Inequality and institutional quality in a growth model


By applying a growth model of an occupational choice between being a worker and being a criminal, an institutional Kuznets curve is produced. Using the Gini coefficient derived from the model, we investigate how institutional quality affects wealth inequality across agents. In the case of a relatively higher capital share, an institutional Kuznets curve can emerge. In this case, an improvement in institutional quality decreases the possibility of being stolen but increases the amount of wealth stolen. In the early stage of economic development, since the latter effect dominates the former, inequality widens as institutional quality improves. In contrast, since the converse is true in the late stage, inequality shrinks once institutions have sufficiently matured. In the case of a lower capital share, an institutional Kuznets curve cannot emerge. In this case, inequality monotonically shrinks as institutional quality improves. Furthermore, we present government policies that reduce inequality and achieve the first-best outcome.

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  1. 1.

    The scatter plot in Fig. 1 is based on 123 countries. The data on the Gini coefficient are collected from the dataset developed by Solt (2009, 2019). The index of the “Legal System and Property Rights” in the Economic Freedom of the World in Gwartney et al. (2018) is used as a proxy of institutional quality. Both datasets are averaged over 2000–2017.

  2. 2.

    As such, the criminal gang plays a role of insurance in our model. Although one can construct a model such that each member of the gang is rewarded as he or she works for the gang, the analysis becomes complicated, while the main result regarding inequality and institutional quality is unchanged.

  3. 3.

    In our model, a criminal’s earnings are negatively affected by law enforcement, and the opportunity cost of being a criminal is the wage income. As a result, the number of criminals decreases as law enforceability becomes strong. In Japan, the government began to enforce a law that cracks down on gangs’ activities in 1991. According to the report by the National Police Agency in Japan (National Police Agency 2020), the number of gang members and associate members declined from 91, 000 in 1991 to 28,200 in 2019, that is, declined by 69%. This evidence makes it reasonable to assume that the opportunity cost of being a criminal is the wage income.

  4. 4.

    As in the model of Galor and Moav (2004), Asano (2012) also highlights the role of human capital accumulation and derives a nonmonotonic relationship between inequality and economic growth.

  5. 5.

    In other words, bequests can be used for investment projects to produce physical capital and/or for the formation of human/knowledge capital. In any case, physical capital and human/knowledge capital are assumed to be perfect substitutes in this paper.

  6. 6.

    Although knowledge capital or know-how may be sold to the representative firm at a certain price, our model setting can remain unchanged essentially.

  7. 7.

    One can consider various types of crimes agents face, including robbery, fraud, expropriation, embezzlement, etc.

  8. 8.

    One can imagine that by laundering money, the criminal gang can access the capital market without being arrested.

  9. 9.

    It is assumed that if the agent is arrested, the prison term is sufficiently short that the agent is released before his or her lifetime ends.

  10. 10.

    It is possible to introduce punishment into our model without changing the main results provided that the punishment is proportional to the amount of wealth that criminals try to steal. We assume that there is no punishment in our model for simplicity.

  11. 11.

    One may consider that the occupational choice would depend on an agent’s wealth, and agents who receive a large bequest would tend not to commit a crime. To focus our analysis on the relationship between institutional quality and inequality in the stationary state, we assume away such a situation that the occupational choice depends on an agent’s wealth, and accordingly, \(l_t\) and \(q_t\) become constant in equilibrium, as seen in Eqs. (14) and (15).

  12. 12.

    Assumption 1 implies that the labor income share is less than a half. Note that capital includes both physical and human capital as assumed in Sect. 3.1.

  13. 13.

    Here, we examine relatively high capital income shares. This is because we assume that capital broadly includes both physical and human capital, as explained in Sect. 3.1.1, and Mankiw et al. (1992) suggest that the labor income share is 33% when one takes into account both physical and human capital in production, which is approximately the mean of labor income shares in the cases of panels A and B (17% and 45%, respectively).


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The authors would like to express thanks to two anonymous referees for their invaluable comments and suggestions to improve the paper significantly. The authors are also grateful to Koichi Futagami, Akihisa Shibata, and session participants of “Macroeconomic Policy” in the 2019 SWET for their helpful comments. All remaining errors, if any, are ours. This work is financially supported by Kwansei Gakuin University and the Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research (Nos. 16K03685 and 20K01647).

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Proof of Lemma 1

Suppose that \(\Gamma _t\) is a set of agents who earn interest income in the second subperiod of period t without being robbed. Then, it follows that

$$\begin{aligned} \int _{i\in \Omega }I_{it}{\text {d}}i&=\int _{i\in \Gamma _t}(\omega _{it}+r_tk_{it}){\text {d}}i+\int _{i\in \Omega \setminus \Gamma _t}\omega _{it}{\text {d}}i \nonumber \\&=\int _{i\in \Omega }\omega _{it}{\text {d}}i+r_t\int _{i\in \Gamma _t}k_{it}{\text {d}}i. \end{aligned}$$

Define \(\Lambda _t\) as a set of agents who are workers in period t. Then, the first term of the right-hand side of Eq. (A.1) becomes \(\int _{i\in \Omega }\omega _{it}{\text {d}}i=\int _{i\in \Lambda _t}w_{t}{\text {d}}i+\int _{i\in \Omega \setminus \Lambda _t}\tilde{w}_{t}{\text {d}}i\). Furthermore, since \(q_t\) measures the population of agents in \(\Gamma _t\) and since criminals randomly choose targets, the second term of the right-hand side of Eq. (A.1) becomes \(r_t\int _{i\in \Gamma _t}k_{it}{\text {d}}i=r_tq_t\int _{i\in \Omega }k_{it}{\text {d}}i=r_tq_tk_t\), where Eq. (12) has been used to obtain the last equality. Then, using Eqs. (1), (2), (3), (9), (12), and (A.1), we compute \(\int _{i\in \Omega }I_{it}{\text {d}}i\) as follows:

$$\begin{aligned} \int _{i\in \Omega }I_{it}{\text {d}}i&=w_tl_t+\tilde{w}_t(1-l_t)+[1-(1-l_t)(1-p)]r_tk_t \\&=w_tl_t+(1-l_t)(1-p)r_tk_t+[1-(1-l_t)(1-p)]r_tk_t \\&=Ak_t^\alpha l_t^{1-\alpha }. \end{aligned}$$

This is our desired conclusion. \(\square\)

Proof of Lemma 4

According to Lemma 3, any family becomes a crime victim almost surely. Suppose that family i with \(k_{it}\) is robbed in period \(t\ge 0\). Then, the wealth that family i has in period \(t+1\) is given by \(k_{it+1}=\sigma w_t=:k_{t+1}^{(0)}\). Looking forward from period \(t+1\) onward, we note that Eq. (22) implies that \(k_{it+2}=\sigma w_{t+1}+\sigma ^2 r_{t+1}w_t=:k_{t+2}^{(1)}\) with probability q and \(k_{it+2}=\sigma w_{t+1}=:k_{t+2}^{(0)}\) with probability \(1-q\), \(\ldots\), \(k_{it+j+1}=\sigma w_{t+j}+\sigma ^2r_{t+j}w_{t+j-1}+\cdots +\sigma ^{j+1}\left( \Pi _{s=1}^{j}r_{t+s}\right) w_t=:k_{t+j+1}^{(j)}\) with probability \(q^{j}\) and \(k_{it+j+1}=\sigma w_{t+j}=:k_{t+j+1}^{(0)}\) with probability \(q^{j-1}(1-q)\). Since it holds that \(r_{t+s}\rightarrow \bar{r}\), \(w_{t+s}\rightarrow \bar{w}\), and \(\sigma r_{t+s}\rightarrow \alpha\) for \(s=1,\ldots j\) as \(t\rightarrow \infty\), it follows that \(k^{(0)}:=\lim _{t\rightarrow \infty }k_{t+j+1}^{(0)}=\sigma \bar{w}\) for \(j=0, 1, \ldots , \infty\) and \(k^{(j)}:=\lim _{t\rightarrow \infty }k_{t+j+1}^{(j)}=\sigma \bar{w}\sum _{s=0}^j\alpha ^s\) for \(j=1, 2, \ldots , \infty\). Therefore, for sufficiently large t, family i’s wealth takes one of the values in \(\{k^{(j)}\}_{j=0}^\infty\). Because this outcome holds for all families in \(\Omega\), the support of the distribution of individual wealth in the stationary state is given by \(\{k^{(j)}\}_{j=0}^\infty\), where \(k^{(j)}=\sigma \bar{w}\sum _{s=0}^j\alpha ^s\). \(\square\)

Proof of Lemma 5

Suppose that \(P_{t}^{(j)}\) is the population of families that have individual wealth, \(k_t^{(j)}\), in period \(t\ge 1\) (see the proof of Lemma 4 for the definition of \(k_t^{(j)}\)). Equation (22) and the law of large numbers imply that the population of families that become crime victims in period t and have individual wealth, \(k_{t+1}^{(0)}\), in period \(t+1\) is equal to \(P_{t+1}^{(0)}=1-q\). Again, Eq. (22) and the law of large numbers yield \(P_{t+2}^{(1)}=qP_{t+1}^{(0)}=q(1-q)\), \(P_{t+3}^{(2)}=qP_{t+2}^{(1)}=q^2(1-q)\), \(\ldots\), \(P_{t+j+1}^{(j)}=qP_{t+j}^{(j-1)}=q^j(1-q)\), \(\ldots\). Or equivalently, we obtain \(P_{t+j+1}^{(j)}=q^j(1-q)\) for \(j=0, 1, \ldots , \infty\). It follows from the last equation that \(P^{(j)}:=\lim _{t\rightarrow \infty }P_{t+j+1}^{(j)}=q^j(1-q)\) for \(j=0, 1, \cdots , \infty\). Conversely, we can verify that \(\sum _{j=0}^\infty P^{(j)}=\sum _{j=0}^\infty q^j(1-q)=1\). This completes the proof. \(\square\)

Proof of Proposition 3

By definition, the Gini coefficient in the stationary state is given as follows:

$$\begin{aligned} g(q)=1-\sum _{j=0}^{\infty }\left[ G(k^{(j)})-G(k^{(j-1)})\right] \left[ B^{(j)}+B^{(j-1)}\right] , \end{aligned}$$

where \(G(k^{(j)})\) is the cumulative distribution function of \(k^{(j)}\), and \(B^{(j)}\) is the cumulative proportion of wealth relative to the total wealth, which is given by \(B^{(j)}:=\sum _{\ell =0}^j(1-q)q^{\ell }k^{(\ell )}/\bar{k}\) according to Eq. (24). In Eq. (A.2), it is assumed for convenience that \(k^{(-1)}=B^{(-1)}=0\). Using Eq. (24) allows \(B^{(j)}+B^{(j-1)}\) to be computed as

$$\begin{aligned} B^{(j)}+B^{(j-1)}=\left( \frac{1-q}{1-\alpha }\right) \left( \frac{\sigma \bar{w}}{\bar{k}}\right) \left[ 2\left( \frac{1-q^{j}}{1-q}-\frac{\alpha (1-(\alpha q)^{j})}{1-\alpha q}\right) +(1-\alpha ^{j+1})q^{j}\right] . \end{aligned}$$

From Eq. (24), it follows that

$$\begin{aligned} G(k^{(j)})-G(k^{(j-1)})=(1-q)q^j. \end{aligned}$$

Additionally, from Eqs. (14), (19), and (20), it follows that

$$\begin{aligned} \frac{\sigma \bar{w}}{\bar{k}}=1-\alpha q. \end{aligned}$$

Eqs. (A.3)–(A.5) allow \(\left[ G(k^{(j)})-G(k^{(j-1)})\right] \left[ B^{(j)}+B^{(j-1)}\right]\) to be computed as follows:

$$\begin{aligned}&\left[ G(k^{(j)})-G(k^{(j-1)})\right] \left[ B^{(j)}+B^{(j-1)}\right] \nonumber \\&=\frac{(1-q)^2(1-\alpha q)}{1-\alpha }\left[ \frac{2(1-\alpha )}{(1-q)(1-\alpha q)}q^j-\frac{1+q}{1-q}(q^2)^{j}+\frac{\alpha +\alpha ^2 q}{1-\alpha q}(\alpha q^2)^{j}\right] . \end{aligned}$$

Eq. (A.6) yields

$$\begin{aligned} \sum _{j=0}^{\infty }\left[ G(k^{(j)})-G(k^{(j-1)})\right] \left[ B^{(j)}+B^{(j-1)}\right] =2-\frac{1-\alpha q}{1-\alpha }+\frac{\alpha (1+\alpha q)(1-q)^2}{(1-\alpha )(1-\alpha q^2)}. \end{aligned}$$

Eqs. (A.2) and (A.7) lead to our desired conclusion. \(\square\)

Proof of Proposition 4

From Eq. (25), it follows that

$$\begin{aligned} g^\prime (q)=\alpha ^2\left( q-\frac{1-\sqrt{1-\alpha }}{\alpha }\right) \left( q-\frac{1+\sqrt{1-\alpha }}{\alpha }\right) {/}(1-\alpha q^2)^2. \end{aligned}$$

It holds that \(g^\prime (0)>g^\prime ((1-\sqrt{1-\alpha })/\alpha )=0>g^\prime (1)\). Therefore, if \((1-\alpha )/\alpha \ge (1-\sqrt{1-\alpha })/\alpha\) and \(\alpha >1/2 \iff 1/2<\alpha \le (-1+\sqrt{5})/2\), it follows that \(g^\prime (q)<0\) in \(((1-\alpha )/\alpha , 1)\), and thus, g(q) monotonically decreases with \(q\in [(1-\alpha )/\alpha , 1)\). If \((1-\alpha )/\alpha < (1-\sqrt{1-\alpha })/\alpha\) and \(\alpha>1/2 \iff \alpha > (-1+\sqrt{5})/2\), it follows that \(g^\prime (q)>0\) in \([(1-\alpha )/\alpha , (1-\sqrt{1-\alpha })/\alpha )\) and \(g^\prime (q)<0\) in \(((1-\sqrt{1-\alpha })/\alpha , 1)\). Therefore, g(q) increases with \(q\in [(1-\alpha )/\alpha , (1-\sqrt{1-\alpha })/\alpha )\), and g(q) decreases with \(q\in [(1-\sqrt{1-\alpha })/\alpha , 1)\). \(\square\)

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Kunieda, T., Takahashi, M. Inequality and institutional quality in a growth model. Evolut Inst Econ Rev (2021).

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  • Inequality
  • Institutional quality
  • Heterogeneous agents
  • Gini coefficient
  • Institutional Kuznets curve

JEL Classification

  • D23
  • D63
  • O11
  • O41
  • O43