2. The model

We develop a dynamic general equilibrium model with VES between goods. Our model has a similar structure to that of Grossman and Helpman (Reference Grossman and Helpman1991, Ch. 3) and Acemoglu (Reference Acemoglu2008, Ch.13). In this model, the population size is $L$ , which is constant throughout time. Individuals have identical preferences:

(1) \begin{equation} W_{0}=\int _{0}^{\infty }e^{-\rho t}U(\tilde{u}_{t})dt,\quad 0\lt \rho \lt 1, \end{equation}

where $\rho$ is the constant subjective discount rate and $U(\tilde{u}_{t})$ is the instantaneous utility per person at time $t$ . I assume that $U(0)=0$ , $U^{\prime }(\tilde{u}_{t})\gt 0$ , and $U^{\prime \prime }(\tilde{u}_{t})\lt 0$ . $\tilde{u}_{t}$ is given by

(2) \begin{equation} \tilde{u}_{t}={\displaystyle \sum \limits _{i=1}^{n_{t}}} u(q_{t}(i)), \end{equation}

where $q_{t}(i)$ represents consumption of differentiated goods $i$ at time $t$ and $n_{t}$ denotes the number of goods. $u(q_{t}(i))$ is strictly increasing, concave, and four times continuously differentiable. Furthermore, I assume that the Inada condition $lim_{q_{t}(i)\rightarrow 0}u^{\prime }(q_{t}(i))=+\infty$ is satisfied and $u(0)=0$ .Footnote ⁴ The utility maximization problem of an individual can be solved in two steps. The first step is to address the static optimization problem, wherein the budget constraint at time $t$ is

(3) \begin{equation}{\displaystyle \sum \limits _{i=1}^{n_{t}}} p_{t}(i)q_{t}(i)=E_{t}, \end{equation}

where $p_{t}(i)$ is the price of product $i$ and $E_{t}$ is the total expenditure at time $t$ . From the first-order condition, we derive the following inverse demand function of product $i$ at time $t$ :

(4) \begin{equation} p_{t}(i)=\frac{u^{\prime }(q_{t}(i))E_{t}}{{\displaystyle \sum \limits _{i=1}^{n_{t}}} u^{\prime }(q_{t}(i))q_{t}(i)}. \end{equation}

Totally differentiating (4), the relationship between the demand of product $i$ and the expenditure level is (See Appendix for proof)

(5) \begin{equation} \frac{\partial q_{t}(i)}{\partial E_{t}}=\frac{{\displaystyle \sum \limits _{j=1}^{n_{t}}} u^{\prime }(q_{t}(j))q_{t}(j)}{(\!-\!\beta _{i})\left(1-{\displaystyle \sum \limits _{j=1}^{n_{t}}} \frac{\alpha _{j}}{\beta _{j}}\right)E_{t}}, \end{equation}

where

\begin{equation*} \alpha _{j}=u^{^{\prime \prime }}(q_{t}(j))q_{t}(j)+u^{^{\prime }}(q_{t}(j)), \end{equation*}

\begin{equation*} \beta _{j}=\frac {u^{^{\prime \prime }}(q_{t}(j))}{u^{^{\prime }}(q_{t}(j))}{\displaystyle \sum \limits _{k=1}^{n_{t}}} u^{\prime }(q_{t}(k))q_{t}(k). \end{equation*}

The second step is to solve the intertemporal optimization problem. The intertemporal budget constraint is given by

\begin{equation*} \int _{0}^{\infty }e^{-R(t) t}E_{t}dt=H(0)+\int _{0}^{\infty }e^{-R(t) t}w_{t}dt, \end{equation*}

where

\begin{equation*} R(t)=\frac {1}{t}\int _{0}^{t}r_{\omega }d\omega. \end{equation*}

$r_{\omega }$ is the interest rate at time $\omega$ , $w$ is the wage rate, and $H(0)$ is the asset holding at time $0$ . Solving the intertemporal optimization problem, we can obtain the following equation: (See Appendix for proof)

(6) \begin{equation} e^{-\rho t}\frac{A_{t}}{E_{t}}=e^{-R(t) t}, \end{equation}

where

\begin{equation*} A_{t}=U^{\prime }(\tilde {u}_{t}) {\displaystyle \sum \limits _{j=1}^{n_{t}}} u^{\prime }(q_{t}(j))q_{t}(j) {\displaystyle \sum \limits _{k=1}^{n_{t}}} \frac {u^{\prime }(q_{t}(k))}{(\!-\!\beta _{j})\left(1- {\displaystyle \sum \limits _{j=1}^{n_{t}}} \frac {\alpha _{j}}{\beta _{j}}\right)}. \end{equation*}

$A_{t}$ represents the marginal utility of the total expenditure at time $t$ . Taking the logarithm and differentiating $t$ and setting $E_{t}=1$ , I can obtain the following equation:

(7) \begin{equation} r_{t}=\rho -\frac{\dot{A}_{t}}{A_{t}}, \end{equation}

where $\dot{A}_{t}=\frac{dA_{t}}{dt}$ . When the marginal utility of the total expenditure increases in $t$ , individuals reduce the consumption and increase saving at time $t$ . Then, the interest rate is less than the subjective discount rate.

The goods market is monopolistically competitive. To produce one unit of product, firms hire $\psi$ unit of labor. Therefore, the profit of firm $i$ is

(8) \begin{equation} \pi _{t}(i)=p_{t}(i)q_{t}(i)L-w_{t}\psi q_{t}(i)L. \end{equation}

Using (4), profit maximization yields the price of firm $i$ as

(9) \begin{equation} p_{t}(i)=\frac{\psi w_{t}}{1-\mu (q_{t}(i))}, \end{equation}

where

(10) \begin{equation} \mu (q_{t}(i))=-\frac{u^{\prime \prime }(q_{t}(i))q_{t}(i)}{u^{\prime }(q_{t}(i))}\gt 0. \end{equation}

The markup of firm $i$ is equal to the elasticity of marginal utility $\mu (q_{t}(i))$ and $0\lt$ $\mu (q_{t}(i))\lt 1$ following Dhingra and Morrow (Reference Dhingra and Morrow2019). I also assume that $\mu (q_{t}(i))$ is increasing in $q_{t}(i)$ (Dhyne et al. (Reference Dhyne, Petrin and Warzynski2011), De Loecker et al. (Reference De Loecker, Goldberg, Khandelwal and Pavcnik2012), and De Loecker and Warzynski (Reference De Loecker and Warzynski2012)). The output of firms are identical; hence, I can write $q_{t}(i)=q_{t}$ , and the variety label $i$ can be dropped. Substituting (9) into (8), the firms’ profit levels can be obtained as

(11) \begin{equation} \pi _{t}=\frac{w_{t}\psi \mu (q_{t})q_{t}L}{1-\mu (q_{t})}. \end{equation}

The R&D activities of the present model follow that of the model of Grossman and Helpman (Reference Grossman and Helpman1991, Ch. 3). The final good producers enter the R&D race and finance the cost of R&D by issuing equity in the stock market. The stock value of the final good producers at time $t$ is equal to the present discounted value of its profit stream after $t$ as follows:

\begin{equation*} v_{t} = \int _{t}^{\infty } e^{-\int _{t}^{\omega }r_{s}ds}\pi _{\omega } d \omega. \end{equation*}

Differentiation of $v_{t}$ with respect to $t$ yields the following no-arbitrage condition:

(12) \begin{equation} \dot{v}_{t}=r_{t}v_{t} - \pi _{t}. \end{equation}

Goods producers hire relevant labor to generate blueprints. I presume that there exists knowledge spillover in R&D activities. I assume that $L_{t}^{A}$ units of labor for R&D activity for the time interval $dt$ produce a new variety of final good according to:

(13) \begin{equation} dn_{t}=\frac{L_{t}^{A}n_{t}^{\gamma }}{a}dt, \end{equation}

where $\gamma$ represents the degree of knowledge spillover in R&D investment.Footnote ⁵ The cost of the R&D activities is $w_{t}L_{t}^{A}dt$ . The production of innovative blueprints creates the value of $v_{t}dn_{t}$ since each blueprint has a market value of $v_{t}$ . There is free entry into the R&D race; thus, the following free-entry condition must hold:

(14) \begin{equation} v_{t}\leqq \frac{aw_{t}}{n_{t}^{\gamma }}. \end{equation}

The labor market equilibrium condition is

(15) \begin{equation} L=L_{t}^{A}+\psi n_{t}q_{t}L. \end{equation}

Labor demand consists of R&D investment and the production of goods.

3. Equilibrium

In the equilibrium, since $q_{t}(i)=q_{t}$ , the marginal utility of the total expenditure at time $t$ , $A_{t}$ , is

\begin{equation*} A_{t}=U^{\prime }(\tilde {u}_{t})n_{t}u^{\prime }(q_{t})q_{t}. \end{equation*}

Then, (7) becomes

(16) \begin{equation} r_{t}=\rho -\left [ 1+\frac{U^{\prime \prime }(\tilde{u}_{t})\tilde{u}_{t}}{U^{\prime }(\tilde{u}_{t})}\right ] \frac{\dot{n}_{t}}{n_{t}}+\left [ 1-\mu (q_{t})+\eta (q_{t})\frac{U^{\prime \prime }(\tilde{u}_{t})\tilde{u}_{t}}{U^{\prime }(\tilde{u}_{t})}\right ] \frac{\dot{q}_{t}}{q_{t}}, \end{equation}

where $\eta (q_{t})=\frac{u^{\prime }(q_{t})q_{t}}{u(q_{t})}$ is the elasticity of utility and we assume that $0\lt \eta (q_{t})\lt 1$ .

Substituting (4) for (9) and using $q_{t}(i)=q_{t}$ and $E_t=1$ , the following equation can be obtained:

\begin{equation*} \frac {\psi w_{t}}{1-\mu (q_{t})}=\frac {1}{n_{t}q_{t}}. \end{equation*}

Using (14), the above equation becomes

(17) \begin{equation} \frac{\psi v_{t}n_{t}^{\gamma }}{a(1-\mu (q_{t}))}=\frac{1}{n_{t}q_{t}}. \end{equation}

Differentiating (17) with respect to $t$ , we can obtain the following dynamics:

(18) \begin{equation} \frac{\dot{v}_{t}}{v_{t}}=-(1+\gamma )\frac{\dot{n}_{t}}{n_{t}}-\left [ 1+\frac{\mu ^{\prime }(q_{t})q_{t}}{1-\mu (q_{t})}\right ] \frac{\dot{q}_{t}}{q_{t}}. \end{equation}

From (11), (12), and (14), the dynamics of the stock value of the final good producers is

(19) \begin{equation} \frac{\dot{v}_{t}}{v_{t}}=r_{t}-\frac{n_{t}^{\gamma }\psi \mu (q_{t})q_{t}L}{a(1-\mu (q_{t}))}. \end{equation}

Substituting (16) into (19), the dynamics of the value becomes

(20) \begin{align} \frac{\dot{v}_{t}}{v_{t}} & =\rho -\frac{n_{t}^{\gamma }\psi \mu (q_{t})q_{t}L}{a(1-\mu (q_{t}))}-\left [ 1+\frac{U^{\prime \prime }(\tilde{u}_{t})\tilde{u}_{t}}{U^{\prime }(\tilde{u}_{t})}\right ] \frac{\dot{n}_{t}}{n_{t}}\\[5pt] & -\left [ 1-\mu (q_{t})+\eta (q_{t})\frac{U^{\prime \prime }(\tilde{u}_{t})\tilde{u}_{t}}{U^{\prime }(\tilde{u}_{t})}\right ] \frac{\dot{q}_{t}}{q_{t}}.\nonumber \end{align}

From (18) and (20), the following dynamics of $q_{t}$ are obtained:

(21) \begin{equation} B_{t}\frac{\dot{q}_{t}}{q_{t}}=\rho -\frac{n_{t}^{\gamma }\psi \mu (q_{t})q_{t}L}{a(1-\mu (q_{t}))}+C_{t}\frac{\dot{n}_{t}}{n_{t}}, \end{equation}

where

\begin{equation*} B_{t}=\eta (q_{t})\frac {U^{\prime \prime }(\tilde {u}_{t})\tilde {u}_{t}}{U^{\prime }(\tilde {u}_{t})}-\frac {\mu ^{\prime }(q_{t})q_{t}}{1-\mu (q_{t})}-\mu (q_{t})\lt 0, \end{equation*}

\begin{equation*} C_{t}=\gamma -\frac {U^{\prime \prime }(\tilde {u}_{t})\tilde {u}_{t}}{U^{\prime }(\tilde {u}_{t})}\gt 0. \end{equation*}

The second term of (21) represents the return on R&D investment. When $q_{t}$ and $n_{t}$ are given at time $t$ , an increase in the growth rate of the number of firms decreases the consumption per capita.

From (13) and (15), the dynamics of the number of firms becomes

(22) \begin{equation} \dot{n}_{t}=\frac{n_{t}^{\gamma }L}{a}(1-\psi n_{t}q_{t}). \end{equation}

(21) and (22) constitute the dynamic system in this economy.

I investigate how many steady states exist. In the steady state, the consumption per capita and the number of firms are given by

(23) \begin{equation} n^{\ast }=\frac{1}{\psi q^{\ast }}, \end{equation}

(24) \begin{equation} \rho =\frac{(n^{\ast })^{\gamma }\psi \mu (q^{\ast })q^{\ast }L}{a(1-\mu (q^{\ast }))}, \end{equation}

where the superscript “ $*$ ” represents the steady states. Substituting (23) into the right-hand side of (24), the following equation can be obtained:

(25) \begin{equation} RHS(q)\equiv \frac{\mu (q)\psi ^{1-\gamma }L}{a q^{\gamma -1}(1-\mu (q))}. \end{equation}

$RHS(q)$ represents the return on R&D investment. In the steady state, the return on R&D investment is equal to the subjective discount rate, $\rho$ . When $\gamma \lt 1$ ( $\gamma \gt 1$ ) and $q$ goes to zero, $RHS(q)$ goes to zero (infinity). When $\gamma \lt 1$ ( $\gamma \gt 1$ ), and $q$ goes to infinity, $RHS(q)$ goes to infinity (zero). Differentiating $RHS(q)$ with respect to $q$ , I can derive the following equation:

(26) \begin{equation} RHS^{\prime }(q)=\frac{\mu (q)\psi ^{1-\gamma }L}{a q^{\gamma }(1-\mu (q))}\left [ \Psi (q)+1-\gamma \right ], \end{equation}

where

\begin{equation*} \Psi (q)\equiv \frac {\mu ^{\prime }(q)q}{\mu (q)(1-\mu (q))}\gt 0. \end{equation*}

When $\gamma \leqq 1$ , $RHS^{\prime }(q)$ is positive, and the return on R&D investment monotonically increases in the output level, as depicted in Figure 1. Therefore, when $\gamma \leqq 1$ , there exists a unique steady state.

Figure 1.

Steady states when $\mu '(q)\gt 0$ .

Conversely, when $\gamma \gt 1$ , the sign of $RHS^{\prime }(q)$ is ambiguous and there may be multiple steady states. I differentiate $\Psi (q)$ with respect to $q$ as follows:

\begin{equation*} \Psi ^{\prime }(q)=\frac {\mu ^{\prime \prime }(q)q\mu (q)(1-\mu (q))+\mu ^{\prime }(q)\left [ \mu (q)(1-\mu (q))-\mu ^{\prime }(q)q+2\mu (q)q\mu ^{\prime }(q)\right ] }{\mu (q)^{2}(1-\mu (q))^{2}}. \end{equation*}

Notably, the sign of $\Psi ^{\prime }(q)$ is ambiguous. Hereafter, I assume that there exists a unique $q=\hat{q}$ that $\Psi ^{\prime }(\hat{q})=0$ holds. Additionally, I assume that $\Psi (q)$ is inverted U-shaped and $\Psi ^{\prime }(q)\gt 0$ ( $\Psi ^{\prime }(q)\lt 0$ ) when $q\lt \hat{q}$ ( $q\gt \hat{q}$ ) and $\Psi (\hat{q})\gt \gamma -1$ . From these assumptions, there exist $\tilde{q}_{1}$ and $\tilde{q}_{2}$ ( $\tilde{q}_{1}\lt \tilde{q}_{2}$ ) where $RHS^{\prime }(\tilde{q}_{1})=0$ and $RHS^{\prime }(\tilde{q}_{2})=0$ hold. Then, in $0\lt q\lt \tilde{q}_{1}$ and $q\gt \tilde{q}_{2}$ , $RHS(q^{\ast })$ is decreasing in $q$ , and $RHS(q)$ is increasing in $\tilde{q}_{1}\lt q\lt \tilde{q}_{2}$ . In Figure 1, when $RHS(\tilde{q}_{1})\lt \rho \lt RHS(\tilde{q}_{2})$ , there exist three steady states, which are labeled as $q_{1}^{\ast }$ , $q_{2}^{\ast }$ , and $q_{3}^{\ast }$ ( $q_{1}^{\ast }\lt q_{2}^{\ast }\lt q_{3}^{\ast }$ ). When $RHS(\tilde{q}_{1})\gt \rho$ or $RHS(\tilde{q}_{2})\lt \rho$ , there exists a unique steady state.

Next, I investigate the stability of the steady states. The condition that a steady state is saddle path stable is (See Appendix for the proof)

\begin{equation*} \gamma \lt 1+\Psi (q^{\ast }). \end{equation*}

The left-hand side of $\gamma$ is the knowledge spillover in the R&D activity, and the right-hand side signifies the elasticity of the returns on R&D investment. When this condition holds, the steady state is saddle path stable. When $\gamma \leqq 1$ , the steady state is always saddle path stable.Footnote ⁶ When $\gamma \gt 1$ and $RHS(\tilde{q}_{1})\gt \rho$ or $RHS(\tilde{q}_{2})\lt \rho$ at the steady state, the slope of $RHS(q^{\ast })$ is negative and $\Psi (q^{\ast })+1-\gamma \lt 0$ holds. Subsequently, the steady state is unstable. In Figure 1, when $\gamma \gt 1$ and $RHS(\tilde{q}_{1})\lt \rho \lt RHS(\tilde{q}_{2})$ , there are three steady states. In two of the three steady states, $q_{1}^{\ast }$ and $q_{3}^{\ast }$ , the slope of $RHS$ is negative. Therefore, these steady states become unstable. Contrarily, the slope of $RHS(q_{2}^{\ast })$ is positive. Then, at $q_{2}^{\ast }$ , the steady state is saddle path stable. Summarizing the above result, we can obtain the following proposition:Footnote ⁷

In this model, there exist multiple steady states and balanced growth does not exist when the elasticity of substitution is not constant. This outcome differed from those of Grossman and Helpman (Reference Grossman and Helpman1991) and Latzer et al. (Reference Latzer, Matsuyama and Parenti2019). From the right-hand side of (24), the return on the R&D investment is

\begin{equation*} RR\equiv \frac {\pi _{t}}{wL_{t}^{A}/\dot {n}_{t}}=\left ( \frac {n_{t}^{\gamma }}{a}\right ) \left ( q_{t}\psi L\right ) \left ( \frac {\mu (q_{t})}{1-\mu (q_{t})}\right ). \end{equation*}

In this model, when firms enter the market, there are three effects on the return on R&D investment: one is ambiguous and the other two are negative. The ambiguous effect is the spillover effect, which is denoted by the first parentheses of $\frac{n_{t}^{\gamma }}{a}$ and is positive (negative) when $\gamma \gt 0$ ( $\gamma \lt 0$ ). Moreover, the spillover effect outlines that an increase in the number of firms reduces the costs of R&D investment, thus increasing the return on R&D investment when $\gamma \gt 0$ . The first negative effect is the consumption effect, $q_{t}\psi L$ . An increase in the number of firms reduces the consumption per good which reduces the profit of firms. The second negative effect is the markup effect, $\frac{\mu (q_{t})}{1-\mu (q_{t})}$ . Since the elasticity of substitution is variable in this model, an increase in the number of firms reduces the consumption per good and the markup rates of firms as $\mu ^{\prime }(q_{t})$ .Footnote ⁸ This correlation diminishes the profit of firms and the return on R&D investment.

When the knowledge spillover of R&D investment is small, the spillover effect is smaller than the sum of the competition effect and the markup effect, and the return on the R&D investment results in a decrease in the number of firms. Therefore, the return on R&D investment is equal to the subjective discount rate and the economy goes to a steady state. When the knowledge spillover of the R&D investment is sufficiently large, the spillover effect is larger than the sum of the consumption effect and the markup effect, and the return on the R&D investment results in an increase in the number of firms. Similarly, the return on R&D investment is larger than the subjective discount rate and the steady state is unstable. When the knowledge spillover of the R&D investment is in the middle, there are some equilibria in $n$ and $q$ , where the rate of return on R&D investment is equal to the subjective discount rate. At some equilibria, because an increase in the number of firms decreases the return on the R&D investment, the return on the R&D investment is equal to the subjective discount rate and the steady states are stable. At other equilibria, because an increase in the number of firms raises the return on R&D investment, the steady states are unstable.

In contrast, in Grossman and Helpman (Reference Grossman and Helpman1991), the elasticity of the marginal utility of $\mu (q_{t})$ is $\frac{1}{\sigma }$ and $\gamma =1$ . Then, $RR$ becomes

\begin{equation*} RR=\left ( \frac {n_{t}}{a}\right ) \left ( q_{t}\psi L\right ) \left ( \frac {1}{\sigma -1}\right ). \end{equation*}

In Grossman and Helpman (Reference Grossman and Helpman1991), there is no markup effect and the spillover effect is equal to the consumption effect. Thereafter, the total output of $q_{t}n_{t}$ is formulated so that the return on the R&D investment is equal to the subjective discount rate, $RR=\rho$ at time $0$ , and there exists a balanced growth path.

Latzer et al. (Reference Latzer, Matsuyama and Parenti2019) assume an externality to marginal costs that the labor requirement is decreasing in the number of firms, that is, $\psi =\frac{\hat{\psi }}{n_{t}}$ .Footnote ⁹ Due to this assumption, labor engaged in production is $\psi q_{t}n_{t}=\hat{\psi }q_{t}$ and is independent of the number of firms. Besides, an increase in the number of firms does not affect the output level of $q_{t}$ . Then, in Latzer et al. (Reference Latzer, Matsuyama and Parenti2019), $RR$ becomes

\begin{equation*} RR=\left ( \frac {n_{t}}{a}\right ) \left ( q_{t}L\right ) \left ( \frac {\mu (q_{t})}{1-\mu (q_{t})}\right ) \left(\frac {\hat {\psi }}{n_{t}}\right)=\frac {\hat {\psi }}{a}\left ( q_{t}L\right ) \left ( \frac {\mu (q_{t})}{1-\mu (q_{t})}\right ). \end{equation*}

The fourth parenthesis represents the additional externality to marginal costs. As the spillover effect is nullified by the additional eternality to marginal costs, the return on the R&D investment does not depend on the number of firms. The consumption per product $q_{t}$ is determined so that the return on the R&D investment is equal to the subjective discount rate at time $0$ . Likewise, the number of workers engaged in production remains constant. Therefore, a balanced growth path exists in Latzer et al. (Reference Latzer, Matsuyama and Parenti2019).

4. First-best equilibrium

In this section, I inspect the first-best equilibrium whereby the central planner distributes the labor for production and R&D investment and distinguishes the allocation of consumption and saving to maximize the welfare level (1). The resource constraint is

\begin{equation*} a\frac {\dot {n}_{t}}{n_{t}}+\psi n_{t}q_{t}L=L. \end{equation*}

Accordingly, the dynamics of the number of firms and consumption per capita are

(27) \begin{equation} \left(B_{t}+\frac{\mu ^{\prime }(q_{t})q_{t}}{1-\mu (q_{t})}\right)\frac{\dot{q}_{t}}{q_{t}}=\rho -\frac{\gamma n_{t}^{\gamma -1}}{a}-\frac{n_{t}^{\gamma }\psi q_{t}L}{a}\!\left(\frac{1}{\eta (q_{t})}-1-\gamma \right)+C_{t}\frac{\dot{n}_{t}}{n_{t}}, \end{equation}

(28) \begin{equation} \dot{n}_{t}=\frac{n_{t}^{\gamma }L}{a}(1-\psi n_{t}q_{t}). \end{equation}

The dynamics of the number of firms in the first-best equilibrium are the same as that in the competitive equilibrium. When no knowledge spillover exists, $\gamma =0$ , the dynamics of the consumption per capita are

(29) \begin{equation} \left(B_{t}+\frac{\mu ^{\prime }(q_{t})q_{t}}{1-\mu (q_{t})}\right)\frac{\dot{q}_{t}}{q_{t}}=\rho -\frac{\psi q_{t}L}{a}\frac{1-\eta (q_{t})}{\eta (q_{t})}+C_{t}\frac{\dot{n}_{t}}{n_{t}}. \end{equation}

Comparing (29) with (21), the consumption per capita in the competitive equilibrium is equal to that in the first-best equilibrium when $\mu ^{\prime }(q_{t})=0$ , and the following equation holds:

\begin{equation*} \frac {\mu (q_{t})}{1-\mu (q_{t})}=\frac {1-\eta (q_{t})}{\eta (q_{t})}. \end{equation*}

As discussed in Grossman and Helpman (Reference Grossman and Helpman1991) and Dhingra and Morrow (Reference Dhingra and Morrow2019), when the elasticity of substitution is constant, this equation holds and the competitive equilibrium coincides with the first-best equilibrium.

When there exists a knowledge spillover and $\gamma =1$ , the consumption per capita of the steady state in the first-best equilibrium is

(30) \begin{equation} \rho =\frac{1}{a}+\frac{n^{F}\psi q^{F}L}{a}\left(\frac{1}{\eta (q^{F})}-2\right), \end{equation}

where the superscript of $F$ represents the first-best equilibrium. Comparing (24) with (30), the consumption per capita in the first-best equilibrium does not coincide that in the competitive equilibrium. Summarizing the above results, the following proposition can be obtained: