Abstract: This study analyzes cooperative and non-cooperative outcomes in the context of private charity and compares them on efficiency grounds. Analysis is conducted in a model economy where there are two rich individuals and one poor and the rich care about the well-being of the poor. The results of the analysis suggest that, when making donations to the poor, the rich should get together and act cooperatively instead of behaving independently. Given the fact that there are many donors in reality and therefore it is difficult for them to get together for cooperation, this result implies a role for governmet to provide a mechanism for cooperation and justifies the support of the governments for charity organizations.
INTRODUCTION
When the action of a person affects welfare of another person and vise versa, then there is interdependence among the individuals involved. In such a situation noncooperative outcome where each person behaves independently of others and cooperative outcome where individuals get together and behave in a cooperative way differ in terms of efficiency. For example, in the case of a Cournot duopoly in a given market where each firm treats the output of the other firm as given and maximizes its own profit (non-cooperation), the resulting profit per firm is less than the profit each will get if both get together and act as a cartel (cooperation). This means that non-cooperative outcome of Cournot competetion is suboptimal to cooperative outcome of Cartel (Kreps, 1990). It is not, however, always the case that non-cooperative outcome is inferior. For example, we know from the optimal tariff theorem in international trade literature that non-cooperation is better for a large country when the trading partner is small in the sense that it can not influence the terms of trade. In this case it won’t be in the best interest of the large country to cooperate.
This research aims to examine non-cooperative and cooperative outcomes in the context of private charity and find out which outcome is superior on the efficiency grounds. For this purpose an economy is considered where there are three individuals, two identical rich and one poor and both rich care about well-being of the poor, which is reflected by the fact that the consumption of the poor enters the utility functions of the rich positively. As a result, each rich has an incentive to make donation to the poor.
Private charity has been analyzed in some studies. For example, Warr (1982) shows that once donors have determined how much to donate voluntarily, additional gains could be achieved from further incremental transfers. No net transfer is achieved, however, unless incremental fiscal redistribution is continued until the point where the private charity has been driven to zero. Another such study is Roberts (1984) who considers a model where private charity and public transfers are determined simultaneously and shows that there is an overprovision of public transfers in the sense that more public transfers take place than altruistic taxpayers prefer. Taking a game-theoretic view, Glazer and Konrad (1996) offers a new motive for giving by treating observable charitable donations as signals of wealth (status-motivated giving) and examines the signaling equilibrium of charitable donations. They find that the resulting equilibrium has such attractive properties as donations increasing proportionately with population size and a rise in the spread between the poorest and the richest leading to increase in donations. Brooks (2002) empirically investigates in a cross-sectional data how the welfare payments affects charitable giving of those receiving payments and finds that charity is negatively associated with welfare receipt. Andreoni and Payne (2003), in addition to the widely accepted hypothesis in the literature that givers, who are also taxpayers, treat their tax-financed donations as an imperfect substitute for their voluntary donations, offers for the crowding-out effect the explanation that charity organizations, upon receiving a grant, will reduce their fund-raising efforts. Employing panel data from arts and social service organizations, they find that government grants to nonprofit organizations are causing significant reductions in their fund-raising efforts.
MODEL
The model economy used here is adopted from Warr (1982) with the exception that it is assumed in our case that the poor has no income. The economy consists of three individuals, two of whom are rich and one is poor. Two rich individuals are identical and indexed by i = 1, 2. The poor individual has no income and each rich has an exogenously determined income of y. The important property of the model is that the consumption of the poor individual enters the utility functions of the rich. Utility functions of the rich and the poor are given by:
where, Ci: consumption of rich individual i and Cp: consumption of the poor
Since the poor has no income, Cp = d1 + d2 where, di: donation made by individual i = 1, 2.
It is assumed that utility functions are continues, strictly increasing, strictly quasi-concave and everywhere twice differentiable with respect to its arguments.
As we see, Cp is appearing in the utility functions of both individuals 1 and 2. In other words, each rich is able to derive utility from the total quantity of the consumption of the poor. Therefore, Cp can be interpreted as a public good for individuals 1 and 2. Quantity appearing (Cp) in the utility function of each rich differs from his donation due to the fact that he can enjoy the contribution of the other. From the point of view of the utility function, d1 and d2 are perfect substitutes. However, di involves an opportunity cost in terms of Ci forgone whereas the donation made by the other rich involves no such cost.
NON-COOPERATIVE EQULIBRIUM
In determining how much to donate, each rich takes the donation of the other as given. Equilibrium donations d1n and d2n are such that given d2n, d1n maximizes 1's utility and d2n maximizes 2's utility given that 1's donation is d1n. This is a non-cooperative Nash equilibrium.
The problem of rich i is given by :
Given d1 and d2, the poor individual gets the utility of U (Cp) where, Cp = d1 + d2.
The problem of each rich is to determine the levels of Ci and di in such a way that utility is maximized. This occurs at that combination of Ci and di where the indifference curve is tangent to the budget line.
To find out how one individual reacts to a change in the donation of the other by adjusting his own donation, suppose that the rich individual under consideration is individual 1 and that individual 2 increases his donation. When d2 increases, individual 1's budget line shifts out in a parallel way because this is an increase in income paid in kind. It has only income effect due to the fact that it does not cause any change in the price ratio. Given that C1 and Cp are normal goods, as a result of increase in d2, C1 and Cp go up. However, y is fixed and equal to d1 + C1. As C1 increases, it must be the case that d1 decreases. This indicates that reaction curve is downward sloping. Since Cp = (d1 + d2) is increasing, decrease in d1 must be less than increase in d2. Then absolute value of ∂d2/∂d1 is greater than 1.
Assuming that the preferences are homotethic, the line connecting the tangency points (expansion path) corresponding to different levels of d2 will be a straight line. In this case reaction curve will also be a straight line. As a result, given the fact that absolute value of ∂d2/∂d1 is greater than 1, the resulting Nash equilibrium, determined by the intersection of the reaction curves, will be unique and stable. At this Nash equilibrium, since rich individuals are identical, equilibrium levels of donations will be equal to each other, i.e., d1n = d2n.
This Nash equilibrium is not Pareto efficient, however. The inefficiency of the equilibrium can be shown as follows:
At equilibrium,
Now consider a binding contract between individual 1 and 2 so that each rich is supposed to increase his total contribution to the poor by 1 unit. As a result of this arrangment, individual i's welfare will be affected as follows:
Since at non-cooperative Nash equilibrium,
and as a result of binding contract
Hence everyone is made better off, indicating that non-cooperative Nash equilibrium is not Pareto efficient.
COOPERATIVE EQULIBRIUM
As we have seen above, increase in donation makes everyone better off. This means that the Nash equilibrium level of donation is less than the optimal level i.e., there is an underprovision of the public good, Cp. Now let’s suppose that two rich individuals cooperate in the following way; they get together and decide that one of them will increase the level of his donation and the other will have to donate the same amount. Will this cooperation lead to Pareto efficiency?
Let's suppose that individual 1 will decide on the increase in his donation and individual 2 has to donate the same amount that individual 1 does. If individual 1 increases his donation by one unit, the other has to increase his donation by one unit as well. By giving up one unit of C1, he is getting two units of Cp. The effective price of Cp in terms of C1 forgone is 1/2.
At Nash equilibrium,
If he gives up one unit of C1, he gets two units of Cp but to stay as well off as before he needs only one unit of Cp. This means that by increasing Cp through increase in donation, he can increase his utility. He will increase the level of donation up to the point at which because at this point his utility is maximized.
Since individual 1 and 2 are identical and donations must be equal to each
other, it is true that individual 2' utility is also maximized because
When there is a public good, Cp, Pareto efficiency requires that
Where,
This is because if one of the rich increases Cp by one unit, the other rich is also able to consume this one unit increase.
The above condition can be interpreted as
Where,
Since in our case
This means that Pareto efficiency is achieved and as a result, compared to Nash equilibrium, everyone is made better off.
AN EXAMPLE
Utility function of each rich has the following form:
Nash equilibrium: Since both rich are identical, we can take only one of them
First-order conditions are;
These conditions imply that
reaction functions.
Using these reaction functions to solve for Nash equilibrium levels of donations yields
Given y, the higher α, the lower will be the Nash equilibrium level of donation, dn.
The Nash equilibrium levels of the consumption are
Cooperative equilibrium: It must be the case that d1 = d2 = d. This means that Cp = 2d.
subject to: C1 + d = y
First-order conditions are:
From these conditions, cooperative equilibrium levels of donation are found as
Given y, the higher α, the lower will be dc.
And cooperative equilibrium levels of the consumption are
We observe that
This inequality holds because 0<α<1.
This means that under the cooperative equilibrium everybody is made better off than under the Nash equilibrium.
DISCUSSION AND CONCLUSIONS
The purpose of this research has been to examine the non-cooparetive and cooperative outcomes in the context of private charity, compare those outcomes and find out which outcome is superior. A model economy consisting of two rich and one poor individuals where the consumption of the poor enters positively the utility functions of the rich is used for the analysis. It is shown, first theoretically and then with an example, that noncooperative outcome is not Pareto efficient and the efficiency can be achieved through cooperation. In other words, everybody including donors is better off under cooperation. This implies that when donating to the poor, the rich should behave cooperatively instead of independently of each other so that everyone will be at a higher utility level. In real world, however, there are so many donors, not just two like here. Therefore, it will be extremely difficult for the givers to get together and cooperate. Seeing this difficulty and knowing that cooperative outcome is better than non-cooperative one, government can play a role in the sense that it can provide some mechanisms that will lead to cooperation among donors. It is observed in the analysis that donations are higher under cooperation. Therefore, charity organizations, since in their absence donations would be less, can be regarded as institutions providing a cooperating mechanism.