## 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.

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## Notes

Note that this is not the same as to care for their offspring’s utility (i.e. altruist motive). See Abel and Warshawsky (1988) for a discussion of the links between joy of giving motive and altruism.

Probably the best way of interpret it is as a composite index of physical and human capital.

Implicitly we are assuming some kind of congestion. As Barro and Sala-i-Martin (1992) points out most of public goods and services suffer from some kind of congestion, and this is typically the case with roads and education. However, note that because there is no population growth in our model, this is not an important assumption.

Again, it can be verified that the second order condition for a maximum is in place.

Remember that production is the only way of transferring resources from one period to the other.

Of course we can classify electoral systems according to a vector of characteristics; but as Norris (1997, p. 299) points out even when we can include for this classification “district magnitude, ballot structures, effective thresholds, malapportionment, assembly size, and open/closed lists, ...the most important variations concern electoral formulas”.

Colomer (2004, p. 10) also describes the characteristics of these systems along similar lines:

“...electoral systems based on the majority principle, ...tend to produce a single, absolute winner and subsequent absolute losers, ...proportional representation, [is] a principle forged to create multiple partial winners and much fewer losers than majority rule.”

This assumption implies that each citizen votes for the policy (or candidate) that brings him/her the maximum utility, ignoring the possible effects that his/her decision and those of others could have on the election outcome. We can justify the sincere voting assumption by noticing that we have a continuum of (or infinite) agents in our model and therefore, the probability of an agent being pivotal tends to zero. Then voters vote for their first best option without any strategic consideration.

Note that each class under PR can can win up to \(s^{i}\) seats of the parliament.

Note that the number of candidates of class

*i*will be greater than \( s^{i}\delta \). \(s^{i}\delta \) candidates will be elected with probability \( q^{i}=1\), and this will imply that each candidate will get an expected gain of \(Z-C>0\). However, if there is free entry and perfect competition, new candidates will arrive until \(q^{i}Z-C=0\) in which case the number of candidates must be greater than \(s^{i}\delta \). This implies that there are enough candidates as to elect \(s^{i}\delta \).The party that has plurality in parliament is usually the party that is in charge of executive power and usually has a prerogative over budget requests. Therefore it is reasonable to assume that the party that has plurality is the one with agenda-setting power. Posner and Park (2007, p. 5–6) discussing trends in budgeting point out that: “Legislatures themselves delegated powers to the executive, wary of their own instincts to favour particular constituency-based policies at the expense of the broader fiscal wellbeing of the country. Moreover, legislatures did not have expertise to keep up with the growing sophistication and complexity of modern budgets, particularly when compared to the detailed knowledge possessed by executive bureaucracies (Schick, 2002).... The eclipse of the legislative role in budget formulation was reflected in the limited formal roles legislatures were given in developing and approving budgets. Legislatures had little formal power to review or approve overarching budget targets or policies, nor were legislatures generally involved in approving medium-term expenditure frameworks.”

These 2 equilibria are defined by 2 different \(\frac{w_{t}^{p}}{w_{t}^{m}}\) ratios. The equilibrium defined by the smallest ratio is unstable, while the other is stable.

These 2 equilibria are defined by 2 different \(\frac{w_{t}^{p}}{w_{t}^{m}}\) ratios. The equilibrium defined by the smallest ratio is unstable, while the other is stable.

Conditions 2 and 5 are enough to avoid the disappearance of a class (i.e. the case where \(T_{t}\ge w_{t}^{i}\) for some

*i*).If condition 3 is verified with strict inequality there will be 4 equilibria. These 4 equilibria are defined by the 4 possible combinations of 2 equilibrium values of \(\frac{w_{t}^{p}}{w_{t}^{r}}\) and 2 equilibrium values of \(\frac{w_{t}^{m}}{w_{t}^{r}}\). Only the equilibrium defined by the maximum value of both ratios is stable.

It is important to notice that in our model higher growth does not imply necessarily higher welfare, so our theory should not be interpreted as an argument in favor of a pro-rich class government. Distributional considerations are important for welfare. It is easy to show that under an utilitarian social planner the growth will be: \(\mu -1=A_{r}\left[ s^{p}\left( \frac{A_{p}}{A_{r}}\right) ^{1/(1-\alpha )}+s^{m}\left( \frac{ A_{m}}{A_{r}}\right) ^{1/(1-\alpha )}+s^{r}\right] ^{1-\alpha }E-1.\)

\(f(1-d)>1-d\iff -d+b_{rm}(1-d)^{\alpha }>1-d\iff b_{rm}(1-d)^{\alpha }>1\)\(\iff \frac{A_{r}}{A_{m}}(\frac{1+\rho }{1+\rho \alpha })^{\alpha }(\frac{ 1+\rho \alpha }{1+\rho })^{\alpha }>1\iff \frac{A_{r}}{A_{m}}>1\) QED.

\(\frac{A_{p}}{A_{m}}< \frac{1/\alpha +\rho }{1+\rho }\Leftrightarrow \frac{A_{p}}{A_{m}}<\frac{1}{ \alpha }\frac{1+\rho \alpha }{1+\rho }\)

\(\Leftrightarrow \frac{A_{p}}{A_{m}}<\frac{1}{\alpha }\left( \frac{1+\rho \alpha }{1+\rho }\right) ^{1-\alpha }\left( \frac{1+\rho \alpha }{1+\rho } \right) ^{\alpha }\Leftrightarrow \alpha \frac{A_{p}}{A_{m}}\left( \frac{ 1+\rho }{1+\rho \alpha }\right) ^{\alpha }<\left( \frac{1+\rho \alpha }{ 1+\rho }\right) ^{1-\alpha }\)

\(\Leftrightarrow \left( \alpha b_{pm}\right) ^{\frac{1}{^{1-\alpha }}}=\left[ \alpha \frac{A_{p}}{A_{m}}\left( \frac{1+\rho }{1+\rho \alpha }\right) ^{\alpha }\right] ^{\frac{1}{^{1-\alpha }}}<\left( \frac{1+\rho \alpha }{ 1+\rho }\right) =1-d\). Note that the first inequality is verified for \(\frac{ A_{p}}{A_{m}}<1.\)

## References

Abel, A., & Warshawsky, M. (1988). Specification of the joy of giving: Insights from altruism.

*The Review of Economics and Statistics*,*70*(1), 145–149.Acemoglu, D. (2009).

*Introduction to modern economic growth*. Princeton: Princeton University Press.Acemoglu, D. (2008). Oligarchic versus democratic societies.

*Journal of the European Economic Association*,*6*(1), 1–44.Acemoglu, D., & Robinson, J. (2000). Why did the west extend the franchise?

*Democracy, Inequality and Growth in Historical Perspective, Quarterly Journal of Economics*,*115*(4), 1167–1199.Acemoglu, D., Johnson, S., & Robinson, J. (2005). Institutions as the fundamental cause of long-run growth. Chapter 6. In P. Aghion & S. Durlauf (Eds.),

*Handbook of economic growth*. Amsterdam: Elsevier.Aghion, P., Caroli, E., & García-Peñalosa, C. (1999). Inequality and economic growth: The perspective of the new growth theories.

*Journal of Economic Literature*,*37*(4), 1615–1660.Aghion, P., & Howitt, P. (1998).

*Endogenous growth theory*. Cambridge: The MIT Press.Alesina, A., & Rodrik, D. (1994). Distributive politics and economic growth.

*The Quarterly Journal of Economics*,*109*(2), 465–490.Banerjee, A., & Duflo, E. (2008). What is middle class about the middle classes around the world?

*Journal of Economic Perspectives*,*22*(2), 3–28.Banerjee, A., & Duflo, E. (2007). The economic lives of the poor.

*Journal of Economic Perspectives*,*21*(1), 141–167.Barro, R. (1990). Government spending in a simple model of endogenous growth.

*Journal of Political Economy*,*98*(5), S103–S126.Barro, R., & Sala-i-Martin, X. (1992). Public finance in models of economic growth.

*Review of Economic Studies*,*59*(4), 645–661.Battaglini, M., & Coate, S. (2008). A dynamic theory of public spending, taxation, and debt.

*American Economic Review*,*98*(1), 201–36.Benabou, R. (1996). Inequality and growth. In B. Bernanke & J. Rotemberg (Eds.),

*NBER macroeconomic annual*. Cambridge: The MIT Press.Benhabib, J., & Rustichini, A. (1996). Social conflict and growth.

*Journal of Economic Growth*,*1*(1), 129–146.Bertola, G. (1993). Factor shares and savings in endogenous growth.

*The American Economic Review*,*83*(5), 1184–1198.Besley, T., & Coate, S. (1997). An economic model of representative democracy.

*Quarterly Journal of Economics*,*108*(1), 85–114.Besley, T., & Coate, S. (1998). Sources of inefficiency in a representative democracy: A dynamic analysis.

*The American Economic Review*,*88*(1), 139–156.Black, S., Devereux, P., & Salvanes, K. (2005). Why the apple doesn’t fall far: Understanding intergenerational transmission of human capital.

*American Economic Review*,*95*(1), 437–449.Colomer, J. (2004). Strategy and history of electoral system choice. In J. Colomer (Ed.),

*Handbook of electoral system choice*. New York: Palgrave Macmillan.Devereux, M., & Wen, J. (1998). Political instability, capital taxation, and growth.

*European Economic Review*,*42*, 1635–1651.Drazen, A. (2000).

*Political economy in macroeconomics*. Princeton: Princeton University Press.Duflo, E. (2006). Poor but rational? In A. Banerjee, R. Benabou, & D. Mookherjee (Eds.),

*Understanding poverty*. Oxford: Oxford University Press.Economides, G., Philippopoulos, A., & Price, S. (2003). How elections affect fiscal policy and growth: Revisiting the mechanism.

*European Journal of Political Economy*,*19*(4), 777–792.Funk, P., & Gathmann, C. (2013). How do electoral systems affect fiscal policy? Evidence from cantonal parliaments, 1890–2000.

*Journal of the European Economic Association*,*11*(5), 1178–1203.Glomm, G., & Ravikumar, B. (1992). Public versus private investment in human capital: Endogenous growth and income inequality.

*Journal of Political Economy*,*100*(4), 818–834.Iversen, T., & Soskice, D. (2006). Electoral institutions and the politics of coalitions: Why some democracies redistribute more than others.

*American Political Science Review*,*100*(2), 165–181.Krusell, P., & Rios-Rull, J. (1996). Vested interests in a positive theory of stagnation and growth.

*Review of Economic Studies*,*63*(2), 301–329.Leblanc, W., Snyder, J., & Tripathi, M. (2000). Majority-rule bargaining and the under provision of public investment goods.

*Journal of Public Economics*,*75*(1), 21–47.Llavador, H., & Oxoby, R. (2005). Partisan competition, growth, and the franchise.

*The Quarterly Journal of Economics*,*120*(3), 1155–1189.Malley, J., Philippopoulos, A., & Woitek, U. (2007). Electoral uncertainty, fiscal policy and macroeconomic fluctuations.

*Journal of Economic Dynamics & Control*,*31*(3), 1051–1080.Marsiliani, L., & Renström, T. (2007). Political institutions and economic growth.

*Economics of Governance*,*8*(3), 233–261.Milesi-Feretti, G., Perotti, R., & Rostagno, M. (2002). Electoral systems and the composition of government spending.

*Quarterly Journal of Economics*,*117*(2), 609–657.Norris, P. (2004).

*Electoral engineering: Voting rules and political behavior*. Cambridge: Cambridge University Press.Norris, P. (1997). Choosing electoral systems: Proportional, majoritarian and mixed systems.

*International Political Science Review*,*18*(3), 297–312.Osborne, M., & Slivinsky, A. (1996). A model of political competition with citizen-candidates.

*Quarterly Journal of Economics*,*111*(1), 65–96.Perotti, R. (1996). Income distribution, democracy, and growth: What the data say?

*Journal of Economic Growth*,*1*(2), 149–187.Perotti, R. (1993). Political equilibrium, income distribution, and growth.

*Review of Economic Studies*,*60*(4), 755–776.Persson, T. (2005).

*Forms of democracy, policy and economic development*. New York: Mimeo.Persson, T. (2004). Presidential address: Consequences of constitutions.

*Journal of the European Economic Association*,*2*(2–3), 139–161.Persson, T., Roland, G., & Tabellini, G. (2007). Electoral rules and government spending in parliamentary democracies.

*Quarterly Journal of Political Science*,*2*(1), 155–188.Persson, T., Roland, G., & Tabellini, G. (2000). Comparative politics and public finance.

*Journal of Political Economy*,*108*(6), 1121–1161.Persson, T., & Tabellini, G. (2009). Democratic capital: The nexus of political and economic change.

*American Economic Journal: Macroeconomics*,*1*(2), 88–126.Persson, T., & Tabellini, G. (2006). Democracy and Development: The Devil in the Details, American Economic Review 96, Papers and Proceedings, pp. 319–324.

Persson, T., & Tabellini, G. (2004). Constitutional rules and fiscal policy outcomes.

*American Economic Review*,*94*(1), 25–45.Persson, T., & Tabellini, G. (2000).

*Political economics. Explaining economic policy*. Cambridge: MIT Press.Persson, T., & Tabellini, G. (1994). Is inequality harmful for growth?

*American Economic Review*,*84*(3), 600–621.Posner, P., & Park, C. (2007). Role of the legislature in the budget process: Recent trends and innovations.

*OECD Journal on Budgeting*,*7*(3), 1–26.Ticchi, D., & Vindigni, A. (2010). Endogenous constitutions.

*Economic Journal*,*120*, 1–39.Tornell, A. (1997). Economic growth and decline with endogenous property rights.

*Journal of Economic Growth*,*2*(3), 219–250.Tornell, A., & Velasco, A. (1992). The tragedy of the commons and economic growth: Why does capital flow from poor to rich countries?

*Journal of Political Economy*,*100*(6), 1208–1231.

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### Publisher's Note

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I would like to thank Michele Boldrin, Alison Booth, Ken Burdett, Nauro Campos, Christian Ghiglino, Victor Lapuente, Abhinay Muthoo, Sebastian Torres and seminar participants at the RSS, University of Essex, and ESNIE 2009, for very helpful comments and suggestions. The usual disclaimer applies.

## 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

and this expression can be rewritten as

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 \(\frac{ A_{r}}{A_{m}}>1\),^{Footnote 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 \(\frac{A_{p}}{A_{m}}<1\),^{Footnote 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

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

### 1.5 Growth rate of \(k_{t}^{i}\)**for**
\(i=p,m,r\)

From (7) we have that

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

### 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

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)

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.

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.

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 \(g_{t-1}^{p}\).

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Aboal, D. Electoral systems and economic growth.
*Econ Polit* **37**, 781–805 (2020). https://doi.org/10.1007/s40888-020-00185-6

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DOI: https://doi.org/10.1007/s40888-020-00185-6