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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
2.5. Second-best policies
We now turn to deal with the welfare effect of monetary and fiscal policies. Along the balanced growth equilibrium, with a given initial private capital stock k , private consumption grows at a 0 common rate γ~ (which is a function of π and τ ) and labor supply stays at its steady-state value n~ (which is a function of π and τ). The time path of private consumption and labor supply can thus be expressed as
t t
t c e xk e
c = 0 γ~ =~ 0 γ~ , (2.24a)
n
nt =~, (2.24b)
where c is endogenously determined as shown below by the economy’s structure. By dividing 0 equation (2.17d) by k , it is quite easy to infer that along the balanced growth equilibrium the balanced growth rate γ~ can be written as:
x n
A~ ~
~= η+β −
γ . (2.24c)
Substituting equations (2.24a), (2.24b) and (2.24c) into equation (2.8), the welfare function (i.e., indirect lifetime utility function) is given by the following expression:
15 The inflation rate is an endogenous variable under the regime of money growth targeting, while it is an exogenous variable under the regime of inflation targeting. So, we cannot examine whether the comparative results of adjusting the inflation rate reported in (23a)-(23c) are robust in the regime of money growth targeting.
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We can then use equation (2.25) to deal with how the fiscal and monetary authorities choose a package of the income tax rate τ and inflation targeting π to maximize welfare.
Differentiating the welfare function with respect to τ and π yields:
τ employment and the consumption-capital ratio associated with second-best optimal policies τ* and π*, namely, n and ~* ~ : x∗
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Then, by substituting equations (2.27a), (2.27b), (2.28a) and (2.28b) into (2.22) and (2.17c) , we can further infer the steady-state values of the nominal interest rate and the economic growth rate associated with second-best optimal policies τ* and π*, namely, R~∗ and γ~*:
We then discuss the second-order conditions for the welfare-maximizing problem. The second-order sufficient conditions for welfare maximization require that Uττ <0, Uππ <0 and
2 >0
16 Appendix B provides a detailed derivation for equations (2.30a)-(2.30c).
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Equations (2.30a)-(2.30c) indicate that the fiscal and monetary authorities could maximize the welfare function only when the degree of the labor externality is relatively small, i.e.,
ε β
η+ <1+ . This result can be summarized by the following proposition:
Proposition 4. Only when the degree of the labor externality is relatively low, i.e., ε
β
η+ <1+ , could the second-best optimal policies be chosen to fulfill the second-order sufficient condition for welfare maximization.
Provided that the second-order sufficient condition for welfare maximization ε
β
η+ <1+ is fulfilled, the second-best optimal policies reported in (2.27a), (2.27b) and (2.29a) reveal some interesting findings, as follows.
The first scenario we deal with can be regarded as a benchmark case, since it can be conveniently used to draw comparisons with other cases. The benchmark case considers the situation where the intermediate goods market is perfectly competitive (θ =0) and both capital and labor externalities are absent (α =1 and η=0). To be more specific, the economy the benchmark case takes care of is a Walrasian market-clearing economy. It is clear from equations (2.27a), (2.27b) and (2.29a) with θ =0, α =1 and η=0 that the optimal income tax, inflation and nominal interest rates are τ∗=0 , π∗=−A(βA/ρ)β/(1+ε−β) , and R~∗ =0 , respectively. This result indicates that, in a Walrasian economy without any distortions, there is no need to tax/subsidize income at all. Moreover, the inflation rate targeting that generates a zero nominal interest rate is optimal, implying that the Friedman rule is optimal. The economic intuition behind the optimality of the Friedman rule is quite simple. Because there is a disparity between the private marginal cost of holding money (which is the nominal interest rate) and the social marginal cost of producing money (which is essentially negligible), a positive nominal interest rate will generate inefficiency losses for the society. Accordingly, to achieve a welfare
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maximum, the monetary authority should adopt inflation rate targeting that generates a zero nominal interest rate in order to eliminate the private opportunity costs.
One point should be noted here. Most of the existing studies on the Friedman rule, e.g., Guidotti and Végh (1993), Correia and Teles (1999), Ireland (2003), and Ho et al. (2007), confine their analysis to the implementation of the monetary growth rate.17 In departing from these existing studies, this chapter instead deals with the optimality of the Friedman rule via the implementation of inflation targeting.18 The above result regarding the benchmark case leads to the following proposition:
Proposition 5: If the intermediate goods market is perfectly competitive (θ =0) and both capital and labor externalities are absent (α =1 and η=0), the second-best policy for the income tax rate is zero and the second-best policy for inflation rate targeting is chosen so as not to deviate from the Friedman rule of a zero nominal interest rate.
Second, if the intermediate goods market is imperfectly competitive (0<θ <1) and both capital and labor externalities are absent (α =1 and η=0), then optimal income tax, inflation and nominal interest rates are τ∗=−θ/(1−θ)<0, π∗ =−A(βA/ρ)β/(1+ε−β) , and R~∗ =0, respectively. This result reveals that the optimal fiscal policy (i.e., optimal income tax) plays a unique and decisive role in correcting production inefficiency stemming from market imperfections. Due to the fact that the optimal tax rate can completely remedy production
17 In his well-cited survey paper, Woodford (1990, p. 1071) defines the Friedman rule (in strong from) as follows:
“The best monetary policy, from the point of view of maximizing the welfare of consumers (the welfare of some representative consumer or, presumably, some measure of average welfare in the case of heterogeneous consumers), would be to maintain a rate of growth of the money supply that is so low (quite likely involving steady contraction of the money supply) as to make the nominal interest rate (on assets with riskless nominal return, such as a Treasury security) as low as possible (typically, zero).”
18 To the best of our knowledge, the only exception which mentions the Friedman rule via the implementation of inflation targeting is Mansoorian and Mohsin (2004). However, their analysis is based on an exogenous growth model, and hence the growth effect of consumption in the indirect utility (welfare) function is excluded in their inference.
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inefficiency caused by imperfect competition, optimal inflation rate targeting is thus maintained at the rate that is conformable to the Friedman rule (i.e., R~∗ =0).
Why can the single second-best policy for the income tax rate completely remedy market imperfections? The intuition can be explained as follows. Given that the intermediate goods market is monopolistically competitive, the private rates of return on capital investment and labor input will be lower than their social rates in the same proportions (referring to equations (2.7a) and (2.7b)). This leads the firm to employ too little capital and labor relative to the optimum. To restore the optimal levels of capital and labor, a subsidy on income (including labor income and capital income) will increase the household’s net-of-tax reward for providing capital and labor in the same proportions. This motivates the household to raise capital investment and the level of employment in the same proportions. Accordingly, a uniform subsidy on capital and labor income is required to correct the same proportionate underemployment of capital and labor arising from imperfect competition.
In their recent paper, Schmitt-Grohé and Uribe (2004) suggest that the Friedman rule is optimal under perfect competition, while it is unlikely to be optimal under imperfect competition.
Our result reveals that, in the presence of imperfect competition, the Friedman rule is still optimal when a second-best uniform tax rate is levied on labor and capital incomes. However, we should emphasize that this result is known in the literature due to Shaw et al. (2006). As explained previously, a single second-best policy for the income tax rate can completely eliminate both distortions on labor and capital caused by imperfect competition. There is therefore no need to utilize the policy of inflation rate targeting to remedy the distortions stemming from imperfect competition, and hence deviation from the Friedman rule is not optimal. Based on the above discussion, the following proposition is immediately established:
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Proposition 6: If the intermediate goods market is imperfectly competitive (0<θ <1) and both capital and labor externalities are absent (α =1 and η=0), the second-best policy for the income tax rate is negative and the second-best policy for the inflation rate is selected to implement the Friedman rule of a zero nominal interest rate.
Third, if the intermediate goods market is perfectly competitive (θ =0) and the labor externality is absent (η=0), but the capital externality is present (0<α<1), then the optimal income tax, inflation and nominal interest rates are τ∗ =−(1−α)/α <0, R~∗ =1/α−1>0, and
1 )
/ ( /
1 − /(1 ) −
= + −
∗ α β ρ β ε β
π A A , respectively. This result indicates that, when the production sector is subject to capital inefficiency in the form of learning-by-doing, both instruments of the income tax rate and inflation targeting should be simultaneously used to improve the level of welfare. To be more specific, in the presence of distortion arising from the capital externality, the Friedman rule fails to be optimal.
Given the above result regarding capital inefficiency, one question naturally arises. Why should the second-best income tax and inflation targeting policies both be simultaneously utilized (i.e., τ∗<0 and π∗ >−A(βA/ρ)β/(1+ε−β) in association with R~∗>0) to correct a single distortion caused by the capital externality? With the capital externality in the form of learning-by-doing, the private rate of return on capital investment will be lower than its social rate, but the private rate of return on the labor input will be equal to its social rate. This leads the firm to hire too little capital relative to the second-best optimum. In order to correct the undue level of capital, a subsidy on income (including labor income and capital income) should be utilized to remove the distortion. However, a subsidy on income (including labor income and capital income) raises the private rates of return on capital investment and labor at the same time, and hence leads to an increase in capital investment and the level of employment. Accordingly,
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although a subsidy on income corrects the capital investment to the optimal level, it also leads employment to exceed the social optimal level. Faced with such a dilemma, the second-best inflation rate that exceeds the rate associated with the Friedman rule should be adopted to correct the unduly high level of employment.19 Summing up the discussion, we can establish the following proposition:
Proposition 7: If the intermediate goods market is perfectly competitive (θ =0) and the labor externality is absent (η=0), but the capital externality is present (0<α <1), the second-best policy for the income tax rate is negative and the second-best policy for inflation rate targeting is chosen to deviate from the Friedman rule.
Fourth, if the intermediate goods market is perfectly competitive (θ =0) but both capital and labor externalities are present (0<α<1 and η>0), then the optimal income tax, inflation , and nominal interest rates are τ∗ =−(1−α)/α <0 , R~*={β/[α(η+β)]}−1≥0 , and
1 ]
/ ) [(
)]}
( /[
{ + − + ( )/(1 )−
= + +− −
∗ β α η β η β ρ η β ε η β
π A A , respectively.20 This result indicates that,
when both labor and capital externalities are present at the same time, both instruments of the income tax rate and inflation targeting should be simultaneously used to improve the level of welfare. To be more specific, the Friedman rule may be no longer optimal when the distortions arise from labor and capital externalities.
This economic intuition behind the optimality of the Friedman rule is related to that of Proposition 7. Proposition 7 indicates that, when the intermediate goods market is perfectly competitive (θ =0), to remedy the inefficiency arising from the single capital externality the
19 A rise in the inflation rate raises the opportunity cost of holding money. Accordingly, the household will reduce its money holdings and, via the cash-in-advance constraint, consumption will be lowered as well. Then, the household will tend to lower it labor supply (increase its leisure) in order to cope with the decrease in consumption (referring to (2.23a)).
20 In this chapter, we rule out the possibility of R~∗<0.
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second-best policy for the income tax rate is negative and the second-best policy for inflation rate targeting involves generating a positive nominal interest rate. However, in this case the labor externality is also present. Faced with such an externality, the private rate of return on labor input will be lower than its social rate. This leads the firm to employ too little labor compared the second-best optimum. In order to correct the undue level of labor, the monetary authority will lower the inflation rate (and hence lower the nominal interest rate) as its response. Consequently, the second-best policy for inflation rate targeting will generate a positive nominal interest rate if the adjustment of R stemming from capital externality is greater than that stemming from labor externality. Otherwise, the second-best policy for inflation rate targeting will generate a zero nominal interest rate. The main result of the above discussion can be summarized by the following proposition:
Proposition 8: If the intermediate goods market is perfectly competitive (θ =0) and both capital and labor are present at the same time (0<α <1 and η >0), the second-best policy for the income tax rate is negative and the second-best policy for the inflation rate may be chosen to deviate from the Friedman rule.
Fifth, if the intermediate goods market is imperfectly competitive (0<θ <1) and the capital externality is present (0<α <1), but the labor externality is absent (η=0), then the optimal income tax, inflation and nominal interest rates are τ∗ =1−1/[α(1−θ)]<0, R~∗ =1/α−1>0,
and π∗ =1/α−A(βA/ρ)β/(1+ε−β)−1, respectively. The economic intuition for the mix of the second-best policies for the income tax rate and inflation targeting can be easily inferred from both Proposition 6 and Proposition 7, and thus we do not repeat them here to save space.
Finally, if the intermediate goods market is imperfectly competitive (0<θ <1) and both capital and labor externalities are present (0<α <1 and η>0), then the optimal income tax
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rate, the inflation rate, and the nominal interest rate are τ∗ =1−1/[α(1−θ)]<0 , 1
] / ) [(
)]}
( /[
{ + − + ( )/(1 )−
= + +− −
∗ β α η β η β ρ η β ε η β
π A A , and R~* ={β/[α(η+β)]}−1≥0 , respectively. The economic intuition regarding the mix of second-best policies can be easily understood from both Proposition 6 and Proposition 8, and we thus do not repeat it here.