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## 经济代写|宏观经济学代写Macroeconomics代考|The balanced growth path and the Euler equation

We are ultimately interested in the dynamic behaviour of our control and state variables, $c_{t}$ and $k_{t}$. How can we turn our FOCs into a description of that behaviour (preferably one that we can represent graphically)? We start by taking (3.7) and differentiating both sides with respect to time:
$$u^{\prime \prime}\left(c_{t}\right) \dot{c}{t} e^{n t}+n u^{\prime}\left(c{t}\right) e^{n t}=\dot{\lambda}{t} .$$ Divide this by (3.7) and rearrange: $$\frac{u^{\prime \prime}\left(c{t}\right) c_{t}}{u^{\prime}\left(c_{t}\right)} \frac{\dot{c}{t}}{c{t}}=\frac{\dot{\lambda}{t}}{\lambda{t}}-n .$$

Next, define
$$\sigma \equiv-\frac{u^{\prime}\left(c_{t}\right)}{u^{\prime \prime}\left(c_{t}\right) c_{t}}>0$$
as the elasticity of intertemporal substitution in consumption. ${ }^{6}$ Then, (3.12) becomes
$$\frac{\dot{c}{t}}{c{t}}=-\sigma\left(\frac{\dot{\lambda}{t}}{\lambda{t}}-n\right) .$$
Finally, using (3.10) in (3.14) we obtain
$$\frac{\dot{c}{t}}{c{t}}=\sigma\left[f^{\prime}\left(k_{t}\right)-\rho\right] .$$
This dynamic optimality condition is known as the Ramsey rule (or Keynes-Ramsey rule), and in a more general context it is referred to as the Euler equation. It may well be the most important equation in all of macroeconomics: it encapsulates the essence of the solution to any problem that trades off today versus tomorrow. ${ }^{7}$

But what does it mean intuitively? Think about it in these terms: if the consumer postpones the enjoyment of one unit of consumption to the next instant, it will be incorporated into the capital stock, and thus yield an extra $f^{\prime}(\cdot)$. However, this will be worth less, by a factor of $\rho$. They will only consume more in the next instant (i.e. $\frac{c_{t}}{c_{t}}>0$ ) if the former compensates for the latter, as mediated by their proclivity to switch consumption over time, which is captured by the elasticity of intertemporal substitution, $\sigma$. Any dynamic problem we will see from now on involves some variation upon this investment) against the discount rate.

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

The household’s problem is to maximise (3.1) subject to (3.5) for given $k_{0}$. If you look at our mathematical appendix, you will learn how to solve this, but it is instructive to walk through the steps here, as they have intuitive interpretations. You will need to set up the (current value) Hamiltonian for the problem, as follows:
$$H=u\left(c_{t}\right) e^{n t}+\lambda_{t}\left[f\left(k_{t}\right)-n k_{t}-c_{t}\right] .$$
Recall that $c$ is the control variable (jumpy), and $k$ is the state variable (sticky), but the Hamiltonian brings to the forefront another variable: $\lambda$, the co-state variable. It is the multiplier associated with the intertemporal budget constraint, analogously to the Lagrange multipliers of static optimisation.

Just like its Lagrange cousin, the co-state variable has an intuitive economic interpretation: it is the marginal value as of time $t$ (i.e. the current value) of an additional unit of the state variable (capital, in this case). It is a (shadow) price, which is also jumpy.
First-order conditions (FOCs) are
$$\begin{gathered} \frac{\partial H}{\partial c_{t}}=0 \Rightarrow u^{\prime}\left(c_{t}\right) e^{n t}-\lambda_{t}=0, \ \dot{\lambda}{t}=-\frac{\partial H}{\partial k{t}}+\rho \lambda_{t} \Rightarrow \dot{\lambda}{t}=-\lambda{t}\left[f^{\prime}\left(k_{t}\right)-n\right]+\rho \lambda_{t}, \ \lim {t \rightarrow \infty}\left(k{t} \lambda_{t} e^{-\rho t}\right)=0 . \end{gathered}$$
What do these optimality conditions mean? First, (3.7) should be familiar from static optimisation: differentiate with respect to the control variable, and set that equal to zero. It makes sure that, at any given point in time, the consumer is making the optimal decision – otherwise, she could obviously do better… The other two are the ones that bring the dynamic aspects of the problem to the forefront. Equation (3.9) is known as the transversality condition (TVC). It means, intuitively, that the consumer wants to set the optimal path for consumption such that, in the “end of times” (at infinity, in this case), they are left with no capital. (As long as capital has a positive value as given by $\lambda$, otherwise they don’t really care…) If that weren’t the case, I would be “dying” with valuable capital, which I could have used to consume a little more over my lifetime.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|The balanced growth path and the Euler equation

$$u^{\prime \prime}\left(c_{t}\right) \dot{c} t e^{n t}+n u^{\prime}(c t) e^{n t}=\dot{\lambda} t .$$

$$\frac{u^{\prime \prime}(c t) c_{t}}{u^{\prime}\left(c_{t}\right)} \frac{\dot{c} t}{c t}=\frac{\dot{\lambda} t}{\lambda t}-n .$$

$$\sigma \equiv-\frac{u^{\prime}\left(c_{t}\right)}{u^{\prime \prime}\left(c_{t}\right) c_{t}}>0$$

$$\frac{\dot{c} t}{c t}=-\sigma\left(\frac{\dot{\lambda} t}{\lambda t}-n\right) .$$

$$\frac{\dot{c} t}{c t}=\sigma\left[f^{\prime}\left(k_{t}\right)-\rho\right] .$$

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

$$H=u\left(c_{t}\right) e^{n t}+\lambda_{t}\left[f\left(k_{t}\right)-n k_{t}-c_{t}\right]$$

$$\frac{\partial H}{\partial c_{t}}=0 \Rightarrow u^{\prime}\left(c_{t}\right) e^{n t}-\lambda_{t}=0, \dot{\lambda} t=-\frac{\partial H}{\partial k t}+\rho \lambda_{t} \Rightarrow \dot{\lambda} t=-\lambda t\left[f^{\prime}\left(k_{t}\right)-n\right]+\rho \lambda_{t}, \lim t \rightarrow \infty\left(k t \lambda_{t} e^{-\rho t}\right)=0$$

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