经济代写|宏观经济学代写Macroeconomics代考|ECOS3007

Doug I. Jones

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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
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经济代写|宏观经济学代写Macroeconomics代考|Overlapping generations in continuous time

The trick to model the OLG model in a continuous-time framework is to include an age-independent probability of dying $p$. By the law of large numbers this will also be the death rate in the population. Assume a birth rate $n>p$. Together these two assumptions imply that population grows at the rate $n-p .^5$ This assumption is tractable but captures the spirit of the OLG model: not everybody is the same at the same time.

As in Blanchard (1985), we assume there exist companies that allow agents to insure against the risk of death (and, therefore, of leaving behind unwanted bequests). This means that at the time of death all of an individual’s assets are turned over to the insurance company, which in turn pays a return of $p$ on savings to all agents who remain alive. If $r_t$ is the interest rate, then from the point of view of an individual agent, the return on savings is $r_t+p$.

We will also assume logarithmic utility which will make the algebra easier. As of time $t$ the representative agent of the generation born at time $\tau$ maximises
$$\int_t^{\infty} \log c_{s, \tau} e^{-(\rho+p)(s-t)} d s$$
subject to the flow budget constraint
$$\dot{a}{t, \tau}=\left(r_t+p\right) a{t, \tau}+y_{t, \tau}-c_{t, \tau},$$
where $a_{t, \tau}$ is the stock of assets held by the individual and $y_{t, \tau}$ is labour income. The other constraint is the no-Ponzi game condition requiring that if the agent is still alive at time $s$, then
$$\lim {s \rightarrow \infty} a{s, r} e^{-\int_i^2\left(r_v+p\right) d v} \geq 0 .$$
If we integrate the first constraint forward (look at our Mathematical Appedix!) and use the second constraint, we obtain
$$\int_t^{\infty} c_{s, \tau} e^{-\int_t^x\left(r_\tau+p\right) d v} d s \leq a_{t, \tau}+h_{t, \tau}$$
where
$$h_{t, \tau}=\int_t^{\infty} y_{s, \tau} e^{-\int_t^x\left(r_v+p\right) d v} d s,$$
can be thought of as human capital. So the present value of consumption cannot exceed available assets, a constraint that will always hold with equality.
With log utility the individual Euler equation is our familiar
$$\dot{c}{s, \tau}=\left(r_s-\rho\right) c{s, \tau},$$
which can be integrated forward to yield
$$c_{s, \tau}=c_{t, \tau} e^{\int_t^x\left(r_v-\rho\right) d v} .$$

经济代写|宏观经济学代写Macroeconomics代考|The closed economy

We have not taken a stance on what kind of asset $a_t$ is. We now do so. In the closed economy we assume that $a_t=k_t$, and $k_t$ is per-capita productive capital that yields output according to the function $y_t=k_t^a$, where $0<\alpha<1$. In this context profit maximisation dictates that $r_t=\alpha k_t^{a-1}$, so that our two differential equations become \begin{aligned} &\dot{c}_t=\left(\alpha k_t^{a-1}-\rho\right) c_t-n(p+\rho) k_t, \ &\dot{k}_t=(1+\alpha) k_t^a-(n-p) k_t-c_t . \end{aligned} In steady state we have $$\begin{gathered} \frac{c}{k^2}=\frac{n(\rho+\rho)}{\alpha k^{-1}-\rho}, \ (1+\alpha) k^{ \alpha-1}-(n-p)=\frac{c^2}{k^v} \end{gathered}$$ Combining the two yields $$(1+\alpha) k^{* \alpha-1}=(n-p)+\frac{n(p+\rho)}{\alpha k^{* a-1-\rho}},$$ which pins down the capital stock. For given $k^$, the first SS equation yields consumption. Rewrite the last equation as $$\alpha k^{ \alpha-1}-\rho=\frac{n(p+\rho)}{(1+\alpha) k^{* \alpha-1}-(n-p)}>0 .$$
So the steady-state level of the (per capita) capital stock is smaller than the modified golden rule level that solves $\alpha k^{a-1}=\rho$, implying under-accumulation of capital. ${ }^7$ This is in contrast to the NGM, in which the modified golden rule applies, and the discrete-time OLG model with two-period lives, in which over-accumulation may occur. Before examining that issue, consider dynamics, described in Figure 8.5.

Along the saddle-path $c_t$ and $k_t$ move together. If the initial condition is at $k>k^$, then consumption will start above its SS level and both $c_t$ and $k_t$ will gradually fall until reaching the steady-state level. If, by contrast, the initial condition is at $k$, then consumption will start below its steadystate level and both $c_t$ and $k_t$ will rise gradually until reaching the steady state.

统计推断代考

经济代写|宏观经济学代写宏观经济学代考|连续时间的重叠代

$$\int_t^{\infty} \log c_{s, \tau} e^{-(\rho+p)(s-t)} d s$$

$$\dot{a}{t, \tau}=\left(r_t+p\right) a{t, \tau}+y_{t, \tau}-c_{t, \tau},$$
，其中$a_{t, \tau}$是个人持有的资产的股票，$y_{t, \tau}$是劳动收入。另一个约束是无庞氏博弈条件，要求如果代理在$s$时仍然活着，那么
$$\lim {s \rightarrow \infty} a{s, r} e^{-\int_i^2\left(r_v+p\right) d v} \geq 0 .$$

$$\int_t^{\infty} c_{s, \tau} e^{-\int_t^x\left(r_\tau+p\right) d v} d s \leq a_{t, \tau}+h_{t, \tau}$$
，其中
$$h_{t, \tau}=\int_t^{\infty} y_{s, \tau} e^{-\int_t^x\left(r_v+p\right) d v} d s,$$

$$\dot{c}{s, \tau}=\left(r_s-\rho\right) c{s, \tau},$$
，它可以向前积分得到
$$c_{s, \tau}=c_{t, \tau} e^{\int_t^x\left(r_v-\rho\right) d v} .$$

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MATLAB代写

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