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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 dynamics of the capital stock

The capital accumulation equation (8.12), together with the factor market equilibrium conditions (8.9) and (8.10), implies the dynamic behaviour of the capital stock:
$$k_{t+1}=\frac{s\left[w\left(k_t\right), r\left(k_{t+1}\right)\right]}{1+n}$$

or
$$k_{t+1}=\frac{s\left[f\left(k_t\right)-k_t f^{\prime}\left(k_t\right), f^{\prime}\left(k_{t+1}\right)\right]}{1+n} .$$
This last equation implies a relationship between $k_{t+1}$ and $k_{\mathrm{r}}$. We will describe this as the savings locus. The properties of the savings locus depend on the derivative:
$$\frac{d k_{t+1}}{d k_t}=\frac{-s_W\left(k_t\right) k_t f^{\prime \prime}\left(k_t\right)}{1+n-s_r\left(k_{t+1}\right) f^{\prime \prime}\left(k_{t+1}\right)} .$$
The numerator of this expression is positive, reflecting the fact that an increase in the capital stock in period $t$ increases the wage, which increases savings. The denominator is of ambiguous sign because the effects of increases in the interest rate on savings are ambiguous. If $s_r \geq 0$, then the denominator in (8.15) is positive, and then so is $d k_{t+1} / d k_t$.

The savings locus in Figure $8.2$ summarises both the dynamic and the steady-state behaviour of the economy. The 45-degree line in Figure $8.2$ is the line along which steady states, at which $k_{t+1}=k_t$, must lie. Any point at which the savings locus $s$ crosses that line is a steady state. We have drawn a locus that crosses the 45-degree line only once, and hence guarantees that the steady state capital stock both exists and is unique. But this is not the only possible configuration. The model does not, without further restrictions on the utility and/or production functions, guarantee either existence or uniqueness of a steady-state equilibrium with positive capital stock.

If there exists a unique equilibrium with positive capital stock, will it be stable? To answer this, evaluate the derivative around the steady state:
$$\left.\frac{d k_{t+1}}{d k_t}\right|_{S s}=\frac{-s_w k^* f^{\prime \prime}\left(k^\right)}{1+n-s_v f^{\prime \prime}\left(k^\right)} .$$

## 经济代写|宏观经济学代写Macroeconomics代考|A workable example

In this sub-section, we analyse the properties of the OLG model under a fairly simple set of assumptions: $\log$ utility (i.e. the limit case where $\sigma=1$ ) and Cobb-Douglas production. (This is sometimes referred to as the canonical OLG model.) This permits a simple characterisation of both dynamics and the steady state.
With this assumption on preferences, the saving function is
$$s_t=\left(\frac{1}{2+\rho}\right) w_t,$$
so that savings is proportional to wage income. Notice that the interest rate cancels out in the case of log utility, but not otherwise. This is a case in which the savings rate will be constant over time (as in the Solow model), though, once again, here this is the result of an optimal choice (as in the version of the AK model that we studied in Chapter 5 ).
With Cobb-Douglas technology, the firm’s rules for optimal behaviour (8.9) and (8.10) become
$$r_t=\alpha k_t^{a-1}$$
and
$$w_t=(1-\alpha) k_t^a=(1-\alpha) y_{t^}$$ Using (8.17) and (8.19) in (8.12) yields $$k_{t+1}=\left(\frac{1-\alpha}{2+\rho}\right)\left(\frac{1}{1+n}\right) k_t^s,$$ which is the new law of motion for capital. Define as usual the steady state as the situation in which $k_{t+1}=k_t=k^$. Equation (8.20) implies that the steady state is given by
$$k^*=\left(\frac{1-\alpha}{2+\rho} \frac{1}{1+n}\right)^{\frac{1}{1-a}},$$
so that we have a unique and positive steady-state per-capita capital stock. This stock is decreasing in $\rho$ (the rate of discount) and $n$ (the rate of population growth). Note the similarities with the NGM and the Solow model.

# 宏观经济学代考

## 经济代写|宏观经济学代写宏观经济学代考|资本存量的动态

$$k_{t+1}=\frac{s\left[w\left(k_t\right), r\left(k_{t+1}\right)\right]}{1+n}$$

$$k_{t+1}=\frac{s\left[f\left(k_t\right)-k_t f^{\prime}\left(k_t\right), f^{\prime}\left(k_{t+1}\right)\right]}{1+n} .$$

$$\frac{d k_{t+1}}{d k_t}=\frac{-s_W\left(k_t\right) k_t f^{\prime \prime}\left(k_t\right)}{1+n-s_r\left(k_{t+1}\right) f^{\prime \prime}\left(k_{t+1}\right)} .$$

$$\left.\frac{d k_{t+1}}{d k_t}\right|_{S s}=\frac{-s_w k^* f^{\prime \prime}\left(k^\right)}{1+n-s_v f^{\prime \prime}\left(k^\right)} .$$

## 经济代写|宏观经济学代写宏观经济代考|一个可行的例子

$$s_t=\left(\frac{1}{2+\rho}\right) w_t,$$
，因此储蓄与工资收入成比例。注意，在日志效用的情况下，利率抵消了，但在其他情况下没有。在这种情况下，储蓄率将随着时间的推移保持恒定(就像在索洛模型中那样)，尽管如此，这里这是一个最优选择的结果(就像我们在第5章中学习的AK模型的版本)。使用Cobb-Douglas技术，公司的最佳行为规则(8.9)和(8.10)变成
$$r_t=\alpha k_t^{a-1}$$

$$w_t=(1-\alpha) k_t^a=(1-\alpha) y_{t^}$$在(8.12)中使用(8.17)和(8.19)产生$$k_{t+1}=\left(\frac{1-\alpha}{2+\rho}\right)\left(\frac{1}{1+n}\right) k_t^s,$$，这是资本的新运动定律。像往常一样将稳定状态定义为$k_{t+1}=k_t=k^$。式(8.20)表明稳态由
$$k^*=\left(\frac{1-\alpha}{2+\rho} \frac{1}{1+n}\right)^{\frac{1}{1-a}},$$

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