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# 经济代写|宏观经济学代考Macroeconomics代写|ECON311 Crusoe’s Production Possibilities

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## 经济代写|宏观经济学代考Macroeconomics代写|Crusoe’s Production Possibilities

Crusoe uses factors of production in order to make output $y$. We can think of this output as being coconuts. Two common factors of production, and those we consider here, are capital $k$ and labor $l$. Capital might be coconut trees, and labor is the amount of time Crusoe works, measured as a fraction of a day. How much Crusoe produces with given resources depends on the type of technology $A$ that he employs. We formalize this production process via a production function.

We often simplify our problems by assuming that the production function takes some particular functional form. As a first step, we often assume that it can be written: $y=A f(k, l)$, for some function $f(\cdot)$. This means that as technology $A$ increases, Crusoe can get more output for any given inputs. It is reasonable to require the function $f(\cdot)$ to be increasing in each argument. This implies that increasing either input $k$ or $l$ will increase production. Another common assumption is that output is zero if either input is zero: $f(0, l)=0$ and $f(k, 0)=0$, for all $k$ and $l$.

One functional form that has these properties is the Cobb-Douglas function, for example:

$y=A k^{1-\alpha} l^\alpha$, for some $\alpha$ between zero and one. This particular Cobb-Douglas function exhibits constant returns to scale, since $(1-\alpha)+(\alpha)=1$. Figure $2.1$ is a three-dimensional rendering of this function for particular values of $A$ and $\alpha$.

We will not be dealing with capital $k$ until Chapter 9, so for now we assume that capital is fixed, say, at $k=1$. This simplifies the production function. With a slight abuse of notation, we redefine $f(\cdot)$ and write production as $y=f(l)$. This is like what Barro uses in Chapter 2.
If the original production function was Cobb-Douglas, $y=A k^{1-\alpha} l^\alpha$, then under $k=1$ the production function becomes: $y=A l^\alpha$. The graph of this curve is just a slice through the surface depicted in Figure 2.1. It looks like Barro’s Figure 2.1.

## 经济代写|宏观经济学代考Macroeconomics代写|.2 Crusoe’s Preferenc

Crusoe cares about his consumption $c$ and his leisure. Since we are measuring labor $l$ as the fraction of the day that Crusoe works, the remainder is leisure. Specifically, leisure is $1-l$. We represent his preferences with a utility function $u(c, l)$. Take note, the second argument is not a “good” good, since Crusoe does not enjoy working. Accordingly, it might have been less confusing if Barro had written utility as $v(c, 1-l)$, for some utility function $v(\cdot)$. We assume that Crusoe’s preferences satisfy standard properties: they are increasing in each “good” good, they are convex, etc.

We will often simplify the analysis by assuming a particular functional form for Crusoe’s preferences. For example, we might have: $u(c, l)=\ln (c)+\ln (1-l)$. With such a function in hand, we can trace out indifference curves. To do so, we set $u(c, l)$ to some fixed number $\bar{u}$, and solve for $c$ as a function of $l$. Under these preferences, we get:
$$c=\frac{e^{\bar{u}}}{1-l} .$$
As we change $\bar{u}$, we get different indifference curves, and the set of those looks like Barro’s Figure 2.6. These should look strange to you because they are increasing as we move to the right. This is because we are graphing a “bad” good (labor $l$ ) on the horizontal axis. If we graph leisure $(1-l)$ instead, then we will get indifference curves that look like what you saw in your microeconomics courses.

# 宏观经济学代写

## 经济代写|宏观经济学代考Macroeconomics代写|Crusoe’s Production

Possibilities 本可能是椰子树，而劳动是克鲁索工作的时间，以一天的一小部分来衡量。克鲁索用给定的栥源生产多少取决于技术的类型 $A$ 他雇 用的。我们通过生产函数形式化这个生产过程。

$y=A k^{1-\alpha} l^\alpha$ ，对于一些 $\alpha$ 在零和一之间。这个特殊的 Cobb-Douglas 函数表现出规模报酬不变，因为 $(1-\alpha)+(\alpha)=1$. 数字 $2.1$ 是这个函数对于特定值的三维渲染 $A$ 和 $\alpha$.

## 经济代写|宏观经济学代考Macroeconomics代写|.2 Crusoe’s Preferenc

$$c=\frac{e^u}{1-l} .$$

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## MATLAB代写

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