Might Tobin be Right?

Abstract

When search frictions are present on financial markets, money demand arises endogenously as not all savings will be seamlessly invested in productive capacity. In such a situation, monetary and fiscal policies affect households’ portfolio decisions through their impact on the value of real balances. With exogenous growth, money continues to be neutral. In case of endogenous growth, however, money-induced portfolio shifts allow for the existence of an optimal inflation rate. Similarly, portfolio shifts induced by governments’ fiscal interventions allow to enhance growth as public debt helps to overcome search externalities by providing additional assets on the financial market. Finally, when monetary and fiscal policy interact, growth is maximum in comparison to each of the two policies taken individually. The results of the paper suggest that balance-sheet recessions can be overcome by targeted monetary and fiscal interventions that provide additional assets and induce portfolio shifts whereby these additional assets are invested in productive capacity.

Appendices

Appendix 1: The Household Program

To solve the problem, the Hamiltonian is set up as follows:

$$\displaystyle\begin{array}{rcl} \mathcal{H}& =& \frac{\left (c \cdot N^{-\gamma }\right )^{1-\eta }} {1-\eta } +\nu \left [r_{W}W - c + \left (1 -\tau _{w}\right )wN\right ]. {}\\ \end{array}$$

The household’s first-order condition can then be determined as:

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{H}} {\partial c} & =& 0 \Leftrightarrow c^{-\eta }N^{-\gamma \left (1-\eta \right )} =\nu {}\end{array}$$

(16)

The co-state variable W evolves according to:

$$\displaystyle\begin{array}{rcl} \dot{\nu }_{1}& =& \rho \nu _{1} - \frac{\partial \mathcal{H}} {\partial W} =\rho \nu -\nu r_{W}{}\end{array}$$

(17)

Deriving (16) with respect to time, noting that $\dot{N} = 0$ in steady state and substituting $\dot{\nu }_{1}$ and ν ₁ in (17), the household’s optimal consumption path can be determined as:

$$\displaystyle\begin{array}{rcl} -\eta \dot{c}c^{-\eta -1}& =& c^{-\eta }\left (\rho -r_{ W}\right ) \\ \Leftrightarrow \hat{ c} \equiv \frac{\dot{c}} {c}& =& \frac{1} {\eta } \left (r_{W}-\rho \right ){}\end{array}$$

(18)

Appendix 2: Equilibrium Investment and Hiring

The firm maximises profits selecting vacancies, $\mathcal{V}\geq 0$, and issues of bonds, $\mathcal{B}\geq 0$, subject to costs of job vacancy creation, $\zeta \cdot w \cdot \mathcal{V}$, and finding a financial investor, $\kappa \cdot r_{K} \cdot \mathcal{B}$. Accordingly, its optimal program writes as:

$$\displaystyle{\max _{N,K,\mathcal{V},\mathcal{B}}\int _{0}^{\infty }\nu _{ 1}\left (t\right )\left (F\left (K,AN\right ) - wN - r_{K}K -\zeta \cdot w\mathcal{V}-\kappa \cdot r_{K}\mathcal{B}\right )e^{-\rho t}dt}$$

where ρ stands for the intertemporal preference rate and $\nu _{1}\left (t\right )$ for the household’s shadow variable related to its accumulation constraint,¹¹ which firms maximise against the constraints of employment and capital accumulation, Eqs. (5) and (6).

Capital is predetermined with respect to the wage bargaining process and firms will take the impact of their capital decision on wages into account. The Hamiltonian therefore writes as:

$$\displaystyle{\begin{array}{ll} \mathcal{H} =&\nu _{1}\left (t\right )\left [F\left (K,N\right ) - w\left (K\right )N - r_{K}K -\zeta \cdot w\mathcal{V}-\kappa \cdot r_{K}\mathcal{B}\right ] +\lambda \left [q\left (\theta \right )\mathcal{V}-\sigma N\right ] \\ & +\mu \left [p\left (\phi \right )\mathcal{B}-\delta K\right ] \end{array} }$$

leading to the first-order conditions:

$$\displaystyle\begin{array}{rcl} \frac{\partial \mathcal{H}} {\partial \mathcal{V}}& =& 0 \Leftrightarrow \nu _{1}\left (t\right )\zeta \cdot w =\lambda q\left (\theta \right ) \Leftrightarrow \lambda =\nu _{1}\left (t\right ) \frac{\zeta \cdot w} {q\left (\theta \right )} {}\\ \frac{\partial \mathcal{H}} {\partial \mathcal{B}}& =& 0 \Leftrightarrow \nu _{1}\left (t\right )\kappa \cdot r =\mu p\left (\phi \right ) \Leftrightarrow \mu =\nu _{1}\left (t\right )\frac{\kappa \cdot r_{K}} {p\left (\phi \right )} {}\\ \end{array}$$

and the evolution of the co-state variables:

$$\displaystyle\begin{array}{rcl} \dot{\lambda }& =& \rho \lambda -\frac{\partial \mathcal{H}} {\partial N} =\lambda \left (\rho +\sigma \right ) -\nu _{1}\left (t\right )\left (F_{N} - w\right ) {}\\ \dot{\mu }& =& \rho \mu -\frac{\partial \mathcal{H}} {\partial K} =\mu \left (\rho +\delta \right ) -\nu _{1}\left (t\right )\left (F_{K} - N \frac{\partial w^{{\ast}}} {\partial K} - r_{K}\right ) {}\\ \end{array}$$

Deriving the first-order conditions with respect to time, one obtains:

$$\displaystyle\begin{array}{rcl} \dot{\lambda }& =& \frac{\zeta \cdot \dot{w}} {q\left (\theta \right )} +\dot{\nu } _{1}\left (t\right ) \frac{\zeta \cdot w} {q\left (\theta \right )} {}\\ \dot{\mu }& =& \frac{\kappa \cdot \dot{r}_{K}} {p\left (\phi \right )} +\dot{\nu } _{1}\left (t\right )\frac{\kappa \cdot r_{K}} {p\left (\phi \right )} {}\\ \end{array}$$

and hence:

$$\displaystyle\begin{array}{rcl} \frac{\zeta \cdot \dot{w}} {q\left (\theta \right )}\nu _{1}\left (t\right ) +\dot{\nu } _{1}\left (t\right ) \frac{\zeta \cdot w} {q\left (\theta \right )}& =& \nu _{1}\left (t\right ) \frac{\zeta \cdot w} {q\left (\theta \right )}\left (\rho +\sigma \right ) -\nu _{1}\left (t\right )\left (F_{N} - w\right ) {}\\ \frac{\kappa \cdot \dot{r}_{K}} {p\left (\phi \right )} \nu _{1}\left (t\right ) +\dot{\nu } _{1}\left (t\right )\frac{\kappa \cdot r_{K}} {p\left (\phi \right )} & =& \nu _{1}\left (t\right )\frac{\kappa \cdot r_{K}} {p\left (\phi \right )} \left (\rho +\delta \right ) -\nu _{1}\left (t\right )\left (F_{K} - N \frac{\partial w^{{\ast}}} {\partial K} - r_{K}\right ) {}\\ \end{array}$$

which can be rewritten as:

$$\displaystyle\begin{array}{rcl} F_{N} - w + \frac{\zeta \cdot w} {q\left (\theta \right )}\left [\hat{w} + \frac{\dot{\nu }_{1}\left (t\right )} {\nu _{1}\left (t\right )} -\rho -\sigma \right ]& =& 0 {}\\ F_{K} - N \frac{\partial w^{{\ast}}} {\partial K} - r_{K} + \frac{\kappa \cdot r_{K}} {p\left (\phi \right )} \left [\hat{r_{K}} + \frac{\dot{\nu }_{1}\left (t\right )} {\nu _{1}\left (t\right )} -\rho -\delta \right ]& =& 0 {}\\ \end{array}$$

where $\hat{w} \equiv \frac{\dot{w}} {w}$ and $\hat{r_{K}} \equiv \frac{\dot{r_{K}}} {r_{K}}$.

Using the wage- and interest-rate curves, Eqs. (7) and (8), and noting that in the steady-state equilibrium $\hat{r}_{K} = 0$ and $\nu _{1}\left (t\right ) = c^{-\eta }\Rightarrow \frac{\dot{\nu }_{1}\left (t\right )} {\nu _{1}\left (t\right )} = -\eta \frac{\dot{c}} {c}$ will give the steady state conditions for labour and financial market liquidity (9) and (10) as given in the proposition.

Appendix 3: Changes in Growth Following Supply and Demand Shocks

Instead of assuming simultaneous negotiations of wages and interest rates, a more realistic assumption would be to consider wages to be set after the capital stock has been built. In this case, interest payment will take into account the effect of the capital stock development on wages, in addition to the marginal productivity of capital and the gain of matching issued debt:

$$\displaystyle{r_{K} =\gamma \left (F_{K} - N \frac{\partial w^{{\ast}}} {\partial K} +\phi \kappa _{0}\right ).}$$

Hence the equilibrium interest rate writes as:

$$\displaystyle\begin{array}{rcl} r_{K}& =& \gamma \left (F_{K} - w_{K}^{{\ast}}\left (\theta \right ) \cdot N +\phi \kappa \cdot r_{ K}\right ) \Leftrightarrow r_{K} = \frac{\gamma } {1-\gamma \phi \kappa }\left (F_{K} - w_{K}^{{\ast}}\left (\theta \right ) \cdot N\right ) {}\\ r_{K}& =& \frac{\gamma } {1-\gamma \phi \kappa }\left (\,\,f^{{\prime}}\left (k\right ) - w_{ K}^{{\ast}}\left (\theta \right ) \cdot N\right ) {}\\ \end{array}$$

When endogenous growth is introduced, the rate of return on capital writes as¹²:

$$\displaystyle\begin{array}{rcl} & & r_{K} =\gamma \left (F_{K} - w_{K}^{{\ast}}\left (\theta \right ) \cdot N +\phi \kappa _{ 0}\right ) {}\\ & \Leftrightarrow & r_{K} = \frac{\gamma } {1-\gamma \phi \kappa }\left (A\left (1-\alpha \right ) - w_{K}^{{\ast}}\left (\theta \right ) \cdot N\right ) {}\\ & \Leftrightarrow & r_{K} = \frac{A\gamma } {1-\gamma \phi \kappa }\left (1 -\alpha - \frac{\beta \alpha } {1 -\beta \theta \zeta -\left (1-\beta \right )R}\right ) {}\\ \end{array}$$

and we have:

$$\displaystyle{\frac{\partial r_{K}} {\partial \theta } < 0, \frac{\partial r_{K}} {\partial \phi } > 0}$$

The simultaneous equilibrium on labour and financial markets therefore writes as:

$$\displaystyle\begin{array}{rcl} \left (1-\beta \right )\left (1 - R\right ) -\beta \theta \zeta - \frac{\zeta \beta } {q\left (\theta \right )}\left (\sigma +\rho + \left (\eta -1\right )g\right )& =& 0 {}\\ 1 -\gamma \left (1+\phi \kappa \right ) - \frac{\kappa \gamma } {p\left (\phi \right )} \cdot \left (\delta +\rho +\eta g\right )& =& 0 {}\\ \end{array}$$

Using (3), the equilibrium condition for returns to investment, r _W  = r _B  = r _K and inputting the equilibrium value for r _K , we obtain:

$$\displaystyle\begin{array}{rcl} & & \left (1-\beta \right )\left (1 - R\right ) {}\\ & & \quad -\beta \theta \zeta - \frac{\zeta \beta } {q\left (\theta \right )}\left (\sigma +\frac{\rho } {\eta }+\left (\frac{\eta -1} {\eta } \right )\left [ \frac{A\gamma } {1-\gamma \phi \kappa }\left (1 -\alpha - \frac{\beta \alpha } {1 -\beta \theta \zeta -\left (1-\beta \right )R}\right )\right.\right. {}\\ & & \left.\left.\qquad \left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right ]\right ) = 0 {}\\ & & \quad 1 -\gamma \left (1+\phi \kappa \right ) - \frac{\kappa \gamma } {p\left (\phi \right )} \cdot \left (\delta + \frac{A\gamma } {1-\gamma \phi \kappa }\left (1 -\alpha - \frac{\beta \alpha } {1 -\beta \theta \zeta -\left (1-\beta \right )R}\right )\right. {}\\ & & \qquad \left.\left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right ) = 0 {}\\ \end{array}$$

and, hence, after rearranging terms:

$$\displaystyle\begin{array}{rcl} & & \left (\begin{array}{c} LL = 0\\ FF = 0 \end{array} \right ) {}\\ & & \Leftrightarrow \!\!\left (\begin{array}{c} \Theta \left (\theta,\beta,\zeta,R\right ) - \frac{A\gamma } {1-\gamma \phi \kappa }\left (1 -\alpha - \frac{\beta \alpha } {1-\beta \theta \zeta -\left (1-\beta \right )R}\right )\left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right ) = \frac{\eta \sigma +\rho } {\eta -1} -\pi u^{\,f} \\ \Phi \left (\phi,\gamma,\kappa \right ) - \frac{A\gamma } {1-\gamma \phi \kappa }\left (1 -\alpha - \frac{\beta \alpha } {1-\beta \theta \zeta -\left (1-\beta \right )R}\right )\left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right ) =\delta -\pi u^{\,f} \end{array} \right ){}\\ \end{array}$$

These equilibrium conditions are no longer characterised by a triangular structure. In order to determine the effect of inflation and multi-factor productivity on the labour and financial market liquidity, we will make use of Cramer’s rule. Writing $x \in \left \{A,\pi \right \}$ and fully differentiating LL and FF we obtain:

$$\displaystyle{\left (\begin{array}{ll} \frac{\partial LL} {\partial \theta } &\frac{\partial LL} {\partial \phi } \\ \frac{\partial FF} {\partial \theta } &\frac{\partial FF} {\partial \phi } \end{array} \right )\left (\begin{array}{l} d\theta \\ d\phi \end{array} \right ) = -\left (\begin{array}{l} \frac{\partial LL} {\partial x} \\ \frac{\partial FF} {\partial x} \end{array} \right )dx}$$

Applying Cramer’s rule allows to write:

$$\displaystyle{ \frac{d\theta } {dx} = -\frac{\left \vert \begin{array}{ll} \frac{\partial LL} {\partial x} &\frac{\partial LL} {\partial \phi } \\ \frac{\partial FF} {\partial x} &\frac{\partial FF} {\partial \phi } \end{array} \right \vert } {\left \vert \begin{array}{ll} \frac{\partial LL} {\partial \theta } &\frac{\partial LL} {\partial \phi } \\ \frac{\partial FF} {\partial \theta } &\frac{\partial FF} {\partial \phi } \end{array} \right \vert }, \frac{d\phi } {dx} = -\frac{\left \vert \begin{array}{ll} \frac{\partial LL} {\partial \theta } &\frac{\partial LL} {\partial x} \\ \frac{\partial FF} {\partial \theta } &\frac{\partial FF} {\partial x} \end{array} \right \vert } {\left \vert \begin{array}{ll} \frac{\partial LL} {\partial \theta } &\frac{\partial LL} {\partial \phi } \\ \frac{\partial FF} {\partial \theta } &\frac{\partial FF} {\partial \phi } \end{array} \right \vert }}$$

Independently of the value of η we have:

$$\displaystyle{\frac{\partial FF} {\partial \phi } < 0, \frac{\partial FF} {\partial \theta } > 0, \frac{\partial LL} {\partial \phi } < 0, \frac{\partial LL} {\partial A} = \frac{\partial FF} {\partial A} < 0, \frac{\partial LL} {\partial \pi } = \frac{\partial FF} {\partial \pi } > 0}$$

and

$$\displaystyle{\left \vert \frac{\partial LL} {\partial \theta } \right \vert > \left \vert \frac{\partial FF} {\partial \theta } \right \vert,\left \vert \frac{\partial LL} {\partial \phi } \right \vert < \left \vert \frac{\partial FF} {\partial \phi } \right \vert }$$

Furthermore:

$$\displaystyle{\frac{\partial LL} {\partial \phi } = \left \{\begin{array}{l} > 0\mbox{ for }\eta < 1\\ < 0\mbox{ for } \eta > 1 \end{array} \right.}$$

and

$$\displaystyle{\frac{\partial LL} {\partial \theta } = \left \{\begin{array}{l} < 0\mbox{ for }\eta < 1\\ < 0 \mbox{ for } \eta > 1\mbox{ at the stable root} \end{array} \right.}$$

Hence, the determinant is positive at the stable root for θ for η > 1:

$$\displaystyle{\left \vert \begin{array}{ll} \frac{\partial LL} {\partial \theta } &\frac{\partial LL} {\partial \phi } \\ \frac{\partial FF} {\partial \theta } &\frac{\partial FF} {\partial \phi } \end{array} \right \vert > 0}$$

Moreover, we have:

• For x = A: 

  $$\displaystyle\begin{array}{rcl} \left \vert \begin{array}{ll} \frac{\partial LL} {\partial A} &\frac{\partial LL} {\partial \phi } \\ \frac{\partial FF} {\partial A} &\frac{\partial FF} {\partial \phi }\end{array} \right \vert & =& \stackrel{(-)}{\frac{\partial LL} {\partial A} }\stackrel{(-)}{\frac{\partial FF} {\partial \phi } } -\stackrel{(-)}{\frac{\partial FF} {\partial A} }\stackrel{(-)}{\frac{\partial LL} {\partial \phi } } = \stackrel{(-)}{\frac{\partial LL} {\partial A} }\left [\stackrel{(-)}{\frac{\partial FF} {\partial \phi } } -\stackrel{(-)}{\frac{\partial LL} {\partial \phi } }\right ] > 0 {}\\ \left \vert \begin{array}{ll} \frac{\partial LL} {\partial \theta } &\frac{\partial LL} {\partial A} \\ \frac{\partial FF} {\partial \theta } &\frac{\partial FF} {\partial A}\end{array} \right \vert & =& \stackrel{(+)}{\frac{\partial LL} {\partial \theta } }\stackrel{(-)}{\frac{\partial FF} {\partial A} } -\stackrel{(+)}{\frac{\partial FF} {\partial \theta } }\stackrel{(-)}{\frac{\partial LL} {\partial A} } < 0 {}\\ \end{array}$$

• For x = π: 

  $$\displaystyle\begin{array}{rcl} \left \vert \begin{array}{ll} \frac{\partial LL} {\partial \pi } &\frac{\partial LL} {\partial \phi } \\ \frac{\partial FF} {\partial \pi } &\frac{\partial FF} {\partial \phi }\end{array} \right \vert & =& \stackrel{(+)}{\frac{\partial LL} {\partial \pi } }\stackrel{(-)}{\frac{\partial FF} {\partial \phi } } -\stackrel{(+)}{\frac{\partial FF} {\partial \pi } }\stackrel{(-)}{\frac{\partial LL} {\partial \phi } } < 0 {}\\ \left \vert \begin{array}{ll} \frac{\partial LL} {\partial \theta } &\frac{\partial LL} {\partial \pi } \\ \frac{\partial FF} {\partial \theta } &\frac{\partial FF} {\partial \pi }\end{array} \right \vert & =& \stackrel{(-)}{\frac{\partial LL} {\partial \theta } }\stackrel{(+)}{\frac{\partial FF} {\partial \pi } } -\stackrel{(+)}{\frac{\partial FF} {\partial \theta } }\stackrel{(+)}{\frac{\partial LL} {\partial \pi } } < 0{}\\ \end{array}$$

The above conclusions regarding the effect of monetary and fiscal policies therefore carry over to the Stackelberg case.

Appendix 4: Monetary Policy and the Optimal Inflation Rate

Proof

As seen from (13), inflation has a first-order negative impact on the growth rate: $\frac{\partial g} {\partial \pi } = -\frac{u^{\,f}} {\eta }$ which is independent from the inflation rate. However, inflation also affects financial market liquidity, as an increase in inflation leads to a portfolio shift by households away from savings towards consumption; totally differencing the financial market equilibrium condition (14) yields (note that in the following equations $u^{\,f} \equiv u^{\,f}\left (\phi ^{{\ast}}\right )$ and $r_{K} \equiv r_{K}\left (\phi ^{{\ast}}\right )$):

$$\displaystyle{\frac{d\phi ^{{\ast}}} {d\pi } \,=\, \frac{p\left (\phi ^{{\ast}}\right )u^{\,f}} {p\left (\phi ^{{\ast}}\right )\left (p\left (\phi ^{{\ast}}\right ) + \left (1 - u^{\,f}\right )r_{K}^{{\prime}}-\left (r_{K}+\pi \right )u^{\,f}{}^{{\prime}}\right ) - p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{K}\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right )}\,>\,0}$$

As shown before in Proposition 3.2, both the equilibrium interest rate and the growth rate increase with financial market tightness as less funds are left unused in the household’s deposits (u ^f decreases with ϕ). Hence, inflation has a positive second-order impact on the growth rate via its impact on the financial market liquidity.

Combining the impact of inflation on financial market liquidity and financial market liquidity on growth, the second-order positive effect writes as:

$$\displaystyle\begin{array}{rcl} \!\!& & \frac{\partial g} {\partial \phi } \cdot \frac{\partial \phi } {\partial \pi } {}\\ \!\!& & = \frac{p\left (\phi ^{{\ast}}\right )u^{\,f}\left (\left (1 - u^{\,f}\right )r_{K}^{{\prime}}- r_{K}u^{\,f}{}^{{\prime}}\right )} {\eta \left (p\left (\phi ^{{\ast}}\right )\left (p\left (\phi ^{{\ast}}\right ) + \left (1 - u^{\,f}\right )r_{K}^{{\prime}}-\left (r_{K}+\pi \right )u^{\,f}{}^{{\prime}}\right ) - p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{K}\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right )\right )} {}\\ \!\!& & = \frac{u^{\,f}\left (\phi \right )} {\eta } \frac{p\left (\phi ^{{\ast}}\right )u^{\,f}\left (\left (1 - u^{\,f}\right )r_{K}^{{\prime}}- r_{K}u^{\,f}{}^{{\prime}}\right )} {p\left (\phi ^{{\ast}}\right )\left (p\left (\phi ^{{\ast}}\right ) + \left (1 - u^{\,f}\right )r_{K}^{{\prime}}-\left (r_{K}+\pi \right )u^{\,f}{}^{{\prime}}\right )\! - p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{K}\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right )} {}\\ \end{array}$$

The growth-maximising inflation rate, therefore, can be determined as:

$$\displaystyle\begin{array}{rcl} & & \qquad \qquad \qquad \qquad \qquad \overline{\pi } \in \left \{\pi \left \vert \frac{\partial g} {\partial \phi } \cdot \frac{\partial \phi } {\partial \pi }\right. = \left \vert \frac{\partial g} {\partial \pi } \right \vert \right \} {}\\ & \Leftrightarrow & \frac{u^{\,f}} {\eta } \frac{p\left (\phi ^{{\ast}}\right )\left (\left (1 - u^{\,f}\right )r_{K}^{{\prime}}- r_{K}u^{\,f}{}^{{\prime}}\right )} {p\left (\phi ^{{\ast}}\right )\left (p\left (\phi ^{{\ast}}\right ) + \left (1 - u^{\,f}\right )r_{K}^{{\prime}}-\left (r_{K}+\pi \right )u^{\,f}{}^{{\prime}}\right ) - p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{K}\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right )}\,=\,\frac{u^{\,f}} {\eta } {}\\ & \Leftrightarrow & \frac{p\left (\phi ^{{\ast}}\right )\left (\left (1 - u^{\,f}\right )r_{K}^{{\prime}}- r_{K}u^{\,f}{}^{{\prime}}\right )} {p\left (\phi ^{{\ast}}\right )\left (p\left (\phi ^{{\ast}}\right ) + \left (1 - u^{\,f}\right )r_{K}^{{\prime}}-\left (r_{K}+\pi \right )u^{\,f}{}^{{\prime}}\right ) - p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{K}\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right )}\,=\,1 {}\\ & \Leftrightarrow & p\left (\phi ^{{\ast}}\right )^{2} = p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{ K}\left (1 - u^{\,f}\right ) -\pi u^{\,f}\right ) {}\\ & \Leftrightarrow & \overline{\pi } = \frac{p\left (\phi ^{{\ast}}\right ){}^{2} - p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{K}\left (1 - u^{\,f}\right )\right )} {-p^{{\prime}}\left (\phi ^{{\ast}}\right )u^{\,f}} \,>\,0. {}\\ \end{array}$$

The impact on employment can be determined in a straightforward way by totally differentiating (15). ■

Appendix 5: Optimal Debt and Growth

In the following we adopt the convention that $\phi \left (\tau _{K}\right ) =\phi +\tau$ for convenience only. The financial market equilibrium is unaffected by capital taxes (only indirectly through impact on savings and consumption decision). The negative direct impact of taxes on growth increases with the equilibrium value of ϕ:

$$\displaystyle{ \frac{\partial g} {\partial \tau _{K}} = -\frac{1} {\eta } r_{K}\left (1 - u^{\,f}\right ) \leq 0,\text{ } \frac{\partial ^{2}g} {\partial \tau _{K}\partial \phi } = -\frac{1} {\eta } \left [\frac{\partial r_{K}} {\partial \phi } \left (1 - u^{\,f}\right ) - r_{ K}\frac{\partial u^{\,f}} {\partial \phi } \right ] < 0}$$

This has to be weighted against the positive effect of government bonds on the financial market equilibrium, which increases the growth rate:

$$\displaystyle\begin{array}{rcl} g& =& \frac{\gamma \left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right )} {\eta } r_{K}\left (\phi \right ) -\frac{\rho +\pi u^{\,f}} {\eta } {}\\ \frac{\partial g} {\partial \phi } & =& \frac{1 -\tau _{K}} {\eta } \left [\left (1 - u^{\,f}\right )\frac{\partial r_{K}} {\partial \phi } - r_{K}\frac{\partial u^{\,f}} {\partial \phi } \right ] -\frac{\pi } {\eta }\frac{\partial u^{\,f}} {\partial \phi } {}\\ \frac{\partial ^{2}g} {\partial \phi ^{2}} & =& \frac{1 -\tau _{K}} {\eta } \left [-2\frac{\partial u^{\,f}} {\partial \phi } \frac{\partial r_{K}} {\partial \phi } - r_{K}\frac{\partial ^{2}u^{\,f}} {\partial \phi ^{2}} \right ] -\frac{\pi } {\eta }\frac{\partial ^{2}u^{\,f}} {\partial \phi ^{2}} {}\\ \end{array}$$

given that $\frac{\partial ^{2}r_{ K}} {\partial \phi ^{2}} = 0$. Under our functional assumption regarding the constant-elasticity-to-scale property of the matching process, $\frac{\partial ^{2}g} {\partial \phi ^{2}}$ will be negative except for very small values of ϕ < ϕ with ϕ > 0. Moreover, we have that at ϕ = 0:

$$\displaystyle{ \frac{\partial g} {\partial \tau _{K}} = 0\text{, }\frac{\partial g} {\partial \phi } \rightarrow \infty }$$

Hence, a zero-tax policy cannot be optimal at ϕ = 0. Given that $\left \vert \frac{\partial g} {\partial \tau _{K}}\right \vert$ is monotonically increasing with ϕ and $\frac{\partial g} {\partial \phi }$ is monotonically decreasing with ϕ, at least from ϕ > ϕ onwards. Hence there will be a $\overline{\phi } >\underline{\phi }> 0$ such that $\frac{\partial g} {\partial \phi } + \frac{\partial g} {\partial \tau _{K}} = 0$ for τ _K  > 0.

When only ϖ % of public spending is financed through corporate taxation, then the growth rate writes as:

$$\displaystyle{g = \frac{\gamma \left (1 -\varpi \cdot \tau \right )\left (1 - u^{\,f}\right )} {\eta } r_{K}\left (\phi \right ) -\frac{\rho +\pi u^{\,f}} {\eta } }$$

and consequently, the direct negative effect of a tax-financed increase in public debt is reduced:

$$\displaystyle{\frac{\partial g} {\partial \tau } = -\varpi \frac{1} {\eta } r_{K}\left (1 - u^{\,f}\right ) \leq 0\text{, }\frac{\partial ^{2}g} {\partial \tau \partial \varpi } = -\frac{1} {\eta } r_{K}\left (1 - u^{\,f}\right ) \leq 0}$$

hence $\frac{\partial g} {\partial \phi } + \frac{\partial g} {\partial \tau } \left (\varpi \right ) = 0$ is satisfied for a financial market equilibrium $\overline{\overline{\phi }}\left (\varpi \right ) > \overline{\phi } > 0$.

Appendix 6: Interaction Between Monetary and Fiscal Policy

Proof of Lemma

The inflation rate does not affect the direct negative impact of corporate taxation on growth. However, it has an impact on the way financial market liquidity affects growth. Recall that the growth rate increases with financial market liquidity at rate:

$$\displaystyle{\frac{\partial g} {\partial \phi } = \frac{1 -\tau _{K}} {\eta } \left [\left (1 - u^{\,f}\right )\frac{\partial r_{K}} {\partial \phi } - r_{K}\frac{\partial u_{f}} {\partial \phi } \right ] -\frac{\pi } {\eta }\frac{\partial u^{\,f}} {\partial \phi } }$$

This derivative depends positively on the inflation rate:

$$\displaystyle\begin{array}{rcl} \frac{\partial ^{2}g} {\partial \phi \partial \pi } & =& \frac{\partial ^{2}g} {\partial \phi ^{2}} \cdot \frac{\partial \phi } {\partial \pi } -\frac{1} {\eta } \frac{\partial u^{\,f}} {\partial \phi } {}\\ & =& \frac{\left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right )u^{\,f}\frac{\partial r_{K}} {\partial \phi } -\left (\left (1 -\tau _{K}\right )r_{K}-\rho \right )\frac{\partial u^{\,f}} {\partial \phi } } {\left (\rho -\left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right )r_{K} +\pi \cdot u^{\,f}\right )^{2}} > 0 {}\\ \end{array}$$

which is unambiguously positive for non-negative growth rates (for which $r_{K}\left (1 -\tau _{K}\right ) > r_{K}\left (1 -\tau _{K}\right )\left (1 - u^{\,f}\right ) >\rho$ holds). ■

Proof of Proposition

This first part follows directly from the lemma above. Moreover, note that

$$\displaystyle{\overline{\pi } = \frac{p\left (\phi ^{{\ast}}\right )^{2} - p^{{\prime}}\left (\phi ^{{\ast}}\right )\left (\delta +r_{K}\left (1 - u^{\,f}\right )\right )} {-p^{{\prime}}\left (\phi ^{{\ast}}\right )u^{\,f}} }$$

depends positively on ϕ ^∗ when the matching function is characterised by constant returns to scale. ■