Financial development, income and income inequality

“The varying expectations of business men... and nothing else, constitute the immediate cause and direct causes or antecedents of industrial fluctuations.”

– Arthur C. Pigou, 1927.

Abstract

The aim of this paper is twofold. Firstly, we present a model in which both income and income inequality are jointly determined in a counter-cyclical manner via self-fulfilling expectation. We argue that multiple equilibria can arise in the presence of inelastic labor demand, a minimum investment requirement, and imperfections in the credit market. In one equilibrium, the market wage and labor income are both low. Young agents who become entrepreneurs work harder and save more than young agents who become depositors. As a result, the equilibrium is characterized by low-income and high-income inequality. In another equilibrium, the market wage and labor income are both high. Young agents supply the same amount of labor and save the same. As a result, the equilibrium is characterized by high-income and low-income inequality. Secondly, we present different dynamic scenarios predicted by the model and analyze the role of self-fulfilling expectations. The paper ends by providing some policy recommendations on how the coordination of agents’ expectations about labor market conditions and how improvements in financial development may affect the long-run income and income inequality.

Appendix

Lemma 1

Suppose population consists of I groups. Let the relative size of group $i=1,2,...,I$ is $p_i$ and everyone in that group earns the income $y_i$. Then, the Gini index of income inequality in the entire population is

$$\begin{aligned} G=\frac{\sum _{i=1}^{I}\sum _{j=1}^{I}p_ip_j|y_i-y_j|}{2\sum _{i=1}^{I}p_i y_i}. \end{aligned}$$

(32)

Proof of Lemma 1: can be found in Stuart and Ord (1963).

Proof of Proposition 1

Entrepreneur’s optimization problem is (time subscripts are eliminated for notational convenience):

$$\begin{aligned} V^{e}=\max _{\begin{array}{c} 0 \le \ell \le 1 \end{array}} \left\{ (1-\ell )\left( \displaystyle \frac{i}{w}\displaystyle \frac{\rho -r}{r}+\ell \right) \left| \right. \ m \le i \le \displaystyle \frac{\ell w r}{r-(1-\lambda )\rho }\right\} . \end{aligned}$$

(33)

If $\rho >r$, then the entrepreneur’s objective function is strictly increasing with respect to i. As a result, $i=\frac{\ell w r}{r -(1-\lambda )\rho }$, $\frac{i}{w}\frac{\rho -r}{r}+\ell =\frac{\lambda \rho }{r-(1-\lambda )\rho }\ell$, and thus, the entrepreneur’s optimization problem is

$$\begin{aligned} \max _{\begin{array}{c} 0 \le \ell \le 1 \end{array}} \left\{ (1-\ell )\ell \left| \right. \ m \le i \le \frac{\ell w r}{r-(1-\lambda )\rho }\right\} . \end{aligned}$$

(34)

The solution of (34) is

$$\begin{aligned} \ell ^{e}=\min \left\{ \max \left\{ \displaystyle \frac{1}{2},\left( 1-\frac{(1-\lambda )\rho }{r}\right) \frac{m}{w}\right\} ,1\right\} . \end{aligned}$$

(35)

If $\rho =r$, then the entrepreneur’s objective function is given by (34), the solution of which is given by (35). This implies that the optimal investment is

$$\begin{aligned} i^{e}=\max \left\{ \displaystyle \frac{r}{r-(1-\lambda )\rho }\displaystyle \frac{w}{2},m\right\} . \end{aligned}$$

(36)

Since $m=\frac{\ell w r}{r-(1-\lambda )\rho }$ $\Leftrightarrow$ $\ell =\left( 1-\frac{(1-\lambda )\rho }{r}\right) \frac{m}{w}$, we consider the following three cases separately.

• (a) If $\left( 1-\frac{(1-\lambda )\rho }{r}\right) \frac{m}{w} \in [0,\frac{1}{2}]$ $\Leftrightarrow$ $m \le \frac{\ell w r}{r-(1-\lambda )\rho }$, then $\ell ^e=\frac{1}{2}$, $i^e=\frac{r}{r-(1-\lambda )\rho }\frac{w}{2}$, and thus $V^e=\frac{1}{4}\frac{\lambda \rho }{r-(1-\lambda )\rho }$.

• (b) If $\left( 1-\frac{(1-\lambda )\rho }{r}\right) \frac{m}{w} \in (\frac{1}{2},1)$ $\Leftrightarrow$ $m>\frac{\ell w r}{r-(1-\lambda )\rho }$, then $\ell ^e=(1-\frac{(1-\lambda )\rho }{r})\frac{m}{w}$, $i^e=m$, and thus $V^e=(1-\frac{m}{w}+\frac{(1-\lambda )m}{w})\frac{\rho }{r})\frac{\lambda m}{w}\frac{\rho }{r}$.

• (c) If $\left( 1-\frac{(1-\lambda )\rho }{r}\right) \frac{m}{w} \ge 1$ $\Leftrightarrow$ $m>\frac{\ell w r}{r-(1-\lambda )\rho }$, then $\ell ^e=1$, $i^e=m$, and thus $V^e=0$.

$\square$

Proof of Proposition 2

We drop time subscript for notational convenience. We consider two cases separately.

• (a) If $(1-\frac{(1-\lambda )\rho }{r})\frac{m}{w} \in [0,\frac{1}{2}]$, then it follows from Proposition 1 that $V^e=V^d$ is equivalent to $\frac{\lambda \rho }{r-(1-\lambda )\rho }=1$ $\Leftrightarrow$ $\frac{\rho }{r}=1$. If $\frac{\rho }{r}=1$, then $(1-\frac{(1-\lambda )\rho }{r})\frac{m}{w} \in [0,\frac{1}{2}]$ becomes $\frac{\lambda m}{w} \le \frac{1}{2}$ $\Leftrightarrow$ $w \ge 2\lambda m$.

• (b) If $(1-\frac{(1-\lambda )\rho }{r})\frac{m}{w} \in (\frac{1}{2},1)$, then it follows from Proposition 1 that $V^e=V^d$ is equivalent to $(1-\frac{m}{w}+\frac{(1-\lambda )m}{w})\frac{\rho }{r})\frac{\lambda m}{w}\frac{\rho }{r}=\frac{1}{4}$. This is a quadratic equation solution of which is

  $$\begin{aligned} \displaystyle \frac{\rho }{r}=\frac{1}{1-\lambda }\left( \frac{1}{2}-\frac{w}{2m}+\sqrt{\frac{1}{4}-\frac{w}{2m}+\frac{w^2}{4\lambda m^2}}\right) . \end{aligned}$$

  (37)

  If the ratio $\frac{\rho }{r}$ is given by (37), then $(1-\frac{(1-\lambda )\rho }{r})\frac{m}{w}=\frac{1}{2}+\frac{m}{2w}-\sqrt{\frac{m^2}{4w^2}-\frac{m}{2w}+\frac{1}{4\lambda }}$ and thus $(1-\frac{(1-\lambda )\rho }{r})\frac{m}{w} \in (\frac{1}{2},1)$ $\Leftrightarrow$ $w<2\lambda m$. $\square$

1.1 Alternative specifications

In the main text of the paper, we made several simplifying assumptions in order to minimize the dimension of the parameter space and avoid unnecessary complications while analyzing the model. Of course, some of the results obtained in this paper depend on these simplifying assumptions; however, the main results are robust to alternative specifications as well. The main features of the model would remain as long as (a) agents face the minimum investment requirement for setting up a firm, (b) financial market is imperfect, (c) labor demand is sufficiently inelastic, and (d) there is a self-employment opportunity for young agents. As long as these features of the model are maintained, alternative assumptions about the consumer utility function or the firm’s production function would not invalidate the key results, although they might considerably complicate the analysis. This is so because (a) and (b) imply a backward bending (i.e., an inverted “U” shaped) aggregate labor supply curve. This along with (c) implies the multiplicity of equilibria, and (d) guarantees a minimum aggregate saving so that the corner steady state is not globally attractive. This section gives a brief sketch of how the analysis needs to be modified to alternative specifications.

1.1.1 CES production function

In the basic model of Sect. 3, we assumed that a firm’s production function to be one of a Leontief’s type. Under this assumption, we demonstrate the possibility of multiple equilibria and determined analytically equilibrium wages, employment, incomes, and income inequalities. In this section, we demonstrate numerically that multiplicity of equilibria persists even under alternative specification of the production function. Suppose firm’s production function is one of CES type

$$\begin{aligned} F(k_{ft},n_{ft})=T\left( \alpha k_{ft}^{\frac{\sigma -1}{\sigma }}+(1-\alpha )n_{ft}^{\frac{\sigma -1}{\sigma }}\right) ^{\frac{\sigma }{\sigma -1}}, \end{aligned}$$

(38)

where parameter $\alpha \in (0,1)$ measures the importance of capital stock in production, while parameter $\sigma \in (0,1)$ represents the elasticity of substitution between capital and labor inputs. This specification of the production function includes a Leontief technology (considered above) as a limit case when $\sigma \downarrow 0$. Firm’s profit maximization implies that for a given pair $(w_t,k_t)$, the aggregate demand of labor at time t is $k_tN^d(w_t)$, where

$$\begin{aligned} N^d\left( w\right) =\left\{ \begin{array}{lll} \in \left[ \left( \frac{1}{\alpha }\left( \frac{T(1-\alpha )}{A}\right) ^{1-\sigma }-\frac{1-\alpha }{\alpha }\right) ^\frac{\sigma }{1-\sigma },\infty \right) &{} \text {if} &{} w=A \\ \left( \frac{1}{\alpha }\left( \frac{T(1-\alpha )}{w}\right) ^{1-\sigma }-\frac{1-\alpha }{\alpha }\right) ^\frac{\sigma }{1-\sigma } &{} \text {if} &{} w \in \left( A,\left( 1-\alpha \right) ^{-\frac{\sigma }{1-\sigma }}T\right) \\ 0 &{} \text {if} &{} w>\left( 1-\alpha \right) ^{-\frac{\sigma }{1-\sigma }}T. \end{array} \right. \nonumber \\ \end{aligned}$$

(39)

One can easily verify that the elasticity of labor demand for $w \in \left( A,\left( 1-\alpha \right) ^{-\frac{\sigma }{1-\sigma }}T\right)$ is

$$\begin{aligned} \epsilon ^d(w)=\displaystyle \frac{w[N^d(w)]^{\prime }}{N^d(w)}=-\displaystyle \frac{\sigma }{\alpha }\left( \displaystyle \frac{T(1-\alpha )}{w}\right) ^{1-\sigma } \left( \displaystyle \frac{1}{\alpha }\left( \displaystyle \frac{T(1-\alpha )}{w}\right) ^{1-\sigma }-\displaystyle \frac{1-\alpha }{\alpha }\right) ^{-1}.\nonumber \\ \end{aligned}$$

(40)

It follows from (40) that if the production function is Leontief, then the elasticity of labor demand is $\lim _{\sigma \downarrow 0}\epsilon ^d(w)=0$, while if the production function is Cobb–Douglas, then the elasticity of labor demand is $\lim _{\sigma \uparrow 1}\epsilon ^d(w)=-\frac{1}{\alpha }$. Figure 10 visualizes the configurations of the aggregate labor demand and its elasticity for different values of parameter $\sigma$. It follows from (39) and (40) that the aggregate demand for labor and the elasticity of the aggregate demand for labor both decrease with $w_t$. This monotonicity property of the labor demand elasticity plays an important role in the existence of multiple equilibria.

Fig. 10

Labor demand and its elasticity: Case with CES production function. Both figures are constructed for different values of parameter $\sigma$. Both figures are constructed when $A=0$, $T=2$, and $\alpha =0.50$

Proposition 3

If $\sigma \in (0,\lambda )$, $A<2\lambda m<T<2m$, and

$$\begin{aligned} T>2\lambda m \left( \displaystyle \frac{\lambda (1-\alpha )^{\sigma }}{\lambda -\sigma }\right) ^{\frac{1}{1-\sigma }}, \end{aligned}$$

(41)

then there exist multiple equilibria for intermediate values of $k_t$.

Proof of Proposition 3

We drop the time subscript for notational convenience. Multiple equilibria exist when $kN^d(w)=N^s(w)$ admits multiple solutions. This is possible when $w \mapsto \frac{N^s(w)}{N^d(w)}$ is a non-monotonic function. Since $w \mapsto 1/N^d(w)$ is a strictly increasing function, while $w N^s(w)$ has inverted “U” shape, it follows that the function $w \mapsto N^s(w)/N^d(w)$ may be either a strictly increasing or may have an inverted “U” shape. $w \mapsto N^s(w)/N^d(w)$ has an inverted “U” shape if an only if

$$\begin{aligned} \begin{array}{lll} \lim _{w \uparrow 2\lambda m}\left( \displaystyle \frac{N^s(w)}{N^d(w)}\right) ^{\prime }<0\Leftrightarrow & {} \lim _{w \uparrow 2\lambda m}\epsilon ^s(w)<\lim _{w \uparrow 2\lambda m}\epsilon ^d(w). \end{array} \end{aligned}$$

(42)

On the one hand, it follows from (15), (16), (18) and (19) that $\lim _{w \uparrow 2\lambda m}L^e(w)=1/2$, $\lim _{w \uparrow 2\lambda m}\epsilon ^e(w)=-1$, $\lim _{w \uparrow 2\lambda m}N^s(w)=1/2$, implying $\lim _{w \uparrow 2\lambda m}\epsilon ^s(w)=-\lambda$. On the other hand, it follows from (40) that

$$\begin{aligned} \lim _{w \uparrow 2\lambda m}\epsilon ^d(w)=-\displaystyle \frac{\sigma }{\alpha }\left( \displaystyle \frac{T(1-\alpha )}{2\lambda m}\right) ^{1-\sigma } \left( \displaystyle \frac{1}{\alpha }\left( \displaystyle \frac{T(1-\alpha )}{2\lambda m}\right) ^{1-\sigma }-\displaystyle \frac{1-\alpha }{\alpha }\right) ^{-1}. \end{aligned}$$

(43)

This with (42) implies that the multiple equilibria exist if and only if $k_t$ takes an intermediate value, $\sigma <\lambda$, and parameters satisfy the inequality given in (41). $\square$

Expression (41) deserves some interpretations. First, the multiplicity of equilibria cannot occur when the firm’s production function is Cobb–Douglas, $\sigma =1$, and the possibility of multiple equilibria widens as $\sigma$ becomes smaller.¹⁸ Second, $A<2\lambda m<T<2m$ and (41) can hold simultaneously only if

$$\begin{aligned} \begin{array}{lll} \lambda \left( \displaystyle \frac{\lambda (1-\alpha )^{\sigma }}{\lambda -\sigma }\right) ^{\frac{1}{1-\sigma }}<1\Leftrightarrow & {} \alpha >1-\left( \displaystyle \frac{\lambda -\sigma }{\lambda ^{2-\sigma }}\right) ^{\frac{1}{\sigma }}. \end{array} \end{aligned}$$

(44)

Berndt (1976) and Hamermesh (1996) surveyed a number of studies and reported the elasticity of factor substitution in the USA is in the ranging from 0.32 to 1.16. Chirinko (2008) reported a considerable cross-country variation of elasticity parameter in a range of 0.4 to 0.6. Figure 11a displays the plots of $\alpha =1-\left( \frac{\lambda -\sigma }{\lambda ^{2-\sigma }}\right) ^{\frac{1}{\sigma }}$ for different values of parameter $\sigma$. As the figure indicates, the existence of multiple equilibria is possible when parameters $\alpha$ and $\lambda$ both take sufficiently large values. Parameter $\alpha$ takes a large value when physical and human capital plays an important contribution to the production process, while parameter $\lambda$ takes a large value when the credit market is highly imperfect. It follows from Proposition 3 that is the parameter pair $(\alpha ,\lambda )$ satisfies (44) and then the multiple equilibria is possible for a sufficiently large value of parameter $\frac{T}{m}$ and for an intermediate value of $k_t$.

Fig. 11

Multiplicity of equilibria under alternative specifications

1.1.2 Consumption during the first period

In the basic model of Sect. 3, we assumed that young agents do not value the first period consumption and they have no time discount for the utility from the second period consumption. In this section, we demonstrate that the inverted “U” shape of the aggregate labor supply curve persists even under alternative specification. Suppose agent’s lifetime utility is given by

$$\begin{aligned} (\ell _t,c_{1t},c_{2t+1}) \mapsto \ln (1-\ell _t)+\ln {c_{1t}}+\beta \ln {c_{2t+1}}, \end{aligned}$$

where $1-\ell _t$ is the first period leisure, while $c_{1t}$ and $c_{2t+1}$ are the first and second period consumptions of the final commodity. Parameter $\beta \in (0,1)$ represents the time discount.

Young agents optimal behavior Suppose a young agent supplies $\ell _t$ units of labor and saves $s_t w_t$ units of the final commodity during the first period. If the agent becomes a depositor during second period, then her optimization problem is

$$\begin{aligned} V^{d}=\max _{\ell _{t}\in [0,1]\text { and }s_t \in [0,\ell _t]}(1-\ell _t)(\ell _t-s_t)s_t^{\beta }, \end{aligned}$$

where $s_t \in [0,\ell _t]$ is the saving rate. First-order condition for optimality implies that

$$\begin{aligned} \begin{array}{lll} s^{d}=2\ell ^d-1&\text {and}&\ell ^{d}=\frac{1+\beta }{2+\beta }. \end{array} \end{aligned}$$

(45)

Depositor’s lifetime utility is $\ln V^d + (1+\beta )\ln w_t+\beta \ln r_{t+1}$, where

$$\begin{aligned} V^d=\frac{\beta ^{\beta }}{(2+\beta )^{2+\beta }}. \end{aligned}$$

(46)

If an agent becomes an entrepreneur during the second period, then her optimization problem is

$$\begin{aligned} V^e_t=\max _{\ell _{t}\in [0,1]\text { and }s_t \in [0,\ell _t]}(1-\ell _t)(\ell _t-s_t)\left( \frac{i_t}{w_t}\frac{\rho _{t+1}-r_{t+1}}{r_{t+1}}+s_{t}\right) ^{\beta }, \end{aligned}$$

subject to the borrowing constraint, $m \le i_t \le \frac{s_t w_t r_{t+1}}{r_{t+1}-(1-\lambda )\rho _{t+1}}$. First-order condition for optimality implies that $s^{e}_{t}=2\ell _t^e-1$,

$$\begin{aligned} \ell ^e_t=\left\{ \begin{array}{lll} \frac{1+\beta }{2+\beta } &{} \text {if} &{} \left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}\le \frac{\beta }{2+\beta } \\ \frac{1}{2}+\frac{1}{2}\left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t} &{} \text {if} &{} \left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}>\frac{\beta }{2+\beta }, \end{array} \right. \end{aligned}$$

(47)

and

$$\begin{aligned} i^e_t=\left\{ \begin{array}{lll} \frac{\beta }{2+\beta }\frac{w_tr_{t+1}}{r_{t+1}-(1-\lambda )\rho _{t+1}} &{} \text {if} &{} \left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}\le \frac{\beta }{2+\beta } \\ m &{} \text {if} &{} \left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}>\frac{\beta }{2+\beta }, \end{array} \right. \end{aligned}$$

(48)

Entrepreneur’s lifetime utility is $\ln V^e_t + (1+\beta )\ln w_t+\beta \ln r_{t+1}$, where

$$\begin{aligned} V^e_t=\left\{ \begin{array}{lll} \frac{\beta ^\beta }{(2+\beta )^{2+\beta }}\left( \frac{\lambda \rho _{t+1}}{r_{t+1}-(1-\lambda )\rho _{t+1}}\right) ^{\beta } &{} \text {if} &{} \left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}\le \frac{\beta }{2+\beta } \\ \left( \frac{1}{2}-\frac{1}{2}\left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}\right) ^2\left( \displaystyle \frac{\lambda m}{w_t}\frac{\rho _{t+1}}{r_{t+1}}\right) ^{\beta } &{} \text {if} &{} \left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}>\frac{\beta }{2+\beta }. \end{array} \right. \nonumber \\ \end{aligned}$$

(49)

If $\left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}>1$, then the young agent cannot simultaneously overcome the borrowing and the minimum investment requirements and thus $V_t^e=0$.

Equilibrium in the financial market As above, let $\frac{k_{t+1}}{m} \in [0,1]$ denote the share of young agents who become entrepreneurs in period $t+1$. Since young agents who become entrepreneurs borrow $m-s_t^ew_t$ units of the final commodity, while young agents who become depositors lend $s^dw_t=\frac{\beta }{2+\beta }w_t$ units of the final commodity, it follows that the financial market equilibrium is established when

$$\begin{aligned} \begin{array}{lll} V_t^e=V^d&\text {and}&\left( m-s_t^ew_t\right) \displaystyle \frac{k_{t+1}}{m}=\displaystyle \frac{\beta }{2+\beta }w_t\left( 1-\displaystyle \frac{k_{t+1}}{m}\right) . \end{array} \end{aligned}$$

(50)

If $\frac{w_t}{m} \ge \frac{2+\beta }{\beta }\lambda$, then it follows from (50) that the equilibrium in the financial market is established when $\frac{\rho _{t+1}}{r_{t+1}}=0$. As a result, $s_t^e=s^d=\frac{\beta }{2+\beta }$, $\ell _t^e=\ell ^d=\frac{1+\beta }{2+\beta }$, and $i_t^e=\frac{\beta }{2+\beta }\frac{w_t}{\lambda }$. This implies that the aggregate labor supply is $\frac{1+\beta }{2+\beta }$. This case is possible when $k_{t+1} \ge \lambda m$.

If $\frac{w_t}{m}<\frac{2+\beta }{\beta }\lambda$, then the equilibrium ratio $\frac{\rho _{t+1}}{r_{t+1}}$ satisfies

$$\begin{aligned}&\left( \frac{1}{2}-\frac{1}{2}\left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}\right) ^2\left( \displaystyle \frac{\lambda m}{w_t}\frac{\rho _{t+1}}{r_{t+1}}\right) ^{\beta }=\frac{\beta ^\beta }{(2+\beta )^{2+\beta }}. \end{aligned}$$

(51)

Since $s_t^e=\left( 1-(1-\lambda )\frac{\rho _{t+1}}{r_{t+1}}\right) \frac{m}{w_t}$, it follows from (50) that

$$\begin{aligned} \begin{array}{lll} \left( m-s_t^ew_t\right) \displaystyle \frac{k_{t+1}}{m}=\displaystyle \frac{\beta }{2+\beta }w_t\left( 1-\displaystyle \frac{k_{t+1}}{m}\right)&\text {is equivalent}&\displaystyle \frac{\rho _{t+1}}{r_{t+1}}=\displaystyle \frac{\beta }{2+\beta }\displaystyle \frac{w_t}{k_{t+1}}\displaystyle \frac{m-k_{t+1}}{(1-\lambda )m}. \end{array}\nonumber \\ \end{aligned}$$

(52)

After substituting (52) into (51), we obtain that equilibrium wage satisfies

$$\begin{aligned} \frac{m}{w_t}=\frac{2k_{t+1}+\beta m}{(2+\beta )k_{t+1}}-\frac{2}{2+\beta }\left( \frac{k_{t+1}}{\lambda m}\frac{(1-\lambda )m}{m-k_{t+1}}\right) ^{\frac{\beta }{2}}. \end{aligned}$$

(53)

(47), (52), and (53) imply that the labor supply and the saving rate of young agents who become entrepreneurs are

$$\begin{aligned} \ell _t^e=1-\frac{1}{2+\beta }\left( \frac{k_{t+1}}{\lambda m}\frac{(1-\lambda )m}{m-k_{t+1}}\right) ^{\frac{\beta }{2}} \end{aligned}$$

(54)

and $s_t^e=2\ell _t^e-1$, respectively. Since young agents who become depositors supply $\ell ^d$ units of labor, while young agents who become entrepreneurs supply $\ell ^e_t$ units of labor, it follows that aggregate labor supply is

$$\begin{aligned} \ell _t^e \frac{k_{t+1}}{m}+\ell ^d\left( 1-\frac{k_{t+1}}{m}\right) =\frac{1+\beta }{2+\beta }+\frac{k_{t+1}}{(2+\beta )m}\left( 1-\left( \frac{k_{t+1}}{\lambda m}\frac{(1-\lambda )m}{m-k_{t+1}}\right) ^{\frac{\beta }{2}}\right) .\nonumber \\ \end{aligned}$$

(55)

This case is possible when $k_{t+1}<\lambda m$.

To summarize the above-considered two cases, the aggregate labor supply is ${\widehat{N}}^s(k_{t+1})$, where

$$\begin{aligned} {\widehat{N}}^s(k)=\frac{1+\beta }{2+\beta }\times \left\{ \begin{array}{lll} 1+\displaystyle \frac{k}{(1+\beta )m}\left( 1-\left( \displaystyle \frac{k}{\lambda m}\displaystyle \frac{(1-\lambda )m}{m-k}\right) ^{\frac{\beta }{2}}\right) &{} \text {if} &{} k \in [0,\lambda m) \\ 1 &{} \text {if} &{} k \in [\lambda m,m]. \end{array} \right. \end{aligned}$$

Figure 11b displays the plots of $k \mapsto {\widehat{N}}^s(k)$ for different values of parameter $\lambda$. As the figure indicates, ${\widehat{N}}^s$ sustains its inverted “U” shape even under alternative specification of the agents’ utility function. Let $k_{t+1}=K(w_t)$ denote a solution of

$$\begin{aligned} \frac{m}{w_t}=\left\{ \begin{array}{lll} \displaystyle \frac{2k_{t+1}+\beta m}{(2+\beta )k_{t+1}}-\displaystyle \frac{2}{2+\beta }\left( \displaystyle \frac{k_{t+1}}{\lambda m}\displaystyle \frac{(1-\lambda )m}{m-k_{t+1}}\right) ^{\frac{\beta }{2}} &{} \text {if} &{} k_{t+1} \in (0,\lambda m) \\ \displaystyle \frac{\beta }{2+\beta }\displaystyle \frac{m}{k_{t+1}} &{} \text {if} &{} k_{t+1} \in [\lambda m,m]. \end{array} \right. \end{aligned}$$

(56)

Then, the aggregate labor supply, consistent with the equilibrium in the financial market, is $N^s(w_t)={\widehat{N}}^s[K(w_t)]$.

Proposition 4

The aggregate labor supply curve $w \mapsto N^s(w)$, consistent with the equilibrium in the financial market, is an inverted “U”-shaped curve for $w \in (0,\frac{2+\beta }{\beta }\lambda m)$ and $N^s(w)=\frac{1+\beta }{2+\beta }$ for $w \in [\frac{2+\beta }{\beta }\lambda m,\frac{2+\beta }{\beta }m]$. In addition, $N^s$ has the following boundary behavior $\lim _{w \downarrow 0}N^s(w)=\lim _{w \uparrow \frac{2+\beta }{\beta }\lambda m}N^s(w)=\frac{1+\beta }{2+\beta }$.

Proof of Proposition 4

It can be easily verifies that

$$\begin{aligned} k \mapsto \left\{ \begin{array}{lll} \displaystyle \frac{2k+\beta m}{(2+\beta )k}-\displaystyle \frac{2}{2+\beta }\left( \displaystyle \frac{k}{\lambda m}\displaystyle \frac{(1-\lambda )m}{m-k)}\right) ^{\frac{\beta }{2}} &{} \text {if} &{} k \in [0,\lambda m) \\ \displaystyle \frac{\beta }{2+\beta }\displaystyle \frac{m}{k} &{} \text {if} &{} k \in [\lambda m,m] \end{array} \right. \end{aligned}$$

(57)

is a continuous and a strictly decreasing function. This implies that for any $w \in [0,\frac{2+\beta }{\beta }m]$, there exists a unique and strictly increasing function K such that $k=K(w)$ solves (56). If $w \in [0,\frac{2+\beta }{\beta }\lambda m)$ then $k=K(w)\in [0,\lambda m)$ and if $w \in [\frac{2+\beta }{\beta }\lambda m,\frac{2+\beta }{\beta }m]$, then $k=K(w)=\frac{\beta }{2+\beta }w \in [\lambda m,m]$. If $w \in [\frac{2+\beta }{\beta }\lambda m,\frac{2+\beta }{\beta } m]$, then $N^s(w)=\frac{1+\beta }{2+\beta }$. Suppose $w \in [0,\frac{2+\beta }{\beta }\lambda m)$ then K(w) is a strictly increasing function while

$$\begin{aligned} {\widehat{N}}^s(k)=\frac{1+\beta }{2+\beta }+\frac{k}{(2+\beta )m}\left( 1-\left( \frac{k}{\lambda m}\frac{(1-\lambda )m}{m-k)}\right) ^{\frac{\beta }{2}}\right) \end{aligned}$$

(58)

is an inverted “U”-shaped curve. As a result, one can conclude that $N^s(w)={\widehat{N}}^{s}[K(w)]$ is also an inverted “U”-shaped curve. In order to demonstrate boundary behavior of $N^s$, we observe that

$$\begin{aligned} \begin{array}{lll} \lim _{w \downarrow 0}N^s(w)=\lim _{k \downarrow 0}{\widehat{N}}^s(k)=\frac{1+\beta }{2+\beta }&\text {and}&\lim _{w \uparrow \frac{2+\beta }{\beta }\lambda m}N^s(w)=\lim _{k \uparrow \lambda m}{\widehat{N}}^s(k)=\frac{1+\beta }{2+\beta }. \end{array}\nonumber \\ \end{aligned}$$

(59)

$\square$

Proposition 4 confirms that the inverted “U” shape of the aggregate labor supply curve persists even under alternative specifications of agent’s utility function. This implies that the main results of the paper hold even in a more general setting as well.