This Draft: August 2017
Allocation Contests for Publicly - Provided Goods Arijit Sen
In many countries, the government provides goods and services that are rival in consumption – essential commodities like potable water, public transportation, and primary health-care, and merit goods like higher education, and tertiary healthcare. For such goods, the government has to specify allocation rules under which citizens can access them. Affluent citizens often have the incentive and the ability to influence public allocation rules by engaging in contests. This paper presents simple models of such ‘allocation contests’ for a divisible essential commodity and for an indivisible merit good, and studies contest equilibria and their implications for social outcomes. We show that: (a) in the context of public provision of an essential commodity like potable water, a fall in public supply can have a magnified negative impact on social welfare; and (b) in the context of designing a reservation policy for a merit good like higher education, raising the reservation quota for a set of disadvantaged citizens might effectively lower their access to the good. Keywords: Publicly-provided goods, rival goods, merit goods, allocation contests. JEL Classification: C72, D61, H42.
1. Introduction In most economies, the government (i.e., the public sector) spends substantial resources to provide its citizenry with many kinds of goods and services that are ‘rival in consumption’. Examples include essential commodities like drinking water, electricity, public transportation, and basic healthcare, and merit goods like professional / higher education, and tertiary healthcare. In the United Kingdom, for example, in each of the last five financial years, the annual average expenditures on healthcare, education, and transportation have been 20%, 10% and 3% of aggregate annual public spending (source: ukpublicspending.co.uk). Recognize that whenever such rival goods are publicly-provided, the government needs to specify – implicitly or explicitly – an ‘allocation rule’ under which different citizens can access (different quantities of) such goods. In many countries, potable water is publicly provided under the implicit ________________________________ Author’s Coordinates: Indian Institute of Management Calcutta, Kolkata 700104, India;
[email protected]
allocation rule that every citizen has equal access to water at zero price. Access to higher education is, in many countries, publicly provided on the basis of merit and / or socio-economic background. Advanced medical care is often publicly provided on the basis of referrals by general physicians. In fact, for essential commodities like safe drinking water and merit goods like higher education, the government’s ability to implement allocation rules that are socially (and morally) superior to price-based allocation rules provides the raison d’etre for public provision of such goods; see Sandel (2012) for an elaboration of this argument. But it is often a challenge to properly implement non-market allocation rules for public delivery of rival goods. If the public system is weak and / or corruptible, private citizens often have the incentive and sometimes the ability to contest public allocation rules in an attempt to bend the rules in their favour. For example, affluent households in urban India install booster pumps to divert public water supply to their homes, especially when the volume of such supply is low. Rich parents bid in auction-like schemes to secure admission for their children in lucrative professional education.1 Well-to-do families use social and political connections to ensure that their wards get access to specialist medical care in public research hospitals. Corrupt bureaucracies facilitate such contests and make them more effective. In their article on the ‘economic jungle’, Piccione and Rubinstein (2007) write: “Throughout the history of mankind, it has been quite common that economic agents, individually or collectively, use power to seize control of assets held by others. The exercise of power is pervasive in every society and takes several forms.” Our contention is that such exercise of economic power is routinely practiced by privileged citizens in various economies in order to improve their access to public provision. We designate such competition to be ‘allocation contests’ for publicly-provided rival goods, and aim to study the causes and the consequences of the competitive bribery inherent in such contests. The extant literature on contest theory provides an appropriate and useful lens through which to view such competition.2 In reviewing the ambit of contest theory, Jia et.al.(2013) write that contests are “games in which each player exerts effort in order to increase his or her probability of winning a prize”. We emphasize that the theoretical framework developed in the contest literature can easily be adapted to study allocation contests in which players exert effort to modify allocation rules in their favour.3 Reports like the following are commonplace in the Indian press: “Post-graduate medical seats auctioned for Rs. 4 crores,” Times of India, September 13, 2013. 1
2
See Corchón (2007), Konrad (2009), and Long (2015) for recent surveys on contest theory.
3
In the contest literature, Corchón (2000) studies how contests can change allocation rules. The issue of allocation efficiency under corruption has also been studied by Lien (1996) and Clark and Riis (2000).
2
In this paper, we study simple models of allocation contests over two kinds of publicly-provided rival goods – divisible essential commodities like potable water, and indivisible merit goods like higher education. In each case, we formulate the citizens’ competitive attempts to perturb public allocations via contests, and identify the consequences of such behaviour for social welfare. Our analysis delineates following features of allocation contests: Contests over essential commodities like water become more intense precisely when the public provision regime deteriorates and the supply volume goes down. As the fall in public supply increases the number of contestants, resource scarcity goes hand in hand with less-cooperative individual behaviour. As a result, the deteriorating public provision system has a magnified negative impact on social welfare. However, since some well-off citizens can perversely be better off from contesting in a worsened public provision regime, there need not arise a unanimous societal call for a ban on such allocation contests. In the context of allocation contests over merit goods like higher education, our analysis uncovers the following subtlety in the design of optimal reservation policies: If allocation contests by affluent citizens are possible, then raising the reservation quota for the less-privileged need not automatically translate to their gaining greater access to the merit good. This is because a higher reservation quota can cause the affluent to contest the allocation policy with greater vigour, thereby reducing the effective chance that the less-privileged have to access the merit good. Then it might well be the case that a lowering of the reservation quota helps raise the ex ante access opportunity of the disadvantaged members of society. The rest of the paper is organized as follows. Section 2 formulates and studies allocations contests over a publicly-provided essential commodity. A model of allocation contests over a publiclyprovided merit good is presented and analyzed in Section 3. Section 4 contains some concluding remarks. 2. Contesting for an Essential Commodity In our first model, we study the scenario of public provision of a divisible essential commodity. There is a continuum of households in the economy. Some of them are affluent – each with after-tax income YA, while the others are middle-class – each with after-tax income YB, with 0 < YB < YA < . In what follows, a household h (respectively, i) will refer to a household with income Yh (respectively Yi) for h, i = A, B. Further, HA (respectively, HB) will denote the set of affluent (respectively, middle-class) households; for simplicity we take these sets to be of equal measure: (HA) = (HB) = ½.
3
Each household consumes two divisible goods W and X. Here, good X is the market-provided numeraire private good, and good W is a divisible publicly-provided rival good – e.g., an essential commodity like potable water. The households have identical preferences over W and X; if household h consumes q ≥ 0 units of W and x ≥ 0 units of X then its utility is [v(q) + u(x)], where v(.) and u(.) are strictly increasing and strictly concave functions of their arguments. Let Q denote the quantity of W provided publicly by the government; we assume that the public supply Q can vary in the range [Q –, Q +], with 0 < Q – < Q + < . Regarding the desired allocation of Q, we posit that the government’s objective is to ensure costless egalitarian access to W for all households. Specifically, given that the measure of all households in the economy is unity, the government aims to ensure that each household receives Q volume of water at no charge.4 If the households do not contest for W, then the government’s desired egalitarian allocation will indeed be implemented. But we allow for the possibility that households can contest for W with the aim of diverting some W from non-contesting households. We posit that each household can engage in an allocation contest for W by making a lumpy investment – say, by installing a booster pump, the investment cost being z (0, YB). The gains from contesting arise from the fact that the investment enables diversion of some amounts of W from non-contestants to contestants in the following way. Let C H denote the set of households who sink the contest investment z, with the measure of the set of contestants C being (C) [0, 1]. There exists an amount D (0, Q –) such that [(C).D] volume of W will be diverted away from each non-contestant, leaving each of them with the amount [Q – (C).D] of W. Then the total volume of W diverted from non-contestants will be [(1– (C)).(C).D], and this amount will be shared equally by all contestants. Thus each contestant will end up with [Q + (1– (C)).D] amount of W. Consequently, for a given level of public supply Q and for a given set of contestants C, we can calculate the utility of a household h (for h = A, B) as follows. If household h becomes a contestant c
then its utility will be: Uh (Q, C) v([Q + (1– (C)).D]) + u(Yh – z), while if household h remains a n
non-contestant then its utility will be: Uh (Q, C) v([Q – (C).D]) + u(Yh). We are interested in characterizing the equilibrium outcome of the allocation contest for W when all households simultaneously decide whether or not to sink the contest investment z (with each household being aware of all other aspects of the contest scenario). To that end, for each household h {A, B} and given a set of contestants C of measure (C), we define household h’s gains-fromcontesting function h(Q, C) as follows:
4
Our analysis will be unaffected even if there is a nominal lump-sum charge/tax on each household for accessing public water supply; we will then simply define the incomes {YA, YB} to be net of that charge.
4
c
n
h(Q, C) ≡ Uh (Q, C) – Uh (Q, C) {v([Q + (1– (C)).D]) – v([Q –(C).D]} – {u(Yh) – u(Yh – z)}. Then C* will be a ‘Nash equilibrium set of contestants’ (contestants being defined as those households that contest with a positive probability) if and only if h (Q, C*) ≥ 0 for all ‘contestant households h’ and i (Q, C*) 0 for all ‘non-contestant households i’.5 The strict concavity of the u(.) and the v(.) functions imply the following three features of the equilibrium outcome of the allocation contest in the current model (which are easy to establish): FEATURE [I]: An affluent household will have a strictly greater incentive to contest relative to a middle-class household: A(Q, C) > B(Q, C) for all Q, C. FEATURE [II]: An increase in the measure of contestants will raise every household’s incentive to contest: (C ) > (C ) implies h(Q, C ) > h(Q, C ) for all Q and h{A, B}. FEATURE [III]: A decrease in the public supply of W will raise every household’s incentive to contest: Q > Q implies h(Q , C) < h(Q , C) for all C and h {A, B}. Note the implications of features [II] and [III] above: In our ‘discrete contest model’, deterioration in the public supply regime unambiguously expands the set of contestants due to a direct effect (of fall in W given C) and an indirect effect (of an expansion of C resulting from the fall in W).6 In what follows, we explore the consequences of this fact. For h = A, B, and [0, 1], we define the public supply threshold Qh() implicitly as follows: h(Q = Qh(), C | (C) = ) = 0. Note that facing a measure of contestants , household h will sink the contest investment if and only if the volume of public supply Q is less than Qh(). Features [I] – [III] regarding contest incentives imply the following: (a) for the affluent households the public supply threshold QA() is strictly greater than the corresponding threshold QB() for the middle-class households irrespective of the measure of contestants (i.e., for all [0, 1]); and (b) for each household h (for h = A, B) the threshold Qh() strictly increases in the measure of contestants . We assume, for simplicity, that Q+ QA(=1) and that QB(= 0) Q–. 5
A Nash equilibrium C* will exist as long as households can randomize in their contest decisions.
6
In a large class of contest models, especially in models of winner-take-all contests, the following result holds: individual equilibrium contest effort increases as the ‘size of the prize’ grows. The opposite is the case in our model of discrete contest for a shared prize. Recently Dickson et.al. (2017) have shown that in a more general model of contest over a shared prize (where contest intensity is endogenously chosen and the contestants can be risk-averse) it is possible that contest intensity will increase precisely when the size of the aggregate prize shrinks.
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We are now in a position to characterize the nature of contest equilibria for different levels of public supply Q of W in two different scenarios of income inequality – one in which [YA – YB] is sufficiently small and the other in which [YA – YB] is sufficiently large. PROPOSITION 1. [a] Consider the case where [YA – YB] is sufficiently small so that QB(1) > QA(0). Here the following results hold. (i) For Q > QB(1), no household is a contestant in the unique Nash equilibrium (i.e., C* is the null set). (ii) For Q (QA(0), QB(1)) there are two Nash equilibria: one in which no household is a contestant (i.e., C* is null), and one in which all households are contestants (i.e., C* = H); while for Q (QB(½), QA(½)) there is an additional Nash equilibrium in which only the affluent households are contestants (i.e., C* = HA). (iii) For Q < QA(0) all households are contestants in the unique Nash equilibrium (i.e., C* = H). [b] Consider the case where [YA – YB] is sufficiently large so that QA(0) > QB(1). Then: (i) For Q > QA(½), C* is the null set in the unique Nash equilibrium; (ii) for Q (QA(0), QA(½)), there are two Nash equilibria: {C* null} and {C* = HA}; (iii) for Q (QB(1), QA(0)), {C* = HA} is the unique Nash equilibrium; (iv) for Q (QB(½), QB(1)), there are two Nash equilibria: {C* = HA} and {C* = H}; (v) for Q < QB(½),{C* = H} is the unique Nash equilibrium. Proposition 1 delineates the following features of the allocation contest for W. Firstly, the strategic complementarity inherent in the allocation contest generates the possibility of multiple equilibria. Secondly, irrespective of the magnitude of income inequality, both the ‘minimal equilibrium set of contestants’ and the ‘maximal equilibrium set of contestants’ (weakly) expand as the public provision regime deteriorates (i.e., as the public supply volume Q falls). Given Proposition 1, we investigate the following issues: the impact of a fall in public supply on social welfare, and the possible emergence of a societal demand for a ban on allocation contests. Negative Magnification Effect of a Deteriorating Public System First, consider the case where the public system deteriorates, causing a fall in public supply from some volume Q to a lower volume Q. We will refer to such a fall as ‘drastic’ if it causes the equilibrium set of contestants C* to expand from the null set to the set of all households H. From Proposition 1, note that such a ‘drastic Q-fall’ will occur if either income inequality is small and Q > QB(1) > QA(0) > Q, or income inequality is large and Q > QA(½) > QB(½) > Q.7 For such a drastic Q-fall, the equilibrium utility of each household h (for h = A, B) will fall by the amount: {[v(Q) – v(Q)] + [u(Yh ) –u(Yh – z)]}. Recognize that for each household h, utility would have fallen by the amount {v(Q) – v(Q)} if the households’ behaviour had not worsened as a result of 7
These are sufficient conditions for a drastic Q-fall; even smaller drops in Q can cause a drastic Q-fall.
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the fall in Q (i.e., if all households could have committed themselves not to contest for W after the fall in public supply). It is in that sense that the additional utility drop of {u(Yh ) –u(Yh – z)} for each household h that is associated with a drastic Q-fall can be taken to be the measure of the ‘negative magnification effect of a deteriorating public system’; this magnification effect is larger for poorer households. Next, consider two distinct cases of ‘non-drastic Q-fall’ of the public supply volume from some Q to Q such that the set of equilibrium contestants expands in one of two ways: Case [1]: C* goes from being the null set to the set of affluent households HA; or Case [2]: C* goes from the set of affluent households HA to the set of all households H.8 It is easy to verify that in both these cases of non-drastic Q-fall, at least one set of households will unambiguously suffer the negative consequences of the magnification effect. In case [1] each middle-class household will suffer a utility drop that exceeds {v(Q) – v(Q)} by the amount {v(Q) – v(Q– ½.D)}, and in case [2] each affluent household will suffer a utility drop that exceeds {v(Q) – v(Q)} by the amount {v(Q+ ½.D) – v(Q)}. We thus conclude as follows: PROPOSITION 2. In all cases where deterioration in public supply causes an expansion in the set of contesting households, at least one group of households will necessarily experience a utility drop that is subject to a ‘negative magnification effect’; the larger is the expansion in the set of contesting households due to the fall in public supply, the bigger will be the set of households who are subject to this magnification effect. On the Call for Banning Allocation Contests Given the recognition that falling public supply of W can lead to magnified welfare losses, a natural societal response might be a call for banning allocation contests. But the following analysis argues that: (a) such a call may not be unanimous, and (b) if citizens foresee that individual households can circumvent such a ban by bribing corruptible administrators, the call for a ban might not even be voiced by anyone. First, consider the case where QA(0) > QB(1) and where the public supply volume intermediate level such that QB(1)
v( ) + u(YA), and v( – D) + u(YB)) > v( ) + u(YB – z). Given that, it is easy to see that a ban on contesting will cause the post-ban utility of every middle-class household to be {v( ) + u(YB )} and the post-ban utility of 8
Case [1] will occur if income inequality is large and Q > QA(½) > QA(0) > Q > QB(1); case [2] will occur if income inequality is large and QA(0) > Q > QB(1) > QB(½) > Q.
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every affluent household to be {v( ) + u(YA )}. Thus, while the middle-class households will gain from such a ban, the affluent will be strictly worse-off and will thus be unlikely to join a societal call for a ban on allocation contests. Next, consider the case of a low level of public supply < min{QA(0), QB(½)}, generating {C* = H } as the unique Nash equilibrium. Starting from such a situation, a ban on contesting will indeed raise the utility of every household h by the amount {u(Yh ) – u(Yh – z)}, with the utility gain being larger for poorer households. In this case, it is reasonable to expect that a unanimous social call for a ban on allocation contests will emerge. However when
< min{QA(0), QB(½)}, suppose that the households become aware that each
individual household can bypass a contest ban by paying a bribe of > 0 to corruptible officials. If the required bribe amount is not too large, then the following inequalities can hold: v( + D) + u(YA – z – ) > v( ) + u(YA), and v( + ½.D) + u(YB – z – )) > v( – ½.D) + u(YB). In that case, every household will circumvent the contest ban by paying the bribe in the ensuing unique equilibrium. As a result the equilibrium utility of each household h will be {v( ) + u(YB – z – )}, while it would have been {v( ) + u(YB – z)} if no ban was in place. On foreseeing the fact that a circumventible ban on contesting would further reduce individual utility by the amount [u(Yh–z) – u(Yh–z–)], it is reasonable to expect that the call for such a ban will not be raised by anybody. 3. Contesting for a Merit Good In our second model, we consider the case of public provision of an indivisible merit good. As in the model in Section 2, there is a continuum of risk-neutral households in the economy. Each household in the set HA of affluent households has after-tax income YA, while each household in the set HB of middle-class households has after-tax income YB, with 0 < YB < YA < . The set of affluent households and the set of middle-class households are of equal measure half. Each household has one offspring who aspires to acquire a merit good M, say higher education, that is publicly provided. If a household acquires M for its offspring (respectively, does not acquire M ) and consumes x ≥ 0 units of a market-provided numeraire private good X, then its utility will be [V + u(x)] (respectively, u(x)). Here, V is a positive constant representing the value of M to a household, and u(.) is a strictly increasing and strictly concave function of its argument. The aggregate capacity of publicly-provided merit good M is (0, 1), implying that exactly measure of households in the economy will be able to access M (heuristically, a total amount of seats are available in the public institutes of higher education).9 We take M to be sufficiently 9
It is this capacity constraint that makes higher education a rival good in the current model.
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scarce: < ½, and assume that the government wants to provide M for free.10 More importantly, we posit that the government wants to ensure that each middle-class household has a sufficiently high chance of accessing M. To that end, the government sets a ‘reservation quota’ , where (0, ) is the measure of the part of the M-capacity that is reserved for middle-class households (i.e., seats out of the total seats for higher education are reserved for the middle-class). If there is no administrative corruption, then a reservation quota will generate the following access opportunities for M : First, measure of middle-class students will be chosen for the reserved seats (with each student having equal chance of being selected). Then, all other students in the economy, of measure (1– ), will compete for the measure ( – ) of unreserved seats, with each student having an equal chance of succeeding in the competition. Each A-offspring will then have ‘ex ante probability of access’: A0() = [( – )/(1– )], while each B- offspring will have ex ante probability of access: B0() = 2 + (1– 2).[(– )/(1– )].11 Things will, however, be different in a scenario where there is bureaucratic corruption. We posit that in such a scenario, each household can improve its access opportunity of M by engaging in an allocation contest that involves paying a non-refundable bribe of a fixed amount x (0, YB) to the bureaucracy.12 When a household pays the bribe it ‘stochastically succeeds’ in raising its access opportunity with probability (0, ½), while bribery is ineffective with probability (1– ); the precise ways in which bribes affect access probabilities are described below. Here is a measure of the extent / prevalence of corruption in the system – it determines the likelihood that a bribe can ‘permeate through the system’ and improve the bribe-giver’s access opportunity.13 We posit that bribery works in the following ways in a ‘-corrupt system’. We start from the situation where a measure of B-households has been randomly selected to fill the reservation quota (we assume that corruption has no impact on filling the quota). Then each of the remaining measure of (1– ) ‘unreserved households’ (A-households of measure ½ and 10
Our analysis will be essentially unchanged even if the government charged a (small) admission fee to access M (i.e., to enroll in higher education) which could be paid by taking a student loan. Then the value V of the merit good M could simply be redefined by netting out the loan repayment amount. From the perspective of an individual B-offspring, she will have probability 2 of accessing M ‘via the reservation’ (since a population of measure ½ will be trying to avail of the quota of measure ), and if that did not happen then she will have probability [(– )/(1– )] of accessing M through open competition. 11
12
To keep things simple, the bribe amount x is taken to be exogenously given (and not a choice variable).
In the real-world, a bribe-giver is never quite sure whether the bribe will ‘make a difference’ in her favour. At the same time, a bribe is indeed more likely to be effective when the regime is more corrupt. It is in this sense that the magnitude of captures the extent of corruption in the system in our model, where every bribe-giver has an identical and independent chance (of ) of having her bribe be effective. 13
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B-households of measure (½ – )) simultaneously decide whether or not to contest for the ‘open M-capacity’ ( – ) by paying a bribe x. If a subset C – of measure C – of these households choose to contest, then a randomly-picked subset CA C of measure [max{C, ( – )}] gets access to M for sure, while each of the other households (the non-contestants and the failed contestants) accesses the remaining measure [max{ – – C, 0}] of M with equal probability. c
n
Under this specification, defining h (, C) (resp., h (, C)) to be the ‘post-reservation probability of access’ for an individual contesting (resp., non-contesting) household h when C is the measure of contesting households, we have: n
[( – – C) / (1 – – C)] 0
for .C ( – ), for .C > ( – );
c
+ (1– )[( – – C) / (1 – – C)]
for .C < ( – ),
[( – ) / (C)]
for .C ( – ).14
h (C, ) = h (C, ) =
The above probabilities allow us to identify each unreserved household’s gains from contesting, c
n
defined by h(, C) ≡ {[h (C, ) – h (C, )].V – [u(Yh) – u(Yh – x)]}, as follows: h(C, ) =
{1– [( – – C) / (1– – C)]}.V – {u(Yh) – u(Yh – x)}
for .C < ( – ),
[( – )/(C)].V – {u(Yh) – u(Yh – x)}
for .C ( – ).
In the allocation contest over the merit good M, C* will be a ‘Nash equilibrium contestants set’ (contestants being those households which contest with a positive probability) with measure C*, if and only if h(C*, ) ≥ 0 for all contestant households h and i (C*, ) 0 for all non-contestant households i. [C* will exist when households can randomize in contest decisions.] The structure of the gains-from-contest function specified above implies the following features of the allocation contest (which are derived by straightforward differentiation): FEATURE [i]: An affluent household will have a greater incentive to contest for M relative to a unreserved middle-class household: A(C, ) > B(C, ) for all C, . After a measure of B-households has been randomly selected to fill the quota, consider an unreserved household that does not pay the bribe. When .C < ( – ), this household will be among a population of measure (1 – – C) who have neither accessed M via reservation nor via bribery, and will have an equal chance of being picked for the ‘remaining M-capacity’ ( – – C). Further, when .C < ( – ), for a household that has bribed, its bribe will be effective with probability in which case it will access M for sure, while its bribe will fail with probability (1– ) in which case it will be just like an unreserved nonbribing household. These arguments explain the values of hn(, C) and hc(, C) when .C < ( – ). Analogous logic allows us to derive hn(, C) and hc(, C) when .C ( – ). 14
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FEATURE [ii]: When few households contest for M so that .C < ( – ), each unreserved household’s incentive to contest rises with the size of the quota and with the measure of contestants C; the rise is greater if corruption is more pervasive (i.e., if is higher). These features of the allocation contest for M have important implications for how each middleclass household’s ex ante access opportunity of M changes in response to a change in the reservation quota . In what follows, we present a simple example to illustrate that such changes can be non-monotonic, and as a result, the ex ante probability of accessing M for each middle-class household might be lower when the reservation quota is set at a higher level. On Corruption, Quotas, and Access We now consider a scenario in which it is feasible for the government to set a reservation quota [0, /2], and study the impact of such a quota in two regimes – a ‘no-corruption regime [NR]’ and a ‘corruption regime [CR]’.15 In [NR] with = 0 it will be pointless for any household to engage in bribery; as a result, the ex ante probability of access to M for an A-household will be A0() = [( – )/(1– )] while that for a B-household will be B0() = 2 + (1– 2)[(– )/(1– )]. In contrast, will be positive in [CR] and there we consider the case in which the following conditions hold: (i) 0 < < , (ii) B(C, ) < 0 for all C[0, 1] and [0, /2], and (iii) A(C = ½, = 0) < 0 and A(C = 0, = /2) > 0. Note that (ii) implies that the middle-class will never contest for M, and (iii) implies that while the affluent households will not contest for M when there is no reservation, they will do so when maximal reservation is imposed.16 Let us now focus on the government’s quota choices in the two regimes, when its objective is to maximize the ex ante probability of access for each B-household subject to the feasibility constraint that must belong in [0, /2]. In [NS] this probability will be B0(), and since B0() increases monotonically in the government will set = /2. In contrast, the following results hold in the corruption-regime. In [CR], define (0, /2) such that A(C = 0, = ) = 0. Then the ex ante access probability for each B-household under corruption, denoted by B(), will be:
It is quite plausible to consider a cap ( /2) on the reservation quota, especially one that is congruent with the population share of the disadvantaged agents in the economy; note that (HB) = ½ in our model. 15
The assumption that < is made to keep the analysis simple, and to emphasize that limited amount of corruption can be enough to generate the outcome that raising the reservation quota is counter-productive. Note that B(C, ) will be negative when YB sufficiently small and x is non-trivial. Further, the conditions A(C = ½, = 0) < 0 and A(C = 0, = /2) > 0 require that: .[2(1– )/(2– )].V > {u(YA) – u(YA – x)} > .[(1– )/(1– 0.5)].V. 16
11
B() =
2 + (1– 2)[( – )/(1– )]
for [0, )
2 + (1– 2)[( – – 0.5)/(1– – 0.5)]
for (, /2].17
The following results are then immediate: PROPOSITION 3. [a] As the reservation quota is raised in the corruption regime, there will be a discontinuous fall in the ex ante access probability for middle-class households as crosses a threshold value. [b] Setting the maximal reservation quota = /2 in the corruption regime will lead to a lower ex ante access opportunity for middle-class households than it would in the no-corruption regime. [c] In the corruption regime, it is possible that for a middle-class household the ex ante probability of accessing M will be higher if the government reduces from [/2] to a value just below ; this is more likely to be true when is ‘close enough’ to [/2].18 The important issue to recognize in this context is the following: When a reservation quota for the less-affluent is in place, some (meritorious) middle-class students succeed in accessing M over and above the quota. It is these students who get squeezed out due to more intense contesting by the affluent when the quota is raised. Then the net effect of an higher quota can very well be that in the aggregate, fewer middle-class households manage to access M .19 For the same reason, if corruption becomes more pervasive then the government might want to optimally lower reservation quotas in order to ensure that disadvantaged members of society continue to enjoy a reasonably high degree of access to certain merit goods. 4. Concluding Remarks For all rival goods that the public sector provides in an economy, the government has to specify allocation rules for public distribution. In many cases, especially in situations of limited public supply of essential commodities and merit goods, it is in the interest of private citizens to contest over the allocation rules. Corruption in the public system facilitates such contests. In this paper, we have provided two simple models of allocation contests. We have considered
17
The ex ante access probability of a household is calculated before the reservation quota is filled, but is based on the equilibrium contesting behaviour that the unreserved households will subsequently pursue. Note that for < and /2, the inequality {.C < ( – )} will hold for all C ½. 18
A little reflection should convince the reader that Proposition 3 might hold under the weaker parameter restriction: A(C = 0, = 0) < 0 and A(C = 0, = /2) > 0, which is satisfied when .[2(1– )/(2– )].V > {u(YA) – u(YA – x)} > .(1– ).V . 19
Our analysis provides a set of sufficient conditions for this pathology to emerge; we do not argue that the pathology must necessarily arise. 12
scenarios where the citizens are either affluent or middle-class, and where contesting is a binary {0 - 1} choice. Given this simple structure, we have shown the following possibilities: The deteriorating public provision of an essential commodity can have magnified negative impact on social welfare. Further, in the context of designing reservation policies, raising the reservation quota of a publicly-provided merit good for a set of disadvantaged citizens might effectively lower their access to the good. In future research, it will be fruitful to generalize our simple model specifications – especially regarding the structure of income heterogeneity, and endogenous choice of contest intensity – in order to determine the robustness of our conclusions as well as to derive additional implications of allocation contests for social welfare. References Clark, D. and C. Riis (2000) “Allocation Efficiency in a Competitive Bribery Game,” Journal of Economic Behavior & Organization 42(1), 109 –24. Corchón, L. (2000) “On the Allocative Effects of Rent-seeking,” Journal of Public Economic Theory 2(4), 483– 91. Corchón, L. (2007) “The Theory of Contests: A Survey,” Review of Economic Design 11(2), 69–100. Dickson, A., I. MacKenzie, and P. Sekeris (2017) “Sharing Contests with General Preferences,” mimeo, University of Strathclyde. Jia, H., S. Skaperdas, and S. Vaidya (2013) “Contest Functions: Theoretical Foundations and Issues in Estimation,” International Journal of Industrial Organization 31(3), 211–22. Konrad, K. (2009) Strategy and Dynamics in Contests, Oxford University Press, Oxford. Lien, D. (1990) “Corruption and Allocation Efficiency,” Journal of Development Economics 33(1), 153–64. Long, N.-V. (2015) “The Theory of Contests: A Unified Model and Review of the Literature,” in R. Congleton and A. Hillman (eds.) Companion to the Political Economy of Rent Seeking, Edward Elgar, Cheltenham. Piccione, M. and A. Rubinstein (2007) “Equilibrium in the Jungle,” Economic Journal 117(522), 883– 96. Sandel, M. (2012) What Money Can’t Buy: The Moral Limits of Markets, Penguin Books: London.
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