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Inequality, Market Imperfections, and the Voluntary Provision of Collective Goods.

Inequality, Market Imperfections, and the Voluntary Provision of Collective Goods.?
Pranab Bardhan Department of Economics, University of California, Berkeley bardhan@econ.berkeley.edu Maitreesh Ghatak Department of Economics, University of Chicago mghatak@midway.uchicago.edu Alexander Karaivanov Department of Economics, University of Chicago akkaraiv@midway.uchicago.edu April, 2002

Abstract We analyze the e?ect of inequality in the distribution of endowment of private inputs (e.g., land, wealth) that are complementary in production with collective inputs (e.g., contribution to public goods such as irrigation and extraction from common-property resources) on e?ciency in a simple class of collective action problems. In an environment where transaction costs prevent the e?cient allocation of private inputs across individuals, and the collective inputs are provided in a decentralized manner, we characterize the optimal second-best distribution of the private input. We show that while e?ciency increases with greater equality within the group of contributors and non-contributors, in some situations there is an optimal degree of inequality between the groups.
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We thank Jean-Marie Baland, Abhijit Banerjee, Timothy Besley, Pierre-Andre Chiappori, Avinash Dixit, Paul

Seabright, and workshop participants in Chicago, London School of Economics, the Conference on Inequality, Cooperation and Environmental Sustainability at the Santa Fe Institute, 2001, and the NEUDC 2001 Conference at Boston University for helpful comments. However, the responsibility for all errors and shortcomings lies only with us.

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1

Introduction

The recent growth literature has revived interest in the age-old question in economics on the relationship between the initial inequality of the distribution of wealth and the pace and pattern of economic development. While the empirical evidence on this question is mixed (see Benabou, 1996 and Banerjee and Du?o, 2000), several theoretical models have suggested inequality may have a negative e?ect on economic performance. The main reasons that are advanced for this connection are, ?rst, inequality increases agency costs in the labor and credit markets (e.g., Galor and Zeira, 1993; Banerjee and Newman 1993) and second, inequality encourages redistributive policies that discourage capital accumulation (e.g., Alesina and Rodrik, 1994).1 This paper suggests a possible link between asset inequality and economic e?ciency that has been largely neglected in this literature: inequality may help or hinder the resolution of certain types of collective action problems. While the literature on collective action in political science and economics is large, its interrelationship with economic inequality is a relatively underresearched area. Typical questions, taken from a broad range of ?elds, for which our paper may be relevant are as follows: (a) Does reduction of land inequality through, say, land reform a?ect agricultural productivity by changing the voluntary provision of collective goods like irrigation? (b) How does an increase in the inequality of ownership of di?erent boat sizes of ?shermen a?ect their total catch and pro?ts in an unregulated ?shery? (c) Does inequality in property holdings in an urban residential area a?ect the performance of neighborhood crime watch groups? (d) Does the inequality of prior assets that a husband and a wife bring to a marriage a?ect their joint creation of public goods inside the family?2 (e) How does the expansion of NATO to poorer countries in Eastern Europe (thereby increasing the inter-member inequality) a?ect the provision of collective defense? While our paper will be relevant to most such cases, for the sake of a common concrete anchor we shall often use the illustration of land reform in our subsequent exposition. Producers use as inputs one private good (say, land) and one collective good (say, irrigation water) to produce a private good (say, rice). The private and collective inputs are complements in the production function. This collective good may be a public good (with positive externalities) like a public irrigation canal, or a common property resource (henceforth, CPR), like a community pond or forest, with negative
1

Agency costs refers to all costs that arise from monitoring, screening and contract enforcement when the owner

of an asset (land, capital, a machine) and its user are not the same person. 2 See Browning and Lechene (2001) for a discussion on the relationship between inequality of earnings of family members and the provision of public goods within a household.

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externalities. The assumption of decreasing returns, a standard one in most economic contexts, implies that the more scarce an input is in a given production unit, the higher is its marginal return. As a result, one would expect a more equal distribution of this input across production units to improve e?ciency. If the market for this input operated well, then the forces of arbitrage would make sure it is allocated equally to maximize e?ciency. However there is considerable evidence that suggests that the market for inputs such as land or capital does not operate frictionlessly and the private endowment of an individual determines how much of that input she can use in her production unit.3 There is a large literature showing that small farms are more e?cient than large farms in agricultural sector of developing countries. This is typically advanced as one of the main arguments for land reform in terms of e?ciency.4 Some authors (e.g., Bardhan, 1984, Boyce, 1987) have gone one step further and argued that a more egalitarian agrarian structure is also more likely to solve collective action problems, especially those related to irrigation.5 But in the presence of collective action problems inequality of private endowments such as land or wealth may pull in the opposite direction. Indeed, in his pioneering work on collective action, Olson (1965) makes the following case in favor of inequality: “In smaller groups marked by considerable degrees of inequality – that is, in groups of members of unequal “size” or extent of interest in the collective good – there is the greatest likelihood that a collective good will be provided; for the greater the interest in the collective good of any single member, the greater the likelihood that the member will get such a signi?cant proportion of the total bene?t from the collective good that he will gain from seeing that the good is provided, even if he has to pay all of the cost himself.” (The Logic of Collective Action, p. 34). We can interpret the “size” of a player with her endowment of the private input if it is complementary with the collective good in production.6 Olson considered pure public goods only, which indeed have the property that only the largest (richest) player contributes. Our paper is concerned with the following questions. Is this property pointed out by Olson true for a more general class of collective goods that include both impure public goods and CPRs? If we look at welfare instead of
3

Evans and Jovanovic (1989) analyzed panel data from the National Longitudinal Survey of Young Men (NLS),

which surveyed a sample of 4000 men in the US between the ages of 14-24 in 1966 almost every year between 1966-81, and found that entrepreneurs are limited to a capital stock of no more than one and one-half times their wealth when starting a new venture.
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According to Berry & Cline (1979) an equitable distribution of land could increase food production by 20-30%. Indeed, the limited evidence that is available on the e?ect of land tenure reform suggests that the productivity In Olson this is the share of the total bene?t to the group that accrues to an individual player.

gains can be large (Banerjee, Gertler and Ghatak, 2002).
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the level of the provision of the collective good, is it possible that some degree of inequality may yield a higher level of joint surplus than perfect equality? Furthermore, is it possible for the allocation under some degree of inequality to Pareto-dominate the allocation under perfect equality? If more than one player is involved in the provision of the public good, how would inequality within the class of contributors a?ect e?ciency and how would inequality between the class of contributors and non-contributors a?ect e?ciency? The public economics literature has addressed the question of inequality among contributing players in some detail. A key ?nding is the surprising “distribution-neutrality” result for a particular class of collective action problems, namely the provision of pure public goods.7 These are public goods where individual contributions are perfect substitutes in the production of the public good and everyone gets the same bene?t from the public good irrespective of the level of their contributions. Then in a Nash equilibrium the wealth distribution within the set of contributors does not matter for the amount of public goods provision. The intuition behind this result is explained very clearly by Bergstorm, Blume and Varian (1986). Suppose after the redistribution every player adjusts his contribution to the public good by exactly the same amount as his change in wealth and leaves the consumption of the private good unchanged. In that case the amount of the public good is the same as before and so the initial allocation is still available to all players. Those who have lower wealth because of redistribution have a restricted budget set and would clearly prefer the previous allocation if it is still available. The budget set of those who have higher wealth because of redistribution expands, but not in the neighborhood of the original choice. In particular, now the extra options available to the player which are not dominated by options in the previous budget set involve a lower level of the public good compared to what she would receive if she did not contribute before, and higher levels of the private good. But she did contribute before, and so she is also better o? with her previous choice. Subsequent work has shown that the neutrality result depends crucially on the individual contributions being perfect substitutes in the production of the public good, the linearity of the resource constraints, the absence of corner solutions, and the “pureness” of the public good (i.e., the bene?t received by a player must depend only on the total level of contributions, but not on her own contribution (see Cornes and Sandler, 1996, pp. 184-190 and p. 539; Bergstorm, Blume and Varian, 1986, Cornes and Sandler, 1994).8 In this paper we consider three points of departure from the distribution-neutrality framework.
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Some of the contributions to the theoretical literature related to this result are Warr (1983), Cornes and Sandler

(1984), Bergstrom, Blume and Varian (1986), Bernheim (1986) and Itaya, de Meza and Myles (1997). 8 Baland and Ray (1999) consider whether inequality in the shares of the bene?t players receive from a public good is good or bad for e?ciency might depend on whether the contributions of the players are substitutes or complements in the production function of the public good.

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First, we adopt the framework of a generalized collective good of which pure and impure public goods with positive externality (e.g., roads, canal irrigation, law and order, public R&D, public health and sanitation) are particular cases. We also analyze collective goods with negative externality (e.g., forestry, ?shery, grazing lands, surface or groundwater irrigation). Second, another point of departure from the standard literature on voluntary provision of public goods is that we look not only at the level of provision of the collective good in question, but also the total surplus from the good, net of costs. Third, the distribution-neutrality result assumes that the contributions towards the public good and the private input are fully convertible.9 In practice, particularly in the building of rural infrastructure in developing countries, the contribution towards the public good often takes the form of labor. To ?x ideas, let us think of the private input as capital. Then this assumption bypasses an important issue of economic inequality: labor is not freely convertible into capital. Typically, labor and capital are not perfect substitutes in the production technology, and because of credit market imperfections capital does not ?ow freely from the rich to the poor to equate marginal returns. We take this more plausible scenario as our starting point and examine the e?ect of distribution of wealth among members of a given community on allocative e?ciency in various types of collective action problems (involving public goods as well as common property resources or CPR) in the presence of missing and imperfect capital markets. This is particularly important in less developed countries where the life and livelihood of the vast masses of the poor crucially depend on the provision of above-mentioned public goods and the local CPR (particularly when it is not under commonly agreed-upon regulations10 ), and where markets for land and credit are often highly imperfect or non-existent. In poor countries where property rights are often ill-de?ned and badly enforced, even usual private goods have sometimes certain public good features attached to them, and due to ongoing demographic and market changes the traditional norms and regulations on the use of CPR are often getting eroded. In such contexts inequality of the players may play a special role. Our work is also motivated by the growing empirical literature on the relationship between inequality and collective good provision. For example, in an econometric study of 48 irrigation communities in south India Bardhan (2000) ?nds that the Gini coe?cient for inequality of landholding among the irrigators has in general a signi?cant negative e?ect on cooperation on water
9

The distribution neutrality literature is couched in the framework of a consumer choosing to allocate a given level

of income between her private consumption and contribution to a pure public good. We adopt the framework of a ?rm using a private input and a public input to produce some good. While not exactly equivalent, formally these frameworks are very similar and what we call the private input is similar to the private consumption good in the distribution neutrality literature.
10

Agreeing upon such regulations is itself a collective action problem.

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allocation and ?eld channel maintenance but there is some weak evidence for a U-shaped relationship. Similar results have been reported by Dayton-Johnson (2000) from his econometric analysis of 54 farmer-managed surface irrigation systems in central Mexico. In a di?erent context, using survey data on group membership and data on U. S. localities, Alesina and La Ferrara (2000) ?nd that, after controlling for many individual characteristics, participation in social activities is signi?cantly lower in more unequal localities. The plan of the paper is as follows. In the rest of this section we provide a brief preview of our framework and the main results. In sections 2-3 we provide a formal analysis. In section 4 we discuss the implications of relaxing some of our main assumptions. In section 5 we provide some concluding observations. The appendix contains all formal proofs. Since the private and collective inputs are complementary in our framework, the marginal return from contributing to the public good is increasing in the amount of the private input an agent has and which we are going to refer to as “wealth” in the rest of the paper. As a result, there will exist a threshold level of the amount of the private input such that only agents who have a level of wealth higher than this threshold will participate in providing the collective good while those with a lower level will free ride on the former group.11 This means that redistributions that increase the wealth of the richer players at the expense of non-contributing poorer players would achieve a greater amount of the public good, and other things being constant, this should increase joint surplus. In our framework this is how Olson’s original argument shows up. However, his argument focuses only on the total amount of the public good and not on joint surplus. In particular, the gain from increasing the size of the collective input has to be measured against the cost arising from worsening the allocation of the private input in the presence of decreasing returns. We show that the amount of contribution towards the provision (in the case of common property resources this has to be interpreted as extraction) of the collective input is a concave function of the endowment of the private input of the player for most well-known production functions (e.g., the Cobb-Douglas and the CES) and also that the equilibrium level of joint surplus (of both contributing and non-contributing players) is a concave function of the wealth distribution and hence displays inequality aversion. In addition, the total amount of the collective input is a concave function of the wealth distribution among contributing players. This means that initial asset inequality lowers the total provision of (pure and impure) public goods, and lowers the total extraction from the CPR. We provide a precise characterization of what the optimal distribution of wealth that maximizes joint surplus is in the case of imperfect convertibility between the private input and
11

Baland and Platteau (1997) provide some very interesting examples where richer agents tend to play a leading

role in collective action in a decentralized setting. For example, in rural Mexico the richer members of the populaton take the initiative in mobilizing labor to manage common lands and undertake conservation measures such as erosion control.

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contribution to the collective input. We show that the joint surplus maximizing wealth distribution under private provision of the public good involves equalizing the wealth levels within the group of all non-contributing players at some positive level and also within the group of all contributing players. The contrast with the conclusions of both Olson and the distribution neutrality literature is quite sharp. The key assumptions leading to our result are: market imperfections that prevent the e?cient allocation of the private input across production units, and some technical properties of the production function that are shared by widely used functional forms such as Cobb-Douglas and CES under decreasing returns to scale. With constant returns to scale, the joint surplus within the group of contributors is independent of the distribution of wealth as in the distribution neutrality theorem. The above result takes the number of contributors to the collective input as given. It is di?cult to characterize the optimal distribution of wealth when the number of contributors can be chosen. A key question of interest is: does perfect equality among all players maximize joint surplus? We provide a limited answer to this question. It turns out that perfect equality among all players (i.e., inter-group inequality in addition to intra-group inequality) is not always optimal. If wealth was equally distributed among all players, the average wealth of contributing players is low and this could reduce the level of the collective good. In contrast concentrating all wealth in the hand of one player will maximize the average wealth of contributors, but will involve signi?cant losses due to the assumed decreasing returns in the individual pro?t function with respect to wealth. The optimal distribution of wealth characterized above achieves a compromise between these two di?erent forces. Under constant returns to scale, where the latter factor is absent, we actually provide an example where an allocation with some degree of inequality can Pareto-dominate the allocation under perfect equality, i.e. it not only achieves higher joint surplus but also higher individual surplus for each agent.

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The Model

There are n > 1 players. Each player uses two inputs, ki and zi , to produce a ?nal good. The input ki is a purely private good, such as land, capital, or managerial inputs. We assume that there is no market for this input and so a player is restricted to choose ki ≤ wi where wi is the exogenously given endowment of this input of player i. While we will focus on this interpretation, there is an alternative one which views wi as capturing some characteristic of a player, such as a skill or a taste parameter.12 In contrast, zi is a collective good in the sense that it involves some
12

The assumption that the market for the private input does not exist at all, while stark, is not crucial for our

results. All that is needed is that the amount a person can borrow or the amount of land she can lease in depends positively on how wealthy she is. Various models of market imperfections, such as adverse selection, moral hazard,

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externalities, positive or negative. We assume that each player chooses some action xi which can be thought of as her e?ort that goes into using a common property resource or contributing towards the collective good. Let X ≡
n i=1 xi

be the sum total of the actions chosen by the players. The

individual actions aggregate into the collective input in the following simple way zi = bxi + cX. The pro?t (surplus) function of player i is13 : πi = f (wi , zi ) ? xi Note that the input xi appears twice in the pro?t function, once on its own as a private input, and once in combination with the quantities used or supplied by other agents. This implies that the private return to a player always exceeds the social return as long as b > 0. The input X can be a good (e.g., R&D, education) or a bad (e.g., any case of congestion or pollution). This formulation allows each player to receive a di?erent amount of bene?t from the collective input which depends on the action level they choose. In contrast, for pure public goods every player receives the same bene?ts irrespective of their level of contribution. This case, as well as many others (involving both positive and negative externalities) appear as special cases of our formulation as we will see shortly. Following the distribution neutrality literature we assume that the cost of supplying xi units of the collective input, is simply xi and that the production function, f exhibits non-increasing returns with respect to the private and the collective inputs xi and zi . Let w = (w1 , w2 , .., wn ) denote the vector of wealth levels of the players. Assume that the wi denote the total amount of wealth of the n players. the production function: Assumption 1: f (w, z ) is a strictly increasing, strictly concave function that is twice continuously di?erentiable with respect to both arguments, f12 ≥ 0 for all (w, z ) , lim f2 (w, z ) = 0 and it satis?es the Inada endpoint conditions. In addition f (w, z ) = ?D for z < 0, where D is a very large number.
w →0

are arranged in descending order of magnitude, i.e., w1 > w2 > .. > wn > 0 and let W ≡
14

n i=1 wi

We make the following assumption about

We also make an assumption about the parameters b and c: Assumption 2 b ≥ 0 and b + cn > 0 (A2)

Notice that we allow c to be positive, negative or zero. Only when c < 0 does the second inequality in (A2) become relevant ensuring that |c| is not too large. This implies that if a social
costly state veri?cation or imperfect enforcement of contracts will lead to this property. P 13 We will also refer to Π = n i=1 πi as joint surplus or joint pro?ts later in the paper.
14

Following the literature on distribution-neutrality, we are assuming linear costs.

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planner chooses the level of the collective input, she would choose a positive level of xi for at least one player. In addition the same inequality also turns out to be a condition on the reaction functions of the players which ensures the stability of the equilibrium. When c = 0 we have the case of a pure private good - there are no externalities. For b = 0 and c > 0 we have the case of pure public goods, i.e. the one on which most of the existing literature has focused. For b > 0 and c > 0 we have the case of impure public goods as de?ned by Cornes and Sandler (1996). For b > 0 and c < 0 we have a version of the commons problem: by increasing her action relative to those of the others an individual gains. We begin our analysis by the following result which shows how the choice of xi by player i depends on how much wealth she has: Lemma 1: γ (w) ≡ arg max{f (w, z ) ? z } is strictly positive for w > 0 and is an
z ≥0

increasing function.

This property follows directly from the complementarity between wi and zi and the diminishing returns with respect to z. An increase in wi raises the marginal return of zi relative to its marginal cost which is assumed to be constant and equal to 1. To restore equilibrium at the individual level, the amount of the collective input must increase. This property is not satis?ed if w and zi are substitutes in the production function. We will discuss this case in section 4. Several widely used in economics functional forms that display complementarity among the inputs satisfy the conditions stipulated in this lemma. These include the following: γ (w) = β 1?β w 1?β which is clearly an increasing function; (a) the Cobb-Douglas production function, f (w, z ) = wα z β with α + β ≤ 1. In this case
1 α

parameter restrictions that ensure non-increasing returns to scale (k ≤ 1), and complementarity between w and z (ρ < k ). We work out the details of this case in the appendix. For k = 1 (constant
δ
1 ρ 1

(b) the generalized CES production function, namely, f (w, z ) = (δwρ + (1 ? δ )z ρ ) ρ under

k

returns to scale) it is possible to solve for γ (w) explicitly, which turns out to be δ ) 1?ρ w.
1

{1?(1?δ ) 1?ρ }

(1?

An important question from the economic point of view is that while clearly both a rich and a poor individual would increase their contribution to the collective input if their wealth increases by the same amount, who would want to expand their contribution to the collective input more? The answer to this question is crucial for determining the e?ect of the distribution of wealth on the total amount of the collective good, X and joint pro?ts. This e?ect depends on the curvature of γ, which in turn depends on the third-order properties of the production function. Intuitively, the question is whether diminishing returns with respect to z would kick in at a faster or a slower rate at a higher wealth level. This would determine whether for a richer person a relatively small 9

or large increase in z would restore her individual optimum compared to a poorer person. It turns out that all widely used functional forms in economics where the inputs are complements to each other have the property that diminishing returns kick in at the same or faster rate the higher is the wealth level. This implies that γ is linear or strictly concave. We make the following technical assumption about the production function which we show is equivalent to γ having this property. Assumption 3 h(w, z ) ≡ ?f (w, z ) is quasi-concave. ?z

The following lemma proves that h being (weakly) quasi-concave is equivalent to γ being (weakly) concave: Lemma 2: Suppose Assumption 1 holds. γ (w) is concave if and only if h(w, z ) ≡
?f (w,z ) ?z

is quasi-concave.

For the Cobb-Douglas and CES production functions, γ (w) is strictly concave under decreasing returns to scale (α + β < 1 for Cobb-Douglas and k < 1 for CES) and linear under constant returns to scale (α + β = 1 or k = 1). The following lemma provides additional characterization of the class of production functions for which γ is linear: Lemma 3: Suppose Assumption 1 holds. If f (w, z ) is homogeneous of degree 1 then γ (w) = Aw where A is a positive constant. This result follows from the fact that if the production function is homogeneous of degree 1 then if both the wealth of a player and the amount of the collective input received by her are increased proportionally, the marginal return from contributing remains unchanged. The Cobbk = 1 are examples of production functions that are homogeneous of degree 1 and we have already veri?ed that γ (w) is linear in these cases. Douglas production wα z β with α + β = 1 and the CES production function (δwρ + (1 ? δ )z ρ ) ρ with
k

3

The Decentralized Equilibrium

We characterize the decentralized equilibrium in the following two steps. First, for a given distribthe total contribution X, and the joint surplus, Π. Second, we look for the distributions of wi , which these two variables. ution of the private input w = {wi }i=1..n , we solve for the optimal contributions of each agent, x ?i ,

maximize the total contribution and joint pro?ts to be able to analyze the e?ects of inequality on

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3.1

Characterization of the Nash Equilibrium for a Given Distribution of the Private Input

Let us consider the decentralized Nash equilibrium allocation. Player i takes the contribution of other players, X?i , as given and solves: max π i = f (wi , bxi + cX ) ? xi .
xi ≥0

The ?rst-order conditions are f2 (wi , bxi + cX )(b + c) ? 1 ≤ 0 xi ≥ 0, together with the standard Kuhn-Tucker complementary slackness condition. Let the function g (wi ) > 0 denote the solution to15 f2 (wi , g (wi ))(b + c) = 1. Notice that under Assumptions 1-3 g (w) is increasing and concave in w. wi and zi in the production function, for a given value of zi , f2 (wi , zi ) is increasing in wi . Therefore, g (wi ) is increasing. Also, by Assumption 1,
?zi ?xi

Suppose m ≤ n players contribute in equilibrium. By the assumed complementarity between = b + c > 0. Therefore irrespective of the value of

c (in particular, even if it is negative) for a given level of contribution of other players, X?i , the richest player has the greatest marginal pro?t from contributing, followed by the second richest and so on. For the case of pure public goods, i.e., b = 0, the amount of the collective input enjoyed by each player is the same, whether the player contributes or not. Therefore, only the richest player will contribute, i.e., m = 1. So long as b > 0, the amount of the collective input enjoyed by each player is di?erent, i.e., bxi + cX = bxj + cX . Therefore, even if the second richest player’s marginal return from contributing is less than that of the richest player, she can contribute less and enjoy a lower level of the collective input and thereby attain an interior equilibrium. As a result the set of contributing players will consist of the richest m ones, where n ≥ m ≥ 1. Let us denote by x ?i the optimal contribution of player i and let X =
m ?i . i=1 x

We assume that

individual contributions are as follows: ? ? g (wi ) ? cX , i = 1, .., m b x ?i = ? 0, i = m + 1, .., n.
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the wealth of the richest player exceeds some threshold level so that x1 > 0. Given z1 = g (w1 ) > 0 g (w1 ) and Assumption 2, we will have x1 = > 0. b+c The above inequality implies that m ≥ 1. Then, the equilibrium conditions for the optimal

(1)

This function would be identical to γ (.) de?ned in the previous section if b + c = 1 and it obviously inherits the

properties of γ discussed above.

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. (2) b + mc c m i=1 g (wi ) g (wm ) ≥ > g (wm+1 ). (3) b + mc Condition (1) states that, for all contributing agents, the ?rst order condition must hold as X (m) = equality. Condition (2) is equivalent to X (m) =
m ?i . i=1 x

m i=1 g (wi )

It states that the total contribution must

have a value that is consistent with the individual maximization problems of all m contributors, and is obtained by adding up the m ?rst-order conditions from (1). The third condition is the most interesting one, as it determines the size of the contributing group, m. To see why it should hold, note that the m + 1-th agent would not contribute if f2 (wm+1 , cX (m))(b + c) < 1, since any further contribution has a marginal bene?t that is lower than marginal cost. This condition is equivalent to cX (m) > g (wm+1 ), which is exactly what the second inequality in (3) states. By the same logic, the m-th agent would not be contributing if g (wm ) < cX (m ? 1) = rearranged as g (wm ) <
c
Pm
i=1 g (wi ) . b+mc

c

Pm?1

must be satis?ed. The number of contributing agents, m is the smallest integer for which (3) is satis?ed. If the left inequality in (3) holds for m = 1, 2..n, then all agents contribute. The following lemma, together with the fact that g is increasing, ensures the existence of a unique value of m that solves (3): Lemma 4: If k + 1 ≤ m, then the function s(k ) = c
k i=1 g (wi )

Thus, if she is to contribute, the condition g (wm ) ≥

i=1 g (wi ) b+(m?1)c ,

which can be
c
Pm
i=1 g (wi ) b+mc

k > m the function s(k ) is decreasing in k.

b + kc

is increasing in k . If

Several useful observations follow directly from (1)-(3): Observation 1. For the case of a pure public good (b = 0, c ≥ 0) (3) cannot hold for m > 1 g (wm ) ≥
m i=1 g (wi )

as that would imply

m which is impossible given that w1 > w2 > .. > wn by assumption and Lemma 1 showing that g (.) is increasing. Thus for pure public goods only the richest player contributes. This has the implication that even when the di?erence in the wealth between the richest player and second richest player is arbitrarily small, the former provides the entire amount of the public good. Observation 2. For the case of pure private goods (c = 0), there is no interdependence across players and all of them will contribute. Observation 3. For those player who contribute in equilibrium, the condition (1) can be rewritten in the form of a reaction function: x ?i = 1 {g (wi ) ? cX?i }, i = 1, .., m b+c 12

equilibrium is b + cm > 0 which is ensured by Assumption 2. The contributions of players are strategic complements for c < 0 and strategic substitutes for c > 0. Formally, this follows from the fact that of various players are perfect substitutes in the payo? function, and in the case of public goods (commons) an increase in the contribution of others is similar to an increase (decrease) in the player’s own contribution, which reduces (increases) the marginal return of her contribution due to diminishing returns in the collective input (i.e., z ).
? 2 πi ?xi ?X?i

where X?i ≡

m ?j . We j =1,j =i x

show in the appendix that the condition for stability of the Nash

= cf22 (wi , bxi + cX )(b + c). Intuitively, the reason is that the contributions

3.2

E?ect of Wealth Inequality on Total Contributions and Joint Pro?ts
m i=1 g (wi )

From (2), X=

b + mc

≡g ?(w).

?(w) is the sum of m concave functions and as such is a concave Under Assumptions 1-3 X = g function itself. Moreover, as these functions are identical and receive the same weight, if we hold the number of contributors constant the total contribution is maximized when all contributing agents have equal amounts of the private input. Therefore we have: Proposition 1: Suppose Assumptions 1-3 are satis?ed. If g (wi ) is strictly concave in wi then X is strictly concave in w and is maximized when all contributing agents have equal amounts of the private input. If g (wi ) is linear in wi then X is linear in w. Recall that Assumption 3 implies that diminishing returns with respect to the collective input used by the i-th individual set in at a faster rate at a higher wealth level, and so the optimal level of the collective input is a concave function of the wealth level (Lemma 2). Proposition 1 follows from this assumption, and the fact that the collective input used by the i-th individual is a linear function of the individual’s own contribution and the contribution of other players. To see this more clearly consider the two player version of the game where player 1 has wealth w + ε and player 2 has wealth w ? ε where ε > 0. From the ?rst order condition of the two players: g (w + ε) = (b + c)x1 + cx2 g (w ? ε) = cx1 + (b + c)x2 . Therefore, the reaction functions are: x1 = x2 = 1 {g (w + ε) ? cx2 }. b+c 1 {g (w ? ε) ? cx1 }. b+c 13

Consider the e?ect of an increase in ε. The direct e?ect is to increase x1 and reduce x2 . For the case of positive externalities (c > 0) the indirect e?ects move in the same direction, while for the case of negative externalities, the indirect e?ects move in the opposite direction. For example, in the former case, a reduction in x2 stimulates a further increase in x1 and an increase in x1 leads to a further decrease in x2 . The stability condition ensures that indirect e?ects in the successive rounds shrink in terms of size. Linearity of the reaction functions implies that the total e?ects of a change in ε on x1 and x2 are linear combinations of the direct e?ects on the two players. The fact that the reaction functions of both players have the same slope in our set up implies additionally that the direct e?ects of redistribution on the contribution of the two players receive the same weight in
dX dε

and so it follows directly from the concavity of g (.) that X is decreasing in ε.

The e?ect of wealth inequality on X has implications which are quite di?erent from those available so far in the public economics literature. Our analysis shows that greater equality among those who contribute towards the collective good will increase the value of X. Therefore a more equal wealth distribution among contributors will increase the equilibrium level of the collective input. In addition, any redistribution of wealth from non-contributors to contributors that does not a?ect the set of contributors will also increase X.16 In terms of the two-player example, this implies that as long as both players contribute, any inequality in the distribution of wealth reduces X . But with su?cient inequality if one player stops contributing then any further increases in inequality will increase X. Let us now turn to the normative implications of changes in the distribution of wealth. Under the ?rst-best, which can be thought of a centralized equilibrium where players choose their contributions to maximize joint surplus, the ?rst-order condition for player i is: f2 (wi , bxi + cX )(b + nc) ≤ 1. The di?erence with the decentralized equilibrium is that now individuals look at the social marginal product of their contribution to the collective input, i.e., f2 (wi , bxi + cX )(b + nc) as opposed to the private marginal product, i.e., f2 (wi , bxi + cX )(b + c). Then it follows directly that those who will contribute will contribute more (less) than in the decentralized equilibrium if c > 0 (c < 0). Also, the number of contributors will be higher (lower) than in the decentralized equilibrium if c > 0 (c < 0). Therefore, for the case of positive externalities, the total contribution in a decentralized equilibrium is less than the e?cient (i.e., joint surplus maximizing) level. Conversely, for the case c < 0, total contributions exceed the socially e?cient level. From this one might want to conclude that greater inequality among contributors increases e?ciency in the presence of negative externalities
16

In the above formula for X , holding m constant a redistribution from non-contributors to contributors will

increase wi (i = 1, 2, .., m) with the increase being strict for some i.

14

and reduces e?ciency if there are positive externalities.17 Indeed, the literature on the e?ect of wealth (or income) distribution on collective action problems have typically focussed on the size of total contributions. However, that is inappropriate as the correct welfare measure is joint pro?ts. In the presence of decreasing returns to scale the distribution of the private input across ?rms will have a direct e?ect on joint pro?ts irrespective of its e?ect on the size of the collective input. In particular, greater inequality will reduce e?ciency by increasing the discrepancy between the marginal returns to the private input across di?erent production units. In the case of negative externalities, these two e?ects of changes in the distribution of the private input work in di?erent directions, while in the case of positive externalities, they work in the same direction. Now we proceed to formally analyze this issue. Using the conditions (1)-(3) agent i’s surplus is: πi (wi , xi , X ) = f (wi , g (wi )) ? g (wi ) ? cX , i = 1..m (contributors) b

πi (wi , xi , X ) = f (wi , cX ), i = m + 1..n (non-contributors) Joint surplus is given by:
n m

Π=
i=m+1

f (wi , cX ) +
i=1

f (wi , g (wi )) ?

m i=1 g (wi )

b + mc

.
Pm

that of non-contributors by

Let us denote the joint surplus of contributing players by Πc ≡ Πn ≡
n i=m+1

m i=1 f (wi , g (wi ))

f (wi , cX ).

?

i=1

g (wi ) b+mc

and

First consider the e?ect of distribution of wealth among non-contributors. This is trivial, since f (wi , cX ) is concave, and hence
n i=m+1

f (wi , cX ) is concave as well. Therefore perfect equality

of wealth among non-contributors will maximize their joint pro?ts. Note that even if f (w, z ) is homogeneous of degree 1, this is still true. Next, let us consider the e?ect of distribution of wealth among contributors. Let ? (w) ≡ f (w, g (w)) ? π Notice that Πc = on Πc : Lemma 5: Suppose Assumptions 1-3 hold and c ≥ 0, or c < 0 but |c| small. If g (w) is ? (w) is linear in w. one then π
m ? (wi ). i=1 π

g (w) . b + mc

The following lemma helps characterize the e?ect of wealth inequality

(w) concave then π ? (w) ≡ f (w, g (w)) ? bg +mc is concave. If f (w, z ) is homogeneous of degree
17

Note however that a su?ciently large degree of inequality among contributors may reduce X below the ?rst-best

level in the c < 0 case.

15

The intuition behind this result is the following. In the absence of externalities (i.e., c = 0) if we want to ?nd the e?ect of a change in w on the pro?t of a player, we can focus only on the direct e?ect and ignore the indirect e?ect via the envelope theorem. As a result, the second derivative of the pro?t function also depends only on the direct e?ect through w. In the presence of externalities, we must take into account the indirect e?ect of w on X that a?ects other players. This residual term, which is a fraction of X (namely,
1 b+c

and reduces them for c < 0 compared to the case where c = 0. Since we already know that X is concave, in the former case this reinforces the concavity of the joint pro?t function but in the latter case the e?ect goes the other way. As a result, for c < 0 a su?cient condition to ensure concavity of π ? (w) is |c| to be small. Lemma 5 implies immediately that for c ≥ 0 and for c < 0 but |c| small, Πc =
m ? (wi ) i=1 π

?

1 b+mc

=

(m?1)c (b+c)(b+mc) )

increases joint pro?ts for c > 0

is

concave in the wealth of contributors so that greater equality will increase joint pro?ts. As a result, perfect equality of wealth among contributors maximizes their joint surplus. For the special case where f (w, z ) is homogeneous of degree one Πc is linear in the wealth of the contributors. In this case a redistribution of wealth among contributors will not a?ect joint surplus. However, equalizing satisfy Assumption 1) we cannot determine the curvature of Πc in general.
α

wealth among non-contributors will still maximize Πn . For c < 0 but |c| large (while continuing to It turns out that for the Cobb-Douglas case, with decreasing returns, π ? (w) is strictly concave

In order for total contribution to be positive, we need π ? (w1 ) > 0. This inequality holds if and only - otherwise no player ever contributes. Hence π ? (w) is concave in w for α + β < 1. In the case of ? (w) is linear. constant returns, i.e., α + β = 1, π w ≥ wm contribute and those with w ≤ wm+1 do not. For this value of m, it is clear by the concavity of Π, that the wealth distribution maximizing joint surplus should have wi = w for all i = 1, .., m and wj = w ? < w for all j = m + 1,..,n. subject to the following two conditions: (n ? m)w ? + mw = W cmg (w) > g (w ? ). b + mc The ?rst of the above conditions can be rewritten as: g (w) ≥ w ?= W ? mw n?m (4) Given the initial wealth distribution w, there is some m ≥ 1 such that players with wealth if b(1 ? β ) > (β ? m)c, which holds for c ≥ 0 or c < 0 and |c| <
b(1?β ) m?β .

? (w) = w 1?β [(b+c)β ] 1?β even if c < 0 and |c| not necessarily very small. In this case, π

1

1 β (b+c)

?

1 b+mc

.

We assume this to be true

Using this, the expression for joint pro?ts becomes: Π = (n ? m) f ( W ? mw cmg (w) g (w ) , ) + m f (w, g (w)) ? . n ? m b + mc b + mc 16

(w b) Also, the total contribution is X = m bg +mc . The following result characterizes the joint surplus

maximizing wealth distribution for a given m. Proposition 2: Suppose Assumptions 1-3 are satis?ed, c ≥ 0 and if c < 0, |c| is small. the public good involves equalizing the wealths of all non-contributing players to w ?>0 and also those of all contributing players to w > w. ? This result shows that maximum joint surplus is achieved for both contributors and noncontributors, if there is no intra-group inequality. This is a direct consequence of joint pro?t of each group being concave in the wealth levels of the group members. The contrast with the conclusions of both Olson and the distribution neutrality literature is quite sharp. The key assumptions leading to the result are, market imperfections that prevent the e?cient allocation of the private input across production units, and some technical properties of the production function that are shared by widely used functional forms such as Cobb-Douglas and CES under decreasing returns to scale. With constant returns to scale, the joint pro?ts within the group of contributors are independent of the distribution of wealth as in the distribution neutrality theorem. In the above result we did not talk about inter-group inequality. Formally, we took m as given while considering alternative wealth distributions. An obvious question to ask is, what is the jointpro?t maximizing distribution of wealth when we can also choose the number of contributors, m. For example, does perfect equality among all players maximize joint surplus? This turns out to be a di?cult question. Below we provide a partial answer to this question for the case of both positive and negative externalities. Let us ?rst look at the case of positive externalities (c > 0). Suppose all players are contributing when wealth is equally distributed. Then from Proposition 2 we know that limited redistribution that does not change the number of contributors cannot improve e?ciency. This immediately suggests the following result: Corollary to Proposition 2: Suppose all players contribute under perfect equality. Then if after a redistribution all players continue to contribute joint pro?ts cannot increase. But suppose we redistribute wealth from one player to the other n ? 1 players up to the point

For a given m the joint pro?t maximizing wealth distribution under private provision of

b) (w where this player stops contributing. Recall that when the group size is m < n, X = m bg +mc . It

is obvious that an increase in the average wealth of contributing players keeping the number of contributors ?xed will increase X . It turns out that an increase in m holding the average wealth of contributors constant will always increase X.18 However, if we simultaneously decrease m from n to
18

Formally, this is because

m b+mc

is increasing in m. The intuition is, the new entrant to the group of contributor

will contribute a positive amount, which would reduce the incentive of existing contributors to contribute due to

17

X goes down then we can unambiguously say that joint pro?ts are lower due to this redistribution (for c > 0) since the e?ect of this policy on the e?ciency of allocation of the private input across production units is de?nitely negative. However, if X goes up then there is a trade o?: the increase in X bene?ts all players (since c > 0), including the player who is too poor to contribute now, but this has to be balanced against the greater ine?ciency in the allocation of the private input. To analyze the e?ect of wealth distribution on joint pro?ts when some players do not contribute we restrict attention to the comparison between joint pro?ts under perfect equality (i.e., when all players have wealth w ≡
W n )

n ? 1 and increase the average wealth of contributors, it is not clear whether X will go up or not. If

and the wealth distribution that is obtained by a redistribution that

under our assumptions all players contribute under perfect equality. We focus on studying only the

leads to k contributing and n ? k non-contributing players. From the discussion above, we know that

e?cient wealth distributions, i.e. ones which achieve maximum joint surplus. Since any intra-group inequality among the contributors and non-contributors reduces joint surplus we assume that all k contributors have wealth w +
ε k

redistribution. Let us denote by ΠE the joint surplus under perfect equality and with ΠI (ε) the one under the unequal wealth distribution of the above type. Let also total wealth be normalized to nw. A player stops contributing if g (w ? solution to
ε kcg (w + k ) ε ) < cX = n?k b + kc

and all n ? k non-contributors have wealth w ?

ε n?k

after the

(5)

Let ε? be de?ned as the indi?erence point between contributing and not contributing, i.e. the ε? ) k g (w )= (6) ?? n?k b + kc Let ε ? denote the degree of inequality maximizing ΠI (ε) subject to ε ≥ ε? , i.e. when there are ε? kcg (w ?+ with wealth w +
ε k

non-contributors in equilibrium19 . The level of joint surplus when there are k contributors each and (n ? k ) non-contributors each with wealth w ? w?
ε ) kg (w + k kc ε ε , g (w + ) ? . n ? k b + kc k b + kc ε n? k

is:

ε ε ΠI (ε) = kf w + , g (w + ) + (n ? k )f k k Di?erentiating with respect to ε we get:

dΠI (ε) kc ε ε ε ε = f1 w + , g (w + ) ? f1 w ? , g (w + ) dε k k n ? k b + kc k (n ? k )c kc ε ε ε +f2 w ? , g (w + ) g (w + ) + n ? k b + kc k b + kc k ε 1 1 +g (w ?+ ) ? k b + c b + kc
have been an equilbrium.
19

(7)

diminishing returns. However, in the new equilibrium X must go up, as otherwise the original situation could not Clearly, if ε < ε? (5) cannot hold (since g is an increasing function and all players contribute).

18

where we used the fact that f2 (w ?+

ε 1 ε ? + )) = by the de?nition of g . The following , g (w k k b+c lemma helps us characterize the optimal degree of inequality: ? ΠI (ε? ) ? ΠI (ε) < 0 implies that < 0 for all ε ≥ ε? and so ε ? = ε? . ?ε ?ε ? ΠI (ε? ) > 0 then ε ? > ε? . Conversely, if ?ε Lemma 6:

? ΠI (ε? ) < 0 then we have a corner solution, i.e. ΠI (ε) is ?ε maximized at ε ? = ε? . Now we are ready to prove: The above lemma implies that if Proposition 3 (a) For pure public goods (b = 0 and c > 0) perfect equality among the agents is never joint pro?t maximizing. (b) For pure private goods (b > 0 and c = 0) perfect equality is always joint pro?t maximizing. We noted a special property of pure public goods in the previous section ( Observation 1), namely, even if the di?erence in the wealth between the richest player and the second richest player is arbitrarily small, the former provides the entire amount of the public good with everyone else free riding on her. This property is the key to explain why perfect equality is not joint pro?t maximizing in this case. Start with a situation where all players except for one have the same wealth level, and this one player has a wealth level which is higher than that of others by an arbitrarily small amount. As a result this player is the single contributor to the public good. A small redistribution of wealth from other players to this player, keeping the average wealth of the other players constant, will have three e?ects on joint pro?ts: the e?ect due to the worsening of the allocation of the private input, the e?ect of the increase in X on the payo? of the non-contributing players, and the e?ect of the increase in X on the payo? of the single contributing player. The result in the proposition follows from the fact that the ?rst e?ect is negligible since by assumption the extent of wealth inequality is very small, the second e?ect is positive, and the third e?ect can be ignored by the envelope theorem. It should also be noted that this result goes through for both constant and decreasing returns to scale. The second part of Proposition 3 follows from the fact that when c = 0 a player will always choose xi > 0 however small her wealth level. Then all players are contributors so long as they have non-zero wealth and it follows directly from Proposition 2 that perfect equality will maximize joint pro?ts. For the case of impure public goods (b, c > 0 under decreasing returns to scale we can provide only a local characterization:

19

Proposition 4 Consider the case of impure public good subject to decreasing returns to scale, i.e. c > 0 and suppose that Assumptions 1-3 hold. Then: b2 , ∞) perfect equality (a) Given c, n there exist some b1 , b2 > 0 such that for all b ∈ [? is always joint pro?t maximizing, whereas for b ∈ [0, ? b1 ] perfect equality is never joint pro?t maximizing. (b) Given b, n there exists some c ? > 0 such that for all c ∈ (0, c ?] perfect equality is always

joint pro?t maximizing.

Two opposing forces are at work in this case - the “decreasing returns to scale” e?ect calling for equalizing the wealth of agents and the “dominant player” e?ect due to the positive externality calling for re-distribution towards the richest players as there is a positive e?ect on the payo?s of the non-contributing players. Each of the two e?ects can dominate the other depending on the parameter values. The direct e?ect of an increase in the richer player contribution on her own payo? can once again be ignored by the envelope theorem. While we cannot provide a full characterization of the case of decreasing returns, due to the existence of two opposing forces, we can provide some illustrative examples using the Cobb-Douglas production function f (w, z ) = wα z β for a two player game. In Figures 1 and 2 we plot how the di?erence between joint pro?ts under perfect equality and under inequality (where the degree of inequality is chosen to maximize joint pro?ts given than only one player contributes) vary with b and c for several alternative sets of values of α and β. As we can see from the ?gures: (a) there is a unique ? b such that ΠE ≥ ΠI (? ε) for b ≥ ? ε) for b < ? b; and (b) there is a unique c b and ΠE < ΠI (? ? ε) for c ≤ c such that ΠE ≥ ΠI (? ? and ΠE < ΠI (? ε) for c > c ?. Under constant returns to scale, we know from Lemma 5 that joint pro?ts are linear in the total

wealth of contributors. Given that joint pro?ts are higher under some degree of inequality for pure public goods compared with joint pro?ts under perfect equality, in this particular case one would expect this property to be true for c > 0 and b > 0. This conjecture turns out to be correct: Proposition 5: If the production function displays constant returns to scale then: (a) In the pure private good case (c = 0) joint pro?ts are independent of the wealth distribution. (b) Perfect equality is never joint pro?t maximizing for impure public goods (i.e., c > 0). Moreover, it is possible to have inequality Pareto-dominate perfect equality. The ?rst part of the proposition follows from the fact that under constant returns to scale joint pro?ts are linear in total wealth (see Lemma 5). The logic of the result in (b) is similar to that of the 20

pure public goods case. In that case the richest person is the only contributor even when the wealth di?erence between him and the second richest player is very small and therefore a small amount of inequality does not result in large losses due to the ine?cient allocation of the private input. With constant returns to scale and for impure public goods, the di?erence between the wealth level of the contributors and non-contributors need not be very small. However, joint pro?ts of contributors depend only on their total wealth and not how it is distributed. As a result, creating some inequality from the point where only player is exactly indi?erent between contributing and not, to the point where she strictly prefers not to contribute, involves a small loss due the ine?cient allocation of the private input. Unlike the pure public goods case, this could involve a signi?cant amount of inequality with respect to the perfectly equal wealth distribution. The second part of Proposition 5 (b) demonstrates the striking possibility that under some circumstances it is possible to have some degree of inequality among agents Pareto-dominate the allocation under perfect equality. If we think of a two player set up, starting with perfect equality if we redistribute wealth from one player to the other, the poorer player is initially strictly better o? than the rich player because she is free-riding on the rich player who contributes most of the good and bears a large share of the costs. This is the starkest possible demonstration of what Olson called the “exploitation of the great by the small”. However, if we continue increasing inequality eventually the loss of the private input o?sets the gain from free riding on the provision of the public good for the poorer player. This makes it possible that the two players get the same level of surplus at some positive level of inequality and that this surplus is higher than the level they get at perfect equality. Finally, we turn to the case of negative externalities, i.e., c < 0. We show that: Proposition 6 Consider the case of commons, i.e. c < 0 and suppose that Assumptions 1-3 hold. Then there exist two critical values of c, c1 < c0 < 0 such that: (a) For c ∈ [c0 , 0) perfect equality is a local maximum of the joint pro?t function. b (b) For c ∈ (? , c1 ) perfect equality is never joint pro?t maximizing. n In this case, Assumption 1 implies that all agents contribute. Notice that then we can write the joint pro?ts function as: ? (w1 ) + (n ? k )? π (w2 ) + Π = kπ where π ? (w) ≡ f (w, g (w)) ? (n ? 1)c X (b + c)

g (w ) , w1 is the wealth of rich players and w2 is the wealth level of poor b+c 1 g (w) = f1 (w, g (w)) from the ? (w) = f1 (w, g (w))+ f2 (w, g (w)) ? players. Notice also that π b+c de?nition of g (w). Therefore given the de?nition of g (w) and the concavity of f, π ? is strictly concave. 21

(n ? 1)c X is convex given that c < 0 and Proposition 1. Intuitively, (b + c) joint pro?ts is the sum of individual pro?ts ignoring the externality of a player’s action on others, On the other hand, the term and the sum total of the externality terms. The former is concave in the wealth distribution but in the case of negative externalities, the latter is convex. For c small enough (in absolute value) the decreasing returns to scale e?ect dominates, i.e. joint pro?ts are maximized at perfect equality but for c large (in absolute value) the “cost of negative externality” term, which is convex, dominates and so greater inequality leads to higher joint pro?ts.

4
4.1

Extensions
Convertibility Between the Private Input and the Contribution to the Collective input

It is important for our result that xi and wi are di?erent types of goods and one cannot be freely converted into the other. Suppose the individual can freely allocate a ?xed amount of wealth between two uses, namely, as a private input and as her contribution to the collective input. This is the formulation chosen by the literature on distribution-neutrality (e.g., Warr (1983), Bergstrom, Varian and Blume (1986), Cornes and Sandler (1996) and Itaya et al (1997)). This literature focuses on pure public goods, i.e., where zi = cX . For ease of comparability, let us consider this case ?rst. Let ki denote the amount of the private input chosen by player i. Then player i’s decision problems is to maximize f (ki , cX ) with respect to ki and xi subject to the budget constraint ki + xi ≤ wi . The ?rst-order condition of an individual who contributes a positive amount in equilibrium is f1 (ki , cX ) = cf2 (ki , cX ), i = 1, 2, .., m.

this condition implicitly de?nes the following function:

As ki = wi ? xi from the budget constraint of the individual, and xi + X?i = X for all i = 1, 2, .., m,

wi ? xi = h(X ). Summing across all players who contribute in equilibrium, we get X + mh(X ) = W. This equation can be solved for X which therefore depends only on total wealth, W and not on its distribution. Joint pro?ts will also be independent of the distribution of wealth. The above formulation is similar to that of a consumer allocating a ?xed amount of money to alternative goods in order to maximize utility. An alternative formulation to capture free convertibility between ki and xi is to pose the problem as that of a ?rm maximizing pro?ts by choosing inputs which can be sold or purchased from the market at a given price. One could think of ki as capital which has a given price r such that a ?rm that has an excess of it (relative to its endowment 22

wi ) can sell it to other ?rms, and a ?rm that has a shortage of it can buy it at the same price, say r. Similarly, one can think of xi as labor that can be used to contribute towards the collective input, or sold in the labor market at price w.20 Now the ?rst order condition of a contributing player, i, is f1 (ki , cX ) = r cf2 (ki , cX ), w i = 1, 2, .., m.
r w

This condition is the same as in the previous formulation, except for the multiplicative constant and so the distribution neutrality result goes through.

Turning now to impure public goods, i.e., where b > 0, the ?rst order condition for player i according to the ?rst formulation is: f1 (ki , bxi + cX ) = (b + c)f2 (ki , bxi + cX ), i = 1, 2, .., m.

It is clear that in general the distribution neutrality result will not go through now. It will go through for some special cases, such as the case where f (w, z ) is homothetic. In this case, the values of ki and zi at a point of individual optimum satis?es the condition ki =A bxi + cX where A is a positive constant. It is readily veri?ed that the distribution neutrality result holds in this case. Our analysis shows that in this case, relaxing the assumption of perfect convertibility of the private input and the contribution to the collective input implies that the distribution neutrality result no longer holds. Speci?cally, greater equality among contributors always improves e?ciency for impure public goods (i.e., c > 0) while for collective inputs subject to negative externalities, the e?ect of inequality on e?ciency is ambiguous. In the latter case, we characterize conditions under which we can sign the e?ect of inequality on e?ciency. Our results do not depend on the production functions being homothetic, but in the general case even with free convertibility, distribution neutrality can break down if the collective input is not a pure public good, as is well recognized in the literature (see for example, Bergstrom, Varian and Blume (1986) and Cornes and Sandler (1996)).

4.2

Substitutability Between the Private and the Collective Input

Above, we assumed that the private input and the public good are complements in the production function. In this section we examine the implications of these two inputs being substitutes. For simplicity, we examine the case where w and z are perfect substitutes: π (wi , xi , X ) = f (w + bxi +
20

In our framework labor is not directly used in production. We can think of another sector which uses labor.

Alternatively, we can extend the basic model by adding labor as a third input. The distribution neutrality result will go through.

23

cX ) ? xi , where f is increasing and strictly concave and b and c satisfy Assumption 2. The ?rst order conditions for the agent’s problem is: (b + c)f (wi + bxi + cX ) ≤ 1 with strict equality when xi > 0. Let us denote by w? the solution to f (w) =
1 b+c ,

which exists and

is unique given the above assumptions. In contrast to the complements case, it is now the poorest player who has the highest marginal product of contributing. In the pure public good case (b = 0) be provided at all. the poorest player will be the only contributor if wn < w? and if wn ≥ w? the public good will not As before, joint surplus goes up if wealth is equally distributed among non-contributors. Also, we cannot say for sure whether the optimal distribution of wealth involves perfect equality, or some inequality among the contributor (the poorest agent) and the rest. This is clearly seen for the case of the pure public good (b = 0). For simplicity, suppose there are two players with wealth levels w1 = w + ε and w2 = w ? ε and, in addition assume for simplicity that c = 1. Now joint pro?ts are: so Π (ε) = 2f (2ε + w? ) ? 1. We know that f (2ε + w? ) ? 1 ≤ 0 since by de?nition f (w? ) ? 1 = 0 Π(ε) = f (w? ) ? {w? ? (w ? ε)} + f (w + ε + w? ? (w ? ε)) = f (w? ) ? w? + w + f (2ε + w? ) ? ε and

but whether 2f (2ε + w? ) ? 1 ≤ 0 or > 0 cannot be determined a priori. For the intuition behind

this, notice that, those who choose xi > 0, i.e., the poorest players, use the e?cient amount of the input. Other players have more than the e?cient level of the input in their production units. Any redistribution from the poor to the rich players does not a?ect the pro?t of the former as they exactly compensate for this by increasing their contribution. Since rich players have more than the e?cient level of the input in their ?rms, normally a transfer of an additional unit of wealth would reduce joint pro?ts since the marginal gain to the rich player is less than the marginal cost to the poor player. But every extra unit of wealth received by the rich player increases the input received by her ?rm by twice the amount because of the increase in the e?ort by the poor player and as a result it is not clear whether joint pro?ts increase or decrease.

4.3

Complementarity Between the Individual Relative Contribution and the Total Contribution

Above we studied the case where the player’s own contribution and the total contribution of all players are perfect substitutes in determining the bene?t from the collective input enjoyed by a player, zi . In this section we consider an alternative formulation where they can be complements: zi = xi X
θ



the total contribution, but her gains are greater, the her contribution is relative to the total. This 24

where 0 ≤ θ ≤ 1 and 0 ≤ γ ≤ 1. According to this speci?cation, each player not only gains from

induces people to choose a higher level of xi which bene?ts others through the term X γ . But it also reduces how much others can enjoy the collective good by a congestion e?ect captured by the term
xi θ X

. If the latter e?ect is unimportant compared to the former, then we have a public good and

indeed for θ = 0 we have the textbook case of a pure public good. But if it is the other way round then the congestion e?ect dominates the bene?cial externality e?ect and in the limit, for θ = 1 we have the textbook case of the commons. When these two e?ects exactly balance each other out (θ = γ ), we have the case of the a pure private good. Analytically, this case turns out to be quite hard to characterize even when we assume a speci?c form of the production function, namely Cobb-Douglas, and consider a two player game. We show that if we compare the allocations under perfect equality (both players have the same level of wealth) and perfect inequality (one player has all the wealth and the other player has nothing) joint surplus there are substantial negative externalities then under some parameter values joint surplus will be higher under perfect inequality. The intuition for this result lies in the fact that when the negative externality problem is very severe then under perfect equality the players choose their actions related to the collective input at too high a level relative to the joint surplus maximizing solution. Perfect inequality converts the model to a one player game and hence eliminates this problem. On the other hand due to joint diminishing returns to the private input and the collective input, joint surplus is lower under perfect inequality compared to perfect equality if there were no externalities. What this result tells us is that perfect inequality is desirable only when the negative externality problem is severe and when the extent of diminishing returns is not too high. If, instead of comparing the allocations under perfect equality and perfect inequality, we consider the e?ects of a continuous change in inequality on total contributions and joint pro?ts, the results are not clear-cut. We prove that in the case of commons its total use (X ) decreases with increasing wealth inequality and joint pro?ts per unit of total contributions (i.e., Π/X ), or what one may call the average rate of return on the collective input, increases with inequality. But the absolute level of joint pro?ts may increase or decrease with inequality. Numerical simulations suggest that joint pro?ts in general decrease with inequality, except for the case of substantial negative externalities. In the case of public goods (pure and impure), we prove that the average rate of return on the public good input decreases with inequality. But as the extent of positive externalities become large (approaching the pure public goods case) the total amount of public good provision (and the absolute amount of the joint pro?ts) may increase with inequality. However there exists a range of moderate presence of positive externalities such that total contributions as well as joint pro?ts decrease with inequality. is always higher under perfect equality for non-negative externalities (i.e., θ ≥ γ ). However, if

25

5

Concluding Remarks

In this paper we analyze the e?ect of inequality in the distribution of endowment of private inputs that are complementary in production with collective inputs (e.g., contribution to public goods such as irrigation and extraction from common-property resources) on e?ciency in a simple class of collective action problems. In an environment where transaction costs prevent the e?cient allocation of private inputs across individuals, and the collective inputs are provided in a decentralized manner, we characterize the optimal second-best distribution of the private input. We show that while e?ciency increases with greater equality within the group of contributors and non-contributors, in some situations there is an optimal degree of inequality between the groups. The limitations of our model suggest several directions of potentially fruitful research. Our model is static. It is important to extend to the case where both the wealth distribution and the e?ciency of collective action are endogenous. For example, it is possible to have multiple stationary states with high (low) wealth inequality leading to low (high) incomes to the poor due to low (high) level of provision of public goods, which via low (high) mobility can sustain an unequal (equal) distribution of wealth. Also, in the dynamic case it will be interesting to analyze the e?ects of inequality on the sustainability of cooperation in a situation of repeated games. Second, technological non-convexities and di?erential availability of exit options seriously a?ect collective action in the real world, and our model ignores them.21 For example, the public good may not be generated if the total amount of contribution is below a certain threshold. This is the case for renewable resources like forests or ?shery where a minimum stock is necessary for regeneration, or in the case of fencing a common pasture. Third, the empirical literature suggests that even when the link between inequality and collective action is consistent with the results in our model, the mechanisms involved may be quite di?erent in some cases. For example, transaction costs in con?ict management and costs of negotiation may be higher in situations of higher inequality. Fourth, following the public economics literature in this paper we focus mainly on the free-rider problem arising in a collective action setup. Here, the issue is the sharing of the costs of collective action. But there is another problem, often called the bargaining problem, whereby collective action breaks down because the parties involved cannot agree on the sharing of the bene?ts.22 Inequality matters in this problem as well. For example, bargaining can break down when one party feels that the other party is being unfair in sharing the bene?ts (there is ample evidence for this in the experimental literature on ultimatum games). More generally, social norms of cooperation and
21

The model of Dayton-Johnson and Bardhan (1999) examines the e?ect of inequality on resource conservation

with two periods and di?erential exit options for the rich and the poor in the case when technology is linear. Baland and Platteau (1997) discuss the e?ect of non-convexities of technology in a static model.
22

See for example, Elster (1989).

26

group identi?cation may be di?cult to achieve in highly unequal environments. Putnam (1993) in his well-known study of regional disparities of social capital in Italy points out that “horizontal” social networks (i.e., those involving people of similar status and power) are more e?ective in generating trust and norms of reciprocity than “vertical” ones. Knack and Keefer (1997) also ?nd that the level of social cohesion (which is an outcome of collective action) is strongly and negatively associated with economic inequality. Finally, we focus only on the voluntary provision of public goods and do not consider the possibility that the players might elect a decision maker who can tax them and choose the level of provision of the collective good. The role of inequality in such a framework is an important topic for future research.23

6

Appendix

f2 (w, z ) > 0 for all w > 0 and limw→0 f2 (w, z ) = 0. Therefore γ (w) > 0 for all w > 0. By concavity, a global maximum exists and f22 (w, z ) < 0. By de?nition, f2 (w, γ (w)) ? 1 = 0. Notice that under our assumptions γ (w) is di?erentiable, and hence continuous. In particular,
dγ (w) dw
12 = ?f f22 > 0.

Proof of Lemma 1: Consider the ?rst order condition, f2 (w, z ) ? 1 = 0. By Assumption 1,

Proof of Lemma 2: By the de?nition of h(w, z ), h(w, γ (w)) = 1. Totally di?erentiating with
(w ) respect to w we get h1 + h2 dγ dw = 0, or,

z are complements) and h2 = respect to w we get:

? 2 f (w,z ) ?z 2

dγ (w) dw

< 0 (by strict concavity). Di?erentiating once again with

1 = ?h h2 . Notice that h1 =

? 2 f (w,z ) ?z?w

> 0 (as w and

d2 γ (w) h2 h22 + h2 2 h11 ? 2h1 h2 h12 . =? 1 3 2 dw h2 0 h1 h2 being ≤ 0 which in turn

The condition

d 2 γ (w ) dw2

≤ 0 is equivalent to the determinant

h1 h11 h12

h2 h12 h22 is equivalent to h(w, z ) being quasi-concave (see Theorem 21.20 of Simon and Blume (1994)). Proof of Lemma 3: Since f (w, z ) is homogeneous of degree 1, f2 (w, z ) is homogeneous of degree 0. If λ > 0, f2 (λw, λγ (w)) = f2 (w, γ (w)). Since by de?nition f2 (w, γ (w)) = 1, so f2 (λw, λγ (w)) = f2 (w, γ (w)) = 1. Then it must be true that γ (λw) = λγ (w) which means γ (w) = Aw where A > 0 is a constant. Proof of Lemma 4: Since agent k +1 contributes a positive amount by assumption, g (wk+1 ) > k+1 k c i =1 g (wi ) i=1 g (wi ) . Straightforward algebra shows that this is equivalent to the inequality > b + kc b + (k + 1)c
23

c

Olszewski and Rosenthal (1999) address this question for pure public goods within the framework of the distrib-

ution neutrality literature.

27

c

k i=1 g (wi )

b + kc

. The second part of the lemma can proved in the same way.

Proof of Lemma 5: Totally di?erentiating with respect to w we get: 1 ?π ? (w ) ≡ f1 (w, g (w)) + f2 (w, g (w)) ? ?w b + mc g (w).
1 b+c .

From the de?nition of g (w) and the ?rst-order condition of a contributing player, f2 (w, g (w)) = Substituting in we get ?π ? (w) (m ? 1)c g (w). ≡ f1 (w, g (w)) + ?w (b + c)(b + mc) Totally di?erentiating once again with respect to w : ?2π ? (w ) (m ? 1)c g (w). ≡ f11 (w, g (w)) + f12 (w, g (w))g (w) + 2 ?w (b + c)(b + mc)
12 From the proof of Lemma 1, g (w) = ? f f22 . Therefore f11 f11 + f12 g (w ) = 2 f11 f22 ?f12 f22

< 0 since

f (w, z ) is concave. Therefore

? (w ) ?2π ?w2

is negative if one of the following holds: (i) g (w) is concave
g (λw) b+mc

and c > 0; (ii) c = 0 or (iii) c < 0 and |c| small. The second part of the lemma follows from the fact λ f (w, g (w)) ? that if f (w, z ) is homogeneous of degree one then g (.) is linear and π ? (w) = f (λw, λg (w)) ?
g (w) b+mc

=

is linear as well.

Proof of Lemma 6: From Lemma 5, we know that the joint pro?t of contributing players is concave in ε. Also, it can be directly veri?ed that the joint pro?t of non-contributors is concave in ε. Di?erentiating the terms in (7) that relate to non-contributing players and using the superscript n to denote these players we get c ε n 1 n f11 ? g (w + )f12 + n?k b + kc k kc ε ε ε 1 (n ? k )c n , g (w + ) g (w + ) w? f2 n ? k b + kc k k b + kc k 1 c (n ? k )c ε ε n + g (w + ) ? f n + f22 g (w + ) . b + kc k n ? k 21 b + kc k This expression is negative since all the terms are negative. Therefore ΠI (ε) is concave in ε and so ? ΠI (ε) ?ε 0 as ε ε ?.

The claim in the lemma follows directly from the above. Proof of Proposition 2: For a given value of m it follows from the concavity of the pro?t functions of both contributors and non-contributors that there should not be any intra-group het? given that contributors must be richer than non-contributors (see (1)-(3)). erogeneity. Also, w > w 28

It is never optimal to set w ? at a very low level given the Inada endpoint conditions, namely, limw ? , it would never be optimal to make w arbitrarily small, since ? cX ) = ∞. Since w > w ? →0 f1 (w, that would mean w ? would be even smaller and almost all of W would be left unused. Proof of Proposition 3: (a) If b = 0 (6) implies that ε? = 0 i.e. any degree of inequality can be sustained in an equilibrium with non-contributors. Consider the derivative in (7) evaluated at ε = 0 (i.e. around the point of perfect equality). We have that: dΠI (0) dΠI (ε? ) kc = = f2 w, g (w) dε dε b + kc (n ? k )c g (w)+ b + kc c(k ? 1) ?) +g (w >0 (b + c)(b + kc)

as all the terms are positive. Then by Lemma 6, ε ? > ε? = 0, i.e., perfect equality is never joint pro?t maximizing in the case of pure public goods. (b) In the case of pure private goods (c = 0), (3) is clearly satis?ed for any redistribution of wealth among the agents, i.e. all of them always contribute. But then it follows directly from Proposition 2 that greater inequality reduces joint pro?ts. Proof of Proposition 4: (a) Di?erentiating both sides of (6) with respect to b we get: ε? n?k 1 = ) n ? k ?b ?ε? kcg (w ?+ ε? ε? 1 ?ε? ) (b + kc) ? kcg (w ?+ ) k k ?b k (b + kc)2

g (w ?? i.e.

)(?

ε? ) k = >0 ? ε ε? 1 ?b ?? cg (w ? + )(b + kc) + g (w ) (b + kc)2 k n?k n?k ? ε? ε Therefore, w1 ≡ w ?? is decreasing in b. Given the de?nition ? + is increasing in b and w2 ≡ w k n?k 1 of ε? , and the fact that f2 (z, g (z )) = we get b+c ?ε? kcg (w ?+ f2 w ? ε? kc ε? , g (w + ) n ? k b + kc k = f2 w ? ε? ε? , g (w ? ) n?k n?k = 1 b+c

Therefore (7) evaluated at ε? can be written as:

(n ? 1)c dΠI (ε? ) (8) = g (w1 ) + f1 (w1 , g (w1 ) ? f1 (w2 , g (w2 )). dε (b + c)(b + kc) n Note that when b → ∞, ε? → (n ? 1)w. ? Therefore, w1 → w ? and w2 → 0. From the Inada k dΠI (ε? ) conditions f1 (w2 , g (w2 )) → ∞ and thus = ?∞ < 0 while the other two terms in the above dε 29

expression are ?nite and non-negative. Since the function ΠI (ε? ) is continuous, this proves that there exists some ? b2 , ∞), b2 > 0 (which would, in general, depend on n, k and c) such that for b ∈ [? I ? dΠ (ε ) < 0, implying that ΠI (ε) is maximized at ε? (the minimum degree of inequality needed to dε have non-contributing agents in equilibrium), i.e. the problem has a corner solution. Let
ε g (w ?+k ) ε ε ε ε ΠC (w ? + , g (w ?? ? + )) ? ? + ,w ) ≡ k [f (w ]+ k n?k k k b + nc ε g (w ? ? n? ε ε k) +(n ? k )[f (w ?? , g (w ?? )) ? ] n?k n?k b + nc ε denote joint pro?ts when there are k agents with wealths w ? + and n ? k agents with wealths k ε ? ? I ? w ?? , all of whom contribute. From the de?nition of ε? , ΠC (w ? ? nε ? + εk , w ?k ) = Π (ε ). Also, n?k ? ? ΠE > ΠC (w ? + εk , w ? ? nε ?k ) from Proposition 2. Thus perfect equality maximizes joint surplus

for large enough values of b. From Proposition 3, part (a) and the continuity of the joint pro?t function, we know that there exists b1 > 0 such that if b ∈ [0, ? b1 ] such that perfect equality is never joint pro?t maximizing.

(b) The proof is very similar to that of part (a). Di?erentiating both sides of (6) with respect to c we get: ε? ε? ε? 1 ?ε? ? + )](b + kc) ? k 2 cg (w ?+ ) ) + kg (w k k ?c k k (b + kc)2

g (w ?? i.e.

ε? n?k

)(?

1 ) = n ? k ?c

?ε?

[kcg (w ?+

ε? ) k = < 0. ? ε? 1 ε ?c ?? kcg (w ) (b + kc)2 ? + )(b + kc) + g (w k n?k n?k ? ε? ε The above implies that w1 ≡ w ?? is increasing in c. At ?+ is decreasing in c and w2 ≡ w k n?k n c = 0 we have (assuming g (0) = 0), ε? = (n ? k )w, ? w1 = w ? and w2 = 0. Now let us look at (8) k dΠI (ε? ) once again. From the Inada conditions f1 (w2 , g (w2 )) = ∞ and thus = ?∞ < 0 as the dε other two terms in (8) are ?nite and non-negative (the ?rst term is actually equal to 0). Since f is ?ε? ?bkg (w ?+ concave by assumption, f1 (z, g (z )) is decreasing in z and as w1 is decreasing in c the second term third term is increasing in c as well. The latter imply that f1 (w1 , g (w1 ) ? f1 (w2 , g (w2 )) < 0 and thus above is increasing in c. Similarly, as ?f1 (z, g (z )) is increasing in z and w2 is increasing in c, the

the left hand side increases towards zero for c → ∞. By a continuity argument, the above shows dΠI (ε? ) that there exists some c ? > 0 (depending on n, k, b) such that for c ∈ [0, c ?], < 0, i.e. ΠI (ε) dε is maximized at ε? (the minimum degree of inequality needed to have non-contributing agents) as. 30

I ? ? ? nε But ΠE > ΠC (w ? + εk , w ?k ) = Π (ε ) and so perfect equality maximizes joint surplus for small

?

?

c.

Proof of Proposition 5: (a) The result follows immediately from Lemma 5. (b) Since we assume constant returns to scale, it follows from Lemma 3 that g (w) = Aw, where A is a positive constant. A player stops contributing if A(w ?
ε kcA(w + k ) ε ) < cX = n?k b + kc

Consider the derivative in (7) evaluated at ε? . The second and third term are clearly positive. Because of constant returns to scale, f1 (λw, λz ) = f1 (w, z ) for λ > 0. Notice also that from the de?nition of ε? : ε? kc w+ b + kc k ? From this equation we can solve for ε : ε? = But then the ?rst term is: f1 w + = f1 ε? , g (w + k ε? w + , g (w + k ε? ε? kc ε? ) ? f1 w ? , g (w + ) = k n ? k b + kc k ? ? ε ε kc ε? kc ) ? f1 w+ , g (w + ) k b + kc k b + kc k =w? ε? . n?k

bw(n ? k ) . cn + b

=0

This implies that ΠI (ε) is increasing in a neighborhood of ε? and thus it achieves a maximum for
I ? I some ε ? > ε? , i.e. maxε∈[ε? ,w ? (n?k)] Π (ε) > Π (ε ). Now let us prove that inequality is always joint

pro?t maximizing. Joint pro?ts under perfect equality when f (w, z ) is homogeneous of degree one is ? ) = n[f (w, ? )) ? ΠE (w ? g (w g (w ?) ] b + nc A = nw ? [f (1, A) ? ]. b + nc

Also, joint pro?ts under an unequal wealth distribution such that there are k agents with wealths ε? ε? and n ? k agents with wealths w ?+ ?? in which only the former group contributes is: w k n?k ?+ ΠI (ε? ) = k [f (w g (w ? + εk ) ε? ε? kc ε? ε? ? + )) ? ? + )) , g (w ] + (n ? k )f (w ?? , g (w k k b + kc n ? k b + kc k ? ? ? ? ? kA(w ? + εk ) ε ε ε ε = kf (w ? + , A(w ,A w ? )? ? + )) + (n ? k )f (w ? k k n?k n?k b + kc ? ? ε A ε ?+ f (1, A) ? + (n ? k ) w ? f (1, A) = k w k b + kc n?k ε? A = nwf ? (1, A) ? k w ?+ . k b + kc 31
?

Using the value of ε? we get:
(n?k) kw ? + bw n(b + kc) n k (b + cn) + b(n ? k ) cn+b = w= w= w. b + kc (b + kc)(b + cn) (b + kc)(b + cn) (b + cn)

i.e. ΠI (ε? ) = ΠE . Therefore, max ΠI (ε) > ΠI (ε? ) = ΠE and thus some degree of inequality (with
ε

ε>

b(n?k) b+nc w )

is joint pro?t maximizing.

For the second part of Proposition 5 (b), it is su?cient to provide an example. Suppose f (w, z ) has the Cobb-Douglas, constant returns to scale form f (w, z ) = wα z 1?α and there are two agents in the economy24 with endowments of the private input w + ε and w ? ε, where ε ∈ [0, w]. By 1 1 α ? Lemma 3 we have g (w) = Aw and f (w, g (w)) = A w, where A = [(b + c)(1 ? α)] α . Under perfect equality each player obtains a surplus of: π E = wA1?α [1 ? Let, as in the proof of Proposition 5, ε? = Aα c + α(b + c) ] = wA1?α [ ] b + 2c b + 2c

(9)

bw be the degree of inequality at which the poorer b + 2c player is just indi?erent between contributing and not contributing. Thus, for ε ∈ (ε? , w] only the by: π rich (ε) = A1?α (w + ε) ? A(w + ε) = A1?α α(w + ε) b+c Ac 1?α π poor (ε) = (w ? ε)α (w + ε)1?α ( ) b+c

richer player (i.e. the one with endowment w + ε) would contribute and her pro?ts would be given

using the expression for πi obtained previously. First notice that, evaluating the above expressions at ε = ε? it is possible to have: π rich (ε? ) < π poor (ε? ) as it is equivalent to α < (10) c . Clearly, for ε > ε? π rich (ε) is increasing in ε. We can verify directly b+c ?π poor (ε) c that . In addition as ε → w, π rich (ε) goes to some positive < 0 for ε > ε? if α > ?ε b + 2c value, while π poor (ε) goes to 0. Combining these results with (10) we see that there can exist some level of inequality ε0 ∈ (ε? , w) such that: π rich (ε0 ) = π poor (ε0 ) Using the expressions obtained above we can solve for ε0 to get: ε0 =
24

(11)

w(1 ? B ) 1+B

We have actually proven the proposition for any f (w, z ) satisfying Assumptions 1-3 and any redistribution in

which k agents obtain w + ε and n ? k obtain w ? ε but the expressions corresponding to (??) and (??) are much less tractable which is why we chose to present the result for a Cobb-Douglas function.

32

1?α 1 b+c α where B = α ( ) α . Finally, it is easy to verify that the condition c π rich (ε0 ) = π poor (ε0 ) > π E c 2α(b + 2c) c > 1 + B. As long as this condition, and < α < hold c + α(b + c) b + 2c b+c simultaneously, we have an example where inequality Pareto-dominates perfect equality. For the is equivalent to case α = 1/2 the ?rst condition is equivalent to c2 > b2 , and the second one is equivalent to c > b > 0, i.e. if the latter is true inequality is Pareto dominating. Proof of Proposition 6: From Assumption 1 we know that f2 (w, z ) = ∞ as z approaches 0

agents contribute in equilibrium, i.e. m = n. Also we know that X is maximized when wealths are g (wi ) . The individual contributions then equal: equalized as it is equal to b + nc xi = g (wi ) ? cX (b + (n ? 1)c) c = g (wi ) ? b b(b + nc) b(b + nc) g (wj ) > 0
j =i

from above and also, that f (w, z ) = ?D for z < 0, where D is a very large number. Therefore all

as c < 0 and b + (n ? 1)c > b + nc > 0 by Assumption 2. Since all agents contribute joint surplus equals: Π= f (wi , g (wi )) ? g (wi ) . b + nc

Let us start at perfect equality, i.e. wi = w and consider a redistribution giving k of the agents ε ε w + and the rest w ? , ε > 0. We then have: k n?k ε ε g (w ? ) g (w + ) ε ε ε ε n ? k ]. k ] + (n ? k )[f (w ? Π = k [f (w + , g (w + )) ? , g (w ? )) ? k k b + nc n?k n?k b + nc ?Π ?ε ε ε = f1 (w + , g (w + )) + k k ε ε ε 1 +g (w + )[f2 (w + , g (w + )) ? ] k k k b + nc ε ε ?f1 (w ? , g (w ? )) ? n?k n?k ε ε ε 1 g (w ? )[f2 (w ? , g (w ? )) ? ] n?k n?k n?k b + nc
1 b+c

Let us see how a change in ε a?ects joint pro?ts:

We have f2 (z, g (z )) = ?Π ?ε

from the ?rst-order conditions. So: (12)

ε ε ε ε = [f1 (w + , g (w + )) ? f1 (w ? , g (w ? ))] + k k n?k n?k (n ? 1)c ε ε + [g (w + ) ? g (w ? )] (b + c)(b + nc) k n?k 33

Evaluating the above at ε = 0, we have that ?Π |ε=0 = 0 ?ε ε and w2 = i.e. ε = 0 is a critical point for the joint surplus function. Denote w1 = w + k ε w? .The second derivative of Π is: n?k ?2Π ?ε2 = 1 [f11 (w1 , g (w1 )) + f12 (w1 , g (w1 ))g (w1 )] + k 1 + [f11 (w2 , g (w2 )) + f12 (w2 , g (w2 ))g (w2 )] n?k 1 (n ? 1)c 1 g (w2 )]. + [ g (w1 ) + (b + c)(b + nc) k n?k

At ε = 0 the above equals:

n (n ? 1)c [f11 (w) + f12 (w)g (w) + g (w)]. k (n ? k ) (b + c)(b + nc) The ?rst term within the brackets is negative (recall from the proof of Lemma 5 that f11 f11 + f12 g (w) =
2 f11 f22 ?f12 f22

< 0 as f (z, w) is concave) but the second term is positive. Therefore we

the concavity of f , i.e. ε = 0 is a local maximum. Recall that by Assumption 2, b + nc > 0, or
b . Suppose c is large enough in absolute value such that b + nc is close enough to 0. Then the c > ?n ?2Π last term within the square brackets becomes arbitrarily large and so > 0 i.e. ε = 0 is a local ?ε2 minimum. Therefore by a continuity argument, if c is close to zero, i.e. for all c in some interval

cannot sign the derivative in general. For c → 0, however, we know it is going to be negative by

[c0 , 0) perfect equality is locally joint pro?t maximizing. If however c is large in absolute value, i.e. b c ∈ [? , c1 ) and so b + cn close to 0, the second term above is arbitrarily large and therefore joint n pro?ts are maximized at some positive degree of inequality. Stability of Equilibrium can be derived from a simple adjustment mechanism of the following form: dxi xi ? xi (t)), i = 1, 2, .., m = ?i (? dt ?i is the reaction where ?i are positive constants, xi (t) is the actual value of xi at time t, and x function as given by (1). Given that reaction functions are linear in our model, the condition for stability is equivalent to the following determinant of order m b+c c . c c b+c . c 34 .. c c . b+c Following Cornes and Sandler (1984), the stability condition in Assumption 2 that b + nc ≥ 0

being positive de?nite. Performing some simple operations to make all elements in the ?rst row (or column) except for the ?rst two to be equal to zero, we can prove by induction that the value of this determinant is equal to bm?1 (b + mc). The CES Example For the CES production function: k f (w, z ) = (δwρ + (1 ? δ )z ρ ) ρ we show that if 0 < ρ < k ≤ 1 then γ (w) is increasing and concave. First we need to ensure that f

f (λw, λz ) = λk f (w, z ). The condition for f12 > 0 is k > ρ. The ?rst order condition of maximization is: (δwρ + (1 ? δ )γ (w)ρ ) ρ ?1 γ (w)ρ?1 =
k

is concave and w and z are complements. The condition for non-increasing returns is k ≤ 1, since

Di?erentiating with respect to w and using the ?rst order condition we get: γ (w ) = (k ? ρ)δwρ?1 γ (w) . (1 ? k )(1 ? δ )(γ (w))ρ + δ (1 ? ρ)wρ Observe that
wγ (w) γ (w)

1 . k (1 ? δ )(b + c)

As k > ρ by assumption the numerator is positive. Also, the denominator is positive as 1 ? k ≥ 0 and ρ ∈ (0, 1) and δ ∈ (0, 1). Therefore γ (.) is increasing.
(k?ρ)δwρ (1?k)(1?δ )(γ (w))ρ +δ (1?ρ)wρ

=

out to be the same as that of the following expression:

nator (which follows from k ≤ 1). Di?erentiating the expression for γ (w), the sign of γ (w) turns (1 ? ρ) (1 ? k )(1 ? δ )wρ?2 γ (w)ρ+1 + δw2ρ?2 γ (w) wγ (w) ?1 . γ (w )
wγ (w) γ (w)

≤ 1 since the numerator is less than the second term in the denomi-

This expression is non-negative under our assumptions, and the fact that 1,
wγ (w) γ (w )

= 1 and so the expression is equal to 0. Therefore γ (w) is concave, and strictly so for

≤ 1. For k =

k < 1.

References
[1] Alesina, A. and D. Rodrik (1994) : “Distributive Politics and Economic Growth” Quarterly Journal of Economics, vol. 109, no. 2, pp. 465-490. [2] Alesina, and E. La Ferrara (2000): “Participation in Heterogeneous Communities”, Quarterly Journal of Economics; vol. 115, no. 3, pp. 847 – 904.

35

[3] Baland, J.-M. and Platteau, J.-P. (1997) : Wealth inequality and e?ciency in the commons : I. The unregulated Case, Oxford Economic Papers 49, pp. 451-482. [4] Baland, J.-M. and Ray, D. (1999) : “Inequality and E?ciency in Joint Projects”, Mimeo. Facult? es Universitaires Notre-Dame de la Paix, Namur and New York University. [5] Banerjee, A., P. Gertler, and M. Ghatak: “Empowerment and E?ciency: Tenancy Reform in West Bengal” Journal of Political Economy, Vol. 110, No. 2, pp. 239-280. [6] Banerjee, A. and A. Newman (1993) : “Occupational Choice and the Process of Development” Journal of Political Economy, vol. 101, no. 2, pp. 274-298. [7] Banerjee, A. and E. Du?o (2000): “Inequality and Growth: What Can the Data Say?”, Mimeo. M.I.T. [8] Bardhan, P. (1984) : Land, Labor and Rural Poverty, New York, Columbia University Press. [9] Bardhan, P. (2000) : “Irrigation and Cooperation: An Empirical Analysis of 48 Irrigation Communities in South India”. Economic Development and Cultural Change. [10] Benabou, R. (1996) : “Inequality and Growth”, NBER Macroeconomics Annual. [11] Bergstrom, T., L. Blume and H. Varian (1986) : “On the Private Provision of Public Goods”, Journal of Public Economics, 29, p.25-49. [12] Bernheim, B.D. (1986) : “On the Voluntary and Involuntary Provision of Public Goods”, American Economic Review, September. [13] Boyce, J. K. (1987) : Agrarian Impasse in Bengal - Institutional Constraints to Technological Change. Oxford: Oxford University Press. [14] Browning, M. and V. Lechene (2001): “Caring and Sharing: Tests Between Alternative Models of Intra-household Allocation”, Mimeo. University of Copenhagen and Oxford University. [15] Cornes, R. and T. Sandler (1984) : “The Theory of Public Goods : Non-Nash Behavior” Journal of Public Economics, 23, p.367-79. [16] Cornes, R. and T. Sandler (1984) : “Easy Riders, Joint Production and Public Goods”, Economic Journal, 94, p.580-98. [17] Cornes, R. and T. Sandler (1994) : “The Comparative Static Properties of the Impure Public Good Model” Journal of Public Economics, 54, p.403-21.

36

[18] Cornes, R. and T. Sandler (1996) : The Theory of Externalities, Public Goods and Club Goods, Cambridge University Press, Second Edition. [19] Dayton-Johnson, J. (2000), “The Determinants of Collective Action on the Local Commons: A Model with Evidence from Mexico”, Journal of Development Economics. [20] Dayton-Johnson, J. and P. Bardhan (2001) : “Inequality and Conservation on the Local Commons : A Theoretical Exercise”. Forthcoming, Economic Journal. [21] Elster, J. (1989): The Cement of Society, Cambridge University Press, Cambridge. [22] Evans, D. and B. Jovanovic [1989] : “An Estimated Model of Entrepreneurial Choice under Liquidity Constraints”. Journal of Political Economy. [23] Galor, O. and J. Zeira (1993) : “Income Distribution and Macroeconomics.” Review of Economic Studies, Vol. 60, No. 1, pp.35-52. [24] Itaya, J., D. de Meza and G.D. Myles (1997) : “In Praise of Inequality : Public Good Provision and Income Distribution”, Economics Letters, 57, p.289-296. [25] Knack, S. and P. Keefer (1997) : “Does social capital have an economic payo?? A cross-country investigation”, Quarterly Journal of Economics; vol. 112, no. 4, pp. 1251-1288. [26] Olson, M. (1965) : The Logic of Collective Action: Public Goods and the Theory of Groups. Cambridge Mass.: Harvard University Press. [27] Olszewski, W. and H. Rosenthal (1999): “Politically Determined Income Inequality and the Provision of Public Goods”, Mimeo. Northwestern University and Princeton University. [28] Putnam, R. (1993) : Making Democracy Work: Civic Traditions in Modern Italy, Princeton University Press, Princeton, NJ. [29] Simon, C.P. and L. Blume (1994): Mathematics for Economists. New York and London: W.W. Norton. [30] Warr, P. G. (1983) : “The Private Provision of a Public Good is Independent of the Distribution of Income”, Economics Letters, 13, p.207-211.

37

Figure 1: Difference in Joint Surplus Under Perfect Equality and Optimal Inequality alpha = 0.2, beta = 0.6 0.3 0.2 0.1 0 0 ?0.1 ?0.01 ?0.02 0 ?0.02 ?0.04 0.03 0.02 0.01 alpha = 0.6, beta = 0.2 0.06 0.04 0.02 alpha = 0.4, beta = 0.4

0

b 1 alpha = 0.1, beta = 0.4

2

0

b 1 alpha = 0.4, beta = 0.1

2

0

b1 alpha = 0.25, beta = 0.25

2

0.4 0.3 0.2

0.15 0.1 0.05

0.2 0.15 0.1 0.05

0.1 0 ?0.1 0 b 1 alpha = 0.05, beta = 0.15 0.4 0.3 0.2 0.2 0.1 0 ?0.1 0 1 2 0.1 0 0.4 0.3 2 0 ?0.05

0 0 b 1 alpha = 0.15, beta = 0.05 0.4 0.3 0.2 0.1 0 0 1 2 ?0.1 0 1 2 2 ?0.05 0 b 1 alpha = 0.1, beta = 0.1 2

b

b

b

Figure 2: Difference in Joint Surplus Under Perfect Equality and Optimal Inequality alpha = 0.2, beta = 0.6 0.2 0 0.06 ?0.2 ?0.4 0 ?0.6 0 2 c 4 6 ?0.02 0 2 c 4 6 ?0.1 0 2 c 4 6 0.04 0.02 ?0.05 0 0.1 0.08 0.05 alpha = 0.6, beta = 0.2 0.1 alpha = 0.4, beta = 0.4

alpha = 0.1, beta = 0.4 0.1 0.08 0.06 0.04 0.02 0 ?0.02 0 2 c 4 6 0.1 0 ?0.1 0 0.4 0.3 0.2

alpha = 0.4, beta = 0.1 0.2 0.15 0.1 0.05 0 2 4 6 ?0.05 0

alpha = 0.25, beta = 0.25

c

2

c

4

6

alpha = 0.05, beta = 0.15 0.4 0.3 0.2 0.1 0 0.8 0.6 0.4

alpha = 0.15, beta = 0.05 0.5 0.4 0.3 0.2 0.2 0 0.1 0 2 4 6 0 0

alpha = 0.1, beta = 0.1

0

2

c

4

6

c

2

c

4

6



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