Discussion Paper Series

Human capital and income distribution in a model of

corruption By

Humna Ahsan and Keith Blackburn

Centre for Growth and Business Cycle Research, Economic Studies, University of Manchester, Manchester, M13 9PL, UK

June 2015

Number 208

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Human capital and income distributionin a model of corruption

Humna Ahsan and Keith BlackburnCentre for Growth and Business Cycles ResearchDepartment of Economics, University of Manchester

AbstractThis paper studies the role of corruption in determining the distri-

bution of income and, with this, the degree of poverty and inequality.The analysis is based on an overlapping generations model in whichindividuals may seek to improve their productive e¢ ciency by sup-plementing or substituting publicly-provided services (education andhealth care) with their own expenditures on human capital forma-tion. Financial market imperfections mean that their ability to dothis depends on their initial wealth status, implying the possibilityof persistent inequality in multiple long-run equilibria. We show howcorruption may exacerbate this by compromising public service provi-sion. This occurs through the double whammy of both reducing theearnings and increasing the population of those who rely most on suchservices. Higher levels of corruption are associated with higher levelsof poverty and may result in a complete polarisation between the richand poor through the elimination of any middle class.

Keywords: Corruption, human capital, inequality, poverty.

JEL Classi�cation: D31, D73, H41, O15.

1 Introduction

Recent years have witnessed a burgeoning literature on the role of corrup-tion in determining economic and social development.1 Underlying this has

1The most widely-used de�nition of corruption is the abuse of authority by publico¢ cials for personal gain. There are many excellent surveys of the existing literature,including Aidt (2003), Bardhan (1997), Jain (2001), Rose-Ackerman (1999), Svensson(2005) and Tanzi (1998).

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been a growing appreciation of the importance of governance for the func-tioning of society�s public institutions.2 Both theoretically and empirically,it has been shown how corrupt practices on the part of public o¢ cials cancompromise growth and exacerbate inequalities by distorting incentives, de-stroying opportunities, squandering resources and perverting public policy.In many countries the scale of such practices is often quite staggering, as isthe ingenuity of those who perpetrate them. Signi�cantly, these countries areamongst the poorest of the world, leading many development experts to viewcorruption as one of the greatest obstacles to alleviating global poverty. Thispaper seeks to make a further contribution to the literature in a theoreticalanalysis of corruption, inequality and income distribution.The thriving research on corruption and development has been stimulated

in large part by the construction and re�nement of various datasets thathave become increasingly accepted as providing reliable measures of corruptactivity.3 Armed with such data, a number of authors - including Gyimah-Brempong (2002), Keefer and Knack (1997), Knack and Keefer (1995), Liet al. (2000), Mauro (1995) and Sachs and Warner (1997) - have estimatedsigni�cant adverse e¤ects of corruption on growth. These and other studiesalso provide evidence on various ways in which corruption might take hold,such as lowering rates of investment (e.g., Mauro 1995), creating obstacles todoing business (e.g., Brunetti et al. 1997; Fisman and Svensson 2007; Kauf-mann 1997; World Bank 2002), reducing in�ows of foreign investment (e.g.,Wei 2000) and causing misallocations of public expenditures (e.g., Haque andKneller 2014; Mauro 1997; Tanzi and Davoodi 1997). Further evidence sug-gests that the direction of causation could go the opposite way, meaning thatthe incidence of corruption is, itself, determined by the level of per capitaincome (e.g., Ades and Di Tella 1999; Fisman and Gatti 2002; Montinolaand Jackman 1999; Paldam 2002; Rauch and Evans 2000; Treisman 2000).Whilst some of these �ndings have been challenged, the broad consensusis that corruption and growth are linked in a relationship that is generallynegative and possibly two-way causal.4 There are many theoretical analyses

2The concept of governance is broader than that of corruption, though there is an inti-mate connection between the two: just as bad governance fosters corruption, so corruptionundermines good governance.

3Known as corruption perception indices, these datasets provide rankings of countries interms of the extent to which corruption is perceived to exist based on questionnaire surveyssent to networks of correspondents around the world. For discussions and appraisals of theindices, see Jain (1998), Tanzi and Davoodi (1997), Treisman (2000). Further remarks,together with a review of the empirical literaure, can be found in Lambsdor¤ (2005).

4Of course, the strength of this relationship can vary across countries and regions, andvarious factors to explain this have been suggested, such as the quality of institutions, thedegree of �nancial openness and the way in which corruption is practised (e.g., Aidt et al.

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which provide explanations for this relationship, together with addressingvarious other issues relating to the macroeconomics of migovernance (e.g.,Acemoglu amd Verdier 1998, 2000; Blackburn et al. 2006, 2010; Blackburnand Forgues-Puccio 2007, 2009, 2010; Blackburn and Powell 2011; Black-burn and Sarmah 2008; Ehrlich and Lui 1999; Murphy et al. 1991, 1993;Rivera-Batiz 2001; Sarte 2000).The e¤ects of corruption are not con�ned to aggregate outcomes alone.

They extend to distributional outcomes as well, and it is this further as-pect of development which occupies our principal concern in this paper. Asabove, research in the area has �ourished over recent years, partly becauseof the quality improvements in data and partly because of the innovations inmodelling corrupt behaviour.At the empirical level, several studies have identi�ed a strong positive

correlation between the incidence of corruption and the degree of incomeinequality. Gyimah-Brempong (2002), using a panel of African countries,estimates sizeable increases in the Gini coe¢ cient as the level of corruptionincreases. Dincer and Gunalp (2008), employing data on US states, observesimilar e¤ects of corruption on di¤erent measures of inequality. Analogous�ndings appear in the contributions of Chong and Calderon (2000a), Guptaet al. (2002) and Gyimah-Brempong and de Camacho (2008), each of whichis based on a broader sample of both developed and developing countries.Foellmi and Oechslin (2007) present additional results which suggest that anincrease in the level of corruption leads to an increase in the income shareof the wealthiest members of the population. Finally, evidence presented byChong and Gradstein (2007) suggests that corruption and inequality may bemutually dependent, an increase in either causing an increase in the other.With due care and consideration, these observations may be given addedsign�cance by translating them to mean that corruption has the e¤ect of ex-acerbating the degree of poverty, as established explicitly in some of the fore-going studies (e.g., Chong and Calderon 2000a; Dincer and Gunalp 2008).5

At the theoretical level, it has been shown how corruption may impact oninequality through various diverse channels. Ahlin (2001) and Foellmi andOechslin (2007) develop occupational choice models in which at least someprivate agents must bribe public o¢ cials (bureaucrats) in order to engage

2007; Neeman et al. 2006; Svensson 2005).5Caution is needed since inequality and poverty are two di¤erent concepts. In Chong

and Calderon (2000b) and Li et al. (2000) the relationship between corruption and incomeinequality is found to be an inverted U-shape. An increase in corruption is associated witha decrease in inequality when inequality is initially high, but this does not necessarily meanthat there is a decrease in poverty: a higher level of corruption may imply a lower level ofinequality precisely because more of the population are made poor.

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in entrepreneurial activity that would make them better o¤. Both analysespredict that an increase in the size of bribe payments leads to a redistri-bution of wealth among the population. In the case of Ahlin (2001) thisresult is re�ected in an inverted u-shaped relationship between corruptionand inequality. In the case of Foellmi and Oechslin (2007) the result has theimplication that a high incidence of corruption can cause a polarisation inincome distribution. From a di¤erent perspective, Blackburn and Forgues-Puccio (2007) analyse the implications of corruption for redistributive policyin a model of bribery and tax evasion. It is shown how such behaviour canundermine attempts to reduce inequality by allowing the rich to bene�t at theexpense of the poor. Another avenue is explored by Alesina and Angeletos(2005) who develop a politco-economic model of corruption and redistribu-tion. The main result is that there are multiple equilibria in the extent ofgovernment intervention, the level of rent-seeking and the degree of incomeinequality because of mutual interactions which reinforce each other. Finally,Glaeser et al. (2003) study the distributional consequences of legislative (asopposed to bureaucratic) corruption in a model of institutional subversion.The narrative in this case is that corruption can lead to greater inequality byprejudicing the judiciary in a way that favours the wealthier, more powerfulmembers of the population to the detriment of the poorer, less in�uentialsections of society.In what follows we present an analysis of corruption and income distrib-

ution from a further public policy perspective. To many observers, the sub-version of public policy is one of the major obstacles to reducing inequality,causing both a bias in the tax system in favour of the rich and a deteriora-tion of social programmes designed to bene�t the poor. Tax evasion by thewealthy, in collusion with bureaucrats, reduces the tax base and makes thetax system more regressive so that the burden of taxation falls disproportion-ately on the non-wealthy. Moreover, for any given tax system, tax evasionimplies a loss of revenue to the government which may be forced to cut backon its expenditures targeted to the same group of low-income citizens (suchas payments of subsidies, spending on health, and funding of education).The availability, provision and quality of social programmes may be threat-ened even further through the increased costs of accessing these programmeswhen bribes are demanded, through the diversion of resources towards otheractivities that o¤er greater scope for rent-seeking, or through a more blatantappropriation of public funds in a manner that amounts to pure theft. Allof these pitfalls have been widely observed in practice and the literature oncorruption is replete with examples of them.The speci�c policy focus of our analysis is the government�s provision of

public goods and services designed to improve human development, especially

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amongst the poor. The cornerstones of this provision are public expenditureson education and health which are presumed to enhance human capital and,with this, the functionality and productivity of individuals.6 Signi�cantly,empirical support for this presumption is quite mixed. An extensive anddiverse body of evidence produces some fairly ambivalent conclusions aboutthe e¤ects of social spending programmes on various economic and socialindicators.7 Whilst the e¤ects are largely positive in the case of growth (e.g.,Baldacci et al. 2004; Barbiero and Cournede 2013; Blankenau et al. 2007;Kneller et al. 1999), they are much more ambiguous for other outcomes, suchas inequality and poverty (e.g., Chu et al. 2000; Dollar and Kraay 2002; Fanet al. 2002; Li et al. 1997; World Bank 2004), and education and healthstatus (Anand and Ravallion 1993; Bidani and Ravallion 1997; Filmer andPritchett 1999; Gupta et al. 2001; Harbison and Hanushek 1992; Pritch-ett 1996). One of the most widely-accepted explanations of this con�ictingevidence is that the e¢ cacy of social programmes is often compromised bypoor quality governance. This argument �nds strong support in a numberof empirical studies which seek to measure the e¤ects of corruption on edu-cation and health status through its e¤ects on the provision and quality ofpublic education and health programmes (e.g., Azfar 2001; Azfar and Gurgur2001; Baldacci et al. 2003; Dreher and Herzfeld 2005; Gupta et al. 1999,2001; Kaufmann et al. 1999, 2004; Lewis 2006; Reinikka and Svensson 2005;Rajkumar and Swaroop 2008). Without exception, these e¤ects are foundto be signi�cant and negative. Corruption undermines the e¤ectiveness ofsocial programmes by causing a wastage of programme funding through em-bezzlement and overspending. This compounds another well-known e¤ect ofcorruption, which is the reduction of such funding to begin with due to themisallocation of public expenditures, the composition of which is distortedaway from pro-development areas (like education and health) towards lessproductive areas (such as defence and infrastructure) (e.g., De la Croix andDalavallade, 2009; Delavallade 2006; Gupta et al. 2002; Mauro 1997). Forthese reasons, corruption can signi�cantly impede human development, es-pecially amongst the poor who may �nd themseleves denied of basic public

6Note that these expenditures may cover not only the obvious items (school/hospitalbuildings and equipment, teachers�/doctors�wages and salaries, etc.), but also targetedareas of infrastructure (such as road, electricity and water supplies to schools/hospitals).For a wide-ranging discussion of the many issues involved (based speci�cally on a humandevelopment perspective), see Mehrotra and Delamonica (2007).

7There is, of course, a related, but distinct, body of research which focuses on thelinkages between the indicators, themselves, such as the correlation of income and growthwith education and health status. For overviews of this research, see, for example, Kruegerand Lindahl (2001), Lopez-Casasnovas et al. (2005) and Strauss and Thomas (1998).

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services which may o¤er their only means of escaping from their plight. Inthis way, corruption may foster both inequality and poverty. The objectiveof our analysis is to provide a simple illustration of this.We present an overlapping generations model in which agents earn income

according to their human capital which we interpret broadly to include botheducation and health status. The government pursues a social programmeof providing education and health care using whatever public funds are atits disposal. Agents may enhance their human capital beyond the level im-plied by this programme alone through various types of personal expenditurewhich may either partially supplement or wholly substitiute their use of pub-lic services. In the case of the former, this might include spending on tuition,medication, nutrition, location, lifestyle, sanitation and basic utilities.8 Inthe case of the latter, it would mean paying for schooling and medical carein the private sector. An agent may or may not be able to a¤ord these ex-penditures depending on her idiosyncratic inheritance of wealth. If not, thenthe agent must borrow under the terms and conditions of loan contracts in�nancial markets. Imperfections in these markets mean that loans are ex-tended only to those agents with su¢ cient wealth to serve as collateral. Thisleads to a limiting wealth distribution that depends on the initial distrib-ution, together with public policy. Against this background, we introducecorruption in the form of the embezzlement of public funds. The immediateconsequence of this is to reduce public service provision which impacts ondistributional outcomes in two ways: the �rst - a wealth e¤ect - is that anyagent who relies on this provision to any extent is made worse o¤; the second- a credit e¤ect - is that the number of agents who rely exclusively on thisprovision is increased. Together, these e¤ects constitute a double-whammyfor inequality and poverty. An extreme outcome of this is the elimination ofany middle-income class of agents and the polarisation of the population into the rich and poor.The remainder of the paper is orgnised as follows. In Section 2 we set out

our basic model of public policy and human capital acquisition. In Section3 we analyse the distributional implications of this model. In Section 4 weincorporate corruption into the model and study the consequences of this. InSection 5 we discuss some extensions of our analysis. In Section 6 we makea few concluding remarks.

8These expenditures can work both directly and indirectly, and their e¤ectiveness isenhanced by the complementarities between education and health. For a review of theliterature on the links between education and health, see Cutler and Lleras-Muney (2006).

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2 The Basic Set-up

We consider a small open economy in which there is a constant population, N ,of two-period lived agents belonging to overlapping generations of dynasticfamilies connected through altruism. Each agent has one parent and onechild, inheriting wealth from the former when young and bequeathing wealthto the latter when old. Young agents occupy themselves with acquiringhuman capital in one way or another. Old agents use their human capitalto work for �rms in the production of output. All agents have identicalpreferences and all markets are competitive.

2.1 Households

Each agent derives lifetime utility from her own old-age consumption andthe bequests that she leaves to her o¤spring.9 The utility of an agent bornat time t is given by

ut = c1� t+1 b

t+1; (1)

( 2 (0; 1)) where ct+1 denotes consumption and bt+1 denotes bequests. Letxt+1 be the total lifetime income of the agent so that ct+1 + bt+1 = xt+1.Then the allocations of consumption and bequests that maximise (1) arect+1 = (1� )xt+1 and bt+1 = xt+1, implying ut = �xt+1 (� = (1� )1� ).Accordingly, the agent�s �nal payo¤ is determined directly by the value ofher �nal income.In the �rst period of life an agent receives her inheritance of wealth and

acquires human capital, the latter of which determines her future produc-tivity and, with this, her future labour income. The agent acquires humancapital in one of three ways: the �rst is by relying solely on the government�sprovision of public goods and services in education and health care; the sec-ond is by exploiting this provision as well, but supplementing it with her ownexpenditures on education and health; and the third is by foregoing such pro-vision entirely and substituting it with more costlier purchases of educationand health services from the private sector. Let e denote the agent�s personalexpenditure on human capital acquisition. In order of the aforementioned al-ternatives, we specify e = 0, e = p > 0 and e = P > p. Thus we assume thatsupplementing or substituting public goods provision entails a �xed cost foran agent. This assumption is used for the purposes of simplifying our mainanalysis. As we subsequently demonstrate, our key results are unchanged

9As in other models, we account for intergenerational altruism in the simplest way byassuming that parents derive utility from the size of their bequests, as opposed to theutility of their o¤spring. For further discussion, see Andreoni (1989).

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in an extension of the model that allows agents to choose optimally theirpreferred personal expenditures on human capital.10 Whatever the case, forany given level of these expenditures and any given inheritance of wealth,an agent is either a net borrower or a net lender, where both borrowing andlending take place at the exogenous world rate of interest, r > 0.In the second period of life an agent supplies one unit of labour to �rms

and earns a wage that depends on her human capital. Our basic assumptionis that the aforementioned alternatives for acquiring education and health atsuccessively higher costs deliver successively higher levels of human capital.This is a revealed presumption of individuals who are generally willing toincur personal expenditures on their own (or their children�s) education andhealth if they can a¤ord to do so. For example, it has been found that wealth-ier households spend more on various school fees and learning activities, andhave a greater inclination towards using more costly health services (e.g.,Foko et al. 2012; Gertler and van der Gaag 1990; Tilak 2002). Evidence todirectly support the assumption - especially the relative e¤ectiveness of pub-lic and private provision - is also available. For example, a number of studiesfor di¤erent countries conclude that private schooling has a positive e¤ecton measures of educational attainment and labour market performance (e.g.,Bedi and Garg 2000; Binelli and Rubio-Codina 2012; Brown and Bel�eld2001; Calonico and Nopo 2007; Cox and Jimenez 1991; Jimenez et al. 1991a,1991b). In the context of our model, the assumption is re�ected in threepossible levels of wages - a wage of w1 for agents who rely solely on publicgoods provision, a wage of w2 > w1 for agents who supplement this provisionwith personal expenditures and a wage of w3 > w2 for agents who substitutesuch provision entirely with private services.Based on the above, we may write the �nal income of an agent as

xt+1 =

8<:(1 + r)bt + w1 if e = 0;(1 + r)(bt � p) + w2 if e = p;(1 + r)(bt � P ) + w3 if e = P:

(2)

Throughout our analysis, we assume the parameter restriction w1 < w2 �(1 + r)p < w3 � (1 + r)P . This has two implications: �rst, for any givenlevel of bequests, an agent always prefers to spend more, rather than less, onacquring human capital; second, if such expenditure requires borrowing, anagent is always able to repay her loan. As we shall see, these features do not

10We also assume that relying solely on publicly-provided education and health servicesentails no cost for an individual. Of course, there may be out-of-pocket expenses (such asthe costs of school uniforms, basic equipment and travel), but to the extent that these arealways incurred regardless of other choices, we may normalise them to zero.

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trivialise the problem facing agents: whether or not preferences are actuallyrealised when borrowing is needed depends on the terms and conditions ofloan contracts in the presence of �nancial market imperfections.It is evident at this stage that the population of households is potentially

distributed across three groups. Each of these di¤er in terms of personalexpenditures on human capital and wage earnings from labour. Speci�cally,there is a group for which expenditures are zero and wages are w1, a groupfor which expenditures are p and wages are w2, and a group for which expen-ditures are P and wages are w3. We denote the corresponding populations ofthese cohorts by N1, N2 and N3. As is apparent from (2), we abstract fromany taxes that the government might levy on one or more of these cohortsin order to �nance its provision of public goods and services. In a subse-quent extension of our analysis, we show how the model can be modi�edstraightforwardly to accommodate this without altering its main results. Inview of this, we prefer to keep matters simple by assuming that the govern-ment �nances its social programme through windfalls of public funds (e.g.,from foreign aid or natural resources). This assumption also serves to illus-trate how corruption can compromise a purely bene�cial public policy thatis costless to any member of society.

2.2 Firms

There is a unit population of �rms, each of which produces output, y, usinglabour from all cohorts of households. The production technology is sum-marised by

y = A1N1 + A2N2 + A3N3; (3)

A1 = a1(g); A2 = a2(g; p); A3 = a3(P ); (4)

where g denotes public expenditures on education and health services. Fol-lowing our previous discussion, we distinguish between the productivities ofdi¤erent groups of households according to their use of public services andtheir own personal expenditures on human capital (i.e., either p or P ). This isre�ected in the functions ai(�), which are understood to be increasing in eachof their arguments and to deliver the productivity ranking A3 > A2 > A1.Given the above, pro�t maximisation implies

wi = Ai: (5)

Thus, consistent with our prior assumption, w3 > w2 > w1.

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2.3 Banks

As indicated earlier, some agents may need to borrow in order �nance theirpersonal expenditures on human capital. Whilst there are no problems ofbankruptcy, capital market frictions exist because of imperfect enforcementof loan contracts. Following others (e.g., Banerjee and Newman 1993; Galorand Zeira 1993), we model this as follows.Borrowing is undertaken via competitive �nancial intermediaries which

have access to a perfectly elastic supply of loanable funds at the world interestrate, r. Suppose that an agent puts up all of her inherited wealth, bt, ascollateral against a loan of either p or P for human capital investment. Theagent has an opportunity of strategically defaulting by absconding with someproportion of this loan, 1�� 2 (0; 1), where � provides a measure of contractenforcment. Doing so, however, means that she foregoes any wage earningsand loses all of her collateral. Naturally, the payo¤ from defaulting must beno greater than the payo¤ from non-defaulting if defaulting is not to occur:formally, either (1��)p � (1+r)(bt�p)+w2 or (1��)P � (1+r)(bt�P )+w3,depending on the size of the loan. When holding with equality, each of theseincentive conditions de�nes a critical level of wealth - bc or bcc - below (above)which loans are denied (granted). That is,

bc =(2 + r � �)p� w2

1 + r; (6)

bcc =(2 + r � �)P � w3

1 + r: (7)

Under the parameter restriction (2+r��)p�w2 < (2+r��)P�w3, we havebc < bcc. In summary, only if an agent inherits a level of wealth above bc (bcc)is she able to acquire a loan of p (P ) to �nance her personal expenditures oneducation and health. Otherwise, she is denied such credit and is forced torely on her next best alternative for acquiring human capital.

3 The Dynamics of Wealth Distribution

Our �rst step in determining the evolution of income distribution is to de-termine the dynamics of wealth for each dynastic household. Then, givenany initial distribution of income, we may use these dynamics to deduce thechanges in the relative fortunes of dynasties and thereby establish long-rundistribution outcomes.As we have seen, an agent�s ability to make personal investments in her

human capital depends on her inherited level of wealth, bt. Only if this

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inheritance is at least equal to some critical value - bc in (6) - is she able tomake an investment of p, and only if the inheritance is at least equal to somehigher critical value - bcc in (7) - is she able to make a greater investment of P .As we have also seen, each agent of each generation bequeaths an inheritancethat is a constant fraction, , of her realised income, xt+1 in (2). Based onthese observations, we may conclude that the intergenerational evolution ofwealth for an individual dynasty satis�es

bt+1 =

8<: [(1 + r)bt + w1] � f1(bt) if bt < bc; [(1 + r)(bt � p) + w2] � f2(bt) if bc � bt < bcc; [(1 + r)(bt � P ) + w3] � f3(bt) if bt � bcc:

(8)

Assuming that (1 + r) 2 (0; 1), each of these lineage transition equations isstable and implies convergence to a unique steady state, as given by

b�1 = w1

1� (1 + r) ; b�2 =

[w2 � (1 + r)p]1� (1 + r) ; b�3 =

[w3 � (1 + r)P ]1� (1 + r) : (9)

Evidently, b�1 < b�2 < b

�3 by virtue of our earlier restriction on parameters.

The above results are depicted in Figure 1 for the most interesting andleast trivial scenario in which b�1 < b

c < b�2 < bcc < b�3. This ordering ensures

that both of the critical levels of wealth are relevant in determining thelong-run distribution of income.11 The characteristics of this distribution aresummarised by three di¤erent groups of agents. First, any agent for whombt < b

c is denied entirely of any credit for funding personal expenditures onhuman capital. Such an agent is forced to rely solely on the public provisionof education and health services, implying a relatively low level of wages,w1, and a relatively low long-run level of wealth, b�1. Second, any agent forwhom bt 2 (bc; bcc) is able to acquire a loan of size p to �nance her own humancapital expenditures. Such an agent supplements her use of publicly-providedservices with value-added personal spending, implying a higher wage of w2and a higher limiting wealth of b�2. Third, any agent for whom bt > bcc iseligible to borrow a larger amount, P , for human capital investment. Suchan agent substitutes public services with private services entirely, implyinga further improvement of wages to w3, and a further improvement of steadystate wealth to b�3.

11For example, if the only di¤erence in the ordering is b�1 > bc, then all agents for whom

bt < bc to begin with would end up at b�2, the same as all agents for whom bt 2 (bc; bcc)

to begin with; hence bc is irrelevant in the long-run. Similarly, if the only di¤erence isb�2 > b

cc, then all agents who start o¤ with bt 2 (bc; bcc) would converge to b�3, the same asall agents who start o¤ with bt > bcc; hence bcc is irrelevant in the long-run. Other (moreextreme) degenerate cases are also ruled out.

11

Based on the above, we may conclude that the population is dividedinto three income classes - a low-income class of size N1, a middle-incomeclass of size N2 and a high-income class of size N3 = N � N1 � N2. Thisdivision re�ects the initial distribution of wealth, meaning that any inequal-ities to begin with tend to persist, rather than vanish, over time. LetHt(bt) =

Rht(bt)dbt denote the cumulative distribution function of wealth

at time t so thatR bbht(bt)dbt provides a measure of the population with

bt 2 (b; b). In the absence of any class mobility the measure of each incomegroup is time-invariant. That is, the sizes of these groups are �xed by theinitial distribution, H0(b0), together with the wealth thresholds, bc and bcc.Speci�cally, we have

N1 =

Z bc

0

h0(b0)db0; N2 =

Z bcc

bch0(b0)db0; N3 = N �

Z bcc

0

h0(b0)db0: (10)

The precise extent to which the population is divided depends on two mainfactors - the degree of capital market imperfections and the scope of publicpolicy. From (6) and (7), together with (4) and (5), each of these is seento a¤ect the threshold levels of wealth. As regards the former, a strongerenforcement of loan contracts (i.e., an increase in �) implies a reduction inboth bc and bcc such that N1 decreases and N3 increases, whilst N2 couldgo either way. As regards the latter, a greater provision of public goods andservices (i.e., an increase in g) implies a reduction in bc such thatN1 decreasesandN2 increases. In each case there are fewer agents on low incomes and moreagents on higher incomes, which means not only that the average income ofthe population is higher, but also that the degree of poverty in the economyis lower. In the case of public policy, there is also a further e¤ect, as re�ectedin (8) - namely, an increase in the wealth transition paths f1(bt) and f2(bt)(because of the increase in w1 and w2); this means that both low-income andmiddle-income classes end up at higher long-run levels of wealth, as given in(9).12

A �nal observation worth noting is that our analysis implies a link be-tween distributional and aggregate outcomes. This is evident from (3) whichshows how the division of the population into di¤erent cohorts of agents isimportant for determining the total output of the economy. Thus, for thesame comparative exercises as performed above, one observes that total out-put increases in each of the cases of an increase in � and an increase in g.

12These comparative exercises can be used to realise more extreme results in which oneor more income classes vanish entirely. This may occur if changes in � and/or g alter theordering of b�1 < b

c < b�2 < bcc < b�3.

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4 Corruption

We introduce corruption into the model by assuming that the public fundsearmarked for the government�s social programme of education and healthprovision can be appropriated by public o¢ cials using their discretionarypowers of administration. In other words, corruption takes the form of pureembezzlement, meaning the theft by an individual of resources that she issupposed to administer. This type of o¤ence can be especially di¢ cult to dealwith when it entails the pilfering of public funds: whilst everyone in societymay be a¤ected, the fact that no private property is stolen or exchangedmeans that individuals have no legal rights by which to protest and seekcompensation. We do not delve into issues of who perpetrates such malfea-sance (bureaucrats or politicians), what factors may motivate it (economic orcultural), or how it may be curtailed (rewards or punishments). Such issuesare dealt with elsewhere in the literature and are not the main concern ofthis paper. Rather, our focus is on studying the consequences of corruption,given that corruption exists as an endemic feature of the economy (which issadly the case in many less developed countries).Our formal modelling of corruption, and the immediate impacts thereof,

are described as follows. The appropriation of public funds by public o¢ cialsmanifests itself in a reduction in g, the provision of public education andhealth services. By virtue of (4), the consequence of this is that there is adecrease in both A1 and A2, the productivities of those agents who rely onsuch provision. In turn, this is re�ected via (5) in a decrease in both w1 andw2, the wages of these agents.Two main implications follow from the above. The �rst is that any lineage

of agents which depends on publicly-provided education and health servicessu¤ers a reduction in its intergenerational transfers of wealth and, with this,a reduction in its long-run level of wealth: that is, there is a decrease in f1(bt)and f2(bt) in (8), together with a decrease in b�1 and b

�2 in (9). The second is

that these lineages are faced with a higher threshold level of wealth which isneeded in order to supplement public services with credit-�nanced personalexpenditures: that is, there is an increase in bc in (6). Diagramatically, theseimplications are illustrated by the arrows of movement in Figure 2.Having established as much, one is able to see how corruption a¤ects

distributional outcomes through two channels - a wealth e¤ect and a credite¤ect. The former (i.e., the shifts in f1(bt) and f2(bt)) means that there is apopulation of agents, N1 +N2, who are made strictly worse o¤. The second(i.e., the shift in bc) means that the composition of this population changesas N2 falls whilst N1 rises in accordance with (10). Together, these e¤ectsconstitiute a double-whammy for the degree of poverty and inequality in the

13

economy: not only are the poor made poorer, but also there is a greaternumber of such individuals. At the aggregate level, there are correspondingdeclines in average income and total output.Naturally, the above e¤ects are enhanced as the level of corruption in-

creases. In principle the e¤ects could be so pronounced as to deliver theextreme scenario in which b�2 < bc, meaning that b�2 vanishes as a possiblesteady state as all agents whose wealth falls below bcc will experience (eitherimmediately or eventually) a decline in their wealth below bc as well so thattheir ultimate wealth status is b�1. This is the case in which corruption leadsto the disappearance of a middle-class and the polarisation of the populationinto the rich and poor.

5 Some Extensions

The foregoing analysis establishes our main results. In what follows we in-troduce some further considerations to illustrate how these results may bestrengthened when viewed within a broader context and how they may sur-vive under extensions of the model.

5.1 Governance and Institutions

Corruption is part of the broader concept of governance which, in turn, is anaspect of the wider issue of institutional quality. Recent years have witnesseda �ourishing literature on the importance of institutions - economic, politi-cal and judicial - in determining the fortunes of an economy through theirfundamental role in governing the incentives and opportunities of citizensvia market structures, democratic processes and legislative mechanisms.13

Our analysis has appealed to two sources of institutional imperfection in theeconomy - an imperfection in �nancial markets (i.e., weak powers of contractenforcement) and an imperfection in governance (i.e., corruption amongstpublic o¢ cials). Whilst we have treated these aspects separately, one mayenvisage them as being connected in the sense that both of them have theirorigins in the general quality of institutions. A deteroriation in this qualitymay therefore manifest in a poorer functioning of both markets and govern-ment.With the foregoing in mind, consider the case in which g (public goods

provision) and � (banks�retrieval of loans) depend positively on some com-mon measure of institutional quality. A reduction in this quality has a credit

13This literature has its origins in North (1990). A �avour of recent research by can befound in Acemoglu et al. (2005) and Acemoglu and Robinson (2010).

14

e¤ect which is both more pronounced and more widespread than before: notonly is there a decrease in g which increases bc, but also there is a simulta-neous decrease in � which both raises bc further and causes an increase in bcc

as well. Compared to our previous �ndings, there is a larger lower-incomeclass, a smaller upper-income class, and either a larger or smaller middle-income class. Regardless of the last ambiguity, average income is lower asthere are more agents who are credit-constrained, and total output is lowerfor the same reason. Accordingly, our previous results may be seen as beingstrengthened when one links corruption to the broader context of institutions.

5.2 Personal Choice of Human Capital Expenditures

In our main analysis we assumed that agents could supplement publicly-provided education and health services by incurring a �xed, exogenous ex-penditure of p. A natural extension of the model is to endogenise this ex-penditure by allowing agents to choose it optimally for themselves. A simpleillustration of this is as follows.Suppose that personal spending augments an agent�s productivity in (4)

according toa2(g; p) = a1(g) + �(p); (11)

where �0(p) > 0 and �00(p) < 0. In principle p and g could be either comple-ments or substitutes for each other as a low (high) quality of public serviceprovision may motivate individuals to spend either less (more) or more (less)themselves in response to their discontent (satisfaction) with such provision.The speci�cation in (11) may be viewed as abstracting from these interac-tions altogether, or assuming that the net e¤ect of them is zero. Given thisspeci�cation, it follows from (5) that the corresponding wage is

w2 = w1 + �(p): (12)

In turn, this implies a corresponding level of income from (2),

xt+1 = (1 + r)(bt � p) + w1 + �(p) if e = p: (13)

The decision problem for an agent is to choose a p so as to maximise xt+1 in(13). The solution to this is given by a p that satis�es

�0(p)� (1 + r) = 0: (14)

From (6) and (12), the lower critical level of wealth may be written as

bc =(2 + r � �)p� w1 � �(p)

1 + r� �(p): (15)

15

This expression implies a u-shaped relationship between bc and p. To besure, observe that sgn�0(p) = sgn[2 + r � � � �0(p)]. Using (14), one maythen infer the following: if p < p, then �0(p) > (1 + r) so that �0(p) ? 0;conversely, if p � p, then �0(p) � (1 + r) so that �0(p) > 0. Accordingly,there exists a bp < p such that �0(bp) � (2 + r � �) = 0, implying �0(bp) = 0,�0(p) < 0 for any p < bp, and �0(p) > 0 for any p > bp.In recognition of the above, an agent who is free to choose p realises that

she can in�uence bc, which may give her the opportunity of acquiring at leastsome amount of loan for human capital expenditure (an opportunity that wasprecluded in our main analysis). To see this, let b

c= �(p) and bbc = �(bp),

where bc> bbc. If bt � bc, then an agent is able to obtain the required amount

of loan to �nance her optimal expenditure, p, which she will obviously select.If bt < b

c, however, then the agent is not able to do this, meaning that her

optimal choice is not feasible. Nevertheless, the agent might be able to obtaina smaller loan to �nance a smaller amount of expenditure: that is, the agentcould choose a p < p in order to reduce bc until bt � bc. Such an opportunityis available for any agent with bt > bbc, and the second-best choice of p wouldbe the value that delivers bt = bc. The opportunity is not available, however,for agents with bt � bbc since reducing p below bp causes bc to increase (ratherthan decrease), in which case these agents are still denied loans.Compared to (8), the dynamics of wealth distribution in this extended

version of the model are summarised as follows: for each of the cases bt < bbc,bc � bt < bcc and bt � bcc, there is a single transition equation, as given byf1(bt) (corresponding to zero personal expenditures), f2(bt) (corresponding topersonal expenditures of p) and f3(bt) (corresponding to expenditures of P );for the remaining case bbc � bt < bc, there is a series of transition equations,as described by f2(bt) for di¤erent values of p 2 (bp; p) (associated withdi¤erent values of bc). Diagrammatically, the dynamics are illustrated inFigure 3. The principle di¤erence from our previous analysis is that there isa distribution of long-run wealth amongst middle-income agents (i.e., agentsfor whom bbc � bt < bcc). Other than this, the implications of the model arethe same, and our main results are unaltered.

5.3 Tax-�nanced Public Expenditures

Another of our simplifying assumptions has been that the government �-nances its social programme of education and health care using windfallsof public funds, such as receipts of foreign aid or revenues from natural re-sources. Again, it is possible to re-work our analysis using a modi�ed versionof the model in which public expenditures are �nanced by taxation. The

16

following provides an illustration of this.Suppose that taxes are levied only on high-income agents at a lump-sum

rate of � . From (2), we have

xt+1 = (1 + r)(bt � P ) + w3 � � if e = P: (16)

A trivial modi�cation of our previous parameter restriction continues to en-sure that agents always prefer to spend more (rather than less) on acquiringhuman capital, and that any loans obtained to �nance this spending canalways be repaid.14

The only other notable change from our previous analysis relates to theupper critical level of wealth, bcc in (7). Assuming that an agent who abscondswith her loan also evades paying taxes, this threshold is now given by

bcc =(2 + r � �)P � w3 + �

1 + r: (17)

Given the above, the dynamics of wealth distribution are essentially thesame as those displayed in Figure 1, with bcc determined according to (17) andf3(bt) adjusted from (8) to include � in conformance with (16). The e¤ectsof corruption are also the same. Under our assumption of balanced budget�nancing, g = N3� . The pilfering of public funds by public o¢ cials means areduction in tax revenues, N3� , implying a reduction in public expenditures,g, which produces the e¤ects indicated in Figure 2.

6 Conclusions

This paper has sought to contribute to the literature on corruption and de-velopment by showing how public sector malfeasance can undermine the ef-fectiveness of social programmes designed to reduce inequality and povertyin an economy. It�s emphasis has been on the acquisition of human capital- de�ned broadly to include both education and health status - as a meansby which individuals become productive and earn income. Education andhealth services are provided by both the public and private sectors, the lat-ter of which may be accessed to either supplement or substitute the use ofthe former through various types of personal expenditure. The ability to dothis depends on an individual�s wealth status which determines whether ornot she requires a loan to �nance such expenditure, and whether or not sheis eligible for a loan as a consequence of capital market imperfections. Cor-ruption manifests as the appropriation of public funds by public o¤cials, the

14That is, w1 < w2 � (1 + r)p < w3 � � � (1 + r)P .

17

immediate impact of which is a reduction in the provision of public educa-tion and health services. Individuals who rely on such provision for acquiringhuman capital are then faced with the prospect of lower wage earnings be-cause of their lower productivity. This is the wealth e¤ect of corruption.In addition, the same individuals face a higher threshold level of wealth forloans to be granted, meaning that more of them are constrained in theirability to borrow. This is the credit e¤ect of corruption. Our analysis showshow these two e¤ects combine to adversely a¤ect both distributional andaggregate outcomes in an economy.

18

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Figure 1

Wealth Distribution Without Corruption

450

𝑓3(𝑏𝑡)

𝑓2(𝑏𝑡)

𝑓1(𝑏𝑡)

𝑏𝑡+1

𝑏𝑡 𝑏𝑐 𝑏𝑐𝑐 𝑏1∗ 𝑏2∗ 𝑏3∗

Figure 2 The Effects of Corruption

450

𝑓3(𝑏𝑡)

𝑓2(𝑏𝑡)

𝑓1(𝑏𝑡)

𝑏𝑡+1

𝑏𝑡 𝑏𝑐 𝑏𝑐𝑐 𝑏1∗ 𝑏2∗ 𝑏3∗

Figure 3 Endogenising Human Capital Expenditures

450

𝑓3(𝑏𝑡)

𝑓2(𝑏𝑡)|𝑝=�̅�

𝑓1(𝑏𝑡)

𝑏𝑡+1

𝑏𝑡 𝑏�𝑐 𝑏𝑐𝑐 𝑏�𝑐

𝑓2(𝑏𝑡)|𝑝∈(𝑝�,�̅�))

𝑏𝑐|𝑝∈(𝑝�,�̅�)