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Proposition 1. There is a unique balanced growth equilibrium if labor externalities are relatively low (i.e.,η+β <1+ε), whereas there are two balanced growth equilibria if labor externalities are relatively high (i.e., η+β >1+ε). When labor externalities are relatively high, one of the balanced growth equilibria (the low-growth equilibrium associated with
Φ +
>
+β (1 ε)/
η ) is characterized by local determinacy and the other (the high-growth equilibrium associated with η+β <(1+ε)/Φ) is characterized by local indeterminacy.12
Based on Proposition 1 and Figure 2.2, it is quite clear that the Benhabib and Farmer (1994) condition (i.e., η+β >1+ε ) is a necessary but not sufficient condition for dynamic indeterminacy.
2.4. Balanced growth equilibrium
We are now in a position to examine how the anchor of inflation targeting and the income tax rate will affect macroeconomic performance under an inflation targeting regime. Given that
k c
x= / and along balanced growth n~ and x~ are constant (referring to (2.19a) and (2.19b)), the capital growth rate is identical to the real consumption growth rate. Moreover, given the production technology y=Aknη+β and n=n~ along balanced growth, we can infer that the output growth rate is equal to the capital growth rate. This implies that all c , k , and y grow at a common growth rate γ~ along the balanced growth path.
Given R&=0 in the balanced growth equilibrium, the non-arbitrage condition between physical capital and government bonds reported in equation (2.17c), and the Keynes-Ramsey rule reported in equation (2.17a), the common growth rate γ~ can be expressed as:
12 From (2.18a) and (2.18b), the slopes of loci n&=0 and AA are: n x
n J J
n
x/ /
0 =−
∂
∂ &= and
x
AA Fn F
n
x/∂ =− /
∂ . It is quite easy from equation (2.21) to infer that in Figure 2.2 η+β >(1+ε)/Φ holds at the low-employment equilibrium Q , where the 0 n&=0 schedule is flatter than the AA schedule. By contrast,
Φ +
<
+β (1 ε)/
η holds at the high-employment equilibrium Q′0, where the n&=0 schedule is steeper than the AA schedule.
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tend to encourage current capital accumulation, which in turn will stimulate the balanced growth rate.Differentiating equations (2.19a), (2.19b), and (2.22) with respect to π and τ and using Cramer’s rule gives rise to:
~ ] consumption-capital ratio, and the common growth rate. Equations (2.23d), (2.23e), and (2.23f)
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reveal that in the balanced growth equilibrium a rise in the income tax rate has ambiguous effects on labor employment, the consumption-capital ratio, and the common growth rate.13
A graphical presentation will be helpful to our understanding of the comparative results reported in equations (2.23a)-(2.23f). Figures 2.3 and 2.4 present the phase diagrams corresponding to the cases where labor externalities are relatively low (η+β <1+ε) and labor externalities are relatively high (η+β >1+ε), respectively. In Figure 2.3, in response to a rise in the inflation rate from π0 to π1, the AA(π0) schedule shifts leftward to AA(π1), and the balanced growth equilibrium changes from Q to 0 Q . At the new stationary equilibrium, both 1
n~ and x~ fall from n to 0 n and 1 x to 0 x , respectively, and, based on equation (2.22), the 1
common balanced growth rate is lowered from its initial level to a new level.
In Figure 2.4, following a rise in the inflation rate from π0 to π1, the AA(π0) schedule shifts rightward to AA(π1). Under the situation η+β >(1+ε)/Φ in which the economy is at the low-growth equilibrium Q initially, the balanced growth equilibrium changes from point 0
Q to 0 Q . At the new stationary equilibrium, both n1 ~ and x~ rise from n to 0 n and 1 x to 0 x , respectively, and, based on equation (2.22), the common balanced growth rate increases from 1
its initial level to a new level. On the contrary, under the situation η+β <(1+ε)/Φ in which the economy is at the high-growth equilibrium Q′0 initially, the balanced growth equilibrium
changes from point Q′0 to Q′1. At the new stationary equilibrium, both n~ and x~ fall from
n′0 to n′1 and x′0 to x′1, respectively, and, based on equation (2.22), the balanced growth rate
13 To make the comparative static results reported in equations (2.23a)-(2.23f) meaningful, we rule out the possibility of (η+β)Φ=1+ε in this chapter.
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is lowered from its initial level to a new level in response. The results in Figure 2.3 and Figure 2.4 can be described by the following proposition:
Proposition 2. When labor externalities are relatively low (η+β <1+ε), a rise in the anchor of the inflation rate definitely lowers the balanced growth rate. When labor externalities are relatively high (η+β >1+ε), at the low-growth equilibrium (η+β >(1+ε)/Φ) a rise in the inflation rate target boosts the balanced growth rate, while at the high-growth equilibrium (η+β <(1+ε)/Φ) a rise in the inflation rate target depresses the balanced growth rate.
The economic intuition behind Proposition 2 can be illustrated as follows. A rise in the inflation rate will induce the household to reduce its money holdings and, via the cash-in-advance constraint, consumption is lowered as well. Then, the household tends to decrease its labor supply (increase its leisure) in order to cope with the reduction in consumption.
If the degree of labor externality is relatively low, the lower labor supply will reduce the level of employment and depress the marginal product of physical capital. Following a reduction in the marginal product of physical capital, the household is inclined to cut down its saving, and hence the balanced growth rate will be lowered as a response. As a result, a rise in the anchor of inflation targeting is associated with a reduction in the balanced economic growth rate. By contrast, based on similar reasoning, we can infer that, if the degree of labor externality is large enough to make the labor demand curve positively sloped and steeper than the labor supply curve, the lower labor supply will raise the level of employment and boost the marginal product of physical capital. Accordingly, a rise in the anchor of inflation targeting is associated with an increase in the balanced economic growth rate.
Figures 2.5 and 2.6 illustrate the relationship between the income tax rate and the balanced growth rate. Figure 2.5 describes the situation where labor externalities are relatively low
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(η+β <1+ε), while Figure 2.6 describes the situation where labor externalities are relatively high (η+β >1+ε). In Figure 2.5, in association with a rise in the income tax rate from τ0 to
τ1, both AA(τ0) and n&=0(τ0) move leftward to AA(τ1) and n&=0(τ1). The balanced growth equilibrium changes from Q to 0 Q . At the new stationary equilibrium, n1 ~ falls from
n to 0 n but x1 ~ may either rise or fall depending upon the relative horizontal distance between )
(τ0
AA to AA(τ1) and n&=0(τ0) to n&=0(τ1). Based on equation (2.22), the balanced growth rate is lowered from its initial level to a new level.
In Figure 2.6, following a rise in the income tax rate from τ0 to τ1, the AA(τ0) schedule shifts rightward to AA(τ1), while the n&=0(τ0) schedule shifts leftward to n&=0(τ1). Under the situation η+β >(1+ε)/Φ in which the economy is at the low-growth equilibrium Q 0 initially, the balanced growth equilibrium moves from point Q to 0 Q . At the new stationary 1
equilibrium, both n~ and x~ rise from n to 0 n and 1 x to 0 x , respectively, and, based on 1 equation (2.22), the balanced growth rate is increased from its initial level to a new level. By contrast, under the situation η+β<(1+ε)/Φ in which the economy is at the high-growth equilibrium Q′0 initially, the balanced growth equilibrium moves from point Q′0 to Q′1. At the
new stationary equilibrium, both n~ and x~ fall from n′0 to n′1 and x′0 to x′1, respectively, and, based on equation (2.22), the balanced growth rate is lowered from its initial level to a new level. The results revealed in Figure 2.5 and Figure 2.6 can be summarized in the following proposition:
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Proposition 3. When labor externalities are relatively low (η+β <1+ε), a rise in the income tax rate definitely lowers the balanced growth rate. When labor externalities are relatively high (η+β >1+ε), at the low-growth equilibrium (η+β >(1+ε)/Φ) a rise in the income tax rate raises the balanced growth rate, while at the high-growth equilibrium (η+β <(1+ε)/Φ) a rise in the income tax rate lowers the balanced growth rate.
The economic intuition behind Proposition 3 can be understood as follows. A rise in the income tax rate will lower the net-of-tax real wage rate. In response to a fall in the net-of-tax real wage rate, the household tends to decrease its labor supply. If the degree of labor externality is relatively low, the lower labor supply will lead to a fall in the level of employment. The higher income tax rate and the lower employment will depress the net-of-tax marginal product of physical capital. In response to a fall in the net-of-tax marginal product of physical capital, the household is inclined to reduce its saving, and hence the balanced economic growth rate is lowered as a response. On the contrary, if the degree of labor externality is large enough so as to make the labor demand curve positively sloped and steeper than the labor supply curve, the lower labor supply will raise the level of employment and boost the net-of-tax marginal product of physical capital. As a consequence, a rise in the income tax rate is accompanied by an increase in the balanced economic growth rate.
Before ending this section, one point should be noted here. In their previous studies on monetary endogenous growth, Itaya and Mino (2003) and Suen and Yip (2005) find that, in the regime of money growth targeting, the dynamic adjustment of relevant macro variables is governed by two characteristic roots.14 Our analysis finds that, in the regime of inflation targeting, the dynamic adjustment of relevant macro variables is governed by a single characteristic root. As is evident, both inflation targeting and money growth targeting do not
14 If the monetary authority implements money growth targeting, the dynamic system of our model is also characterized by two characteristic roots. A detailed proof is available upon request to the authors.
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exhibit the same equilibrium dynamics. Moreover, equations (2.23a)-(2.23f) and the graphical analysis displayed in Figures 2.3-2.6 indicate that the single characteristic root plays the crucial role in determining the comparative results of adjusting inflation targeting and the income tax rate. We thus do not expect that our comparative results in (2.23d)-(2.23f) are robust when the monetary authority implements money growth targeting (rather than inflation targeting) since both regimes is characterized by different types of equilibrium dynamics.15