2. Model

To analytically examine the effects of money growth on economic growth and welfare, we incorporate the CIA constraint of consumption and endogenous labor supply into the standard R&D-based growth model of Grossman and Helpman (Reference Grossman and Helpman1991, Ch.3). Under the CIA constraint, people require money in advance to consume goods. The nominal interest rate is the cost of consumption and consequently, its rise reduces consumption, which is the benefit of labor supply. Higher money growth induces higher inflation and nominal interest rates, which reduce the benefit of labor supply and reduce labor supply, thus impeding economic growth. Conversely, money growth also increases government revenue and lowers taxes including corporate income tax, thereby enhancing economic growth. Through this mechanism, we examine how money growth affects economic growth and social welfare.

2.1 Households

There exists one representative household in the economy. The household supplies labor $\ell (t) (\!\in\![0, \bar {\ell }])$ elastically. The lifetime utility of the household is given by

(1) \begin{eqnarray} U=\int _0^\infty e^{-\rho t} \left [ \ln c(t) + \gamma \ln \left ( \bar {\ell }-\ell (t) \right ) \right ] dt, \end{eqnarray}

where $\rho$ denotes the discount rate, $c(t)$ denotes the consumption of goods, and $\gamma$ represents the weight of the utility from leisure $(\bar {\ell }-\ell (t))$ . We let $\tau _w$ denote the labor income tax rate. The household budget constraint is

\begin{eqnarray*} \dot {a}(t) &=& r(t) a(t) +(1-\tau _w) w(t) \ell (t) -c(t)- \pi (t) m(t)-T(t) - s(t), \\ \dot {m}(t) &=& s(t), \end{eqnarray*}

where $r(t)$ , $w(t)$ , $\pi (t)$ , $a(t)$ , $m(t)$ , $T(t)$ , and $s(t)$ denote the real interest rate, real wage rate, inflation rate, asset holdings, real money holdings, lump-sum tax, and the increase in money holdings, respectively.

The household must have money in advance to consume the good; that is, we assume a CIA constraint. Moreover, to consider a general situation, we assume that a fraction of consumption expenditure $\xi (\!\in\!(0,1])$ is subject to the following CIA constraint:

(2) \begin{eqnarray} \xi c(t) \le m(t). \end{eqnarray}

We let $\lambda (t)$ denote the costate variable for $a(t)$ , as in Appendix A. The optimality condition for consumption is

(3) \begin{eqnarray} \frac {1}{c(t)} =\left [ 1+\xi i(t) \right ] \lambda (t), \end{eqnarray}

where $i(t)$ denotes the nominal interest rate and satisfies the Fisher equation, $i(t)=r(t)+\pi (t)$ . The optimality condition for labor supply is

(4) \begin{eqnarray} \gamma \frac {1}{\bar {\ell }-\ell (t)}=\frac {(1-\tau _w)w(t)}{\left [1+ \xi i(t) \right ] c(t)}. \end{eqnarray}

Additionally, we obtain

(5) \begin{eqnarray} \frac {\dot {\lambda }(t)}{\lambda (t)} =\rho -r(t). \end{eqnarray}

2.2 Firms

The consumption good can be produced by utilizing intermediate goods. Following Benassy (Reference Benassy1998), we assume the production function of the good as follows.

(6) \begin{eqnarray} Y(t) =n(t)^{\nu -\left (\frac {1}{\alpha }-1\right )} \left [ \int _0^{n(t)} x(i, t)^\alpha di \right ]^\frac {1}{\alpha }, \quad 0\lt \alpha \lt 1, \end{eqnarray}

where $x(i,t)$ denotes the volume of intermediate good $i$ for $i \in [0,n(t)]$ , $\frac {1}{1-\alpha } (\gt 1)$ represents the elasticity of substitution among the intermediate goods, and $\nu (\gt 0)$ represents the degree of the contribution of variety $n$ to productivity. The higher the $\nu$ , the greater the contribution of variety to productivity. Hereafter, we refer to this parameter as the contribution of R&D to productivity.Footnote ⁶ By using this specification of production function, we can disentangle the elasticity of substitution and the contribution of R&D to productivity. We let $P(t)$ denote the price of the good. We assume that perfect competition prevails in the good market. By solving the cost-minimization problem for firms, given the production volume of good $Y(t)$ , we obtain the demand for the intermediate good $i$ as follows:

(7) \begin{equation} x(i, t) = \left ( \frac {P(i, t)}{P(t)} \right )^{-\frac {1}{1-\alpha }} Y(t) n(t)^{\frac {\alpha }{1-\alpha }\nu -1}. \end{equation}

where $P(i,t)$ denotes the price of intermediate good $i$ . Moreover, we obtain the unit cost, that is, the minimized cost of producing one unit of the good as $n(t)^{-(\nu - (\frac {1}{\alpha }-1 ))} [ \int _0^{n(t)} P(i, t)^{-\frac {\alpha }{1-\alpha }} di ]^{-\frac {1-\alpha }{\alpha }}.$ Perfect competition prevails, and thus the price of the final good equals this unit cost:

(8) \begin{eqnarray} P(t)=n(t)^{-\left(\nu -\left (\frac {1}{\alpha }-1\right )\right)} \left [ \int _0^{n(t)} P(i, t)^{-\frac {\alpha }{1-\alpha }} di \right ]^{-\frac {1-\alpha }{\alpha }}. \end{eqnarray}

2.2.1 Intermediate goods

One unit of each intermediate good is produced using one unit of labor. Each intermediate good is supplied monopolistically by a firm with a patent.

We must consider how governments protect patents. Generally, governments control the degree of patent protection through patent length and patent breadth. However, for simplicity, we assume that patent length is fixed and infinite and governments control the degree of patent protection using only patent breadth.Footnote ⁷ Following Goh and Olivier (Reference Goh and Olivier2002), we assume that when the patent breadth is broader, it is more difficult to produce imitative goods. We let $W(t)$ denote the nominal wage rate. We specify the unit cost of producing imitative goods as $\beta W(t) \ (\beta \in (1, 1/\alpha ])$ ; where strengthening patent protection increases $\beta$ . A firm that produces each intermediate good charges its price such that producers of imitative goods cannot earn a positive profit as follows:

(9) \begin{eqnarray} \displaystyle P(i,t)=\beta W(t). \end{eqnarray}

When $\beta$ obtains the highest value $1/\alpha$ , that is, when the patent breadth is the broadest, the price is equal to the monopoly price.

By substituting (9) into (8), we obtain the relationship between price of final good $P(t)$ and nominal wage $W(t)$ as $P(t)=n(t)^{-\nu } \beta W(t)$ . From this, we obtain the real wage rate $w(t) \equiv W(t)/P(t)$ as follows:Footnote ⁸

(10) \begin{eqnarray} w(t)=\frac {1}{\beta } n(t)^\nu . \end{eqnarray}

Because intermediate goods firms charge the same price, they produce the same volume. Letting $x(t)$ denote the symmetric production volume, from (6), we obtain

(11) \begin{equation} x(t) = n(t)^{-(\nu +1)} Y(t). \end{equation}

The real profit of the intermediate goods firm $i$ is $ \left ( p(i,t) -w(t) \right ) x(i, t),$ where $p(i, t) \equiv P(i, t)/P(t)$ is the real price of intermediate good $i$ . Therefore, we obtain the real profit of the intermediate goods firm $\Pi (t)$ as follows:

(12) \begin{equation} \Pi (t) =(p(i,t)-w(t))x(t) =(\beta -1) \frac {Y(t)}{\beta n(t)}. \end{equation}

2.2.2 R&D

Finally, we consider the behavior of R&D firms. We assume that R&D firms can invent a new intermediate good by devoting $\bar {a}(t)$ units of labor. Letting $\ell _R(t)$ denote the labor devoted to R&D activities, then, we can express the R&D technology as $\dot {n}(t) = \frac {1}{\bar {a}(t)} \ell _R(t)$ . We assume the presence of knowledge spillovers from the past R&D activities and thus assume that an increase in $n(t)$ reduces the R&D cost $\bar {a}(t)$ . Moreover, we assume $\bar {a}(t) = a/n(t)$ , as in the first-generation endogenous growth models such as Romer (Reference Romer1990). Thus, the R&D technology becomes

(13) \begin{eqnarray} \dot {n}(t) = \frac {n(t)}{a} \ell _R(t). \end{eqnarray}

We let $v(t)$ denote the real stock value of the intermediate goods. Then, the profit of an R&D firm is $v(t) dn - w(t) (a/n(t)) dn$ = $[v(t) -w(t) (a/n(t))] dn$ . The R&D equilibrium condition is given by

(14) \begin{eqnarray} v(t) n(t) =w(t) a. \end{eqnarray}

The real stock value $v(t)$ must satisfy the no-arbitrage condition, as follows:

(15) \begin{eqnarray} r(t) = \frac {(1-\tau _\Pi )\Pi (t)}{v(t)} +\frac {\dot {v}(t)}{v(t)}, \end{eqnarray}

where $\tau _\Pi$ denotes the corporate income tax rate.

2.3 Money supply

In this study, we consider the case in which the growth rate of the nominal money stock $M(t)$ is a constant $\mu$ ; that is, $\dot {M}{(t)}/M(t)=\mu$ . As Appendix B shows, by using the Fisher equation, we derive the nominal interest rate as follows:

(16) \begin{eqnarray} i(t)=\rho +\mu . \end{eqnarray}

2.4 Government

In fact, governments can obtain revenue by increasing the inflation rate; that is, they earn revenue by issuing new money. In this study, we consider a situation where the government can reduce distortionary tax rates by issuing new money and generating revenue. Specifically, we assume that governments satisfy the bala nced budget constraint at each time point. The government hires $\ell _G$ units of labor to supply public services at each point in time. The government must satisfy the following budget constraint: $\frac {\dot {M}(t)}{P(t)}+\tau _\Pi \Pi (t) n(t) + \tau _w w(t) \ell (t) +T(t)= w(t) \ell _G$ . The CIA constraint is binding; that is, $m(t)=\xi c(t)$ . Therefore, by rewriting this, we obtain

(17) \begin{eqnarray} \mu \xi c(t)+\tau _\Pi \Pi (t) n(t) + \tau _w w(t) \ell (t) +\frac {T(t)}{c(t)}= w(t) \ell _G. \end{eqnarray}

The government can choose the tax rates for corporate and labor income considering the budget constraint above.

3. Market equilibrium

In this section, we derive the market equilibrium path.

In this economy, due to knowledge spillovers, the number of varieties of goods grows at a positive constant rate. As mentioned above, $n(t)^\nu$ represents productivity in this economy. As will be shown later, there exists a balanced growth path along which productivity $n(t)^\nu$ and consumption $c(t)$ grow at the same rate.Footnote ⁹

On the balanced growth path, the ratio of consumption to $n(t)^\nu$ is constant, and thus, we define $z(t) \equiv \frac {c(t)}{n(t)^\nu }$ : $Y(t)=n(t)^\nu [n(t)x(t)]$ where $n(t)x(t)$ represents the volume of labor input devoted to production and $n(t)^\nu$ represents the total factor productivity (TFP) of the economy. Thus, $z(t)$ denotes the consumption-to-productivity ratio. We characterize the market equilibrium path using $z(t)$ .

First, we derive the equilibrium labor supply: Substituting (10) and (16) into (4) and rewriting it yields

(18) \begin{eqnarray} \ell (t) =\bar {\ell } - \gamma \frac {1+ \xi (\rho +\mu )}{1-\tau _w} \beta z(t), \end{eqnarray}

which is determined by $z(t)$ . Subsequently, the labor market equilibrium condition is $\ell (t) = \ell _G + \frac {a}{n(t)} \dot {n}(t) + n(t) x(t)$ . From (11), $n(t)x(t)= \frac {c(t)}{n(t)^\nu } =z(t)$ . Using these expressions, we obtain the growth rate of $n(t)$ as follows:

(19) \begin{eqnarray} \displaystyle \frac { \dot {n}(t) }{n(t)}= \frac {1}{a} \left [ \ell (t) - \ell _G -z(t) \right ], \end{eqnarray}

which is determined by $z(t)$ . From (3), (5), and (16), we obtain the growth rate of $c(t)$ as follows:

(20) \begin{eqnarray} \frac {\dot {c}{(t)}}{c(t)} =r(t)-\rho . \end{eqnarray}

We derive the real interest rate to determine the growth rate of $c(t)$ . From (10), (12), (14), and (15), we obtain

(21) \begin{eqnarray} r(t) = \frac {(1-\tau _\Pi ) \Pi (t)n(t)}{a w(t)} +\frac {\dot {w}(t)}{w(t)} - \frac {\dot {n}(t)}{n(t)} = \frac {(1-\tau _\Pi )(\beta -1)}{a}z(t) +\nu \frac {\dot {n}(t)}{n(t)} - \frac {\dot {n}(t)}{n(t)}. \end{eqnarray}

Finally, we derive the differential equation for only $z(t)$ . Considering the logarithm of both sides of $z(t) \equiv \frac {c(t)}{n(t)^\nu }$ , we obtain $ \frac {\dot {z}(t)}{z(t)} = \frac {\dot {c}(t)}{c(t)} - \nu \frac {\dot {n}(t)}{n(t)}.$ Substituting (3), (19), (20), and (21) into the RHS of this equation yields the growth rate of $z(t)$ as follows:

(22) \begin{eqnarray} \frac {\dot {z}(t)}{z(t)} &=& \frac {(1-\tau _\Pi )(\beta -1)}{a}z(t) - \frac {\dot {n}(t)}{n(t)} -\rho \nonumber \\ &=& \frac {(1-\tau _\Pi )(\beta -1)}{a}z(t) - \frac {1}{a} \left [ \ell (t) - \ell _G -z(t) \right ]-\rho , \end{eqnarray}

where $\ell (t)$ is expressed by (18).

We assume that the government is unable to employ lump-sum taxation and therefore must rely on distortionary taxes to finance public expenditure. In particular, we consider a public finance regime in which the labor income tax rate is constant and the corporate income tax rate is endogenously determined to satisfy the budget constraint (17). By substituting $T(t)=0$ and (10) into (17) and dividing both sides by $c(t)$ , we obtain $ \mu \xi +\tau _\Pi \frac {\Pi (t) n(t)}{c(t)} + \tau _w \frac {n(t)^\nu \ell (t)}{\beta c(t)} = \frac {n(t)^\nu \ell _G}{ \beta c(t)}.$ Using $z(t)$ , we can rewrite the equation above as follows: $ \mu \xi + \tau _\Pi ( 1-\frac {1}{\beta } ) + \tau _w \frac {\ell (t)}{\beta z(t)} = \frac {\ell _G}{\beta z(t)}.$ Subsequently, the corporate income tax rate is endogenously determined asFootnote ¹⁰

(23) \begin{eqnarray} \tau _\Pi (t) \left ( 1-\frac {1}{\beta } \right ) = \frac {\ell _G-\tau _w \ell (t)}{\beta z(t)}-\mu \xi . \end{eqnarray}

By substituting (18) and (23) into (22), we obtain the differential equation for $z(t)$ as follows:

\begin{eqnarray*} \frac {\dot {z}(t)}{z(t)} &=& \left \{ \mu \xi +1+\gamma [1+\xi (\rho +\mu )] \right \}\frac {\beta }{a}z(t) - \left [(1-\tau _w) \frac {\bar {\ell }}{a} +\rho \right ]. \end{eqnarray*}

A unique interior steady state exists. Letting $z^\ast$ denote the steady-state level of $z$ , we obtain

(24) \begin{eqnarray} z^\ast = \frac {(1-\tau _w) \bar {\ell } +\rho a}{\beta \left \{ \mu \xi +1+\gamma [1+\xi (\rho +\mu )] \right \}}. \end{eqnarray}

The unique interior steady state, $z^\ast$ is unstable. If $z(0)$ is not equal to $z^\ast$ , $z(t)$ diverges to a negative or positive infinity. Thus, $z(t)$ is equal to $z^\ast$ from the initial point in time. $z(t)$ is constant, and from (18) $\ell (t)$ is constant. Substituting these into (19), the growth rate of $n(t)$ is determined as follows:

(25) \begin{eqnarray} g_n^\ast \equiv \frac {\dot {n}(t)}{n(t)} = \frac {1}{a} \left \{ \bar {\ell } - \ell _G - \left [ 1+\gamma \beta \frac {1+\xi (\rho +\mu )}{1-\tau _w} \right ] z^\ast \right \}. \end{eqnarray}

All the other macroeconomic variables are constant or growing at constant rates. Thus, the balanced equilibrium path is obtained. Hereafter, we limit our analysis to cases in which the growth rate is nonnegative. Differentiating (25) with respect to $\mu$ yields

(26) \begin{eqnarray} \displaystyle \frac {\partial g_n^\ast }{\partial \mu } &=& \xi \gamma \beta \frac {z^\ast }{a} \left \{ -\frac {1}{1-\tau _w} + \left [ \frac { 1+\xi (\rho +\mu )}{1-\tau _w} + \frac {1}{\gamma \beta } \right ] \frac {1+\gamma }{\gamma [1+\xi (\rho +\mu )] +1+\xi \mu } \right \} \nonumber \\ &=& \xi \gamma \beta \frac {z^\ast }{a} \frac {\frac {1}{1-\tau _w} \xi \rho +\frac {1+\gamma }{\gamma \beta } }{\gamma [1+\xi (\rho +\mu )] +1+\xi \mu } \gt 0. \end{eqnarray}

Therefore, the growth rate is an increasing function of $\mu$ , and we can summarize the results as follows:

The reasons for this result are as follows. As shown in (25), an increase in the nominal money stock growth rate has two opposing effects: On the one hand, it raises the inflation rate and nominal interest rates. As shown in (18), an increase in the nominal interest rate reduces labor supply and consequently lowers the growth rate. On the other hand, it increases seigniorage, which lowers the corporate income tax rate, and thus promotes R&D and economic growth. As shown in the results, in this model, the latter’s positive effect overwhelms the former’s negative effect.Footnote ¹¹

To understand the direct effect of an increase in money growth, we consider the case in which an increase in money growth – an increase in seigniorage – does not reduce the corporate income tax rate. In other words, we consider the case in which the government is able to employ lump-sum transfer, and money growth does not lower the distortionary tax rate. In this lump-sum transfer regime, an increase in money growth has only the aforementioned negative effect on economic growth. As shown in Appendix C, stronger patent protection strengthens this negative effect. We summarize the results in the following proposition:

The intuition for the second result is as follows. An increase in money growth raises the nominal interest rate, which reduces labor supply. Stronger patent protection has a negative effect on the wage rate, as shown in (10). This is because it enables firms to raise the price of goods by reducing the production volume, which consequently reduces the demand for labor and wages. Hence, it strengthens the labor supply reducing effect of money growth.

4. Welfare analysis of money growth

4.1 Welfare-maximizing money growth rate

Subsequently, we examine how an increase in the growth rate of the nominal money stock affects welfare.

First, we derive welfare as a function of the growth rate of nominal money stock $\mu$ . The economy is on a balanced growth path, where $z(t)(\equiv c(t)/n(t)^\nu )$ is equal to $z^\ast$ , as previously shown. Therefore, consumption can be rewritten as $c(t)=z^\ast n(t)^\nu$ . Moreover, from (18), leisure is constant over time and is determined by $z^\ast$ . Substituting these into the utility function, we obtain

(27) \begin{eqnarray} U &=&\int _0^\infty e^{-\rho t} \left \{ \ln c(t) + \gamma \ln \left [ \bar {\ell } -\ell (t) \right ] \right \} dt \nonumber \\ &=& \int _0^\infty e^{-\rho t} \left \{ \ln z^\ast + \nu \left [ g_n^\ast t +\ln n(0) \right ] + \gamma \ln \left ( \bar {\ell } -\ell ^\ast \right ) \right \} dt \nonumber \\ &=& \frac {1}{\rho } \left [ \ln z^\ast +\gamma \ln (\bar {\ell }-\ell ^\ast ) + \nu \frac {g_n^\ast }{\rho } + \nu \ln n(0) \right ]. \end{eqnarray}

In order to derive the welfare-maximizing money growth rate, we differentiate (27) with respect to $\mu$ . The derivative of $U$ is given by:

(28) \begin{align} \displaystyle \frac {\partial U}{\partial \mu } = \frac {1}{\rho } \Bigl [ \underbrace { \frac {1}{z^\ast } \frac {\partial z^\ast }{\partial \mu }} _{\stackrel {\mbox{consumption-reducing effect}}{\mbox{(-)}}} \quad + \quad \underbrace { \gamma \frac {1}{\bar {\ell }-\ell ^\ast } \frac {\partial (\bar {\ell }-\ell ^\ast )}{\partial \mu }} _{\stackrel {\mbox{leisure-saving effect}}{\mbox{(+)}}} \quad + \quad \underbrace {\frac {1}{\rho } \nu \frac {\partial g_n^\ast }{\partial \mu }} _{\stackrel {\mbox{tax-reducing effect}}{\mbox{(+)}}} \Bigr ]. \end{align}

An increase in money growth affects welfare though the following channels. First, an increase in money growth raises the inflation rate and the nominal interest rate. Because the nominal interest is the relative cost of consumption to leisure under the CIA constraint, this consequently reduces consumption and increases leisure. The first effect reduces welfare and the second raises welfare. We refer to the first and second effects as the consumption-reducing effect and leisure-saving effect respectively. They are indicated by the first and second terms on the right-hand side (RHS) of (28). Finally, an increase in money growth raises revenue from issuing new money. This allows a reduction of corporate income tax through the balanced budget constraint and enhances innovation and thus raises welfare. We refer to this effect as the tax-reducing effect. This positive welfare effect is indicated by the third term on the RHS of (28).

The former two effects relate to the static allocation between consumption and leisure. We refer to the sum of these two effects as the static allocation effect. Using (18), we can rewrite the part related to static allocation in the utility as follows:

(29) \begin{eqnarray} U &=& \frac {1}{\rho } \left [ (1+\gamma ) \ln z^\ast +\gamma \ln \left [ 1+\xi (\rho +\mu ) \right ] +\nu \frac {g_n^\ast }{\rho } + \nu \ln n(0) \right ]. \end{eqnarray}

Differentiating this again, we decompose the welfare effect into the following two parts:

(30) \begin{eqnarray} \displaystyle \frac {\partial U}{\partial \mu } = \frac {1}{\rho } \Bigl [ \underbrace { (1+\gamma ) \frac {1}{z^\ast } \frac {\partial z^\ast }{\partial \mu } + \gamma \frac {\xi }{ 1+\xi (\rho +\mu )} } _{\stackrel {\mbox{static-allocation effect}}{\mbox{(-)}}} \quad + \quad \underbrace {\frac {1}{\rho } \nu \frac {\partial g_n^\ast }{\partial \mu }} _{\stackrel {\mbox{tax-reducing effect}}{\mbox{(+)}}} \Bigr ]. \end{eqnarray}

From (24), we obtain

(31) \begin{eqnarray} \displaystyle \frac {1}{z^\ast } \frac {\partial z^\ast }{\partial \mu } = -\frac {(1+\gamma )\xi }{\gamma [1+\xi (\rho +\mu )] +1+\xi \mu }\lt 0. \end{eqnarray}

Rewriting the static allocation effect, we obtain

(32) \begin{align} \displaystyle (1+\gamma ) \frac {1}{z^\ast } \frac {\partial z^\ast }{\partial \mu } + \gamma \frac {\xi }{ 1+\xi (\rho +\mu )} = -\frac {(1+\gamma )\xi }{\gamma [1+\xi (\rho +\mu )] +1+\xi \mu } \left [ 1+ \frac {\gamma }{1+\gamma } \frac {\xi \rho }{ 1+\xi (\rho +\mu )} \right ] \lt 0. \end{align}

Thus, the static allocation effect is negative. This is because an increase in money growth increases nominal interest rates and distort the allocation between consumption and leisure activities. This static allocation effect is the marginal cost (MC) of raising money growth, while the tax-reducing effect is the marginal benefit (MB) of raising money growth. Thus, the welfare-maximizing growth rate of money is determined so that MB is equal to MC.

As shown in Appendix D, the welfare-maximizing growth rate of money is uniquely determined, and that this level is negatively correlated with the patent protection strength. We summarize the results in the following proposition:

Proposition 3. Under a regime in which an increase in seigniorage – revenue from issuing new money – reduces corporate income tax, there is a welfare-maximizing growth rate for nominal money stock.

ting $\mu ^\ast$ denote the money growth rate, we can analytically derive the following:

(33) \begin{eqnarray} \displaystyle \mu ^\ast = \frac {\rho }{2(1+\gamma )} \left [ \frac {\Omega }{\xi \rho } + 1-\gamma +\sqrt {\left (\frac {\Omega }{\xi \rho } +1-\gamma \right )^2 +4 \gamma } \right ] - \left (\frac {1}{\xi }+\rho \right ), \end{eqnarray}

where

(34) \begin{eqnarray} \displaystyle \Omega \equiv \frac {1}{\rho } \nu \left ( \frac {\bar {\ell }}{a} +\frac {\rho }{1-\tau _w} \right ) \left (\frac {1-\tau _w}{\beta }+\frac {\gamma }{1+\gamma }\xi \rho \right ). \end{eqnarray}

From ( 33 ) and ( 34 ), stronger patent protection (higher $\beta$ ) reduces the welfare-maximizing money growth rate.

The reason for this result is as follows. First, in a regime in which the corporate income tax rate is constant, that is, where an increase in money growth does not lower the corporate income tax rate, as shown in Appendix C, stronger patent protection strengthens the negative effect of an increase in money growth on the output growth rate. The intuition is as mentioned below Proposition2: A rise in money growth raises the nominal interest rate, which reduces labor supply. Here, stronger patent protection induces a lower wage rate and strengthens the labor-supply-reducing effect of money growth. Owing to this effect, under the tax-reducing regime, stronger patent protection reduces the total positive growth effect of a rise in money growth and subsequently reduces the MB of a rise in money growth. Therefore, stronger patent protection reduces the welfare-maximizing money growth rate.

As we can see in Figure 1 in the introduction, the cross-country data show that the relationship between patent protection and inflation rates is negative. Therefore, the theoretical results of this study are consistent with actual tendencies.

4.2 Optimal monetary policy and patent protection

4.2.1 Optimal nominal interest rate

Finally, we examine whether the Friedman rule is optimal or not in this model. The Friedman rule is to set money growth rate such that the nominal interest rate equals zero, that is, $i(t)=0$ . In this case, the money growth rate is given by $\mu =-\rho$ . As is well known, the Friedman rule is likely to be optimal in the case where money growth is not used to reduce income taxes.Footnote ¹² However, in the case where money growth can reduce the income tax rate such as the present setting, the Friedman rule tends to be suboptimal, as discussed by Phelps (Reference Phelps1973). At the end of this section, we examine how whether the Friedman rule ( $i=0$ ) is optimal or not depends on patent protection. As shown in Appendix E, we obtained the following results:

Proposition 4. Suppose that the nominal interest rate is subject to the zero lower bound. We let $\bar {\beta }$ denote the patent breadth that satisfies $\frac {\partial U}{\partial \mu }\Big |_{i=\mu +\rho =0}=0$ . $\bar {\beta }$ is given by

\begin{eqnarray*} \displaystyle \bar {\beta } \equiv \left [ \frac {(1+\gamma ) -(1-\gamma )\xi \rho + \frac {\gamma }{1+\gamma } \xi ^2 \rho ^2}{\frac {1}{\rho } \nu \left ( \frac {\bar {\ell }}{a} +\frac {\rho }{1-\tau _w} \right ) } - \frac {\gamma }{1+\gamma }\xi \rho \right ]^{-1}. \end{eqnarray*}

Whether the welfare-maximizing nominal interest rates $i^\ast$ is positive or zero depends on the strength of patent protection $\beta$ as follows:

If (i) $\displaystyle \bar {\beta } \le 1$ , $i^\ast =0$ for all $\beta \in (1, \frac {1}{\alpha }]$ .

If (ii) $\displaystyle 1\lt \bar {\beta } \le \frac {1}{\alpha }$ , whether the welfare-maximizing nominal interest rate is positive or zero depends on the patent breadth as follows.

(35) \begin{align} i^\ast \begin{cases} \gt 0 \quad & \text{if} \ \ 1\lt \beta \lt \bar {\beta }, \\ =0 & \text{if} \ \ \bar {\beta } \le \beta \le \frac {1}{\alpha }, \end{cases} \end{align}

as depicted in Figure 2.

If (iii) $\displaystyle \frac {1}{\alpha }\lt \bar {\beta }$ , $i^\ast \gt 0$ for all $\beta \in (1, \frac {1}{\alpha }]$ .

Figure 2.

The welfare-maximizing interest rate $i^\ast$ in case (ii).

As shown in cases (i) and (ii), if $\bar {\beta }$ is lower, the Friedman rule can be optimal even in the present model where money growth can reduce income tax. For instance, if $\nu$ is lower, $\bar {\beta }$ is lower, then the Friedman rule tends to be optimal. The reason is as follows. $\nu$ represents the degree of contribution of R&D to TFP. If $\nu$ is lower, the benefit of R&D is lower from social point of view, and thus R&D tends to be eccessive. Hence raising corporate income tax rate and reducing money growth raise welfare.

4.2.2 Numerical examples

We demonstrate the welfare-maximizing nominal interest rate and inflation rate quantitatively. As mentioned above, whether the Friedman rule ( $i=0$ ) maximizes welfare or not depends heavily on R&D contribution to TFP $\nu$ . Thus we show the case where $\nu$ is high and that where it is low.

First, we assume $\nu =0.25$ , which is relatively high. For the quantitative evaluation, we set the parameters as follows. We set the discount rate $\rho$ to $0.05$ following Acemoglu and Akcigit (Reference Acemoglu and Akcigit2012). We set $\alpha$ to $0.625$ so that the range of patent breadth $[1, 1/\alpha ]$ includes $[1.05, 1.4]$ , which is the range of the markup rate estimated by Basu (Reference Basu1996). Following Chu and Wang (Reference Chu and Wang2022), we set the labor income tax to $0.238$ . We set the strength of CIA to one and normalize $\bar {\ell }=1$ . Finally, we consider $\beta =1.4$ as a benchmark case and we set $\gamma =1.6$ , $\ell _G=0.091$ , and $a=0.925$ . By using these values, we can make the ratio of the labor supply to $\bar {\ell }$ close to $1/3$ , which is a general value by used in earlier studies such as Zheng, Huang, and Yang (2021), and the ratio of government spending to GDP $w \ell _G/Y$ close to $0.36$ , which is the value of the US in 2022 reported by the IMF, and the growth rate of output $\nu g_n^\ast$ close to $1.8 \%$ , which is the average growth rate of the US. In this case, the welfare-maximizing nominal interest rate and inflation rate are relatively high as in Panel (a) of Figure 3. Even when the patent protection is so strong that patent breadth is $1.6$ , the optimal interest rate is positive, namely, the Friedman rule is not optimal.

Figure 3.

The welfare-maximizing nominal interest rate and infaltion rate.

Second, we assume $\nu =0.005$ , which is relatively low. Here, in order to make the ratio of the labor supply, the government spending GDP ratio, and the growth rate of output, close to the respective actual values, we set $\gamma =1.89$ , $\ell _G=0.09$ , and $a=0.01835$ . In this case, the welfare-maximizing nominal interest rate and inflation rate are relatively low as in Panel (b) of Figure 3. In particular, when patent protection is so strong that patent breadth is higher than $1.5$ , the optimal nominal interest rate is zero, namely, the Friedman rule is optimal.

5. Lump-sum tax regime

So far, we assume that the government is unable to employ lump-sum taxation. In this section, we assume that the government is able to employ lump-sum taxation and can set all the distortionary taxes to zero, that is, $\tau _\Pi =\tau _w=0$ . In this case, the government’s budget is given by $\frac {\dot {M}(t)}{P(t)}+T(t)= w(t) \ell _G$ , and thus, the lump-sum tax $T(t)$ is determined as $T(t)=w(t)\ell _G-\mu \xi c(t)$ . This means that the seigniorage by money growth only reduces the lump-sum tax. The lump-sum tax has no impact on the resource allocations such as consumption $c(t)$ and labor supply $\ell (t)$ . In this regime, the ratio of consumption to the productivity, $z(t)$ and $\ell (t)$ are determined as follows:Footnote ¹³

(36) \begin{eqnarray} z^\ast =\frac { \bar {\ell }-\ell _G +a \rho }{\{1 + \gamma [1+\xi (\rho +\mu )] \} \beta }, \end{eqnarray}

(37) \begin{eqnarray} \ell ^\ast = \bar {\ell } - \gamma [ 1+ \xi (\rho +\mu ) ] \beta z^\ast . \end{eqnarray}

Subsequently, the growth rate of $n(t)$ is as follows.

(38) \begin{eqnarray} g_n^\ast = \frac {1}{a} \left [ \ell (t) - \ell _G -z^\ast \right ] = \frac {1}{a} \left ( \bar {\ell } - \ell _G - \left \{ 1+\gamma \beta \left [1+\xi (\rho +\mu ) \right ] \right \} z^\ast \right )\!. \end{eqnarray}

Differentiating (38) with respect to $\mu$ yields

\begin{align*} \displaystyle \frac {\partial g_n^\ast }{\partial \mu } &= \frac {1}{a} \left ( -\gamma \xi \beta z^\ast - \left \{ 1+\gamma \beta \left [ 1+\xi (\rho +\mu ) \right ] \right \} \frac {\partial z^\ast }{\partial \mu } \right ) \\[6pt] &= -\gamma \xi \beta \frac {z^\ast }{a} \left [ 1- \frac {1+\gamma \beta [1+\xi (\rho +\mu )] }{\{1 + \gamma [1+\xi (\rho +\mu )] \} \beta } \right ] \\[6pt] &= -\gamma \xi \beta \frac {z^\ast }{a} \frac { \beta -1 }{\{ 1+ \gamma [1+\xi (\rho +\mu )] \}\beta }\lt 0. \end{align*}

In the lump-sum tax regime, there is no positive effect on growth, and thus, the growth effect of raising money growth rate is necessarily negative.

Next, we examine the welfare effect of raising money growth rate. The utility of the household is given by (29). Differentiating this with respect to $\mu$ , we obtain

(39) \begin{eqnarray} \displaystyle \frac {\partial U}{\partial \mu } = \frac {1}{\rho } \Bigl [ \underbrace { (1+\gamma ) \frac {1}{z^\ast } \frac {\partial z^\ast }{\partial \mu } + \gamma \frac {\xi }{ 1+\xi (\rho +\mu )} } _{\stackrel {\mbox{static-allocation effect}}{\mbox{(-)}}} \quad + \quad \underbrace {\frac {1}{\rho } \nu \frac {\partial g_n^\ast }{\partial \mu }} _{\stackrel {\mbox{growth effect}}{\mbox{(-)}}} \Bigr ]. \end{eqnarray}

As is shown before, the growth effect is negative in the lump-sum tax regime. From (36), we can rewrite the term of the static allocation effect as follows.

\begin{align*} \displaystyle (1+\gamma ) \frac {1}{z^\ast } \frac {\partial z^\ast }{\partial \mu } + \gamma \frac {\xi }{ 1+\xi (\rho +\mu )} &= - (1+\gamma ) \frac {\gamma \xi }{1+\gamma [ 1+\xi (\rho +\mu )]} + \gamma \frac {\xi }{ 1+\xi (\rho +\mu )} \\ &= \gamma \xi \left \{ -\frac {1+\gamma }{1+\gamma [ 1+\xi (\rho +\mu )]} + \frac {1}{ 1+\xi (\rho +\mu )} \right \} \\ &= \gamma \xi \frac {- \xi (\rho +\mu )}{\left \{1+\gamma [ 1+\xi (\rho +\mu )] \right \} [1+\xi (\rho +\mu )]}\lt 0. \end{align*}

Consequently, the static allocation effect is negative also. The total welfare effect of raising money growth rate is negative, and thus, lower money growth rate increases welfare. Therefore, if the nominal interest rate is subject to the zero lower bound, $\mu =-\rho$ maximizes welfare in the lump-sum tax regime.Footnote ¹⁴