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

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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 steady state consumption and current account

Now that you have the level of income you should be able to compute the level of consumption. How do we do that? By solving the differential equation that is the budget constraint (4.9), which we can rewrite as
$$\dot{a}_t-r a_t=f\left(k^\right)-r k^-c^,$$ using the solutions for optimal consumption and capital stock. Using our strategy of integrating factors, we can multiply both sides by $e^{-r t}$, and integrate the resulting equation between 0 and $t$ : $$a_t e^{-r t}-a_0=\frac{c^+r k^-f\left(k^\right)}{r}\left(e^{-r t}-1\right) .$$
Now evaluate this equation as $t \rightarrow \infty$. Considering the NPC and the TVC, it follows that:
$$c^=r a_0+f\left(k^\right)-r k^* .$$
We can also find the optimal level of debt at each time period. It is easy to see that $a_t$ is kept constant at $a_0$, from which it follows that $b_t=b_0+k_0-k^*$. The current account is zero. In other words, the NGM delivers a growth model with no growth, as we saw in the last chapter, and a model of the current account dynamics without current account surpluses or deficits.

Not so fast, though! We saw that the NGM did have predictions for growth outside of the BGP. Let’s look at the transitional dynamics here as well, and see what we can learn.

## 经济代写|宏观经济学代写Macroeconomics代考|The inexistence of transitional dynamics

There are no transitional dynamics in this model: output per capita converges instantaneously to that of the rest of the world!

Suppose that initial conditions are $k_0<k^$ and $b_0>0$. But, condition (4.19) says that capital must always be equal to $k^$. Hence, in the first instant, capital must jump up from $k_0$ to $k^*$. How is this accomplished? Domestic residents purchase the necessary quantity of capital (the single good) abroad and instantaneously install it. Put differently, the speed of adjustment is infinite.

How do the domestic residents pay for this new capital? By drawing down their holdings of the bond. If $\Delta k_0=k^-k_0$, then $\Delta b_0=-\Delta k_0=-\left(k^-k_0\right)$. Note that this transaction does not affect initial net national assets, since
$$\Delta a_0=\Delta k_0+\Delta b_0=\Delta k_0-\Delta k_0=0 .$$

Suppose now that the production function is given by
$$f\left(k_t\right)=A k_t^a, A>0,0 \leq \alpha \leq 1 .$$
This means that condition (4.19) is
$$\alpha A\left(k^\right)^{\alpha-1}=r$$ so that the level of capital on the BGP is $$k^=\left(\frac{\alpha A}{r}\right)^{\frac{1}{1-\alpha}},$$
which is increasing in $A$ and decreasing in $r$.
Using this solution for the capital stock we can write $y^$ as $$y^=A k^{* \alpha}=A\left(\frac{\alpha A}{r}\right)^{\frac{\alpha}{1-\alpha}}=A^{\frac{1}{1-\alpha}}\left(\frac{\alpha}{r}\right)^{\frac{a}{1-\alpha}} \equiv z(A),$$
with $z(A)$ increasing in $A$.
It follows that consumption can be written as
$$c^=r u_0-r k^+z(A)=r u_0+(1-\alpha) z(A),$$
with $z^{\prime}(A)>0$.

## 经济代写|宏观经济学代写Macroeconomics代考|The steady state consumption and current account

$c^{\wedge}=r^{\prime} a_{-} 0+f$ lleft(k^\ight) $-\mathrm{k}^{\wedge} \mathrm{k}^{\star}$

## 经济代写|宏观经济学代写Macroeconomics代考|The inexistence of transitional dynamics

$$c^{=} r u_0-r k^{+} z(A)=r u_0+(1-\alpha) z(A)$$

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## MATLAB代写

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