# 经济代写|宏观经济学代写Macroeconomics代考|ECON102

## 经济代写|宏观经济学代写Macroeconomics代考|Financing the transition by issuing debt

Now let’s think about how things would change if the transition is financed by issuing debt. (Maybe that is a politically more palatable option!) In this case, for the current young there are no taxes, and debt is another asset that they can purchase:
$$c_{1 t}+s_t+d+g_{\text {debt }}=w_t,$$
so consumption in old age can be
$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t+\left(1+r_{t+1}\right) d+\left(1+r_{t+1}\right) g_{\text {debt }} .$$
Following the same logic as before, private savings are
$$s_t=\frac{w_t}{(2+\rho)}-d-g_{\text {debt }} .$$
How about aggregate savings? Note that contributions to the fully funded system $d$, work as savings from an aggregate perspective: they are available to finance the accumulation of capital. However, the amount of debt issued by the government is in fact not used for capital accumulation, but rather for consumption, because it is a transfer to the old. As such, aggregate savings are given by
$$s_t^{a g g}=s_t+d=\frac{w_t}{(2+\rho)}-g_{\text {debt }}=\frac{w_t}{(2+\rho)}-d,$$
where in the last step we use the fact that (under no population growth) the government issues $g_{\text {debt }}=d$ of debt to pay benefits to the current old.

Let’s see how this compares to the pay-as-you-go savings. Rewriting equation (9.22) which shows the savings rate in a pay-as-you-go system
$$s_t^{a g g}=s_t=\left(\frac{1}{2+\rho}\right) w_t-d \frac{\left(1+r_{t+1}\right)+(1+\rho)}{(2+\rho)\left(1+r_{t+1}\right)} .$$
Notice that if
$$\frac{\left(1+r_{t+1}\right)+(1+\rho)}{(2+\rho)\left(1+r_{t+1}\right)}<1,$$ then in this case savings is even lower than in the pay-as-you-go system, which happens because the government now pays $r$ on its debt, which in this case is higher than $n$.

Another way to see this is if the government imposed a fully funded system but then makes the pension firms purchase government debt that is used for the current old (i.e. for consumption). There is no way this type of reform can increase the capital stock.

## 经济代写|宏观经济学代写Macroeconomics代考|Do people save enough

The above setup assumes that agents optimise their savings to maximise intertemporal utility, with or without pensions. However, there are several reasons why this may not be the case. We will get into more detail in Chapters 11 and 12, but it is worth going over some of the possibilities here, in the context of pensions.

First and foremost, people may believe that if they don’t save, the government will step in and bail them out. If you believe that, why would you save? This time inconsistency feature (the government may tell you to save but, if you don’t, will feel tempted to help you out) may lead to suboptimal savings.

Even in the case of the U.S., where these considerations may not be so relevant, there has been ample discussion about the intensity with which people save for retirement. On the one hand, Scholz et al. (2006) shows that the accumulation of assets of people roughly matches the life cycle hypothesis (which we will see in detail in Chapter 11); on the other hand, there is evidence that suggests that consumption levels drop upon retirement (Bernheim et al. 2001), which is inconsistent with optimal savings. One possible reconciliation of these two facts is given by including the dimension of what type of assets people use for savings. Beshears et al. (2018) show that people save sizable amounts, but they tend to save in illiquid assets. Illiquid assets may provide unusually high returns (for example, owning your house provides steady rental income). Kaplan et al. (2014) estimate that housing services provides an after-tax risk adjusted rate of return of close to $8 \%$. In such a world agents hold a large share of illiquid assets but consumption tracks income while they use some potentially expensive mechanisms to partially smooth consumption.

Present bias has also been mentioned as a reason why people tend to save less than the optimal level. In this case, imposing a pension system that forces people to save may be a ex ante optimal commitment device. We will discuss present bias in detail in Chapter 12.

Finally, recent research on savings for retirement has delivered some interesting new ideas and policy suggestions. One typical way of saving in the U.S. is the $401 \mathrm{~K}$ programs, where you save with the benefit of a tax deferral: your income is taxed when withdrawing the funds. These programs are typically arranged with your employer, which matches the contributions with a vesting period to entice labour stability. Yet it has been found that matching is a fairly inefficient way to stimulate savings. Madrian (2013) finds that a matching contribution of $25 \%$ increases savings by $5 \%$. In contrast default setting seems, to have a much stronger effect. Madrian and Shea (2001) show that when a company shifted from a default where, unless the worker would opt out, it would start contributing $3 \%$ of its salary to a $401 \mathrm{~K}$ program, they found that fifteen months after the change, $85 \%$ of the workers participated, and $65 \%$ contributed $3 \%$ of their wages. This compared with only $49 \%$ participation for those workers hired previously in which only $4 \%$ contributed $3 \%$. In short, default standards may be powerful (and cheap) tools for incentivizing savings.

## 经济代写|宏观经济学代写宏观经济学代考|通过发行债务为转型融资

$$c_{1 t}+s_t+d+g_{\text {debt }}=w_t,$$

$$c_{2 t+1}=\left(1+r_{t+1}\right) s_t+\left(1+r_{t+1}\right) d+\left(1+r_{t+1}\right) g_{\text {debt }} .$$

$$s_t=\frac{w_t}{(2+\rho)}-d-g_{\text {debt }} .$$

$$s_t^{a g g}=s_t+d=\frac{w_t}{(2+\rho)}-g_{\text {debt }}=\frac{w_t}{(2+\rho)}-d,$$

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

$$\frac{\left(1+r_{t+1}\right)+(1+\rho)}{(2+\rho)\left(1+r_{t+1}\right)}<1,$$，那么在这种情况下，储蓄甚至低于现收现付系统，这是因为政府现在支付$r$的债务，在这种情况下高于$n$

## 经济代写|宏观经济学代写宏观经济学代考|人们是否储蓄足够

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