The drect and ndrect effects of corrupton on nequaly Ratbek Dzhumashev Department of Economcs, Monash Unversy Abstract Emprcal studes have establshed the followng regulares n the relatonshp between corrupton and nequaly: () the nomnal amount of brbes pad ncreases wh the ncome of agents; () the burden of brbes as a share of ncome decreases; and () at the macro-level, corrupton and nequaly have a nonlnear relatonshp; that s, n countres wh ntermedate levels of corrupton, the effect of corrupton on nequaly s hgher than n countres where corrupton levels are low or hgh. Ths study proposes a theoretcal model that jontly explans the aforementoned emprcal fndngs by showng that corrupton affects nequaly drectly, by creatng ncome and productvy dspary across agents, and ndrectly, by reducng prvate productvy through curtalng the posve externaly of government spendng. Key words: nequaly, corrupton, publc spendng JEL classfcaton: H54, O4
. INTRODUCTION Corrupton occurs when publc agents, abuse ther publc posons, seek and extract llc ncome from the government or prvate agents; as a result, dstorts the qualy and quanty of the servces and nfrastructure provded by the government. The servces and nfrastructure provded by the government are an essental nput to prvate producton, (Barro, 990; Barro and Sala--Martn, 99); hence, by creatng neffcences n the publc sector actves, corrupton affects economc growth of the economy as well as the welfare of the socety. Although, a sgnfcant body of research s dedcated to understandng the causes and the growth mplcatons of corrupton, much less research has been done to study the factors that may affect the relatonshp between corrupton and ncome nequaly. Analysng the relatonshp between corrupton and nequaly s mportant because leads to a better understandng of how corrupton affects economc growth. There s emprcal evdence that ndcates exstence of negatve relatonshps between nequaly, ncome and economc growth (Barro, 000; Galor and Zera, 993; Galor and Moav, 004; Persson and Tabelln, 994; Aleshna and Rodrk, 994). The prmary objectve of ths study s to obtan new nsghts nto the relatonshp between corrupton and nequaly that help to reconcle analytcally several emprcal fndngs about ths relatonshp. Specfcally, a few emprcal papers that examne ths problem at the macro level have found that corrupton ncreases ncome nequaly. For example, Gupta et al. (00), have presented emprcal evdence to support the hypothess that there s a posve relatonshp between corrupton and ncome nequaly. Alongsde Gupta et al. (00), L et al. (000), Gymah-Brempong (00) and Gymah-Brempong and de Camacho (006) usng cross-country and panel data analyses fnd that corrupton has a sgnfcant posve effect on nequaly. See Rajkumar and Swaroop (00), Mauro (998), Tanz and Davood (000), Shlefer and Vshny (993), Adt (003); Alam (989), Dreher and Gassebner (007), Méon and Well (00), and Dzhumashev (04a,b). Lerature on the effect of corrupton on economc growth by lowerng ncentves to nvest (see, for example, Mauro, 995; We, 000; Lambsdorff, 003; and Ln and Zhang, 009), on creatng neffcences n publc spendng (see, for example, Mauro, 998; Tanz and Davood, 000, and Shlefer and Vshny, 993), and on mposng burden on frms (see, for example. Kaufmann and We, 999; Gurev, 004; and Fsman and Svensson, 007). It should be noted that the relatonshp between corrupton and nequaly does not run n just one drecton. For example, You and Kharam (005) hghlght that s not only that corrupton leads to hgher nequaly, but also nequaly n turn affects the socal norms about corrupton and helps perpetuate. They also fnd emprcal evdence that the adverse effect of nequaly on corrupton depends on the strength of democracy n a country. Analogously, Alam (995) reasons that the ably of prvate agents to resst corrupton depends on ther ncome levels as any counter-acton s costly, thus greater nequaly encourages corrupton.
The emprcal fndngs of the corrupton-nequaly relatonshp based on mcro-economc data are less conclusve. For example, Hunt and Lazlo (0a) fnd that the burden of brbes as a share of ncome decreases wh the agent s ncome. Dncer and Gunalp (0), usng an objectve measure of corrupton (the number of government offcals convcted of corrupton n the US), fnd robust evdence that ncome nequaly ncreases wh hgher corrupton. On the other hand, studes based on mcro-level data such as Hunt (007b, 004), Mocan (008) and Svensson (003) fnd that people wh hgher ncomes bear the brunt of corrupton, whch appears to be nconsstent wh the posve relatonshp between corrupton and nequaly establshed by the above-mentoned studes. The nequaly-corrupton nexus s complcated further by the evdence ndcatng that the relatonshp between corrupton and nequaly mght be of a non-lnear nature. For example, L et al. (000) fnd that the effect of corrupton on nequaly s nverted U-shaped; that s, the effect of corrupton on nequaly s less pronounced when the level of corrupton s at the two extremes, hgh or low, but s stronger when corrupton ncdence s n the ntermedate range. The results by Dobson and Ramlogan-Dobson (00) and Andres and Ramlogan- Dobson (0) also support the nonlneary of the relatonshp between corrupton and nequaly. They show that lower corrupton s assocated wh hgher ncome nequaly for a subset of economes where the corrupton level s n the ntermedate range. The man ponts of these emprcal fndngs can be summarzed as follows: ) at the macro-level corrupton and nequaly are posvely correlated (Gupta et al., 00; 00; Gymah-Brempong and de Camacho, 006 ), and ths relatonshp s nonlnear, as the mpact of corrupton on nequaly s hgher n countres wh an ntermedate level of corrupton compared to countres wh low or hgh levels of corrupton (L et al., 000; and Andres and Ramlogan-Dobson, 0); ) at the mcro level, the nomnal amount of brbes pad ncreases wh the agent s ncome (Hunt,007b, 004; Mocan, 008; and Svensson, 003); ) the burden of brbes as a share of ncome decreases wh the agent s ncome ( Hunt and Lazlo, 0a). Naturally, one may wonder whether there exsts a theoretcal explanaton for all these stylzed facts. In the extant lerature, there are some theoretcal studes that try to obtan nsghts nto how corrupton affects nequaly. The frst strand of these models, relate eher the qualy of nstutons or the level of corrupton to the allocaton of productve factors, 3
whch n turn leads to ncome nequaly. In partcular, Spnes (009) develops a Schumpeteran growth model where nstutonal qualy drves nequaly and growth. In ths model, hgher publc sector neffcency dverts a larger share of both sklled and unsklled labour from productve to unproductve actves. Ths leads to hgher wage nequaly, and negatvely affects the aggregate rate of nnovaton and growth. In the same manner, n Echer et al. (009), corrupton affects the decson of prvate agents n regards to obtanng educaton, whch then drves the aggregate dstrbuton of sklled and unsklled types of producton. Smlarly to Spnes (009), n the model of Mandal and Marj (00), corrupton dverts resources from productve sectors, whch contrbutes to the wage gap n dfferent sectors. Combnng both these mechansms, Chong and Gradsten (007a) lnk the qualy of nstutons to the rents captured by prvate agents, whch n turn, drve the productvy of prvate agents. These authors show that the varably of ncome across agents ncreases wh hgher rates of rent-seekng whch are n turn fuelled by weak nstutons. Even more straghtforward, n Alesna and Angeletos s (005) paper, corrupt rent-seekng by creatng unequal access to ncome flows drectly generates ncome nequaly. The aforementoned models show that at the macro level, nequaly and corrupton are posvely correlated due to ncome and productvy effects caused by corrupton or nstutonal qualy. However, the mechansms leadng to those effects do not ncorporate brbery and the productvy effect of government spendng (as n Barro, 990). It s well known that corrupton creates both ncome and productvy effects through the burden of brbes (Hunt, 007b, 004; Mocan, 008; and Svensson, 003) and dstortons n the productve externaly generated by government spendng (Mauro, 998; Tanz and Davood, 000; and Shlefer and Vshny, 993). Obvously, whout accountng for brbery one cannot reconcle the fact that brbes pad by the prvate agents ncrease wh ther level of ncome, albe does not reduce nequaly. Moreover, the exstng models cannot explan why the relatonshp between corrupton and nequaly s nonlnear at the macro-level (L et al., 000). Ths paper addresses these ssues by accountng for the burden of brbes and dstortons caused by corrupton n the productvy effect stemmng from government spendng. In order to ncorporate the effects of brbes and productvy effects of government spendng nto the relatonshp between corrupton and nequaly, s helpful to establsh an ntuve lnk between these factors. Frst, gven that the burden of corrupton s a decreasng functon of ncome, can be postulated that the amount of brbes pad ncreases wh ncome 4
at a decreasng rate. That s, n the nequaly-nducng settng, the poor pay more brbes as a share of ther ncome than the rch, although the rch pay more brbes n nomnal terms. The latter outcome s que straghtforward as, f we consder ths from the effcency pont of vew; the demand for publc servces s greater amongst the agents wh a hgher level of productvy (Leff, 964; Huntngton, 968; Lu, 985; Svensson, 003; and Mocan, 008). On the other hand, the fact that the burden of brbes s regressve mples that the relatonshp between brbes and the agent s ncome level must be convex. An explanaton for ths may be that the gans from corrupton on the ndvdual level should be ncreasng faster than the burden of brbes. Then establshng the mechansm of how these gans from corrupton are obtaned by the brbe payng ndvduals wll be requred. In ths regards, seems reasonable to lnk corrupton outcomes at the ndvdual level to the agent s level of productvy, as the above-mentoned lerature suggests the demand for corrupt transactons ncreases wh the prvate agent s level of productvy. Therefore, as the corrupton-nduced productvy gans ncrease at a hgher rate than the burden of brbes, the relatve burden of corrupton on the wealthy s fallng. The above ratonale stll cannot explan why the relatonshp between corrupton and nequaly s nonlnear at the macro-level. Specfcally, s stll unclear why n case when the ncdence of corrupton s hgh (or low) s effect on the level of nequaly s weaker than when the ncdence of corrupton s of an ntermedate level (L et al., 000). To understand the causes of ths nonlneary n the nequaly-corrupton nexus, one needs to explan that why the relatonshp between corrupton-nduced productvy and the level of corrupton s of the concave form. To do that, recall that corrupton dstorts the posve externaly provded by government spendng (Barro, 990) and hence, an ncrease n the ncdence of corrupton leads to a declne n the productve externaly provded by government spendng (see, for example, Mauro, 998; Tanz and Davood, 000, Shlefer and Vshny, 993; and Dzhumashev 04a). Thus, follows that as long as wh hgher levels of corrupton, an ndvdual gan from corrupton s fallng due to ncreasng declne n the productve externaly provded by government, one can expect that the nequaly-corrupton nexus wll be a concave functon. In lght of the above dscusson, one can conjecture that the non-lneary of dependence between corrupton and nequaly s drven by the effect of corrupton on governmentnduced productvy n two ways. Specfcally, one can postulate that corrupton affects the ncome level of agents through the followng channels: () the burden of brbes as a share of 5
ncome declnes wh hgher corrupton-nduced productvy gans, () hgher corrupton leads to ncreasng negatve effects by reducng the average amount of productve publc nputs for the prvate agents. 3 In other words, there s a drect effect of corrupton on nequaly by creatng dspary n terms of the burden of brbes and productvy gans at the ndvdual level. In addon, there s an ndrect effect of corrupton on nequaly as the magnude of corrupton-nduced ndvdual gans extracted from the nteractons wh the publc sector s condonal on the overall effcency of the publc sector. Thus, wh hgher levels of corrupton, relatve gans from corrupton for the rch segments of populaton fall. 4 Takng nto account the above arguments, ths paper develops a model where corrupt bureaucrats and prvate agents nteract, whch produces both the drect and ndrect effects of corrupton on the ncome levels of prvate agents. The results of the analyss suggest that as soon as there s dspary n the productvy of agents that stems from publc sector nvolvement, the drect burden of corrupton s dsproportonately borne by the poor. In addon to the burden of brbes, corrupton generates an ndrect burden by affectng the productvy of prvate agents through dstortons n the posve externales provded by publc servces. The current study demonstrates that when corrupton levels are low, the negatve mpact of the declnng average publc sector productve externales s less than the benef of corrupton obtaned by an agent. However, wh the levels of corrupton ncreasng beyond ntermedate levels, the loss of productvy stemmng from reduced publc externales becomes hgh enough to offset most of the benefs n the form of corruptonnduced ndvdual productvy gans. Therefore, these fndngs suggest a theoretcal explanaton for that why at the macro level the relatonshp between corrupton and nequaly s nverted U-shaped (L et al., 000) whle the burden of brbes s regressve (Hunt and Lazlo, 0a). Overall, ths study adds to the growng lerature on corrupton by showng the 3 One can also recall that not only do the productvy gans decrease wh hgher corrupton, but s possble that the burden of brbes falls as well. For example, Mller (006), Kngston (007, 008), Çule and Fulton (009), Mshra (006) and Hunt (004) fnd that a hgher acceptance of corrupton through socal connectons decreases s burden. For example, Hunt (004) fnds that the development of a blateral trust between the prvate and publc agent leads to the substuton of an mplc qud pro quo for a brbe, thus reducng brbery. In smlar ven, Kngston (008) consders nformal relatonshps among brbe payers that help them to enforce agreements aganst payng brbes. These arrangements also lead to a reducton n the ncdence of brbe payments. However, payng less n brbes, for example to conceal non-complance to regulatons or tax evason, does not mean a reducton of the ncdence of corrupton. 4 Accordng to Bardhan et al. (007), there s also a negatve feedback effect from nequaly to government spendng that contrbutes to a decrease of productve externaly provded. Chong and Gradsten (007b) also fnd that a hgher nequaly results n a larger nformal economy, whch mples that the tax revenue collected by the government falls, hence, government spendng also declnes. 6
mportance of both ndvdual productvy dfferentals (drect effect) and the overall publc sector posve externales (ndrect effect) n determnng the effect of corrupton on nequaly. The rest of the paper s structured as follows: n Secton, the related lerature s dscussed n more detal, and n Secton 3, the theoretcal model s presented and mplcatons are drawn. Secton 4 concludes the study.. MODEL OF THE ECONOMY WITH CORRUPTION.. The Envronment There are two types of agents: bureaucrats and producers. It s assumed that there s no populaton growth and no socal mobly, so that those who start as a producer, reman a producer. Workers pay tax at a fxed rate,. Tax revenue s used to pay salares of the bureaucrats that are provdng publc servces. Gven that a fracton of populaton,, s workers and the rest are bureaucrats, the followng equaly holds: where yt s the average ncome of the worker agents and y w( ), () rate. It s assumed that the bureaucrats are pad the market wage rate, t t wt s the market determned wage w t. Ths mples that n the absence of corrupton, the expected (statutory) amount of servces to be provded to each prvate agent, s gven by qˆ y. () In ther nteractons, the publc and prvate agents play a smple sequental game. The publc agent moves frst by shrkng and provdng less than the statutory level of servces. The prvate agent observes the move of the publc agent and then moves eher by acceptng the sub-standard servces or by offerng a brbe to obtan hgher than the nal level of servces offered by the corrupt publc agent. Then the game ends. That s, by decreasng the supply of servces to an admssble mnmum, q, the suppler effectvely creates rents and captures them. In ths respect, the ratonale of such an assumpton s based on the fndng of Gurev (004), who shows that corrupton rases the equlbrum level of red tape; and of Shlefer and Vshny (993), who demonstrate that hgher red tape results n shortages of the servces beng provded. In the gven context, corrupton results n the mposon of more red t 7
tape on the producer, whch leads to a reducton n the amount of publc servces beng offered to the admssble mnmum, q... Workers The worker-agents, ndexed by [0,] productvy gans from publc servces, captured by coeffcent, are dentcal except n ther capacy of. Ths coeffcent s nterpreted as an ndvdual s ably to ncrease hs or her prvate productvy usng the publc sector through corrupt deals wh the bureaucrats. It s assumed that ths productvy coeffcent s normally dstrbuted across the worker agents as N(, ), where s the mean and s the varance of ths dstrbuton. Each agent s self-employed and has an dentcal endowment of nal capal, k0 and a un of tme n each perod. Thus, nally there s no wealth nequaly; hence, any change n nequaly wll be explaned by changes n the productve factors that may be caused by corrupton. The producer combnes capal and government goods to produce the sngle fnal good. In ths sense, the producton functon s smlar to one employed n Barro (990). It s assumed that the output of a self-employed agent, at each date t, s gven by the followng functon: k, y ( ) k g, (3) where g q ( q q) q, q s the average per worker amount of publc servces, and 0. It s assumed that the prvate agent pays a brbe proportonal to the amount of addonal servces he or she s tryng to obtan. That s, b( q q), where b s the brbe rate. Ths constutes as the response by the producer to the nal move of the bureaucrat. Snce, the valuaton of the addonal servce,, may dffer dependng on the ndvdual attrbutes of the agent and the level of corrupton, those who attach a hgher value to the servces obtaned are wllng to pay more n brbes to obtan more of the publc servces. Ths assumpton s justfed by the fndngs of Lu (985), who has shown that those who value the servce to be obtaned from the corrupt bureaucrat have a hgher wllngness to pay brbes on the effcency grounds. Moreover, emprcal studes by Svensson (004), Mocan (008) and Hunt (007) ndcate that those who have hgher ncomes also demand more servces. 8
Supportng ths lne of reasonng, there s evdence showng that government spendng benefs low and mddle ncome groups dfferently (Rone et al., 009). By lnkng together the level of productvy that results n a hgher ncome and the demand for publc servces suppled by corrupt bureaucrats, allows us to explan why brbe payment ncrease wh ncome. The dfference n the corrupton-nduced productvy gans may also stem from havng socal networks entwned n the publc sector operatons. The stronger the role of these networks n corrupton transactons the hgher s the ncdence of corrupton, as these connectons can faclate enforcement of collusve arrangements (Choe et al., 0; Kngston, 007; 008; and Hunt, 004). Therefore, n some envronments the ncdence of corrupton s hgher because the socal norms n place perm corrupton, creatng greater ndvdual benefs compared to envronments where corrupton s less wdespread (Mshra, 006; and Mller, 006). Ths ratonale leads one to postulate that wh a rse n corrupton, the prvate benefs n terms of ndvdual productvy gans ncrease for all prvate agents. Ths mples the value of s ncreasng n the overall level of corrupton. In other words, f there s an ndex, that measures the extent of corrupton n the economy, then 0. However, may be more realstc to assume that ths relatonshp s concave n the level of corrupton, as the gans n ndvdual productvy through corrupton are most lkely to be dmnshng. That s, 0.... Workers optmal choces Let us consder agent s optmal choces. The objectve of the worker agent s to maxmze ther ntertemporal utly by choosng ther consumpton level, c, and the level of government goods, q, gven ther socal status and nal capal. That s, max cq, t u ln c e dt (4) 0 s.t. k Ak g b q q c, (5) ( ) ( ) 9
and the nal endowment of capal, k0 k. It s assumed that a brbe s only pad f q that the transversaly condon, gven as q. To exclude non-optmal cases, s assumed ( rt ) lm ke 0 t, (6) holds. The amount of publc servces obtaned after payng brbes s g q ( q q) q (7) and whout payng brbes s gnt qq. The Hamltonan of the workers problem s gven by H u( c ) t e ( ) Ak g b( q q) c. (8) Further, for smplcy and where does not obscure the presentaton, tme and agent ndces are dropped. The frst-order condons of the above optmal control problem are gven by: H c t u( c) e 0, (9) H k ( )( ) qa b 0, (0) q q ( q q) H q ( q q) ( ) A. () k k From (0), the followng equlbrum expresson for the amount of publc servces demanded s obtaned: ( )( ) qa q k q. () b Combnng (9) and (), one can fnd the growth rate of consumpton, 0
c g ( ) A. (3) c k Analysng the equlbrum amount of publc servces demanded yelds the followng proposon. Proposon. An agent s demand for publc servces rses wh wealth, and falls wh the q q brbe rate. That s, 0 and 0. k b q q Proof. It s straghtforward to show that 0 and 0, by takng the frst-order k b dfferental of () wh regards to k and b : q ( )( ) qa k b 0 (4) q k ( )( ) qa b b 0. (5) The above results are que ntuve. Wealther people demand a greater amount of productve publc servces, whch s an emprcally observed fact (Svensson, 004; Mocan, 008; Hunt and Laszlo, 0b and Hunt, 007). On the other hand, the more extortve bureaucrats become through the mposon of hgher brbe rates, the lower s the demand for publc servces, as ncreases the margnal cost of obtanng publc servces for the prvate agents. Next, s mportant to ascertan how the amount of publc servces demanded by the prvate agents depends on the productvy of the ndvdual agent nduced by the publc sector. The followng proposon summarzes the lnk between the productvy of the agents and the amount of publc servce demanded. Proposon. The amount of publc servces s a concave functon of the corrupton-nduced productvy. Proof. To establsh the curvature of the functon, let us determne the frst-order and the second-order dervatves of q wh regards to. It can be verfed that
q ( )( ) qa q k 0 b and 3 q ( )( ) qa k 3 b q 0 hold. Therefore, the amount of q s ncreasng n, at a decreasng rate. The man fndng here s that a one un ncrease n the corrupton-nduced productvy measure wll lead to less than one un ncrease n the productve publc servces obtaned. The fndng stated n Proposon, allows one to draw a concluson about the average value of the amount of publc servces demanded and the average agent s productvy. Ths concluson s stated as the followng lemma. Lemma. The rato of the average amount of publc servces demanded to the average q productvy benef of the prvate agents s a decreasng functon. That s, 0. Proof. Let us consder agent, who experences one un ncrease n hs or her corruptonnduced productvy measure, whch results n an ncrease n the publc servces obtaned by less than one un, due to Proposon. That s,. Then the ncremental ncrease n the average productvy nduced by the publc sector wll be equal to q, whereas the n q ncremental ncrease n the average publc servces wll be gven by q. Clearly, n q. Snce, ths outcome s true for an arbrary agent, s also true for all of them smultaneously..3. The Bureaucrats For smplcy, s assumed that the bureaucrats do not save. Ths assumpton does not change the general structure of the analyss beng conducted. An ndvdual bureaucrat deals wh n workers, and the average amount of publc servces the workers requre s gven by qˆ q qdf( q), (6) q
where Fqs ( ) the cumulatve dstrbuton of the level of publc servces demanded by the prvate agents. There s a cost to the bureaucrat for beng corrupt, whch s ncreasng n the dfference between the statutory and mnmum amounts of publc servces offered by the bureaucrat. Ths cost s specfed as ( qˆ q), where s a cost parameter, ˆq s defned as n () and q s determned through the optmzng behavour of the agents. 5 The bureaucrat maxmzes her or hs utly gven by max ln c, q t ct e dt (7) 0 s.t. the budget constrant, ( ) c w nb q q q q. (8) ˆ t t The Hamltonan of ths problem s gven by H ˆ u( c ) t t e wnb( q q) ( q q) c t. (9) The frst-order condons (FOCs) are gven as follows: H c 0 u( c ) e t, (0) H q nb ( q q) 0. () ˆ Solvng for q from () we obtan that n equlbrum the bureaucrat sets the mnmum level of servces at the expected level. That s, nb q qˆ. () By analysng the above result, the followng lemma s formulated. Lemma. The mnmum level of publc servces offered falls wh an ncrease n the brbe rate, b and rses wh the cost of bureaucratc corrupton,. 5 Ths assumpton follows Chen (003) and Dzhumashev (04a,b). 3
Proof. Gven that the expected amount of publc servces, ˆq and the number of prvate agents, n to deal wh are gven to the bureaucrat, hs or her behavour only alters the second term on the rght-hand sde of (). Then, clearly, an ncrease n the brbe rate, b results n lower values, whereas an ncrease n the cost of corrupton leads to hgher values of the mnmum level of publc servces, q. In the next secton, the equlbrum condons are stated and the dynamcs of the model consdered..4. Equlbrum Equlbrum n the economy s defned as the streams of consumpton { c( t), cj( t)} 0, physcal capal, { k ( t)} 0, prces, { w( t), r( t)} 0, the tax rate, () t, the actual level of publc servces obtaned, { q ( t)} 0, the mnmum level of publc servces offered, { qt ( )} 0, and the brbe rates, { bt ( )} 0, such that they satsfy the optmaly condons obtaned for the household's and bureaucrat's problems, as descrbed above. Gven the producton functon by equaton (3), the productvy of a prvate agent depends on the actual government servces obtaned relatve to the stock of capal owned. Combnng () and (7) one can fnd ths rato as g q( )( ) A g k b. (3) Equaton (3) leads to the followng lemma. Lemma 3. The effectve government servces for an agent, gven by (3) are ncreasng n the corrupton-nduced productvy gan,, and decreasng n the brbe rate, b. Proof. The result s straghtforward from (3). The dynamcs of ths system are descrbed by two dfferental equatons n c and k gven by (3) and (5) correspondngly. The growth rate of consumpton, gven by (3) as c g c ( ) A, has the followng general soluton: c k 4
c c e [( ) Ag ] t 0, (4) where c0 s the exogenously gven nal level of consumpton. Then the growth rate of capal, gven by equaton (5), can be wrten as k Ag k b q q c e [( ) Ag ] t ( ) ( ) 0. (5) By accountng for (), one can further smplfy (5) as bq( ) k k c e, (6) [( ) Ag ] t 0 where ( ) Ag ( ) b. The general soluton of equaton (6) s bq( ) c k C e e, (7) t 0 [( ) Ag ] t where ( )( ) A. Now, substute for k from (7) nto the b transversaly condon (6) and obtan, bq( ) 0 lm t C t e e 0 t c. Clearly, wh tme, the last two terms n the square brackets converge to zero. Then, for the transversaly condon to hold, must bec 0. 6 Therefore, combnng (7) and (4) one can establsh that c k bq( ). (8) 6 See Barro and Sala--Martn (004, p.08). 5
Ths mples that consumpton and capal grow at the same rate. Therefore, the followng lemma s stated. Lemma 4. The growth rates of consumpton and capal satsfy k. c Proof. Due to the equlbrum condon, gven by (8), the level of consumpton of an agent s proportonal to the capal stock he or she owns. Ths mples that on the balanced growth path, both consumpton and capal grow at the same rate. c g Snce, the growth rate of consumpton s gven as ( ) A c k, s straghtforward to see that ths growth rate s constant and depends on the values of parameters reflectng the ncdence of corrupton, the effect of socal status, and the productvy of the economy. 7 3. Evoluton of nequaly The next step s to show how the dfference n the productvy levels due to the dfference n obtanng publc servces wll lead to an ncrease n nequaly. Snce, the productvy of each agent depends on the value of the rato of publc servces and capal, the magnude of ths measure across the agents wll drve the level of ncome and g k, the dfference n consumpton to dverge. Ths s how nequaly evolves between agents of dfferent socal statuses and hence, dfferent exposure to corrupton and productve publc sector externales. That s, the growth rate of consumpton of agent,, s ncreasng n the publc servces-to- capal rato, g k. Ths leads to the followng lemma. Lemma 5. The growth rate of consumpton and wealth of an agent posvely depends on the g effectve productve publc nput gven as the rato k. 7 A constant growth rate s a standard feature of the AK-type growth models. See, for example, Barro (990) for dscussons on ths ssue. 6
g Proof. Recall the growth rate of consumpton, ( ) A. Clearly, for any k g gven tax rate, hgher values of the rato, k, lead to hgher growth rates. The ntuon underlyng ths result s not very dfferent from what has been stated by Barro (990): output (ncome) s ncreasng wh productve publc nputs, g. The only dfference, n ths case s that the agents are heterogeneous n terms of demand for publc nputs, so that those who are more productve wll be gettng more of the publc nputs due to dstortons n the delvery of publc servces as a result of corrupton. Based on ths ntuon one can state the followng lemma. g Lemma 6. The rato k. s ncreasng n the corrupton-nduced productvy of agent, Proof. Usng the defnon of g k g, can be verfed that 0. Ths result demonstrates how the dversy n the corrupton-nduced productvy gans results n dfferences n the publc servces obtaned by prvate agents. The mportant pont here s that the dfferences n the productvy wll not lead to dfferences n the level of publc servces obtaned unless there s corrupton n the process of delverng the publc servce. Furthermore, the effect of corrupton s reflected on the dsperson of growth rates of consumpton and wealth across agents, as the followng proposons states. Proposon 3. The growth rate of consumpton and wealth s ncreasng n the corruptonnduced productvy of the agent and decreasng n the brbe rate. Proof. It can be verfed that t 0 and b 0. The detals are gven n Appendx A. 7
In lght of the above fndngs, one can state that heterogeney of prvate agents n terms socal or polcal status mmnently leads to economc nequaly. Importantly, the evoluton of ths nequaly s possble only when there s corrupton n the delvery of publc servces. It s reasonable to expect that nequaly wll grow n an envronment where the agents have dfferent growth rates of capal accumulaton. A more rgorous way of showng the evoluton of nequaly s to demonstrate the dynamcs usng a measure of nequaly. As a measure, one can adopt the logarhm of the rato of mean to medan capal stock per worker, followng Bandyopadhyay and Tang (0), Chong and Gradsten (007), and Benabou (00). That s, f capal has a lognormal dstrbuton, then the measure of nequaly s expressed as obtans kt t log k medan var log k. (9) Combnng (4) and (8), and takng a logarhm of both sdes of the equaton, one c0 log k Ag t log. (30) Clearly, the varance of log k depends on the varance of publc servces, consderaton allows one to state the followng proposon. g, and tme. Ths Proposon 4. Corrupton nduced dstortons n the dstrbuton of publc servces ncrease nequaly. Proof. It follows from (30) that c0 var log log ( ) var( ) k A g t. (3) Thus, wh tme var log k ncreases and hence, the level of nequaly, t, also rses. The detals are n Appendx A. 8
Gven that q( )( ) A g( ) b, from (3) s evdent that an ncrease n the average benef of corrupton for prvate agents,, results n a hgher value of the varance of wealth. Thus, when an ncrease n the ncdence of corrupton leads to the larger average productve gan,, ths would mply that corrupton s posvely correlated wh nequaly. However, when the ncdence of corrupton ncreases, results n the reducton of the average level of publc servces, q. Ths latter effect reduces the magnude of g( ), hence, that of the varance. It s only when the ncdence of corrupton s n the ntermedate range wll the average level of publc servces also be ntermedate; hence, the effectve publc servce, g( ), as a product of these two elements wll be the hghest. Therefore, nequaly grows faster when corrupton s ncreasng n the ntermedate range as the emprcal evdence suggests (L et al., 000). Ths ntuon leads us to the followng proposon. Proposon 5. If an ncrease n corrupton reduces the average level of publc servces more than ncreases the average level of corrupton-nduced productvy, the relatonshp between corrupton and nequaly s of an nverted U-shaped form (.e. concave). Proof. Based on Proposon 4 and usng equaton (3), one can see that the mpact of the varance of the productvy, nduced by the publc sector, on nequaly s drven by ths term: c 0 Alog g ( ). It s straghtforward to establsh that the varance of the publc sector-nduced productvy depends on g ( ) q( )( ) A ( ). Snce, b accordng to () the lower value of publc servces, q, s reduced when corrupton levels rse, the average value of publc servces, q, also falls. On the other hand, a hgher level of corrupton mples a hgher level of prvate benefs; hence, the average level of the corrupton benef,, s hgher. Snce, accordng to Lemma, q wll be of an nverted U-shaped form. 3.. Numercal llustraton holds, ( ) Although the fndngs of Proposon 5 are ntuve, a numercal smulaton of ths part of the model can be helpful to vsualze the concavy of the relatonshp between the level of g 9
corrupton and nequaly, and to see f the proposon s confrmed. In partcular, ascertanng f the expresson by ( ) g ( ) q( )( ) A, b exhbs a concave form wll also be useful. In the latter expresson, one can consder only the part that depends on the level of corrupton. Snce,, and A are exogenous parameters, one can dsregard ther contrbuton to the non-lnear relatonshp between the ncdence of corrupton and ncome volatly, and focus only on the mpact of the average value of publc servces, q, the average level of the corrupton-nduced productvy gans,, and the brbe rate, b. Ths leads to analysng the behavour of the measure g q by changng the b ( ) level of corrupton. Here, followng Chen (003), one can assume that 0.9. Fgure. Smulaton results for the element of varance of the ncome dstrbuton that depends on the level of corrupton, g 7.8 7.6 7.4 7. 7 6.8 6.6 6.4 6. 6 5.8 5.6 level of corrupton, ξ 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 Corrupton-dependent ncome nequaly Recall that both the average level of the corrupton-nduced productvy gans,, and the brbe rate, b, posvely depend on the level of corrupton n the economy. As aforementoned n the setup of the model, the level of ndvdual benef nduced by corrupton s a concave functon of the ncdence of corrupton. For practcal purposes, ths 0
functon can be assumed to be of the followng form: b = x p,0 < p <. From Proposon, s known that q s a concave functon of. To capture ths dea, we agan smply assume that q,0. In Fgure gven below, can be seen that the relatonshp between ncome nequaly and the level of corrupton has an nverted-u shape. In ths partcular case, 0.35 the smulaton s done assumng that, q 0.6 and a corrupton measure s defned as [0.,0.8]. Addonal calculatons show that the varance of the ncome dstrbuton wll have a concave form n the level of corrupton as soon as the curvature of the average publc nputs and the ndvdual corrupton-nduced productvy gans are sgnfcantly dfferent. In ths partcular smulaton, ths means that as long as the parameter values assumed for and are gven on the oppose sdes of pont 0.5, one wll obtan a varance for the ncome dstrbuton that s concave n the level of corrupton. 4. Concluson Ths study proposes an analytcal model that explans some of the emprcal evdence that relates corrupton to nequaly. In partcular, has been shown by several emprcal studes that the nomnal amount of brbes pad ncreases wh the ncome of the agents, whereas the burden of brbes as a share of ncome decreases. Thus, nequaly s posvely correlated wh corrupton. However, has also been demonstrated that at the macro-level, corrupton and nequaly have a nonlnear relatonshp: as n countres wh ntermedate levels of corrupton, the effect of corrupton on nequaly s stronger than n countres where corrupton levels are low or hgh. To reconcle these stylsed facts, ths paper develops a model buldng on the model of Barro (990) where government servces enter the prvate producton functon as an nput. The extenson proposed n the present study s that the effect of publc actves affects the prvate productvy not only through publc sector externales a la Barro (990), but also through the agents ndvdual productvy nduced by corrupton. The possbly of heterogeney of prvate agents n demandng publc servces depends on ther productvy as well as ther socal status, as hghlghted n the lerature, so the same ratonale s employed n the present paper. Gven the costs and benefs of corrupton for both prvate agents and bureaucrats, has been shown that for any gven level of corrupton, the exstence of heterogeney across agents n terms of corrupton-nduced productvy gans leads to an ncrease n nequaly.
In addon, more corrupton mples a greater dstorton n the delvery of publc sector servces. Thus, wh hgher levels of corrupton the overall publc generated productve externaly falls (ndrect effect), whereas the ndvdual gans (drect effect) from corrupton ncrease. The ndrect effect of corrupton tends to reduce, whereas the drect effect ncreases nequaly. Therefore, when corrupton s low the ndvdual gan from corrupton s also low, whle the overall publc sector posve externaly s hgh. As the level of corrupton ncreases, the prvate gans from corrupton rse whle the publc posve externaly falls. The latter effect grows stronger than the ndvdual gans from corrupton; hence, the varaton of ncome and wealth due to corrupton declnes after the corrupton level reaches some threshold value. Ths explans the observed non-lneary of the relatonshp between the corrupton level and nequaly. Appendx ( )A g and A. Proof of Proposon 3. Gven that q( )( ) A g b, one can obtan the followng: ( )( ) Ag g ( ), (3) ( ) s the frst-order dervatve of g wh respect to. Snce ( ) 0, mples that g 0. A. Proof of Proposon 4. Recall that 0 g c log k Ag t log. For a gven pont n tme, the varance of the rght-hand sde of ths expresson s as follows: var log var c c log log var 0 0 k Ag t At g. (33)
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