# 经济代写|宏观经济学代写Macroeconomics代考|ECOS3007

#### Doug I. Jones

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• Statistical Inference 统计推断
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## 经济代写|宏观经济学代写Macroeconomics代考|The time profile of tax distortions

The combination of (18.5) and (18.6) implies that tax revenue as a share of output should be constant along a perfect foresight path:
$$\ell^{\prime}\left(\frac{\tau_t}{y_t}\right)=\lambda \text { for all } t \geq 0 .$$
We call this tax smoothing. The intuition is the same as in consumption smoothing. With the rate of interest equal to the rate at which loss is discounted, there is no incentive to have losses be higher in one moment than in another. The intuition is that, because the marginal distortion cost per unit of revenue raised is increasing in the tax rate (the ratio $\frac{\tau_f}{y_t}$ ), a smooth tax rate minimises distortion costs.
Denote the implicit tax rate by
$$\phi_t=\frac{\tau_t}{y_t}$$
Expression (18.8) says that along a perfect foresight path, the tax rate should be constant:
$$\phi_t=\frac{\tau_t}{y_t}=\phi \text { for all } t$$
This is known as the “tax smoothing” principle, and is the key result of the paper by Barro (1979). ${ }^1$ Notice also that, if $\lambda$ is constant and non-zero, the TVC (18.7) implies that
$$\lim _{T \rightarrow \infty}\left(d_T e^{-r T}\right)=0$$
That is, the solvency condition will hold with equality. Since the shadow value of debt is always positive, the government will choose to leave behind as much debt as possible (in present value) – that is to say, zero.

## 经济代写|宏观经济学代写Macroeconomics代考|The level of tax distortions

Solving (18.1) forward starting from some time 0 yields
$$d_T e^{-r T}=d_0+\int_0^T\left(g_t-\tau_t\right) e^{-r t} d t$$
Next, apply to this last equation the transversality/solvency condition (18.11) to get
$$\lim _{T \rightarrow \infty}\left(d_T e^{-r T}\right)=d_0+\int_0^{\infty}\left(g_t-\tau_t\right) e^{-r t} d t=0 .$$
Rearranging, this becomes
$$\int_0^{\infty} \tau_t e^{-r t} d t=d_0+\int_0^{\infty} g_t e^{-r t} d t$$
But, given (18.10), this can be rewritten as
$$\phi \int_0^{\infty} y_t e^{-r t} d t=d_0+\int_0^{\infty} g_t e^{-r t} d t$$
or
$$\phi=\frac{d_0+\int_0^{\infty} g_t e^{-r t} d t}{\int_0^{\infty} y_t e^{-r t} d t}$$
That is to say, the optimal flat tax rate equals the ratio of the present value of the revenue the government must raise to the present value of output.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|The time profile of tax distortions

(18.5) 和 (18.6) 的组合意味着税收收入占产出的比例应该 沿着完美的预见路径保持不变:
$$\ell^{\prime}\left(\frac{\tau_t}{y_t}\right)=\lambda \text { for all } t \geq 0 .$$

$$\phi_t=\frac{\tau_t}{y_t}$$

$$\phi_t=\frac{\tau_t}{y_t}=\phi \text { for all } t$$

$$\lim _{T \rightarrow \infty}\left(d_T e^{-r T}\right)=0$$

## 经济代写|宏观经济学代写Macroeconomics代考|The level of tax distortions

$$d_T e^{-r T}=d_0+\int_0^T\left(g_t-\tau_t\right) e^{-r t} d t$$

$$\lim _{T \rightarrow \infty}\left(d_T e^{-r T}\right)=d_0+\int_0^{\infty}\left(g_t-\tau_t\right) e^{-r t} d t=0$$

$$\int_0^{\infty} \tau_t e^{-r t} d t=d_0+\int_0^{\infty} g_t e^{-r t} d t$$

$$\phi \int_0^{\infty} y_t e^{-r t} d t=d_0+\int_0^{\infty} g_t e^{-r t} d t$$

$$\phi=\frac{d_0+\int_0^{\infty} g_t e^{-r t} d t}{\int_0^{\infty} y_t e^{-r t} d t}$$

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## MATLAB代写

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