Individual welfare and the group size paradox

Abstract

How does an increase in group size affect individual welfare in the presence of Olson’s group size paradox? Under the standard approaches to modeling the group size paradox, individual welfare declines as the size of the group expands. However, this may not be true if we apply the group size paradox to politics. In particular, if a smaller group attempts to extract a fixed income transfer from a larger group, the welfare of individuals in the larger group is increasing monotonically in the size of their group under a standard contest success function. Even though the probability of the transfer taking place is rising monotonically in the size of the group, the expected transfer per member shrinks because more members share the burden of financing the transfer. A similar result is obtained if the transfer is modeled as a fixed entitlement to be received by each member of the smaller group. By contrast, if the transfer imposes a fixed cost per member of the larger group, the result that individual welfare in the larger group is falling monotonically in its own group size is restored.

Appendices

Appendix

In “Group size and welfare with a rival, but nonexcludable good” section in Appendix, I will briefly develop a model in which a single group attempts to provide itself with a fully rival, but nonexcludable good. In this setting, the group size paradox will hold and individual welfare is decreasing in group size. Similarly, in “Group size and welfare in a contest for a positive rent” section in Appendix, I will show that when two groups compete for a fully rival positive rent that the welfare of both groups is decreasing in the size of their own group and increasing in the size of the other group. This establishes for some standard models that, when the group size paradox holds, individual welfare is decreasing in own group size. This contrasts with Results 1 and 2 of the paper. In “The derivatives for Result 1, A derivative for Result 2 and The derivatives for Result 3” sections in Appendix, I provide the some of the comparative statics underlying Results 1–3.

Group size and welfare with a rival, but nonexcludable good

Pure public goods are nonrival and nonexcludable, but obtaining the group size paradox generally requires that the good in question be rivalrous, which is the assumption employed here. There are n > 1 identical agents. Person i’s consumption of the private good is x _i and her contribution towards the public good is s _i where x _i + s _i = w. The parameter w is an exogenously given level of income. Provision of the rival and nonexcludable good is $g = (1/n)\sum\limits_{i = 1}^{n} {s_{i} }$. Utility is given by U(x _i, g), where there is a diminishing marginal rate of substitution between g and x _i. The first order condition at an interior equilibrium implies

$$\frac{{U_{g} (x_{i} ,g)}}{{U_{{x_{i} }} (x_{i} ,g)}} = n.$$

(13)

Since agents are identical, a symmetric Nash equilibrium exists in which s _i  = s, i. We thus have g = ns/n = s. Using the budget constraint and the result g = s, (13) may be expressed as follows:

$$\frac{{U_{g} (w - s,s)}}{{U_{{x_{i} }} (w - s,s)}} = n.$$

(14)

Because of a diminishing marginal rate of substitution, it must be the case that as n increases, s = g decreases, i.e., dg/dn < 0. Thus, individual consumption of g is decreasing in group size; in this setting we have a strong form of the group size paradox.

If we differentiate the utility function we obtain $dU = U_{{x_{i} }} dx_{i} + U_{g} dg$. From (13) $U_{{x_{i} }} = (1/n)U_{g}$ and from the budget constraint, dx _i  = −ds. Since s = g, we have ds = dg. Combining these we can write

$$\frac{dU}{dn} = U_{g} \frac{dg}{dn}\left( {1 - \frac{1}{n}} \right) < 0.$$

Recall that dg/dn < 0. Individual utility is decreasing in the size of the group. This contrasts with Results 1 and 2 of this paper.

Group size and welfare in a contest for a positive rent

The main body of the paper lays out a rent seeking contest in which group 2 seeks a positive transfer from group 1. Thus, while group 1 seeks a positive rent, group 2 seeks to avoid a possible loss. This is the key to the result that the welfare of members of group 2 is increasing in group 2’s size. The only difference with the analysis in the main body of the paper is that now group 2 also is competing for a positive rent R with each member receiving R/m ₂ if the group is successful in obtaining the rent. The problem for group 1 remains as in (3a) with R ₁ = R/m ₁. The problem for person j in group 2 becomes

$$\mathop {Max}\limits_{{x_{2}^{j} }} w_{2}^{j} = \,\,\frac{{X_{2}^{r} }}{{aX_{1}^{r} + X_{2}^{r} }}\frac{R}{{m_{2} }} - x_{2}^{j} ,$$

(15)

It can be shown that the solutions to the problems in (3a) and (15) imply the same values of X ₁ , X ₂ and p as found in (4a–4c) with the substitutions R ₁ = R/m ₁ and R ₂ = R/m ₂. Substituting these values into $w_{2}^{j}$ we have the following:

$$w_{2}^{j} = \left( {\frac{{m_{1}^{r} }}{{m_{1}^{r} + am_{2}^{r} }}} \right)\left( {\frac{R}{{m_{2} }}} \right)\left[ {1 - \frac{{ram_{2}^{r} }}{{m_{2} \left( {m_{1}^{r} + am_{2}^{r} } \right)}}} \right].$$

(16)

By inspection we can see that welfare of an individual in group 2 is increasing in the size of group 1. The derivative of the welfare function of individual j with respect to the size of group 2 may be expressed as follows:

$$\frac{{dw_{2}^{j} }}{{dm_{2} }} = - \frac{{m_{1}^{r} R}}{{\left( {\left[ {m_{1}^{r} + am_{2}^{r} } \right]m_{2} } \right)^{3} }}\left( {am_{1}^{r} m_{2}^{r} [(2 + r)m_{2} - 2r] + (am_{2}^{r} )^{2} [(1 + r)m_{2} - r(2 + r)] + m_{1}^{r} m_{2}^{r} [m_{1}^{r} m_{2}^{1 - r} + ar^{2} ]} \right) < 0$$

(17)

Because r ≤ 1 and m ₂ ≥ 2, we know that the term in the large parentheses is positive, making the entire expression negative. Thus, in a standard model of rent seeking, the welfare of members of group 2 is increasing in the size of group 1 and decreasing in the size of group 2. The results for group 1 are found from the analysis in the main body of the paper: welfare is decreasing in m ₁ and increasing in m ₂. These results correspond to the standard intuition that if one belongs to a group subject to the group size paradox and that group’s size expands, individuals in that group will be made worse off. For group 2, this conclusion stands in contrast to Results 1 and 2.

The derivatives for Result 1

Taking the derivative of (10a), we have the following:

$$\frac{{dw_{1}^{i} }}{{dm_{1} }} = - \frac{{am_{2}^{r} R}}{{\left( {\left[ {m_{1}^{r} + am_{2}^{r} } \right]m_{1} } \right)^{3} }}\left( {rm_{1}^{r} \left[ {m_{1}^{r} (m_{1} - 1 - r) + am_{2}^{r} (m_{1} - 1) + ram_{2}^{r} } \right] + m_{1} (m_{1}^{r} + am_{2}^{r} )\left[ {m_{1}^{r} - r + am_{2}^{r} } \right]} \right) < 0$$

(18)

Since r ≤ 1 and m ₁ ≥ 2, we know the term in the large parentheses is positive, making the derivative overall negative. Thus, the welfare of members of group 1 is decreasing in own group size.

Next, consider the effect of an increase in the size of group 1 on the welfare of an individual in group 2. From (10b) we obtain the following:

$$\frac{{dw_{2}^{j} }}{{dm_{1} }} = \left( {\frac{{raRm_{1}^{r - 1} m_{2}^{r - 1} }}{{\left( {m_{1}^{r} + am_{2}^{r} } \right)^{2} }}} \right)\left( {1 + r\left[ {\frac{{m_{1}^{r} - am_{2}^{r} }}{{m_{2} \left( {m_{1}^{r} + am_{2}^{r} } \right)}}} \right]} \right) > 0$$

(19)

The term in brackets inside the second parentheses is less than 1 in absolute value. As a result, the expression inside the second parentheses is positive, implying that $dw_{2}^{j} /dm_{ 1} > 0$. Thus, the welfare of group 2 is increasing in the size of group 1.

Next, in determining the sign of $dw_{2}^{j} /dm_{ 2}$ note from (10b) that $w_{2}^{j}$ may be expressed as follows:

$$w_{2}^{j} = - AB,$$

where

$$A = \left( {\frac{{am_{2}^{r} }}{{m_{1}^{r} + am_{2}^{r} }}} \right)\left( {\frac{R}{{m_{2} }}} \right)\quad {\text{and}}\quad B = \left[ {1 + \frac{{rm_{1}^{r} }}{{m_{2} \left( {m_{1}^{r} + am_{2}^{r} } \right)}}} \right].$$

(20)

Since there is a negative sign out front, if the product of A times B falls when m ₂ rises, and $w_{2}^{j}$ will increase. By inspection, it is clear that B is decreasing in m ₂. Taking the derivative of A with respect to m ₂ yields,

$$\frac{dA}{{dm_{2} }} = \left( {\frac{{aRm_{2}^{r} }}{{\left( {m_{2} [m_{1}^{r} + am_{2}^{r} ]} \right)^{2} }}} \right)\left[ {(r - 1)m_{1}^{r} - am_{2}^{r} } \right] < 0.$$

(21)

Since A and B both decrease in m ₂, welfare is increasing in m ₂.

A derivative for Result 2

Taking the derivative of (11a) yields the following:

$$\frac{{dw_{1}^{i} }}{{dm_{1} }} = - \frac{{rm_{1}^{r} am_{2}^{r} R_{1} }}{{\left( {m_{1}^{r} + am_{2}^{r} } \right)^{3} \left( {m_{1} } \right)^{2} }}\left[ {m_{1}^{r} (m_{1} - 1 - r) + am_{2}^{r} (m_{1} - 1 + r)} \right] < 0.$$

(22)

We are able to sign the term in brackets, because r ≤ 1 and m ₁ ≥ 2. Once again, the welfare of members of group 1 is decreasing in the size of group 1.

The derivatives for Result 3

Notice that the solution for $w_{1}^{i}$ from (12a) is m ₂ times the solution for $w_{1}^{i}$ from (10a). Thus, the derivative of welfare with respect to m ₁ is m ₂ times the expression in (18), which is negative. As before, the welfare of group 1 is decreasing in own group size.

Next we turn to $w_{2}^{j}$. Note that the solution in (12b) is m ₂ times the solution in (10b).Thus when we take the derivative of $w_{2}^{j}$ in (12b) with respect to m ₁, it is simply m ₂ times the derivative in (19), which is positive. The welfare of group 2 is increasing in the size of group 1. Taking the derivative of (12b) with respect to m ₂, we have

$$\frac{{dw_{2}^{i} }}{{dm_{2} }} = - \frac{{arm_{1}^{r} m_{2}^{r} R}}{{m_{2}^{2} \left( {m_{1}^{r} + am_{2}^{r} } \right)^{3} }}\left[ {m_{1}^{r} (m_{2}^{r} + r - 1) + am_{2}^{r} (m_{2} - 1 - r)} \right] < 0$$

(23)

Because m ₂  > 2 > 1 + r, we know that the term in the big brackets is positive, making the entire expression negative. In contrast with Results 1 and 2, the welfare of group 2 is decreasing in its own group size.