2.1. Representative household’s problem

The economy is populated by a continuum of identical infinitely lived “large” households of a unit mass. The large household framework allows for modeling capital accumulation under a dynamic general equilibrium setting in a tractable manner while taking into consideration market frictions highlighted in labor search and matching models. The setup of large households is convenient in that all family members pool resources regardless of their labor market status. This useful method of modeling perfect consumption insurance in general equilibrium search models has been common since Merz (Reference Merz1995) and Andolfatto (Reference Andolfatto1996).

A large household consists of a continuum of family members (of measure one). Family members may be (i) workers, who engage in productive activities with the wage rate at $w_{t}$ , (ii) job seekers, who undertake job search activities, or (iii) leisure takers, who are involved in nonmarket activities. Employment is a predetermined state in each period. Let $ n_{t}\in\! \left ( 0,1\right )$ and $s_{t}\in\! \left ( 0,1\right )$ denote the fraction of household members working and searching for jobs, respectively, with the remaining fraction $1-n_{t}-s_{t}$ $\in\! \left ( 0,1\right )$ being in leisure.Footnote ⁵ If an individual member is employed, then he will supply his time endowment inelastically to the market. When an unemployed agent searches for jobs, he will be matched with a job vacancy with a certain probability. It is thus possible that an unemployed member remains unemployed for some time. As a result, individuals face an uncertainty in income, consumption and leisure. Following Lucas (Reference Lucas1990), Merz (Reference Merz1995) and Andolfatto (Reference Andolfatto1996), we assume that all members in a large household pool their resources in order to maximize the household’s utility.

The representative household is assumed to derive utility from consumption and disutility from both working and job search. Its lifetime utility is described by

(1) \begin{equation} \sum _{t=0}^{\infty }\beta ^{t}u(c_{t},n_{t}+s_{t})=\sum _{t=0}^{\infty }\beta ^{t}\left [ \log c_{t}-\epsilon \frac{\left ( n_{t}+s_{t}\right ) ^{1+\sigma }}{ 1+\sigma }\right ], \end{equation}

where $c_{t}\gt 0$ is the consumption, $\sigma \gt 0$ denotes the inverse of the labor supply elasticity, $\epsilon$ is a preference parameter attached to leisure, and $\beta \in\! \left ( 0,1\right )$ is the subjective discount factor. Heer (Reference Heer2003) also used this additively separable logarithmic preference, which has been shown to be consistent with the balanced growth path (BGP) (King et al., Reference King, Plosser and Rebelo1998).

Let $a_{t}$ and $M_{t}$ be the amount of non-monetary real asset and nominal money, respectively, owned by the representative household in period $t$ .Footnote ⁶ Since households are the owners of the firms, they receive the profits $\! \left ( \psi _{t}\right )$ remitted from firms in each period. The household’s budget constraint in period $t$ is then given by

(2) \begin{equation} (1+\tau _{c})c_{t}+a_{t+1}-a_{t}+(1+\pi _{t+1})m_{t+1}-m_{t}=(1-\tau _{w})w_{t}n_{t}+(1-\tau _{a})r_{t}a_{t}+\psi _{t}+\phi _{t}, \end{equation}

where $\tau _{c}$ , $\tau _{w}$ , and $\tau _{a}$ are the tax rate for consumption, labor income, and capital income, respectively, $r_{t}$ denotes the rental rate of assets, and $\phi _{t}$ is a lump sum transfer from the government, which includes tax revenues collected and the money injection. As a result, there is a wealth effect from redistribution of seigniorage revenues, like that in Wang and Xie (Reference Wang and Xie2013), Bhattacharya et al. (Reference Bhattacharya, Haslag and Martin2009), and Ghossoub and Reed (Reference Ghossoub and Reed2019). Variable $m_{t}=M_{t}/P_{t}$ is real money holdings in period $t$ , where $P_{t}$ is the aggregate price level in period $t$ . The inflation rate is $\pi _{t+1}=P_{t+1}/P_{t}-1$ in period $t$ .

As in Heer (Reference Heer2003), the representative household faces the CIA constraint on consumption as follows:

(3) \begin{equation} h(1+\tau _{c})c_{t}\leq m_{t}, \end{equation}

where $h\in (0,1]$ is the fraction of the household’s consumption expenditure that must be paid by real money balances. Thus, an $h$ fraction of consumption is cash goods, and the remaining fraction $(1-h)$ is credit goods. Taxes due for the purchase of cash goods are paid in cash, and thus $ h\tau _{c}c_{t}$ is a part of the CIA constraint.

As to the evolution of employment over time, some of the agents who are searching for jobs become employed in the next period, but some currently employed individuals may lose their jobs in the next period. Denote $\eta _{t}$ as the success rate of job search in period $t$ , which will be endogenously determined. Let $\theta \gt 0$ be the exogenous job separation rate. The number of employed individuals in the next period is given by

(4) \begin{equation} n_{t+1}=\eta _{t}s_{t}+\left ( 1-\theta \right ) n_{t}. \end{equation}

The problem of the representative household is to maximize the lifetime utility in (1), subject to (2), (3) and (4), the no-Ponzi-game condition, initial values $a_{0}$ , $m_{0}$ , $n_{0}$ , and the feasibility conditions $c_{t}\geq 0,$ $n_{t}\in\! \left ( 0,1\right ),\ s_{t}\in (0,1)$ . Let $\lambda _{t},$ $\zeta _{t}$ and $\varphi _{t}$ be the Lagrange multipliers for (2), (3) and (4), respectively. The first-order conditions with respect to $c_{t},$ $s_{t}$ , $ n_{t+1}$ , $a_{t+1}$ and $m_{t+1}$ are, respectively,

(5) \begin{equation} \frac{1}{c_{t}}=(1+\tau _{c})(\lambda _{t}+h\zeta _{t}), \end{equation}

(6) \begin{equation} \epsilon \!\left ( n_{t}+s_{t}\right ) ^{\sigma }=\varphi _{t}\eta _{t}, \end{equation}

(7) \begin{equation} \beta \epsilon\! \left ( n_{t+1}+s_{t+1}\right ) ^{\sigma }=\beta (1-\tau _{w})w_{t+1}\lambda _{t+1}+\beta\! \left ( 1-\theta \right ) \varphi _{t+1}-\varphi _{t}, \end{equation}

(8) \begin{equation} \lambda _{t}=\beta \lbrack 1+(1-\tau _{a})r_{t+1}]\lambda _{t+1}, \end{equation}

(9) \begin{equation} (1+\pi _{t+1})\lambda _{t}=\beta\! \left ( \lambda _{t+1}+\zeta _{t+1}\right ). \end{equation}

In these conditions, (5) equates the marginal utility of consumption to the marginal cost of consumption, and (6) equates the marginal disutility of job search to the expected marginal gain of a successful job match. Equation (7) is the optimal condition for employment tomorrow, which equates the discounted marginal disutility of employment tomorrow to the discounted marginal benefit of employment tomorrow, the latter being the after-tax wage income plus the adjusted shadow value of remaining in employment later. Finally, (8) and (9) equate the marginal cost to the marginal benefit of holding non-monetary assets and money, respectively.

Combining (5), (8) and (9) gives the following consumption Euler equation:

(10) \begin{equation} \frac{1}{c_{t}}\frac{1}{1+h[\pi _{t}+(1+\pi _{t})(1-\tau _{a})r_{t}]}=\frac{ \beta }{c_{t+1}}\frac{1+(1-\tau _{a})r_{t+1}}{1+h[\pi _{t+1}+(1+\pi _{t+1})(1-\tau _{a})r_{t+1}]}. \end{equation}

Note that in the case when $h=0$ , there is no CIA constraint. Then, (10) reduces to the standard Euler equation $\frac{1}{c_{t}}=\frac{\beta \lbrack 1+(1-\tau _{a})r_{t+1}]}{c_{t+1}}.$

2.2. The firms

There is a continuum of identical infinitely lived firms. In each period, the firm uses capital $k_{t}$ and labor $n_{t}$ to produce output $ y_{t}$ according to the following technology:

\begin{equation*} y_{t}=f\!\left ( k_{t},n_{t},\bar {k}_{t}\right ) =A_{t}k_{t}^{\varepsilon }n_{t}^{1-\varepsilon }, \end{equation*}

where $\varepsilon \in (0,1)$ measures the income share of capital and $ A_{t}$ is the technology level in period $t$ . In order for the model to exhibit perpetual economic growth, we follow Bean and Pissarides (Reference Bean and Pissarides1993) and Eriksson (Reference Eriksson1997) and assume that the technology level is $A_{t}=A\bar{k} _{t}^{b}\gt 0$ , where $A\gt 0$ is a productivity coefficient and $\bar{k}_{t}$ is economy-wide average capital in period $t$ , which is taken as given by the firm. In equilibrium, $\bar{k}_{t}$ is endogenous and equals $k_{t}$ . The model can sustain economic growth, when $b=1-\varepsilon$ , as in Romer (Reference Romer1986). Alternatively, the model reduces to a neoclassical growth model, when $b=0$ , as in Heer (Reference Heer2003).

There is also an evolution of employment from the firm’s perspective. Employment is increased by the inflow of workers due to recruitment and is decreased by the outflow of workers due to job separation.

(11) \begin{equation} n_{t+1}=q_{t}v_{t}+\left ( 1-\theta \right )\! n_{t}, \end{equation}

where $v_{t}$ is endogenously created vacancies, and $q_{t}$ is the rate at which a job vacancy matches with job seekers in period $t$ .

In order to hire workers, a firm has to post job vacancies in the labor market. There are costs of creating and maintaining vacancies. We assume that posting and maintaining one job vacancy costs $e_{t}=ew_{t}\gt 0$ units of output in period $t$ , where $e\in\! \left ( 0,1\right )$ is a constant parameter.Footnote ⁷ Hence, the representative firm’s profit flow in period $t$ , $\psi _{t}$ , is equal to the output produced net of the costs of employment, capital, and vacancy creation and maintenance.

(12) \begin{equation} \psi _{t}=y_{t}-w_{t}n_{t}-(r_{t}+\delta )k_{t}-e_{t}v_{t}, \end{equation}

where $\delta$ is the capital depreciation rate.

When computing the firm’s value at time $0,$ the profit in any period $t\geq 0$ is discounted by the market interest rates $z_{t}\equiv \prod \limits _{i=1}^{t}\frac{1}{1+r_{i}},$ with $z_{0}=1$ . As employment is a state variable, the firm’s problem is an optimal control problem. The representative firm maximizes the following discounted sum of profits

\begin{equation*} \underset {\left \{ k_{t},v_{t},n_{t+1}\right \} _{t=0}^{\infty }}{\max } \sum _{t=0}^{\infty }z_{t}\psi _{t}, \end{equation*}

subject to the production technology and the evolution of employment in (11), where the profit flow $\psi _{t}$ is in (12). Let $\xi _{t}$ be the Lagrange multiplier for the evolution of employment. The first-order conditions with respect to $k_{t},$ $v_{t}$ and $n_{t+1}$ are

(13) \begin{equation} A\varepsilon k_{t}^{\varepsilon +b-1}n_{t}^{1-\varepsilon }=r_{t}+\delta, \end{equation}

(14) \begin{equation} e_{t}=\xi _{t}q_{t}, \end{equation}

(15) \begin{equation} A(1-\varepsilon )k_{t+1}^{\varepsilon +b}n_{t+1}^{-\varepsilon }=w_{t+1}-[\left ( 1-\theta \right )\! \xi _{t+1}-\frac{z_{t}}{z_{t+1}}\xi _{t}]. \end{equation}

Equation (13) states that, in optimum, the firm rents capital to the amount where the marginal product of capital equals the marginal cost, the latter being the sum of the rental rate and the depreciation rate. Equation (14) equates the marginal cost of a job vacancy in period $t$ to the expected marginal benefit of new hiring in period $t$ . Equation (15) states that in period $t+1$ , the firm employs workers to the level where the marginal product of labor equals the marginal cost, the latter being the wage rate in period $t+1$ net of the shadow value of remaining in employment in period $t+1$ .

2.3. Job matching

The labor market exhibits search and match frictions with the aggregate flow of matches depending on the masses of job vacancies and seekers. Following Diamond (Reference Diamond1982), we assume pair-wise random matching. The number of successful job matches is determined by the following matching function.

\begin{equation*} M\left ( v_{t},s_{t}\right ) =Bv_{t}^{\alpha }s_{t}^{1-\alpha }, \quad B\gt 0\text { and }\alpha \in\! \left ( 0,1\right ), \end{equation*}

where $B\gt 0$ measures the degree of matching efficiency and $\alpha \in\! \left ( 0,1\right )$ denotes the elasticity that a job vacancy contributes to a match. The matching function facilitates the endogenous determination of job finding rates and recruitment rates.

Define the tightness of the labor market as $x_{t}\equiv \frac{v_{t}}{s_{t}} .$ The job finding rate is

(16) \begin{equation} \eta _{t}=\frac{M\left ( v_{t},s_{t}\right ) }{s_{t}}=B\!\left ( \frac{v_{t}}{ s_{t}}\right ) ^{\alpha }=Bx_{t}^{\alpha }. \end{equation}

By contrast, the recruitment rate is given by

(17) \begin{equation} q_{t}=\frac{M\left ( v_{t},s_{t}\right ) }{v_{t}}=B\left ( \frac{v_{t}}{s_{t}} \right ) ^{-(1-\alpha )}=Bx_{t}^{-(1-\alpha )}. \end{equation}

2.4. Wage determination

In a frictionless Walras world, the wage rate is taken as given, as there is implicitly an auctioneer in the labor market, who sets an equilibrium wage rate so as to equate the labor supply to labor demand. In a frictional labor market, however, there is no auctioneer, and a job seeker would encounter at most one unfilled job at one time, and similarly, an unfilled job would be filled by at most one job seeker at one time. This creates a bilateral monopoly.

Following conventional wisdom, the wage rate $w_{t}$ is determined by a matched worker-job pair through a cooperative Nash bargaining game. Hiring an additional worker at the wage rate $w_{t}$ would create a surplus of $ (\,f_{nt}-w_{t})$ for a firm, where $f_{nt}$ is the marginal product of labor in period $t.$ Moreover, accepting an offer at the wage rate $w_{t}$ would generate a gain of $(w_{t}+\xi \Omega _{t})$ for a worker, where $\xi \in \lbrack 0,1]$ is a fraction, $\Omega _{t}\equiv \frac{(1+\tau _{c})u_{nt}}{(1-\tau _{w})u_{ct}}$ is interpreted as a worker’s outside option (or a worker’s reservation wage), in which $u_{ct}$ and $u_{nt}$ represent the marginal utility of consumption and working, respectively. The expression $\frac{(1+\tau _{c})u_{nt}}{(1-\tau _{w})u_{ct}}$ is the after-tax marginal rate of substitution (hereafter MRS) between consumption and leisure. In the bargaining wage, the worker takes into account the fraction of the outside option. In the case when $\xi =0$ , the worker has zero reservation wage, and thus, the outside option is not taken into account in the wage bargaining.

We assume that all workers have the same bargaining strength $\varrho \in (0,1).$ The outcome of the bargaining game is a wage rate $w_{t}$ that solves the following maximization problem.

\begin{equation*} \max _{w_{t}}\!\left \{ (1-\varrho )\log\! \left ( \,f_{nt}-w_{t}\right ) +\varrho \log\! \left ( w_{t}+\xi \Omega _{t}\right ) \right \}. \end{equation*}

The optimization condition is $\varrho (f_{nt}-w_{t})=(1-\varrho )(w_{t}+\xi \Omega _{t}),$ which gives the following bargaining wage.

(18) \begin{equation} w_{t}=(1-\varrho )\xi \Omega _{t}+\varrho A(1-\varepsilon )k_{t}^{\varepsilon +b}n_{t}^{-\varepsilon }. \end{equation}

Thus, the bargaining is a weighted average of the reservation wage and the marginal product of labor with the weight on the latter being the worker’s bargaining power $\varrho$ . Notice that the reservation wage is proportional to the MRS between leisure and consumption. Thus, with other things being equal, higher consumption $c_{t}$ , employment $n_{t}$ , and job search $s_{t}$ all increase the MRS, the reservation wage, and thus the bargaining wage.

2.5. The government

Finally, the model is closed by setting the policy of the passive government. The nominal money $M_{t}$ is assumed to grow at a constant rate $ \mu \gt 0$ as follows.

(19) \begin{equation} M_{t+1}=(1+\mu )M_{t}. \end{equation}

The government budget is balanced in each period.

\begin{equation*} \phi _{t}=\tau _{c}c_{t}+\tau _{w}w_{t}n_{t}+\tau _{a}r_{t}a_{t}+\mu m_{t}. \end{equation*}

3.1. Equilibrium in the model with sustainable growth

In this case, $b=1-\varepsilon$ and growing variables increase without a bound. To ensure a stationary system, we will transform the equilibrium system by deflating growing variables by capital stock $k_{t}$ .

The equilibrium system is characterized by a system of seven difference equations. The system governs the dynamic properties of $\{\chi _{t},x_{t},n_{t+1},s_{t},\pi _{t+1},g_{t+1},\vartheta _{t}\}$ , where $\chi _{t}$ $\equiv$ $\frac{c_{t}}{k_{t}}$ and $\vartheta _{t}\equiv \frac{m_{t}}{ k_{t}}$ are, respectively, the ratio of consumption to capital and the ratio of real money balances to capital, and $g_{t+1}\equiv$ $\frac{k_{t+1}}{k_{t} }-1$ is the growth rate of capital from period $t$ to period $t+1$ .

First, if we let $\omega _{t}\equiv \frac{w_{t}}{k_{t}}$ denote the ratio of wage to capital, then (18) gives

(20) \begin{equation} \omega _{t}=(1-\varrho )\xi \widetilde{\Omega }_{t}\frac{1+\tau _{c}}{1-\tau _{w}}\frac{\epsilon\! \left ( n_{t}+s_{t}\right ) ^{\sigma }}{1/\chi _{t}} +\varrho A(1-\varepsilon )n_{t}^{-\varepsilon }, \end{equation}

where $\widetilde{\Omega }_{t}\equiv \frac{\Omega _{t}}{k_{t}}=\frac{1+\tau _{c}}{1-\tau _{w}}\frac{\epsilon\! \left ( n_{t}+s_{t}\right ) ^{\sigma }}{ 1/\chi _{t}}.$

Next, the resource constraint, divided by $k_{t}$ , is

(21) \begin{equation} \chi _{t}+g_{t+1}+\delta =An_{t}^{1-\varepsilon }-e\omega _{t}s_{t}x_{t}. \end{equation}

Moreover, substituting (16) into (4), the law of motion of employment in equilibrium is

(22) \begin{equation} n_{t+1}=Bx_{t}^{\alpha }s_{t}+\left ( 1-\theta \right ) n_{t}. \end{equation}

Using (13), we can rewrite the Euler equation in (10). If we multiply both sides of the equation by $c_{t+1}$ , with some manipulation, the Euler equation is rewritten as

(23) \begin{equation} \frac{\chi_{t+1}}{\chi_{t}}\frac{1+g_{t+1}}{\Psi (\pi_{t},n_{t})}=\frac{ \beta \lbrack 1+(1-\tau _{a})(A\varepsilon n_{t+1}^{1-\varepsilon }-\delta )] }{\Psi (\pi _{t+1},n_{t+1})}, \end{equation}

where $\Psi (\pi _{t},n_{t})\equiv 1+h[\pi _{t}+(1+\pi _{t})(1-\tau _{a})(A\varepsilon n_{t}^{1-\varepsilon }-\delta )].$

Furthermore, using (19), the inflation rate in period $t+1$ is given by

(24) \begin{equation} \pi _{t+1}=(1+\mu )\frac{\vartheta _{t}}{\vartheta _{t+1}}\frac{1}{1+g_{t+1}} -1. \end{equation}

In addition, combining (14) and (15) and substituting (6) into (7) yield firms’ demand for labor and households’ supply of labor, respectively, are as follows.

(25) \begin{equation} A(1-\varepsilon )n_{t+1}^{-\varepsilon }=\left [ 1-\frac{e(1-\theta )}{ Bx_{t+1}^{\alpha -1}}\right ] \omega _{t+1}+(1+A\varepsilon n_{t+1}^{1-\varepsilon }-\delta )\frac{e\omega _{t}}{Bx_{t}^{\alpha -1}(1+g_{t+1})}, \end{equation}

(26) \begin{equation} \frac{\epsilon\! \left ( n_{t}+s_{t}\right ) ^{\sigma }}{Bx_{t}^{\alpha }}=\beta \left [ \frac{(1-\tau _{w})\omega _{t+1}}{(1+\tau _{c})\chi _{t+1}}\frac{1}{ \Psi (\pi _{t+1},n_{t+1})}-\epsilon (n_{t+1}+s_{t+1})^{\sigma }[1-\frac{ (1-\theta )}{Bx_{t+1}^{\alpha }}]\right ]. \end{equation}

Finally, the binding CIA constraint, divided by $k_{t}$ , is

(27) \begin{equation} h(1+\tau _{c})\chi _{t}=\vartheta _{t}. \end{equation}

Thus, the equilibrium system consists of the seven difference equations (21) - (27), which determine the seven variables $\chi _{t},x_{t},n_{t+1},s_{t},\pi _{t+1},\vartheta _{t}$ and $g_{t+1}$ .

In the model without sustainable growth, Shi and Wen (Reference Shi and Wen1997) have shown that, given constant search intensity $s$ , the steady state is locally stable and thus the equilibrium path toward the steady state is a saddle, if the intertemporal elasticity of substitution is sufficiently large. Heer (Reference Heer2003) is otherwise identical to Shi and Wen (Reference Shi and Wen1997) except for endogenous search intensity $s$ . Setting the intertemporal elasticity of substitution at $ \frac{1}{2}$ , Heer (Reference Heer2003) numerically showed that the steady state is a saddle. Our model is otherwise identical to Heer (Reference Heer2003) except for $ b=1-\epsilon \gt 0$ , and thus the model exhibits sustainable growth. We have followed Heer (Reference Heer2003) and numerically shown that the BGP is a saddle.

The equilibrium is on a BGP, when $\chi _{t}$ , $x_{t}$ , $n_{t}$ , $s_{t}$ , $ \pi _{t}$ , $\vartheta _{t}$ , $w_{t},$ and $g_{t}$ are constant over time, denoted by $\chi$ , $x$ , $n,$ $s,$ $\pi,$ $\vartheta,$ $w,$ and $g$ , respectively. Note that in the BGP, $c_{t}$ , $y_{t}$ and $k_{t}$ all grow at the common growth rate $g$ . To simplify the algebra, without loss of generality, below we let $\tau _{c}=\tau _{w}=\tau _{a}=0.$ The conditions that determine the BGP are as follows.

First, in the BGP, the resource constraint in (21) becomes

(28) \begin{equation} \chi +g+\delta =An^{1-\varepsilon }-e\omega sx, \end{equation}

and the ratio of wage to capital in (20) in the BGP is

(29) \begin{equation} \omega =\omega (n,s,\chi )\equiv \xi (1-\varrho )\widetilde{\Omega }+\varrho A(1-\varepsilon )n^{-\varepsilon }, \end{equation}

where $\widetilde{\Omega }=\epsilon \!\chi\! \left ( n+s\right ) ^{\sigma }.$

Next, along the BGP, the law of motion of employment in (22) and the consumption Euler equation in (23) become, respectively,

(30) \begin{equation} Bx^{\alpha }s=\theta n, \end{equation}

(31) \begin{equation} 1+g=\beta (1+A\varepsilon n^{1-\varepsilon }-\delta ). \end{equation}

Moreover, in the BGP, the inflation in (24) becomes

(32) \begin{equation} \pi =\frac{1+\mu }{1+g}-1. \end{equation}

Further, in the BGP, the labor demand in (25) is

\begin{equation*} A(1-\varepsilon )n^{-\varepsilon }=\omega -\omega \frac {e(1-\theta )}{ Bx^{\alpha -1}}+(1+A\varepsilon n^{1-\varepsilon }-\delta )\frac {e\omega }{ Bx^{\alpha -1}(1+g)}. \end{equation*}

Substituting $\omega$ in (29) and $1+g$ in (31) into the labor demand equation above yields

(33) \begin{equation} (1-\varrho )=\left(\frac{1}{\beta }-1+\theta \right)\frac{\varrho ex^{1-\alpha }}{B} +\xi \left [ \frac{(1-\varrho )\epsilon \chi\! \left ( n+s\right ) ^{\sigma }}{ A(1-\varepsilon )n^{-\varepsilon }}\left ( 1-\left(1-\frac{1}{\beta }-\theta \right) \frac{ex^{1-\alpha }}{B}\right ) \right ]. \end{equation}

Furthermore, the labor supply in (26) is

\begin{equation*} \frac {\epsilon\! \left ( n+s\right ) ^{\sigma }}{\beta Bx^{\alpha }}=\left [ \frac {\omega }{\chi }\frac {1}{\Psi (\pi,n)}+\left(\frac {1-\theta }{Bx^{\alpha }} -1\right)\epsilon\! \left ( n+s\right ) ^{\sigma }\right ]. \end{equation*}

Substituting $\omega$ in (29) into the labor supply equation above gives

(34) \begin{equation} \frac{1}{\beta Bx^{\alpha }}=\left [ \frac{\varrho A(1-\varepsilon )}{\chi } \frac{n^{-\varepsilon }}{\Psi (\pi,n)\epsilon\! \left ( n+s\right ) ^{\sigma }} +\left(\frac{1-\theta }{Bx^{\alpha }}-1\right)\right ] +\xi \left [ \frac{1-\varrho }{ \Psi (\pi,n)}\right ], \end{equation}

where $\Psi (\pi,n)\equiv 1+h[\pi +(1+\pi )(A\varepsilon n^{1-\varepsilon }-\delta )].$

Finally, in the BGP, the CIA constraint in (27) is

(35) \begin{equation} h\chi =\vartheta. \end{equation}

The BGP system includes eight equations (28)-(35) that determine eight variables $\{\omega,x,n,s,\pi,\vartheta,\chi,g\}$ . Once $ \{\omega,x,n,s,\pi,\vartheta,\chi,g\}$ are determined, as $c$ , $y$ , and $ k$ all grow at the common growth rate $g$ along the BGP, the values of $c$ , $y$ and $k$ over time along the BGP are in turn determined. Hence, all variables along the BGP are solved.

3.2. Existence and uniqueness of the BGP

To analyze the existence and the uniqueness of BGP, in the Appendix we use other conditions to simplify the system into two equations in only two variables: employment $n$ and the tightness of the labor market $x.$ They are the firms’ labor demand equation (33) and the households’ labor supply equation (34) as follows.

(36) \begin{equation} (1-\varrho )=\left(\frac{1}{\beta }+\theta -1\right)\frac{\varrho ex^{1-\alpha }}{B} +\xi\! \left [ \frac{(1-\varrho )\epsilon }{A(1-\varepsilon )}\frac{P(n,x)X(n,x) }{1+\xi Z(n,x)}\right ], \end{equation}

(37) \begin{equation} 1+\frac{1-\beta (1-\theta )}{\beta B}\frac{1}{x^{\alpha }}=\frac{\varrho A(1-\varepsilon )}{\Lambda (\mu )\epsilon }\frac{n^{-(\varepsilon +\sigma )} }{(1+\frac{\theta }{B}x^{-\alpha })^{\sigma }}\frac{1}{X(n,x)}+\xi \!\left [ \frac{1-\varrho }{\Lambda (\mu )}N(n,x)\right ], \end{equation}

where $\Lambda (\mu )\equiv \frac{h(1+\mu )+\beta (1-h)}{\beta };$

$P(n,x)\equiv (1+\frac{\theta }{B}x^{-\alpha })^{\sigma }n^{(\varepsilon +\sigma )}[1+(\frac{1}{\beta }+\theta -1)\frac{e}{B}x^{1-\alpha }];$

$X(n,x)\equiv n^{1-\varepsilon }[A(1-\beta \varepsilon )-e\varrho A(1-\varepsilon )\frac{\theta }{B}x^{1-\alpha }+(1-\beta )(1-\delta )n^{-(1-\varepsilon )}];$

$Z(n,x)\equiv e(1-\varrho )\epsilon \frac{\theta }{B}\!\left ( 1+\frac{\theta }{ B}x^{-\alpha }\right ) ^{\sigma }n^{1+\sigma }x^{1-\alpha };$

$N(n,x)\equiv 1+\frac{\varrho A(1-\varepsilon )e\theta }{B}\frac{x^{1-\alpha }}{A(1-\beta \varepsilon )-e\varrho A(1-\varepsilon )(\theta/B)x^{1-\alpha }+(1-\delta )(1-\beta )n^{-(1-\varepsilon )}};$

$ P_{n}(n,x)\gt 0,P_{x}(n,x)\lt 0;\ X_{n}(n,x)\gt 0,X_{x}(n,x)\lt 0;\ Z_{n}(n,x)\gt 0,Z_{x}(n,x)\gt 0;$

$N_{n}(n,x)\gt 0,N_{x}(n,x)\gt 0$ .

Note that the left-hand side of (36) is the marginal cost of the labor demand and the right-hand side is the marginal benefit. In (37), the left-hand side is the marginal cost of the labor supply and the right-hand side is the marginal benefit. These two equations depend on the signs of the derivative of $P(n,x),X(n,x),Z(n,x)$ and $N(n,x)$ with respect to $n$ and $x$ .

To simplify the notation, we have used subscripts $n$ and $x$ to denote the derivative of $P(n,x),X(n,x),Z(n,x)$ and $N(n,x)$ with respect to $n$ and $ x,$ respectively. To understand how these derivatives in the above are signed, we consider

Condition 1 requires that the reciprocal of the labor supply elasticity $ \sigma$ be larger than the ratio of the contribution of a job seeker and the contribution of a job vacancy to a job match, $\frac{1-\alpha }{\alpha }$ . It is clear that Condition 1 is easily met.Footnote ⁸

Moreover, if we let the largest and the smallest possible value of $x$ be denoted by $x_{\max }$ and $x_{\min }$ , respectively, we consider

$(ii)$ $x_{\min }\gt\! \left ( \frac{ \theta }{B}(\frac{\sigma \alpha }{1-\alpha }-1)\right ) ^{\frac{1}{\alpha }},$

In Condition 2, as the upper bound is set at a very large value and the lower bound is set at a very small value, the condition is easily met.Footnote ⁹ We are ready to understand the signs of the derivative of $P(n,x),X(n,x),Z(n,x)$ and $N(n,x)$ with respect to $n$ and $x$ .

First, we examine the sign of $P_{x}(n,x)$ and $P_{n}(n,x).$ Combining Condition 1 with part $(i)$ of Condition 2, it is clear that the effect of $ x$ on $P(n,x)$ is negative; that is, $P_{x}(n,x)\lt 0.$ Footnote ¹⁰ Moreover, it is straightforward to verify that $ P_{n}(n,x)\gt 0.$

Next, investigating the sign of $X_{x}(n,x)$ and $X_{n}(n,x),$ part $(i)$ of Condition 2 implies $(1-\beta \varepsilon )\gt e\varrho (1-\varepsilon )\frac{ \theta }{B}x^{1-\alpha },$ so $X_{n}(n,x)\gt 0.$ In addition, it is straightforward to see that the effect of $x$ on $X(n,x)$ is negative, and thus, $X_{x}(n,x)\lt 0.$

Third, envisaging the signs of $Z_{x}(n,x)$ and $Z_{n}(n,x),$ part $(ii)$ of Condition 2 implies $x\gt\! \left ( \frac{\theta }{B}\left(\frac{\sigma \alpha }{ 1-\alpha }-1\right)\right ) ^{\frac{1}{\alpha }},$ and thus, $Z_{x}(n,x)\gt 0.$ Footnote ¹¹ Moreover, it is straightforward to show that $Z_{n}(n,x)\gt 0.$

Finally, it is clear to verify that $N_{n}(n,x)\gt 0.$ In addition, it is obvious that the positive effect of $x$ in the numerator of $N(n,x)$ dominates the negative effect of $x$ in the denominator, and thus, $ N_{x}(n,x)\gt 0.$

Now, we analyze the slope of equations (36) and (37) in the $(n,x)$ plane. First, we envisage the special case when the workers have no reservation wage, and thus, $\xi =0.$ The labor demand and the labor supply equations (36) and (37) in the case of $\xi =0$ are, respectively, as follows.

(38) \begin{equation} 1-\varrho =\left[\frac{1}{\beta }+\theta -1\right]\frac{e\varrho x^{1-\alpha }}{B} \equiv MB^{D}(\underset{(0)}{n},\underset{(+)}{x}), \end{equation}

(39) \begin{equation} 1+\frac{1-\beta (1-\theta )}{\beta B}\frac{1}{x^{\alpha }}=\frac{\varrho A(1-\varepsilon )}{\Lambda (\mu )\epsilon }\frac{n^{-(\varepsilon +\sigma )} }{(1+\frac{\theta }{B}x^{-\alpha })^{\sigma }}\frac{1}{X(n,x)}\equiv MB^{S}( \underset{(-)}{n},\underset{(+)}{x}). \end{equation}

It is obvious from (38) that the marginal cost of the labor demand in the left-hand side is constant, while the marginal benefit of the labor demand in the right-hand side $MB^{D}$ is increasing in $x$ but is independent of $n$ . Thus, the labor demand is a horizontal locus in the $ (n,x)$ plan. See Locus $L^{D}$ in Figure 1.

Figure 1. Existence and uniqueness of BGP without outside options.

Moreover, it is clear from (39) that the marginal cost of the labor supply in the left-hand side is decreasing in $x$ . As for the marginal benefit of the labor supply in the right-hand side, notice that the term $ \frac{1{\large \ }}{X(n,x)}$ is increasing in $x$ and decreasing in employment $n.$ Further, the term $\frac{n^{-(\varepsilon +\sigma )}}{ (1+(\theta/B)x^{-\alpha })^{\sigma }}$ is also increasing in $x$ and decreasing in $n.$ Hence, the marginal benefit of the labor supply $MB^{S}$ is increasing in $x$ and decreasing in $n.$ As a result, the labor supply locus is positively sloping in the $(n,x)$ plane. See Locus $ L^{S}$ in Figure 1.

With a horizontal labor demand locus and a positively sloping labor supply locus, it is clear that there exists a unique value of $x_{0}^{\ast }$ and a unique value of $n_{0}^{\ast }$ in the BGP.

Next, we investigate the general case when there is the reservation wage, and thus, $\xi \in (0,1],$ and focus on the case $\xi =1.$ In this case, it serves to rewrite the system of two equations in (36) and (37) as follows.

(40) \begin{equation} (1-\varrho )=MB^{D}(\underset{(0)}{n},\underset{(+)}{x})+\xi MB^{D\xi }( \underset{(+)}{n},\underset{(-)}{x})\equiv MB^{LD}(\underset{(+)}{n}, \underset{(+)}{x}), \end{equation}

(41) \begin{equation} 1+\frac{1-\beta (1-\theta )}{\beta B}\frac{1}{x^{\alpha }}=MB^{S}(\underset{ (-)}{n},\underset{(+)}{x})+\xi MB^{S\xi }(\underset{(+)}{n},\underset{(+)}{x} )\equiv MB^{LS}(\underset{(-)}{n},\underset{(+)}{x}), \end{equation}

where $MB^{D\xi }(n,x)\equiv \frac{(1-\varrho )\epsilon }{A(1-\varepsilon )} \frac{P(n,x)X(n,x)}{1+\xi Z(n,x)}$ and $MB^{S\xi }(n,x)\equiv \frac{ 1-\varrho }{\Lambda (\mu )}N(n,x).$

We start by the slope of the labor demand curve in (40). First, note that in the marginal benefit of the labor demand $MB^{LD},$ the term $ MB^{D}$ is not affected by $n$ . Yet, in the term $MB^{D\xi },$ the numerator $P(n,x)X(n,x)$ and the denominator $1+\xi Z(n,x)$ both are positively affected by $n$ , and the net effect of $n$ on $MB^{D\xi }(n,x)$ is dictated by the term $\frac{(1-\beta )(1-\delta )/n^{1-\varepsilon }}{ 1/n^{1+\sigma }},$ whose power is $\sigma +\varepsilon,$ and is increasing in $n.$ Footnote ¹² As the effect of $n$ on the marginal benefit of the labor demand $MB^{LD}$ is dictated by the positive effect of $n$ on $MB^{D\xi },$ it follows that $MB^{LD}$ is unambiguously increasing in $n$ .

Next, to see the effect of $x$ on $MB^{LD}$ , note that $MB^{D}$ is increasing in $x,$ but $MB^{D\xi }$ is decreasing in $x$ , since $ P_{x}(n,x)\lt 0,$ $X_{x}(n,x)\lt 0,$ and $Z_{x}(n,x)\gt 0.$ We do not know whether the positive effect of $x$ on $MB^{LD}$ is dominated by, or dominates, the negative effect of $x$ on $MB^{D\xi }.$ Nevertheless, we are sure that when $ n=0,$ (40) gives $x=\left[\frac{(1-\varrho )B}{(\frac{1}{\beta }+\theta -1)\varrho e}\right]^{\frac{1}{1-\alpha }}\equiv x_{0}^{\ast }.$ We may also investigate what the value $n$ is when $x=0$ in the locus (40), but it is impossible to do so, as $MB^{LD}$ then goes to infinity. As an alternative, we study what the value $n$ is in the locus (40) when $ x=1$ . Denote $\Delta \equiv \frac{A(1-\varepsilon )}{\xi (1-\varrho )\epsilon }\left[\frac{(1-\varrho )-(\frac{1}{\beta }+\theta -1)\frac{\varrho e}{B }}{1+(\frac{1}{\beta }+\theta -1)\frac{e}{B}}\right].$ Consider

Condition 3 requires that the unit vacancy cost $e$ be sufficiently small. These conditions are easily met, if the matching coefficient $B$ is sufficiently large.Footnote ¹³

First, under the first element of Condition 3, we know $x_{0}^{\ast }\gt 1$ and $\Delta \gt 0.$ Next, under the second element of Condition 3, we get $A\left [ (1-\beta \varepsilon )-e\varrho (1-\varepsilon )\frac{\theta }{B}\right ] \gt \xi \Delta e(1-\varrho )\epsilon \frac{\theta }{B}.$ Finally, under the third element of Condition 3, we obtain $(1-\beta )(1-\delta )+A\left ( (1-\beta \varepsilon )-e\varrho (1-\varepsilon )\frac{\theta }{B}\right ) -\left ( \xi \Delta e(1-\varrho )\epsilon \frac{\theta }{B}\right ) \gt \frac{ \Delta }{(1+\frac{\theta }{B})^{\sigma }}.$

Then, if we set $x=1,$ (40) gives $\frac{n^{1+\sigma }(1+\frac{ \theta }{B})^{\sigma }[A(1-\beta \varepsilon )-e\varrho A(1-\varepsilon ) \frac{\theta }{B}+(1-\beta )(1-\delta )n^{-(1-\varepsilon )}]}{1+\xi e(1-\varrho )\epsilon \frac{\theta }{B}\left ( 1+\frac{\theta }{B}\right ) ^{\sigma }n^{1+\sigma }}=\Delta \gt 0,$ according to the first element of Condition 3. This equation is a function of $n$ and thus can determine the value of $n.$ We can rewrite the equation as follows.

(42) \begin{equation} (1-\beta )(1-\delta )+\left ( A\!\left[(1-\beta \varepsilon )-e\varrho (1-\varepsilon )\frac{\theta }{B}\right]-\Delta \xi e(1-\varrho )\epsilon \frac{ \theta }{B}\right ) n^{(1-\varepsilon )}=\frac{\Delta }{(1+\frac{\theta }{B} )^{\sigma }n^{\sigma +\varepsilon }}. \end{equation}

The left-hand side of (42), denoted by $LH(n)$ , is increasing in $n,$ whose value is equal to $LH(0)\ =\ (1-\beta )(1-\delta )\gt 0$ when $n=0,$ and equal to $LH(1)\ =\ LH(0)+\left ( A[(1-\beta \varepsilon )-e\varrho (1-\varepsilon )\frac{\theta }{B}]-\xi \Delta e(1-\varrho )\epsilon \frac{ \theta }{B}\right ) \gt 0,$ when $n=1.$ Note that $LH(1)\gt LH(0),$ according to the second element of Condition 3. See locus $LH(n)$ in Figure 2. Moreover, the right-hand side of (42), denoted by $RH(n)$ , is decreasing in $n,$ whose the value is equal to $RH(0)=\infty$ when $n=0,$ and equal to $RH(1)= \frac{\Delta }{(1+\frac{\theta }{B})^{\sigma }}\gt 0$ when $n=1.$ It is clear that $LH(0)\lt RH(0).$ Moreover, the third element of Condition 3 ensures that $ LH(1)\gt \frac{\Delta }{(1+\frac{\theta }{B})^{\sigma }}.$ See locus $RH(n)$ in Figure 2. As a result, there exists a unique value of an interior $n_{1}\gt 0.$

Figure 2. Determination of labor demand $n$ , given tightness of the labor market $x=1$ .

We thus discover that (40) gives $x=$ $x_{0}^{\ast }\gt 1$ when $n=0$ and gives $x=1\lt x_{0}^{\ast }$ when $n_{1}\gt n=0.$ The results indicate that, in the plane $(n,x),$ (40) is a negatively sloping locus in the plane $(n,x),$ decreasing from the value of $x_{0}^{\ast }\gt 1$ to the value $ x=1$ as the value of $n$ increases from $n=0$ to $n_{1}\gt 0.$ See Locus $L^{D}$ in Figure 3.

Figure 3. Existence and uniqueness of BGP with outside options and effects of seigniorage taxes.

Now, we envisage the slope of the labor supply curve in (41). Note that, due to $N_{x}(n,x)\gt 0,$ the term $MB^{S\xi }(n,x)$ is increasing in $x,$ which re-enforces the positive effect of $x$ on $MB^{S}(n,x),$ thereby making the marginal benefit of the labor supply $MB^{LS}$ to increase in $x$ . Moreover, due to $N_{n}(n,x)\gt 0,$ the positive effect of $n$ on $MB^{S\xi }(n,x)$ offsets the negative effect of $n$ on $MB^{S}(n,x),$ but it is easy to show that the negative effect of $n$ on $MB^{S}(n,x)$ dominates.Footnote ¹⁴ Therefore, the marginal benefit of the labor supply $MB^{LS}$ is decreasing in $n.$ As a result, the labor supply locus is positively sloping in the $(n,x)$ plan. See Locus $L^{S}$ in Figure 3.

With a negatively sloping labor demand locus and a positively sloping labor supply locus, it is clear that there exists a unique value of $x^{\ast }$ and a unique value of $n^{\ast }$ in the BGP. Thus, we have established:

With the existence and the uniqueness for the values of $x^{\ast }$ and $ n^{\ast }$ , when we substitute the values of $x^{\ast }$ and $n^{\ast }$ to other equilibrium conditions, we can determine the values of $w,$ $s^{\ast },\pi ^{\ast },\vartheta ^{\ast },\chi ^{\ast }$ , and $g^{\ast }$ in the BGP. As $c$ , $y$ , and $k$ all grow at the common growth rate $g$ along the BGP, the values of $c$ , $y$ , and $k$ over time are in turn determined. Hence, all variables are solved in the case with the reservation wage.

3.3. Comparative-static effects of the seigniorage tax in the BGP

To analyze the optimal seigniorage tax, we note that the discounted sum of the lifetime utility of the representative household is a function of $c$ , $n$ , and $s$ , according to (1). Thus, it is useful to study the comparative-static analysis of the effect of an increase in the money growth rate $\mu$ on $n$ and $s$ by using Figure 3. Notice that an increase in the money growth rate does not affect Locus $L^{D}$ , but Locus $L^{S}$ is influenced by an increase in $\mu$ , as seen by (37). Clearly, a higher $\mu$ decreases the marginal benefit of the labor supply. As a result, the labor supply $n$ needs to increase in order to lower the marginal benefit of the labor supply, so Locus $L^{S}$ shifts rightwards. The new BGP is at $E^{\mu }$ in Figure 3, and thus, the employment increases to $n^{\mu }$ , while the vacancy to search ratio decreases to $x^{\mu }.$

As the result of an increase in employment, via (31) the economic growth rate $g^{\mu }$ is unambiguously increased in the new BGP. Yet, as seen from (28)–(32) and (35), the effects on other variables $\chi ^{\mu },$ $s^{\mu },$ $\pi ^{\mu },$ $\vartheta ^{\mu }$ and $w^{\mu }$ are ambiguous. Thus, we need to resort to numerical solution.

Indeed, as observed by Heer (Reference Heer2003), it impossible to analyze his model, which is the case of $b=0$ in our model. As a result, Heer (Reference Heer2003) noted that the effects of a change in the growth rate of money supply cannot be studied analytically but only numerically in his model. Different from Heer (Reference Heer2003), we can study the existence and the uniqueness of the BGP and carry out the comparative-static analysis of the effect of the seigniorage tax in the BGP in terms of a change in the growth rate of money supply. However, to compare the effect of the optimal seigniorage tax in the BGP in our endogenous growth model and the exogenous growth model in Heer (Reference Heer2003), we still need to rely on numerical solution. In the next section, we follow Heer (Reference Heer2003) and calibrate our model in order to investigate the optimal money growth rate.

4.1. Calibration

In the model economy, $16$ parameters require values: preference $ (\beta$ , $\epsilon$ , and $\sigma )$ , production $(A,$ $\varepsilon$ , $ \delta$ , and $e)$ , labor market $(B,\alpha,$ $\theta$ , and $\varrho )$ , monetary parameters $(\mu$ and $h),$ and government $(\tau _{a}$ , $\tau _{w}$ , and $\tau _{c})$ . The time frequency is quarters.

First, we follow Cooley (Reference Cooley1995) and set the quarterly discount factor at $ \beta =0.99$ , which corresponds to an annual discount rate of 4%. Next, we set $\sigma =2.25$ . This parameter value indicates the labor supply elasticity of $0.4$ , which is consistent with the estimates reported in MaCurdy (Reference MaCurdy1981) and Killingsworth (Reference Killingsworth1983). Following Kydland and Prescott (Reference Kydland and Prescott1982), we set the capital share in income at $\varepsilon =0.36$ . The capital depreciation rate is set to be $\delta =0.025$ , which indicates an annual depreciation rate of $10$ %. According to the Job Openings and Labor Turnover Survey (JOLTS), the average quarterly separation rate was $10.45$ % during 2001-2015. We abide by the JOLTS and set $\theta =0.1045$ .

Moreover, Hall and Milgrom (Reference Hall and Milgrom2008) have estimated and found the daily cost of opening a job vacancy at about $43$ % of daily wage, which we follow and set $e=0.43$ . As to the wage bargaining, we follow Albrecht and Vroman (Reference Albrecht and Vroman2002) and simply set an equal share for workers and firms, and thus, $\varrho =0.5$ . This value is within the range of $0.3$ and $0.6$ that is commonly used.Footnote ¹⁵ To ensure that the Hosios condition is met, we set $\alpha =0.5$ . We follow Wang and Xie (Reference Wang and Xie2013) and set the tax rates at $\tau _{a}=\tau _{w}=0.2$ and $ \tau _{c}=0.05$ , which are commonly chosen in the dynamic tax incidence literature calibrating the US economy. Following Heer (Reference Heer2003), we set $ h=0.84$ , so $84$ % of the consumption expenditure goes to cash goods.

Finally, the remaining four parameters $(\mu$ , $A$ , $B$ , $\epsilon )$ are calibrated simultaneously in order for the steady state of the model economy to match four key statistics in the US. The first target is a $0.5$ % average quarterly real economic growth rate in 1947–2015. The second target is a $0.881$ % average quarterly inflation rate during 1947–2015. The third and fourth targets are the labor force participation rate and the unemployment rate, which on average were $62.9$ % and $ 5.8$ %, respectively from 1948 to 2015. The calibration gives $\mu =0.0139$ , $A=0.1706$ , $B=1.027$ , and $\epsilon =2.7862$ .

The baseline parameter values are summarized in Table 1. When $b=0$ , the model reduces to Heer (Reference Heer2003) and does not exhibit sustainable growth. By contrast, when $b=1-\epsilon$ , the model exhibits sustainable growth. Given the baseline parameter values, we have numerically shown that there exists a unique BGP. Moreover, the equilibrium path toward the BGP exhibits saddle-path stability.

Table 1. Calibration of parameter values for the US economy (quarter)

Our calibration indicates that in the long run, the size of employed agents is $n=59.25$ %, and the size of unemployed agents is $s=3.65$ %, while the remaining fraction of agents out of the labor force is $(1-n-s)=37.1$ %. These values imply that the unemployment rate is $5.80$ %.Footnote ¹⁶ In addition, the job finding rate per quarter is $\eta =1.697$ . A larger-than-one job finding rate means that, on average, a job seeker has more than one job match per quarter.

Moreover, in the model with sustainable growth, the firm’s recruitment rate per quarter is $q=0.622$ . In order to match the inflation rate in the model with sustainable growth, the money growth rate per quarter is calibrated at $ \mu =1.39$ %. Our calibration indicates that the quarterly consumption-capital ratio is $\chi =8.68$ %, and the quarterly economic growth rate is $g=0.5$ % in the model with sustainable growth.

Alternatively, in the model without sustainable growth, the firm’s recruitment rate per quarter is $q=0.646$ . In order to match the inflation rate in the model without sustainable growth, the money growth rate per quarter is calibrated at $\mu =0.881$ %. Our calibration indicates that equilibrium consumption is $c=1.5186$ and capital stock is $k=20.194$ in the steady state in the model without sustainable growth.

4.2. Numerical results

Now, we offer the numerical effects of a change in the money growth rate. We focus on the effects on the consumption-capital ratio, the inflation rate, employment, unemployment, the economic growth rate, and the welfare in the long run.

4.2.1. Model without sustainable growth

We begin with the model without sustainable growth, which is the model studied by Heer (Reference Heer2003). The baseline of the money growth rate is $\mu =0.881$ % in the model without sustainable growth. We explore the effects on key endogenous variables in the steady state when the money growth rate is changed in the range of $\mu \in \lbrack \!-0.5$ % $,2.5$ % $]$ that covers the baseline rate. The results are illustrated in Figure 4.

Figure 4. Effects of changes in money growth rates in the model without sustainable growth.

Figure 4 highlights the following. Firstly, when the money growth rate $ \mu$ decreases, the rate of inflation goes down. In the presence of the CIA constraint, households substitute consumption for leisure. As consumption $c$ increases, savings decrease, which reduces capital stock $k$ . As agents decrease leisure, they increase their search effort. More agents searching for jobs enlarges the size of unemployment $s$ . More job seekers would increase the rate $q$ at which firms fill a vacancy, which encourages firms to create more vacancies $v$ . However, as consumption increases and leisure decreases, the MRS between leisure and consumption increases, which increases the reservation wage and hence the bargaining wage $w$ . A higher wage would discourage firms’ recruitment activities. For the baseline calibration, the latter effect dominates, and thus, the vacancy decreases.

Next, there are also two offsetting effects on the size of employment $n$ . On the one hand, since $n=\eta s/\theta$ , if the job finding probability $ \eta$ is unchanged, an increase in the size of unemployment $s$ would enlarge the size of employment $n$ . On the other hand, as firms decrease their posted vacancies $v$ , the job finding probability $\eta$ will decrease, which will in turn shrink the size of employment $n$ . For the baseline calibration, the former effect dominates, and thus, the employment decreases in the steady state.

Our quantitative exercises indicate that a decrease in the money growth rate from the baseline $0.881$ % to $0$ % increases consumption by $0.32$ percentage points (from $1.5186$ to $1.5218$ ) and unemployment by $0.19$ percentage points (from $0.0365$ to $0.0384$ ), while employment falls by $ 0.03$ percentage points (from $0.5925$ to $0.5922$ ), and capital also decreases by $1.1$ percentage points (from $20.194$ to $20.183$ ). As a consequence, the wage rate increases by $0.75$ percentage points (from $ 2.1149$ to $2.1224$ ) and posted vacancies decrease by about $0.48$ percentage points (from $0.0959$ to $0.0911$ ). We find that the household’s welfare increases as the money growth rate decreases from the baseline rate to $0$ %, as the increase in utility from considerably higher consumption compensates for the decrease in utility from a higher labor force and thus, lower leisure.

4.2.2. Model with sustainable growth

Next, we examine the model with sustainable growth. The baseline of the money growth rate is $\mu =1.39$ % in the model with sustainable growth. We analyze the effects on key endogenous variables in the BGP when the money growth rate is changed in the range of $\mu \in \lbrack 0$ % $,4.5$ % $],$ which covers the baseline rate. The results are illustrated in Figure 5.

Figure 5. Effects of changes in money growth rates in the model with sustainable growth.

Figure 5 highlights the following. Like the model without sustainable growth, an increase in the money growth rate $\mu$ raises the inflation rate. The presence of a CIA constraint on consumption induces households to substitute leisure for consumption and to replace real money balances $m$ by capital $k$ . As a result, the ratio of consumption to capital $\chi$ decreases. Moreover, as agents increase leisure, they reduce the search effort. Hence, the size of unemployment $s$ decreases.

Two offsetting effects on firm’s hiring activities $v$ are at work. On the one hand, since there are fewer agents searching for jobs, the firms’ recruitment rate $q$ is reduced, which deters firms from hiring activities $ v$ . Conversely, as the consumption to capital ratio falls, the MRS between leisure and consumption decreases. Thus, the reservation wage decreases (c.f. (20)), which leads to a lower wage to capital ratio $\omega$ . A fall in the wage-capital ratio increases firms’ recruiting activities $v$ . The net effect depends on whether the former effect or the latter effect dominates. For the baseline calibration, the latter effect dominates, and thus, the posted vacancy increases in the BGP.

Moreover, two counteracting effects influence the size of employment $n$ . At the outset, as $n=\eta s/\theta$ , with a given job finding probability $ \eta$ , a decrease in the unemployment size $s$ will reduce the employment size $n$ . In contrast, as recruiting activities $v$ increase, the job finding probability $\eta$ is enhanced, which in turn raises the employment size. For the baseline calibration, the latter effect dominates the former effect. Thus, the size of employment increases in the BGP. As both employment and capital increase, the real interest rate (i.e. the marginal product of capital) is pushed up and the economic growth rate $g$ increases.

In our quantitative exercise, if the quarterly money growth rate is decreased from the baseline $1.39$ % to $0$ %, unemployment would increase by $0.29$ percentage points (from $0.0365$ to $0.0394$ ) and employment would decrease by $0.04$ percentage points (from $0.5925$ to $0.5921$ ), while the consumption to capital ratio would increase by $0.4$ percentage points (from $0.868$ to $0.872$ ). As a result, the wage to capital ratio would increase by $0.07$ percentage points (from $0.1219$ to $0.1226$ ) and vacancies posted by firms would decrease by $0.75$ percentage points (from $ 0.0996$ to $0.0921$ ). In the BGP, we find that as the money growth rate is decreased to $0$ %, the household’s welfare decreases, since the increase in utility from a higher consumption to capital ratio does not compensate for the decrease in utility from a higher labor force and thus lower leisure. By contrast, when the quarterly money growth rate is increased modestly from $ 0$ %, the increase in utility from more leisure serves as compensation for a lower consumption to capital ratio. We find that the welfare is increasing in the money growth rate, until the money growth rate reaches around $1.02$ %. Thus, the optimal money growth rate is positive. Using (c.f. (32)), the optimal money growth rate at $1.02$ % implies an optimal inflation rate of $0.5$ % per quarter, or equivalently, an inflation rate of $2$ % per year.

4.3. Dynamic welfare effects of inflation

We have analyzed the long-run effects of changes in the rate of money growth in models with and without sustainable growth. Changes in the rate of money growth also have transitional dynamic effects. In this subsection, we calculate the welfare effect of moderate rates of inflation, taking the transition dynamics into account.

Recall that, to generate the same quarterly inflation rate of $0.881$ % in the US data, the quarterly baseline rate of money growth is calibrated at $ \mu =0.881$ % for the model without sustainable growth, whereas it is calibrated at $\mu =1.39$ % in the model with sustainable growth. Now, we carry out the exercise of unexpected permanent changes in the rate of money growth from the baseline level to different levels. As the money growth rate changes, the equilibrium path will shift and gradually moves toward a new long-run equilibrium. Specifically, when there is a permanent change in the money growth rate from the benchmark rate to a new rate, the allocation will move from the baseline path { $c_{t}$ , $s_{t}$ , $n_{t}$ } $_{t=0}^{t=\infty }$ toward a new equilibrium path { $c_{t}^{\ast }$ , $s_{t}^{\ast }$ , $ n_{t}^{\ast }$ } $_{t=0}^{t=\infty }$ . The resulting welfare change in terms of the consumption equivalence $\kappa$ is calculated as follows.

(43) \begin{equation} \sum _{t=0}^{\infty }\beta ^{t}u((1+\kappa )c_{t},n_{t}+s_{t})=\sum _{t=0}^{ \infty }\beta ^{t}u(c_{t}^{\ast },n_{t}^{\ast }+s_{t}^{\ast }). \end{equation}

Thus, if $\kappa \gt 0$ , the representative household has a welfare gain in that the money growth rate increases consumption at the growth rate of $ \kappa$ . By contrast, if $\kappa \lt 0$ , there is a welfare loss because of a decrease in the growth rate of consumption. The dynamic welfare effects are illustrated in Figures 6 and7.

Figure 6. Welfare gains of changes in money growth rates in the model without sustainable growth.

Figure 6 is the welfare change in consumption equivalence in the model without sustainable growth. As seen from the figure, when the quarterly money growth rate decreases from $2.5$ %, the welfare gain measured in consumption equivalence monotonically increases. This pattern is also illustrated in Table 2 in terms of selective money growth rates that are smaller and larger than the baseline rate of $0.881$ %. The welfare is maximized when the money growth rate is $\mu =-0.5$ %. Our computation indicates that a reduction in the growth rate of the money supply from the baseline rate of $0.881$ % to the optimal rate of $-0.5$ % gives rise to a welfare gain that is equivalent to a consumption growth rate of $0.095$ % per quarter. Even for a reduction in the rate of money supply to zero, there is a welfare gain equivalent to a consumption growth rate of $0.064$ % per quarter. Thus, as in Heer (Reference Heer2003), the optimal monetary policy in the model without sustainable growth consists of a deflation consistent with a zero or near-zero nominal interest rate as advocated by Friedman (Reference Friedman and Friedman1969).

Table 2. Welfare effect of inflation in the model without sustainable growth (%)

Note: Welfare is measured in terms of the consumption equivalence $\kappa$ (per quarter).

Table 3. Welfare effect of inflation in the model with sustainable growth (%)

Note: Welfare is measured in terms of the consumption equivalence $\kappa$ (per quarter).

Figure 7. Welfare gains of changes in money growth rates in the model with sustainable growth.

Figure 7 is the welfare change in consumption equivalence in the model with sustainable growth. By contrast, it is obvious to see that the figure displays a pattern that is very different from Figure 6. The figure is not monotonic in the money growth rate, and a monotonic reduction in the money supply from the baseline rate of $1.39$ % does not necessarily increase the welfare. As seen from the figure, it exhibits an inverted U shape. When the growth rate of the money supply decreases from the baseline rate of $1.39$ %, the welfare first increases and then decreases. Such a non-monotonic pattern is also illuminated in Table 3 in terms of selective money growth rates that are smaller and larger than the baseline rate. Our quantitative exercises suggest that the welfare is maximized when the money growth rate is decreased to $\mu =1.02$ %. Such a decrease in the money supply growth rate from the baseline to the optimal one raises the welfare to the level equivalent to a consumption growth rate at $0.005$ % per quarter. Thus, even along the transitional dynamic path, the Friedman rule does not hold in our economy with sustainable growth.

4.4. Sensitivity analysis

This subsection carries out sensitivity analysis.

4.4.1. Changes of key parameter values

This sub-subsection confirms that our results of dynamic welfare analysis in the previous subsection are robust with regard to the choice of the parameter values. We will change four key parameter values: the inverse of the labor supply elasticity $\sigma$ , workers’ bargaining power $\varrho$ , vacancy cost parameter $e$ , and the CIA constraint parameter $h$ .

In our analysis, we perform an analysis of increasing and decreasing each of these four parameter values from the baseline. For each new set of parameter values, we have found a new BGP which is unique. Moreover, the equilibrium path toward the new BGP exhibits saddle-path stability. Using each set of parameter values as a new baseline, we then analyze numerical effects of unexpected permanent changes in the rate of money growth to different levels in each set of parameter values. The results are reported in Figure 8 and.

Figure 8. Welfare gains of changes in money growth rates in the model with sustainable growth when parameter values are changed from the baseline.

From Figure 8, it is clear that all charts are not monotonic in the money growth rate. Indeed, they all exhibit an inverted U shape. As a consequence, for all cases of these changes in parameter values, the optimal money growth rate is positive, departing from the Friedman rule.

Table 4. Robustness: Welfare effect of inflation in the model with sustainable growth (%)

Note: 1. ${\mu}^{\ast}$ is the optimal money growth rate, and $\kappa $ is the resulting welfare gain in consumption equivalence (per quarter). 2. Baseline parameter values are in Table 1.

Moreover, according to Table 4, when the labor supply elasticity $\frac{1}{ \sigma }$ increases from the baseline to $\frac{1}{2}$ or decreases to $ \frac{1}{2.5}$ , the optimal money growth rate $\mu ^{\ast }$ decreases to $ 0.38$ % per quarter or increases to $1.62$ % per quarter. When workers’ bargaining power $\varrho$ increases from the baseline to $0.55$ or decreases to $0.45$ , the optimal money growth rate $\mu ^{\ast }$ increases to $1.892$ % per quarter or decreases to $0.06$ % per quarter. In addition, when the fraction of household consumption paid by cash $h$ increases from the baseline to $1$ or decreases to $0.7$ , the optimal money growth rate $ \mu ^{\ast }$ decreases to $0.72$ % per quarter or increases to $1.386$ % per quarter. Finally, a change in the vacancy cost parameter $e$ does not change the optimal money growth rate. Table 4 also reports the corresponding welfare gains in consumption equivalence $\kappa$ when one of these parameter values changes from the baseline and the optimal money growth rate also changes from the baseline to the corresponding rate.

4.4.3. Endogenous growth model without externalities

Finally, can an endogenous growth model without capital externalities yield positive optimal money growth? To understand whether or not this is possible, we modify our production function to $y_{t}=Ak_{t}n_{t}^{1- \varepsilon }$ , so there is no capital externality. To calibrate the model, we maintain our baseline parameter values as those in Table 1, except for $ b=0.$ Note that, even with $b=0$ , to match the quarterly inflation rate, the calibrated money growth rate is still $\mu =0.0139$ and thus, the same as that in Table 1.

For the new set of parameter values, we have found the existence of a new BGP which is unique. Moreover, the equilibrium path toward the new BGP exhibits saddle-path stability. Using the new baseline, we analyze numerical effects of unexpected permanent changes in the rate of money growth to different levels. The quantitative results are illustrated in.

As is clear from Figure 9, the results are the same as those in Figure 5. In particular, the welfare is not monotonic but exhibits an inverted U shape in the money growth rate. See the plots in the bottom panel of Figure 9. Therefore, an endogenous growth model without capital externalities also yields a positive optimal money growth rate at $0.3$ % per quarter.

Figure 9. Effects of changes in money growth rates in an Ak model without externalities.

4.5. Is the model with sustainable growth closer to the reality?

Readers may be curious as to whether our model with sustainable growth is closer to the reality than Heer (Reference Heer2003)’s model without sustainable growth. To answer the question, this subsection compares these two models’ predictions with the observations in the US data. We compare the transition dynamics generated by our baseline endogenous growth model and the transition dynamics generated by Heer (Reference Heer2003)’s baseline exogenous growth model with the observations in the US data.

To compute the transition dynamics of these two models, first, we solve for the initial steady state and the final steady state, given the values of the quarterly money series $\{M_{t}\}_{t=1}^{T}$ .Footnote ¹⁹ Output corresponds to gross domestic product, and consumption is the personal consumption expenditures. The data are quarterly and have been seasonally adjusted and deflated by the GDP deflators. The data are obtained from the Federal Reserve Economic Data published by the Federal Reserve Bank of St. Louis.

We compare the model’s predictions with the observations in the US data over 1947Q1-2015Q4. Without loss of generality, we normalize $M_{1}=1$ and assume $t=1$ for 1947Q1, $t$ = $T$ for 2015Q4, and $t$ = 2, $\ldots$ , $T$ -1 for quarters in transitional periods. Since $T=276$ is sufficiently large, the transition dynamics between 1947Q1 and 2015Q4 in these models are not affected by small variations in $T$ . On the basis of the solutions for the initial and the final steady states in the model, we compute the transition dynamics using a non-linear solution method in accordance with Chen et al. (Reference Chen, İmrohoroğlu and İmrohoroğlu2006), He and Liu (Reference He and Liu2008), and Chen and Liao (Reference Chen and Liao2015), among others.

Figure 10 draws a distinction between the transition dynamics in Heer (Reference Heer2003)’s model without sustainable growth (red dashed line) and the actual US data (green solid line). Figure 11 contrasts the transition dynamics in our model with sustainable growth (blue dashed line) with the actual US data (green solid line).

Figure 10. Transition dynamics in the model without sustainable growth.

Figure 11. Transition dynamics in the model with sustainable growth.

Apparently, our endogenous growth model well matches the growth trend of output in the US data over 1947-2015. Yet, the transition dynamics in Figure 10 cannot catch the growth trend of output in the US data over 1947-2015. Therefore, the model with sustainable growth is closer to the data than the model without sustainable growth.