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经济代写|微观经济学代写Microeconomics代考|Continuity: Survival
A possible cause of non-convexity of consumption sets, which may cause a discontinuity of demand for many consumers at the same price, is that a minimum level of consumption is necessary in order to survive. Let me quote here a passage from Debreu (1959), which accompanies a drawing from which $\mathbf{\bullet i g} .6 .5$ has been copied:
) …consider the case where there are one location and two dates; a certain foodstuff at the first date defines the first commodity, the same foodstuff at the second date defines the second commodity. Let the length of $[\mathrm{O}, \mathrm{O}]$ be the minimum quantity of the first commodity which that consumer must have available in order to survive until the end of the first elementary time-interval. If his input of the first commodity is less than or equal to this minimum, it might seem, on first thought, that his input of the second commodity must be zero. The set $X_i$ [i.e. the consumption set of consumer $i$, F.P.] would therefore consist of the closed segment $[\mathrm{O}, \mathrm{O}]$ and a subset of the closed quadrant $1, O^{\prime}, 2^{\prime}$. Such a set has the disadvantage of not being convex in general. However, if both commodities are freely disposable, the set $X_i$ is the closed quadrant $1, \mathrm{O}, 2$, which is convex: if the consumer chooses (perhaps because he is forced to) a consumption $x_i$ in the closed strip $2, \mathrm{O}^{\prime}, \mathrm{O}^{\prime}, 2^{\prime}$, it means that $x_{i 1}$ of the first commodity is available to him and he will actually consume at most that much of it, and that $x_{i 2}$ of the second commodity is available to him and he will actually consume none of it.
The choice by the $i$ th consumer of $x_i$ in $X_i$ determines implicitly his life span (Debreu, 1959, pp. 51-2).
(The last sentence in this quotation may appear disconcerting, but it refers to the fact that Debreu at this stage of his treatise is assuming no uncertainty, so consumers know the date when they will die and how that depends on their choices.)
Once again, the distinction between availability and actual consumption (plus the free disposal assumption) is used to avoid the non-convexity of the consumption set. But Debreu does not seem to perceive that this solution does not avoid another danger: of discontinuity in the excess demands for the goods after the initial period. Many people’s endowment consists only of their labour/leisure. If the real wage gets sufficiently low for a number of periods, they may find it impossible to survive beyond those periods, and their supply of labour and demand for goods for the subsequent periods drops discontinuously to zero. ${ }^7$
Debreu has an intertemporal consumption set in mind, connected with the notion of intertemporal equilibrium which will be studied in depth here in -Chap. 8; but the idea should be clear, the consumer determines simultaneously her demands for today and for tomorrow, and too low a demand for consumption today implies a sudden jump to zero of demand tomorrow (and also of labour supply tomorrow) because the consumer does not survive to tomorrow. And intertemporal equilibria too need a continuity of intertemporal excess demands in order for existence not to be endangered.
经济代写|微观经济学代写Microeconomics代考|The Zero-Income Problem
Even if survival poses no problem, another kind of discontinuity due to income falling too much is possible. I call it the zero-income problem.
When income depends on the prices of endowments, a zero price can cause discontinuities of excess demand. Consider the following example. In a two-goods exchange economy a consumer has a monotonic utility function $u=x_1^{1 / 2}+x_2^{1 / 2}$ and has a positive endowment consisting only of good 1, i.e. $\omega_1>0, \omega_2=0$. Let us determine the demand function for good 1. Indifference curves have slope MRS $=-x_2^{1 / 2} / x_1^{1 / 2}$. Utility maximization requires tangency between indifference curve and budget line, i.e.
$$
x_2^{1 / 2} / x_1^{1 / 2}=p_1 / p_2
$$
under the budget constraint that we can initially specify in the Marshallian way:
$$
p_1 x_1+p_2 x_2=m, \quad \text { where } m=p_1 \omega_1 .
$$
Solving these two equations in two variables, and assuming $p_2=1$ one obtains (check it!) $x_1=m /\left(p_1+p_1^2\right)$. Now we use $m=p_1 \omega_1$ to obtain
$$
x_1=\omega_1 /\left(1+p_1\right) .
$$
Thus as long as $p_1$ is positive, $x_1$ is positive and less than $\omega_1$, the consumer is a net supplier of good 1. Let $p_1$ tend to zero: income tends to zero and $x_1$ tends to $\omega_1$; as long as $p_1>0$ the consumer remains a net supplier of good 1 , but when $p_1$ is zero the demand for good 1 is infinite. This means that $\lim _{p 1 \rightarrow 0} x_1=\omega_1$ but at $p_1=0$ the demand for good 1 jumps discontinuously to $+\infty$ as $\boldsymbol{F i g} .6 .6$ illustrates.
This is called a zero-income problem because the discontinuity would not arise if the consumer’s income did not tend to zero as $p_1$ tends to zero.
Then a general equilibrium of exchange can fail to exist. As long as $p_1>0$, our consumer is a net supplier of good 1 to the market. Suppose that no one else is interested in good 1 ; then at any positive $p_1$ there is excess supply of good 1 and $p_1$ decreases; as long as $p_1>0$ the excess supply does not disappear and there isn’t equilibrium; when $p_1$ becomes zero, our consumer’s excess demand for good 1 shoots up to $+\infty$, and there isn’t equilibrium. There is no price at which there is equilibrium on the market for good 1; a general equilibrium does not exist.
To separate this problem from the survival problem, we can assume that even a zero income ensures survival (the consumption set coincides with $R_{+}^n$ ).
经济代写|微观经济学代写微观经济学代考|连续性:生存
消费集的非凸性可能会导致许多消费者在相同价格下的需求不连续，造成这种情况的一个可能原因是，为了生存，必须有最低的消费水平。让我在这里引用Debreu(1959)的一段话，其中附有一幅图，$\mathbf{\bullet i g} .6 .5$已被复制:
)……考虑一个地点和两个日期的情况;第一次约会时的某种食物定义了第一种商品，第二次约会时同样的食物定义了第二种商品。设$[\mathrm{O}, \mathrm{O}]$的长度是消费者为了生存到第一个基本时间区间结束所必须拥有的第一种商品的最小数量。如果他对第一种商品的投入小于或等于这个最小值，乍一想，他对第二种商品的投入可能是零。集合$X_i$[即消费者$i$的消费集，F.P.]因此将由封闭段$[\mathrm{O}, \mathrm{O}]$和封闭象限$1, O^{\prime}, 2^{\prime}$的一个子集组成。这样的集合的缺点是通常不是凸的。然而，如果两种商品都是可自由丢弃的，则集合$X_i$为封闭象限$1, \mathrm{O}, 2$，其为凸:如果消费者选择(可能是因为他被迫)在封闭的带$2, \mathrm{O}^{\prime}, \mathrm{O}^{\prime}, 2^{\prime}$中的消费$x_i$，这意味着第一种商品的$x_{i 1}$对他来说是可用的，他实际上最多消费那么多，而第二种商品的$x_{i 2}$对他来说是可用的，他实际上不会消费它。$i$第一个消费者在$X_i$中对$x_i$的选择隐式地决定了他的寿命(Debreu, 1959, pp. 51-2)。
(这段引语中的最后一句话可能看起来令人不安，但它指的是Debreu在他的论文的这一阶段是假设没有不确定性的，因此消费者知道他们的死亡日期，以及这如何取决于他们的选择)
再次，可用性和实际消费(加上自由处置假设)之间的区别被用来避免消费集的非凸性。但德布鲁似乎并没有意识到，这种解决办法并不能避免另一种危险:在最初阶段之后，对商品的超额需求会中断。许多人的禀赋只来源于他们的劳动/闲暇。如果实际工资在若干时期内处于足够低的水平，他们可能会发现无法在这些时期之后继续生存下去，他们的劳动力供应和随后时期的商品需求就会不连续地降为零。${ }^7$
Debreu在脑海中设定了跨期消费，与跨期平衡的概念相联系，这一概念将在本章中深入研究。8;但这个想法应该很清楚，消费者同时决定了她对今天和明天的需求，今天的消费需求过低意味着明天的需求(以及明天的劳动力供应)突然跳到零，因为消费者活不到明天。跨期平衡也需要跨期超额需求的连续性，以使其存在不受威胁
经济代写|微观经济学代写微观经济学代考|零收入问题
即使生存不成问题，由于收入下降太多而导致的另一种不连续也是有可能的。我称之为“零收入问题”
当收入依赖于禀赋的价格时，零价格会导致超额需求的不连续。考虑下面的例子。在两种商品交换经济中，消费者具有单调效用函数$u=x_1^{1 / 2}+x_2^{1 / 2}$，且具有仅由商品1构成的正禀赋，即$\omega_1>0, \omega_2=0$。我们来确定商品1的需求函数。无差异曲线有斜率MRS $=-x_2^{1 / 2} / x_1^{1 / 2}$。效用最大化需要无差异曲线和预算线之间的切线，即
$$
x_2^{1 / 2} / x_1^{1 / 2}=p_1 / p_2
$$
在预算约束下，我们最初可以用马歇尔方法指定:
$$
p_1 x_1+p_2 x_2=m, \quad \text { where } m=p_1 \omega_1 .
$$
用两个变量解这两个方程，假设$p_2=1$得到(检查它!)$x_1=m /\left(p_1+p_1^2\right)$。现在我们用$m=p_1 \omega_1$得到
$$
x_1=\omega_1 /\left(1+p_1\right) .
$$
因此只要$p_1$为正数，$x_1$为正数且小于$\omega_1$，消费者就是商品1的净供应商。让$p_1$趋于零:收入趋于零，$x_1$趋于$\omega_1$;只要$p_1>0$消费者仍然是商品1的净供应商，但当$p_1$为零时，对商品1的需求是无限的。这意味着$\lim _{p 1 \rightarrow 0} x_1=\omega_1$，但在$p_1=0$，商品1的需求不连续地跳到$+\infty$，如$\boldsymbol{F i g} .6 .6$所示。
这被称为零收入问题，因为当$p_1$趋于零时，如果消费者的收入不趋于零，不连续就不会出现
那么一般交换均衡可能不存在。只要$p_1>0$，我们的消费者就是市场的净供给者。假设没有人对商品1感兴趣;然后在任何正值$p_1$处，商品1供给过剩，$p_1$减少;只要$p_1>0$过剩供给不消失，不存在均衡;当$p_1$为0时，消费者对商品1的超额需求激增到$+\infty$，不存在均衡。对于商品1，市场上不存在均衡价格;一般均衡不存在。
为了将这个问题从生存问题中分离出来，我们可以假设即使是零收入也能确保生存(消费集与$R_{+}^n$一致)
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