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## 经济代写|微观经济学代写Microeconomics代考|Aggregability of Consumers: Gorman Aggregability

When one is interested in Marshallian demand, the existence of a representative consumer requires different conditions from those guaranteeing the existence of a representative consumer in the determination of general equilibrium. The difference is seldom stressed but is of some importance.
Let us start with the Marshallian perspective.
Suppose there are $\mathrm{S}$ consumers, distinguished by index $\mathrm{s}=1, \ldots, \mathrm{S}$, each one with a Marshallian vectorial demand function $\mathbf{x}^{\mathrm{s}}\left(\mathbf{p}, m^{\mathrm{s}}\right), \mathbf{x} \in \mathrm{R}^{\mathrm{k}}$. For each consumer, income is given, so the analysis does not examine what determines incomes; the prices and rentals of the goods and factors in the endowments of consumers are implicitly treated as given; vector $p$ includes only a restricted set of prices, the prices of the consumption goods and services consumers demand. Indicate the market (vectorial) demand function deriving from the choices of these $\mathrm{S}$ consumers as
$$\mathbf{X}\left(\mathbf{p}, \mathrm{m}^1, \ldots, \mathrm{m}^{\mathrm{s}}\right):=\sum_{s=1}^s \mathbf{x}^s\left(\mathbf{p}, \mathrm{m}^{\mathrm{s}}\right)$$
where $\mathbf{p} \in \mathrm{R}_{+}^{\mathrm{k}}{ }^{57}$ Four properties of individual demand functions are inherited by this market Marshallian demand function.
First, $\mathbf{X}\left(\mathbf{p}, \mathrm{m}^1, \ldots, \mathrm{m}^{\mathrm{s}}\right)$ is homogeneous of degree zero.
Second, $\mathrm{X}\left(\mathrm{p}, \mathrm{m}^1, \ldots, \mathrm{m}^{\mathrm{s}}\right)$ is continuous if the individual demand functions are continuous.

## 经济代写|微观经济学代写Microeconomics代考|General Equilibrium of Pure Exchange

The issue of aggregability of consumers in general equilibrium requires a description of the latter. For the moment we can only discuss the general equilibrium of the exchange economy since we have not discussed production. This is often called general equilibrium of pure exchange in order to stress that the model admits no production at all. We need to generalize the simple economy discussed in $>$ Sect. 4.7. The intuition is provided by the ‘forest economy’ of $-$ Chap. $3 .$

The economy consists of a finite number of consumers, indexed $1, \ldots, s, \ldots, S$, and of a finite number of perfectly divisible perishable commodities indexed $1, \ldots, i, \ldots, N$. Each consumer $s$ has given preferences and a given $n$-vector of endowments $\omega^s=\left(\omega_1^s, \ldots, \omega_n^s\right)$.

Consumers are price-takers; given a vector of prices $\mathbf{p}$, for each consumer utility maximization can determine a unique vector $\mathbf{x}^3$ of consumption demands that maximizes her utility under the budget constraint ‘value of consumption basket $\leq$ value of endowments’, or more than one such vector, among which the consumer is indifferent; let us indicate this best-choice set as $d^{\mathrm{s}}(\mathbf{p})$, it can contain one vector $\mathbf{x}^{\mathbf{s}}$, or a finite, or even an infinite, number of vectors. The consumer solves a Walrasian UMP. Note that we can imagine consumer $s$ first selling all her endowment and obtaining $\mathbf{p} \omega^5$ and then using this sum of purchasing power to buy the (or an) optimal $\mathbf{x}^3$; therefore the consumer’s optimal consumption basket at prices $\mathbf{p}$ can also be obtained from a Marshallian UMP with $m r^s=\mathbf{p} \omega^5$, therefore $\mathbf{p} \omega^5$ can be called the consumer’s income or wealth at prices $\mathbf{p}$.

If $d^5(\mathbf{p})$ contains more than one vector, each one of them is a possible demand vector of the consumer, ${ }^{58}$ to which corresponds a vector of excess demands $\mathbf{x}^s(\mathbf{p})-\omega^s$, whose elements can be positive or negative. The set of excess demands $d^5(\mathbf{p})-\omega^s$ changes with $\mathbf{p}$, generating the excess demand correspondence of consumer $s$. For each p, unless each consumer’s demand vector is unique, the aggregate or market excess demand for a good can be the sum of any combination of choices of the several consumers, so it can be considerably indeterminate: if at the given $\mathbf{p}$ all consumers but five have a uniquely determined demand for good 1 and those five different consumers have each two possible demands for good 1 , there are 32 possible market demands for good 1 . Clearly, to reach any definite result the theory of general equilibrium must assume that non-uniquely determined demands are an extremely rare occurrence; this assumption is universally tacitly made, with no discussion of its legitimacy, and I will conform to this habit, but the issue would deserve more discussion.

# 微观经济学代考

## 经济代写|微观经济学代写Microeconomics代考|Aggregability of Consumers: Gorman Aggregability

$$\mathbf{X}\left(\mathbf{p}, \mathrm{m}^1, \ldots, \mathrm{m}^{\mathrm{s}}\right):=\sum_{s=1}^s \mathbf{x}^s\left(\mathbf{p}, \mathrm{m}^{\mathrm{s}}\right)$$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师