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## 经济代写|宏观经济学代写Macroeconomics代考|The AK model

The model in the previous section, just like the Solow model, was not micro-founded in terms of individual decisions. Let us now consider whether its lessons still hold in a framework with optimising individuals.

We have seen that the key aspect to obtaining long-run growth in the previous model is to have constant returns to reproducible factors when taken together. Including human capital as one such factor is but one way of generating that. To keep things as general as possible, though, we can think of all reproducible factors as capital, and we can subsume all of these models into the so-called AK model.

Consider once again a model with one representative household living in a closed economy, members of which consume and produce. There is one good, and no government. Population growth is 0 , and the population is normalised to 1 . All quantities (in small-case letters) are per-capita. Each consumer in the representative household lives forever.
The utility function is
$$\int_0^{\infty}\left(\frac{\sigma}{\sigma-1}\right) c_t^{\frac{\sigma-1}{\sigma}} e^{-\rho t} d t, \rho>0$$
where $c_t$ denotes consumption, $\rho$ is the rate of time preference and $\sigma$ is the elasticity of intertemporal substitution in consumption.
We have the linear production function from which the model derives its nickname:
$$Y_t=A k_t, A>0 .$$
Again, think of household production: the household owns the capital and uses it to produce output. The resource constraint of the economy is
$$\dot{k}_t=A k_t-c_t$$

## 经济代写|宏观经济学代写Macroeconomics代考|At long last, a balanced growth path with growth

Take (5.24) and differentiate both sides with respect to time, and divide the result by (5.24) to obtain
$$-\frac{1}{\sigma} \frac{\dot{c}_t}{c_t}=\frac{\dot{\lambda}_t}{\lambda_t} .$$
Multiplying through by $-\sigma,(5.27)$ becomes
$$\frac{\dot{c}_t}{c_t}=-\sigma\left(\frac{\dot{\lambda}_t}{\lambda_t}\right) .$$
Finally, using (5.25) in (5.28) we obtain
$$\frac{\bar{c}_t}{c_t}=\sigma(A-\rho),$$
which is the Euler equation. Note that here we have $f^{\prime}(k)=A$, so this result is actually the same as in the standard Ramsey model. The difference is in the nature of the technology, as now we have constant returns to capital.

Define a BGP once again as one in which all variables grow at a constant speed. From (5.22) we get
$$\frac{\dot{k}_t}{k_t}=A-\frac{c_t}{k_t} .$$
This implies that capital and consumption must grow at the same rate – otherwise we wouldn’t have $\frac{k_t}{k_1}$ constant. And since $y_t=A k_t$, output grows at the same rate as well. From (5.29) we know that this rate is $\sigma(A-\rho)$. Hence,
$$\frac{\dot{c}_t}{c_t}=\frac{\dot{k}_t}{k_t}=\frac{\dot{y}_t}{y_t}=\sigma(A-\rho) .$$
Note, there will be positive growth only if $A>\rho$ that is, only if capital is sufficiently productive so that it is desirable to accumulate it.
Second, from (5.30) we see that along a BGP we must have
$$y_t-c_t=\sigma(A-\rho) k_t \Rightarrow c_t=[(1-\sigma) A+\sigma \rho\rceil k_t=\left[\frac{(1-\sigma) A+\sigma \rho}{A}\right] y_t .$$
In words, consumption is proportional to capital. Or, put differently, the agent consumes a fixed share of output every period. Notice that this is much like the assumption made in Solow. If $s$ is the savings rate, here $1-s=\frac{(1-\sigma) A+\sigma \rho}{A}$, or $s=\sigma\left(\frac{A-\rho}{A}\right)$. The difference is that this is now optimal, not arbitrary. There are no transitional dynamics: the economy is always on the BGP.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|The AK model

$$\int_0^{\infty}\left(\frac{\sigma}{\sigma-1}\right) c_t^{\frac{\sigma-1}{\sigma}} e^{-\rho t} d t, \rho>0$$

$$Y_t=A k_t, A>0 .$$

$$\dot{k}_t=A k_t-c_t$$

## 经济代写|宏观经济学代写Macroeconomics代考|At long last, a balanced growth path with growth

$$-\frac{1}{\sigma} \frac{\dot{c}_t}{c_t}=\frac{\dot{\lambda}_t}{\lambda_t} .$$

$$\frac{\dot{c}_t}{c_t}=-\sigma\left(\frac{\dot{\lambda}_t}{\lambda_t}\right)$$

$$\frac{\bar{c}_t}{c_t}=\sigma(A-\rho)$$

$$\frac{\dot{k}_t}{k_t}=A-\frac{c_t}{k_t}$$

$$\frac{\dot{c}_t}{c_t}=\frac{\dot{k}_t}{k_t}=\frac{\dot{y}_t}{y_t}=\sigma(A-\rho)$$

$$y_t-c_t=\sigma(A-\rho) k_t \Rightarrow c_t=[(1-\sigma) A+\sigma \rho\rceil k_t=\left[\frac{(1-\sigma) A+\sigma \rho}{A}\right] y_t .$$

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