# 经济代写|宏观经济学代写Macroeconomics代考|ECON1120

## 经济代写|宏观经济学代写Macroeconomics代考|Pay-as-you-go pension system

To see the effect of this program on savings, insert $(9.2)$ into $(9.7)$ (paying attention to the appropriate time subscripts) to get
$$s_t=\left(\frac{1}{2+\rho}\right) w_t-\frac{\left(1+r_{t+1}\right) d_t+(1+\rho)(1+n) d_{t+1}}{(2+\rho)\left(1+r_{t+1}\right)} .$$
This is a rather complicated expression that depends on $d_t$ and $d_{t+1}$ – that is, on the size of the contributions made by each generation. But there is one case that lends itself to a simple interpretation. Assume $d_t=d_{t+1}=d$, so that contributions are the same per generation. Then equation (9.13) becomes
$$s_t=\left(\frac{1}{2+\rho}\right) w_t-d\left[\frac{\left(1+r_{t+1}\right)+(1+\rho)(1+n)}{(2+\rho)\left(1+r_{t+1}\right)}\right] .$$

Note that, from an individual’s perspective, the return on her contributions is given by $n$, and not $r$. This return depends on there being more individuals to make contributions to the pension system in each period – you can see how demographic dynamics play a crucial role here! From (9.14) we have
$$\frac{\partial s_t}{\partial d_t}=-\frac{\left(1+r_{t+1}\right)+(1+\rho)(1+n)}{(2+\rho)\left(1+r_{t+1}\right)}<0 .$$
We can see contributions decrease individual savings – and, in principle, aggregate savings, as here they coincide (see the caveat below). Why do private and aggregate savings coincide? Because the pension system here is a transfer scheme from young to old, and not an alternative savings scheme. The only source of capital is private savings $s_t$.

## 经济代写|宏观经济学代写Macroeconomics代考|How do pensions affect the capital stock

So far we have asked what happens to savings holding interest and wages constant – that is to say, the partial equilibrium effect of pensions. In the case of a fully funded system, that is of no consequence, since changes in contributions leave savings – and hence, capital accumulation, wages, and interest rates – unchanged. But it matters in the case of a pay-as-you-go system.

To examine the general equilibrium effects of changes in contributions within the latter system, recall that capital accumulation is given by
$$k_{t+1}=\frac{s_t}{1+n} .$$
Substituting (9.14) into this equation we have
$$k_{t+1}=\left(\frac{1}{2+\rho}\right) \frac{w_t}{1+n}-h\left(k_{t+1}\right) d,$$
where
\begin{aligned} h\left(k_{t+1}\right) &=\frac{1+(1+\rho)(1+n)\left(1+r_{t+1}\right)^{-1}}{(1+n)(2+\rho)}, \ &=\frac{1+(1+\rho)(1+n)\left(1+\alpha k_{t+1}^{a-1}\right)^{-1}}{(1+n)(2+\rho)}, \end{aligned}
and where $h^{\prime}\left(k_{t+1}\right)>0$. (Note the use of $(9.8)$ above.)
Next, totally differentiating ( $9.17)$, holding $k_t$ constant, and rearranging, we have
$$\frac{d k_{t+1}}{d d_t}=-\frac{h\left(k_{t+1}\right)}{1+h^{\prime}\left(k_{t+1}\right) d}<0 .$$
Therefore, the effect of an increase in contributions in a pay-as-you-go system is to shift down the savings locus. The consequences appear in Figure 9.1. The new steady-state capital stock is lower. If the capital stock at the time of the policy shock is to the left of the new steady state, the economy continues to accumulate capital, but at a rate slower than before the change.

## 经济代写|宏观经济学代写宏观经济学代考|现收现付养老金制度

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

$$s_t=\left(\frac{1}{2+\rho}\right) w_t-d\left[\frac{\left(1+r_{t+1}\right)+(1+\rho)(1+n)}{(2+\rho)\left(1+r_{t+1}\right)}\right] .$$

$$\frac{\partial s_t}{\partial d_t}=-\frac{\left(1+r_{t+1}\right)+(1+\rho)(1+n)}{(2+\rho)\left(1+r_{t+1}\right)}<0 .$$

## 经济代写|宏观经济学代写宏观经济学代考|养老金如何影响资本存量

$$k_{t+1}=\left(\frac{1}{2+\rho}\right) \frac{w_t}{1+n}-h\left(k_{t+1}\right) d,$$
where
\begin{aligned} h\left(k_{t+1}\right) &=\frac{1+(1+\rho)(1+n)\left(1+r_{t+1}\right)^{-1}}{(1+n)(2+\rho)}, \ &=\frac{1+(1+\rho)(1+n)\left(1+\alpha k_{t+1}^{a-1}\right)^{-1}}{(1+n)(2+\rho)}, \end{aligned}

$$\frac{d k_{t+1}}{d d_t}=-\frac{h\left(k_{t+1}\right)}{1+h^{\prime}\left(k_{t+1}\right) d}<0 .$$因此，在现收现付制度中，增加缴款的影响是向下移动储蓄轨迹。结果如图9.1所示。新的稳定状态资本存量更低。如果政策冲击时的资本存量处于新的稳定状态的左侧，那么经济将继续积累资本，但速度比政策冲击前要慢

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