2.1 Bad projects are not available

Consider the case where the banking system does not invest in the bad technology, $V_{t+1}=0$ . Given that the left-hand side of (1) is increasing in $z$ , agents with low initial endowments prefer to become depositors and those with high initial endowments prefer to become entrepreneurs. Market clearing requires that the aggregate amount of loans is equal to aggregate deposits,

\begin{equation*} \int \limits _{z_{t+1}^{\ast }}^{\bar {z}}(1+\gamma )\left ( m-1\right ) W(k_{t})zg(z)dz=\int \limits _{\underline {z}}^{z_{t+1}^{\ast }}W(k_{t})zg(z)dz. \end{equation*}

On the left hand side, we have the allocation of funds between loans and expenditure to cover the cost of banking services. On the right-hand side, we have aggregate deposits. The above condition implies that $z_{t+1}^{\ast }=z^{\ast }$ for every $t$ and is the solution of

(4) \begin{equation} 1=\left ( m+\gamma m-\gamma \right ) \int \limits _{z^{\ast }}^{\bar {z} }zg(z)dz. \end{equation}

The banking system’s zero-profit condition when $V_{t+1}=0$ is given by

\begin{equation*} r_{t+1}^{b}\int \limits _{z^{\ast }}^{\bar {z}}\left ( m-1\right ) W(k_{t})zg(z)dz=r_{t+1}^{d}\int \limits _{\underline {z}}^{z^{\ast }}W(k_{t})zg(z)dz. \end{equation*}

Using the market clearing condition, the zero-profit condition implies that

(5) \begin{equation} r_{t+1}^{b}=r_{t+1}^{d}\left ( 1+\gamma \right ) \text{ for every }t\text{ such that }V_{t+1}=0. \end{equation}

Each entrepreneur invests her endowment $zW(k_{t})$ plus her funds borrowed from banks, $(m-1)zW(k_{t})$ , to produce $R(m)$ units of physical capital. Using (4) we can write (3) as

(6) \begin{equation} k_{t+1}\equiv F\left ( k_{t}\right ) =\frac {R(m)m}{m+\gamma m-\gamma }W(k_{t}). \end{equation}

Dynamic stability requires that $F^{\prime }\left ( k_{t}\right ) \gt 0$ , $ F^{\prime \prime }\left ( k_{t}\right ) \lt 0$ and $F^{\prime }(k)\rightarrow \infty$ as $k\rightarrow 0$ , (see Figure 4).Footnote ¹⁵ Lastly, substituting (4) and (5) in (1) determines implicitly the equilibrium interest rates, $r_{t+1}^{d}$ and $r_{t+1}^{b}$ , such that an agent of type $z^{\ast }$ is indifferent between investing and saving (depositing).

Figure 4.

Financial repression.

To obtain the steady state, we set $k_{t+1}=k_{t}=k^{\ast }$ in (6). Proposition1 describes comparative statics at the steady state:

The qualification on $\gamma$ is necessary only for the first result, as $m$ influences the steady-state capital-labor ratio through two distinct channels. First, by relaxing the financing constraint, an increase in $m$ promotes growth by enhancing project productivity. Second, a higher $m$ leads to a greater volume of lending, which, given the costs associated with loans, negatively impacts capital accumulation. Notably, $z^{\ast }$ is independent of $y$ . The economic intuition is that an increase in $y$ requires a lower interest rate to ensure the marginal old agent remains indifferent. Furthermore, $k^{\ast }$ decreases with $\gamma$ (see Figure 4). In economies with high intermediation costs, financial repression leads to a lower steady-state capital-labor ratio, as a significant share of potentially productive capital is absorbed by the operational and administrative expenses of granting new loans.

2.2 Investment in bad projects: the unconstrained case

The derivations in the previous section apply as long as $ r_{t+1}^{b}\geqslant B\left ( 1+\gamma \right )$ . However, if $ r_{t+1}^{b}\lt B\left ( 1+\gamma \right )$ the banking system will invest some of the deposits in the bad technology, $\ V_{t+1}\gt 0$ . For the moment we concentrate on the case where the investment in bad projects is not constrained, which is $\hat {V}(k_{t})\approx \infty$ .

The market clearing condition now is given by (2). For the banking system to be indifferent between lending and investing in the bad technology, we must have $r_{t+1}^{b}=B(1+\gamma )$ . The banking system’s zero-profit condition is given by $r_{t+1}^{b}\int \nolimits _{z_{t+1}^{\ast }}^{\bar {z}}\left ( m-1\right ) W(k_{t})zg(z)dz+BV_{t+1}^{\ast }=r_{t+1}^{d}\int \nolimits _{\underline {z}}^{z_{t+1}^{\ast }}W(k_{t})zg(z)dz$ , which implies (using equation (2)) that

(7) \begin{equation} r_{t+1}^{b}=B(1+\gamma )\gt r_{t+1}^{d}=B\quad \text{ for every }t\text{ such that }V_{t+1}\gt 0. \end{equation}

The stock of physical capital at $k_{t+1}$ is given by (3). Given that in equilibrium an agent of type $z_{t+1}^{\ast }$ must be indifferent between investing and saving (depositing) (1) implies that the following equality must hold:

(8) \begin{equation} z_{t+1}^{\ast }W(k_{t})(f^{^{\prime }}(k_{t+1})R(m)m-B(m+\gamma m-\gamma )) =y. \end{equation}

Then, for given $W(k_{t})$ , we can solve (3) and (8) together for $k_{t+1}$ and $z_{t+1}^{\ast }$ and then by substituting the solution for $ z_{t+1}^{\ast }$ in (2) we can solve for $V_{t+1}^{\ast }$ .

As noted in the introduction, a key distinction between our paper and Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016) is that we move beyond the representative agent framework by introducing heterogeneity in initial endowments. In Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016), when investment in bad projects is unconstrained, the capital-to-labor ratio remains independent of its previous-period value. This is because it is solely determined by the condition that marginal productivity equals the constant return on bad projects. In contrast, our model determines equilibrium returns endogenously. Specifically, when the loan market clears, the marginal agent remains indifferent between becoming an entrepreneur and depositing their endowment in the bank and becoming self-employed.

Let

(9) \begin{equation} k_{t+1}\equiv H(k_{t})=R(m)mW(k_{t})\int \limits _{z_{t+1}^{\ast }}^{\bar {z} }zg(z)dz \end{equation}

denote the evolution of the capital/labor ratio when investment in bad projects is unconstrained.

When investment in bad projects is not constrained, the evolution of the capital/labor ratio follows an upward path, once we allow for heterogeneous agents. There are two effects responsible for that result. In contrast to Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016), the last period’s capital stock matters when investment in bad projects is not constrained. The reason is that agents with higher endowments benefit more from being entrepreneurs. To restore indifference the return of good projects has to decline which means the capital/labor ratio must go up. There is also a second adjustment that takes place through a change in the cut-off endowment level that separates the two types of old agents. More specifically, the number of entrepreneurs decreases, i.e., $z_{t+1}^{\ast }$ increases, if $\frac {d^{2}f\left ( k_{t+1}\right ) }{dk_{t}^{2}}$ is sufficiently low.

Figure 5.

Phase map of the capital/labor ratio.

Notes: This figure shows the phase map of the capital/labor ratio. The parameter values used are $R=2.5$ , $A=2.5$ , $m=1.5$ , $ \gamma =0.03$ , $y=1$ , $B=1.25$ , $k^{FG}=6$ , and $ \delta =-1.25$ . $F(k_{t})$ , $G(k_{t})$ , and $H(k_{t})$ refer to the respective branch of the map where $k^{FH}$ , $k^{FG}$ , and $k^{FH}$ denote the respective thresholds.

This result is driven by a direct and an indirect effect (as can be seen from (8)): An increase in $k_{t}$ , that increases $W(k_{t})$ and thus the incentive to become an entrepreneur, reduces—ceteris paribus, i.e., keeping $k_{t+1}$ constant—the cut-off (and the number of entrepreneurs would increase). However, there is an indirect effect due to the increase in next period’s capital/labor ratio ( $H^{\prime }(k_{t})\gt 0$ ). $ k_{t+1}$ increases, $f^{^{\prime }}(k_{t+1})$ declines and the incentive to become an entrepreneur is reduced (the cut-off increases). Only if the indirect effect dominates the direct effect, the overall result will be an increase in the cut-off and a reduced number of entrepreneurs (see Appendix A for the exact condition). These adjustments not only result in a richer dynamic behavior of our model relative to Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016) but also have important implications for the variations of income inequality in both the long run and within the business cycle (see Section 4).

In Figure 5 below, we can compare—for a fixed $\gamma$ —the evolution of the capital/labor ratio $k_{t+1}$ when bad projects are not available (the higher dark blue line depicting $k_{t+1}=F(k_{t})$ ) with the case when bad projects are available and investment in bad projects is not constrained, $ k_{t+1}=H(k_{t})$ , in light blue. For future reference, we denote the point of intersection $k^{FH}$ . For $k_{t}=k^{FH}$ we have $r_{t+1}^{b}=B\left ( 1+\gamma \right )$ so that the banking system is indifferent between offering loans to entrepreneurs and investing in bad projects. Banks begin to invest in bad projects when $H(k_{t})\lt F(k_{t})$ , meaning that the capital/labor ratio $k_{t+1}$ is lower than it would be if such investments were not available.

Importantly, $H(k_{t})$ is upward sloping, which is in significant contrast to Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016), where the corresponding branch of the function is flat. In this model there is a representative agent, who adjusts her capital/labor ratio. In our model with heterogeneous agents, physical capital adjusts along two margins. Borrowing terminology from the international trade literature, over time the capital/labor ratio not only adjusts along the intensive margin as entrepreneurs of the same type adjusts their output levels (as in Matsuyama et al. Reference Matsuyama, Sushko and Gardini2016), but also along the extensive margin as the proportion of entrepreneurs changes.

As soon as $k_{t}$ grows to the level where the return to good projects is equal to the return of the bad projects all additional savings are invested in bad projects so that the two returns stay equal. Thus, in Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016), the function corresponding to $H(k_{t})$ is flat. In our model, agents are heterogeneous and therefore adjustments can happen on both margins. Our simulations shown below confirm that even with investments in bad projects, there is still a positive relationship between $k_{t}$ and $k_{t+1}$ (on $ H(k_{t})$ ). They illustrate the case in which as $k_{t}$ and $ k_{t+1}$ increase the threshold $z_{t+1}^{\ast }$ increases, that is, the mass of entrepreneurs decreases.

Note that this is also different from the evolution of capital/labor along $ F(k_{t})$ . In the absence of bad projects as $k_{t}$ increases the return to capital, and hence the two interest rates, decline. In fact the decline in the interest rate just compensates for the increase in the capital/labor ratio so that the marginal agent is still indifferent between becoming an entrepreneur or depositing her wages in the bank and becoming self-employed, so there is no adjustment on the extensive margin and $z_{t+1}^{\ast }$ remains constant. However, on $H(k_{t})$ the interest rate on deposits is equal to the return of the bad projects (adjusted for the cost of loans) and does not drop as some of the extra capital is invested in bad projects. There is also an adjustment on the extensive margin.

Finally, comparing the evolution of the capital-to-labor ratio in the case where bad projects are unavailable, as given by equation (6), with the case where investment in bad projects is unconstrained, implicitly represented by equation (9), we find that the latter is indirectly influenced by changes in the level of self-employment $y$ and the return on bad projects, $B$ . This influence occurs through their effect on the threshold level $z_{t+1}^{\ast }$ (see equation (8)), whereas these variables do not affect equation (6). This observation is also important for understanding the main methodological contribution of our paper relative to Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016).

To obtain the steady state we set $k_{t+1}=k_{t}=k^{\ast }$ and $ z_{t+1}^{\ast }=z^{\ast }$ in (8) and (9).

Any increase in $y$ or $B$ by encouraging investment in bad projects has a negative effect on $k^{\ast }$ . This is because an increase in $y$ makes it more attractive to agents depositing their endowments at the bank, thus increasing the supply of funds and depressing the interest rate. This favors investment in bad projects which can also be the result of a direct increase in the return of bad projects. Therefore, either of the above two changes shifts $H(k_{t})$ downwards moving the intersection with the $45^{\circ }$ -line to the left and below the original intersection and as a result $k^{\ast }$ will decrease (see Figure 5). When bad projects are unavailable, we previously established that an increase in $m$ positively affects $k^{\ast }$ as long as $\gamma$ remains small. Now, an additional effect emerges: banks have less incentive to invest in bad projects, as the rising return on capital leads to higher interest rates. As a result, the positive impact of $m$ on $k^{\ast }$ is even stronger. The comparative statics of $z^{\ast }$ depend on both direct and indirect effects, making the overall impact indeterminate unless further restrictions are imposed on the functional forms of production functions and the distribution of endowments.

2.3 Investment in bad projects: the constrained case

Finally, consider the case when the investment in bad projects is constrained. The following two conditions must hold: (a) $r_{t+1}^{b}\lt B\left ( 1+\gamma \right )$ when there is no investment in bad projects, and (b) $V_{t}\gt \hat {V} \left ( k_{t}\right )$ , where $V_{t}$ denotes the level of investment in bad projects that would have been obtained had such investment been unconstrained. Exactly because this investment is constrained the borrowing rate and the return to bad projects are not equalized. Now market clearing implies that $\int \nolimits _{z_{t+1}^{\ast }}^{\bar {z}}(1+\gamma )\left ( m-1\right ) W(k_{t})zg(z)dz+\hat {V}\left ( k_{t}\right ) =\int \nolimits _{\underline {z}}^{z_{t+1}^{\ast }}W(k_{t})zg(z)dz$ , which can be written as:

(10) \begin{equation} 1=\left ( m+\gamma m-\gamma \right ) \int \limits _{\underline {z}}^{z_{t+1}^{\ast }}zg(z)dz+\frac {\hat {V}\left ( k_{t}\right ) }{W\left (k_{t}\right ) }. \end{equation}

Let

(11) \begin{equation} k_{t+1}\equiv G(k_{t})=R(m)mW(k_{t})\int \limits _{z_{t+1}^{\ast }}^{\bar {z} }zg(z)dz \end{equation}

denote the evolution of the capital/labor ratio when the investment in bad projects is constrained.

$\hat {V}\left ( k_{t}\right )$ indicates the availability of bad projects. As we argued above, strong economies of scale implies that after the economy has reached a threshold level financial innovation and, hence, the availability of bad projects, accelerates. For sufficiently high values of $ \hat {V}^{\prime }\left ( k_{t}\right )$ we have $G^{\prime }(k_{t})\lt 0$ which is necessary for the existence of cycles. To obtain the steady state, we set $k_{t+1}=k_{t}=k^{\ast }$ and $ z_{t+1}^{\ast }=z^{\ast }$ in (10) and (11).

An increase in the cost of banking $\gamma$ depresses the capital/labor by reducing the availability of loans. Other things equal, this has a positive effect on the number of producers of good projects even if each one is offered a smaller loan. However, as bad projects offer a better return now this increases the incentives to deposit endowments at the bank having the opposite effect on the threshold. Once more, without imposing further restrictions on the functional forms of the distribution and production functions the effect of changes in financial constraints on $z^{\ast }$ remains ambiguous.

3.1 The dynamic map and its functional specification

In Figure 5, we present the complete phase map of the capital/labor ratio. The dark blue line, defined in (6) as $F(k_{t})$ , corresponds to that part of the map of the capital/labor ratio when there is no investment in bad projects. Banks would like to invest in bad projects when the light blue line falls below the dark blue line, i.e., for $k_{t}\gt k^{FH}$ .Footnote ¹⁶ Beyond that point investment in bad projects offers a higher return to the banks than lending to entrepreneurs. However, as the level of economic development does not support financial innovation, they cannot do so because such investment is not available. As soon as the dark blue line intersects the red line, defined in (11) as $G(k_{t})$ , banks begin to invest in bad projects which become available as a result of financial innovation. Here we assume that the availability increases sufficiently rapid to generate a negative slope of $G(k_{t})$ . We denote the intersection of the two segments, $F(k_{t})=G(k_{t})$ , as $k^{FG}$ . As long as the red line is above the light blue line, which is defined by (9) as $H(k_{t})$ , the investment in bad projects is constrained. As soon as $H(k_{t})$ intersects $ G(k_{t})$ , investment in bad projects is unconstrained and the evolution of the capital/labor ratio is captured by the light blue line. We denote the intersection of the last two segments of the map at $G(k_{t})=H(k_{t})$ as $ k^{GH}$ .

Then, we can define the whole dynamic map as:

(12) \begin{equation} k_{t+1}=\left \{ \begin{array}{l@{\quad}ll} F(k_{t}) & \text{if} & k_{t}\leq k^{FG} \\ G(k_{t}) & \text{if} & k^{FG}\lt k_{t}\lt k^{GH} \\ H(k_{t}) & \text{if} & k^{GH}\leq k_{t} \end{array} \right . . \end{equation}

The phase map of the evolution of the capital/labor ratio, and, hence, its dynamic path, is very similar to the one obtained by Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016). However, there is an added complexity due to the heterogeneity of agents according to their initial endowments. In this model, physical capital adjusts along two margins: (a) the intensive margin as entrepreneurs of the same type adjusts their output levels, and (b) the extensive margin as the proportion of entrepreneurs changes.

One of the implications of this added complexity is that the part of the map that corresponds to the case when investment in bad projects is unconstrained has always a positive slope while the corresponding part in Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016) is flat. The reason is that in the in this model as long as the capital/labor ratio is sufficiently high, so that the interest on borrowing (which in turn depends on the marginal product of physical capital) is less than the return on bad projects, the following period’s capital/labor ratio will drop to ensure that the two returns are equal. However, in our paper there is a second margin of adjustment. The interest rate that banks charge to entrepreneurs will still be equal to the return of the bad projects but not the marginal productivity of capital which also depends on the fraction of entrepreneurs (which changes as well, see Proposition2).

This added complexity limits our ability to study the dynamics of the full model analytically, and therefore, we explore its properties by mostly relying on numerical analysis. In order to do so, we have to specify functional forms for the dynamic map. In particular, we use a Cobb-Douglas production function and we linearize the $G(k_{t})$ function. The specifications are summarized in the following Proposition:

Proposition 6. Suppose that the production function is symmetric Cobb-Douglas, $f(k)=Ak^{ \frac {1}{2}}$ , and the distribution of endowments is uniform, $g(z)\sim U[0,2]$ .Footnote ¹⁷

(a) If $k_{t}\leq k^{FG}$ then

\begin{equation*} k_{t+1}=F(k_{t})=\frac {1}{2}\frac {R(m)mA}{m+\gamma m-\gamma }k_{t}^{\frac {1}{2} } \end{equation*}

with the corresponding steady-state solution given by:

\begin{equation*} k^{\ast }=\left ( \frac {\frac {1}{2}R(m)A}{1+\gamma (1-\frac {1}{m})}\right ) ^{ \frac {1}{2}}. \end{equation*}

(b) For $k^{FG}\lt k_{t}\lt k^{GH}$ we impose the following linear functional form on $G(k_{t})$ :

\begin{equation*} k_{t+1}=G(k_{t})=\delta \left ( k_{t}-k^{FG}\right ) +F\left ( k^{FG}\right ) =\delta \left ( k_{t}-k^{FG}\right ) +\frac {R(m)mA}{m+\gamma m-\gamma }\frac { \sqrt {k^{FG}}}{2}. \end{equation*}

This specification ensures that $G(k_{t})$ passes through the point $\left ( k^{FG},F\left ( k^{FG}\right ) \right )$ and $\delta$ approximates its slope. The corresponding steady-state solution is given by:

\begin{equation*} k^{\ast }=\frac {1}{1-\delta }\left ( -\delta k^{FG}+\frac {R(m)A}{1+\gamma (1- \frac {1}{m})}\frac {\sqrt {k^{FG}}}{2}\right ) . \end{equation*}

(c) If $k^{GH}\leq k_{t}$ then the solution for $H(k_{t})$ is implicitly defined by the fourth degree polynomial:

\begin{equation*} \Phi _{4}x^{4}+\Phi _{3}x^{3}+\Phi _{2}x^{2}+\Phi _{1}x+\Phi _{0}=0, \end{equation*}

where $x=\sqrt {k_{t+1}}$ , $\Phi _{4}\equiv B^{2}(m+\gamma m-\gamma )^{2}k_{t}^{\frac {1}{2}}$ , $\Phi _{3}\equiv -AR(m)mB(m+\gamma m-\gamma )k_{t}^{ \frac {1}{2}}$ , $\Phi _{2}\equiv \frac {\left ( AR(m)m\right ) ^{2}}{4}k_{t}^{\frac { 1}{2}}-B^{2}(m+\gamma m-\gamma )^{2}\frac {AR(m)m}{2}k_{t}+\frac {R(m)my^{2}}{2A}$ , $ \Phi _{1}\equiv \frac {\left ( AR(m)m\right ) ^{2}}{2}B\left ( m+\gamma m-\gamma \right ) k_{t}$ and $\Phi _{0}\equiv -\frac {\left ( AR(m)m\right ) ^{3}}{8}k_{t}$ . The corresponding steady-state solution is determined by the following third degree polynomial:

\begin{equation*} \Psi _{3}\left ( x^{\ast }\right ) ^{3}+\Psi _{2}\left ( x^{\ast }\right ) ^{2}+\Psi _{1}x^{\ast }+\Psi _{0}=0,\end{equation*}

where $\Psi _{3}\equiv B^{2}(m+\gamma m-\gamma )^{2}$ , $\Psi _{2}\equiv -mR(m)AB(m+\gamma m-\gamma )\left ( 1+\frac {1}{2}(m+\gamma m-\gamma )B\right )$ , $\Psi _{1}\equiv m^{2}R(m)^{2}A^{2}\left ( \frac {1}{4}+\frac {1}{2}(m+\gamma m-\gamma )B\right )$ and $\Psi _{0}\equiv -R(m)m\left ( \frac {1}{8} m^{2}R(m)^{2}A^{3}-\frac {1}{2A}y^{2}\right )$ .

Before proceeding to numerical simulations, we provide complete analytical results for the case when the investment in bad projects is always constrained, that is $F(k^{FG})\lt k^{GH}$ . The inequality implies that the dynamics in the interval $[0,k^{GH}]$ involve only the first two branches $F$ and $G$ . Despite these restrictions, we show that the dynamics of the model are still very complex involving a rich menu of oscillations from periodic cycles to chaotic attractors.

It is not surprising that the numerical analysis below confirms that, broadly, the dynamic behavior of our model is similar to the one in Matsuyama et al. (Reference Matsuyama, Sushko and Gardini2016) although the introduction of agent heterogeneity has added some, from an economic point of view, interesting complexities. In particular, the inclusion of both margins in our model offers a much richer picture of the behavior of income inequality over the business cycle. The model is consistent with a great variety of equilibrium dynamic paths that depend on the values of particular parameters. There are cases where such paths converge to a unique steady state. But more often we find that the paths converge to cycles that are either regular or irregular and of various degrees of periodicity including chaotic dynamics.

However, in the following numerical explorations, we focus on low-period cycles, as these allow for a more transparent economic interpretation in terms of boom-and-bust dynamics. This choice supports our goal of linking endogenous fluctuations to economically meaningful mechanisms and revealing structural dynamics, rather than emphasizing complex or chaotic behavior per se.Footnote ²⁰

3.2 Numerical solution

Our model is intentionally stylized and not suited for exact calibration; instead, we explore its behavior across a wide range of economically plausible parameterizations, aiming to remain as consistent as possible with key stylized facts. The debt-to-total-assets ratio for most non-financial firms is between $0$ and $1/2$ , corresponding to $m$ values of $1$ and $2$ . Thus, we set $m=1.5$ . The value of $R$ is equal to the inverse price of capital formation, given that we use the final good as the numeraire and assume full capital depreciation at the end of each period. According to the Penn World Tables (Feenstra et al. Reference Feenstra, Inklaar and Timmer2015, PWT Version 10.01), the average price level of capital formation relative to the price level of U.S. GDP during the period 1950–2019 is approximately $0.4$ , implying $R=2.5$ . The cost of loans, $\gamma$ , in a competitive banking system can be approximated by the net interest rate margin, i.e., the difference between bank lending and deposit rates. Over the last 10 years, the margin for U.S. banks has been around 3 percentage points.Footnote ²¹ The value of $y$ is set to 1 because, for this value, our simulations align with empirical evidence. Specifically, Hamilton (Reference Hamilton2000) finds that, over a 10-year period, average earnings from self-employment were 35% lower than wages. On average, across all our simulations, the ratio of depositors’ income (which includes both savings income and self-employment income) to wages closely matches this value.

In our bifurcation simulations below, we focus on three core parameters: $A$ , $\delta$ , and $\gamma$ , which we vary over a broad range. Parameters $\delta$ and $\gamma$ directly influence the bank’s project choices, thereby shaping the mechanism that generates cycles in our model. Changes in $\delta$ affect the availability of bad projects, while variations in $\gamma$ reflect the cost of financing good projects. We find that increases in the loan cost $\gamma$ produce qualitatively similar results to increases in the return of bad projects, $B$ , and thus we do not vary the latter. Similarly, variations in $\gamma$ have effects opposite to those of changes in the benefits of intermediation, represented by $m$ . We prefer to present results in terms of $\gamma$ rather than $m$ , as the latter depends on the specification of $R(m)$ . While our qualitative results remain robust to alternative specifications, the quantitative results are sensitive. Therefore, we select fixed values for $m$ and $R$ that align with their long-run averages. Finally, variations in $A$ capture changes in economic productivity. By setting $k_{FG} = 6$ (implying $A^{BC} = 1.97$ ) and varying $A$ between $2$ and $3.5$ we illustrate the analytic results beyond the border collision (see Property 3 in Proposition7 and Remark 2). The full set of parameter values for our benchmark calibration is as follows:

\begin{equation*} R = 2.5, \quad A = 2.5, \quad m = 1.5, \quad \gamma = 0.03, \quad \delta = -1.25, \quad y = 1, \quad B = 1.25, \quad k^{FG} = 6. \end{equation*}

3.2.1 Numerical analysis: variations in $\delta$

In this section, we fix all other parameters at their benchmark values mentioned above and focus on $\delta$ . Recall that $\delta$ is the slope of the $G(k_{t})$ branch, when investment in bad projects is constrained. More rapid generation of bad projects is reflected in a steeper $G(k_{t})$ branch, thus in a lower $\delta$ value. We let $\delta$ vary over a wider range to produce the bifurcation diagram shown in Figure 6. The diagram shows the values of the capital/labor ratio approach asymptotically fixed points and periodic cycles as we vary $\delta$ . A small interval with chaotic attractors is also visible. We provide a detailed description of the economic mechanism behind these dynamics.

Figure 6.

Bifurcation on $\delta$ .

Notes: This figure shows the bifurcation diagram of $k$ on parameter $\delta$ . The parameter values are: $-3\leq \delta \leq 0$ , $A=2.5$ , $\gamma =0.03$ , $R=2.5$ , $m=1.5$ , $y=1$ , $B=1.25$ , $k^{FG}=6$ .

Given that $\delta$ takes negative values it is more convenient to describe what happens as its values decrease (increase in absolute terms). For very high values of $\delta$ (close to zero) we have convergence to a unique steady state. Availability of bad projects is relatively low. In terms of Figure 5, as $\delta$ decreases the red line gets steeper by rotating clockwise. As the economy grows it begins to oscillate eventually converging to a unique steady state (on the red line).

For slightly lower values of $\delta$ the fixed point loses stability and we obtain stable period- $2$ cycles as shown in Figure 7. In this case, the economy rotates between a high capital/labor ratio state with no investment in bad projects, which we denote by A, and a low capital/labor ratio state with constrained investment in bad projects, denoted by B. We refer to this period- $2$ cycle as Type 1. When in some period $\left ( t-1\right )$ the capital/labor ratio and, hence, wages are high, at period $t$ there will be a lot of funds available and banks will invest some of them in bad projects (but not as much as they would if the availability of bad projects was unconstrained). The reason is that if all funds were lent to entrepreneurs, the marginal productivity of capital, and hence the equilibrium borrowing interest rate would drop well below the return to bad projects. However, in the new equilibrium the interest rate will still be below the return of bad projects given that the level of financial innovation is not extensive enough to create a quantity of bad projects consistent with the equalization of the two rates. As some funds are invested in bad projects the new equilibrium capital/labor in period $t$ , and hence the wage rate, is below that of period $\left ( t-1\right )$ . Then, at $\left ( t+1\right )$ the new level of deposits would be too low to support financial innovation and all funds will now be lent to entrepreneurs. The new higher capital/labor ratio marks the beginning of a new cycle.

Figure 7.

Type 1 period-2 cycle.

Notes: This figure shows the phase map and the corresponding period-2 cycle, which we refer to as Type 1. The parameter values used are $R=2.5$ , $A=2.5$ , $m=1.5$ , $ \gamma =0.03$ , $y=1$ , $B=1.25$ , $k^{FG}=6$ and $ \delta =-1.25$ .

Figure 8.

Period-4 mixed cycle.

Notes: This figure shows the phase map and the corresponding period-4 mixed cycle. The parameter values used are $R=2.5$ , $A=2.5$ , $m=1.5$ , $ \gamma =0.03$ , $y=1$ , $B=1.25$ , $k^{FG}=6$ and $ \delta =-2$ .

For still lower values of $\delta$ we enter a small region where high periodicity cycles appear including chaotic attractors. But eventually the economy settles into a period- $4$ cycle (see Figure 8). Now in one of the low states the investment in bad projects is constrained while in the other low state this investment is unconstrained. We refer to this cycle as a Mixed Period- $4$ Cycle. Suppose that at $\left ( t-1\right )$ the economy has reached the high capital/labor ratio state. Next period the level of deposits will allow some investment in bad projects but the level of financial innovation constrains that investment. Because at $t$ the investment in bad projects was constrained the capital/labor ratio and, thus, wages did not drop as much as they would have done had the investment in bad projects being unconstrained. Thus, at $\left ( t+1\right )$ the economy moves to a very high capital/labor ratio. This is because (a) the previous period the investment in bad projects was constrained thus allowing for more lending to entrepreneurs, and (b) this period there is no investment in bad projects. Next, the good performance of the economy in period $\left ( t+1\right )$ means that at $\left ( t+2\right )$ there are plenty of deposits, intensive financial innovation and unconstrained investment in bad projects which, in turn, implies that the economy has moved to the very low capital/labor ratio state. Next, period there will not be any investment in bad projects and the economy enters a new cycle.

Lastly, for very low values of $\delta$ , financial innovation is very efficient and the economy follows a new period- $2$ cycle shown in Figure 9. We will refer to this cycle as Type 2 Period- $2$ Cycle. In terms of Figure 5 the red line is much steeper. Now investment in bad projects is unconstrained given that the interest rate banks receive on their loans, after adjusting for the costs of creating new loans, is equal to the return on bad projects.

Figure 9.

Type 2 period-2 cycle.

Notes: This figure shows the phase map and the corresponding period-2 cycle, which we refer to as Type 2. The parameter values used are $R=2.5$ , $A=2.5$ , $m=1.5$ , $ \gamma =0.03$ , $y=1$ , $B=1.25$ , $k^{FG}=6$ and $ \delta =-3$ .

Figure 10.

Bifurcation on $A$ .

Notes: This figure shows the bifurcation diagram of $k$ on parameter $A$ . The parameter values are: $1.5\leq A\leq 3.5$ , $ \gamma =0.03$ , $R=2.5$ , $m=1.5$ , $ \delta =-1.25$ , $y=1$ , $B=1.25$ , $k^{FG}=6$ . Vertical lines denote from left to right, $2.5$ , $2.66$ , and $2.8$ , respectively.

3.2.2 Numerical analysis: variations in $A$

Figure 10 shows the bifurcation diagram for $A$ , where there are five distinct regions.Footnote ²² For very low values of $A$ , we observe a unique steady state. Since the productivity of good projects is low, the capital-labor ratio and wages are also low. Consequently, the banking sector remains too small to support financial innovation, and no investment occurs in bad projects. As $A$ increases, the economy enters a region of Type 1 Period- $2$ Cycles (A and B). Here, the economy alternates between periods where the capital-labor ratio is high with no investment in bad projects and periods where the capital-labor ratio is low, leading banks to invest in bad projects, albeit in a constrained manner.

Further increases in $A$ move the economy into a region of Type 2 Period- $2$ Cycles (C and D). In this regime, the productivity of good projects is sufficiently high so that, in periods with no investment in bad projects, wages and savings rise. As a result, in the following period, the banking sector expands enough to support financial innovation at a level that allows unconstrained investment in bad projects. Even higher values of $A$ introduce a new type of cycle where investment in bad projects occurs in every period. However, in one state, investment remains constrained, while in the other state, it is unconstrained. We refer to this type of cycle, illustrated in Figure 11, as the Type 3 Period- $2$ Cycle (E and F). In this case, even when investment in bad projects is unconstrained, the economy performs well enough to sustain financial innovation in the subsequent period, albeit at a lower level.

Finally, for very high values of $A$ , the economy converges once again to a unique steady state, where investment in bad projects is always unconstrained. Despite the large allocation of funds to bad projects, the capital-labor ratio remains sufficiently high. The high productivity of good projects supports the level of financial innovation necessary to set the equilibrium interest rate equal to the return on bad projects.

Figure 11.

Type 3 period-2 cycle.

Notes: This figure shows the phase map and the corresponding period-2 cycle, which we refer to as Type 3. The parameter values are: $R=2.5$ , $ \gamma =0.03$ , $A=2.8$ , $m=1.5$ , $ \delta =-1.25$ , $y=1$ , $B=1.25$ , and $k^{FG}=6$ .

3.2.3 Numerical analysis: variations in $ \gamma$

Finally, we turn to the cost of banking and vary $\gamma$ over a wide range (while keeping the other parameters at their benchmark). Figure 12 shows the bifurcation diagram for $\gamma$ .

For very high values of the cost of lending, the economy always converges to a unique steady state where there is no financial innovation and no investment in bad projects. The banking system becomes highly inefficient, allocating too many resources to its operations rather than offering funds to entrepreneurs. As the cost of lending declines, the economy eventually enters a region characterized by Type 1 Period- $2$ Cycles. However, despite the decline in lending costs, too many resources are still absorbed by the cost of providing loans to entrepreneurs. As a result, aggregate economic activity can only support a moderate level of financial innovation, and investment in bad projects remains constrained.

Figure 12.

Bifurcation on $\gamma$ .

Notes: This figure shows the bifurcation diagram of $k$ on parameter $\gamma$ . The parameter values are: $0\leq \gamma \leq 1$ , $R=2.5$ , $A=2.5$ , $m=1.5$ , $ \delta =-1.25$ , $y=1$ , $B=1.25$ , and $k^{FG}=6$ .

4.1 Numerical analysis

We present numerical results for each of the three Period-2 Cycles defined above. What we find is that the qualitative effects of business cycles on income inequality are robust not only to changes in the parameter values but also as we move from one type of cycle to another. The following parameter values are the same in all three numerical examples:

\begin{equation*} \gamma =0.03,R=2.5,m=1.5,y=1,B=1.25\text{ and }k^{FG}=6. \end{equation*}

The Type 1 period-2 Cycle (A and B) is presented for $A=2.5$ and $\delta =-1.25$ (as used for Figure 7); the Type 2 period-2 Cycle (C and D) is presented for $A=2.5$ and $\delta =-3$ (as used for Figure 9); the Type 3 period-2 Cycle (E and F) is presented for $A=2.8$ and $\delta =-1.25$ (as used for Figure 11). Table 1 shows, for each type of cycle, the proportions of each type of agent, the corresponding aggregate income for each group of agents, and the average income within each group (for workers who have a unit measure, aggregate and average income are the same). In all three types of cycles, the proportion of entrepreneurs increases as the economy moves from the low states (points B, D, and F) to the high states (A, B, and C) because investment in bad projects decreases, and thus banks lend more funds to entrepreneurs. Those who gain in high states are workers, as the higher capital/labor ratio implies that the marginal productivity of labor increases. Entrepreneur income is countercyclical because of the decline in the marginal productivity of capital during booms. However, as a group, they earn more because their numbers significantly increase during booms. Our numerical results also show that the income of depositors, after subtracting the part earned from bad projects, is procyclical.

Table 1.

Impact of cycles on income distribution dynamics

Notes: The parameter values are as follows. Type 1: $A=2.5$ and $\delta =-1.25$ ; Type 2: $A=2.5$ and $\delta =-3$ ; Type 3: $A=2.8$ and $\delta =-1.25$ ; the other parameters are at their baseline values ( $\gamma =0.03,R=2.5,m=1.5,y=1,B=1.25$ and $k^{FG}=6$ ).

We are also interested in understanding the impact of cycles on the relative shares of entrepreneurs and workers, as well as on inequalities within groups. Table 2 below shows the income shares of each group of agents along the two phases of each of the three types of cycle, using the same parameter values as those in Table 1. We find that entrepreneurial shares are procyclical, while average entrepreneurial income is countercyclical. This is because the higher proportion of entrepreneurs during periods of expansion more than compensates for the decline in productivity. While we have set the share factor in the Cobb-Douglas function equal to $0.5$ , the share of wage income is below $0.5$ even at points A and C, where there is no investment in bad projects. This is because depositors earn an additional income from self-employment. In the other states, some funds are also allocated to bad projects. Thus, the aggregate production function is a hybrid of the Cobb-Douglas function, the self-employment technology, and, when available, the technology producing bad projects. In fact, when banks invest in bad projects, both workers and entrepreneurs involved in the production of good projects find that their income shares decline.

Table 2.

Impact of cycles on income shares

Notes: The parameter values are as follows. Type 1: $A=2.5$ and $\delta =-1.25$ ; Type 2: $A=2.5$ and $\delta =-3$ ; Type 3: $A=2.8$ and $\delta =-1.25$ ; the other parameters are at their baseline values ( $\gamma =0.03,R=2.5,m=1.5,y=1,B=1.25$ and $k^{FG}=6$ ).