Electoral systems and economic growth

Abstract

Electoral systems are rules through which votes translate into seats in parliament. The political economy literature tells us that alternative electoral systems can generate different distributions of power among different social groups in the legislature and therefore lead to different equilibrium economic policies. On the other hand, we know from the endogenous economic growth literature that economic policy can affect growth. What the literature is lacking is a clear link between electoral systems and economic growth. The main objective of this paper is to establish a connection between them. Two main results emerge from our model. First, electoral systems matter for economic growth. Second, the way in which they matter is not straightforward. A precise ranking of these political institutions in terms of economic growth requires the knowledge of the distribution of people among different social classes in society.

Appendices

No social mobility property

We have to prove that $w_{t}^{p}<w_{t}^{m}<w_{t}^{r}$$\forall t$. This is true for period 0 by assumption. By complete induction we will prove that this is also true for any $t>0$.

First, let us prove that if $w_{0}^{p}<w_{0}^{m}<w_{0}^{r}$ then $w_{1}^{p}<w_{1}^{m}<w_{1}^{r}$. From (10) we have that $\frac{w_{t+1}^{i}}{ w_{t+1}^{m}}=\frac{A_{i}}{A_{m}}\left( \frac{w_{t}^{i}-g_{t}}{w_{t}^{m}-g_{t} }\right) ^{\alpha }$, then $\frac{w_{1}^{i}}{w_{1}^{m}}=\frac{A_{i}}{A_{m}} \left( \frac{w_{0}^{i}-g_{0}}{w_{0}^{m}-g_{0}}\right) ^{\alpha }$. Since $A_{p}<A_{m}<A_{r}$ and $w_{0}^{p}<w_{0}^{m}<w_{0}^{r}$, then $\frac{w_{1}^{r} }{w_{1}^{m}}=\frac{A_{r}}{A_{m}}\left( \frac{w_{0}^{r}-g_{0}}{w_{0}^{m}-g_{0} }\right) ^{\alpha }>1$ and $\frac{w_{1}^{p}}{w_{1}^{m}}=\frac{A_{p}}{A_{m}} \left( \frac{w_{0}^{p}-g_{0}}{w_{0}^{m}-g_{0}}\right) ^{\alpha }<1$. Therefore $w_{1}^{p}<w_{1}^{m}<w_{1}^{r}$.

Second, following the same steps as before it can be proved that if $w_{t-1}^{p}<w_{t-1}^{m}<w_{t-1}^{r}$, given that $A_{p}<A_{m}<A_{r}$, then $w_{t}^{p}<w_{t}^{m}<w_{t}^{r}$.

Then by complete induction, $w_{t}^{p}<w_{t}^{m}<w_{t}^{r}$$\forall t$.

Propositions 2 and 4

1.1 Existence of a steady-growth equilibrium

First note that because [from (13)] $w_{t}^{m}$ is always growing at rate $\mu _{m}-1$, if the ratio $\frac{w_{t}^{m}}{w_{t}^{i}}$ is constant then $w_{t}^{i}$ must also be growing at rate $\mu _{m}-1$. Using this argument we will prove that under some conditions an equilibrium exist.

From (13) and (14) we have that

$$\begin{aligned} \frac{\frac{w_{t+1}^{i}}{w_{t}^{i}}}{\frac{w_{t+1}^{m}}{w_{t}^{m}}}=\frac{ cA_{i}}{\mu _{m}}\left( 1-\frac{\rho (1-\alpha )}{1+\rho }\frac{w_{t}^{m}}{ w_{t}^{i}}\right) ^{\alpha }\left( \frac{w_{t}^{m}}{w_{t}^{i}}\right) ^{1-\alpha }, \end{aligned}$$

(17)

and this expression can be rewritten as

$$\begin{aligned} x_{t+1}^{i}=-d+b_{im}\left( x_{t}^{i}\right) ^{\alpha }, \end{aligned}$$

(18)

where $d\equiv \frac{\rho (1-\alpha )}{1+\rho }$, $b_{im}\equiv \frac{cA_{i} }{\mu _{m}}=\frac{A_{i}}{A_{m}}\left( \frac{1+\rho }{1+\rho \alpha }\right) ^{\alpha }$ and $x_{t}^{i}\equiv \frac{w_{t}^{i}}{w_{t}^{m}}-d$. Note that $0<d<1/2$, and $b_{im}>0$.

Let us analyze this non-linear first order difference equation. First note that because $0<\alpha <1$, this function is concave, and so we can have from 0 to 2 equilibria. Call $f(x_{t}^{i})=-d+b_{im}\left( x_{t}^{i}\right) ^{\alpha }$. The bifurcation point (where we pass from 0 to 2 equilibria), is defined by $f^{\prime }(x_{t}^{i})=1$, and takes the values: $(x_{t}^{i},x_{t+1}^{i})=\left( (\alpha b_{im})^{\frac{1}{1-\alpha }},(\alpha b_{im}{})^{\frac{1}{1-\alpha }}\right)$. Therefore, the condition for existence of an equilibrium is $f\left( (\alpha b_{im})^{\frac{1}{1-\alpha } }\right) \ge (\alpha b_{im})^{\frac{1}{1-\alpha }}$, or $-d+b_{im}(\alpha b_{im})^{\frac{\alpha }{1-\alpha }}\ge (\alpha b_{im})^{\frac{1}{1-\alpha } }$. In terms of parameters this condition is equivalent to: $\frac{A_{m}}{ A_{i}} \le \frac{1+\rho }{\rho }\left( \frac{\rho \alpha }{1+\rho \alpha } \right) ^{\alpha }$ for $i=r,p$.

Note that because $A_{r}>A_{p}$, it is enough to impose the condition $\frac{ A_{m}}{A_{p}} \le \frac{1+\rho }{\rho }\left( \frac{\rho \alpha }{1+\rho \alpha }\right) ^{\alpha }$ to guarantee the existence of an equilibrium in our model (Condition 1). Note that for $\rho ,\alpha \in (0,1)$, $1< \frac{1+\rho }{\rho }\left( \frac{\rho \alpha }{1+\rho \alpha }\right) ^{\alpha }$. Therefore, for any value of $\rho ,\alpha$ it will exist $A_{m}$ and $A_{p}$ such that $A_{m} >A_{p}$ and $\frac{A_{m}}{A_{p}} \le \frac{1+\rho }{\rho }\left( \frac{\rho \alpha }{1+\rho \alpha }\right) ^{\alpha }$. Thus, for $A_{m}$ sufficiently close to $A_{p}$ we know that an equilibrium exists.

Assuming that this condition is satisfied with strict inequality, we will have two balanced growth equilibria, $x^{i1}$ and $x^{i2}$, both defined by the solution to the equation $x^{i}=-d+b_{im}\left( x^{i}\right) ^{\alpha }$. Because the function $f(\cdot )$ is concave, the first one, $x^{i1}$, is unstable and second one, $x^{i2}$, is stable (note that $f^{\prime }(x^{i1})>1$ while $f^{\prime }(x^{i2})<1$).

1.2 Convergence

It is easy to verify that if $x_{0}^{i}$ is at the right of $x^{i1}$, the economy will converge to $x^{i2}$. If the initial distribution of wealth implies a $x_{0}^{i}$ that is at the left of $x^{i1}$, then there is no convergence.

Note now that $x_{0}^{r} \equiv \frac{w_{0}^{r}}{w_{0}^{m}}-d$ is always at the right of $x^{r1}$. To see this note first that $\frac{w_{0}^{r}}{ w_{0}^{m}}>1$, in which case $x_{0}^{r}>1-d$. Now, if we can prove that $f\left( 1-d\right) >1-d$, then we will know that $1-d$ is at the right of $x^{r1}$ (more precisely $x^{r1}<1-d <x^{r2}$) and because $x_{0}^{r}>1-d$ then $x_{0}^{r}$ will be as well at the right of $x^{r1}$ . It can be verified that $f\left( 1-d\right) >1-d$ if and only if $\frac{ A_{r}}{A_{m}}>1$,¹⁷ and the later condition is true by definition. Thus, $x_{0}^{r}>x^{r1}$ and the relative wealth of the rich class will always converge to the stable equilibrium $x^{r2}$.

Unfortunately the same cannot be said about $x_{0}^{p}$. Note that $0<\frac{ w_{0}^{p}}{w_{0}^{m}}<1,$ therefore $x_{0}^{p}$ is restricted by definition to the interval $(-d,1-d),$ i.e. $-d<x_{0}^{p}\equiv \frac{w_{0}^{p}}{ w_{0}^{m}}-d<1-d$. To ensure convergence it is necessary to impose additionally the condition $x^{p1}<x_{0}^{p}$ (Condition 2). The question is, does such $x_{0}^{p}$ exist? The answer is yes it does. We must prove that exist $x_{0}^{p}$ such that $x^{p1}\le x_{0}^{p}<1-d$. In other words we must prove that $x^{p1}<1-d$ or that the interval $(x^{p1},1-d)$ is not empty.

First, it can be verified that $f(1-d)<1-d$ if $\frac{A_{p}}{A_{m}}<1$ (and this is the case here), then $1-d<x^{p1}$ or $1-d>x^{p2}$. Second, the following inequality $(\alpha b_{pm})^{\frac{1}{1-\alpha }}<1-d$ is also always verified for $\frac{A_{p}}{A_{m}}<1$,¹⁸ and $x^{p1}< (\alpha b_{pm})^{\frac{1}{1-\alpha } }$ (since we are assuming $f\left( (\alpha b_{pm})^{\frac{1}{1-\alpha } }\right) >(\alpha b_{pm})^{\frac{1}{1-\alpha }}$ and therefore $x^{p1}<(\alpha b_{pm})^{\frac{1}{1-\alpha }}<x^{p2}$ ), then $x^{p1}< 1-d$ (in fact $x^{p1}<x^{p2}< 1-d$). Therefore, the interval $(x^{p1},1-d)$ is not empty. QED

1.3 Growth rates of other variables

All the variables are linked to wealth. We will now prove that if wealth is growing at rate $\mu _{m}-1$, then all the other variables of the economy will grow at the same rate.

1.4 Growth rate of $g_{t}$

Note that

$$\begin{aligned} g_{t}=g_{t}^{m}=\frac{\rho }{1+\rho }(1-\alpha )w_{t}^{m}. \end{aligned}$$

(19)

Dividing this expression by the equation lagged one period, and using $\frac{ w_{t}^{m}}{w_{t-1}^{m}}=\mu _{m},$ we have that

$$\begin{aligned} \frac{g_{t}}{g_{t-1}}=\frac{w_{t}^{m}}{w_{t-1}^{m}}=\mu _{m}. \end{aligned}$$

(20)

1.5 Growth rate of $k_{t}^{i}$for $i=p,m,r$

From (7) we have that

$$\begin{aligned} k_{t}^{i}=\frac{\rho \alpha }{1+\rho \alpha }(w_{t}^{i}-g_{t}). \end{aligned}$$

(21)

Now, using the fact that $g_{t}=\mu _{m}g_{t-1}$ [Eq. (20)] and $w_{t}^{i}=\mu _{m}w_{t-1}^{i}$, the 1 + growth rate of the investment is

$$\begin{aligned} \frac{k_{t}^{i}}{k_{t-1}^{i}}=\frac{(w_{t}^{i}-g_{t})}{(w_{t-1}^{i}-g_{t-1})} =\mu _{m}\frac{(w_{t-1}^{i}-g_{t-1})}{(w_{t-1}^{i}-g_{t-1})}=\mu _{m}. \end{aligned}$$

(22)

1.6 Growth rate of $y_{t}^{i}$for $i=p,m,r$

Because the production functions have constant returns to scale with respect to $k_{t}^{i}$ and $g_{t}$ and both variables are growing at the same rate $\mu _{m}-1$, $y_{t}^{i}$ will grow at this rate.

Properties of the indirect utility functions

The indirect utility function for an individual of class i for $w_{t}^{i}>g_{t}$ is

$$\begin{aligned} v_{t}^{i}(w_{t}^{i},g_{t})=D_{i}+(1+\rho \alpha )\log (w_{t}^{i}-g_{t})+\rho (1-\alpha )\log (g_{t}). \end{aligned}$$

(23)

Because $D_{p}<D_{m}<D_{r}$ (since $A_{p}<A_{m}<A_{r}$) and $w_{t}^{p}<w_{t}^{m}<w_{t}^{r}$ we will have that $v_{t}^{p}(w_{t}^{p},g_{t})<v_{t}^{m}(w_{t}^{m},g_{t})<v_{t}^{r}(w_{t}^{r},g_{t})$, i.e. for a given level of $g_{t}$ the utility is higher as higher is the wealth.

We already know that the indirect utility functions are concave and have a maximum at $g_{t}^{i}=\frac{\rho (1-\alpha )}{1+\rho }w_{t}^{i}$, then $g_{t}^{p}<g_{t}^{m}<g_{t}^{r}$. Additionally, $\underset{g_{t} \longrightarrow w_{t}^{i}}{\lim }v_{t}^{i}(w_{t}^{i},g_{t})=-\infty$, and when $g_{t}=0$, $u_{t}^{i}=\log (w_{t}^{i})$ (since $k_{t}^{i}=y_{t}^{i}=0$).

$\bar{g_{t}^{p}}< \bar{g_{t}^{m}} <\bar{g_{t}^{r}}$

We will prove that ${\bar{g_{t}^{p}}}<{\bar{g_{t}^{m}}}<{\bar{g_{t}^{r}}}$.

Totally differentiating $\log (w_{t}^{i})=D_{i}+(1+\rho \alpha )\log (w_{t}^{i}-g_{t})+\rho (1-\alpha )\log (g_{t})$ we can find the effect of an increase in wealth on ${\bar{g_{t}^{i}}}$ (this is valid even when the increase is from $w_{t}^{p}$ to $w_{t}^{m}$ or to $w_{t}^{r}$ because the indirect utility functions are identical, except for the term $D_{i}$, then we just have to take into account the additional increase due to changes in $D_{i}$ when we change classes)

$$\begin{aligned} \frac{dg_{t}}{dw_{t}^{i}}_{g_{t}={\bar{g_{t}^{i}}}}=\frac{\left( \frac{ 1+\rho \alpha }{w_{t}^{i}-{\bar{g_{t}^{i}}}}-\frac{1}{w_{t}^{i}} \right) }{\left( \frac{1+\rho \alpha }{w_{t}^{i}-{\bar{g_{t}^{i}}}}- \frac{\rho (1-\alpha )}{{\bar{g_{t}^{i}}}}\right) }>0. \end{aligned}$$

(24)

Note that the numerator is always positive since $1+\rho \alpha >1$ and $w_{t}^{i}-{\bar{g_{t}^{i}}}<w_{t}^{i}$, and the denominator is also positive because ${\bar{g_{t}^{i}}}>g_{t}^{i}=\frac{\rho (1-\alpha )}{ 1+\rho }w_{t}^{i}$ (note that at $g_{t}=g_{t}^{i}$ the denominator is zero, from there up, i.e. for ${\bar{g_{t}^{i}}}>g_{t}^{i}$, is positive). When we go from $i=p$ to $i=m$ , i.e. from $w_{t}^{p}$ to $w_{t}^{m}$, we have then that ${\bar{g_{t}^{m}}}>{\bar{g_{t}^{p}}}$ (as mentioned above the increase will be larger than (24) suggests because we have to add an additional positive effect due to the increase in $D_{i}$).

Proposition 5

Similar to Propositions 2 and 4.

$$\begin{aligned} w_{t}^{p}<w_{t}^{m}<w_{t}^{r}\quad \forall t. \end{aligned}$$

See no social mobility property above.

1.1 Existence

Now the condition will be $\frac{A_{p}}{A_{m}} \le \frac{1+\rho }{\rho } \left( \frac{\rho \alpha }{1+\rho \alpha }\right) ^{\alpha }$. But note that this condition is always verified since $\frac{A_{p}}{A_{m}}<1$ and $\frac{ 1+\rho }{\rho }\left( \frac{\rho \alpha }{1+\rho \alpha }\right) ^{\alpha }>1$ for $\rho ,\alpha \in (0,1)$.

1.2 Convergence

Note that both $\frac{w_{0}^{m}}{w_{0}^{p}},\frac{w_{0}^{r}}{w_{0}^{p}}>1$. Thus, following the same lines of reasoning that in the proof of Propositions 2 and 5 it can be shown that we always begin at the right of the unstable equilibrium. Therefore, there is always convergence to the stable equilibrium.

1.3 Growth rates of other variables

All variables are growing at rate $\mu _{p}-1$. Proof: same as in Propositions 2 and 4.

Proposition 6

Similar to Propositions 2 and 4.

This case is the most demanding in terms of the conditions that are necessary to verify for existence and convergence to equilibrium.

$$\begin{aligned} w_{t}^{p}<w_{t}^{m}<w_{t}^{r}\quad \forall t. \end{aligned}$$

See no social mobility property above.

1.1 Existence

Now the condition will be $\frac{A_{r}}{A_{p}} \le \frac{1+\rho }{\rho } \left( \frac{\rho \alpha }{1+\rho \alpha }\right) ^{\alpha }$.

1.2 Convergence

Note that both $\frac{w_{0}^{p}}{w_{0}^{r}},\frac{w_{0}^{m}}{w_{0}^{r}}<1$. Therefore for convergence we must impose the condition that both $\frac{ w_{0}^{p}}{w_{0}^{r}}-d$ and $\frac{w_{0}^{m}}{w_{0}^{r}}-d$ are at the right of the unstable equilibrium.

1.3 Growth rates of other variables

All variables are growing at rate $\mu _{r}-1$. Proof: same as in Propositions 2 and 4.

Alternative default policy

Assume that the middle class has plurality. For period 0 this class will be able to choose $g_{0}=g_{0}^{m}$, because for the rich class $g_{0}^{m}$ is better than the default policy $g_{0}=0.$ It is easy to prove this, just note that because we are in the increasing region of the utility function, any public investment $g_{0}$ strictly greater than 0 and smaller than $g_{0}^{r}$ is better than the default public policy $g_{0}=0$. Note that 0 $<g_{0}^{m}< g_{0}^{r}$ then $g_{0}^{m}$ is preferred to 0.

Next for $t=1$, note that because we are assuming that the parameters are such that there is economic growth, i.e. $w_{1}^{m}>w_{0}^{m},$ then $g_{1}^{m}=\frac{\rho (1-\alpha )}{1+\rho }w_{1}^{m}>g_{0}^{m}=\frac{\rho (1-\alpha )}{1+\rho }w_{0}^{m}.$ Additionally, by the no-social-mobility property $w_{1}^{r}>w_{1}^{m}$ (Appendix A), therefore $g_{1}^{r}=\frac{\rho (1-\alpha )}{1+\rho }w_{1}^{r}>g_{1}^{m}=\frac{\rho (1-\alpha )}{1+\rho } w_{1}^{m}$. Therefore $g_{0}^{m}<g_{1}^{m}<g_{1}^{r}$ and then for the same argument as before the rich class will prefer $g_{1}^{m}$ to $g_{0}^{m}$.

Following the same steps as before we can prove that the rich class will prefer $g_{t}^{m}$ to $g_{t-1}^{m}$ as long as $g_{t-1}^{m}<g_{t}^{m}<g_{t}^{r}$ (and this is true by the no-social-mobility property).

By complete induction this will be true for any t.

Following the same reasoning as before, we can prove that if the poor class has plurality, then for both m and r is better to accept $g_{t}^{p}$ in period t than the default policy $g_{t-1}^{p}$.