2.1. Firms

Final good sector. The final good is a composite of differentiated intermediate goods indexed by $\omega \in [0, 1]$ . The production technology is given by

\begin{equation*} Y_t= Z_t \exp \left [ \int _0^1 \ln \left ( \lambda ^{K_t(\omega )} x_t(\omega ) \right ) d\omega \right ]\! , \end{equation*}

where $Y_t$ is the output of the final good, $x_t(\omega )$ is the demand for variety $\omega$ , $K_t(\omega ) (\!=1, 2, \ldots )$ represents the highest quality of variety $\omega$ in period $t$ , and $\lambda \gt 1$ represents the size of the quality improvement achieved by an innovation. Without loss of generality, I assume the initial condition $K_0(\omega )=1$ for all $\omega$ . Then, $K_t(\omega )-1$ is the number of occurrences of quality-upgrading innovations for $\omega$ before period $t$ . The term $Z_t$ is the exogenous technology level, growing at a constant rate of $g_Z\gt 0$ . Even if this term were not present, the qualitative results would not change at all. I introduce the term $Z_t$ to capture the fact that there are other contributing factors to productivity growth in addition to R&D activities.Footnote ⁷

Following Grossman and Helpman (Reference Grossman and Helpman1991b, Ch.4), I take the final expenditure as the numeraire: $P_t Y_t=1$ , where $P_t$ is the price of the final good. Thus, in this model, all prices are evaluated in terms of the final expenditure. Let $p_t(\omega )$ denote the price of variety $\omega$ . Profit maximization yields the demand function for variety $\omega$ : $ x_t(\omega )=1/p_t(\omega )$ and the zero-profit condition:

(1) \begin{equation} P_t= \dfrac{1}{Z_t} \exp \left [ \int _0^1 \ln \left ( \dfrac{p_t(\omega )}{ \lambda ^{K_t(\omega )}} \right )d\omega \right ]\!. \end{equation}

Intermediate good sector. Producing $x_t(\omega )$ units requires the same units of labor as inputs, implying that the wage rate $W_t$ is the unit cost of production. As in Grossman and Helpman (Reference Grossman and Helpman1991a, Reference Grossman and Helpman1991b: Ch.4), each variety has several potential suppliers that can produce the good with a quality of less than $K_t(\omega )$ . The leader firm determines its price as $p_t( \omega ) =\lambda W_t$ to monopolize the market, and it sells $x_t(\omega ) = 1/ (\lambda W_t)$ units of the good. The resulting profit is $\pi _t(\omega )=\pi \equiv 1-1/\lambda$ .

Let $Q_t$ denote the end-of-period stock price of the leader firm. Here, “end-of-period” has two meanings. First, $Q_t$ is ex-dividend, that is, $Q_t$ is evaluated after the dividend in period $t$ has been paid. Second, $Q_t$ is evaluated after it turns out that the innovation did not occur in period $t$ .Footnote ⁸ Let $R^e_{t+1}$ denote the one-period gross rate of return from holding the equity from the end of period $t$ to $t+1$ .

(2) \begin{equation} R^e_{t+1} \equiv \dfrac{\pi +(1-I_{t+1})Q_{t+1} }{Q_t}, \end{equation}

where $I_{t+1} \in [0, 1]$ denotes the probability that an innovation by potential entrants succeeds in period $t+1$ and the current leader loses its market power. $I_t$ is determined endogenously from the resource constraint in this economy. As in the literature on quality-ladder growth, $I_t$ is independent and identically distributed (i.i.d.) across varieties. Then, from the law of large numbers, $I_t$ is equal to the ex-post measure of varieties in which innovation occurs.

If each potential entrant hires $\kappa I_t$ units of labor in period $t$ , then it can succeed in innovation with probability $I_t$ , where $\kappa \gt 1$ is the labor requirement to obtain 100% success in innovation. If the innovation succeeds, then the entrant becomes the new leader firm for one variety from period $t+1$ . Consequently, the new leader faces the idiosyncratic risk of the next innovation and other aggregate risks. Therefore, the expected benefit of innovation in period $t$ is given by $I_t Q_t$ . Then, the free entry condition of R&D activities for a variety is $Q_t \leq W_t \kappa$ , the equality of which holds if $I_t\gt 0$ , that is, R&D is conducted. Throughout this study, I focus on the equilibrium with $I_t \gt 0$ .

(3) \begin{equation} Q_t= W_t \kappa. \end{equation}

2.2. Households

I formulate this sector in a similar way as Gertler et al. (Reference Gertler, Kiyotaki and Prestipino2020). There is a continuum of households, and each household in turn consists of a continuum of family members with measure $1+f\gt 0$ , where $f\in (0, 1)$ is constant. Within a household, members are classified into workers and bankers. The measure of workers is 1, while that of bankers is $f$ . I normalize the measure of households to 1 so that the total population is constant at $1+f$ .Footnote ⁹ Each worker supplies labor to earn wages, and each banker manages a bank. The detail of the bankers’ behavior is explained in Section 2.3.

As seen in Section 2.1, the measure of profit-earning intermediate good firms is unity. Let $S^h_t$ be the number of firms whose equities are held directly by the households and $S^b_t$ be the number of firms whose equities are intermediated by the bankers. Then,

\begin{equation*} S^h_t+S^b_t=1. \end{equation*}

Within the household, the members consume the same amount of the final good. Let $C_t$ denote the amount of aggregate real consumption. Each member consumes $C_t/(1+f)$ . Each worker is endowed with one unit of time. Since the population of workers is normalized to 1, $L_t$ also represents the total labor supply. The representative household’s utility is given byFootnote ¹⁰

\begin{equation*} E_0 \left \{ \sum _{t=0}^\infty \beta ^t \Big [ \ln C_t + \zeta \ln\! (1-L_t) - \Gamma \big(S^h_t\big) \Big ] \right \}\! , \end{equation*}

where $\beta \in (0, 1)$ is the discount factor, $ \zeta \gt 0$ is the weight of the utility from leisure, and $ E_t(\!\cdot\! )$ is the expectation operator conditioned on the information available in period $t$ . Function $\Gamma$ represents the disutility from the household’s direct equity holding. Following Gertler et al. (Reference Gertler, Kiyotaki and Prestipino2020), I introduce this disutility function to simply capture the household’s lower efficiency in handling equity investments compared to banks.Footnote ¹¹ I assume that function $\Gamma$ satisfies

\begin{equation*} \Gamma ^{\prime}(S^h)\gt 0, \Gamma ^{\prime\prime}(S^h)\gt 0 \text { for } S^h\gt 0, \Gamma ^{\prime}(0) = 0. \end{equation*}

By the law of large numbers, the fraction $I_t$ of the leader firms are leapfrogged at the end of period $t$ ; the stock price of these firms then becomes zero. As in the existing studies employing the quality-ladder growth model, I assume that the household can diversify equity investments. Thus, the households are not exposed to any risk other than the aggregate financial shocks that we will see in Section 2.3. Therefore, the budget constraint is given byFootnote ¹²

\begin{equation*} R^d_t D_{t-1}+R^e_t Q_{t-1} S^h_{t-1} +W_t L_t +\Pi ^{bank}_t -T_t = P_t C_t +D_t +Q_t S^h_t , \end{equation*}

where $D_t$ represents deposits, $R^d_t$ is the gross rate of return on the deposits, $T_t$ represents lump-sum taxes, and $\Pi ^{bank}_t$ shows the transfers from bankers; how $\Pi ^{bank}_t$ is determined is explained in Section 2.3.

The household chooses $C_t, L_t, S^h_t$ , and $D_t$ to maximize the utility subject to the budget constraint. The conditions for utility maximization are given by

\begin{align*} &\dfrac{\zeta }{ 1-L_t}=\dfrac{W_t}{P_t C_t},\\ &\dfrac{1}{P_t C_t}=\beta E_t \left (\dfrac{1}{P_{t+1}C_{t+1}} R^d_{t+1}\right )\!, \\ &\dfrac{\Gamma ^{\prime}\big(S^h_t\big)}{Q_t}+\dfrac{1}{P_t C_t}= \beta E_t \left ( \dfrac{1}{P_{t+1} C_{t+1} } R^e_{t+1} \right )\!. \end{align*}

Since the market equilibrium of the final good implies $P_tC_t=P_t Y_t (\!=1)$ , these conditions can be rewritten as

(4) \begin{align} &L_t = 1-\dfrac{\zeta }{W_t}, \end{align}

(5) \begin{align} &E_t R^d_{t+1} =\dfrac{1}{\beta }, \end{align}

(6) \begin{align} &E_t \!\left ( R^e_{t+1}-R^d_{t+1} \right )= \dfrac{\Gamma ^{\prime}\big(S^h_t\big)}{\beta Q_t}. \end{align}

2.3. Banks

Each banker manages a bank. Hereafter, I use bankers and banks interchangeably. The aggregate net revenue of bankers in period $t$ is given by $R^e_t Q_{t-1}S^b_{t-1}-R^d_t D_{t-1}$ . At the end of each period, each banker faces an idiosyncratic risk of exit that occurs with probability $1-\delta \in (0, 1)$ . Throughout this study, I assume the following inequality:

Each bank, if it is hit by the exit shock, gives its net revenue to the household. Since the exit probability is i.i.d. across bankers, the $1-\delta$ share of the aggregate net revenue is transferred to the household:

\begin{equation*} \Pi ^{bank}_t= (1-\delta )\big(R^e_t Q_{t-1}S^b_{t-1}-R^d_t D_{t-1}\big). \end{equation*}

After exiting, a banker becomes a worker starting in the next period. To keep the populations of both workers and bankers constant over time, the workers with mass $(1-\delta )f \in (0, 1)$ are randomly chosen at the end of each period to act as bankers starting in the next period.

Consider a bank with its net revenue given by $n_t = R^e_t Q_{t-1} s^b_{t-1} -R^d_t d_{t-1}$ , where $s^b_{t-1}$ is the measure of firms purchased by this bank and $d_{t-1}$ is the issued deposits. If this bank is not hit by the exit shock, it then finances equity purchases $Q_t s^b_t$ with this revenue and newly issued deposits:

(7) \begin{equation} Q_t s^b_t=n_t+d_t. \end{equation}

Then, this bank’s net revenue in period $t+1$ is given by $n_{t+1}=R^e_{t+1}Q_t s^b_t -R^d_{t+1} d_t$ . Note that (7) represents the banker’s balance sheet and $n_t$ corresponds to the banker’s net worth. Henceforth, I simply call $n_t$ net worth. In Appendix A.1, it is shown that the banker’s objective function is given by

\begin{equation*} \tilde V_t \equiv E_t\! \left \{\sum _{j=1}^\infty \beta ^j (1-\delta ) \delta ^{j-1} n_{t+j} \right \}\!. \end{equation*}

The term $(1-\delta )\delta ^{j-1}$ is the conditional probability of exit in period $t+j$ given that the bank does not exit in period $t$ . Let $V_t(n_t) \equiv \max \tilde V_t$ denote the value function. The banker’s optimization problem is written as the Bellman equation:

\begin{align*} V_t(n_t)=\max _{s^b_t, d_t} E_t \big \{ \beta \big [(1-\delta )n_{t+1}+\delta V_{t+1}(n_{t+1}) \big ] \big \}. \end{align*}

The bank faces the balance sheet condition (7) and the following constraint:

(8) \begin{equation} \widetilde V_t \geq \theta _t Q_t s^b_t, \end{equation}

which comes from the potential moral hazard problem. After buying equities, the bank has the following two options. One is to hold the assets, receive dividends, and then meet its deposit obligations in period $t+1$ . The other is to secretly sell the assets to obtain the funds for its own use. To remain undetected, the bank can sell only up to fraction $\theta _t$ of the assets. Inequality (8) is a constraint in which the bank has no incentive to choose the latter option. In this model, a change in $\theta _t$ generates a financial shock. $\theta _t$ changes according to

\begin{equation*} \ln\! (\theta _{t+1}/\theta )=\rho \ln\! (\theta _t/\theta )+\varepsilon _{t+1}, \end{equation*}

where $\theta$ is the baseline value of $\theta _t$ , $\varepsilon _t$ is an i.i.d. shock, and $\rho \in (0, 1)$ is the parameter specifying the persistence of shocks.

To solve the problem, we can use the guess and verify method. Guess $V_t(n_t)$ as a linear function of $n_t$ : $V_t(n_t)=\psi _t n_t$ . The Bellman equation is rewritten as

\begin{equation*} \psi _t n_t = E_t \left \{ \beta (1-\delta +\delta \psi _{t+1}) \max _{s^b_t} \left [ R^d_{t+1} n_t + \left (R^e_{t+1}-R^d_{t+1}\right )Q_t s^b_t\right ] \right \}, \end{equation*}

subject to $\psi _t n_t \geq \theta _t Q_t s^b_t.$ Then, as long as $R^e_{t+1}-R^d_{t+1}\gt 0$ , the bank invests as much as it can:

(9) \begin{equation} Q_t s^b_t = \dfrac{\psi _t n_t}{\theta _t}. \end{equation}

Substituting this result into the Bellman equation yields the dynamic equation of $\psi _t$ :

(10) \begin{align} \psi _t= \dfrac{ \beta E_t\! \left [ (1-\delta +\delta \psi _{t+1} ) R^d_{t+1} \right ]}{\left \{ 1- \frac{\beta }{\theta _t} E_t\left [ (1-\delta +\delta \psi _{t+1} ) \left (R^e_{t+1}-R^d_{t+1}\right ) \right ] \right \} }. \end{align}

The denominator is assumed to be positive:

Consider a banker that newly enters the market. Let $e_t$ denote the new banker’s initial net worth and assume that this is fully subsidized by the government. The new banker’s behavior is then given by (7) and (9), with $n_t$ replaced by $e_t$ . The same equation as (10) is then implied for the new banker. Let $N$ denote the aggregate net worth of banks. To obtain the equilibrium of the model, I assume that the subsidies to each new banker are proportional to the average net worth in the previous period. Since the measure of bankers is always $f$ ,

\begin{equation*} e_t= \mu N_{t-1}/f, \end{equation*}

where I assume that $\mu \gt 0$ is not so large:

This assumption is required to obtain the uniqueness of the equilibrium. $N_t$ is then given by the sum of the incumbent banks’ net worth as well as that of the new entrants:

\begin{align*} N_t = \delta \big(R^e_t Q_{t-1} S^b_{t-1}-R^d_t D_{t-1}\big) + (1-\delta ) \mu N_{t-1}. \end{align*}

Since each bank’s decision about equity holding $Q_t s^b_t$ is linear in its state variable $n_t$ or $e_t$ , these decisions are easily aggregated over all banks. Given $N_t$ , $Q_t S^b_t$ is given by

\begin{equation*} Q_t S^b_t=\dfrac {\psi _t N_t}{\theta _t}. \end{equation*}

Thus, $\psi _t/\theta _t$ represents the banks’ leverage.

2.4. Government

The government’s budget constraint is given by

\begin{equation*} T_t= (1-\delta ) \mu N_{t-1}, \end{equation*}

from which $T_t$ is determined.

2.5. Market-clearing conditions

The timing of events during a given period is summarized as follows.

1. 1. Aggregate financial shocks are realized. The workers determine their labor supply, the final and intermediate good firms produce the goods, and the intermediate good firms pay dividends to the equity owners.

2. 2. The outcomes of R&D are realized. By the law of large numbers, the fraction $I_t$ of the leader firms is leapfrogged and the stock price of these firms becomes zero. Since the shareholders have diversified equity investments, their total values of equity change from $Q_{t-1}S^{h(b)}_{t-1}$ to $Q_t(1-I_t)S^{h(b)}_{t-1}$ . Their gross interest income from holding equities is $\pi +Q_t(1-I_t)S^{h(b)}_{t-1} =R^e_t Q_{t-1}S^{h(b)}_{t-1}$ . In this stage, the households also obtain the gross interest income from their deposits, $R^d_t D_{t-1}.$

3. 3. Each bank exits in this stage with an i.i.d. probability of $1-\delta \in (0, 1)$ . Upon exit, the profits of such banks, $\Pi ^{bank}_t$ , are transferred to the households. The workers with mass $(1-\delta )f$ become new bankers and enter the financial market with their initial net worth subsidized by the government. The households pay the lump-sum taxes $T_t$ to the government.

4. 4. The asset markets open. The households consume the final good and determine their portfolios, $Q_t S^h_t$ and $D_t$ , respectively. The bankers buy the equities $Q_t S^b_t.$

The market-clearing condition for the final good is $Y_t=C_t =1/P_t$ . The labor market clears as

(11) \begin{equation} L_t = \dfrac{1}{\lambda W_t}+\kappa I_t. \end{equation}

The market-clearing condition of equities is $S^h_t+S^b_t=1$ . Finally, the deposits $D_t$ must satisfy

(12) \begin{equation} D_t+N_t= Q_t S^b_t. \end{equation}

From these market-clearing conditions, together with the agents’ behavior, the household’s budget constraint is automatically satisfied from Walras’ law.

3.1. Equilibrium conditions

This subsection derives key equations in characterizing the equilibrium. Substituting (13) and (14) into (10) without the expectation operator yields

(15) \begin{equation} \psi _t = (1-\delta +\delta \psi _{t+1}) \left (1+ \dfrac{\psi _t}{\theta } \dfrac{\Gamma ^{\prime}\big(S^h_t\big)}{Q_t}\right ). \end{equation}

Substituting (12)–(14) and $Q_t S^b_t= \psi _t N_t/\theta$ into the banks’ aggregate net worth in period $t+1$ , we can obtain the dynamic equation of $N_t$ as follows:

(16) \begin{equation} N_{t+1}= \left [ \dfrac{\delta }{\beta } \left (1+ \dfrac{\psi _t}{\theta }\dfrac{\Gamma ^{\prime}\big(S^h_t\big)}{Q_t}\right ) +(1-\delta )\mu \right ] N_t. \end{equation}

Substituting (3) and (4) into the labor market equilibrium (11) and evaluating the resulting equation in period $t+1$ ,

(17) \begin{equation} I_{t+1}= \dfrac{1}{\kappa } -\dfrac{1+\lambda \zeta }{\lambda } \dfrac{1}{Q_{t+1}}. \end{equation}

Substituting (14) and (17) into (2), we can obtain the dynamic equation of $Q_t$ as follows:

(18) \begin{equation} Q_t=\beta (1-1/\kappa )Q_{t+1}+\beta (1+\zeta )-\Gamma ^{\prime}\big(S^h_t\big). \end{equation}

Equations (15), (16), and (18) include $S^h_t$ . Since $S^h_t +S^b=1$ , $S^h_t$ is given by the functions of $\psi _t$ , $N_t$ , and $Q_t$ :

(19) \begin{equation} S^h_t= 1- \dfrac{\psi _t N_t}{\theta Q_t}. \end{equation}

Thus, the autonomous dynamical system in the deterministic economy is given by (15), (16), and (18) together with (19). Note that $N_t$ is a state variable, whereas $\psi _t$ and $Q_t$ are forward-looking variables whose initial values are determined endogenously.

3.2. Balanced growth path

In this section, I examine the equilibrium where $Q_t, \psi _t, N_t, S^h_t$ , and $I_t$ become stationary. I call such an equilibrium the balanced growth path (BGP) equilibrium, since in that case consumption grows at a constant rate as shown below.

Equation (16) with $N_t=N_{t+1}$ implies

(20) \begin{equation} 1+ \dfrac{\psi }{\theta }\dfrac{\Gamma ^{\prime}(S^h)}{Q}=B^\ast, \end{equation}

where

\begin{equation*} B^\ast \equiv \dfrac {\beta [1-(1-\delta )\mu ]}{\delta }\gt 1. \end{equation*}

Note that $B^\ast$ depends only on the exogenous parameters and Assumption 3 ensures $B^\ast \gt 1$ . Hereafter, a superscript asterisk over a variable represents its stationary value. For example, $\psi ^*$ denotes the stationary value of $\psi$ . Equation (15) with $\psi _t=\psi _{t+1}$ provides $\psi ^*$ :

\begin{equation*} \psi ^\ast =\dfrac {(1-\delta )B^\ast }{1-\delta B^\ast }\gt 0. \end{equation*}

Substituting the obtained $\psi ^\ast$ back into equation (20) yields the following relationship between the stock price $Q$ and the households’ equity purchases $S^h$ :

(21) \begin{equation} Q = \dfrac{\delta \psi ^\ast }{\left [ \beta -\delta - \beta (1-\delta )\mu \right ] \theta }\Gamma ^{\prime}(S^h), \end{equation}

where the sign of the denominator is positive from Assumption 3. Since $\Gamma ^{\prime\prime}(S^h)\gt 0$ for $S^h\gt 0$ , this equation shows a positive relationship between $q$ and $S^h$ . The intuition is explained as follows. When $S^h$ becomes larger, households become more reluctant to hold equities directly unless their rate of return becomes sufficiently higher. Indeed, $R^e-R^d=\Gamma ^{\prime}(S^h)/(\beta Q)$ experiences upward pressure. This upward pressure in turn has a positive impact on the banks’ aggregate net worth $N$ , and hence, they want to purchase more of these equities. In the stationary equilibrium in which $N$ is constant, such an increase in their equity demand puts upward pressure on the stock price $Q$ . As equation (20) shows, the upward pressure on $S^h$ is offset by a rise in $Q$ such that $R^e-R^d$ remains constant.

There is the other relationship between $q$ and $S^h$ . From (18) with $Q_t=Q_{t+1}$ , we can obtain

(22) \begin{equation} Q= \dfrac{\beta (1+\zeta )-\Gamma ^{\prime}(S^h)}{1-\beta (1-1/\kappa )}, \end{equation}

where the sign of the denominator is positive. Since $\Gamma ^{\prime\prime}(S^h)\gt 0$ for $S^h\gt 0$ , this equation shows a negative relationship between $q$ and $S^h$ . The intuition is straightforward. The increase in $S^h$ makes households less willing to hold equities unless their rate of return becomes sufficiently higher. Therefore, this unwillingness depresses the unit cost of the equity purchase, which is $Q$ .

In Fig. 3, the upward- and downward-sloping curves represent equations (21) and (22), respectively. These two curves have only one intersection. $Q^\ast$ is explicitly obtained as

\begin{equation*} Q^\ast = \dfrac { \beta (1+\zeta ) \delta \psi ^\ast }{[1-\beta (1-1/\kappa )]\delta \psi ^\ast + [\beta -\delta -\beta (1-\delta )\mu ]\theta }. \end{equation*}

By its definition, $S^{h*}$ must be in $(0, 1)$ . Since I assume $\Gamma ^{\prime}(0)=0$ , the value of $Q$ in (21) is necessarily smaller than $Q$ in (22) for $S^h=0.$ Thus, $S^{h*}\gt 0$ is guaranteed. Throughout this study, I also assume that $S^{h*}\lt 1$ is satisfied, and hence, equity holdings are diversified between banks and households. For example, given $Q^*$ , $S^{h\ast }\lt 1$ is satisfied if

(23) \begin{equation} \Gamma ^{\prime}(1)\gt \dfrac{Q^\ast }{\delta \psi ^\ast }[\beta -\delta -\beta (1-\delta )\mu ] \theta. \end{equation}

Substituting $Q_{t+1}=Q^*$ into (17) yields $I^\ast$ :

(24) \begin{equation} I^\ast = \dfrac{1}{\kappa }-\dfrac{1+\lambda \zeta }{\lambda Q^\ast }. \end{equation}

By its definition, $I^*$ must be in (0, 1). Note that $I^\ast \lt 1$ is guaranteed because of $\kappa \gt 1$ . From (24), $I^*\gt 0$ if and only if

(25) \begin{equation} Q^\ast \gt \dfrac{\kappa (1+\lambda \zeta )}{\lambda }. \end{equation}

The results obtained so far can be summarized as the following Proposition:

Figure 3. Determination of $S^{h*}$ and $Q^*$ .

Then, I derive the growth rate of consumption, which always grows at the same rate as the inverse of the final good price. Since $p_t(\omega )=\lambda W_t$ for all $\omega$ , equation (1) implies

\begin{equation*} P_t= \dfrac {W_t} {(1+g_Z)^t \lambda ^{\int _0^1 (K_t(\omega )-1)d\omega }}. \end{equation*}

On the BGP, the wage rate is constant at $Q^\ast/\kappa$ . Recall that $K_t(\omega )$ is the index of highest quality for variety $\omega$ and increases by one for each successful innovation. Thus, $\int _0^1 (K_{t+1}(\omega )-K_t(\omega ))d\omega$ is equal to the measure of varieties in which successful innovation occurs in period $t$ . By the law of large numbers, this is equal to $I^\ast$ . Therefore, we can obtain

\begin{align*} \ln P^*_{t+1} - \ln P^*_t = -\ln\! (1+g_Z)- I^\ast \ln \lambda, \end{align*}

which implies that the final good price declines over time. Then, the growth rate of consumption is given by

\begin{equation*} g^\ast = g_Z+ I^\ast \ln \lambda, \end{equation*}

where $g^\ast \simeq \ln (1+g^\ast )$ and $ g_Z \simeq \ln (1+g_Z)$ are used. Since the wage rate, stock price, and banks’ net worth become stationary, their real values also grow at the rate of $g^*$ . Therefore, I simply call $g^*$ the balanced growth rate.

Finally, we have to check that Assumption 2 is satisfied on the BGP. In this non-stochastic economy, this assumption is rewritten as

\begin{equation*} \theta \gt (1-\delta +\delta \psi _{t+1}) \dfrac {\Gamma ^{\prime}\big(S^h_t\big)}{Q_t}. \end{equation*}

Since $\Gamma ^{\prime}(S^{h\ast })/Q^\ast =\theta (B^\ast-1)/\psi ^\ast$ holds from (20) and $1-\delta +\delta \psi ^\ast =\psi ^\ast/B^\ast$ holds from (15), we can rewrite the inequality above as $\theta \gt \theta (B^\ast-1)/B^\ast$ , which is necessarily satisfied.

3.3. Comparative statics

Since the model is tractable, we can easily conduct a comparative statics analysis of the BGP and can gain insight on the inner workings of the model. Suppose that $\theta$ increases and, hence, the banks’ leverage ratio decreases. This decreases the equities held by banks $S^{b*}$ . Thus, as illustrated in panel (a) of Fig. 4, an increase in $\theta$ shifts the curve representing equation (21) to the right and increases $S^h$ . However, because of their utility costs, the households are less efficient at purchasing equities than are banks. Therefore, the households demand a high premium $R^{e*}-R^{d*}=\Gamma ^{\prime}(S^{h*})/(\beta Q^*)$ to hold additional equities. Consequently, the stock price $Q^*$ becomes low in an economy with a large $\theta$ .

Figure 4. Comparative statics of the BGP equilibrium.

The decline in $Q^*$ in turn makes R&D activities less profitable for potential entrants. In fact, (24) clearly shows that $I^\ast$ decreases. Since $g^\ast = g_Z+ I^\ast \ln \lambda$ , an increase in $\theta$ results in a decrease in the BGP growth rate. From the above results, we can obtain the following proposition.

Appendix A.2 provides the results of comparative statics for other variables. It should be noted here that the result stated in Proposition 2 never occurs when R&D costs alone increase. To understand why, suppose that $\kappa$ increases. In an economy in which $\kappa$ is large, it becomes more costly for a potential entrant to conduct R&D activities. Then, through the entrants’ free entry condition, the benefit of R&D must be high. As panel (b) of Fig. 4 shows, this induces upward shifts in the curve representing equation (22). Thus, as shown in the Appendix A.2, in this case, the stock price rises even though the innovation rate falls.

This section concludes with an analysis of the long-run impact on the banks’ net worth. From (19), $N^\ast$ is given by

\begin{equation*} N^\ast =\dfrac {\theta Q^\ast S^{b \ast }}{\psi ^\ast }. \end{equation*}

Suppose that $\theta$ increases. From Proposition 2, both $Q^\ast$ and $S^{b\ast } = 1-S^{h\ast }$ decrease while $\psi ^*$ is constant. Simultaneously, an increase in $\theta$ has the direct effect of increasing $N^\ast$ . The reason for this direct effect is simple: When banks are no longer able to leverage sufficiently, they must have a higher net worth to purchase equities. Appendix A.2 shows

\begin{equation*} \dfrac {dN^\ast }{N^\ast }=\dfrac {1}{1+a}\left (1 -\dfrac {\Gamma ^{\prime}}{(1-S^{h\ast }) \Gamma ^{\prime\prime} } \right ) \dfrac {d\theta }{\theta }, \end{equation*}

where $a \equiv \frac{\Gamma ^{\prime} }{\beta (1+\zeta )-\Gamma ^{\prime}}\gt 0$ . Briefly speaking, the first and second terms within the parentheses correspond to the direct and indirect effects, respectively. The magnitude of the indirect effect depends on the extent to which $S^{h*}$ and $Q^*$ decrease, which further depends on the shape of the cost function of households’ equity holdings. For example, if function $\Gamma (S^h)$ is specified as

\begin{equation*} \Gamma (S^h)=\frac {\gamma (S^h)^{1+\eta }}{1+\eta }, \end{equation*}

with $\eta \gt 0$ , the elasticity of marginal cost is given by $\eta$ . $dN^*/N^*$ is rewritten as

\begin{equation*} \dfrac {dN^\ast }{N^\ast }=\dfrac {1}{1+a}\left (1 -\dfrac {S^{h*}}{(1-S^{h\ast }) }\dfrac {1}{\eta } \right ) \dfrac {d\theta }{\theta }. \end{equation*}

If $\eta$ is small (large), $S^h$ decreases more (less) sharply. Then, given the value of $S^{h*}$ before the change in $\theta$ , the banks’ net worth decreases (increases) when $\eta$ is small (large). It is worth emphasizing, however, that whichever way $N^*$ changes, the share of banks’ equity holdings invariably declines and the stock price of intermediate goods firms drops.