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• Statistical Inference 统计推断
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经济代写|宏观经济学代写Macroeconomics代考|Solving for the time profile and level of consumption

Take (11.5) and differentiate both sides with respect to time
$$u^{\prime \prime}\left(c_t\right) \dot{c}_t=\dot{\lambda}_t .$$

Divide this by (11.5) and rearrange
$$\frac{u^{\prime \prime}\left(c_t\right) c_t}{u^{\prime}\left(c_t\right)} \frac{\dot{c}_t}{c_t}=\frac{\dot{\lambda}_t}{\lambda_t} .$$
Now, as we’ve seen before, define
$$\sigma \equiv\left[-\frac{u^{\prime \prime}\left(c_t\right) c_t}{u^{\prime}\left(c_t\right)}\right]^{-1}>0$$
as the elasticity of intertemporal substitution in consumption. Then, (11.9) becomes
$$\frac{\dot{c}_t}{c_t}=-\sigma \frac{\dot{\lambda}_t}{\lambda_t} .$$
Finally using (11.6) in (11.11) we obtain (what a surprise!):
$$\frac{\dot{c}_t}{c_t}=\sigma(r-\rho)=0 .$$
Equation (11.12) says that consumption is constant since we assume $r=\rho$. It follows then that
$$c_t=c^,$$ so that consumption is constant. We now need to solve for the level of consumption $c^$. Using (11.13) in (11.2) we get
$$\dot{b}_t=r b_t+w_t-c^,$$ which is a differential equation in $b$, whose solution is, for any time $t>0$, $$b_t=\int_0^t w_s e^{r(t-r)} d s-\left(e^{n t}-1\right) \frac{c^}{r}+b_0 e^{n t} .$$
where time $v$ is any moment between 0 and $t$. Evaluating this at $t=T$ (the terminal period) we obtain the stock of bonds at the end of the agent’s life:
$$b_T=\int_0^T w_s e^{r(T-s)} d s-\left(e^{r T}-1\right) \frac{c^}{r}+b_0 e^{r T} .$$ Dividing both sides by $e^{r T}$ and rearranging, we have $$b_T e^{-r T}=\int_0^T w_t e^{-r s} d s-\left(1-e^{-r T}\right) \frac{c^}{r}+b_0 .$$
Notice that using (11.5), (11.7), and (11.13), the TVC can be written as
$$u^{\prime}\left(c^*\right) b_T e^{-r T}=0 .$$

经济代写|宏观经济学代写Macroeconomics代考|Consumption with uncertainty

The analysis of consumption under uncertainty is analogous to that under certainty with the difference that now we will assume that consumers maximise expected utility rather than just plain utility. As it turns out, it is more convenient to analyse the case with uncertainty in discrete, rather than continuous, time. The utility that the consumer maximises in this case is
\begin{aligned} &\operatorname{maxE}\left[\sum_{t=0}^T \frac{1}{(1+\rho)^t} u\left(c_t\right)\right], \ &\text { s.t. } b_{t+1}=\left(w_t+b_t-c_t\right)(1+r) . \end{aligned}
The uncertainty comes from the fact that we now assume labour income $w_t$ to be uncertain. ${ }^1$ How do we model individual behaviour when facing such uncertainty? When we impose that individuals use the mathematical expectation to evaluate their utility we are assuming that they have rational expectations: they understand the model that is behind the uncertainty in the economy, and make use of all the available information in making their forecasts. (Or, at the very least, they don’t know any less than the economist who is modelling their behaviour.) As we will see time and again, this will have very powerful implications.

Let us start with a two-period model, not unlike the one that we used when analysing the OLG model. As you will recall and can easily verify, the FOC looks like this:
$$u^{\prime}\left(c_1\right)=\left(\frac{1+r}{1+\rho}\right) E_1\left[u^{\prime}\left(c_2\right)\right] .$$
This FOC generalises to the case of many periods, with exactly the same economic intuition:
$$u^{\prime}\left(c_t\right)=\left(\frac{1+r}{1+\rho}\right) E_t\left[u^{\prime}\left(c_{t+1}\right)\right] .$$
This is our Euler equation for optimal consumption.
To see how this helps us find the consumption level in a multiperiod framework, we use the tools of dynamic programming, which you can briefly review in the math appendix at the end of the book. We show there that intertemporal problems can be solved with the help of a Bellman equation. The Bellman equation rewrites the optimisation problem as the choice between current utility and future utility. Future utility, in turn, is condensed in the value function that gives the maximum attainable utility resulting from the decisions taken today.

宏观经济学代考

经济代写|宏观经济学代写宏观经济学代考|求解时间剖面和消费水平

.

$$u^{\prime \prime}\left(c_t\right) \dot{c}_t=\dot{\lambda}_t .$$

$$\frac{u^{\prime \prime}\left(c_t\right) c_t}{u^{\prime}\left(c_t\right)} \frac{\dot{c}_t}{c_t}=\frac{\dot{\lambda}_t}{\lambda_t} .$$

$$\sigma \equiv\left[-\frac{u^{\prime \prime}\left(c_t\right) c_t}{u^{\prime}\left(c_t\right)}\right]^{-1}>0$$

$$\frac{\dot{c}_t}{c_t}=-\sigma \frac{\dot{\lambda}_t}{\lambda_t} .$$

$$\frac{\dot{c}_t}{c_t}=\sigma(r-\rho)=0 .$$
(11.12)表示，由于我们假设$r=\rho$，消耗是恒定的。因此可以得出
$$c_t=c^,$$，因此消费是恒定的。我们现在需要解出消费水平$c^$。利用(11.13)在(11.2)中我们得到
$$\dot{b}_t=r b_t+w_t-c^,$$，这是$b$中的一个微分方程，它的解是，对于任何时间$t>0$, $$b_t=\int_0^t w_s e^{r(t-r)} d s-\left(e^{n t}-1\right) \frac{c^}{r}+b_0 e^{n t} .$$
，其中时间$v$是0到$t$之间的任何时刻。在$t=T$处计算这个值(终止期)，我们得到代理生命周期结束时的债券存量:
$$b_T=\int_0^T w_s e^{r(T-s)} d s-\left(e^{r T}-1\right) \frac{c^}{r}+b_0 e^{r T} .$$两边除以$e^{r T}$并重新排列，我们得到$$b_T e^{-r T}=\int_0^T w_t e^{-r s} d s-\left(1-e^{-r T}\right) \frac{c^}{r}+b_0 .$$

$$u^{\prime}\left(c^*\right) b_T e^{-r T}=0 .$$

经济代写|宏观经济学代写宏观经济代考|消费与不确定性

\begin{aligned} &\operatorname{maxE}\left[\sum_{t=0}^T \frac{1}{(1+\rho)^t} u\left(c_t\right)\right], \ &\text { s.t. } b_{t+1}=\left(w_t+b_t-c_t\right)(1+r) . \end{aligned}

$$u^{\prime}\left(c_1\right)=\left(\frac{1+r}{1+\rho}\right) E_1\left[u^{\prime}\left(c_2\right)\right] .$$

$$u^{\prime}\left(c_t\right)=\left(\frac{1+r}{1+\rho}\right) E_t\left[u^{\prime}\left(c_{t+1}\right)\right] .$$

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