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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

经济代写|宏观经济学代写Macroeconomics代考|The TVC and the consumption function

We must now ask the following question. Are we sure the BGP is optimal? If $A>\rho$, the BGP implies that the capital stock will be growing forever. How can this be optimal? Would it not be better to deplete the capital stock? More technically, is the BGP compatible with the TVC? Since we did not use it in constructing the BGP, we cannot be sure. So, next we check the BGP is indeed optimal in that, under some conditions, it does satisfy the TVC.
Using (5.24) the TVC can be written as
$$\lim {t \rightarrow \infty}\left(k_t c_t^{-\frac{1}{\pi}} e^{-\rho t}\right)=0 .$$ Note next that equation (5.29) is a differential equation which has the solution $$c_t=c_0 e^{\sigma(A-\rho) t} .$$ Combining the last two equations the TVC becomes $$\lim {t \rightarrow \infty}\left(k_t c_0^{-\frac{1}{\sigma}} e^{-A t}\right)=0 .$$
From the solution to expression (5.31) we have
$$k_t=k_0 e^{\sigma(A-\rho) t} .$$
Using this to eliminate $k_T$, the TVC becomes
$$\lim {t \rightarrow \infty}\left(k_0 c_0^{-\frac{1}{\sigma}} e^{\sigma(A-\rho) t} e^{-A t}\right)=\lim {t \rightarrow \infty}\left(k_0 c_0^{-\frac{1}{\sigma}} e^{-\theta t}\right)=0,$$
where
$$\theta \equiv(1-\sigma) A+\sigma \rho .$$
Hence, for the TVC we need $\theta>0$, which we henceforth assume. Note that with logarithmic utility $(\sigma=1), \theta=\rho$

经济代写|宏观经济学代写Macroeconomics代考|The permanent effect of transitory shocks

In the AK model, as we have seen, growth rates of all pertinent variables are given by $\sigma(A-\rho)$. So, if policy can affect preferences $(\sigma, \rho)$ or technology $(A)$, it can affect growth.

If it can do that, it can also affect levels. From the production function, in addition to (5.31) and (5.32), we have
$$k_t=k_0 e^{\sigma(A-\rho) t},$$
Clearly, changes in $\sigma_{,} \rho$ and $A$ matter for the levels of variables.

Notice here that there is no convergence in per capita incomes whatsoever. Countries with the same $\sigma, \rho$, and $A$ retain their income gaps forever. ${ }^4$

Consider the effects of a sudden increase in the marginal product of capital $A$, which suddenly and unexpectedly rises (at time $t=0$ ), from $A$ to $A^{\prime}>A$. Then, by (5.31), the growth rate of all variables immediately rises to $\sigma\left(A^{\prime}-\rho\right)$.

What happens to the levels of the variables? The capital stock cannot jump at time 0 , but consumption can. The instant after the shock $\left(t=0^{+}\right)$, it is given by
$$c_{0^{+}}=\left[(1-\sigma) A^{\prime}+\sigma \rho\right] k_{0^{+}}>c_0=[(1-\sigma) A+\sigma \rho] k_{0^{+}},$$
where $k_{0^{+}}=k_0$ by virtue of the sticky nature of capital.
So, consumption rises by $(1-\sigma)\left(A^{\prime}-A\right) k_0$. But, output rises by $\left(A^{\prime}-A\right) k_0$. Since output rises more than consumption, growth picks up right away.

It turns out that the $\mathrm{AK}$ model has very different implications from the Neoclassical Growth Model when it comes to the effects of transitory shocks. To see that, consider a transitory increase in the discount factor, i.e. suppose $\rho$ increases for a fixed interval of time; for simplicity, assume that the new $\rho$ is equal to $A$.

Figure $5.2$ shows the evolution of the economy: the transitory increase in the discount rate jolts consumption, bringing growth down to zero while the discount factor remains high. When the discount factor reverts, consumption decreases, and growth restarts. But there is a permanent fall in the level of output relative to the original path. In other words, there is full persistence of shocks, even if the shock itself is temporary. You may want to compare this with the Neoclassical Growth Model trajectories (Figure 5.3), where there is catch-up to the original path and there are no long-run effects.

宏观经济学代考

经济代写|宏观经济学代写Macroeconomics代考|The TVC and the consumption function

$$\lim t \rightarrow \infty\left(k_t c_t^{-\frac{1}{\pi}} e^{-\rho t}\right)=0$$

$$c_t=c_0 e^{\sigma(A-\rho) t} .$$

$$\lim t \rightarrow \infty\left(k_t c_0^{-\frac{1}{\sigma}} e^{-A t}\right)=0$$

$$k_t=k_0 e^{\sigma(A-\rho) t} .$$

$$\lim t \rightarrow \infty\left(k_0 c_0^{-\frac{1}{\sigma}} e^{\sigma(A-\rho) t} e^{-A t}\right)=\lim t \rightarrow \infty\left(k_0 c_0^{-\frac{1}{\sigma}} e^{-\theta t}\right)=0$$

$$\theta \equiv(1-\sigma) A+\sigma \rho .$$

经济代写|宏观经济学代写Macroeconomics代考|The permanent effect of transitory shocks

$$k_t=k_0 e^{\sigma(A-\rho) t},$$

$$c_{0^{+}}=\left[(1-\sigma) A^{\prime}+\sigma \rho\right] k_{0^{+}}>c_0=[(1-\sigma) A+\sigma \rho] k_{0^{+}},$$

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