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# 经济代写|宏观经济学代考Macroeconomics代写|ECON1002 An Infinite-Period Model

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## 经济代写|宏观经济学代考Macroeconomics代写|An Infinite-Period Model

The version of the model in which the representative household lives for an infinite number of periods is similar to the two-period model from the previous section. The utility of the household is now:
$$U\left(c_1, c_2, \ldots\right)=u\left(c_1\right)+\beta u\left(c_2\right)+\beta^2 u\left(c_3\right)+\cdots .$$
In each period $t$, the household faces a budget constraint:
$$P y_t+b_{t-1}(1+R)=P c_t+b_t .$$
Since the household lives for all $t=1,2, \ldots$, there are infinitely many of these budget constraints. The household chooses $c_t$ and $b_t$ in each period, so there are infinitely many choice variables and infinitely many first-order conditions. This may seem disconcerting, but don’t let it intimidate you. It all works out rather nicely. We write out the maximization problem in condensed form as follows:
$$\begin{gathered} \max {\left{c_t, b_t\right}{t=1}^{\infty}} \sum_{t=1}^{\infty} \beta^{t-1} u\left(c_t\right), \text { such that: } \ P y_t+b_{t-1}(1+R)=P c_t+b_t, \forall t \in{1,2, \ldots} . \end{gathered}$$
The ” $\forall$ ” symbol means “for all”, so the last part of the constraint line reads as “for all $t$ in the set of positive integers”.

To make the Lagrangean, we follow the rules outlined on page 15. In each time period $t$, the household has a budget constraint that gets a Lagrange multiplier $\lambda_t$. The only trick is that we use summation notation to handle all the constraints:
$$\mathcal{L}=\sum_{t=1}^{\infty} \beta^{t-1} u\left(c_t\right)+\sum_{t=1}^{\infty} \lambda_t\left[P y_t+b_{t-1}(1+R)-P c_t-b_t\right] .$$
Now we are ready to take first-order conditions. Since there are infinitely many of them, we have no hope of writing them all out one by one. Instead, we just write the FOCs for period-t variables. The $c_t \mathrm{FOC}$ is pretty easy:
$$\left(\text { FOC } c_t\right) \quad \frac{\partial \mathcal{L}}{\partial c_t}=\beta^{t-1} u^{\prime}\left(c_t^{\star}\right)+\lambda_t^{\star}[-P]=0$$

## 经济代写|宏观经济学代考Macroeconomics代写|A Present-Value Budget Constrai

Now we turn to a slightly different formulation of the model with the infinitely-lived representative household. Instead of forcing the household to balance its budget each period, now the household must merely balance the present value of all its budgets. (See Barro’s page 71 for a discussion of present values.) We compute the present value of all the household’s income:
$$\sum_{t=1}^{\infty} \frac{P y_t}{(1+R)^{t-1}} .$$

This gives us the amount of dollars that the household could get in period 1 if it sold the rights to all its future income. On the other side, the present value of all the household’s consumption is:
$$\sum_{t=1}^{\infty} \frac{P c_t}{(1+R)^{t-1}} .$$
Putting these two present values together gives us the household’s single present-value budget constraint. The household’s maximization problem is:
\begin{aligned} \max {\left{c_t\right}{t=1}^{\infty}} & \sum_{t=1}^{\infty} \beta^{t-1} u\left(c_t\right), \text { such } \ & \sum_{t=1}^{\infty} \frac{P\left(y_t-c_t\right)}{(1+R)^t-1}=0 . \end{aligned}
We use $\lambda$ as the multiplier on the constraint, so the Lagrangean is:
$$\mathcal{L}=\sum_{t=1}^{\infty} \beta^{t-1} u\left(c_t\right)+\lambda\left[\sum_{t=1}^{\infty} \frac{P\left(y_t-c_t\right)}{(1+R)^t-1}\right] .$$
The first-order condition with respect to $c_t$ is:
$\left(\mathrm{FOC} c_t\right)$
$$\beta^{t-1} u^{\prime}\left(c_t^{\star}\right)+\lambda^{\star}\left[\frac{P(-1)}{(1+R)^{t-1}}\right]=0 .$$
Rotating this forward and dividing the $c_t$ FOC by the $c_{t+1}$ FOC yields:
$$\frac{\beta^{t-1} u^{\prime}\left(c_t^{\star}\right)}{\beta^t u^{\prime}\left(c_{t+1}^{\star}\right)}=\frac{\lambda^{\star}\left[\frac{P}{(1+R)^t-T}\right]}{\lambda^{\star}\left[\frac{P}{(1+R)}\right]},$$
which reduces to:
$$\frac{u^{\prime}\left(c_t^{\star}\right)}{u^{\prime}\left(c_{t+1}^{\star}\right)}=\beta(1+R),$$
so we get the same Euler equation once again. It turns out that the problem faced by the household under the present-value budget constraint is equivalent to that in which there is a constraint for each period. Hidden in the present-value version are implied bond holdings. We could deduce these holdings by looking at the sequence of incomes $y_t$ and chosen consumptions $c_t^{\star}$.

# 宏观经济学代写

## 经济代写|宏观经济学代考Macroeconomics代写|An Infinite-Period Model

$$U\left(c_1, c_2, \ldots\right)=u\left(c_1\right)+\beta u\left(c_2\right)+\beta^2 u\left(c_3\right)+\cdots .$$

$$P y_t+b_{t-1}(1+R)=P c_t+b_t .$$

\left 䄴少或无法识别的分隔符

$$\mathcal{L}=\sum_{t=1}^{\infty} \beta^{t-1} u\left(c_t\right)+\sum_{t=1}^{\infty} \lambda_t\left[P y_t+b_{t-1}(1+R)-P c_t-b_t\right]$$
$c_t$ FOC很简单:
$$\left(\mathrm{FOC} c_t\right) \quad \frac{\partial \mathcal{L}}{\partial c_t}=\beta^{t-1} u^{\prime}\left(c_t^{\star}\right)+\lambda_t^{\star}[-P]=0$$

## 经济代写|宏观经济学代考Macroeconomics代写|A Present-Value Budget Constrai

$$\sum_{t=1}^{\infty} \frac{P y_t}{(1+R)^{t-1}} .$$

$$\sum_{t=1}^{\infty} \frac{P c_t}{(1+R)^{t-1}} .$$

$$\mathcal{L}=\sum_{t=1}^{\infty} \beta^{t-1} u\left(c_t\right)+\lambda\left[\sum_{t=1}^{\infty} \frac{P\left(y_t-c_t\right)}{(1+R)^t-1}\right] .$$

$\left(\mathrm{FOC} c_t\right)$
$$\beta^{t-1} u^{\prime}\left(c_t^{\star}\right)+\lambda^{\star}\left[\frac{P(-1)}{(1+R)^{t-1}}\right]=0 .$$

$$\frac{\beta^{t-1} u^{\prime}\left(c_t^{\star}\right)}{\beta^t u^{\prime}\left(c_{t+1}^{\star}\right)}=\frac{\lambda^{\star}\left[\frac{P}{(1+R)^t-T}\right]}{\lambda^{\star}\left[\frac{P}{(1+R)}\right]},$$

$$\frac{u^{\prime}\left(c_t^{\star}\right)}{u^{\prime}\left(c_{t+1}^{\star}\right)}=\beta(1+R),$$

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