### 经济代写|宏观经济学代写Macroeconomics代考|ECON6002

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## 经济代写|宏观经济学代写Macroeconomics代考|Transitional dynamics

Moreover, the Solow model does generate growth in the transition to the BGP. To see the implications in this regard, note that dividing both sides of (2.11) by $k$ implies that the growth rate of $k$ is given by
$$\gamma_{k} \equiv \frac{\dot{k}}{k}=\frac{s \cdot f(k)}{k}-(n+\delta) .$$
Equation (2.15) says that $\gamma_{k}$ equals the difference between two terms, $s \cdot f(k) / k$ and $(n+\delta)$ which we plot against $k$ in Figure 2.5. The first term is a downward-sloping curve, which asymptotes to infinity at $k=0$ and approaches 0 as $k$ tends to infinity. The second term is a horizontal line crossing the vertical axis at $n+\delta$. The vertical distance between the curve and the line equals the growth rate of capital per person, and the crossing point corresponds to the BGP. Since $n+\delta>0$ and $s \cdot f(k) / k$ falls monotonically from infinity to 0 , the curve and the line intersect once and only once. Hence (except for the trivial solution $k^{}=0$, where capital stays at zero forever), the BGP capital-labour ratio $k^{}>0$ exists and is unique.
Note also that output moves according to
$$\frac{\dot{y}}{y}=\alpha \frac{\dot{k}}{k}=\alpha \gamma_{k} .$$
A formal treatment of dynamics follows. From (2.11) one can calculate
$$\frac{d \dot{k}}{d k}=s \cdot f^{\prime}(k)-(n+\delta) .$$
We want to study dynamics in the neighbourhood of the BGP, so we evaluate this at $k^{}$ : $$\left.\frac{d \dot{k}}{d k}\right|_{k=k^{}}=s \cdot f^{\prime}\left(k^{*}\right)-(n+\delta) .$$

## 经济代写|宏观经济学代写Macroeconomics代考|Policy experiments

Suppose that the economy is initially on a BGP with capital per person $k_{1}^{}$. Imagine that the government then introduces some policy that raises the saving rate permanently from $s_{1}$ to a higher value $s_{2}$. Figure $2.6$ shows that the $s \cdot f(k) / k$ schedule shifts to the right. Hence, the intersection with the $n+\delta$ line also shifts to the right, and the new BGP capital stock, $k_{2}^{}$, exceeds $k_{1}^{*}$. An increase in the saving rate generates temporarily positive per capita growth rates. In the long run, the levels of $k$ and $y$ are permanently higher, but the per capita growth rates return to 0 .

A permanent improvement in the level of the technology has similar, temporary effects on the per capita growth rates. If the production function, $f(k)$, shifts upward in a proportional manner, then the $s \cdot f(k) / k$ curve shifts upward, just as in Figure 2.6. Hence, $\gamma_{k}$ again becomes positive temporarily. In the long run, the permanent improvement in technology generates higher levels of $k$ and $y$, but no changes in the per capita growth rates.

## 经济代写|宏观经济学代写Macroeconomics代考|Transitional dynamics

$$\gamma_{k} \equiv \frac{\dot{k}}{k}=\frac{s \cdot f(k)}{k}-(n+\delta)$$

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## MATLAB代写

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