Regulatory Capital Requirements, Inflation Targeting, and Equilibrium Determinacy

Abstract

This paper studies the stability properties of inflation-targeting interest rate rules in an economy with regulatory capital requirements. We derive the conditions for rational expectations equilibrium determinacy in a sticky-price model augmented with the cost channel of monetary policy transmission. We find that when tightening Basel II-type capital regulations, strict inflation targeting leads to significant expansions in regions of determinacy. This result is attributed to the supply side of credit markets, and especially to the procyclical nature of bank leverage and the restricted interest rate pass-through. However, when banks maintain capital ratios beyond the required thresholds, strict inflation targeting suffers from considerable shrinking regions of determinacy. Moreover, excessive bank capital holdings may give rise to self-fulfilling business cycles. The availability of countercyclical capital buffers, as proposed by Basel III, and/or a flexible inflation targeting regime offer an antidote to these problems.

Appendices

Appendices

1.1 Appendix 1

Banks’ balance sheet constraint states that banks can finance their loans using either deposits or bank capital. Log-linearization of the banks’ balance sheet constraint and solving for deposits yields:

$${d}_{t}=\frac{1}{1-v}{b}_{t}-\frac{v}{1-v}{k}_{t}^{b}$$

In addition, the working capital hypothesis implies that ${b}_{t}={w}_{t}^{r}+{h}_{t}$ where ${w}_{t}^{r}$ denotes the real wage and ${h}_{t}$ the hours worked. Substituting ${y}_{t}={h}_{t}$ and households optimality condition ${w}_{t}^{r}=\left(\sigma +\varphi \right){y}_{t}$ in ${b}_{t}={w}_{t}^{r}+{h}_{t}$ leads to a modified working capital constraint:

$${b}_{t}=\left(\varphi +\sigma +1\right){y}_{t}$$

Substituting ${j}_{t}^{b}=\frac{r+spr+{k}_{kb}{v}^{3}}{rv+spr}{b}_{t}-\frac{r}{rv+spr}{d}_{t}+\frac{v}{rv+spr}{r}_{t}-\frac{{k}_{kb}{v}^{3}}{rv+spr}{k}_{t}^{b}$ into the log-linearised version of Eq. (26), that is, ${k}_{t}^{b}=\left(1-\delta \right){k}_{t-1}^{b}+\delta {j}_{t}^{b}$, we eliminate ${j}_{t}^{b}$.

Then, we substitute the first two equations for ${d}_{t}$ and ${b}_{t}$.

$$\begin{aligned}&{k}_{t}^{b}=\frac{\left(1-v\right)\left(rv+spr\right)}{\left(rv+spr+{\delta k}_{kb}{v}^{3}\right)\left(1-v\right)-\delta rv}\left(1-\delta \right)\\&{k}_{t-1}^{b}+\frac{\left(r+spr+{k}_{kb}{v}^{3}\right)\delta \left(1-v\right)-\delta r}{\left(rv+spr+{\delta k}_{kb}{v}^{3}\right)\left(1-v\right)-\delta rv}\left(\varphi +\sigma +1\right)\\&{y}_{t}+\frac{\left(1-v\right)\delta v}{\left(rv+spr+{\delta k}_{kb}{v}^{3}\right)\left(1-v\right)-\delta rv}{r}_{t}\end{aligned}$$

Substracting the variable ${k}_{t}^{b}$ from (both sides of) ${b}_{t}=\left(\varphi +\sigma +1\right){y}_{t}$ and using the definition ${lev}_{t}\equiv {b}_{t}-{k}_{t}^{b}$ we get that:

$${lev}_{t}=\left(\varphi +\sigma +1\right){y}_{t}-{k}_{t}^{b}$$

Finally, substituting the previous equation for ${k}_{t}^{b}$ and the interest rate rule, Eqs. (44) or (45) leads to Eq. (43) in the text.

1.2 Appendix 2

Equations (39), (40), (42)–(44) is the system of difference equations describing the equilibrium dynamics of our economy. After some algebraic substitutions, we can reduce the system to one involving two variables. In particular, we substitute Eqs. (42) and (43) into Eqs. (40) and (44) into Eqs. (39) and (40) and then write the model in the state space form $A{E}_{t}{z}_{t+1}=B{z}_{t}$ where ${z}_{t}$ is the 2 × 1 vector of the endogenous variables which are non-predetermined ${z}_{t}={\left[{y}_{t},\hspace{0.33em}{\pi }_{t}\right]}^{\prime}$. Τhe 2 × 2 square matrices of the coefficients are defined as:

$$A\equiv \left[\begin{array}{cc}1& \frac{1}{\sigma }\\ 0& \frac{\beta }{1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }}\end{array}\right]$$

and

$${\rm B}\equiv \left[\begin{array}{cc}1& \frac{{\varphi }_{\Pi }}{\sigma }\\ -\frac{k\left(\sigma +\varphi \right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)}{1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }}& 1\end{array}\right]$$

  

Since, under baseline calibration, matrix $A$ is invertible, we get that ${E}_{t}{z}_{t+1}={A}^{-1}B{z}_{t}=\Gamma {z}_{t}$, where ${\Gamma \equiv A}^{-1}B$.

$$\Gamma =\left[\begin{array}{cc}1+\frac{k\left(\sigma +\varphi \right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)}{\sigma \beta }& \frac{\beta {\varphi }_{\Pi }-\left[1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }\right]}{\beta \sigma }\\ -\frac{k\left(\sigma +\varphi \right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)}{\beta }& \frac{1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }}{\beta }\end{array}\right]$$

By simple algebra, we have that the determinant and trace of matrix $\Gamma$ are given by, respectively:

$$\begin{aligned}{det}\left(\mathit\Gamma \right)=&\;\frac{1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }}{\beta }\\&+\frac{k\left(\sigma +\varphi \right)+k\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)}{\sigma \beta }{\varphi }_{\mathit\Pi }\end{aligned}$$

$$\begin{aligned}trace\left(\mathit\Gamma \right)=&\;1+\frac{k\left(\sigma +\varphi \right)+k\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)}{\sigma \beta }\\&+\frac{1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }}{\beta }\end{aligned}$$

For determinacy, the number of eigenvalues of Γ outside the unit circle must equal the number of non-predetermined endogenous variables (Blanchard and Kahn 1980). In our case, there are two non-predetermined endogenous variables, inflation and output. Following Woodford (2003), this condition is satisfied if and only if either Case I or Case II below is true.

Case I:

$${det}\;\mathit\Gamma >1$$

(A.1)

$${det}\;\mathit\Gamma -tr\mathit\Gamma >-1$$

(A.2)

$${det}\;\mathit\Gamma +tr\mathit\Gamma >-1$$

(A.3)

Case II:

$${det}\;\mathit\Gamma -tr\mathit\Gamma <-1$$

(A.4)

$${det}\;\mathit\Gamma +tr\mathit\Gamma <-1$$

(A.5)

Consider Case I. Let us first focus on (A.1) which translates into $k{\varphi }_{\Pi }\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma \right)-\mathit\Xi {\rm K}_{2}-\sigma \left(F-\mathit\Xi {\rm K}_{3}\right)\right]>-\sigma (1-\beta )$. To isolate ${\varphi }_{\Pi }$ on the LHS, we need to divide both sides of the inequality by $\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma \right)-\mathit\Xi {\rm K}_{2}-\sigma \left(F-\mathit\Xi {\rm K}_{3}\right)$. Thus, for $\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma \right)-\mathit\Xi {\rm K}_{2}-\sigma \left(F-\mathit\Xi {\rm K}_{3}\right)>0$, ${v>v}_{1},$ we get ${\varphi }_{\mathit\Pi }>-\sigma (1-\beta )/k\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma \right)-\mathit\Xi {\rm K}_{2}-\sigma \left(F-\mathit\Xi {\rm K}_{3}\right)\right]$, The resulting condition is nested in

$${\varphi }_{\mathit\Pi }>0$$

(A.6)

In the alternative case, i.e. for ${v<v}_{1}$ and thus $\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma \right)-\mathit\Xi {\rm K}_{2}-\sigma \left(F-\mathit\Xi {\rm K}_{3}\right)<0$, we get:

$$\begin{aligned}{\varphi }_{\Pi }<{\varphi }_{2}\equiv& \;\sigma (1-\beta )\\&/k[-\sigma -\varphi -\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}\right)+\sigma F]\end{aligned}$$

(A.7)

We now consider (A.2). This condition leads to ${\varphi }_{\mathit\Pi }k\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)\right]>k\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)\right]$. For every value of $v$ we have that $\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma \right)-\mathit\Xi {\rm K}_{2}>0$ and thereby (A.2) translates into:

$${\varphi }_{\mathit\Pi }>1$$

(A.8)

Condition (Α.3) leads to $\begin{aligned}{\varphi }_{\mathit\Pi }k&\,\left\{-2\sigma F+\sigma +\varphi +\mathit\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}\right)\right\}>\\&-\left\{2\sigma \left(1+\beta \right)+k\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)\right]\right\}\end{aligned}$. Again, we have to evaluate the sign of $-2\sigma F+\sigma +\varphi +\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}\right)$. This turns out to be positive for every value of $v$ and $\sigma <\varphi$. Otherwise, this expression is negative. The assumption $\sigma >\varphi$ corresponds to a situation in which the weight of the cost channel of monetary policy transmission is relatively larger than the weight of the demand channel. In the latter case, an explicit condition for ${\varphi }_{\Pi }$ is the following:

$${\varphi }_{\mathit\Pi }< {\varphi }_{1}\equiv \frac{2\sigma \left(1+\beta \right)+k\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)\right]}{k\left[2\sigma F-\sigma -\varphi -\mathit\Xi \left(1+\sigma +\varphi -2\sigma {\rm K}_{3}+{\rm K}_{2}\right)\right]}$$

(A.9)

Putting things together, for $\sigma >\varphi$ we can reduce the three inequalities in Case I to $max\left\{0, 1\right\}{<\varphi }_{\Pi }<{\varphi }_{1}$ for ${v>v}_{1}$. Equation (49) from Proposition 2 then follows immediately. Otherwise, for ${v<v}_{1}$ we have that $1{<\varphi }_{\Pi }<min\left\{{\varphi }_{1}, {\varphi }_{2}\right\}$. This results in Eq. (48) in the text.

1.3 Appendix 3

The upper bound on the inflation coefficient ${\varphi }_{2}$ intersects with the lower bound, and thus determinacy is never attained when $\sigma \left(1-\beta \right)/k\left[-\sigma -\varphi -\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}\right)+\sigma F\right]=1$. The latter holds for ${K}^{b}/B=0.33$ and $v<{v}_{2}=0.06$. Note that we concentrate on the upper bound ${\varphi }_{2}$ and not on ${\varphi }_{1}$ since we are interested in empirically plausible values of the inflation coefficient. For instance, under baseline parameterization, ${\varphi }_{1}\in \left(31.73, 37.25\right)$.

1.4 Appendix 4

Considering Eq. (46) with Eqs. (39), (40), (42), and (43), the reduced-form equilibrium system ${E}_{t}{z}_{t+1}={A}^{-1}B{z}_{t}=\Gamma {z}_{t}$ is characterized by the system matrix $\Gamma\!:$

$$\Gamma =\left[\begin{array}{cc}1+\frac{k\left(\sigma +\varphi \right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)}{\sigma \beta }& \frac{\beta {\varphi }_{\Pi }-\left[1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }\right]}{\beta \sigma }\\ -\frac{k\left(\sigma +\varphi \right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)}{\beta }& \frac{1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }}{\beta }\end{array}\right]$$

Its determinant and trace are given by, respectively:

$$\mathit{det}\left(\Gamma \right)=\frac{1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }}{\beta }+\frac{k\left(\sigma +\varphi \right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)}{\sigma \beta }{\varphi }_{\Pi }$$

$$trace\left(\Gamma \right)=1+\frac{k\left(\sigma +\varphi \right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)}{\sigma \beta }+\frac{1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }}{\beta }$$

From an argument similar to that in the proof of Proposition 1, the necessary and sufficient condition for local determinacy of REE is that the number of eigenvalues of Γ outside the unit circle must equal the number of non-predetermined endogenous variables. By Proposition C.1 of Woodford (2003), this is the case if and only if either Case I or Case II is satisfied. We start deriving policy parameter restrictions from (A.1). We can write the latter as ${\varphi }_{\Pi }k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)-\sigma F\right]>-\sigma \left(1-\beta \right)$. We have to distinguish two cases. First, for $\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)-\sigma F>0,$ which holds for ${v<v}_{1}^{*}$, we have that ${\varphi }_{\Pi }>-\sigma \left(1-\beta \right)/k\left[ \sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)-\sigma F\right]$. From the latter inequality, we get:

$${\varphi }_{\Pi }>0$$

(A.10)

Second, for ${v>v}_{1}^{*}$, we get that in terms of the inflation equation:

$${\varphi }_{\Pi }<{\varphi }_{4}\equiv \frac{\sigma \left(1-\beta \right)}{k\left[-\sigma -\varphi -\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)+\sigma F\right]}$$

(A.11)

Next, we derive restrictions for ${\varphi }_{\Pi }$ from the condition (Α.2). This condition implies that ${\varphi }_{\Pi }k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right]>k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right].$ Since $\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)>0$, the term $\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)$ is also positive. Therefore, the parameter restriction derived from fulfillmenting (Α.2) is:

$${\varphi }_{\Pi }>1$$

(A.12)

Condition (A.3) leads to ${\varphi }_{\Pi }k\left\{-2\sigma F+\sigma +\varphi +\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}+{\chi }_{V}\right)\right\}>-\left\{2\sigma \left(1+\beta \right)+k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right]\right\}.$ The term $-2\sigma F+\sigma +\varphi +\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}\right)$ is is negative for every value of $v {\text{and}}$ for $\sigma >\varphi$ (baseline assumption). In this case the upper bound for ${\varphi }_{\Pi }$ is equal to:

$${\varphi }_{\Pi }<{\varphi }_{3}\equiv \frac{2\sigma \left(1+\beta \right)+k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right]}{k\left[2\sigma F-\sigma -\varphi -\Xi \left(1+\sigma +\varphi -2\sigma {\rm K}_{3}+{\rm K}_{2}+{\chi }_{V}\right)\right]}$$

(A.13)

Equation (52) stated in Proposition 5 follows after combining conditions (A.10), (A.12), and (A.13), for example, $max\left\{0, 1\right\}{<\varphi }_{\Pi }<{\varphi }_{3}$, whereas the combination of (A.11)–(A.13) yields $1{<\varphi }_{\Pi }<min\left\{{\varphi }_{3}, {\varphi }_{4}\right\}$. The latter results in Eq. (53) in the text.

Finally, when$\sigma <\varphi$, we obtain $max\left\{0 , 1\right\}< {\varphi }_{\Pi }<{\varphi }_{4}$ for ${v>v}_{1}^{*}$ and ${\varphi }_{\Pi }>1$ for ${v<v}_{1}^{*}$. Thus, the last part of Proposition 5, i.e. Equation (54), follows immediately.

1.5 Appendix 5

The dynamic system ${E}_{t}{z}_{t+1}={A}^{-1}B{z}_{t}=\Gamma {z}_{t}$ is now defined by Eqs. (39), (40), (42), (43), and (45). The matrix $\Gamma$ is given by:

$$\mathit\Gamma =\left[\begin{array}{cc}1+\frac{1}{\sigma }{\varphi }_{\Upsilon}+\frac{k\left(\sigma +\varphi +F{\varphi }_{\Upsilon}\right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}-{\rm K}_{3}{\varphi }_{\Upsilon}\right)}{\sigma \beta }& \frac{\beta {\varphi }_{\Pi }-\left[1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }\right]}{\beta \sigma }\\ -\frac{k\left(\sigma +\varphi +F{\varphi }_{\Upsilon}\right)+k\Xi \left(1+\varphi +\sigma -{\rm K}_{2}-{\rm K}_{3}{\varphi }_{\Upsilon}\right)}{\beta }& \frac{1-\left(F-\Xi {\rm K}_{3}\right)k{\varphi }_{\Pi }}{\beta }\end{array}\right]$$

With

$$\begin{aligned}{det}\left(\mathit\Gamma \right)=&\;\frac{1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }}{\beta }+\frac{{\varphi }_{\mathit\Upsilon}\left[1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }\right]}{\sigma \beta }\\&+\frac{{\varphi }_{\mathit\Pi }k\left[\sigma +\varphi +F{\varphi }_{\mathit\Upsilon}+\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}-{\rm K}_{3}{\varphi }_{\mathit\Upsilon}\right)\right]}{\sigma \beta }\end{aligned}$$

$$\begin{aligned}trace\left(\mathit\Gamma \right)=&\;1+\frac{1}{\sigma }{\varphi }_{\mathit\Upsilon}\\&+\frac{k\left[\sigma +\varphi +F{\varphi }_{\mathit\Upsilon}+\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}-{\rm K}_{3}{\varphi }_{\mathit\Upsilon}\right)\right]}{\sigma \beta }\\&+\frac{1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }}{\beta }\end{aligned}$$

The system has two non-predetermined variables, and therefore, the system will have unique rational expectations equilibrium if, and only if, Case I or Case II is satisfied. Consider Case I. Condition (A.1) corresponds to$\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)-\sigma \left(F-\Xi {\rm K}_{3}\right)\right]k{\varphi }_{\Pi }>-\left[\sigma \left(1-\beta \right)+{\varphi }_{\Upsilon}\right]$.We distinguish two cases. First,$\sigma +\varphi +\Xi \left(1+\varphi +\sigma \right)-\Xi {\rm K}_{2}-\sigma \left(F-\Xi {\rm K}_{3}\right)>0$, which holds for${v>v}_{1}$, condition (A.1) implies ${\varphi }_{\Pi }>-[\sigma \left(1-\beta \right)+{\varphi }_{\Upsilon}]/k\left[\sigma \left(1-F\right)+\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}\right)\right]$. From the latter inequality, we obtain:

$${\varphi }_{\Pi }>0$$

(A.14)

Second, for $\sigma +\varphi +\Xi \left(1+\varphi +\sigma \right)-\Xi {\rm K}_{2}-\sigma \left(F-\Xi {\rm K}_{3}\right)$, i.e. ${v<v}_{1}$, condition (A.1) takes the form:

$$\begin{aligned}{\varphi }_{\mathit\Pi }<{\varphi }_{7}\equiv&\; \left[\sigma \left(1-\beta \right)+{\varphi }_{\Upsilon}\right]/k[\sigma \left(1-F\right)+\varphi \\&+\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}\right)]\end{aligned}$$

(A.15)

Condition (A.2) is true if and only if ${\varphi }_{\Pi }k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma \right)-\Xi {\rm K}_{2}\right]>k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}-{\rm K}_{3}{\varphi }_{\Upsilon}\right)+F{\varphi }_{Y}\right]-{\varphi }_{Y}\left(1-\beta \right).$ The LHS is always positive and thereby (A.2) corresponds to:

$${\varphi }_{\Pi }>{\varphi }_{6}\equiv 1+\frac{kF-\left(1-\beta \right)-k\Xi {\rm K}_{3}}{k\left\{\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)\right\}}{\varphi }_{Y}$$

(A.16)

We derive, next, restrictions for ${\varphi }_{\Pi }$ from the condition (Α.3) which can be written as ${\varphi }_{\Pi }k\left\{-2\sigma F+\sigma +\varphi +\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}\right)\right\}>-\left\{2\sigma \left(1+\beta \right)+k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)\right]+\left(1+\beta +kF-k\Xi {\rm K}_{3}\right){\varphi }_{\Upsilon}\right\}$. The term $-2\sigma F+\sigma +\varphi +\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}\right)$ turns out to be negative for every value of $v$ and $\sigma >\varphi$ (baseline assumption). In this case, the LHS is negative. In this case, another bound for ${\varphi }_{\Pi }$ is:

$${\varphi }_{\Pi }< {\varphi }_{5}\equiv \frac{2\sigma \left(1+\beta \right)+k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}\right)\right]+\left(1+\beta +kF-k\Xi {\rm K}_{3}\right){\varphi }_{Y}}{k\left\{2\sigma F-\sigma -\varphi -\Xi \left(1+\varphi +\sigma +2\sigma {\rm K}_{3}-{\rm K}_{2}\right)\right\}}$$

(A.17)

Considering all the above, we can reduce the three inequalities in Case I to $max\left\{0, {\varphi }_{6}\right\}{<\varphi }_{\Pi }<{\varphi }_{7}$ for ${v<v}_{1}$. Equation (55) from Proposition 7 then follows immediately. Otherwise, for ${v>v}_{1}$ we have that ${\varphi }_{6}{<\varphi }_{\Pi }<min\left\{{\varphi }_{5}, { \varphi }_{7}\right\}$. This results in Eq. (56).

1.6 Appendix 6

Consider the dynamic system defined by Eqs. (39), (40), (43), (45), and (47) and written in space state for $A{E}_{t}{z}_{t+1}=B{z}_{t}$. Since matrix A is invertible we have that ${E}_{t}{z}_{t+1}={A}^{-1}B{z}_{t}=\Gamma {z}_{t}$, where

$$\Gamma =\left[\begin{array}{cc}1+\frac{{\varphi }_{\mathit\Upsilon}}{\sigma }+\frac{k\left[\sigma +\varphi +{\varphi }_{\mathit\Upsilon}F+{\mathit\Xi} \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}-{{\rm K}_{3}\varphi }_{\mathit\Upsilon}\right)\right]}{\sigma \beta }& \frac{\beta {\varphi }_{{\mathit\Pi} }-\left[1-\left(F-{\mathit\Xi} {\rm K}_{3}\right)k{\varphi }_{{\mathit\Pi} }\right]}{\beta \sigma }\\ -\frac{k\left[\sigma +\varphi +{\varphi }_{\mathit\Upsilon}F+{\mathit\Xi} \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}-{{\rm K}_{3}\varphi }_{\mathit\Upsilon}\right)\right]}{\beta }& \frac{1-\left(F-{\mathit\Xi} {\rm K}_{3}\right)k{\varphi }_{{\mathit\Pi} }}{\beta }\end{array}\right]$$

and

$$\begin{aligned}{det}\left(\mathit\Gamma \right)=&\;\frac{1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }}{\beta }+\frac{{\varphi }_{\Upsilon}\left[1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }\right]}{\sigma \beta }\\&+\frac{k\left[\sigma +\varphi +{\varphi }_{\Upsilon}F+\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}-{{\rm K}_{3}\varphi }_{\Upsilon}\right)\right]}{\sigma \beta }\end{aligned}$$

$$\begin{aligned}trace\left(\mathit\Gamma \right)=&\;1+\frac{1}{\sigma }{\varphi }_{\Upsilon}\\&+\frac{k\left[\sigma +\varphi +{\varphi }_{\Upsilon}F+\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}-{{\rm K}_{3}\varphi }_{\Upsilon}\right)\right]}{\sigma \beta }\\&+\frac{1-\left(F-\mathit\Xi {\rm K}_{3}\right)k{\varphi }_{\mathit\Pi }}{\beta }\end{aligned}$$

The system has two non-predetermined variables, and therefore, the system will have unique rational expectations equilibrium if, and only if, Case I or Case II is satisfied. Condition (A.1) from Case I leads to $\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)-\sigma F\right]k{\varphi }_{\mathit\Pi }>-\left[\sigma \left(1-\beta \right)+{\varphi }_{\mathit\Upsilon}\right]$. For $\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)-\sigma F>0$, which holds for ${v<v}_{1}^{*}$, we obtain ${\varphi }_{\mathit\Pi }>-\left[\sigma \left(1-\beta \right)+{\varphi }_{\mathit\Upsilon}\right]/k\left[\sigma +\varphi +\mathit\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)-\sigma F\right]$. The resulting condition is nested in

$${\varphi }_{\Pi }>0$$

(A.18)

In the alternative case, i.e. for ${v>v}_{1}^{*}$ and thus $\varphi >\mathit\Xi \left(\sigma +\varphi -{\chi }_{V}\right)<0$, condition (A.1) results in:

$${\varphi }_{\Pi }<{\varphi }_{10}\equiv \frac{\sigma \left(1-\beta \right)+{\varphi }_{\Upsilon}}{k\left[-\sigma -\varphi -\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+\sigma {\rm K}_{3}+{\chi }_{V}\right)+\sigma F\right]}$$

(A.19)

From condition (A.2) we find ${\varphi }_{\Pi }k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right]>k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right]+{F\varphi }_{\Upsilon}-\left(1-\beta \right){\varphi }_{\Upsilon}.$ Since $\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)$ is always positive, the parameter restriction derived from fulfillmenting (Α.2) is:

$${\varphi }_{\Pi }>{\varphi }_{8}\equiv 1+\frac{kF-\left(1-\beta \right)-k\Xi {\rm K}_{3}}{k\left\{\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right\}}{\varphi }_{Y}$$

(A.20)

Condition (A.3) implies that ${\varphi }_{\Pi }k\left\{-2\sigma F+\sigma +\varphi +\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}+{\chi }_{V}\right)\right\}>-2\sigma \left(1+\beta \right)-k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}{+\chi }_{V}\right)\right]-\left(1+\beta +kF-k\Xi {\rm K}_{3}\right){\varphi }_{\Upsilon}.$ Since the term $-2\sigma F+\sigma +\varphi +\Xi \left(1+\sigma +\varphi +2\sigma {\rm K}_{3}-{\rm K}_{2}+{\chi }_{V}\right)$ is negative for $\sigma >\varphi$, the the upper bound for ${\varphi }_{\Pi }$ is:

$${\varphi }_{\Pi }<{\varphi }_{9}\equiv \frac{2\sigma \left(1+\beta \right)+k\left[\sigma +\varphi +\Xi \left(1+\varphi +\sigma -{\rm K}_{2}+{\chi }_{V}\right)\right]+\left(1+\beta +kF-k\Xi {\rm K}_{3}\right){\varphi }_{Y}}{k\left\{2\sigma F-\sigma -\varphi -\Xi \left(1+\varphi +\sigma +2\sigma {\rm K}_{3}-{\rm K}_{2}+{\chi }_{V}\right)\right\}}$$

(A.21)

Considering all the above, we can reduce the three inequalities in Case I to $max\left\{0, {\varphi }_{8}\right\}{<\varphi }_{\Pi }<{\varphi }_{9}$ for ${v<v}_{1}^{*}$. Otherwise, for ${v>v}_{1}^{*}$ we have that ${\varphi }_{8}{<\varphi }_{\Pi }<min\left\{{\varphi }_{9}, {\varphi }_{10}\right\}$. This results in Eq. (57) in the text.