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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 经济代写|宏观经济学代写Macroeconomics代考|Knowledge as a factor of production

We have seen that the key to obtaining long-run growth in our models is to get constant returns in the reproducible factors. But this begs the question: why do we think that this would actually be the case in reality?

As we have seen, a world of constant returns to reproducible factors is, actually, a world with increusing returns lo scule (IRS) – after all, there is at least labour and techulugy in the production functions as well. But, this is a problem because IRS implies that our simple model with perfect competition doesn’t really work anymore.

To see why, note that with perfect competition, each factor of production gets paid its marginal product – you know that from Econ 101. However, if the production function is
$$F(A, X),$$
where $X$ has constant returns, then we have
$$F(A, X)<A \frac{\partial F}{\partial A}+X \frac{\partial F}{\partial X} .$$
There is not enough output to pay each factor their marginal productivity!
We had sidestepped this discussion up to this point, assuming that technology was there and was left unpaid. But now the time has come to deal with this issue head-on.

In doing so, we will build a bridge between what we have learned about accumulation and what we have talked about when referring to productivity. The crucial insight again is associated with Paul Romer, and can be summarised in one short sentence: economies can grow by accumulating “knowledge”.

But what drives the accumulation of knowledge? Knowledge is a tricky thing because it is difficult to appropriate, i.e. it has many of the properties of a public good. As you may remember, the two distinguishing characteristics of any good are

• Rivalry $\longrightarrow$ if I use it you can’t.
• Excludability $\longrightarrow$ I can prevent you from using it.
Private goods are rival and excludable, pure public goods are neither. Technology/knowledge is peculiar because it is non-rival, although excludable to some extent (with a patent, for example).

The non-rivalry of knowledge immediately gives rise to increasing returns. If you think about it, knowledge is a fixed cost: in order to produce one flying car, I need one blueprint for a flying car, but I don’t need a second blueprint to build a second unit of that flying car. In other words, one doesn’t need to double all inputs in order to double output.

## 经济代写|宏观经济学代写Macroeconomics代考|Learning by doing

This was first suggested by Romer (1987). The idea is that you become better at making stuff as you make it: knowledge is a by-product of production itself. This means that production generates an externality. If each firm does not internalise the returns to the knowledge they generate and that can be used by others, firms still face convex technologies even though there are increasing returns at the level of the economy. It follows that competitive behaviour can be preserved.
Let us model this with the following production function,
$$y=A k^a \bar{K}^\eta,$$
where $\bar{k}$ is the stock of knowledge (past investment). Given this we compute the (private) marginal product of capital and the growth rate:
$$f^{\prime}(k)=A \alpha k^{\alpha-1} k^\eta=A \alpha k^{\alpha+\eta-1},$$
$$\gamma_c=\sigma\left(A \alpha k^{\pi+\eta-1}-\rho\right) .$$
We have endogenous growth if $\alpha+\eta \geq 1$. Notice that we need CRS in the reproducible factors, and, hence, sufficiently strong IRS. It is not enough to have IRS; we need that $\eta \geq 1-\alpha$.
For a central planner who sees through the learning-by-doing exercise:
$$f(k)=A k^{\pi+\eta},$$
$$f^{\prime}(k)=(\alpha+\eta) A k^{\alpha+\eta-1},$$
It follows that the economy does not deliver the right amount of growth. Why? Because of the externality: private agents do not capture the full social benefit of their investment since part of it spills over to everyone else. This is a crucial lesson of the endogenous growth literature. Once we introduce IRS, there will typically be a wedge between the decentralised equilibrium and the optimal growth rate.

# 宏观经济学代考

## 经济代写|宏观经济学代写Macroeconomics代考|Knowledge as a factor of production

$$F(A, X)<A \frac{\partial F}{\partial A}+X \frac{\partial F}{\partial X} .$$

• 对抗 $\longrightarrow$ 如果我使用它，你不能。
• 排他性一我可以阻止你使用它。
私人物品具有竞诤性和排他性，纯粹的公共物品两者都不是。技术/知识是特殊的，因为它是非竞争性的，尽管在某种程度上是排他性的（例 如，有专利)。
知识的非竞争性立即产生递增的回报。如果你想一想，知识是固定成本: 为了生产一辆K行汽车，我需要一个飞行汽车的监图，但我不需要第二张 蓝图来建造第二辆 K行汽车。换句话说，不需要将所有输入加倍以使输出加倍。

## 经济代写|宏观经济学代写Macroeconomics代考|Learning by doing

$$y=A k^a \bar{K}^\eta$$

$$f^{\prime}(k)=A \alpha k^{\alpha-1} k^\eta=A \alpha k^{\alpha+\eta-1},$$
$$\gamma_c=\sigma\left(A \alpha k^{\pi+\eta-1}-\rho\right) .$$

$$f(k)=A k^{\pi+\eta},$$
$$f^{\prime}(k)=(\alpha+\eta) A k^{\alpha+\eta-1}$$

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## MATLAB代写

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