Occupational choice and entrepreneurship: effects of R&D subsidies on economic growth

Abstract

Most R&D-based growth models fail to explicitly account for the role of entrepreneurs in economic growth. By contrast, this study accounts for this factor and constructs an overlapping-generations model that includes entrepreneurial innovation and the occupational choice of becoming an entrepreneur or a worker. For the role of entrepreneurs, even a policy intended to encourage innovation can negatively affect economic growth. For the effect of such policies, I focus on the role of R&D subsidies. I show that while R&D subsidies promote entrepreneurs’ R&D activities, they increase workers’ wages by boosting labor demand. Thus, it is more attractive to be a worker, which reduces the number of entrepreneurs. Subsidies can have both a negative and positive effect on growth, which results in an inverted U-shaped relationship between R&D subsidies and growth. In addition, a growth-maximizing R&D subsidy rate exists, although this rate is too high to maximize the welfare level of any one generation. When individuals are heterogeneous in their abilities, R&D subsidies reduce intra-generational inequalities.

Appendix

1.1 Intermediate goods firms’ problems

There are two steps in the intermediate goods firms’ problem. First, the intermediate goods firms invest in R&D activities and then produce intermediate goods. I solve the problems backward. In the second step, I maximize an intermediate goods firm’s profit from production, given its productivity and R&D outlay. Then, I go back to the first step and decide how much the intermediate goods firm should invest in R&D activities.

The jth intermediate goods firm’s problem in the second step can be written as

$$\begin{aligned} \pi _t(A_t(j)) \equiv \max \bigl \{ p_t(j) x_t(j) - w_t l_t^x(j) \bigr \} , \end{aligned}$$

(36)

subject to (6) and (7) where $\pi (A)$ is a profit function in this step of the problem given its productivity A. The first-order condition yields the optimal price and output of firm j given its productivity and price index as

$$\begin{aligned} p_t(j)&= \frac{\eta }{\eta -1} \frac{w_t}{A_t(j)} , \end{aligned}$$

(37)

$$\begin{aligned} x_t(j)&= \frac{\left[ \frac{\eta }{\eta -1} \frac{w_t}{A_t(j)} \right] ^{-\eta }}{P_t^{1-\eta }} . \end{aligned}$$

(38)

Substituting (37) and (38) into (36) yields the profit function

$$\begin{aligned} \pi _t(A_t(j)) = \frac{(\eta -1)^{\eta -1}}{\eta ^\eta } \left[ \frac{A_t(j)}{w_t} P_t \right] ^{\eta -1} . \end{aligned}$$

(39)

Then, I return to the first step. The objective of this step is to maximize the intermediate goods firm’s net profits, $\varPi _t$. Since R&D activities are subsidized, the net profit of the jth intermediate goods firm is

$$\begin{aligned} \varPi _t(j) \equiv \max \bigl \{ \pi _t(A_t(j)) - (1-\mu ) w_t l_t^R(j) \bigr \} , \end{aligned}$$

(40)

subject to (8) and (39). The first-order condition with respect to $l_t^R(j)$ is

$$\begin{aligned} \gamma \left( \frac{\eta -1}{\eta } \right) ^{\eta } \left[ \frac{\theta K_{t-1}}{w_t} P_t \right] ^{\eta -1} [l_t^R(j)]^{\gamma (\eta -1) -1} = (1-\mu _t )w_t . \end{aligned}$$

(41)

Equation (41) implies that $l_t^R(j)$ is independent of j and so are $A_t(j)$ and $p_t(j)$. In addition, it is necessary to assume $1-\gamma (\eta -1) >0$ to satisfy the second-order condition. Since all firms behave in the same way, the price index becomes

$$\begin{aligned} P_t = N_t^{\frac{1}{1-\eta }} \frac{\eta }{\eta -1} \frac{w_t}{\bar{A}_t} . \end{aligned}$$

(42)

By substituting (42) into (41) and rearranging it, the optimal level of labor input for R&D activities is (13). Moreover, substituting (38) and (42) into (7) yields (12).

1.2 Proof of Proposition 2

Proof

I show that (i) $v_t$ is maximized at $\mu =0$ for $t\le \bar{t}$, (ii) $v_t$ is maximized at an internal point $\mu ^* \in (0,1)$ for $t>\bar{t}$, and (iii) $\mu ^*$ is less than the growth-maximizing rate $\hat{\mu }$.

Differentiating (26) with respect to $\mu$ yields

$$\begin{aligned} \frac{dv_{t} }{d\mu }&= \frac{\beta \gamma }{1-\mu } \Biggl \{ 1 - \frac{\eta -1}{\eta (1-\mu )+\gamma (\eta -1)\mu } \nonumber \\&\quad -\frac{Z}{(\eta -1) [(1+Z)(1-\mu ) + \gamma Z]} \Biggr \} +\beta t \frac{dg/d\mu }{1+g}, \end{aligned}$$

(43)

$$\begin{aligned}&= \frac{\beta \gamma }{(1-\mu ) [\eta (1-\mu )+\gamma (\eta -1)\mu ] [(1+Z)(1-\mu ) + \gamma Z] } \varPsi _t (\mu ) , \end{aligned}$$

(44)

where

$$\begin{aligned} \varPsi _t (\mu )&\equiv \psi _{1,t} \mu ^2 +\psi _{2,t} \mu +\psi _{3,t}, \nonumber \\ \psi _{1,t}&\equiv (1+t)(1+Z)[\eta -\gamma (\eta -1)] >0 ,\nonumber \\ \psi _{2,t}&\equiv - [\eta -\gamma (\eta -1)] \biggl \{ 1+Z -\frac{1-\gamma (\eta -1)}{\eta -1} Z+[1+(1+\gamma -\phi )Z]t \biggr \}\nonumber \\&\qquad -(1+Z)(1+\eta t) ,\nonumber \\ \psi _{3,t}&\equiv 1-\frac{1-\gamma (\eta -1)}{\eta -1} Z+[1+(1+\gamma -\phi )Z]\eta t. \end{aligned}$$

(45)

Note that $\frac{\beta \gamma }{(1-\mu ) [\eta (1-\mu )+\gamma (\eta -1)\mu ] [(1+Z)(1-\mu ) + \gamma Z] } >0$ for all $\mu \in [0,1)$, that is, the sign of $dv_t/d\mu$ is the same as that of $\varPsi _t$. In addition, note that $\varPsi _t$ is a continuous, quadratic, and convex function of $\mu$. By substituting $\mu =0,1$—the lower and upper limits of $\mu$—into $\varPsi _t(\mu )$, I obtain

$$\begin{aligned} \varPsi _t (0)&= 1-\frac{1-\gamma (\eta -1)}{\eta -1}Z +[1+(1+\gamma -\phi )Z] \eta t, \\ \varPsi _t (1)&= -Z\gamma [\eta -\gamma (\eta -1) +(\eta -1) (\phi -\gamma ) t ] <0, \ \forall t. \end{aligned}$$

$\varPsi _t(0)$ is negative if and only if $t < \bar{t} \equiv \frac{(1-z)^{-1/\beta } -1}{[1+(1+\gamma -\phi )Z]\eta }$, whereas $\varPsi _t(1)$ is always negative since I assume $\gamma < \phi$.

• (i) When t is less than or equal to $\bar{t}, \varPsi _t(0)$ is non-positive and $\varPsi _t(1)$ is negative. Since $\varPsi _t$ is quadratic and convex, $\varPsi _t$ is non-positive for all $\mu \in [0,1)$ and so is $dv_t/d\mu$. That is, $v_t$ is a non-increasing function for the interval [0, 1) and is maximized at $\mu =0$.

• (ii) When t is greater than $\bar{t}, \varPsi _t(0)$ is positive and $\varPsi _t(1)$ is negative. Since $\varPsi _t$ is continuous, by the intermediate value theorem, there exists at least one $\mu \in (0,1)$ at which $\varPsi _t$ becomes zero. Let $\mu _t^*$ denote such $\mu$, that is, $\varPsi _t(\mu _t^*)=0$. Because of the functional form of $\varPsi _t$ (quadratic and convex), $\mu _t^*$ is unique and $\varPsi _t$ is downward sloping at $\mu _t^*$, that is, $\varPsi _t (\mu _t^*) =0$ and $\varPsi ^\prime (\mu _t^*)<0$. Then, I show that $\mu _t^*$ satisfies the first- and second-order conditions for maximization because

  $$\begin{aligned} \frac{dv_t}{d\mu } \bigg | _{\mu =\mu _t^* }&= \frac{\beta \gamma }{(1-\mu _t^* ) [\eta (1-\mu _t^* )+\gamma (\eta -1)\mu _t^* ] [(1+Z) (1-\mu _t^*) + \gamma Z] } \varPsi _t (\mu _t^*) =0, \\ \frac{d^2 v_t}{d\mu ^2} \bigg | _{\mu =\mu _t^* }&= \frac{\beta \gamma }{(1-\mu _t^* ) [\eta (1-\mu _t^* )+\gamma (\eta -1)\mu _t^* ] [(1+Z) (1-\mu _t^*) + \gamma Z] } \varPsi ^\prime _t (\mu _t^*) <0. \end{aligned}$$

• (iii) Finally, I show that the welfare-maximizing rates of R&D subsidies, $\mu _t^*$, are less than the growth-maximizing rate. By evaluating (43) at the growth-maximizing R&D subsidy rate, $\hat{\mu }$, I obtain

  $$\begin{aligned} \begin{aligned} \frac{dv_t }{d\mu } \bigg | _{\mu =\hat{\mu } }&= \frac{\beta \gamma }{1-\hat{\mu } } \biggl \{ 1 - \frac{\eta -1}{\eta (1-\hat{\mu } )+\gamma (\eta -1)\hat{\mu } } \\&\quad \quad \quad -\frac{Z}{(\eta -1) [(1+Z)(1-\hat{\mu }) + \gamma Z]} \Biggr \} +\frac{\beta t}{1+g} \frac{dg}{d\mu } \bigg | _{\mu =\hat{\mu } }. \end{aligned} \end{aligned}$$

  (46)

The last term of (46) is zero because $\hat{\mu }$ satisfies the first-order condition for maximization of the growth rate. Therefore $dv_t/d\mu |_{\mu =\hat{\mu }}$ is independent of time t and I obtain,

$$\begin{aligned} \frac{dv_t }{d\mu } \bigg | _{\mu =\hat{\mu } }&= \frac{dv_0 }{d\mu } \bigg | _{\mu =\hat{\mu } } < 0. \end{aligned}$$

The last inequality holds because $v_t$ is a decreasing function of $\mu$ for $t < \bar{t}$. Since $\varPsi _t$ and $dv_t/d\mu$ are negative only for $\mu >\mu _t^*$, the growth-maximizing rate, $\hat{\mu }$, is larger than the welfare-maximizing rate, $\mu _t^*$. $\square$

1.3 Intermediate goods firms’ problems with heterogeneity

I solve the problems backward, as in Sect. 2. Since the final goods firm’s behavior does not change, the demand functions for the intermediate goods are the same as (6). The production functions of the intermediate goods, given the productivity, are also the same as (7). Therefore, the second step in the intermediate goods firms’ problem is (36), subject to (6) and (7). Thus, the optimal price and output and the profit function of firm j, given its productivity and price index, are (37), (38), and (39), respectively.

The first step of the problem marginally changes because R&D efficiency differs by entrepreneur. The objective of this step is to maximize the intermediate goods firms’ net profits (40) subject to (39) and

$$\begin{aligned} A_t(j) = \theta _j K_{t-1} [l_t^R(j)]^\gamma . \end{aligned}$$

(47)

The first-order condition with respect to $l_t^R(j)$ is

$$\begin{aligned} \gamma \left( \frac{\eta -1}{\eta } \right) ^{\eta } \left[ \frac{\theta _j K_{t-1}}{w_t} P_t \right] ^{\eta -1} [l_t^R(j)]^{\gamma (\eta -1) -1} = (1-\mu _t )w_t . \end{aligned}$$

(48)

Since there is $\theta _j$ on the left-hand side of (47), $l_t^R(j)$ differs by the intermediate goods firms and so does $A_t(j)$ and $p_t(j)$. Note that it is necessary to assume $1-\gamma (\eta -1) >0$ to satisfy the second-order condition. Substituting (37) and (47) into the definition of the price index of the intermediate goods, $P_t$, I obtain

$$\begin{aligned} P_t = \frac{\eta }{\eta -1} \frac{w_t}{K_{t-1}} \biggl \{ \int _0^{N_t} \theta _j ^{\eta -1} [l_t^R(j)]^{\gamma (\eta -1)} dj \biggr \} ^{\frac{1}{1-\eta }}. \end{aligned}$$

(49)

I define $\varTheta _t$ as $\int _0^{N_t} \theta _j ^{\eta -1} [l_t^R(j)]^{\gamma (\eta -1)} dj$. Then, by using (49), I obtain

$$\begin{aligned} \varTheta _t \equiv \int _0^{N_t} \theta _j ^{\eta -1} [l_t^R(j)]^{\gamma (\eta -1)} dj = \left[ \frac{\eta -1}{\eta } \frac{P_t}{w_t} K_{t-1} \right] ^{1-\eta }. \end{aligned}$$

(50)

Solving (48) for $l_t^R(j)$ and rewriting it using $\varTheta _t$, I obtain

$$\begin{aligned} l_t^R(j) =\left[ \frac{\gamma }{1-\mu } \frac{\eta -1}{\eta w_t} \varTheta _t^{-1} \theta _j^{\eta -1} \right] ^{\frac{1}{1-\gamma (\eta -1)}}. \end{aligned}$$

(51)

Raising both sides of (51) to the power of $\gamma (\eta -1)$ and multiplying these by $\theta _j ^{\eta -1}$ yields

$$\begin{aligned} \theta _j ^{\eta -1} [l_t^R(j)]^{\gamma (\eta -1)} =\left[ \frac{\gamma }{1-\mu } \frac{\eta -1}{\eta w_t} \varTheta _t^{-1} \right] ^{\frac{\gamma (\eta -1)}{1-\gamma (\eta -1)}} \theta _j^{\frac{\eta -1}{1-\gamma (\eta -1)}} . \end{aligned}$$

(52)

Integrating both sides of (52) with respect to $j \in [0,N_t]$, I obtain

$$\begin{aligned} \int _0^{N_t} \theta _j ^{\eta -1} [l_t^R(j)]^{\gamma (\eta -1)} dj =&\left[ \frac{\gamma }{1-\mu } \frac{\eta -1}{\eta w_t} \varTheta _t^{-1} \right] ^{\frac{\gamma (\eta -1)}{1-\gamma (\eta -1)}} \nonumber \\&\int _0^{N_t} \theta _j^{\frac{\eta -1}{1-\gamma (\eta -1)}} dj. \end{aligned}$$

(53)

The left-hand side of (53) is $\varTheta _j$ and, thus, I solve for $\varTheta _t$ as follows

$$\begin{aligned} \varTheta _t =\left[ \frac{\gamma }{1-\mu } \frac{\eta -1}{\eta w_t} \right] ^{\gamma (\eta -1)} \left[ \int _0^{N_t} \theta _k^{\frac{\eta -1}{1-\gamma (\eta -1)}} dk \right] ^{1-\gamma (\eta -1)}. \end{aligned}$$

(54)

Substituting (54) into (51), I obtain the optimal labor input of the jth intermediate goods firm for the R&D activities as (29). Substituting (29) into (47), the productivity of the jth intermediate goods firm becomes

$$\begin{aligned} A_t(j) = K_{t-1} \Bigl ( \frac{\eta -1}{\eta w_t} \frac{\gamma }{1-\mu } \Bigr ) ^\gamma \frac{\theta _j^{\frac{1}{1-\gamma (\eta -1)}}}{\Bigl [ \int _0^{N_t} \theta _k^{\frac{\eta -1}{1-\gamma (\eta -1)}} dk \Bigr ] ^\gamma }. \end{aligned}$$

(55)

Using (49), (50), and (54), the price index of the intermediate goods can be computed as follows:

$$\begin{aligned} P_t =\left( \frac{1-\mu }{\gamma } \right) ^\gamma \left( \frac{\eta w_t}{\eta -1} \right) ^{1+\gamma } K_{t-1}^{-1} \left[ \int _0^{N_t} \theta _k^{\frac{\eta -1}{1-\gamma (\eta -1)}} dk \right] ^{\frac{1-\gamma (\eta -1)}{1-\eta }}. \end{aligned}$$

(56)

Using (7), (38), (A19), and (56), I obtain the optimal labor input of the jth intermediate goods firm for production as (28). Finally, substituting (29), (39), (55), and (56) into (40), I obtain the net profit of the jth intermediate goods firm as (30).