## 经济代写|宏观经济学代写Macroeconomics代考|Firm’s problem

The firm’s objective funetion is to maximise the diseounted value of profits:
$$\int_0^{\infty} \pi_t e^{-r t} d t,$$
where $\pi_t$ denotes firm profits and $r>0$ is (constant and exogenously-given) real interest rate. The profit function is ${ }^4$
$$\pi_t=y_t-\psi\left(i_t, k_t\right)-i_t,$$
where $y_t$ is output and $\psi\left(i_t, k_t\right)$ is the cost of investing at the rate $i_t$, when the stock of capital is $k_t$. The term $\psi\left(i_t, k_t\right)$ is the key to this model; it implies adjustment costs, or costs of investing. Notice that if there are no costs of adjustment, given our assumption of a constant and exogenous interest rate, firms should go right away to their optimal stock. This would give an instantaneous investment function that is undefined, the investment rate either being zero or plus or minus infinite. However, in reality it seems that investment decisions are smoother and this has to mean that there are costs that make it very difficult if not impossible to execute large and instantaneous jumps in the stock of capital. Why would there be costs? We can think of several reasons. One is time-to-build; it simply takes time to build a facility, a dam, a power plant, a deposit, etc. This naturally smooths investment over time. Now, if you really want to hurry up, you can add double shifts, more teams, squeeze deadlines, etc., but all this increases the cost of investment expansions. We thus introduce the costs of adjustment equation as a metaphor for all these frictions in the investment process.

What is the equation of motion that constrains our firm? It is simply that the growth of capital must be equal to the rate of investment:
$$\dot{k}_t=i_t .$$
The production function is our familiar
$$y_t=A f\left(k_t\right),$$
where $A$ is a productivity coefficient and where $f^{\prime}(\cdot)>0, f^{\prime \prime}(\cdot)<0$, and Inada conditions hold.

## 经济代写|宏观经济学代写Macroeconomics代考|The consumer’s problem

The utility function is
$$\int_0^{\infty} u\left(c_t\right) e^{-\rho t} d t$$
where $c_t$ denotes consumption of the only traded good and $\rho(>0)$ is the rate of time preference. We assume no population growth.
The consumer’s flow budget constraint is
$$\dot{b}_t=r b_t+\pi_t-c_t,$$
where $b_t$ is the (net) stock of the internationally-traded bond; $r$ is the (constant and exogenouslygiven) world real interest rate; and $\pi_t$ is firm profits. Notice that the consumer is small and, therefore, takes the whole sequence of profits as given when maximising his utility.

Notice that the LHS of the budget constraint is also the economy’s current account: the excess of national income (broadly defined) over national consumption.
Finally, the solvency (No-Ponzi game) condition is
$$\lim _{T \rightarrow \infty} b_T e^{-r T}=0 .$$
The Hamiltonian can be written as
$$H=u\left(c_t\right)+\lambda_t\left[r b_t+\pi_t-c_t\right]$$
where $\lambda_t$ is the costate corresponding to the state $b_t$, while control variables is $c_t$.
The first order condition with respect to the control variables is
$$u^{\prime}\left(c_t\right)=\lambda_t$$
The law of motion for the costate is
$$\dot{\lambda}_t=\lambda_t(\rho-r)=0,$$
where the second equality comes from the fact that, as usual, we assume $r=\rho$.
Since $\lambda$ cannot jump in response to anticipated events, equations (13.34) and (13.35) together say that the path of consumption will be flat over time. In other words, consumption is perfectly smoothed over time. Along a perfect foresight path the constant value of $c_t$ is given by
$$c_0=c_t=r b_0+r \int_0^{\infty} \pi_t e^{-r t} d t, \quad t \geq 0 .$$

# 宏观经济学代考

## 经济代写|宏观经济学代写宏观经济代考|公司的问题

$$\int_0^{\infty} \pi_t e^{-r t} d t,$$

$$\pi_t=y_t-\psi\left(i_t, k_t\right)-i_t,$$
，其中$y_t$是产出，$\psi\left(i_t, k_t\right)$是按比率$i_t$投资的成本，而资本存量是$k_t$。术语$\psi\left(i_t, k_t\right)$是这个模型的关键;它意味着调整成本，或投资成本。请注意，如果没有调整成本，假设我们的利率是恒定的和外生的，那么公司应该立即转向他们的最优股票。这将给出一个未定义的瞬时投资函数，投资率要么为零，要么为正负无穷。然而，在现实中，投资决策似乎更加顺利，这必然意味着存在成本，使得在资本存量上执行大幅的瞬时跃升变得非常困难(如果不是不可能的话)。为什么会有成本?我们可以想到几个原因。一个是建造时间;建造一个设施，一个水坝，一个发电厂，一个矿藏等等，都需要时间。随着时间的推移，这自然会使投资变得顺畅。现在，如果你真的想加快进度，你可以增加两班倒，更多的团队，压缩截止日期等，但所有这些都增加了投资扩张的成本。因此，我们引入调整成本方程作为投资过程中所有这些摩擦的隐喻

$$\dot{k}_t=i_t .$$

$$y_t=A f\left(k_t\right),$$
，其中$A$是生产力系数，其中$f^{\prime}(\cdot)>0, f^{\prime \prime}(\cdot)<0$, Inada条件成立

## 经济代写|宏观经济学代写宏观经济学代考|消费者的问题

$$\int_0^{\infty} u\left(c_t\right) e^{-\rho t} d t$$

$$\dot{b}_t=r b_t+\pi_t-c_t,$$

$$\lim _{T \rightarrow \infty} b_T e^{-r T}=0 .$$

$$H=u\left(c_t\right)+\lambda_t\left[r b_t+\pi_t-c_t\right]$$
，其中$\lambda_t$是对应于状态$b_t$的共态，而控制变量是$c_t$。

$$u^{\prime}\left(c_t\right)=\lambda_t$$

$$\dot{\lambda}_t=\lambda_t(\rho-r)=0,$$

$$c_0=c_t=r b_0+r \int_0^{\infty} \pi_t e^{-r t} d t, \quad t \geq 0 .$$

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