Inequality and institutional quality in a growth model

Abstract

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.

Appendix

1.1 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

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

(A.1)

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. $\square$

1.2 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 $k^{(j)}=\sigma \bar{w}\sum _{s=0}^j\alpha ^s$. $\square$

1.3 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. $\square$

1.4 Proof of Proposition 3

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

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

(A.2)

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

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

(A.3)

From Eq. (24), it follows that

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

(A.4)

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

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

(A.5)

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:

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

(A.6)

Eq. (A.6) yields

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

(A.7)

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

1.5 Proof of Proposition 4

From Eq. (25), it follows that

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

(A.8)

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 $q\in [(1-\sqrt{1-\alpha })/\alpha , 1)$. $\square$