Stabilization Policies and Business Cycle Dynamics

Abstract

Stabilization policies interfere with the dynamic workings of the economic system to which they are applied. As a result, they necessarily generate some dynamic repercussions in the process. This chapter analyzes the dynamic interplay between stabilization policies, capital accumulation, and business cycles. Capital accumulation is an integral part of business cycles. It is not just a component of demand but an addition to the economy’s productive capacity, and, as such, has a lasting influence on employment. As a result, by affecting capital accumulation, stabilization policies can modify the entire shape of business cycles. Assuming a feedback-type policy function, it is shown that, due to the crowding-out effect of fiscal expenditures, too intensive implementation of fiscal stabilization policies leads to instability of the dynamics, but that suitably coordinated monetary policy may be capable of recovering stability.

Earlier versions of the paper on which this chapter is based were presented at the workshops held at Keio University and Tokyo University. The author wishes to thank the participants of these workshops for helpful comments and suggestions.

Appendices

Appendix

In appendices dealing with long-run equilibria, all the variables and functions are evaluated at the relevant long-run equilibrium. For notational convenience, however, the asterisks used in the main text to symbolize long-run equilibria are omitted here.

A.1 Some Mathematical Expressions Appearing in the Base Model

3.2.1 A.1.1 Comparative Statics

Let

$$\displaystyle{ D_{0} = z^{{\prime}}\left (\ell_{ 2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right ) + s(1-\tau )\ell_{ 1}(f^{{\prime}})^{2} <0. }$$

(3.94)

Then the partial derivatives of the main endogenous variables with respect to the state variable w are as follows.

$$\displaystyle{\begin{array}{l} \dfrac{dx} {dw} = -\dfrac{\ell z^{{\prime}}f^{{\prime}}} {wD_{0}}> 0, \\ \dfrac{di} {dw} = -\dfrac{s(1-\tau )(f^{{\prime}})^{2}\ell} {wD_{0}}> 0, \\ \dfrac{dp} {dw} = \dfrac{f^{{\prime}}\left (s\left (1-\tau \right )\ell_{1} + z^{{\prime}}\ell_{2}\right )} {D_{0}}> 0. \end{array} }$$

3.2.2 A.1.2 Speed of Wage Adjustment

The upper bound of the speed of wage adjustment for the occurence of cycles is given by

$$\displaystyle{ \bar{\sigma }= \frac{4s\left (1-\tau \right )xD_{0}} {kz^{{\prime}}\ell}. }$$

A.2 Some Mathematical Expressions Appearing in the Balanced-Budget Model

3.3.1 A.2.1 Comparative Statics

Let

$$\displaystyle{ D_{1} = D_{0} + sf^{{\prime}}f\ell_{ 1}kg^{{\prime}} <0, }$$

(3.95)

where D ₀ is defined by (3.94). Then the comparative static partial derivatives in the balanced-budget model are as follows.

$$\displaystyle\begin{array}{rcl} \frac{\partial x} {\partial k}& =& -\frac{sf^{{\prime}}f\ell_{1}g^{{\prime}}} {D_{1}} <0, {}\\ \frac{\partial i} {\partial k}& =& \frac{sxf\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )g^{{\prime}}} {D_{1}} <0, {}\\ \frac{\partial x} {\partial w}& =& -\frac{z^{{\prime}}f^{{\prime}}\ell} {wD_{1}} <0, {}\\ \frac{\partial i} {\partial w}& =& -\frac{f^{{\prime}}\ell\left (s(1-\tau \right )f^{{\prime}} + skfg^{{\prime}})} {wD_{1}}> 0, {}\\ \frac{\partial e} {\partial k}& =& \frac{xD_{0}} {D_{1}}> 0. {}\\ \end{array}$$

3.3.2 A.2.2 Linear Dynamic System

With D ₀ and D ₁ as defined by (3.94) and (3.95), the coefficients ζ _j ⁱ of the linear dynamic system (3.31) are as follows.

$$\displaystyle{\begin{array}{l} \zeta _{1}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial k} = \dfrac{skz^{{\prime}}xf\left (\ell_{2}\left (f^{{\prime}}\right )^{2} -\ell f^{{\prime\prime}}\right )g^{{\prime}}} {D_{1}}> 0, \\ \zeta _{2}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial w} = -\dfrac{kz^{{\prime}}f^{{\prime}}\ell\left (s\left (1-\tau \right )f^{{\prime}} + skfg^{{\prime}}\right )} {wD_{1}} <0, \\ \zeta _{1}^{2} =\sigma w \times \dfrac{\partial e} {\partial k} = \dfrac{\sigma wxD_{0}} {D_{1}}> 0, \\ \zeta _{2}^{2} =\sigma w \times \dfrac{\partial e} {\partial w} = -\dfrac{\sigma kz^{{\prime}}f^{{\prime}}\ell} {D_{1}} <0. \end{array} }$$

3.3.3 A.2.3 Characteristic Polynomial

With D ₁ as defined by (3.95), the coefficients α _j of the characteristic polynomial associated with the linear dynamic system (3.31) are as follows.

$$\displaystyle{\begin{array}{l} \alpha _{1} = \dfrac{kz^{{\prime}}\left (-\sigma f^{{\prime}}\ell + sxf\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )g^{{\prime}}\right )} {D_{1}} \lesseqgtr 0, \\ \alpha _{2} = \dfrac{\sigma s\left (1-\tau \right )(f^{{\prime}})^{2}kz^{{\prime}}x\ell} {D_{1}}> 0. \end{array} }$$

3.3.4 A.2.4 Limit Policy Intensity

The limit policy intensity for the balanced-budget fiscal policy is given by

$$\displaystyle{ \gamma _{1} = \dfrac{\sigma f^{{\prime}}\ell} {sxf\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )}. }$$

(3.96)

A.3 Some Mathematical Expressions Appearing in the Partial Debt Finance Model

3.4.1 A.3.1 Comparative Statics

Let

$$\displaystyle{ D_{2} = D_{0} + kff^{{\prime}}\ell_{ 1}g^{{\prime}}, }$$

(3.97)

where D ₀ is defined by (3.94). Then the comparative static derivatives appearing in the partial debt finance model are as follows.

$$\displaystyle{ \begin{array}{l} \dfrac{\partial x} {\partial k} = -\dfrac{ff^{{\prime}}\ell_{1}g^{{\prime}}} {D_{2}} <0, \\ \dfrac{\partial x} {\partial w} = -\dfrac{z^{{\prime}}f^{{\prime}}\ell} {wD_{2}} <0, \\ \dfrac{\partial x} {\partial b} = -\dfrac{z^{{\prime}}f^{{\prime}}\left (\ell-\ell_{2}f\right )} {kD_{2}} <0 \\ \dfrac{\partial i} {\partial k} = \dfrac{fx\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )g^{{\prime}}} {D_{2}} <0, \\ \dfrac{\partial i} {\partial w} = -\dfrac{f^{{\prime}}\ell\left (s(1-\tau )f^{{\prime}} + kfg^{{\prime}}\right )} {wD_{2}}> 0, \\ \dfrac{\partial i} {\partial b} = -\dfrac{f^{{\prime}}\left (\ell-\ell_{2}f\right )\left (s\left (1-\tau \right )f^{{\prime}} + kfg^{{\prime}}\right )} {kD_{2}}> 0, \\ \dfrac{\partial e} {\partial k} = \dfrac{xD_{0}} {D_{2}}> 0. \end{array} }$$

3.4.2 A.3.2 Linear Dynamic System

Let D ₀ and D ₂ be as defined by (3.94) and (3.97). Then the coefficients ξ _j ⁱ of the linear dynamic system (3.41) are as follows.

$$\displaystyle{ \begin{array}{l} \xi _{1}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial k} = \dfrac{kz^{{\prime}}fx\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )g^{{\prime}}} {D_{2}}> 0, \\ \xi _{2}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial w} = -\dfrac{kz^{{\prime}}f^{{\prime}}\ell\left (s(1-\tau )f^{{\prime}} + kfg^{{\prime}}\right )} {wD_{2}} <0, \\ \xi _{3}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial b} = -\dfrac{z^{{\prime}}f^{{\prime}}\left (\ell-\ell_{2}f\right )\left (s\left (1-\tau \right )f^{{\prime}} + kfg^{{\prime}}\right )} {D_{2}} <0, \\ \xi _{1}^{2} =\sigma w \times \dfrac{\partial e} {\partial k} = \dfrac{\sigma wxD_{0}} {D_{2}}> 0, \\ \xi _{2}^{2} =\sigma w \times \dfrac{\partial e} {\partial w} = -\dfrac{\sigma kz^{{\prime}}f^{{\prime}}\ell} {D_{2}} <0, \\ \xi _{3}^{2} =\sigma w \times \dfrac{\partial e} {\partial b} = -\dfrac{\sigma wz^{{\prime}}f^{{\prime}}\left (\ell-\ell_{2}y\right )} {D_{2}} <0, \\ \xi _{1}^{3} = -fkg^{{\prime}}\times \dfrac{\partial e} {\partial k} = -\dfrac{fkxD_{0}g^{{\prime}}} {D_{2}} <0, \\ \xi _{2}^{3} = -fkg^{{\prime}}\times \dfrac{\partial e} {\partial w} = \dfrac{fk^{2}z^{{\prime}}f^{{\prime}}\ell g^{{\prime}}} {wD_{2}}> 0, \\ \xi _{3}^{3} = -fkg^{{\prime}}\times \dfrac{\partial e} {\partial b} - n = \dfrac{fkz^{{\prime}}f^{{\prime}}\left (\ell-\ell_{2}f\right )g^{{\prime}}} {D_{2}} - n \lesseqgtr 0. \end{array} }$$

3.4.3 A.3.3 Characteristic Polynomial

Let D ₀ be as defined by (3.94). Then the α _ij and β appearing in (3.44) are as follows.

$$\displaystyle{\begin{array}{l} \alpha _{11} = -fk\left [z^{{\prime}}\left (x\left (\ell_{2}\left (f^{{\prime}}\right )^{2} -\ell f^{{\prime\prime}}\right ) + f^{{\prime}}\left (\ell-\ell_{2}f\right )\right ) - n\ell_{1}f^{{\prime}}\right ] \lesseqgtr 0, \\ \alpha _{12} = -\sigma f^{{\prime}}kz^{{\prime}}\ell- nD_{0}> 0, \\ \alpha _{21} = -fkxz^{{\prime}}\left [s(1-\tau )(f^{{\prime}})^{2}\left (\ell-\ell_{2}f\right ) + n\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )\right ]> 0, \\ \alpha _{22} =\sigma f^{{\prime}}kz^{{\prime}}\ell\left (n + s(1 -\tau xf^{{\prime}}\right ) <0, \\ \alpha _{32} = -\sigma xnkz^{{\prime}}\ell s\left (1-\tau \right )\left (f^{{\prime}}\right )^{2}> 0, \\ \beta = kff^{{\prime}}\ell_{1} <0. \end{array} }$$

A.4 An Elementary Proof of the Routh-Hurwitz Stability Conditions for a Three-Variable System

Routh-Hurwitz stability conditions: Let $\varphi \left (\lambda \right ) = -\lambda ^{3} +\alpha _{1}\lambda ^{2} -\alpha _{2}\lambda +\alpha _{3}$  be the characteristic polynomial. Then, in order for the real parts of all the charactersitic roots to be negative, it is necessary and sufficient that

$$\displaystyle{ \alpha _{1} <0,\alpha _{3} <0,\text{ and }\alpha _{1}\alpha _{2} -\alpha _{3} <0. }$$

(3.98)

Proof

Let λ _i ,  i = 1, 2, 3, be the characteristic roots. They are related to the α _j by the identities (1.30) reproduced below:

$$\displaystyle\begin{array}{rcl} \alpha _{1}& =& \lambda _{1} +\lambda _{2} +\lambda _{3}, {}\\ \alpha _{2}& =& \lambda _{2}\lambda _{3} +\lambda _{1}\lambda _{3} +\lambda _{2}\lambda _{3}, {}\\ \alpha _{3}& =& \lambda _{1}\lambda _{2}\lambda _{3}. {}\\ \end{array}$$

(1.30) also yields

$$\displaystyle{ \alpha _{1}\alpha _{2} -\alpha _{3} = \left (\lambda _{1} +\lambda _{2}\right )\left (\lambda _{1}\lambda _{2} +\lambda _{3}\left (\lambda _{1} +\lambda _{2} +\lambda _{3}\right )\right ). }$$

(3.99)

• I] Necessity. If all the λ _i are real and negative, then (3.98) holds due to (1.30) and (3.99). Suppose the characteristic roots contain complex numbers, assume without loss of generality that λ ₁ = a + bi, λ ₂ = a − bi, λ ₃ = c, and assume that a < 0, c < 0, and b ≠ 0. Then α ₁ = 2a + c < 0,  α ₃ = c(a ² + b ²) < 0, and $\alpha _{1}\alpha _{2} -\alpha _{3} = 2a\left (a^{2} + b^{2} + c\left (2a + c\right )\right ) <0.$

• II] Sufficiency. Suppose all the inequalities in (3.98) hold. A contradiction will be derived from the assumption that some of the characteristic roots have non-negative real parts.

  First suppose that all the λ _i are real. If $\lambda _{i} \geq 0\ \forall i,$ then α ₁ ≥ 0. If two of the characteristic roots are negative but one is nonnegative, then α ₃ ≥ 0. If one of the characteristic roots, λ ₃ say, is negative but the other two are nonnegative, then λ ₁ +λ ₂ ≥ 0 and λ ₁ λ ₂ ≥ 0. Moreover, since λ ₁ +λ ₂ +λ ₃ = α ₁ < 0 by the first inequality in (3.98), we also have $\lambda _{3}\left (\lambda _{1} +\lambda _{2} +\lambda _{3}\right )> 0.$ Therefore it follows from (3.99) that α ₁ α ₂ −α ₃ ≥ 0, which contradicts the supposition that all the inequalities in (3.98) hold. Next suppose that the characteristic roots contain complex numbers and assume, without loss of generality, that λ ₁ = a + bi, λ ₂ = a − bi, and λ ₃ = c, where a, b, and c are real. If c ≥ 0, then α ₃ = λ ₁ λ ₂ λ ₃ ≥ 0. If c < 0 and a ≥ 0, then, since λ ₁ +λ ₂ +λ ₃ = α ₁ < 0  by assumption, it follows from (3.99) that $\alpha _{1}\alpha _{2} -\alpha _{3} = 2a\left (c\alpha _{1} + a^{2} + b^{2}\right ) \geq 0$, which is a contradiction.

A.5 Proof that the Inequality γ ₂ < s γ ₁ Holds in the Partial Debt Finance Model

Write $\zeta _{j}^{i}\left (\gamma \right )$ and $\xi _{j}^{i}\left (\gamma \right )$ to make explicit the dependence of ζ _j ⁱ and ξ _j ⁱ on $\gamma = g^{{\prime}}\left (0\right ),$ where $\zeta _{j}^{i}\left (\gamma \right )$ and $\xi _{j}^{i}\left (\gamma \right )$ are the coefficients of the linearized systems in the balanced-budget and partially debt-financed models respectively and are specified in Appendices A.2 and A.3. Then, since D ₂(s γ) = D ₁(γ) by (3.95) and (3.97), and since the long-run equilibria of the balanced-budget and partial debt finance models coincide, we have

$$\displaystyle{ \xi _{j}^{i}\left (s\gamma \right ) =\zeta _{ j}^{i}\left (\gamma \right ),i,j = 1,2. }$$

(3.100)

Let $\bar{\gamma }= s\gamma _{1}$ for brevity, where γ ₁ is the limit policy intensity in the balanced budget model and is defined by $\zeta _{1}^{1}\left (\gamma _{1}\right ) +\zeta _{ 2}^{2}\left (\gamma _{1}\right ) = 0$. Then we have $\xi _{1}^{1}(\bar{\gamma }) +\xi _{ 2}^{2}(\bar{\gamma }) =\zeta _{ 1}^{1}\left (\gamma _{1}\right ) +\zeta _{ 2}^{2}\left (\gamma _{1}\right ) = 0$ by (3.100) and therefore, by the identities relating the α _i and the ξ _j ⁱ,

$$\displaystyle{ \alpha _{1}(\bar{\gamma }) =\xi _{ 3}^{3}(\bar{\gamma }). }$$

(3.101)

Furthermore, by partly using (3.96) for γ ₁ and using the formulae for the α _ij provided in Appendix A.3, we obtain

$$\displaystyle{ \alpha _{2}\left (\bar{\gamma }\right ) = -\left (D_{2}\left (\bar{\gamma }\right )\right )^{-1}kz^{{\prime}}xs\left (1-\tau \right )\left (f^{{\prime}}\right )^{2}\left \{f\left (\ell-\ell_{ 2}f\right )\bar{\gamma }-\sigma \ell\right \}, }$$

(3.102)

$$\displaystyle{ \alpha _{3}\left (\bar{\gamma }\right ) = -\left (D_{2}\left (\bar{\gamma }\right )\right )^{-1}\sigma xnkz^{{\prime}}\ell s\left (1-\tau \right )(f^{{\prime}})^{2}. }$$

(3.103)

Therefore, by (3.101), (3.102), and (3.103),

$$\displaystyle\begin{array}{rcl} A\left (\bar{\gamma }\right )& \equiv & \alpha _{1}\left (\bar{\gamma }\right )\alpha _{2}\left (\bar{\gamma }\right ) -\alpha _{3}\left (\bar{\gamma }\right ) \\ & =& -\left (D_{2}\left (\bar{\gamma }\right )\right )^{-1}kz^{{\prime}}xs\left (1-\tau \right )\left (f^{{\prime}}\right )^{2} \\ & & \times \left \{\xi _{3}^{3}\left (\bar{\gamma }\right )\left (\ell-\ell_{ 2}f\right ) -\sigma \ell\left (\xi _{3}^{3}\left (\bar{\gamma }\right ) + n\right )\right \}.{}\end{array}$$

(3.104)

If $\xi _{3}^{3}\left (\bar{\gamma }\right )> 0,$ then $\alpha _{1}(\bar{\gamma })> 0$ by (3.101). If $\xi _{3}^{3}\left (\bar{\gamma }\right ) \leq 0,$ then $A\left (\bar{\gamma }\right )> 0$ by (3.5) and (3.10) because

$$\displaystyle{ \xi _{3}^{3}\left (\bar{\gamma }\right ) + n = -fk\bar{\gamma } \times \frac{\partial e} {\partial b}> 0 }$$

by (3.39). Thus the long-run equilibrium is unstable in either case if γ = s γ ₁. Combining this with the discussion of γ ₂ given in the main text, we obtain γ ₂ < s γ ₁. 

A.6 Some Mathematical Expressions Appearing in the Full Debt Finance Model

3.7.1 A.6.1 Comparative Statics

Let D ₀, D ₂, and $\Pi$ be defined by (3.94), (3.97), and (3.50). Then the comparative static derivatives in the full debt finance model are as follows.

$$\displaystyle\begin{array}{rcl} \frac{\partial x} {\partial k}& =& -\frac{xff^{{\prime}}\ell_{1}g^{{\prime}}} {D_{2}} <0, {}\\ \frac{\partial x} {\partial w}& =& -\frac{z^{{\prime}}f^{{\prime}}\ell} {wD_{2}} <0, {}\\ \frac{\partial x} {\partial b}& =& -\frac{f^{{\prime}}\ell_{1}z^{{\prime}}\Pi } {kD_{2}} \lesseqgtr 0, {}\\ \frac{\partial i} {\partial k}& =& \frac{xf\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )g^{{\prime}}} {D_{2}} <0, {}\\ \frac{\partial i} {\partial w}& =& -\frac{f^{{\prime}}\ell\left (s(1-\tau )f^{{\prime}} + kfg^{{\prime}}\right )} {wD_{2}}> 0, {}\\ \frac{\partial i} {\partial b}& =& -\frac{f^{{\prime}}\left (\ell-\ell_{2}f\right )\left (s\left (1-\tau \right )f^{{\prime}} + kfg^{{\prime}}\right ) + ci\left (\ell_{2}\left (f^{{\prime}}\right )^{2} -\ell f^{{\prime\prime}}\right )} {kD_{2}}> 0, {}\\ \frac{\partial e} {\partial k}& =& \frac{xD_{0}} {D_{2}}> 0. {}\\ \end{array}$$

3.7.2 A.6.2 Linear Dynamic System

Let ξ _j ⁱ be as in Appendix A.3. Then, with D ₂ and $\Pi$ as defined by (3.97) and (3.50), the coefficients η _j ⁱ of the linear dynamic system (3.53) are as follows.

$$\displaystyle{\begin{array}{l} \eta _{j}^{i} =\xi _{ j}^{i}\text{ for }i = 1,2,3\text{ and }j = 1,2, \\ \eta _{3}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial b} \\ \ \ \; = -\dfrac{z^{{\prime}}\left [f^{{\prime}}\left (\ell-\ell_{2}f\right )\left (s\left (1-\tau \right )f^{{\prime}} + fkg^{{\prime}}\right ) + ci\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )\right ]} {D_{2}} <0, \\ \eta _{3}^{2} =\sigma w \times \dfrac{\partial e} {\partial b} = -\dfrac{\sigma wf^{{\prime}}\ell_{1}z^{{\prime}}\Pi } {D_{2}} \lesseqgtr 0, \\ \eta _{3}^{3} = -fkg^{{\prime}}\times \dfrac{\partial e} {\partial b} + \left (i - n\right ) = \dfrac{f^{{\prime}}fk\ell_{1}z^{{\prime}}\Pi g^{{\prime}}} {D_{2}} + \left (i - n\right ) \lesseqgtr 0. \end{array} }$$

3.7.3 A.6.3 Characteristic Polynomial

Let $\Pi$ and D ₀ be defined by (3.50) and (3.94). Then the α _ij that appear in (3.54) are as follows.

$$\displaystyle{\begin{array}{l} \alpha _{11} = fk\left (xz^{{\prime}}\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right ) + f^{{\prime}}\ell_{1}z^{{\prime}}\Pi -\left (n - i\right )f^{{\prime}}\ell_{1}\right ) \lesseqgtr 0, \\ \alpha _{12} = -\sigma kf^{{\prime}}z^{{\prime}}\ell-\left (n - i\right )D_{0}> 0, \\ \alpha _{21} = -fkxz^{{\prime}}\left (s\left (1-\tau \right )(f^{{\prime}})^{2}\left (\ell-\ell_{2}f\right ) + \left (n - si\right )\left (\ell_{2}\left (f^{{\prime}}\right )^{2} -\ell f^{{\prime\prime}}\right )\right )> 0, \\ \alpha _{22} =\sigma kz^{{\prime}}\ell\left (\left (n - i\right ) + s\left (1-\tau \right )xf^{{\prime}}\right ) <0, \\ \alpha _{32} = -\sigma x(n - i)kz^{{\prime}}\ell s(1-\tau )(f^{{\prime}})^{2}> 0. \end{array} }$$

A.7 Proof that γ ₃ < s γ ₁ in the Full Debt Finance Model if the Crowding-Out Effect of Debt Accumulation Dominates so that $\dfrac{\partial e} {\partial b} <0$

Let $\bar{\gamma }= s\gamma _{1}$ as in Appendix A.5. Then

$$\displaystyle{ \eta _{1}^{1}\left (\bar{\gamma }\right ) +\eta _{ 2}^{2}\left (\bar{\gamma }\right ) =\xi _{ 1}^{1}\left (\bar{\gamma }\right ) +\xi _{ 2}^{2}\left (\bar{\gamma }\right ) =\zeta _{ 1}^{1}\left (\gamma _{ 1}\right ) +\zeta _{ 2}^{2}\left (\gamma _{ 1}\right ) = 0 }$$

by the relations between η _j ⁱ, ξ _j ⁱ, and ζ _j ⁱ provided in Appendix A.3, (3.100), and (3.96). Therefore

$$\displaystyle{ \alpha _{1}\left (\bar{\gamma }\right ) =\eta _{ 3}^{3}\left (\bar{\gamma }\right ). }$$

(3.105)

Partly using (3.96) for γ ₁ and using the formulae provided in Appendix A.6, we also obtain

$$\displaystyle{ \alpha _{2}\left (\bar{\gamma }\right ) = -\left (D_{2}\left (\bar{\gamma }\right )\right )^{-1}kz^{{\prime}}f^{{\prime}}\left [xs(1-\tau )f^{{\prime}}\left \{\bar{\gamma }f\left (\ell-\ell_{ 2}f\right )-\sigma \ell\right \} +\sigma \ell ci\right ] }$$

(3.106)

and

$$\displaystyle{ \alpha _{3}\left (\bar{\gamma }\right ) = -\left (D_{2}\left (\bar{\gamma }\right )\right )^{-1}\sigma x\left (n - i\right )kz^{{\prime}}\ell s(1-\tau )(f^{{\prime}})^{2}. }$$

(3.107)

Thus by (3.105), (3.106) and (3.107),

$$\displaystyle\begin{array}{rcl} A\left (\bar{\gamma }\right )& \equiv & \alpha _{1}\left (\bar{\gamma }\right )\alpha _{2}\left (\bar{\gamma }\right ) -\alpha _{3}\left (\bar{\gamma }\right ) {}\\ & =& -\left (D_{2}\left (\bar{\gamma }\right )\right )^{-1}kz^{{\prime}}f^{{\prime}}\left [\eta _{ 3}^{3}\left (\bar{\gamma }\right )\left \{s(1-\tau )f^{{\prime}}\left (\ell-\ell_{ 2}f\right )\bar{\gamma }xf +\sigma \ell ci\right \}\right. {}\\ & & \left.-\sigma \ell xs\left (1-\tau \right )f^{{\prime}}\left \{\eta _{ 3}^{3}\left (\bar{\gamma }\right ) + n - i\right \}\right ], {}\\ \end{array}$$

where

$$\displaystyle{ \eta _{3}^{3}\left (\bar{\gamma }\right ) + n - i = -\bar{\gamma }fk \times \frac{\partial e} {\partial b}> 0 }$$

(3.108)

by the assumption that ∂ e∕∂ b < 0. If $\eta _{3}^{3}\left (\bar{\gamma }\right )> 0$ then $\alpha _{1}\left (\bar{\gamma }\right )> 0$ by (3.105). If $\eta _{3}^{3}\left (\bar{\gamma }\right ) \leq 0,$ then $A\left (\bar{\gamma }\right )> 0$ by (3.108). Therefore the long-run equilibrium is unstable at $\gamma =\bar{\gamma }$ in either case and thus we have $\gamma _{3} <\bar{\gamma }\equiv s\gamma _{1}.$

A.8 Some Mathematical Expressions Appearing in the Crowding-In Model

In order to show that a long-run equilibrium with b > 0 is unstable, we first report the general expression for α ₃ in A.8.1. The expressions in A.8.2, A.8.3, and A.8.4 below, presenting comparative statics, linear dynamaic system, and characteristic polynomial respectively, assume that b = 0. 

3.9.1 A.8.1 General Expression for α ₃

Let

$$\displaystyle{\begin{array}{l} D_{3} = (cb + kz_{1})\left (\ell_{2}k(f^{{\prime}})^{2} -\ell\left (k + b\right )f^{{\prime\prime}}\right ) \\ + k(k + b)f^{{\prime}}\ell_{1}\left (s(1-\tau )f^{{\prime}}- z_{2}r^{{\prime}} + kfg^{{\prime}}\right ) <0. \end{array} }$$

Then

$$\displaystyle{\begin{array}{l} \phantom{0.}\alpha _{3} = \left (D_{3}\right )^{-1}\left (a_{31}\left (i - n\right ) + a_{32}b\right ), \\ \ \ a_{31} =\sigma k^{2}\left (k + b\right )z_{1}\ell sx\left (1-\tau \right )\left (f^{{\prime}}\right )^{2} <0, \\ \ \ a_{32} = -\sigma k\left (k + b\right )z_{2}f^{{\prime}}\ell r^{{\prime}}x\left (a + ci\right ) <0. \end{array} }$$

3.9.2 A.8.2 Comparative Statics

Let $\Pi$ be defined by (3.75), and let

$$\displaystyle{ D_{40} = z_{1}\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right ) + f^{{\prime}}\ell_{ 1}\left (s(1-\tau \right )f^{{\prime}}- z_{ 2}r^{{\prime}}) <0, }$$

(3.109)

$$\displaystyle{ D_{4} = D_{40} +\ell _{1}kff^{{\prime}}g^{{\prime}} <0. }$$

(3.110)

Then we have

$$\displaystyle\begin{array}{rcl} \frac{\partial x} {\partial k}& =& -\frac{\ell_{1}xff^{{\prime}}g^{{\prime}}} {D_{4}} <0, {}\\ \frac{\partial i} {\partial k}& =& \frac{xf\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )g^{{\prime}}} {D_{4}} <0, {}\\ \frac{\partial x} {\partial w}& =& -\frac{z_{1}f^{{\prime}}\ell} {wD_{4}} <0, {}\\ \frac{\partial i} {\partial w}& =& -\frac{f^{{\prime}}\ell\left (s(1-\tau \right )f^{{\prime}}- z_{2}r^{{\prime}} + kfg^{{\prime}})} {wD_{4}}> 0, {}\\ \frac{\partial x} {\partial b}& =& -\frac{f^{{\prime}}\left (z_{1}\left (\ell-\ell_{2}f\right ) -\ell_{1}(a + ci)\right )} {D_{4}} \lesseqgtr 0\text{ as }\Pi \lesseqgtr 0, {}\\ \frac{\partial i} {\partial b}& =& - \frac{1} {D_{4}}\! \times \!\left [\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right )(a + ci) + f^{{\prime}}\left (\ell-\ell_{ 2}f\right )\left (s(1\,-\,\tau )f^{{\prime}}\!-\! z_{ 2}r^{{\prime}}\,+\,kfg^{{\prime}}\right )\right ]\!>\! 0, {}\\ \frac{\partial e} {\partial k}& =& \frac{xD_{40}} {D_{4}}> 0. {}\\ \end{array}$$

3.9.3 A.8.3 Linear Dynamic System

Let $D_{40},D_{4},\ \Pi,$ and $\Delta$ be as defined by (3.109), (3.110), and (3.75), and (3.70). Then the coefficients θ _j ⁱ of the linear dynamic system (3.84) are as follows.

$$\displaystyle{\begin{array}{l} \theta _{1}^{1} = k\left (z_{1} \times \dfrac{\partial i} {\partial k} + z_{2}r^{{\prime}}\times \dfrac{\partial x} {\partial k}\right ) = -\dfrac{\ell_{1}ff^{{\prime}}\Delta g^{{\prime}}} {D_{4}}, \\ \theta _{2}^{1} = k\left (z_{1} \times \dfrac{\partial i} {\partial w} + z_{2}r^{{\prime}}\times \dfrac{\partial x} {\partial w}\right ) = -\dfrac{z_{1}f^{{\prime}}\ell\left (s\left (1-\tau \right )f^{{\prime}} + kfg^{{\prime}}\right )} {wD_{4}}, \\ \theta _{3}^{1} = k\left (z_{1} \times \dfrac{\partial i} {\partial b} + z_{2}r^{{\prime}}\times \dfrac{\partial x} {\partial b}\right ) \\ \quad \ = - \dfrac{k} {D_{4}} \times \big [\!\left (a + ci\right )\left \{z_{1}\left (\ell_{2}(f^{{\prime}})^{2} -\ell f^{{\prime\prime}}\right ) - z_{2}r^{{\prime}}\ell_{2}\right \} \\ + z_{1}f^{{\prime}}\left (\ell-\ell_{2}f\right )\left (s(1-\tau )f^{{\prime}} + kfg^{{\prime}}\right )\big], \\ \theta _{1}^{2} =\sigma w \times \dfrac{\partial e} {\partial k} = \dfrac{\sigma wxD_{40}} {D_{4}}, \\ \theta _{2}^{2} =\sigma w \times \dfrac{\partial e} {\partial w} = -\dfrac{\sigma kz_{1}f^{{\prime}}\ell} {D_{4}}, \\ \theta _{3}^{2} =\sigma w \times \dfrac{\partial e} {\partial b} = -\dfrac{\sigma wkf^{{\prime}}z_{1}\ell\Pi } {D_{4}}, \\ \theta _{1}^{3} = -kfg^{{\prime}}\times \dfrac{\partial e} {\partial k} + b \times \dfrac{\partial i} {\partial k} = -\dfrac{fD_{30}g^{{\prime}}} {D_{4}}, \\ \theta _{2}^{3} = -kfg^{{\prime}}\times \dfrac{\partial e} {\partial w} + b \times \dfrac{\partial i} {\partial w} = \dfrac{k^{2}ff^{{\prime}}z_{1}\ell g^{{\prime}}} {wD_{4}}, \\ \theta _{3}^{3} = -kfg^{{\prime}}\times \dfrac{\partial e} {\partial b} + \left (i - n\right ) + b \times \dfrac{\partial i} {\partial b} = \dfrac{k^{2}ff^{{\prime}}z_{1}\ell\Pi g^{{\prime}}} {D_{4}} + (i - n). \end{array} }$$

3.9.4 A.8.4 Characteristic Polynomial

With $D_{40},\Pi,$ and $\Delta$ as defined by (3.109), (3.75), and (3.70), the α _ij that appear in (3.86) are as follows.

$$\displaystyle{\begin{array}{l} \alpha _{11} = -kff^{{\prime}}\ell_{1}\left \{x\Delta - z_{1}\Pi -\left (i - n\right )\right \} \lesseqgtr 0, \\ \alpha _{12} = -\sigma kz_{1}f^{{\prime}}\ell + \left (i - n\right )D_{40}> 0, \\ \alpha _{21} = kfxf^{{\prime}}\left [\ell_{1}\left (a + ci -\left (i - n\right )\right )\Delta - s\left (1-\tau \right )f^{{\prime}}z_{1}\left (\ell-\ell_{2}f\right )\right ] \\ \qquad \!\! \equiv kfxf^{{\prime}}\ell_{1} \\ \quad \qquad \times \left [\left \{\left (a + ci -\left (i - n\right )\right )\Delta - s(1-\tau )z_{1}f^{{\prime}}\Pi \right \} - s\left (1-\tau \right )f^{{\prime}}\left (ci + a\right )\right ] \lesseqgtr 0, \\ \alpha _{22} = -\sigma kz_{1}f^{{\prime}}\ell\left \{\left (i - n\right ) - sx\left (1-\tau \right )f^{{\prime}}\right \} <0, \\ \alpha _{32} = \left (i - n\right )\sigma skxz_{1}\ell\left (1-\tau \right )\left (f^{{\prime}}\right )^{2}> 0. \end{array} }$$

A.9 Some Mathematical Expressions Appearing in the Coordinated Monetary Policy Model

3.10.1 A.9.1 Linear Dynamic System

Let D ₀ and D ₂ be defined by (3.94) and (3.97), let η _j ⁱ be the coefficients of the linear dynamic system in the full debt finance model, and let

$$\displaystyle{ D_{5} = D_{0} + kff^{{\prime}}\ell_{ 1}g^{{\prime}}(0) + \frac{z^{{\prime}}k(f^{{\prime}})^{2}} {w} m^{{\prime}}(0) <0. }$$

Then the coefficients ψ _j ⁱ of the linear dyanamic system (3.90) can be written in the following way.

$$\displaystyle{\begin{array}{l} \psi _{1}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial k} = \dfrac{1} {D_{5}}\left [D_{2}\eta _{1}^{1} -\dfrac{kz^{{\prime}}s(1-\tau )\left (f^{{\prime}}\right )^{2}} {w} \times m^{{\prime}}\right ] \lesseqgtr 0, \\ \psi _{2}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial w} = \dfrac{D_{2}\eta _{2}^{1}} {D_{5}} <0, \\ \psi _{3}^{1} = kz^{{\prime}}\times \dfrac{\partial i} {\partial b} = \dfrac{1} {D_{5}}\left [D_{2}\eta _{3}^{1} -\dfrac{z^{{\prime}}k\left (f^{{\prime}}\right )^{2}ci} {w} \times m^{{\prime}}\right ] <0, \\ \psi _{1}^{2} =\sigma w \times \dfrac{\partial e} {\partial k} = \dfrac{D_{2}\eta _{1}^{2}} {D_{5}}> 0, \\ \psi _{2}^{2} =\sigma wk \times \dfrac{\partial x} {\partial w} = \dfrac{D_{2}\eta _{2}^{2}} {D_{5}} <0, \\ \psi _{3}^{2} =\sigma wk \times \dfrac{\partial x} {\partial b} = \dfrac{D_{2}\eta _{3}^{2}} {D_{5}} \lesseqgtr 0, \\ \psi _{1}^{3} = -kfg^{{\prime}}\times \dfrac{\partial e} {\partial k} = \dfrac{D_{2}\eta _{1}^{3}} {D_{5}} <0, \\ \psi _{2}^{3} = -fk^{2}g^{{\prime}}\times \dfrac{\partial x} {\partial w} = \dfrac{D_{2}\eta _{2}^{3}} {D_{5}}> 0, \\ \psi _{3}^{3} = -fk^{2}g^{{\prime}}\times \dfrac{\partial x} {\partial b} + (i - n) \\ \quad \;\, = \dfrac{1} {D_{5}}\left [D_{2}\eta _{3}^{3} + \dfrac{\left (i - n\right )z^{{\prime}}k(f^{{\prime}})^{2}} {w} \times m^{{\prime}}\right ] \lesseqgtr 0.\end{array} }$$

3.10.2 A.9.2 Characteristic Polynomial

The α _ij ^m whose signs are reported in (3.93) in the text are as follows.

$$\displaystyle\begin{array}{rcl} \alpha _{11}^{m}& =& w^{-1}kz^{{\prime}}(f^{{\prime}})^{2}\left (i - n - s\left (1-\tau \right )\right )> 0, {}\\ \alpha _{21}^{m}& =& -w^{-1}kz^{{\prime}}x\left (i - n\right )s(1-\tau )(f^{{\prime}})^{3} <0. {}\\ \end{array}$$