# 经济代写|宏观经济学代写Macroeconomics代考|ECON305

## 经济代写|宏观经济学代写Macroeconomics代考|Moving out of a pay-as-you-go system

There are several transitions associated with the introduction or revamping of pensions systems, and that we may want to analyze. For example, you could move from no pension system and implement a full capitalisation system. As aggregate saving behaviour does not change, we do not expect anything really meaningful to happen from such change in terms of capital accumulation and growth. (That is, of course, to the extent that rational behaviour is a good enough assumption when it comes to individual savings behaviour. We will get back to this pretty soon when we talk about consumption.) Alternatively, as discussed above, if we implement a pay-as-you-go system, the initial old are happy, while the effect for future generations remains indeterminate and depends on the dynamic efficiency of the economy.

However, in recent years it has become fashionable to move away from pay-as-you-go systems to fully funded ones. The reasons for such change is different in each country, but usually can be traced back to deficit and sometimes insolvent systems (sometimes corruption-ridden) that need to be revamped. ${ }^3$ But one of the main reasons was to undo the capital depletion associated with pay-asyou-go systems. Thus, these countries hoped that going for a capitalisation system would increase the capital stock and income over time.

In what remains of this chapter we will show that what happens in such transitions from pay-asyou-go to fully funded systems depends very much on how the transition is financed. There are two options: either the transition is financed by taxing the current young, or it is financed by issuing debt. Both have quite different implications.

To make the analysis simple, in what follows we will keep $n=0$. (Note that this puts us in the region where $r>n$, i.e. that of dynamic efficiency.)
Aggregate savings without pensions or with a fully funded system are
$$s_t^{a g g \rho}=\left(\frac{1}{2+\rho}\right) w_t .$$
With a pay-as-you-go system, they are
$$s_t^{a g g}=s_t=\left(\frac{1}{2+\rho}\right) w_t-\frac{\left(1+r_{t+1}\right) d+(1+\rho) d}{(2+\rho)\left(1+r_{t+1}\right),}$$
which is trivially lower (we knew this already). So now the question is how savings move when going from a pay-as-you-go to a fully funded system. You may think they have to go up, but we need to be careful: we need to take care of the old, who naturally will not be part of the new system, and their retirement income has to be financed. This, in turn, may have effects of its own on capital accumulation.

## 经济代写|宏观经济学代写Macroeconomics代考|Financing the transition with taxes on the young

If the transition is financed out of taxes, the young have to use their wages for consumption $\left(c_{1 t}\right)$, private savings $\left(s_t\right.$ ), to pay for their contributions ( $d$ and also for taxes $\tau_t$ ):
$$c_{1 t}+s_t+d+\tau_t=w_t .$$
Future consumption is in turn given by
$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t+\left(1+r_{t+1}\right) d,$$

as we are in a fully funded system. Because taxes here are charged to finance the old, we have $\tau_t=d$ (remember we have assumed population growth to be equal to zero). If you follow the logic above, it can be shown that in this case we have
$$s_t^{a g g}=\frac{\left(w_t-\tau_t\right)}{(2+\rho)} .$$
You may notice that this is lower than the steady-state savings rate (next period, i.e. in 30 years, there are no more taxes), but you can also show that it is higher than in the pay-as-you-go system. To do so, replace $\tau_t$ with $d$ in (9.25) and then compare the resulting expression with that of (9.22).

So savings goes up slowly, approaching its steady-state value. These dynamics are what supports World Bank recommendations that countries should move from pay-as-you-go to fully capitalised systems. Notice however that the reform hurts the current young that have to save for their own and for the current old generation. Then remember that one period here is actually one generation, so it’s something like 30 years. What do you think would be the political incentives, as far as reforming the system, along those lines?

## 经济代写|宏观经济学代写宏观经济学代考|走出现收现付的体系

$$s_t^{a g g \rho}=\left(\frac{1}{2+\rho}\right) w_t .$$

$$s_t^{a g g}=s_t=\left(\frac{1}{2+\rho}\right) w_t-\frac{\left(1+r_{t+1}\right) d+(1+\rho) d}{(2+\rho)\left(1+r_{t+1}\right),}$$
，略低一些(我们已经知道了这一点)。所以现在的问题是，当储蓄从现收现付体系转变为资金充足的体系时，储蓄将如何变化。你可能认为他们必须提高，但我们需要小心:我们需要照顾老年人，他们自然不会是新体系的一部分，他们的退休收入必须得到资助。反过来，这也可能对资本积累产生影响

## 经济代写|宏观经济学代写宏观经济学代考|用对年轻人征税为过渡提供资金

$$c_{1 t}+s_t+d+\tau_t=w_t .$$

$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t+\left(1+r_{t+1}\right) d,$$

，因为我们处于一个资金充足的系统中。因为这里的税收是用来资助老年人的，所以我们有$\tau_t=d$(记住，我们假设人口增长等于零)。如果你遵循上面的逻辑，在这种情况下我们可以看到
$$s_t^{a g g}=\frac{\left(w_t-\tau_t\right)}{(2+\rho)} .$$

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