经济增长|ECON8050 Economic Growth代写 anu代写

To compute $\beta$, we have to rewrite the growth rate as a function of $\log (\hat{k})$ :
$$\dot{\hat{k}} / \hat{k}=s A \cdot e^{-(1-\alpha) \cdot \log (\hat{k})}-(x+n+\delta)$$
We can take the derivative of equation (1.43) with respect to $\log (\hat{k})$ to get an expression for $\beta$ :
$$\beta=(1-\alpha) \cdot s A \cdot(\hat{k})^{-(1-\alpha)}$$
Notice that the speed of convergence is not constant but, rather, declines monotonically as the capital stock increases toward its steady-state value. At the steady state, $s A \cdot(\hat{k})^{-(1-\alpha)}=$ $(x+n+\delta)$ holds. Therefore, in the neighborhood of the steady state, the speed of convergence equals
$$\beta^{}=(1-\alpha) \cdot(x+n+\delta)$$ During the transition to the steady state, the convergence rate, $\beta$, exceeds $\beta^{}$ but declines over time.

Another way to get the formula for $\beta^{}$ is to consider a log-linear approximation of in the neighborhood of the steady state: $$\dot{\hat{k}} / \hat{k} \cong-\beta^{} \cdot\left[\log \left(\hat{k} / \hat{k}^{*}\right)\right]$$

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