Electoral Competition, Decentralization, and Public Investment Underprovision

1

Electoral Competition, Decentralization, and Public Investment Underprovision by Marco Magnani∗

The paper analyzes the optimal level of decentralization in local-public-good provision. Although all voters pay for such investments, only a subset benefit from them; their rate of return, however, is positive. Through pork-barrel projects, which do not increase welfare, any resource allocation is realizable. Candidates competing in local and national elections therefore face a trade-off between targetability and efficiency, which causes some profitable projects to be discarded. Decentralization affects underinvestment because the share of the electorate who benefit from an investment and the share of total budget absorbed by its costs depend on the size of that electorate. (JEL: D 72, H 41)

1

Introduction

The present paper studies public-capital underprovision and focuses on a specific class of investments, viz., local public goods consisting of infrastructure that serves a limited geographic area. These are, for instance, streets, bridges, and schools, which only benefit people who do not face high access costs and have specific demand for them. Several studies claim that modern democracies underinvest in public capital. David Aschauer provides substantial evidence that infrastructure spending in the U.S. is far below efficient levels (Aschauer [2000]), while the rate of return on public capital greatly exceeds the return on private capital (Aschauer [1989a]); similar results are also found for other countries.1 Why is this? A frequent argument, first introduced by Lizzeri and Persico [2001], concerns the role of politics. If electoral platforms are believed to affect future government policies, underinvestment may result from the dynamics of electoral competition. In fact, parties running for office might discard efficient projects if the resulting benefit distribution does not maximize their vote share. ∗

Universit`a degli Studi di Parma. I would like to thank Elmar Wolfstetter and two anonymous referees for their useful comments. 1 See, for instance, Aschauer [1989b]; Glomm and Ravikumar [1997] also provide an exhaustive survey of the empirical evidence on the effect of public capital stock on aggregate output.

2 The current analysis endorses this point of view and studies the conditions that allow local-public-good provision when two parties competing in an election choose between pure redistributive policies and investment in infrastructure. It further considers the effects of decentralization by studying how parties’ incentives change when the decision over infrastructure spending is the responsibility of different government levels. The main contribution of the paper is the characterization of the one-to-one relationship linking the fraction of the voters who benefit from a local public good to the minimum rate of return, which leads parties to include it in their platforms. If the size of the electorate varies because different government levels are considered, the share of the beneficiaries and the efficiency threshold also change. This shows the effects of decentralization and allows the determination of the conditions that minimize the problem of underinvestment. The efficiency threshold is the lowest when investment benefits accrue to half the voters. Indeed, the minimum rate of return increases monotonically when both lower and higher government levels are considered, i.e., when the electorate size shrinks or expands. This defines an optimal degree of decentralization for a particular local public good. There may be policy implications stemming from this positive analysis: Fostering public-capital provision in fact involves the choice of the optimal degree of decentralization. For a specific investment, flexibility in the realization is enhanced, the set of available options whose rate of return exceeds the efficiency threshold being extended. And with regard to a specific area, the number of prospective investments that parties are willing to provide increases. These results depend on two factors. The first is the trade-off between targetability and efficiency that characterizes budget allocation between public capital and redistributive policies. While redistributive policies can target any subset of the electorate and yield a nil rate of return, investing in a local public good benefits a specific group of voters and offers a positive return.2 The second factor has to do with information. Importantly, investment in public capital needs to be understood as the main element of development strategy hinging on locally concentrated intervention, which is often prescribed for development policies.3 Localization is thus crucial for investment outcomes, both because different areas have different needs for infrastructure and because making an effective effort towards economic improvement requires that the intervention be designed after consultation with beneficiaries. So the investment is self-revealing before the completion of the electoral agenda. 2

See Lizzeri and Persico [2001] and Persson and Tabellini [2000] for definitions, respectively, of redistributive policies and local public goods. In Lizzeri and Persico [2001], redistributive policies make use also of local public goods that are not perfectly targetable per se and must be included in a mix in order to yield a targetable policy. Therefore they need to be small, and constraints on resource allocation need to be negligible. Their aim is to target some voters, and their rate of return is nil. They thus substantially differ from infrastructure. 3 See Stern [2001] for an exhaustive discussion of development strategies.

3 Once the local public good is provided, opposing parties get to know the constraints on resource distribution placed by the reduction in funds available, and also that investment benefits are tied to a specific section of the electorate. They thus have the opportunity to take advantage of this information by deciding not to pledge the infrastructure, and instead they can make direct cash transfers targeting those who do not stand to benefit from it. The paper has the following structure. Section 2 discusses the literature on fiscal decentralization and redistributive policies. Section 3 describes the model. Section 4 derives the equilibrium for the game, defining the efficiency threshold that permits investment provision. Section 5 considers the effects over the previous threshold of variations in the electorate size. Section 6 duly concludes. 2

Related Literature

The present paper joins the debate over the effects of fiscal decentralization on localpublic-good provision, exhaustively surveyed by Lockwood [2005]. Adopting a politicaleconomy perspective, it studies inefficiencies in the policy formation process. There are two approaches to the issue. One branch of the literature focuses on postelection policies where local-public-good provision is financed indirectly, by earmarked funds made available by the central government to local authorities. Politicians choose a distribution across different districts of a common pool of resources with the aim either of maximizing social welfare, as described in the seminal contribution of Oates [1972], or of maximizing the welfare of the particular district that appointed them as representatives, as in models of legislative behavior (Baron and Ferejohn [1989], Lockwood [2002], and Besley and Coate [2003]). In this setting, a benevolent central government does not achieve an optimum result, because it is unable to diversify local-public-good provision across regions. On the other hand, fiscal decentralization produces inefficiencies due to coordination failures that prevent the internalization of tax and expenditure externalities and the exploitation of economies of scale. The centralized bargaining of self-interested regional delegates is also inefficient, because the cost sharing that follows from uniform taxation results in a bias toward minimizing costs instead of maximizing net benefits. More recent literature adopts a different approach, and focuses on the competition between parties who seek to maximize vote shares. In this context, public investments are financed directly out of local government budgets and compete with redistributive policies in the formation of electoral platforms. The present paper belongs to this strand of studies and builds on the seminal contribution of Lizzeri and Persico [2001]. These authors consider the case of two candidates choosing between alternative policies whose costs exhaust the revenues of a uniform tax rate levied on the electorate: redistribution of the public budget and the provision of a pure public good. This represents a variant of the redistributive setting with homogeneous voters first introduced by Myerson [1993]. The trade-off between efficiency and targetability of different types of government spending leads politicians to invest in the public good only if it yields a rate of return above a minimum level. A

4 problem of underprovision then arises because there are welfare projects that candidates are not willing to include in their platforms. Roberson [2008] modifies the model by Lizzeri and Persico [2001] to study the effects of decentralization. The assumption of uniform taxation is dropped, and the discrete choice over a unique public good is replaced with that over a finite number of homogeneous local public goods, each targeting a specific district. Comparing all cases, ranging from uniform to discriminatory taxation and project provision, the author makes the case for centralization. The greater discriminatory ability of a centralized system indeed reduces underinvestment. Roberson’s [2008] model maintains the hypothesis of a homogeneous electorate. Every district in fact benefits equally from the provision of the same local public good. This assumption fits well the case of in-kind transfers, but ignores the heterogeneity introduced by investments in infrastructure that are peculiar to a certain area and selfrevealing. In this setting, potential recipients of the local public good form a privileged group whose members are ex ante different from other voters. The present paper explicitly considers heterogeneity4 and complements the analysis of Roberson [2008]. This requires an extension of Lizzeri and Persico’s [2001] model: we introduce the choice over a local public good that gives the same per capita utility to a group of voters whom each candidate party can identify. It is further assumed that investment costs do not equal fiscal revenues, so both redistributive and investment policies can be included in electoral platforms at the same time. In this framework, decentralization changes the size of the electorate, which also varies accordingly the fraction of voters who are potential recipients of the local public good. The amount of fiscal revenues and the share of the budget absorbed by investment costs also change. These elements jointly affect the trade-off between targetability and efficiency and determine the extent of the underinvestment problem. The analysis of the effects of fiscal decentralization on costs and benefits of localpublic-good provision is novel to the literature. A further contribution of the paper concerns the characterization of an equilibrium for the case where only one party provides the local public good. Redistributive games with asymmetric players are the subject of a recent strand of studies adopting the approach developed by Myerson [1993]. These consider a number of different issues ranging from fiscal inequality to distortionary taxation and the provision of (local) public goods.5 Competition between asymmetric parties is analyzed by Sahuguet and Persico [2006] and Kovenock and Roberson [2008], which both examine election outcomes in terms of fiscal inequality. Sahuguet and Persico [2006] 4

Lizzeri and Persico [1998] study the provision of a public good that delivers a twopart benefit: one part evenly distributed, the other fully targetable. Candidates thus are also involved in redistribution, and it is assumed that if heterogeneity increases, the targetable share of the aggregate benefit increases. The hypothesis of a homogeneous electorate, though, is preserved, and redistribution is made among ex ante identical voters. 5 See Crutzen and Sahuguet [2008] for distortionary taxation, and Roberson [2008] and Lizzeri and Persico [2001] for public good provision.

5 define an equilibrium for a redistributive game where parties investing in a persuasive campaign end up having different budgets. Kovenock and Roberson [2008] study a setting where voter loyalty causes the effects of redistributive policies to vary depending on the party that adopts them. The present model focuses on a further circumstance: it characterizes an equilibrium for an election where one party has a fixed advantage regarding a specific fraction of voters whose number and identity are common knowledge. 3 The Model The model considers electoral competition between two symmetrical parties, 1 and 2, which credibly commit to their platforms. These include the decision over the provision of a local public good, g, and a set of redistributive policies. Voters are of two types: recipients and nonrecipients of g. They observe electoral platforms and vote for the party that pledges the greatest utility. The game consists of four stages played sequentially: (1) (2) (3) (4)

The parties announce simultaneously whether to include g in their platforms. Redistribution plans for the public budget are presented simultaneously. Elections are held, and votes are cast simultaneously. Electoral platforms are implemented.

The decision on the public project is announced before the redistribution plan and reflects the self-revealing nature of the investment. The last stage is modeled as a probabilistic compromise whose features are exhaustively discussed in Sahuguet and Persico [2006]. The probability that party i’s platform (i = {1, 2}) is implemented is an increasing function of its vote share Si . Parties are opportunistic and translate influence over the political process into personal benefits. The functional form linking vote shares to the previous elements is not crucial for the results of the model. It is thus possible to study proportional and majority systems in the same setting and assume that the parties maximize Si without the necessity of developing a detailed description of the implementation stage. In order to eliminate the problems posed by a large finite number of voters, the analysis considers an approximation for this situation: the electorate is a continuum denoted by the interval [0, 1], as in Myerson [1993]. Recipients of the local public good are denoted by l and form a special interest group whose size is fixed. So the group amounts to a share of total voters, which varies with the size of the electorate. The present analysis focuses on a benchmark case where an intermediate government level is considered and the group includes a fraction λ > 0 of total voters. A higher (lower) degree of decentralization results ultimately in an increase (decrease) in λ. It is assumed here that the investment does not produce spillovers and that the special interest group is always fully included in each constituency at different government levels. Nonrecipients of the local public good are standard voters denoted by the letter k and amount to a share 1 − λ of the total. In a finite electorate, this means

6 that while the number of special interest group members is fixed, that of standard voters varies according to the degree of decentralization. Party promises are financed through a lump-sum tax, normalized to 1 dollar, levied on each voter. Using Myerson’s [1993] approach, the infinite budget that results from taxation is also an approximation of a large finite amount of money proportional to the scale of the electorate. Extending this interpretation to local-public-good costs allows us to define a budget allocation between investments and redistribution.6 In order to simplify the analysis dealing with finite quantities, all the important variables are reported in per capita terms. Electoral platforms show how the unit per capita budget is to be used. Party i’s money spent on the local public good is denoted by gi . The remaining money is distributed through direct cash transfers. Each recipient of g gets the utility U when it is provided. The aggregate benefit generated thus is λU > 0. As in Lizzeri and Persico [2001], local-public-good provision is characterized by degrees of indivisibility, and the project has a fixed size. Its costs are a share of λU that depends on the parameter ξ ∈ [0, 1] and amount to λU (1 − ξ) ≤ 1. The investment rate of return then is [λU/λU (1 − ξ)] − 1 = 1/ (1 − ξ) − 1. The redistribution plan specifies, for a generic voter n (n = {l, k}), the ratio xni ∈ [0, +∞) of the sum that party i transfers to the voter to bi = 1−gi (gi ∈ {0, λU (1 − ξ)}), i.e., the redistributable per capita budget. The balanced-budget constraint requires that promises to voters do not average more than 1, i.e., 100% of bi . In other words, the redistribution plan determines, for each voter, the size of their promised slice compared to the per capita size of the cake. Notice that xni can take any value in R+ , since the budget constraint requires only that the average value of the shares of bi not exceed 1. It is thus possible that some voters are promised more than the per capita budget, although this means that others will receive a share of bi smaller than 1. Special-interest-group members have a linear utility function that depends both on money and on the local public good. If party i’s platform is implemented, they get ul = xli · bi + U (gi ), where xli ∈ [0, +∞) is the ratio of the cash transfer received through party i’s redistribution plan to bi , and where U (0) = 0 and U [λU (1 − ξ)] = U hold. Standard voters’ utility is linear in money. If party i’s platform is implemented, they get uk = xki · bi , where xki ∈ [0, +∞) is defined as in the previous case. Consider now players’ strategies. In the first stage, a strategy for party i, ΣA i , is either gi = 0 or gi = λU (1 − ξ). In the next stage, party i announces a redistribution plan; this is a map that, for a generic voter n, defines the share xni of bi that the voter receives if the platform is implemented.7 The strategy ΣB i is a function of party i’s investment and that of the opponent: © l ª k ΣB (g ; g ) : {0, λU (1 − ξ)} × {0, λU (1 − ξ)} → x (l) ; x (k) , i j i i i 6

A fraction of the lump-sum tax finances the local public good; these expenses amount to an infinite quantity and absorb part of the infinite total revenues. This setting approximates the situation where a finite cost decreases the size of a finite public budget. 7 As in Lizzeri and Persico [2001], this definition of party strategies is meant to approximate real-world circumstances where redistribution occurs across many groups in the electorate but does not target every single voter.

7 where xli (l) : [0, λ] → [0, +∞) and xki (k) : [λ, 1] → [0, +∞) are such that Z

λ 0

Z xli

(l) · dl + λ

1

xki (k) · dk = 1.

The vals [0, λ] and [λ, 1] describe the subsets of the electorate containing special-interestgroup members and standard voters. In the last stage, people turn out to vote in the election8 and choose between party 1 and party 2. The standard voters’ strategy, Σk , is a function of the transfer received through the redistribution plan: ¡ ¢ Σk xki · bi ; xkj · bj : [0, +∞) × [0, +∞) → {1, 2} . The special-interest-group members’ strategy, Σl , depends on party transfers and on their decisions on the local public good: ¢ ¡ Σl gi ; gj ; xli · bi ; xlj · bj : {0, λU (1 − ξ)} × {0, λU (1 − ξ)} × [0, +∞) × [0, +∞) → {1, 2} . 4 Equilibrium Analysis The equilibrium concept is that of a subgame-perfect Nash equilibrium obtained by backward induction. The election is analyzed first. In equilibrium, a standard voter chooses party i if k xi bi ≥ xkj bj ; a member of the special interest group chooses party i if xli · bi + U (gi ) ≥ xlj · bj + U (gj ). 4.1 Redistribution Game The second stage is a modified version of the redistribution game studied by Myerson [1993]. The values of g1 and g2 are taken as given and result in two possible settings: one where parties are symmetric and the other where they are not. In fact, there might be a share of voters whose preferences are biased toward the party that only provides the local public good. The game is zero-sum, since S1 = 1 − S2 holds, and does not admit solutions in pure strategies. Lemma 1 No pure-strategy equilibria exist for the subgame where redistribution plans are defined. 8

It is assumed here that this happens even though each voter has zero mass and the probability of being pivotal in the election is nil. This assumption is justified because here the infinite electorate is being used as an approximation for a large finite population.

8 Proof Nonexistence of pure-strategy equilibria when g1 = g2 = 0 is a well-known result.9 Consider then the case where gi > 0 holds, and suppose that party i adopts a pure strategy. When gj = 0, party j ranks the electorate according to the expected utility of the platform of its opponent and pledges nothing to voters at the top. It then defines a redistribution plan that replicates party i’s utility distribution and awards each voter an arbitrarily small additional transfer ε > 0. This is a best response, because party j gets the votes of all those who are promised xnj > 0. Provided that investment costs do not equal the public budget (i.e., gi < 1), party i always has a profitable deviation and pure strategies are never optimal. If gj > 0, party j ranks voters according to the direct transfers defined by party i’s redistribution plan, and an analogous argument applies. Q.E.D. Using an infinite electorate defines a very large pure strategy space and makes mixed strategies complicated to handle. Adopting Myerson’s [1993] approach, this analysis focuses on the cumulative probability functions Fik (xki ) and Fil (xli ) that characterize the distribution of party i’s promises to standard voters and special-interest-group members. A strategy consists thus of the choice of a random variable for each type of agent, implying that a generic voter receives a promise xni that is drawn independently from Fin (xni ). A second effect of having a continuum of voters is that the law of large numbers applies.10 The function Fin (xni ) expresses the fraction of type-n (n = l, k) voters who are pledged a share of bi smaller than xni . Full redistribution being always optimal in equilibrium, party i’s budget constraint is defined by the following equality: Z +∞ Z +∞ l l l xki · dFik (xki ) = 1. xi · dFi (xi ) + (1 − λ) λ 0

0

Party i obtains the votes of people who are promised an expected utility greater than that of its opponent; its maximization problem is defined as follows: ¸ Z ∞ · ¡ ¢ U (gi ) − U (gj ) l l bi max Si = λ F j xi · + dFil xli bj bj Fil ,Fik 0 µ ¶ Z ∞ ¡ ¢ k k bi + (1 − λ) Fj xi · dFik xki bj 0 · ¸ Z ∞ Z ∞ ¡ l¢ ¡ k¢ l l k k + µi 1 − λ xi dFi xi − (1 − λ) xi dFi xi . 0

9

0

See Myerson [1993] and Sahuguet and Persico [2006]. Both the assumption of independence and the application of the law of large numbers pose problems, as discussed by Myerson [1993] and Alos-Ferrer [2002]. However, interpreting the continuum as an approximation for a large finite number of voters allows such technical difficulties to be overcome. 10

9 Resources available for redistribution play an important role in the definition of the equilibrium. The size of bi , in fact, sets party i’s shadow cost of money, which is equal to the value of the Lagrange multiplier attached to the budget constraint, µi . Consider first the asymmetric case, and suppose without loss of generality that g1 > 0 and g2 = 0. The following lemmas show how the parties compete on different sections of the electorate. Lemma 2 The parties pledge positive transfers to special-interest-group members if and only if 1/µ2 ≥ U holds. Proof The quantity 1/µi defines the monetary value that party i assigns to a standard voter. The value of a special-interest-group member is 1/µ1 for party 1.11 In order to contest one of these votes, party 2 must first promise a transfer larger than or equal to U and fill the gap with its opponent. Their net value thus is (1/µ2 ) − U , which is positive if 1/µ2 ≥ U . Q.E.D. The advantage awarded by the local public good works as a deterrent, and weakens the competition for the special-interest-group members. Lemma 3 Party 2’s assessment of standard voters’ worth exceeds that of its opponent Proof Since b2 = 1 ≥ b1 = 1 − λU (1 − ξ) holds, µ1 ≥ µ2 and 1/µ2 ≥ 1/µ1 must also hold. Q.E.D. Party 2 then can pay a higher average cost for these votes and obtain the main share of them. Combining the results of the previous lemmas, it is possible to conclude: Proposition 1 If g1 > 0, g2 = 0, and 1/µ2 < U hold, then in the unique equilibrium for the redistribution game, both parties promise zero to special-interest-group members. Redistribution is limited to the standard voters. Party 1 promises zero to a randomly chosen fraction 1 − (µ2 /µ1 ) of these and allocates its budget to the remaining voters according to a uniform distribution on [0; 1/µ2 ]. Party 2’s promises are uniformly distributed on the support [0; b1 /µ2 ]. The expected vote shares are S1 = λ + (1 − λ) (b1 /2) and S2 = 1 − λ − (1 − λ) (b1 /2). Proof See the Appendix. The votes of the special interest group accrue to party 1 due to the preference bias introduced by the local public good. Moreover, its lower assessment of the worth of voters prevents party 1 from matching the opponent’s average investment in standard voters. This limits the competition to a random fraction. 11

See the connection with all-pay auctions proven in the Appendix.

10 Proposition 2 If g1 > 0, g2 = 0, and 1/µ2 ≥ U , then in the unique equilibrium for the redistribution game, both parties pledge zero to a randomly chosen fraction of the special-interest-group members. Party 1’s promises to these voters are uniformly distributed on the support [0; (1/µ2 ) (1/b1 ) − U/b1 ] with an atom of probability at zero amounting to U · µ2 . Party 2 adopts a uniform distribution on [U ; 1/µ2 ] with an atom at zero amounting to 1 − (µ1 /µ2 ) + U · µ1 . Party 1’s distribution involves pledging zero to a randomly chosen fraction 1−(µ2 /µ1 ) of the standard voters and using a uniform distribution on [0; 1/ (µ1 · b1 )] for the rest. Party 2’s promises are uniformly distributed on the support [0, 1/µ1 ]. The expected vote shares are S1 = λ [1 − (µ1 /µ2 )] + µ1 (1 + λU ξ) and S2 = 1 − λ [1 − (µ1 /µ2 )] − µ1 (1 + λU ξ). Proof See the Appendix. It could be shown that 1/µ1 > (1/µ2 ) − U always holds, meaning that party 1 outspends its opponent and gets the biggest vote share in the special interest group. The advantage deriving from the provision of the local public good allows a fraction of these voters to be subsequently ignored. Since party 1’s average investment exceeds its assessment of special-interest-group members, party 2 cannot compete for all of them on equal terms. So in its redistribution plan, each member of the group has a positive probability of receiving a zero transfer. Dealing with standard voters, party 2 outspends the opponent and gets the main vote share, while party 1 focuses on a fraction of them and promises zero to the others. The case where g1 = g2 boils down to the setting studied by Myerson [1993]. If neither party invests, there are no differences in fact between special-interest-group members and standard voters. In the same way, when both provide the local public good, no subset of voters has an initial preference for either party and the effects of g are canceled out. The unique equilibrium in both cases has the following characteristics:

⇐=Au.: O.K.?

(1) Each voter receives from party i (i = 1, 2) the promise of a fraction of its per capita budget that depends on the realization of a random variable uniformly distributed on the support [0; 1/µ2 ], where 1/µ2 = 1/µ1 = 2. (2) S1 = S2 = 1/2 holds, and no party gets a majority of votes. 4.2 Local-Public-Good Provision The first stage of the platform presentation process is a simultaneous game with the normal-form representation given in Table 1, where Si (ξ) = λ+[(1 − λ) /2] [1 − λU (1 − ξ)], if 1/µj < U , and Si (ξ) = λ [1 − (µi /µj )] + µi (1 + λU ξ) if instead 1/µj ≥ U holds. Investment provision requires Si (ξ) ≥ 1/2; if the previous condition is satisfied, the strategy gi = λU (1 − ξ) is weakly dominant and a Nash equilibrium exists where both parties include g in their platforms. It is thus possible to define a function ξ ∗ (U, λ) such that for every ξ ≥ ξ ∗ (U, λ) we have Si (ξ) ≥ 1/2. This characterizes the minimum efficiency level that, in equilibrium, leads to local-public-good provision. Consider initially

11 Table 1 Local-Public-Good Provision Party 2

Party 1

0

λU (1 − ξ)

0

1/2, 1/2

1 − S2 (ξ); S2 (ξ)

λU (1 − ξ)

S1 (ξ); 1 − S1 (ξ)

1/2, 1/2

the simplest case where the votes of special-interest-group members are not contested, and assume, without loss of generality, that g1 > 0, g2 = 0, and U > 1/µ2 . Proposition 3 When U > 2/ (1 − 2λ) and λ < 1/2, both parties include the local public good in their electoral platforms if ξ ≥1−

1 = ξ ∗ (U, λ) . (1 − λ) U

Proof An equilibrium where S1 (ξ) = 1/2 and U > 1/µ2 hold requires that both the following conditions occur at the same time: S1 (ξ) = λ + and U>

1−λ 1 [1 − λU (1 − ξ)] = 2 2

2 1 1 · = . 1 − λ 1 − λU (1 − ξ) µ2

Solving for ξ in the first equation gives the minimum efficiency level that allows investment provision, ξ ∗ (U, λ) = 1 − 1/U (1 − λ). Substituting this into the second inequality characterizes the region of the plane (U, λ) where ξ ∗ (U, λ) defines an equilibrium such that both parties include g in their platforms. This gives U>

2 1 2 · = . λU 1 − λ 1 − (1−λ)U 1 − 2λ

The condition 1/µ2 ≥ 0 further requires that λ ≤ 1/2.

Q.E.D.

Note that ξ ∗ (U, λ) = 1 − 1/U (1 − λ) is always smaller than 1, implying that investment provision never requires g to have zero cost. Thus, whenever the share of the public budget absorbed by the cost of g is strictly less than 1 − ξ ∗ (U, λ), parties investing in the local public good can do no worse than obtain an equilibrium vote share larger than, or equal to, 50%. Consider now what happens if the individual benefit, U , is not very large and competition takes place for both types of voters.

12 Proposition 4 When U ≤ 2/ |1 − 2λ|, both parties include the local public good in their electoral platforms if p 1 2 1 + λ (1 − λ) U 2 − 1 ξ≥ · = ξ ∗ (U, λ) . 2 λU (1 − λ) Proof See the Appendix. In this case too, ξ ∗ (U, λ) < 1 holds and it is possible to exclude the circumstance where investment provision requires g to have zero cost. It is easy to check that in fact the inequality p 2 1 + λ (1 − λ) U 2 < 1 + 2λU (1 − λ) is always verified. Raising both sides of the inequality to the power of two and reordering the terms then yields [2/ (1 − 2λ)]2 > U , which is always true given that U ≤ 2/ |1 − 2λ| holds. There is one last circumstance to be considered, that where g benefits at least 50% of voters and the individual benefit exceeds the value 2/ (2λ − 1). Proposition 5 If U > 2/ (2λ − 1) and λ ≥ 1/2, every local public good for which ξ ≥ 1 − (1/λU ) holds is affordable and provides a vote share S1 (ξ) ≥ 1/2. Proof Suppose initially that ξ = 1 − (1/λU ) and U = 2/ (2λ − 1) hold; this means that b1 = 1 − λU (1 − ξ) = 0 and party 1 cannot promise any cash transfer. Party 2’s best response then is to pledge each standard voter the quantity xk2 = ε > 0, with ε arbitrarily small, to obtain all these votes. Note further that since (2λ − 1) (U/2) = 1 holds, promising slightly more than U to a fraction (2λ − 1) /2 of the special interest group is optimal, because it allows the budget constraint to be fulfilled with equality and at the same time attracts the votes of all those who receive a promise xl2 > 0. Consider now party 1’s vote share; it is the case that S1 = λ [1 − (2λ − 1) /2] ≥ 1/2. Indeed, through some simple algebra, it is possible to get 2λ2 − 3λ + 1 ≤ 0. Solving for λ shows that the previous inequality is always verified for 1 ≤ λ ≤ 1/2 and must hold a fortiori when ξ > 1 − (1/λU ) and U > 2/ (2λ − 1). Q.E.D. From the previous propositions a further result follows. Proposition 6 In order to provide a local public good, it is not enough, in general, for the investment to be affordable. Proof When λ < 1/2 and U > 2/ (1 − 2λ) do indeed hold, the inequality 1 − 1/U (1 − λ) > 1 − (1/λU ) is verified; if instead, 2/ |1 − 2λ| ≥ U holds, it is easy to check that p 2 1 + λ (1 − λ) U 2 − 1 1 1 ≥1− . ξ ∗ (U, λ) = · 2 λU (1 − λ) λU

13 Raising both sides to the power of 2 and rearranging the terms gives i2 h p 1 − (2λ − 1) 2 1 + λ (1 − λ) U 2 ≥ 0, which is always true.

Q.E.D.

Investments whose costs absorb the entire public budget and do not leave available resources for direct cash transfers are never provided except if λ ≥ 1/2 and U > 2/ (2λ − 1).12 The function ξ ∗ (U, λ) can then be characterized as follows:  1 2  1 − (1−λ)U if U > 1−2λ and λ < 12 ,   √ 2 2 ξ ∗ (U, λ) = 1 · 1+λ(1−λ)U −1 if U ≤ 2 , 2 λU (1−λ) |1−2λ|   1 − 1 if U > 2 and λ ≥ 1 . λU

2λ−1

2

The efficiency threshold that allows local-public-good provision is always positive or at most nil. If U > 2/ (1 − 2λ) and 0 ≤ λ < 1/2 hold, then ξ ∗ (U, λ) = 1 − 1/U (1 − λ) > 0. When 0 ≤ U ≤ 2/ |1 − 2λ| holds, for every 0 < λ < 1 the inequality p 2 1 + λ (1 − λ) U 2 − 1 1 ξ ∗ (U, λ) = · > 0 2 λU (1 − λ) is verified. If λ approaches zero moreover, the condition U ≤ 2/ (1 − 2λ) boils down to U ≤ 2 and also p 1 2 1 + λ (1 − λ) U 2 − 1 U lim · = ≥ 0. λ→0 2 λU (1 − λ) 4 4.3 Pure Public Goods The previous section discarded the case λ = 1, but since other authors have studied that circumstance, a comparison with their results is worth a supplementary analysis. Lizzeri and Persico [2001], in particular, consider pure public-good provision in a setting that differs from the present one, mainly with regard to the timing of the model. They assume that the decisions on investment provision and on redistribution are made simultaneously. The differences in the timing of the model mean that the analyses focus on different specific relationships between the variables. Lizzeri and Persico [2001] in fact consider how parties’ incentives vary when the investment rate of return changes; project efficiency defines the probability that the public good is included in electoral platforms. The present model looks at the problem from a reverse angle and focuses on the case where investment provision is certain. This requires the definition of the minimum rate of return, which characterizes all the public goods that parties are willing to realize. 12

This result partially complements the analysis of Lizzeri and Persico [2001] obtained in a slightly different setting, when λ < 1.

14 The two settings return identical results when, in the simultaneous-decisions model, public-good provision is not stochastic. This happens either when the rate of return is negative and no party invests in the project or when individual benefits exceed 2 and the public good is provided with certainty. Consider, indeed, what happens in the present setting if λ approaches 1. The condition U ≤ 2/ (2λ − 1) boils down to U ≤ 2, and limλ→1 ξ (U, λ) is p 1 2 1 + λ (1 − λ) U 2 − 1 U lim · = . λ→1 2 λU (1 − λ) 4 Note that for U = 2, the condition above requires only that the public good be affordable, since (U − 1) /U = U/4 = 1/2; this finding is consistent with the result of Proposition 6 and shows that when the individual benefit is large enough, the only binding constraint is 1−λU (1 − ξ) ≥ 0. When 1 < U < 2 holds, Lizzeri and Persico [2001] find that an increase in U also increases the probability that the public good is provided; this depends crucially on the assumption that the project has the fixed cost 1. In the present model, however, the cost of the public good depends on the size of the individual benefit; consequently, when U increases, ξ must also increase to compensate for the reduction in resources available for direct cash transfers.13 5 The Effects of Decentralization The present section analyzes the function linking the size of the special interest group and the efficiency threshold, ξ ∗ (U, λ). This allows the effects of decentralization on underinvestment to be studied and the optimal degree of decentralization to be defined. A simple way to approach the issue is to consider the sign of the first derivative of the function ξ ∗ (U, λ) with respect to λ. An increase (decrease) in λ is understood as a reduction (increase) in electorate size following a rise in the level of decentralization (centralization). This causes both an increase in the relative vote share of the beneficiaries and an increase in the fraction of the public budget absorbed by the investment costs. If δ · ξ ∗ (U, λ) /δ · λ ≥ 0 holds, the minimum rate of return that allows investment provision increases. Parties thus are willing to include in their platform a local public good supplying a fixed aggregate benefit λU , only if its cost can be reduced below a certain threshold. Even further cost reductions are necessary as lower government levels are considered, meaning that the underprovision problem widens. The opposite happens if δ · ξ ∗ (U, λ) /δ · λ < 0. In real economies where the electorate size changes discontinuously if different government levels are considered, allowing the parameter λ to vary continuously may seem unrealistic. Nonetheless, the present analysis shows that the effects of decentralization are determined exclusively by the fact that the threshold λ = 1/2 is exceeded. This permits a straightforward application to actual circumstances. The following propositions summarize the main results. 13

Through simple algebra, it is easy to check that δ · ξ ∗ (U, λ) /δ · U ≥ 0 always holds.

⇐=Au.: This use of “δ·” instead of “d” or “d” is nonstandard.

15 Lemma 4 If λ < 1/2 and 2/ (1 − 2 · λ) < U hold, decentralization reduces the underprovision problem. Proof The implementation of g requires that ξ ≥ 1 − 1/ [U · (1 − λ)]; the effects of an increase in λ are described by the first derivative of ξ ∗ (U, λ): δ · ξ ∗ (U, λ) 1 =− < 0. δ·λ (1 − λ)2 Since the above expression is always negative, an increase in the size of the special interest group lowers the efficiency threshold. Q.E.D. Lemma 5 When U ≤ 2/ |1 − 2λ| holds, decentralization reduces the underprovision problem as long as λ ≤ 1/2, but it has the opposite effect when λ > 1/2. Proof The implementation of g requires that p 1 2 1 + λ (1 − λ) U 2 − 1 ξ≥ · . 2 λU (1 − λ) Consider the first derivative of ξ ∗ (U, λ) with respect to λ: " # δ · ξ ∗ (U, λ) 1 1 − 2λ 2 + λ (1 − λ) U 2 = · 1− p . δ·λ 2 λ2 (1 − λ)2 U 2 2 1 + λ (1 − λ) U 2 p Note that the inequality 2 2 1 + λ (1 − λ) U 2 ≤ 2 + λ (1 − λ) U 2 holds given that 4 + 4λ (1 − λ) U 2 ≤ 4 + λ2 (1 − λ)2 U 4 + 4λ (1 − λ) U 2 . Hence the sign of δ·ξ ∗ (U, λ) /δ·λ depends on the quantity 1−2λ and is δ·ξ ∗ (U, λ) /δ·λ ≤ 0 for λ ≤ 1/2 and δ · ξ ∗ (U, λ) /δ · λ ≥ 0 for λ > 1/2. Q.E.D. Proposition 7 The function ξ ∗ (U, λ) has a unique minimum at λ = 1/2, implying that decentralization reduces underprovision as long as the size of the special interest group does not exceed 50% of the electorate. If it does, the underinvestment problem is exacerbated. Proof The proof follows from the previous lemmas and from the fact that, when U > 2/ (2λ − 1) and λ ≥ 1/2 hold, parties are always willing to provide local public goods which are affordable. Reducing the size of the public budget thus requires that ξ ∗ (U, λ) = 1 − (1/λU ) increase, since δ · ξ ∗ (U, λ) /δ · λ = 1/λ2 U ≥ 0. Q.E.D. So an optimal degree of decentralization is defined as the situation where responsibility for the provision of a particular local public good is assigned to the government level where beneficiaries amount to 50% of the electorate.

⇐=Au.: O.K. – not (1/U )(1 − λ)?

16 Consider what happens to the party providing the local public good if the opponent does not invest. Two circumstances are important when redistributive policies are chosen: the party is initially preferred by people benefiting from the investment, but its budget decreases due to the costs of the project. The investment rate of return being positive, overall resources (both monetary and nonmonetary, i.e., the benefits generated by the local public good) increase, but targetable funds decrease. Since the identity of the special interest group is known, the opponent has the opportunity to lower the competition for these voters and increase funds targeting the rest of the electorate; some of the beneficiaries thus receive ex post a zero transfer from it, meaning that the party that invests would in fact be able to obtain their votes by offering much less than the benefit U . Being locked up in a specific fraction of the electorate, some of the nontargetable resources generated by the local public good fail to find useful employment in the electoral contest. As λ increases, the special interest group gets larger and becomes more attractive for the opponent; the probability that these people receive a zero transfer is reduced, implying that investment benefits are better exploited. However, the rise in the budget share absorbed by investment costs decreases the amount of freely targetable funds; the resource distribution of the party that provides the local public good is more constrained than before. The threshold of 50% of total voters is pivotal because it determines whether the positive effect of an increase in total resources prevails over the negative effect of a decrease in targetable funds. If the group of beneficiaries is small, investment costs are small too, targetable funds are abundant, and there are few constraints on resource distribution. In this case, the net effect of decentralization is positive and the minimum efficiency level ξ ∗ (U, λ) decreases along with λ. When the value λ = 1/2 is exceeded, the disbenefit caused by the shrink in targetable funds exceeds the benefit deriving from better exploitation of nontargetable resources. Provision of the local public good imposes bigger constraints on resource distribution and becomes less attractive. An increase in efficiency is required to persuade parties to invest. 6 Final Remarks The present paper considers a tractable model of elections for different government levels in which parties determine an allocation of the public budget between redistributive policies and public-capital provision. The results of the analysis show that fiscal decentralization has ambiguous effects on this allocation. For every infrastructural investment there is a particular government level where the underprovision problem is minimized. This finding clashes with the view that decentralization always improves public funding allocation,14 which is supported by a number of arguments: from a wider scope for adapting policies to “local” preferences, to Tiebout’s “foot voting.” An inefficient low provision of local public goods, though, is a general feature of the present model. Since parties maximize vote shares, the limited targetability of infras14

See Oates [2006] for an overview of these issues.

17 tructure prevents projects whose benefits balance costs from being included in electoral platforms. Such investment in fact absorbs resources and delivers advantages only to a specific share of the electorate, a strategic drawback that necessitates compensation in the form of a positive rate of return. Decentralization is not neutral with respect to the problem of local-public-good underprovision and produces effects that do not depend only on the degree of discriminatory ability of the fiscal system (see Roberson [2008]). The efficiency threshold that persuades parties to include a particular project in their platforms varies with the share of the electorate that is recipient of its benefits and reaches a minimum at 50%. This realization could supply a useful rule of thumb for a national government in deciding the level of administration to which to delegate the decision to provide a specific infrastructure. The distribution of fiscal revenues among different government levels is of course made before this decision. Consider a real-world example: the German system of road building and maintenance. State authorities in Germany autonomously decide the level of spending on their roads, as do local governments on theirs; most of this work is financed from own-source revenues. Classification of a road as state or local thus entails a different level of spending on maintenance and might also be crucial in the decision over provision of the road. The present analysis rests on some assumptions that are worth discussing. The first assumption concerns the timing of the model. Infrastructure provision causes a privileged group to form within the electorate and splits voters into recipients and nonrecipients of its benefits. The presentation of electoral platforms moreover happens in two steps, and thus reveals voters’ types before the end of the campaign. This is because the required degree of institutional cooperation implies self-revelation of the investment. Nonetheless, studying the case where decisions over local-public-good provision and over redistributive policies are presented simultaneously might represent a valuable extension to the analysis of different classes of local public goods. Policy financing is in this model assumed to be exclusively from local government budgets. No earmarked funds are available, which represents the fact that decentralization reduces the amount of resources available to parties. It is meant to approximate real-world circumstances where lower government levels count on smaller budgets, both because the number of taxpayers decreases and because general transfers from central government diminish as the electorate shrinks. Finally, public-capital provision is a problem that is unlikely to involve only one project at a time, since parties usually choose from a menu of possible local public goods. The introduction of multiple alternatives would perhaps make the model more realistic and supply further insight into underinvestment. It could be a fruitful avenue for future research.

18 Appendix A.1 Main Proofs A.1.1 Proof of Proposition 2 In order to prove that the proposed strategies are in fact an equilibrium, we can check whether they are mutual best responses. ¡ ¢ Consider initially party 1’s strategy. F1k∗ xk1 must be an optimal response to ¡ ¢ F2k∗ xk2 . Party 2 in fact does not compete for the special interest group, and it is optimal to set F1l∗ (0) = 1. No resources accrue to the people who vote for party 1 because of the benefits to be generated by the local good. The following inequality ¡ public ¢ thus must hold for every possible choice of F1k xk1 :15 Z +∞ Z +∞ ¡ l ¢ l¡ l¢ ¡ ¢ ¡ ¢ l∗ λ F2 x1 · b1 + U dF1 x1 + (1 − λ) F2k∗ xk1 · b1 dF1k xk1 0 0 Z +∞ ¡ ¢ ≤ λ + (1 − λ) µ2 xk1 · dF1k∗ xk1 . 0

¡ ¢ R +∞ Substituting for the feasibility constraint (1 − λ) 0 xk1 · dF1k∗ xk1 = 1 gives Z +∞ ¡ ¢ ¡ ¢ λ + (1 − λ) F2k∗ xk1 · b1 dF1k xk1 ≤ λ + µ2 . 0

¡ ¢ For every F1k xk1 defined over the support [0; ¡ 1/µ ¢ 2 ] the previous inequality holds with equality; this happens for the proposed F1k∗ xk1 that is an optimal response to party 2’s strategy and gets the maximum expected vote share. Moreover, in order to have an equilibrium, a solution for the following system must exist:  1 1 = (1 − λ) µ2 R µ2 xk · µ , 2 1 µ1 0 1 1 = (1 − λ) R µ2 ·b1 xk · µ2 . 0

2

b1

Solving 1/µ2 in the second equation gives 1 2 1 = · , µ2 1 − λ b1 and party 1’s vote share is thus S1 = λ + (1 − λ) (b1 /2). Substituting the previous result in the first equation, we obtain 2 1 = . µ1 1−λ A symmetric argument applies to party 2’s strategy and completes the proof. Uniqueness derives from the connection with all-pay auctions (proved in the last section of this Appendix) and from the results of Hillman and Riley [1989]. 15

Note that since party 2 randomizes continuously on the support [0; b1 /µ2 ], the probability of a tie is nil.

19 A.1.2 Proof of Proposition 3 The proof exploits the argument used for the previous proposition and requires checking whether the parties’ strategies are mutual best responses. Consider initially party 1, and note that, given the upper bounds of party 2’s distribution, any strategy including xk1 > 1/µ1 b1 or xl1 > (1/µ2 b1 ) − (U/b1 ) is a dominated strategy; it is¡ thus the following inequality always holds for any possible ¢ the case ¡ l ¢that k k l 16 choice of F1 x1 and F1 x1 : Z +∞ Z +∞ ¡ ¢ ¢ ¡ ¡ l¢ ¢ ¡ l l l∗ F2k∗ xk1 · b1 dF1k xk1 λ F2 x1 + U b1 · dF1 x1 + (1 − λ) 0

µ

¶

Z

0

1 · 1 − bU µ2 b1 1

µ1 + U · µ1 + λ · µ1 · b1 · µ2 0 Z 1·1 µ1 b1 ¡ ¢ + µ1 · b1 (1 − λ) xk1 · dF1k∗ xk1

≤ λ 1−

¡ ¢ xl1 · dF1∗l xl1

0

so that, from the feasibility constraint Z Z 1 · 1 −U µ2 b1 b1 ¡ l¢ l ∗l x1 · dF1 x1 + (1 − λ) 1=λ 0

1 ·1 µ1 b1

0

¡ ¢ xk1 · dF1k∗ xk1 ,

it is possible to get Z +∞ Z +∞ ¡ l ¢ ¡ l¢ ¡ ¢ ¡ ¢ l∗ l λ F2 x1 + U b1 · dF1 x1 + (1 − λ) F2k∗ xk1 · b1 dF1k xk1 µ0 ¶ µ ¶0 µ1 µ1 + U · µ1 + µ1 · b1 = λ 1 − ≤ λ 1− + µ1 (1 + λU ξ) . µ2 µ2 ¡ ¢ ¡ ¢ Every F1k xk1 defined over the support [0; 1/µ1 · b1 ] together with every F1l xl1 defined over [0; (1/µ2 · b1 ) − (U/b1 )] is an optimal response to party 2’s strategy and gets the maximum expected vote share, S1 = λ [1 − (µ1 /µ2 )] + µ1 (1 + λU ξ). This is also the case for the strategy proposed here. Optimality further requires that parties’ budget constraints hold with equality and that a solution for the following system exist:  1 1 U 1 1 1 = λ R µ2 · b1 − b1 xl · µ · b + (1 − λ) µ2 R µ1 · b1 xk · µ · b , 2 1 1 1 0 1 1 µ1 0 1 1 R R 1 = λ · µ1 µ2 xl · µ + (1 − λ) µ1 xk · µ . µ2

U

2

2

0

2

1

Solving the integrals and reordering the terms gives  ·³ ´ ¸ ³ ´2 2  λ 1 1−λ 1 1 2  + U + ,  µ2 (b1 + λU ) = 2 µ2 2 µ1 ·³ ´ ¸ ³ ´2 2  1 1−λ 1 2  − U + .  µ11 = λ2 µ2 2 µ1 16

Note that the probability of a tie is again nil. If standard voters are considered, party 2 randomizes continuously on the support [0; 1/µ1 ]. In the case of distribution to special-interestgroup members, the atom at zero cannot produce ties, given that no negative promises are allowed and that party 1 has the advantage U over these voters.

20 Now solve for 1/µ1 as a function of 1/µ2 in the second equation to get17 v  "µ ¶ # u 2   u 1 1 1 2 t 2 . = · 1 + 1 − λ (1 − λ) −U  µ1 1−λ  µ2 Substituting 1/µ1 in the first equation and raising both sides to the power of two allows us to obtain through simple algebra a solution for 1/µ2 : ¡ ¢ q £ ¤¡ ¢ 1 1 1 2 (b + λU ) λ · U + + 2 (b1 )2 + 2 · b1 · λU λ · U 2 + 1−λ 1 1 1−λ 1−λ = . 2 λ µ2 (b1 + λU ) + 1−λ Substituting for 1/µ2 into the equation that defines 1/µ1 gives q 2 ¡ (b1 )2 +2·b1 ·λU 1 ·λU + (b1 + λU ) 2 (b1 ) +2·b λ · U2 + 1 1−λ 1−λ = λ µ1 (b1 + λU )2 + 1−λ

1 1−λ

¢ .

¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ The optimality of F2k∗ xk2 and F2l∗ xl2 in response to F1k∗ xk1 and F1l∗ xl1 is proven following the same steps and concludes the characterization of the equilibrium. Uniqueness derives from the link with all-pay auctions proven in the last section of this Appendix; the results obtained by Hillman and Riley [1989] and by Magnani [2006] complete the argument. A.1.3 Proof of Proposition 5 Investment provision requires that µ ¶ µ1 1 λ 1− + µ1 (1 + λU ξ) = , µ2 2 i.e., it requires that the system of three equations in three unknowns reported below possess a solution:18 ·³ ´ ¸  ³ ´2 2 λ 1 1 1 1−λ  2  (1 + λU ξ) = + U + ,  µ2 2 µ2 2 µ1   s (  ·³ ´ ¸) 2 1 1 = 1−λ · 1 + 2 1 − λ (1 − λ) µ12 − U 2 ,  µ1      1 + λU ξ = 1−2λ · µ11 + µλ2 . 2 17

Note that when λ = 0, the present framework reduces to the problem where 1/µ1 = 1/µ2 = 2 outlined in Sahuguet and Persico [2006]. Hence the solution for the second-order equation where 1/µ1 |λ=0 6= 2 can be discarded. 18 The second and third equations are derived, through simple algebra, from parties’ budget constraints. See the proof of Proposition 3 for further details.

21 Substitute the values of 1/µ1 and 1 + λU ξ obtained from the second and third equations into the first. Simplifying and reordering the terms yields v # "µ ¶ u 2 u 1 − 2λ 1 1 2 · =t 1 − λ (1 − λ) − U2 . 2 µ2 µ2 Raising both sides of the equation to the power of two and solving for 1/µ2 gives p 1 = 2 2 1 − λ (1 − λ) U 2 . µ2 Now substitute 1/µ2 in the second equality to get the equilibrium value of 1/µ1 : h i p 1 1 = · 1 + (1 − 2λ) 2 1 + λ (1 − λ) U 2 . µ1 1−λ The value for ξ ∗ (U, λ) is derived, through some simple algebra, by substituting µ1 and µ2 in the last equation: p 1 2 1 + λ (1 − λ) U 2 − 1 ∗ ξ (U, λ) = · . 2 λU (1 − λ) Consider, finally, the values for U such that the inequality 1/µ2 ≥ U is verified. This condition requires that p 2 2 1 + λ (1 − λ) U 2 ≥ U. Raising both sides of the inequality to the power of two and rearranging the terms gives 4 ≥ U 2, 2 1 − 4λ + 4λ so that

2 ≥ U. |1 − 2λ|

A.2 Connection with All-Pay Auctions In order to exploit results from the literature, the equilibrium for the subgame where g1 6= g2 is studied, adopting Sahuguet and Persico’s [2006] approach. These authors analyze the problem of two candidates who maximize vote shares, redistributing a given budget among ex ante identical voters, and prove a connection with the problem of two players bidding in a first-price all-pay auction with complete information. Assume, without loss of generality, that only party 1 invests. In this case g1 = λU (1 − ξ) and g2 = 0 hold, while the redistributable budgets amount respectively to b1 = 1 − λU (1 − ξ) and b2 = 1. The following equivalence is verified.

22 Proposition 8 When g1 = λU (1 − ξ) and g2 = 0, a unique pair of independent all-pay auctions exists such that the parties’ maximization problem is strategically equivalent to that of two players bidding simultaneously. The first auction is a standard first-price all-pay one; in the second, the auctioneer increases player 1’s bid by a fixed amount U . The players’ valuations of the auctioned item are known; they correspond to the reciprocals of the optimal choices for the Lagrange multiplier attached to the budget constraint in the parties’ maximization problem, 1/µ1 and 1/µ2 . Proof In order to prove the proposition, I show that the maximization problem of the bidders and that of the parties are the same up to a linear transformation. Party 1’s maximization problem is ¶ Z ∞ µ ¡ ¢ b1 U l l F2 x1 · + dF1l xl1 max λ b2 b2 F1l ,F1k 0 µ ¶ Z ∞ ¡ ¢ k b1 k + (1 − λ) F 2 x1 · dF1k xk1 b2 0 · ¸ Z ∞ Z ∞ ¡ l¢ ¡ k¢ l l k k + µ1 1 − λ x1 dF1 x1 − (1 − λ) x1 dF1 x1 . 0

0

¡ ¢ ¡ ¢ Substituting b2 = 1 and defining xl1 · b1 = y1l , xk1 · b1 = y1k , F1l y1l /b1 = Fˆ1l y1l , and ¡ ¢ ¡ ¢ F1k y1k /b1 = Fˆ1k y1k allows the previous formula to be rewritten as Z ∞ ¡ ¢ ¡ ¢ max λ F2l y1l + U dFˆ1l y1l Fˆ1l ,Fˆ1k 0 Z ∞ ¡ ¢ ¡ ¢ + (1 − λ) F2k y1k dFˆ1k y1k 0 ¸ · Z ∞ Z ∞ ¡ k¢ ¡ l¢ k ˆk l ˆl y dF y . + µ1 b1 − λ y dF y − (1 − λ) 0

1

1

1

0

1

1

1

Through simple algebra we obtain ( ) ¸ Z ∞· ¡ ¢ ¡ ¢ 1 l l (λ · µ1 ) max F2 y1 + U − y1l dFˆ1l y1l l ˆ µ1 F1 0 ) ( ¸ Z ∞· ¡ ¢ ¡ ¢ 1 k k F2 y1 − y1k dFˆ1k y1k + b1 · µ1 . + (1 − λ) µ1 max k ˆ µ1 F1 0 Up to a linear transformation this problem is equivalent to that of a risk-neutral player who maximizes expected utility by bidding on two types of independent first-price allpay auctions where identical objects are auctioned. In the first, the auctioneer increases player 1’s bids by the fixed amount U ; in the second, players are symmetric. Player 1’s valuation of the auctioned item, 1/µ1 , corresponds to the optimal choice for the Lagrange multiplier attached to the budget constraint in party 1’s maximization problem. Analogous manipulations of party 2’s maximization problem allow the proof to be completed.

⇐=Au.: O.K.?

23 Q.E.D.

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24 Sahuguet, N., and N. Persico [2006], “Campaign Spending Regulation in a Model of Redistributive Politics,” Economic Theory, 28, 95–124. Stern, N. [2001], “A Strategy for Development,” ABCDE Keynote Adress, World Bank, Washington, D.C. Marco Magnani Department of Economics Universit`a degli Studi di Parma via J.F. Kennedy 6 43100 Parma Italy E-mail: [email protected]