politiians. Journal of Law and Eonomis. XVIII (3), 217-243. Osborne, D., Gaebler, T., 1993. Reinventing Government: How the Entrepreneurial Spirit is Transforming the Publi Setor. Penguin Books USA In., New York. Savas, E.S., 1977. Poliy analysis for loal government: Publi vs private refuse olletion. Poliy Analysis. 3(1), 49-74. Rogerson, William P., 1985. The first-order approah to prinipal-agent problems. Eonometria. 53(6): 1357-1367. Stevens, B.J., 1978. Sale, market struture, and the ost of refuse olletion. Review of Eonomis and Statistis. 60: 395-406. Eonomis, York University E-mail address: Gervan.Fearon@yorku.a Department of Eonomis, University of Waterloo, Waterloo ON E-mail address: lbush@uwaterloo.a N2L 3G1