Steady state analysis

Table 1: Calibrated Parameters

Parameters Description Values

 Elasticity of substitution across individual brands 6

1 Inter-temporal elasticity of substitution 4

2 Elasticity of public consumption 4

 Capital share 0.36

 Discount factor 0.99

 Elasticity of labor supply 1

c,_L Tax rate on consumption and labor income 0.18, 0.2

 Average period of one year between price adjustments 0.75

 Capturing the relative valuation of public consumption 1

r Autocorrelation of interest rate 0.8

 Autocorrelation of inflation target 1.5

y Autocorrelation of output 0.2

G Persistence of the government expenditure 0.9

A Persistence of the productivity 0.9

bb The ratio of public debt to GDP 0.5

3.3 Steady state analysis

Given the parameters values above, we discuss the economic effect of the parameter bb on steady state and focus on the range between 0 and 1. First, we set

bb equal to 0.5 to discuss the main variables on the steady state in Table 2

According to the market clearing condition, the consumption plus government expenditure are equal to output level. The capital share is 0.3, in other words, the labor share is equal to 0.7. According to the production function above, the labor to the power of 0.7 are equal to output level. On the other hand, the tax are not equal to government expenditure are caused by bond issuing. The government expenditure must include the interest payments on bond, so the government consumption plus the interest payments on bond should equal to the tax level.

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Table 2: Primary variables in steady state values when bb0.5.

L Y C T W G Utility Welfare

1.0189 1.0121 0.77052 0.24665 0.52975 0.24154 -24.903 -2490.3

3.4. Long run effect of the debt ratio

Now, we discuss the effect of bb on the steady state. The analytical solutions for output and consumption in the steady state can be stated as below:

     

large amount of debt, in order to repay the interest on public debt, will have to issue more bonds for repayment. Moreover, the increase in the ratio of public debt to GDP will cause the consumption to increase.

Furthermore, the analytical tax and government spending solution on the steady state can be represented by:

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income tax. The effect of bb on consumption tax in the steady state is positive effect, but the income labor tax is negative effect. In the steady state, the taxes are equal to government spending plus the interest of public debt. We rewrite the form of government budget constraint to discuss the ratio of public debt to GDP. According to the Eq. (32), the government spending in the steady state is mainly affected by taxes and output, but Eq. (39) is difficult to see the effect on bb.

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Table 3: The primary numerical solution effect of bb on steady state 0

bb bb0.1 bb0.2 bb0.3 bb0.4 bb0.5 dYss

dbb

-0.00812 -0.00811 -0.00809 -0.00808 -0.00806 -0.00805

dCss

dbb

0.002823 0.002821 0.002819 0.002817 0.002814 0.002812

dTss

dbb

-0.00044 -0.00044 -0.00044 -0.00044 -0.00043 -0.00043

dWelfss

dbb

-232.038 -235.613 -239.254 -242.962 -246.737 -250.583

0.6

0.00281 0.002808 0.002806 0.002804 0.002801

dTss

dbb

-0.00043 -0.00043 -0.00043 -0.00043 -0.00043

dWelfss

dbb

-254.5 -258.489 -262.553 -266.692 -270.91

Following Schmitt-Grohé (2007) we use the expected lifetime utility of the representative household in period t as the welfare measure:

1 2

The analytical steady state solution of welfare can be state as:

1 2 ranging from 0 to 1. From the table 3, we can see that the relationship between taxes

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and bb in the steady state is negative. Consequently, the effect of labor income tax is larger than the effect of consumption tax in this range. We know that bb has the negative effect on output and positive effect on consumption. Moreover, we find the influence of bb on the output will decline with bb rises. In other words, the output will decrease less and less when bb rises. Otherwise, its influence on consumption is also lower with bb rises; it means the consumption will increase less and less when bb rises. There is no significant effect on taxes with the different bb values.

Furthermore, there is negative relationship between welfare and bb on the steady state. The result means the large debt that government accumulates will lead to greater welfare loss. The influence of bb on welfare will increase with bb rises.

3.5. Dynamic

All parameters values are listed in Table 1. We set the ratio of public debt to GDP 0.5

bb to match the facts of long term average level on the US. The exogenous process is assumed to follow first-order autoregressive process, and the persistence of productivity and government expenditure are established 0.9. Both of the standard deviations are set to 0.01.

3.5.1 Productivity Shock

Figure 1 shows the impulse response paths following a shock to the productivity in Eq. (23). Both output and consumption rise, while labor falls. The real wage rises more than the increase in the consumption. The result leads to lower taxes. Under the positive productivity shock, the price index fall (marginal cost lowers) and the interest rates fall (consumption rises for larger income). Marginal cost falls due to the positive productivity shock. The welfare rises intuitively.

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3.5.2 Government Expenditure Shock

Figure 2 shows the impulse response paths for an increase government spending in Eq. (22). The labor and output rise as a result of the expansionary fiscal policy. The labor rise, so that marginal cost of firms rises. The tax rises because the labor income tax rises larger than the decrease in the consumption tax. The consumption fall result from the tradeoff between consumption and labor. The price index rises because of the lower marginal cost. The interest rate rises result from lower consumption. According to the analysis above, the welfare rises consequently.

3.5.3 Welfare analysis

public consumption. The fraction ^aof steady state consumption that the household would decrease to be as well off under the steady state as under a regime a . Consequently, a higher value of ^a means higher welfare cost. Given the parameter value above, we evaluate the welfare criterion under bb ranging between 0 and 1 as below:

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Table 4: The welfare criterion

bb0 bb0.1 bb0.2 bb0.3 bb0.4 bb0.5

a 27.01331 27.34905 27.66804 28.02578 28.36738 28.71266

bb0.6 bb0.7 bb0.8 bb0.9 bb1.0

a 29.0568 29.40194 29.7533 30.1066 30.46062

Table 4 reports the welfare criterion. The ratio of public debt to GDP bb in this range, we find that the variable ^a rises continuously. It means that the welfare cost continuously rises. We find that the means of labor and output change irregularly when the value of bb is less than 0.5. If the value of bb is more than 0.5, the mean of labor and output fall smoothly with the rise in bb. Moreover, the mean of consumption is almost unchanged under different debt ratios to the GDP. The mean of government spending falls when the value of bb rises. The results imply that the welfare level is lowered with the rise in bb. Thus, the optimal debt ratio should be 0.

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4. Conclusion

In this paper, we investigate the optimal ratio of public debt to GDP by using a micro-based dynamic stochastic general equilibrium (DSGE) model. To construct the general equilibrium model, we discuss the optimal ratio of public debt to GDP. The discussions focus on the effects of the public debt to GDP ratio. We analyze the result of steady state in the short run and long run.

In the long run analysis, we find that the ratio of public debt to GDP has the negative effect on output and positive effect on consumption. The results are similar to the Euro debt crisis. The countries which hold large amount of debt, in order to repay the interest on public debt, will have to issue more bonds for repayment. The result shows the large debt that government accumulates will lead the welfare to decline. On the other hand, we discuss the welfare criterion in the short run. We find that the welfare cost rises continuously under the ratio of public debt to GDP ranging between 0 and 1. In other words, the optimal debt ratio should be 0.

The debt ratio to the GDP would be an important issue in the future. We conclude this paper by providing some issues for future research. First, the results are represented under the specification of parameter values and may change for different parameters. Second, the model that we build is a closed economy. The situation of real world is almost open economy. The discussion neglects some important variables under the specification of model. The result may change for the new variables.

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References

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Master’s thesis, University of Donghwa, Hualien, Taiwan. (in traditional Chinese) Dalamages B. (2000), "Public Sector and Economic Growth: The Greek Experience."

Applied Economics, 32(3), 277-288.

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Globalization and Monetary Policy Institute Working Paper No. 44.

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Schmitt-Grohé , S. and M. Uribe (2007), "Optimal Simple and Implementable Monetary and Fiscal Rules," Journal of Monetary Economics, 54(6), 1702-1725.

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Appendix 1: Optimal price setting

When setting a new price in period t firm seeks to maximize the current value of its dividend stream, conditional on that price being effective:

           

¹¹

Subject to the sequence of demand constraints:

 

t

 

Here the problem is written as maximizing real profits discounted by the stochastic discount factor as well as the probability of being able to make price changes.

 

         

The first order condition for this problem is:

         

¹

Finally, we get the flexible price:

     

^{ }

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Appendix 2: The analytical solution on steady state

We discuss the effect of bb on the steady state. The influence of mainly variables is represented below:

In addition, we define:

     

sing of this formula first. According to the parameter values in Table 1, the formula

 is negative. The sign of other formula is not difficult to see and then we can get the result as below:

0

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values to determine the sign of formula. The influence of other mainly variable can be state as:

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Appendix 3: The graph of shocks

Figure 1: Impulse responses following a shock to productivity

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Figure 2: Impulse responses following a shock to government spending

在文檔中 政府公債佔GDP的最適比率 - 政大學術集成 (頁 19-0)