2.1. Government

Assume that government revenue includes only labor and capital income taxes, government purchases are constant, and that the government is subject to a balanced-budget requirement. The government budget constraint is then given by:

(1) \begin{equation} G=\tau _{t}^{l}w_{t}l_{t}+\tau _{t}^{k}\left ( r_{t}-\delta \right ) k_{t}\text{,} \end{equation}

where $G$ denotes government purchases of goods, $\tau _{t}^{l}$ ( $\tau _{t}^{k}$ ) the labor (capital) tax rate, and $w_{t}$ ( $r_{t}$ ) the pretax wage (rental) rate, $l_{t}$ hours worked, and $\delta \geq 0$ the capital depreciation rate. Here, the term $-\tau _{t}^{k}\delta k_{t}$ represents a depreciation allowance.

We assume in this paper that the government adjusts the tax rates to balance the budget according to

(2) \begin{equation} \frac{d\tau _{t}^{k}}{\tau _{t}^{k}}=s\cdot \frac{d\tau _{t}^{l}}{\tau _{t}^{l}}\text{, } \end{equation}

where $s\geq 0$ is a parameter. With endogenous capital and labor income tax rates, SGU (Reference Schmitt-Grohé and ín Uribe1997) restrict to the case with $s=1$ , that is, the two tax rates vary in the same proportion. Instead, we allow the two tax rates to change in distinct proportions, that is, $s\neq 1$ . Note that (2) also nests the two special cases examined in SGU (Reference Schmitt-Grohé and ín Uribe1997): an exogenous capital tax rate ( $s=0$ ) and an exogenous labor tax rate ( $s=\infty$ ).

2.2. Households

The economy is populated by a unit measure of identical infinitely lived households. The representative household chooses paths for consumption, $c_{t}$ , hours worked $l_{t}$ , and capital, $k_{t}$ , so as to maximize the present discounted value of its lifetime utility:

\begin{equation*} \max \int _{0}^{\infty }\left ( \ln c_{t}-\eta l_{t}\right ) e^{-\rho t}dt. \end{equation*}

subject to,

\begin{equation*} \dot {k}_{t}=\big ( 1-\tau _{t}^{k}\big ) \big ( r_{t}-\delta \big ) k_{t}+\big ( 1-\tau _{t}^{l}\big ) w_{t}l_{t}-c_{t},\text { }k_{0}\gt 0\text { given} \end{equation*}

where $\rho \gt 0$ is the subjective discount rate and $\eta \gt 0$ is the a scaling parameter. The first-order conditions associated with this problem are

(3) \begin{equation} c_{t}^{-1}=\lambda _{t}, \end{equation}

(4) \begin{equation} \eta =\lambda _{t}\big ( 1-\tau _{t}^{l}\big ) w_{t}, \end{equation}

(5) \begin{equation} \dot{\lambda }_{t}=\lambda _{t}\left [ \rho -\big ( 1-\tau _{t}^{k}\big ) \left ( r_{t}-\delta \right ) \right ], \end{equation}

where $\lambda _{t}$ denotes the marginal utility of income.

2.3. Firms

The representative firm produces output $y_{t}$ , using capital and labor as inputs, with a constant returns-to-scale Cobb–Douglas production function:

(6) \begin{equation} y_{t}=Ak_{t}^{\alpha }l_{t}^{1-\alpha }\text{, }0\lt \alpha \lt 1. \end{equation}

Perfect competition in factor and product markets implies that factor demands are given by:

(7) \begin{equation} w_{t}=\left ( 1-\alpha \right ) \frac{y_{t}}{l_{t}},\text{ }r_{t}=\alpha \frac{y_{t}}{k_{t}} \end{equation}

Finally, the aggregate resource constraint for the economy is given by:

(8) \begin{equation} c_{t}+G+\dot{k}_{t}+\delta k_{t}=y_{t}\text{.} \end{equation}

2.4. Analysis of dynamics

The model’s equilibrium can be reduced to two differential equations:

(9) \begin{equation} \dot{k}_{t}=y_{t}-\delta k_{t}-c_{t}-G, \end{equation}

(10) \begin{equation} \dot{\lambda }_{t}=\lambda _{t}\left [ \rho +\big ( 1-\tau _{t}^{k}\big ) \left ( \delta -\alpha \frac{y_{t}}{k_{t}}\right ) \right ], \end{equation}

along with (1)–(4), (6), and (7). In what follows, we denote the steady-state value of a variable by dropping its time index. It is straightforward to show that the dynamic system possesses a unique interior steady state. Linearizing (9)–(10) around the steady state, we obtain

(11) \begin{equation} \begin{pmatrix} \dot{k}_{t} \\ \dot{\lambda }_{t}\end{pmatrix}=\mathbf{J}\begin{pmatrix} k_{t}-k \\ \lambda _{t}-\lambda \end{pmatrix}\text{,} \end{equation}

where J is the Jacobian matrix. The elements of $\mathbf{J}$ , denoted as $J_{ij}$ , are given by:

\begin{align*} J_{11}&=-\alpha \epsilon _{yk}\theta _{y\lambda }\big ( 1-\tau ^{k}\big ) +\left ( \alpha \epsilon _{yk}-\delta \right ) \epsilon _{\tau \lambda }s\tau ^{k},\\ J_{12}&=\epsilon _{\lambda k}\left [ \alpha \big ( 1-\tau ^{k}\big ) \left ( 1-\theta _{yk}\right ) \epsilon _{yk}+\left ( \alpha \epsilon _{yk}-\delta \right ) \epsilon _{\tau k}s\tau ^{k}\right ], \end{align*}

\begin{equation*} J_{21}=\epsilon _{yk}\epsilon _{\lambda k}^{-1}\left ( \theta _{y\lambda }+\epsilon _{cy}\right ) \text {, }J_{22}=\epsilon _{yk}\theta _{yk}-\delta , \end{equation*}

where

\begin{equation*} \epsilon _{yk}\equiv \frac {1}{\alpha }\frac {\rho +\big ( 1-\tau ^{k}\big ) \delta }{1-\tau ^{k}}\text {, }\epsilon _{cy}\equiv \left ( 1-\alpha \right ) \big ( 1-\tau ^{l}\big ) +\rho \epsilon _{yk}^{-1}, \end{equation*}

\begin{equation*} \theta _{\tau \lambda }\equiv -\frac {\left [ \left ( 1-\alpha \right ) \tau ^{l}+\alpha \tau ^{k}\right ] \left ( 1-\alpha \right ) \epsilon _{yk}}{\kappa },\theta _{\tau k}\equiv \frac {\delta \alpha \tau ^{k}-\left [ \left ( 1-\alpha \right ) \tau ^{l}+\alpha \tau ^{k}\right ] \alpha \epsilon _{yk}}{\kappa }, \end{equation*}

\begin{equation*} \kappa \equiv \frac {\alpha +\alpha \tau ^{k}-\tau ^{l}}{1-\tau ^{l}}\left ( 1-\alpha \right ) \tau ^{l}\epsilon _{yk}+\left ( \alpha \epsilon _{yk}-\delta \right ) s\alpha \tau ^{k}, \end{equation*}

\begin{equation*} \theta _{y\lambda }\equiv \frac {1-2\tau ^{l}}{1-\tau ^{l}}\frac {1-\alpha }{\alpha }\theta _{\tau \lambda },\text { }\theta _{yk}=1-\frac {\left ( 1-\alpha \right ) \tau ^{l}}{\alpha \big ( 1-\tau ^{l}\big ) }\theta _{\tau k}. \end{equation*}

The expression for $\epsilon _{\lambda k}$ is not given here as its value affects neither the trace nor the determinant of $\mathbf{J}$ .

While the closed-form equilibrium solutions cannot generally be obtained, for the case with $\delta =0$ , analytical results can be derived. To see this, we note that, when $\delta =0$ , the trace and determinant of $\mathbf{J}$ are given by:

\begin{equation*} Tr=\alpha \frac {y}{k}\Xi y_{\lambda },\text { }Det=\frac {\alpha }{1-\alpha }\left ( \frac {y}{k}\right ) ^{2}\frac {\Omega }{\left ( 1-\alpha \right ) \tau ^{l}+s\alpha \tau ^{k}}y_{\lambda }, \end{equation*}

where

\begin{equation*} y_{\lambda }=\frac {1}{\frac {\alpha }{1-\alpha }-\frac {\tau ^{l}}{1-\tau ^{l}}\frac {\left ( 1-\alpha \right ) \tau ^{l}+\alpha \tau ^{k}}{\left ( 1-\alpha \right ) \tau ^{l}+s\alpha \tau ^{k}}},\text { }\Xi \equiv \frac {\alpha }{1-\alpha }+\frac {\left ( 1-\alpha \right ) \left ( 1-s\right ) \tau ^{l}}{\left ( 1-\alpha \right ) \tau ^{l}+s\alpha \tau ^{k}}\tau ^{k}, \end{equation*}

\begin{eqnarray*} \Omega &\equiv &-\left \{ \left [ \left ( 1-\alpha \right ) \tau ^{l}+\alpha \tau ^{k}\right ] ^{2}+\left ( 1-\alpha \right ) \big ( 1-\tau ^{l}\big ) -\tau ^{k}\right \} \alpha \tau ^{k}s \\ &&-\frac{\left \{ \left [ \left ( 1-\alpha \right ) \tau ^{l}+\alpha \tau ^{k}\right ] ^{2}+\left ( 1-\alpha \right ) \big ( 1-\tau ^{l}\big ) -\left [ \left ( 1-\alpha \right ) \tau ^{l}+\alpha \tau ^{k}\right ] \right \} \left ( 1-\alpha \right ) \big ( 1-\tau ^{k}\big ) }{1-\tau ^{l}}\tau ^{l}. \end{eqnarray*}

The following proposition gives a necessary condition for equilibrium indeterminacy.

Proposition 1. If $\Xi \gt 0$ , then a necessary condition for indeterminacy is that

\begin{equation*} \tau ^{l}\gt \underline {\tau }^{l}, \end{equation*}

where, in the interval $(0,1),$ $\underline{\tau }^{l}$ is the unique root of the quadratic equation

\begin{equation*} q\big ( \tau ^{l},\tau ^{k},s\big ) \equiv \big ( \tau ^{l}\big ) ^{2}+ \left [ \left ( 1+\frac {\alpha s}{1-\alpha }\right ) \tau ^{k}-1\right ] \alpha \tau ^{l}-\frac {s\alpha ^{2}}{1-\alpha }\tau ^{k}=0\text {.} \end{equation*}

Proof. A necessary condition for indeterminacy is that the trace is negative, which is equivalent to $y_{\lambda }\lt 0$ if $\Xi \gt 0$ ,Footnote ⁴ that is, $q\big ( \tau ^{l};\,\tau ^{k},s\big ) \gt 0$ . For $\tau ^{k}\gt 0$ , $q\big ( \tau ^{l}=0\big ) =-\frac{s\alpha ^{2}}{1-\alpha }\tau ^{k}\lt 0$ , and $q\big ( \tau ^{l}=1\big ) =1-\alpha +\alpha \tau ^{k}\gt 0$ . Thus, there exists a unique root in the interval $\underline{\tau }^{l}\in \left ( 0,1\right )$ such that $q\big ( \underline{\tau }^{l}\big ) =0$ . If $\tau ^{k}=0$ , $q=(\tau ^{l}-\alpha )\tau ^{l}$ , and $\underline{\tau }^{l}=\alpha$ . Thus, for all $\tau ^{l}\gt \underline{\tau }^{l}$ , we have $q\gt 0$ and hence $y_{\lambda }\lt 0$ .

The following corollary can be readily established.

Proof. From $q\big ( \underline{\tau }^{l};\,\tau ^{k},s\big ) =0$ , we have

\begin{align*} \frac {\partial \underline {\tau }^{l}}{\partial s}=-\frac {\dfrac {dq\big ( \underline {\tau }^{l}\big ) }{ds}}{\dfrac {dq\big ( \underline {\tau }^{l}\big ) }{d\underline {\tau }^{l}}}=\frac {\frac {\alpha ^{2}}{1-\alpha }\tau ^{k}\big ( 1-\underline {\tau }^{l}\big ) }{\left ( \underline {\tau }^{l}\right ) ^{2}+\frac {s\alpha ^{2}}{1-\alpha }\tau ^{k}}\underline {\tau }^{l}\gt 0. \\[-36pt]\end{align*}

We can elaborate the results in Proposition 1 and Corollary 1 in line with the intuition related to the labor market (e.g., SGU (Reference Schmitt-Grohé and ín Uribe1997)), using the log-linearized version of the labor demand function (where a variable with a hat denotes the log deviation of the variable from its steady state):

\begin{equation*} \hat {w}_{t}^{d}=\alpha \left [ 1+\frac {\left ( 1-\alpha \right ) \tau ^{l}+\alpha \tau ^{k}}{\left ( 1-\alpha \right ) \tau ^{l}+s\alpha \tau ^{k}}\frac {\tau ^{l}}{1-\tau ^{l}}\right ] \widehat {k}_{t}-\frac {1-\alpha }{y_{\lambda }}\widehat {l}_{t}. \end{equation*}

The necessary condition for indeterminacy, $y_{\lambda }\lt 0$ , implies that the slope of the labor demand curve is positive. The larger is $s$ , the smaller is $y_{\lambda }$ , thus the flatter is the labor demand curve, which makes it less likely for the labor demand curve to be steeper than the labor supply curve. Hence, indeterminacy is less likely to occur for larger value of $s$ . This nexus of our paper’s results and the conventional labor market elucidation of indeterminacy is consistent with the elaboration of our model’s mechanism outlined in the introduction section based on the two polar cases, one with endogenous labor but fixed capital income tax rates (i.e., $s=0$ ) and the other with endogenous capital but fixed labor income tax rates (i.e., $s=\infty$ ).

To get a quantitative feel, we calibrate the model using the same baseline parameterization as in SGU (Reference Schmitt-Grohé and ín Uribe1997), that is, $\alpha =0.3$ , $\rho =0.04$ , $\delta =0.1$ . Fig. 1 draws the lower bounds for indeterminacy for different values of $s$ . The values of $s=0$ and $1$ correspond to the two cases considered in SGU (1997), where $s=0$ corresponds to the polar case with endogenous labor but fixed capital income tax rates. To get a practical grab of our results, note that, in both cases ( $s=0,$ $1$ ), SGU (Reference Schmitt-Grohé and ín Uribe1997) find that the labor and capital income tax rates for the year 1988 estimated by Mendoza et al. (Reference Mendoza, Razin and Tesar1994) for USA, UK, Germany (G), Canada (C), and Japan (J), which also are the five economies studied by Giannisarou (Reference Giannisarou2007), all fall inside the range of values for which the equilibrium is indeterminate (the instability region). However, as can be seen from Fig. 1, the stability region can get larger as $s$ increases, and it is possible that a country in the instability region when $s=1$ may fall in the stability region for larger values of $s$ depending on its economic characteristics, and especially on the levels of the two tax rates.Footnote ⁶

Table 1. Estimated effective tax rates $^{\textrm a}$ and threshold value of $s$

^aEstimated effective factor income tax rates for 1988 are taken from Mendoza et al. (1994); updated estimates for 1996 areavailable at http://www.sas.upenn.edu/~egme/pp/newtaxdata.pdf.

Figure 1. Adjustments of factor income tax rates and (in)stability. S—stability, I—instability. The estimated tax rates are for 1988 and taken from Mendoza et al. (Reference Mendoza, Razin and Tesar1994).

Table 1 (for the 1988 data as shown on the left side of the table) lists the threshold value of $s$ for each of the five countries ( $s_{\min }$ ), above which the country falls in the stability region. As can be seen from the table, if $s\gt 2.21$ , then except for Germany, all the other four economies fall in the stability region. The table also makes it clear that this threshold value depends on the levels of the two factor income tax rates. For example, for the case of Germany, its labor income tax rate is high whereas its capital income tax rate is low, and hence a large $s$ is called for to ensure determinacy.Footnote ⁷

Table 2 compares our results with those in SGU (Reference Schmitt-Grohé and ín Uribe1997), as well as those in Giannisarou (Reference Giannisarou2007) who shows that the possibility of indeterminacy can be largely mitigated if capital income tax in the model of SGU (Reference Schmitt-Grohé and ín Uribe1997) is replaced with consumption expenditure tax, that is, if the labor income and consumption tax rates are adjusted endogenously to balance government budget.

Table 2. Stability properties for different countries

We also extend our analysis using data for the more recent years. The right side of Table 1 lists our results based on the 1996 tax rate estimates, updated by the same authors of Mendoza et al. (Reference Mendoza, Razin and Tesar1994) (available online from the authors). It is evident that the results concerning the threshold value of $s$ are generally similar to those obtained above using the 1988 tax rates. Our results based on the tax rates for the years 1995–2007 for US and some European countries estimated by Trabandt and Uhlig (Reference Trabandt and Uhlig2011) are broadly consistent: the threshold value is greater than $1$ , while its specific magnitude depends on country specifics and particularly on the levels of their capital and labor income tax rates; for example, $(\tau ^{l},\tau ^{k},$ $s_{\min })$ for USA, UK, Germany, Ireland, Portugal, and Denmark are $(0.281,0.364,1.951)$ , $(0.278,0.456,1.780)$ , $(0.412,0.233,12.413)$ , $(0.268,0.207,1.235)$ , $(0.313,0.234,3.673)$ , and $(0.474,0.506,10.091)$ , respectively. Note that for some European countries because of their tax structure and especially high labor income tax rates, the threshhold values of $s$ for them are relatively high.Footnote ⁸