On the efficiency of equilibria in a legislative bargaining model with particularistic and collective goods

Abstract

This study analyzes the efficiency of equilibria in a multilateral bargaining game in which a legislature divides its budget among collective and particularistic goods. In order to disentangle the causes of inefficiency, we extend the model of Volden and Wiseman (Am Polit Sci Rev 101:79–92, 2007) by considering quasi-linear utility functions, and consensus requirements ranging from simple majority to unanimity. Although unanimous agreements can be reached under weaker consensus requirements, we show that (Pareto) inefficiency is associated with non-unanimous consent. We also study how (i) the endogenous selection of the legislature’s size or (ii) a sequential choice of collective and particularistic spending eliminates or attenuates the inefficiency problem.

Appendices

Appendix

Proofs

Proof (Proof of Lemma 2)

Let ${\overline{u}_{1}}\le \cdots \le {\overline{u}_{q}}\le \cdots \le {\overline{u}_{n}}$ with some strict inequality. Using the no-delay property of the SSPE, expected utilities are given by

$$\begin{aligned} n{\overline{u}_{1}}&= \delta \left[{\sum \nolimits_{i\notin \left\{ 1,n\right\} }} {\overline{u}_{1}^i}+{\overline{u}_{1}^1}+{\overline{u}_{1}^n}\right] \\ n{\overline{u}_{n}}&=\delta \left[ {\sum\nolimits_{i\notin \left\{ 1,n\right\} }} {\overline{u}_{n}^i}+{\overline{u}_{n}^1}+{\overline{u}_{n}^n}\right] \end{aligned}$$

where ${\overline{u}_{j}}^{i}$ denotes the expected utility obtained by player $j$ when $i$ is the proposer.

Since the cheapest to assemble winning coalition is chosen by any proposer $i \notin \{1,n\}$, then $u_{1}^{i}\ge \delta {\overline{u}_{1}}$ if $u_{n}^{i}=\delta {\overline{u}_{n}}$ and $u_{1}^{i}\ge u_{n}^{i}$ otherwise. Thus,

$$n\left[ {\overline{u}_{n}}-{\overline{u}_{1}}\right] \le m\delta \left[ {\overline{u}_{n}}-{\overline{u}_{1}}\right] +\delta \left[ {\overline{u}_{n}^1}+{\overline{u}_{n}^n}-{\overline{u}_{1}^1}-{\overline{u}_{1}^n}\right]$$

for some $m\in \left[ 0,n-2\right]$.

Moreover, either $u_{1}^{n}\ge \delta {\overline{u}_{1}}$ and $u_{n}^{1} \le \delta {\overline{u}_{n}}$ or $1$ is not included in the cheapest to assemble winning coalition. In the last case, the optimal proposal of agent $1$ is also ${\mathbf x^{n}}$ so that ${\overline{u}_{1}^1} = {\overline{u}_{n}^n}$ and ${\overline{u}_{1}}^{n} = {\overline{u}_{n}^1}$ . Thus,

$$\begin{aligned} \left[ n-m\delta \right] \left[ {\overline{u}_{n}}-{\overline{u}_{1}}\right] \le 0\text {,} \end{aligned}$$

which is a contradiction.

In case that $u_{1}^{n}\ge \delta {\overline{u}_{1}}$ and $u_{n}^{1}\le \delta \overline{u}_{n}$ we obtain

$$\begin{aligned} \left[ n-\left( m+1\right) \delta \right] \left[ \overline{u} _{n}-\overline{u}_{1}\right] \le \delta \left[ \overline{u}_{n}^{n} -\overline{u}_{1}^{1}\right] \text {.} \end{aligned}$$

By mimicking the proposal of agent $n$ player $1$ can obtain

$$\begin{aligned} \overline{u}_{1}^{1}\ge \alpha \left( 1-y_{n}-\sum_{i\in L-\{1,n\}} x_{j}^{n}-x_{n}^{1}\right) +y_{n}^{\beta }, \end{aligned}$$

where $x_{n}^{1}=\max \left\{ \frac{\delta \overline{u}_{n}-y_{n}^{\beta }}{\alpha },0\right\}$. Thus, $\overline{u}_{n}^{n}-\overline{u}_{1}^{1}=\alpha \left( x_{n}^{1}-x_{1}^{n}\right)$. We distinguish three cases:

• 1. $x_{n}^{1}=0$, which implies $x_{1}^{n}=0$. In this case, we obtain

  $$\begin{aligned} \left[ n-\left( m+1\right) \delta \right] \left[ \overline{u} _{n}-\overline{u}_{1}\right] \le 0\text {,} \end{aligned}$$

  which is a contradiction.

• 2. $x_{n}^{1}=\frac{\delta \overline{u}_{n}-y_{n}^{\beta }}{\alpha }$ and $x_{1}^{n}=0$. Since $y_{n}^{\beta }\ge \delta \overline{u}_{1}$,

  $$\begin{aligned} \alpha \left( x_{n}^{1}-x_{1}^{n}\right) =\alpha x_{n}^{1}=\delta \overline{u}_{n}-y_{n}^{\beta }\le \delta \left( \overline{u}_{n}-\overline{u}_{1}\right) , \end{aligned}$$

  and therefore

  $$\begin{aligned} \left[ n-\left( m+2\right) \delta \right] \left[ \overline{u} _{n}-\overline{u}_{1}\right] \le 0\text {,} \end{aligned}$$

  which is also a contradiction.

• 3. Similarly, if $x_{n}^{1}=\frac{\delta \overline{u}_{n}-y_{n}^{\beta }}{\alpha }$ and $x_{1}^{n}=\frac{\delta \overline{u}_{1}-y_{1}^{\beta }}{\alpha }$ then $\alpha \left( x_{n}^{1}-x_{1}^{n}\right) =\delta \left( \overline{u}_{n}-\overline{u} _{1}\right)$ and a contradiction is obtained, too.

$\square$

Lemma 4

$\alpha_{1}=\left[ \frac{n(1-\delta )\beta ^{\frac{\beta }{1-\beta }}}{ \delta }+\beta ^{\frac{1}{1-\beta }}\right] ^{1-\beta }$ and $\alpha _{2} = \left[ {\frac{{n\left( {nQ} \right)^{{\frac{\beta }{{1 - \beta }}}} (1 - \delta )\beta ^{{\frac{\beta }{{1 - \beta }}}} }}{\delta } + \left( {nQ} \right)^{{\frac{1}{{1 - \beta }}}} \beta ^{{\frac{1}{{1 - \beta }}}} } \right]^{{1 - \beta }} > \alpha _{1} > \beta .$

Proof

$\alpha_{1}$ solves $f(y_{1},\alpha_{1};\delta )=\frac{\alpha \delta }{n} (1-y_{1})-(1-\delta )y_{1}^{\beta }=0$. That is, $\frac{\alpha_{1}\delta }{n} (1-y_{1})-(1-\delta )y_{1}^{\beta }=0$. Substituting $y_{1}$ and rearranging terms, we obtain

$$\begin{aligned} \alpha_{1}=\left[ \frac{n(1-\delta )\beta ^{\frac{\beta }{1-\beta }}}{\delta }+ \beta ^{\frac{1}{1-\beta }}\right] ^{1-\beta } \end{aligned}$$

Similarly, $\alpha_{2}$ solves $f(y_{q},\alpha_{2};\delta )=\frac{\alpha_{2}\delta }{n}(1-y_{q})-(1-\delta )y_{q}^{\beta }=0$ and therefore substituting $y_{q}=\left( \beta q /\alpha \right) ^{\frac{1}{1-\beta }}$ we obtain

$$\begin{aligned} \alpha_{2}=\left[ \frac{n(1-\delta )(q\beta ) ^{\frac{\beta }{1-\beta }}}{\delta }+ (q\beta ) ^{\frac{1}{1-\beta }}\right] ^{1-\beta } \end{aligned}$$

It is immediate to see that $\alpha_{1}>\beta$ and $\alpha_{2}>\alpha_{1}$. $\square$

Proof (Proof of Proposition 2)

By Proposition 1, $\alpha_{1}\rightarrow \beta$ and $\alpha_{2}\rightarrow \beta nQ$ when $\delta \rightarrow 1$. Thus, in the limiting case ($\delta \rightarrow 1$), SSPE proposals given by Proposition 1 can be written as:

• (a) $\mathbf {x}^{*}=(0,0,1)$ if $n>n^{*}$,

• (b) $\mathbf {x}^{*}=(1-(q-1)x-y_{q},x,y_{q})$ if $n\le n^{*}$ where $x=\frac{1}{\alpha }f\left( y_{q},\alpha ;\delta \right)$.

When $n_{o}\ge n^{*}$ legislators cannot increase their utility by adding new members to the legislature. Thus, $n_{f}\ge n_{o}$ when $n_{o}\ge n^{*}$.

When $n_{o}<n^{*},\, y_{q}=\left( \beta n_{o}Q/\alpha \right) ^{\frac{1}{1-\beta } }$ and $x=\frac{1-\left( \beta n_{o}Q/\alpha \right) ^{\frac{1}{1-\beta }}}{n_{o}}$, thus the legislators’ utility is

$$\begin{aligned} u(n_{o})=\alpha \frac{1-\left( \beta n_{o}Q/\alpha \right) ^{\frac{1}{1-\beta }}}{n_{o}}+\left( \beta n_{o}Q/\alpha \right) ^{\frac{\beta }{1-\beta }}. \end{aligned}$$

Let $W(n)=nu(n)$. Since $W^{\prime }\left( n\right) >0,\, W^{\prime \prime }\left( n\right) >0,\, W\left( 1\right) >0,\, W\left( n^{*}\right) =n^{*}$, and

$$\begin{aligned} W^{\prime }\left( n^{*}\right) =\frac{1}{1-\beta }\left( \frac{\beta Q}{\alpha }\right) ^{\frac{\beta }{1-\beta }}\left[ 1-\beta Q\right] \left( \frac{\alpha }{\beta Q}\right) ^{\frac{\beta }{1-\beta }}=\frac{1-\beta Q}{1-\beta }>1, \end{aligned}$$

then $W\left( n\right) =n$ has exactly one solution $\widehat{n}<n^{*}$. Notice that a fixed point of $W\left( n\right)$ holds that $u(n)=1$. So, by the convexity of $W$, we have that $u(n_{o})<1$ only when $n_{o}\in (\widehat{n},n^{*})$. Moreover, when $n_{o}<\widehat{n} ,\, u(n_{o})>u\left( n_{f}\right)$ for all $n_{f}>n_{o}$, so legislators do not have any incentives to accept new members into the legislature. In consequence, only when $n_{o}\in (\widehat{n},n^{*})$ will the legislators have any incentives to increase the size of their legislature so that $n_{o}<n_{f}$, where $n_{f}\ge n^{*}$.

When $Q=1$, then

$$\begin{aligned} u(n) = \left\{ {\begin{array}{*{20}l}&1 &\qquad {{\text{if }}n \ge \alpha /\beta } \\&{\alpha \frac{{1 - \left( {\beta n/\alpha } \right)^{{\frac{1}{{1 - \beta }}}} }}{n} + \left( {\beta n/\alpha } \right)^{{\frac{\beta }{{1 - \beta }}}} } &\qquad {{\text{otherwise}}{\text{.}}}\\ \end{array} } \right. \end{aligned}$$

Hence, $n_{f}\ge n_{o}$ when $n_{o}\ge \alpha /\beta$ because legislators cannot increase their utility by adding new members to the legislature; when $n_{o}<\alpha /\beta$ the members of the original legislature will not have any incentives to add new members because $W^{\prime }\left( n^{*}\right) =1$, implying $u^{\prime }(n_{o})<0$ (due to the convexity of $W$).

Finally, we show that $\widehat{n}<\alpha /\beta$. This holds if and only if

$$\begin{aligned} u(\alpha /\beta )=\beta \left( 1-Q^{\frac{1}{1-\beta }}\right) +Q^{\frac{\beta }{1-\beta }}\equiv V(Q)<1. \end{aligned}$$

Since $V^{\prime }(Q)>0$ and $V(1)=1$, it can be concluded that $V(Q)<1$ for any $Q\in (0,1)$. $\square$

Proof (Proof of Proposition 3)

[Proof of Proposition 3]In the linear model of VW, where $Qn=\left( n+1\right) /2$, three cases can be distinguished in the limiting case ($\delta \rightarrow 1$): (1) $n_{o}\ge \alpha /\beta Q$, (2) $n_{o}\in \left[ \alpha /\beta ,\alpha /\beta Q\right)$, and (3) $n_{o}<\alpha /\beta$. In the first case, the efficient and equilibrium provisions of the collective good are both equal to $1$. Moreover, notice that adding new members to the legislature does not change the utility of the original legislators, so legislatures within this size range are stable. In case (3), the efficient and equilibrium provisions of the collective good are both equal to $0$ and the expected particularistic share of any legislator is $1/n_{o}$. Moreover, $u\left( 0,0,1\right) <u\left( {\hat{\mathbf{x}}}\right)$ where ${\hat{\mathbf{x}} }=\left( 1/n_{o},...,1/n_{o},0\right)$, so that the legislature is stable. In case (2), the equilibrium outcome $y=0$ is inefficient (under-provision). Within this range, legislators obtain an expected utility $\alpha /n_{o}$ smaller than $\beta$, thus they have incentives to add new members to the legislature until the new size is greater or equal to $\alpha /\beta Q$, where the whole budget will be assigned to the collective good in equilibrium. Hence, this is only possible when $N\ge \alpha /\beta Q$. If this holds, legislatures are not stable when $n_{o}\in \left[ \alpha /\beta ,\alpha /\beta Q\right)$. Otherwise, legislatures of size $n_{o}\in \left[ \alpha /\beta ,\alpha /\beta Q\right)$ are stable and yield inefficient bargaining outcomes. $\square$

Endogenous size of the legislature when $\delta <1$

Consider the notation introduced in Sect. 4.1, where $nQ=q$. When $\delta \rightarrow 1,\, y^{*}\left( n\right) =\min \left\{ 1,\left( \beta Qn/\alpha \right) ^{\frac{1}{1-\beta }}\right\}$ whereas Pareto efficiency requires a provision of the collective good of $y^{0}\left( n\right) =\min \left\{ \left( \beta n/\alpha \right) ^{\frac{1}{1-\beta }},1\right\}$ . By assuming $\delta <1$ the collective good provision displays the four elements represented in Proposition 1, which complicates the analysis and makes less clear predictions. Nevertheless, we next specify the conditions required to obtain Pareto improvements by increasing the size of the legislature.

As noted, adding new members to the legislature may mean that they appropriate part of the budget, so the legislators will be willing to incorporate new members only if the resulting provision of the collective good increases. The size of the legislature affects $\alpha_{1},\, \alpha_{2},\, \widehat{y}$, and $y^{*}\left( n\right)$ defined in Proposition . Specifically, $\frac{\partial \alpha_{1}\left( n\right) }{\partial n }>0$, $\frac{\partial \alpha_{2}\left( n\right) }{\partial n}>0,\, \frac{ \partial \widehat{y}\left( n\right) }{\partial n}<0$ and $\frac{\partial y^{*}\left( n\right) }{\partial n}\ge 0.$

Suppose that $\alpha >\beta,$ so that the provision of the collective good is affected by the (endogenous) size of the legislature. In these cases, the utility of the committee members is given by

$$u\left( n \right) = \left\{ {\begin{array}{*{20}l} {\alpha \frac{{1 - \left( {\beta /\alpha } \right)^{{\frac{1}{{1 - \beta }}}} }}{n} + \left( {\beta /\alpha } \right)^{{\frac{\beta }{{1 - \beta }}}} } & {{\text{if }}\alpha \le \alpha _{1} \left( n \right)} \\ \alpha \frac{{1 - \hat{y}\left( n \right)}}{n} + \hat{y}^{\beta } \left( n \right) & {\text{if }}\alpha \in \left( {\alpha _{1} \left( n \right),\alpha _{2} \left( n \right)} \right) \hfill\\ \alpha \frac{{1 - \left( {\beta nQ/\alpha } \right)^{{\frac{1}{{1 - \beta }}}} }}{n} + \left( {\beta nQ/\alpha } \right)^{{\frac{\beta }{{1 - \beta }}}} & {\text{otherwise}}{\text{.}} \hfill \\ \\ \end{array} } \right.$$

Since $\widehat{y}$ is attained whenever $\alpha \in \left[ \alpha_{1}\left( n\right) ,\alpha_{2}\left( n\right) \right]$ and the collective good is constant when $\alpha <\alpha_{1}\left( n\right),$ it is immediate to see that the original legislature would never increase its size when $\alpha \le \alpha_{2}\left( n_{o}\right)$ since any increase in $n$ can only reduce the provision of the collective good. This is formally stated in the next result.

Lemma 5

For any $\alpha \le \alpha_{2}\left( n\right) ,\, \partial u\left( n\right) /\partial n<0.$

Proof

When $\alpha \le \alpha_{1},$

$$\begin{aligned} \frac{\partial \left( \alpha \dfrac{1-\left( \beta /\alpha \right) ^{\frac{1 }{1-\beta }}}{n}+\left( \beta /\alpha \right) ^{\frac{\beta }{1-\beta } }\right) }{\partial n}<0 \end{aligned}$$

Moreover, when $\alpha \in \left( \alpha_{1},\alpha_{2}\right) ,$

$$\begin{aligned} \frac{{\partial \left( {\alpha \frac{{1 - \hat{y}}}{n} + \hat{y}^{\beta } } \right)}}{{\partial n}} & & & = \alpha \frac{{ - n\frac{{\partial \hat{y}}}{{\partial n}} - \left( {1 - \hat{y}} \right)}}{{n^{2} }} + \beta \hat{y}^{{\beta - 1}} \frac{{\partial \hat{y}}}{{\partial n}} \\ & & & \; = - \frac{{\alpha \left( {1 - \hat{y}} \right)}}{n}\frac{1}{n} + \left[ {\beta \hat{y}^{{\beta - 1}} - \frac{\alpha }{n}} \right]\frac{{\partial \hat{y}}}{{\partial n}} \\ \end{aligned}$$

Since $\frac{\partial {\widehat{y}}}{\partial n}=-\frac{\alpha \delta (1- {\widehat{y}})}{n\alpha \delta +{n^{2}}\beta (1-\delta ){\widehat{y}}^{\beta -1}},$ after some algebra we obtain

$$\begin{aligned} \frac{\partial \left( \alpha \dfrac{1-{\widehat{y}}}{n}+{\widehat{y}}^{\beta }\right) }{\partial n} = -\frac{\alpha \left( 1-{\widehat{y}}\right) \beta }{n\alpha \delta +n^{2}\beta (1-\delta ){\widehat{y}}^{\beta -1}}{\widehat{y}}^{\beta -1}<0. \end{aligned}$$

$\square$

Although legislatures are stable when $\alpha \le \alpha_{2}\left( {n_{o}}\right),$ this is not necessarily the case when $\alpha >\alpha_{2}\left( {n_{o}}\right),$ where increasing the size of the legislature might increase the provision of the collective good. Hence, legislators may be interested in selecting ${n_{f}}>{n_{o}}$ if the individual gains from such an increase outweigh the losses derived from reducing the expected particularistic share of their members.

For any given α, using the definitions of $\alpha_{1}\left( n\right)$ and $\alpha_{2}\left( n\right),$ the utility function can be written in terms of the legislature’s size as follows:

$$\begin{aligned} u\left( n \right) = \left\{ {\begin{array}{*{20}l} {\alpha \frac{{1 - \left( {\beta /\alpha } \right)^{{\frac{1}{{1 - \beta }}}} }}{n} + \left( {\beta /\alpha } \right)^{{\frac{\beta }{{1 - \beta }}}} } & {{\text{if }}n > n_{1} } \\ \alpha \frac{{1 - \hat{y}}}{n} + \hat{y}^{\beta } \hfill & {\text{if }}n \in \left( {n_{2} ,n_{1} } \right) \hfill \\ \alpha \frac{{1 - \left( {\beta nQ/\alpha } \right)^{{\frac{1}{{1 - \beta }}}} }}{n} + \left( {\beta nQ/\alpha } \right)^{{\frac{\beta }{{1 - \beta }}}} \hfill & {\text{otherwise}}{\text{.}} \hfill \\ \\ \end{array} } \right. \end{aligned}$$

where ${n_{1}}=\frac{\delta \left( \alpha ^{\frac{1}{1-\beta }}-\beta ^{\frac{1 }{1-\beta }}\right) }{(1-\delta )\beta ^{\frac{\beta }{1-\beta }}}$ and ${n_{2}}=\frac{\alpha }{\left( Q\beta \right) ^{\beta }}\left( \frac{\delta }{ (1-\delta )+\delta Q\beta }\right) ^{1-\beta}.$

From Lemma 5, we know that ${u^{\prime} }\left( n\right) <0$ for all $n>{n_{2}}.$ Thus, the legislature would be willing to add new members only if ${n_{o}}\le {n_{2}}.$ Figure 5 depicts the utility of the legislature’s members as a function of its size for a specific example. In this case, $u\left( n\right)$ attains a local maximum at $n={n_{2}}=29.085.$ Thus, there exists a range of values of ${n_{o}}<n_{2}$ where increasing the committee size to ${n_{2}}$ would generate a Pareto improvement. The following lemma states the condition required for this property to hold.

Fig. 5

Expected utility of a legislature member as a function of $n$ when $\left( \alpha ,\beta ,Q\right) =\left( 8,0.5,0.5\right)$ and $\delta =0.95.$ The legislature would modify its size when ${n_{o}}\in \left( 11.736,29.085\right).$ Not surprisingly, this range is reduced with respect to the example represented by Fig. 4 in Sect. 4.1 where ${n_{o}}\in \left( 10.667,32\right)$ when $\delta \rightarrow 1$

Lemma 6

Given $Q,\, \alpha$ and β, if $\delta >{\widetilde{\delta} }=\frac{1-\beta}{ 1-Q\beta}$ then ${\widetilde{n}} \equiv {\mathrm{argmin} }_{n\le n_{2}}u(n)<n_{2}.$

Proof

When $n\ge n_{2},$

$$\begin{aligned} \frac{\partial u(n)}{\partial n}&= \frac{1}{n^{2}\left( 1-\beta \right) } \left( -\alpha +\alpha \beta -\alpha \frac{\beta }{\left( Q\frac{n}{\alpha } \beta \right) ^{\frac{1}{\beta -1}}}+n\frac{\beta }{\left( Q\frac{n}{\alpha } \beta \right) ^{\frac{\beta }{\beta -1}}}\right) \\&= \frac{1}{n^{2}\left( 1-\beta \right) }\left( \alpha ^{-\frac{\beta }{ 1-\beta }}n^{\frac{1}{1-\beta }}\beta ^{\frac{1}{1-\beta }}Q^{\frac{\beta }{ 1-\beta }}\left[ 1-Q\beta \right] -\alpha \left( 1-\beta \right) \right) , \end{aligned}$$

which equals zero if

$$\begin{aligned} n^{\frac{1}{1-\beta }}&= \frac{\alpha \left( 1-\beta \right) }{\left( 1-Q\beta \right) }\alpha ^{\frac{\beta }{1-\beta }}\beta ^{\frac{-1}{1-\beta }}Q^{\frac{-\beta }{1-\beta }}=\frac{\alpha \left( 1-\beta \right) }{\left( 1-Q\beta \right) }\alpha ^{\frac{\beta }{1-\beta }}\beta ^{\frac{-\beta }{ 1-\beta }}\beta ^{-1}Q^{\frac{-\beta }{1-\beta }} \\ n&= \widetilde{n}=\left[ \frac{\alpha \left( 1-\beta \right) }{\beta \left( 1-Q\beta \right) }\right] ^{\frac{1}{1-\beta }}\left( \frac{\alpha }{Q\beta } \right) ^{\beta }. \end{aligned}$$

Note that

$$\begin{aligned} {u^{\prime} }\left( n_{2}\right)&= \frac{1}{n_{2}^{2}\left( 1-\beta \right) } \left( -\alpha +\alpha \beta -\alpha \frac{\beta }{\left( Q\frac{n_{2}}{ \alpha }\beta \right) ^{\frac{1}{\beta -1}}}+n_{2}\frac{\beta }{\left( Q \frac{n_{2}}{\alpha }\beta \right) ^{\frac{\beta }{\beta -1}}}\right) \\&= \frac{1}{n_{2}^{2}\left( 1-\beta \right) }\alpha \left( \beta \left( 1-Q\beta \right) \left( \frac{\delta }{Q\beta \delta -\delta +1}\right) -\left( 1-\beta \right) \right) , \end{aligned}$$

which is positive iff $\delta >{\widetilde{\delta} }=\frac{1-\beta }{1-Q\beta }.$

Thus, since $\lim _{n\rightarrow 0}u(n)=\infty$ we obtain that ${\widetilde{n}} \equiv {\mathrm{argmin} }_{n\le n_{2}}u(n)<n_{2}$ iff $\delta >{\widetilde{\delta} }=\frac{1-\beta }{1-Q\beta }.$ $\square$

From the previous lemma, $u{( n) ^{\prime }}>0$ if and only if $n\in \left( {\widetilde{ n}},n_{2}\right),$ which is possible only when $\delta >{\widetilde{\delta}}.$ In the rest of cases, $u\left( n\right) ^{\prime }<0.$ Since $\lim _{n\rightarrow 0}u(n)=\infty,$ by the continuity of u, there exists an ${\widehat{n}}<\tilde{n}$ such that $u({\widehat{n}})=u(n_{2}).$ Thus, legislators would never add new members if $\delta \le {\widetilde{\delta}}$ or if $n_{o}\notin \left( {\widehat{n}},n_{2}\right)$ when $\delta > {\widetilde{\delta}}.$ In contrast, the size of the legislature increases up to $n_{f}=n_{2}$ when $n_{o}\in \left( {\widehat{n}},n_{2}\right)$ and $\delta > {\widetilde{\delta} }$ (whenever this is possible; that is, assuming $N\ge n_{2}$). In the latter case, the collective good provision of the resulting legislature increases. This mitigates the original inefficiency because there is a Pareto improvement. However, neither $y^{0}\left( n_{o}\right)$ nor $y^{0}\left( n_{f}\right)$ is (generically) attained in equilibrium. This contrasts with the limiting case ($\delta \rightarrow 1$) where $y^{0}\left( n_{f}\right) =1$ could be achieved. Nonetheless, as discussed, in that case $y^{0}\left( n_{f}\right)$ does not coincide with $y^{0}\left( n_{o}\right) <1.$