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

2022年9月23日

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## 经济代写|宏观经济学代写Macroeconomics代考|Fully funded and pay-as-you-go systems

Fully funded system Under a fully funded system, the contributions made when young are returned with interest when old:
$$b_{t+1}=\left(1+r_{t+1}\right) d_t .$$
This is because the contribution is invested in real assets at the ongoing interest rate. Pay-as-you-go system Under a pay-as-you-go system, the contributions made by the current young go directly to the current old:
$$b_t=(1+n) d_t .$$
The reason why population growth pops in is because if there is population growth there is a larger cohort contributing than receiving. Notice the subtle but critical change of subscript on the benefit on the left-hand side.

There are many questions that can be asked about the effects of such pension programs on the economy. Here we focus on only one: Do they affect savings, capital accumulation, and growth? ${ }^1$
With pensions, the problem of an individual born at time $t$ becomes
$$\max \log \left(c_{1 t}\right)+(1+\rho)^{-1} \log \left(c_{2 t+1}\right),$$
subject to
$$c_{1 t}+s_t+d_t=w_t,$$
$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t+b_{t+1} .$$
The first-order condition for a maximum is still the Euler equation
$$c_{2 t+1}=\left(\frac{1+r_{t+1}}{1+\rho}\right) c_{1 t} .$$
Substituting for $c_{1 t}$ and $c_{2 t+1}$ in terms of $s, w$, and $r$ implies a saving function
$$s_t=\left(\frac{1}{2+\rho}\right) w_t-\frac{\left(1+r_{t+1}\right) d_t+(1+\rho) b_{t+1}}{(2+\rho)\left(1+r_{t+1}\right)} .$$
Again, savings is an increasing function of wage income, and is a decreasing function of contributions and benefits – leaving aside the link between those, and the general equilibrium effects through factor prices. These will mean, however, that savings will be affected by the pension variables in a complicated way.

## 经济代写|宏观经济学代写Macroeconomics代考|Fully funded pension system

Let us start by looking at the effect of this kind of program on individual savings. (The distinction between individual and aggregate savings will become critical later on.) We can simply insert (9.1) into (9.7) to get
$$s_t=\left(\frac{1}{2+\rho}\right) w_t-d_t .$$
Therefore,
$$\frac{\partial s_t}{\partial d_t}=-1 .$$
In words, holding the wage constant, pension contributions decrease private savings exactly one for one. The intuition is that the pension system provides a rate of return equal to that of private savings, so it is as if the system were taking part of that individual’s income and investing that amount itself. The individual is indifferent about who does the saving, caring only about the rate of return.

Hence, including the pension savings in total savings, a change in contributions $d$ leaves overall, or aggregate savings (and, therefore, capital accumulation and growth) unchanged. To make this clear, let’s define aggregate savings as the saving that is done privately plus through the pension system. In a fully funded system the aggregate savings equals
$$s_t^{\operatorname{agg}}=s_t+d_t=\left(\frac{1}{2+\rho}\right) w_{t^{-}} .$$
This is exactly the same as in Chapter 7 , without pensions.

# 统计推断代考

## 经济代写|宏观经济学代写宏观经济学代考|资金充分和现收现付的系统

$$b_{t+1}=\left(1+r_{t+1}\right) d_t .$$

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

$$c_{1 t}+s_t+d_t=w_t,$$
$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t+b_{t+1} .$$最大值的一阶条件仍然是欧拉方程
$$c_{2 t+1}=\left(\frac{1+r_{t+1}}{1+\rho}\right) c_{1 t} .$$

$$s_t=\left(\frac{1}{2+\rho}\right) w_t-\frac{\left(1+r_{t+1}\right) d_t+(1+\rho) b_{t+1}}{(2+\rho)\left(1+r_{t+1}\right)} .$$同样，储蓄是工资收入的递增函数，是贡献和福利的递减函数——不考虑两者之间的联系，以及通过要素价格产生的一般均衡效应。然而，这将意味着储蓄将以复杂的方式受到养恤金变数的影响

## 经济代写|宏观经济学代写宏观经济学代考|完全基金的养老金制度

$$s_t=\left(\frac{1}{2+\rho}\right) w_t-d_t .$$

$$\frac{\partial s_t}{\partial d_t}=-1 .$$

$$s_t^{\operatorname{agg}}=s_t+d_t=\left(\frac{1}{2+\rho}\right) w_{t^{-}} .$$

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## MATLAB代写

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