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经济代写|宏观经济学代写Macroeconomics代考|Implications for fiscal, financial and monetary policy
As the discussion in the previous section should make clear, bubbles have huge implications for fiscal policy. We saw that in an economy with strong demand for liquidity, private sector agents may be willing to hold government debt even if that debt pays an interest rate that is lower than the rate of population growth, which is also the rate of growth of the economy.
This is good news for treasury officials and fiscal policymakers: the model suggests, they can run a primary deficit forever without ever having to raise taxes to retire the resulting debt. This is not just a theoretical curiosum. Today in most advanced economies, the real rate of interest is below the rate of economic growth (however paltry that rate of growth may be). This fact is motivating a deep rethinking about the limits of fiscal policy and the scope for a robust fiscal response not only to the Covid-19 pandemic, but also to the green infrastructure buildup that global warming would seem to require. Olivier Blanchard devoted his Presidential Lecture to the American Economic Association (2019) to argue that a situation in which $r<g$ for a prolonged period of time opens vast new possibilities for the conduct of fiscal policy ${ }^{10}$.
But, at the same time, a bubbly world also bears bad news for those in charge of fiscal policy because, as we have seen above, bubbly equilibria are inherently fragile. Could it be that an advanced country issues a great deal of debt at very low interest rates and one day investors decide to dump it simply because of a self-fulfilling change in expectations? Hard to say, but it is not a possibility that can be entirely ignored. In fact, Blanchard (2019) acknowledges that arguments based on the possibility of multiple equilibria are “the most difficult to counter” when making the case for the increased fiscal space that low interest rates bring.
Bubbles also have vast implications for financial markets and financial regulation. The obvious concern, mentioned at the outset, is that asset bubbles typically end in tears, with overvaluation abruptly reversing itself and wrecking balance sheets. But here, also, the news is not all bad. Financial markets typically involve inefficient borrowing constraints that keep a subset of agents (especially small and medium enterprises) from undertaking positive net-present-value projects. Therefore, as Martin and Ventura (2018) emphasise, to the extent that bubbles pump up the value of collateral and relax borrowing constraints, they can promote efficiency and raise welfare as long as those bubbles do not burst ${ }^{11}$.
Last but certainly not least, bubbles present difficult dilemmas for central banks and for monetary policy more generally. In the presence of sticky prices, if bubbles affect aggregate demand they also affect output and inflation, giving rise to bubble-driven business cycles. The implication is that standard monetary and interest rate rules need to be modified to take into account this new source of fluctuations. In some cases those modifications are relatively minor, but that is not always the case. Gali (2020) discusses the issues involved in greater detail than we can here.
经济代写|宏观经济学代写Macroeconomics代考|Dynamic optimisation in continuous time
We have described macroeconomic policy problems in discrete and continuous time at different points, depending on convenience. In continuous time, we can solve these problems using the optimality conditions from optimal control theory. ${ }^1$
What kinds of problems fit the optimal control framework? The idea is that you choose a certain path for a choice variable – the control variable – that maximises the total value over time of a function affected by that variable. This would be relatively easy, and well within the realm of standard constrained optimisation, if whatever value you chose for the control variable at a certain moment in time had no implication for what values it may take at the next moment. What makes things trickier, and more interesting, is when it is not the case. That is to say, when what you do now affects what your options are for tomorrow – or, in continuous time, the next infinitesimal moment. That’s what is captured by the state variable: a variable that contains the information from all the previous evolution of the dynamic system. The evolution of the state variable is described by a dynamic equation, the equation of motion.
The simplest way to see all of this is to look at a simple example. Consider a simplified consumer problem: ${ }^2$
$$
\max {\left{c_t\right}{t=0}^T} \int_0^T u\left(c_t\right) e^{-\rho t} d t,
$$ subject to the budget constraint $c_t+\dot{a}_t=y_t+r a_t$ and to an initial level of assets $a_0$. In words, the consumer chooses the path for their consumption so as to maximise total utility over their lifetime, and whatever income (labour plus interest on assets) they do not consume is accumulated as assets. The control variable is $c_t$ that is, what the consumer chooses in order to maximise utility and the state variable is $a_t$ that is, what links one instant to the next, as described by the equation of motion:
$$
\dot{a}_t=y_t+r a_t-c_t
$$
The maximum principle can be summarised as a series of steps:
Step 1 – Set up the Hamiltonian function: The Hamiltonian is simply what is in the integral – the instantaneous value, at time $t$, of the function you are trying to maximise over time plus, “the righthand-side of the equation of motion” multiplied by a function called the co-state variable, which we will denote as $\lambda_t$. In our example, we can write:
$$
H_t=u\left(c_t\right)+\lambda_t\left[y_t+r a_t-c_t\right] .
$$
宏观经济学代考
经济代写|宏观经济学代写Macroeconomics代考|Implications for fiscal, financial and monetary policy
正如上一节的讨论所表明的那样，泡沫对财政政策具有巨大影响。我们看到，在流动性需求强劲的经济体中，私营部门代理人可能愿意持有政府债务，即使该债务的利率低于人口增长率，人口增长率也是经济增长率.
这对财政部官员和财政政策制定者来说是个好消息：模型表明，他们可以永远维持基本赤字，而无需提高税收来偿还由此产生的债务。这不仅仅是理论上的好奇心。如今，在大多数发达经济体中，实际利率低于经济增长率（无论该增长率可能多么微不足道）。这一事实促使人们深入反思财政政策的局限性，以及对 Covid-19 大流行病以及全球变暖似乎需要的绿色基础设施建设做出强有力财政反应的范围。奥利维尔·布兰查德 (Olivier Blanchard) 在美国经济学会 (2019) 的总统演讲中指出，在这种情况下，r<G在很长一段时间内，为实施财政政策开辟了巨大的新可能性10.
但与此同时，泡沫世界也给那些负责财政政策的人带来了坏消息，因为正如我们在上面看到的那样，泡沫均衡本质上是脆弱的。有没有可能一个发达国家以极低的利率发行了大量债务，有一天投资者仅仅因为预期的自我实现变化而决定抛售这些债务？很难说，但也不是完全可以忽略的可能性。事实上，Blanchard（2019 年）承认，在论证低利率带来的财政空间增加时，基于多重均衡可能性的论点“最难反驳”。
泡沫还对金融市场和金融监管产生巨大影响。一开始提到的明显担忧是，资产泡沫通常以泪水告终，高估突然逆转并破坏资产负债表。但在这里，也不全是坏消息。金融市场通常涉及低效的借贷限制，使一部分代理人（尤其是中小型企业）无法进行正的净现值项目。因此，正如 Martin 和 Ventura（2018 年）所强调的那样，只要泡沫不破裂，泡沫会提高抵押品的价值并放松借贷约束，它们可以提高效率并提高福利11.
最后但并非最不重要的一点是，泡沫给中央银行和更广泛的货币政策带来了两难困境。在存在粘性价格的情况下，如果泡沫影响总需求，它们也会影响产出和通货膨胀，从而导致泡沫驱动的商业周期。这意味着需要修改标准的货币和利率规则，以考虑到这种新的波动来源。在某些情况下，这些修改相对较小，但情况并非总是如此。Gali (2020) 比我们在这里更详细地讨论了所涉及的问题。
经济代写|宏观经济学代写Macroeconomics代考|Dynamic optimisation in continuous time
为方便起见，我们在不同时间点描述了离散和连续时间的宏观经济政策问题。在连续时 间内，我们可以使用最优控制理论中的最优性条件来解决这些问题。 1
什么样的问题适合最优控制框架? 这个想法是，你为一个选择变量一控制变量一一选 择了一条特定的路径，它使受该变量影响的函数随时间的总价值最大化。这将相对容 易，并且完全在标准约束优化的范围内，如果您在特定时刻为控制变量选择的任何值都 不会影响下一时刻可能采用的值。让事情变得更棘手、更有趣的是，情况并非如此。也 就是说，当你现在所做的事情影响你明天的选择时―—或者，在连续的时间里，影响下 一个无穷小的时刻。这就是状态变量所捕获的: 一个包含动态系统所有先前演化信息的 变量。
查看所有这些内容的最简单方法是查看一个简单示例。考虑一个简化的消费者问题: ${ }^2$
受制于预算约束 $c_t+\dot{a}_t=y_t+r a_t$ 并达到初始资产水平 $a_0$. 换言之，消费者选择的消 费路径是为了使他们一生的总效用最大化，而他们没有消费的任何收入 (劳动加资产利 息) 都作为资产积累起来。控制变量是 $c_t$ 也就是说，消费者为了最大化效用而选择什 么，状态变量是 $a_t$ 也就是说，是什么将一个瞬间与下一个瞬间联系起来，正如运动方程 所描述的那样:
$$
\dot{a}_t=y_t+r a_t-c_t
$$
最大值原理可以概括为一系列步骤:
第 1 步 – 设置哈密顿函数：哈密顿函数就是积分中的简单值 – 时间的瞬时值 $t$ ，你试图 随时间最大化的函数加上“运动方程的右侧”乘以一个称为共态变量的函数，我们将其表 示为 $\lambda_t$. 在我们的例子中，我们可以这样写:
$$
H_t=u\left(c_t\right)+\lambda_t\left[y_t+r a_t-c_t\right]
$$
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