### 经济代写|宏观经济学代写Macroeconomics代考|EC4505

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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 decentralized equilibrium

Here we present a discrete time model initially developed by Diamond (1965), building on earlier work by Samuelson (1958), in which individuals live for two periods (young and old). The economy lasts forever as new young people enter in every period. We first characterise the decentralised competitive equilibrium of the model. We then ask whether the market solution is the same as the allocation that would be chosen by a central planner, focusing on the significance of the golden rule, which will allow us to discuss the possibility of dynamic inefficiency (i.e. excessive capital accumulation).

The market economy is composed of individuals and firms. Individuals live for two periods. They work for firms, receiving a wage. They also lend their savings to firms, receiving a rental rate.
An individual born at time $t$ consumes $c_{1 t}$ in period $t$ and $c_{2 t+1}$ in period $t+1$, and derives utility
$$\left(\frac{\sigma}{\sigma-1}\right) c_{1 \mathrm{t}}^{\frac{\sigma-1}{\sigma}}+(1+\rho)^{-1}\left(\frac{\sigma}{\sigma-1}\right) c_{2 f+1}^{\frac{\sigma-1}{\sigma}}, \rho \geq 0, \sigma \geq 0 .$$
Note that the subscript ” 1 ” refers to consumption when young, and ” 2 ” labels consumption when old. Individuals work only in the first period of life, inelastically supplying one unit of labour and earning a real wage of $w_t$. They consume part of their first-period income and save the rest to finance their second-period retirement consumption. The saving of the young in period $t$ generates the capital stock that is used to produce output in period $t+1$ in combination with the labour supplied by the young generation of period $t+1$.
The time structure of the model appears in Figure 8.1.
The number of individuals born at time $t$ and working in period $t$ is $L_t$. Population grows at rate $n$ so that $L_t=L_0(1+n)^t$.

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

Consider an individual born at time $t$. His maximisation problem is
$$\max \left{\left(\frac{\sigma}{\sigma-1}\right) c_{1 t}^{\frac{\sigma-1}{\sigma}}+(1+\rho)^{-1}\left(\frac{\sigma}{\sigma-1}\right) c_{2 t+1}^{\frac{\sigma-1}{\sigma}}\right}$$
subject to
$$c_{1 t}+s_t=w_t,$$
$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t,$$
where $w_t$ is the wage received in period $t$ and $r_{t+1}$ is the interest rate paid on savings held from period $t$ to period $t+1$. In the second period the individual consumes all his wealth, both interest and principal. (Note that this assumes that there is no altruism across generations, in that people do not care about leaving bequests to the coming generations. This is crucial.)
The first-order condition for a maximum is
$$c_{1 t}^{-\frac{1}{\sigma}}-\left(\frac{1+r_{t+1}}{1+\rho}\right) c_{2 t+1}^{-\frac{1}{\sigma}}=0,$$
which can be rewritten as
$$\frac{c_{2 t+1}}{c_{1 t}}=\left(\frac{1+r_{t+1}}{1+\rho}\right)^\sigma \text {. }$$
This is the Euler equation for the generation born at time $t$. Note that this has the very same intuition, in discrete time, as the Euler equation (Ramsey rule) we derived in the context of the NGM.
Next, using (8.3) and (8.4) to substitute out for $c_{1 t}$ and $c_{2 t+1}$ and rearranging we get
$$s_t=\left(\frac{1}{\left(1+r_{t+1}\right)^{1-\sigma}(1+\rho)^\sigma+1}\right) w_t .$$
We can think of this as a saving function:
$$s_t=s\left(w_t, r_{t+1}\right), \quad 0<s_w \equiv \frac{\partial s_t}{\partial w_t}<1, s_r \equiv \frac{\partial s_t}{\partial r_{t+1}} \geq 0 \text { or } \leq 0 .$$
Saving is an increasing function of wage income since the assumption of separability and concavity of the utility function ensures that both goods (i.e. consumption in both periods) are normal. The effect of an increase in the interest rate is ambiguous, however, because of the standard income and substitution effects with which you are familiar from micro theory. An increase in the interest rate decreases the relative price of second-period consumption, leading individuals to shift consumption from the first to the second period, that is, to substitute second- for first-period consumption. But it also increases the feasible consumption set, making it possible to increase consumption in both periods; this is the income effect. The net effect of these substitution and income effects is ambiguous. If the elasticity of substitution between consumption in both periods is greater than one, then in this two-period model the substitution effect dominates and an increase in interest rates leads to an increase in saving.

## 经济代写|宏观经济学代写宏观经济学代考|分散均衡

$$\left(\frac{\sigma}{\sigma-1}\right) c_{1 \mathrm{t}}^{\frac{\sigma-1}{\sigma}}+(1+\rho)^{-1}\left(\frac{\sigma}{\sigma-1}\right) c_{2 f+1}^{\frac{\sigma-1}{\sigma}}, \rho \geq 0, \sigma \geq 0 .$$

## 经济代写|宏观经济学代写宏观经济学代考|个人

$$\max \left{\left(\frac{\sigma}{\sigma-1}\right) c_{1 t}^{\frac{\sigma-1}{\sigma}}+(1+\rho)^{-1}\left(\frac{\sigma}{\sigma-1}\right) c_{2 t+1}^{\frac{\sigma-1}{\sigma}}\right}$$

$$c_{1 t}+s_t=w_t,$$
$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t,$$
，其中$w_t$是在$t$期间收到的工资，$r_{t+1}$是在$t$至$t+1$期间持有的储蓄支付的利率。在第二阶段，个人消费他所有的财富，包括利息和本金。(请注意，这是假设没有跨代的利他主义，即人们不关心给下一代留下遗产。

$$c_{1 t}^{-\frac{1}{\sigma}}-\left(\frac{1+r_{t+1}}{1+\rho}\right) c_{2 t+1}^{-\frac{1}{\sigma}}=0,$$
，可以改写为
$$\frac{c_{2 t+1}}{c_{1 t}}=\left(\frac{1+r_{t+1}}{1+\rho}\right)^\sigma \text {. }$$

$$s_t=\left(\frac{1}{\left(1+r_{t+1}\right)^{1-\sigma}(1+\rho)^\sigma+1}\right) w_t .$$

$$s_t=s\left(w_t, r_{t+1}\right), \quad 0<s_w \equiv \frac{\partial s_t}{\partial w_t}<1, s_r \equiv \frac{\partial s_t}{\partial r_{t+1}} \geq 0 \text { or } \leq 0 .$$

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## MATLAB代写

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