# 经济代写|微观经济学代写Microeconomics代考|BU1003

## 经济代写|微观经济学代写Microeconomics代考|The Non-uniqueness of Equilibrium

An equilibrium is composed of two ‘things’, an equilibrium allocation and an equilibrium relative prices vector. $\Lambda \mathrm{n}$ equilibrium is unique if both the equilib rium alluation and the equilibsium eclative prives valus ane miyuc. (Beluw I take it for granted that a numéraire has been chosen, so ‘different price vectors’ means non-proportional price vectors.) A unique equilibrium allocation may be supported by more than one relative prices vector, when indifference curves or isoquants have ‘kinks’ at the equilibrium allocation, such that-within limitsvariations of relative prices or of relative factor rentals do not induce changes in the agents’ decisions. But the more relevant cases are those of multiple equilibria differing in the equilibrium allocation.

Uniqueness of equilibrium requires restrictive assumptions, without which it is easy to produce examples with multiple equilibria. For example it is not difficult to draw an Edgeworth box with three equilibria, that is, three points other than the endowment point where the two choice curves cross, owing to the possible horseshoe-like shape of choice curves. The reader can try as an exercise to confirm, by drawing opportune maps of indifference curves that the choice curves can indeed have the shapes assumed in $\boldsymbol{-}$ Fig. 6.10.

In $\bullet$ Fig. $6.10$ the three slopes of the budget line that cause it to go through a point where the choice curves cross indicate the three equilibrium relative prices.
It is also possible that the two choice curves be tangent to each other, in the endowment point or elsewhere, cf. $\boldsymbol{F i g}$. 6.11; or that the two choice curves overlap over an entire stretch (cf. – Fig. 6.15), in which case there is a continuum of equilibria differing in relative price and in allocation-but these two cases are flukes that disappear with any small change in tastes or endowments.

The figures show that in a two-goods economy, if the choice curves cross each other three times, the equilibria are alternatively locally stable or unstable We can be content here with a simple intuitive description of the price adjustment process: a price tends to rise if a good is in excess demand, to decrease if in excess supply.

## 经济代写|微观经济学代写Microeconomics代考|Uniqueness Through Conditions on Excess Demand

The Sonnenschein-Mantel-Debreu result has caused great discomfort among the followers of the supply-and-demand approach, because of the indeterminateness it shows to be possible in the results of the equilibration process. But the damage would be considerably reduced if assumptions could be found, capable of avoiding multiple equilibria, and such that one could argue that they are nearly always satisfied in actual economies. I proceed to list the main assumptions so far discovered that ensure (i.e. are sufficient for) uniqueness and then to assess their plausibility.

Two types of assumptions have been explored in this respect: assumptions on $\mathbf{z}(\mathbf{p})$; and assumptions on the distribution of characteristics (preferences and endowments) of consumers. I list now the main results of the first type.
The oldest and most widely cited result is:
Proposition $6.5$ (Gross substitutes) Define the gross substitutes condition for consumers as follows: if the price of only one good varies, the consumer’s excess demands for all other goods vary in the same direction, and this is true whichever the price that varies. The gross substitutes condition holds for the market excess demand function of an exchange economy if in the above definition the market excess demands replace the consumer’s excess demands. It is then indicated as GS for brevity. If GS holds, then if the exchange economy has an equilibrium, this is unique.

More formally, GS states: the goods are all gross substitutes if, for any two price vectors $\mathbf{p}$ and $\mathbf{q}$ (not normalized to be in the unit simplex) such that $p_i=q_i$ except for one good, $j$, for which $q_j>p_j$, then $z_i(\mathbf{q})>z_i(\mathbf{p})$, $\forall i \neq j$.

The adjective ‘gross’ distinguishes this property from the usual, or Hicksian, definition of substitutes, which requires negative cross-derivatives of the compensated demand function.

In an exchange economy ${ }^{19} \mathbf{z}(\mathbf{p})$ is homogeneous of degree zero (because income derives from cndowments); then GS implics that the excess demand for a good decreases when its own price alone increases: the reason is that, because of the homogeneity of degree zero of excess demands, the increase of the sole jth price has the same effect on excess demands as a decrease in the same proportion of all other prices; and if we keep the jth price constant and let the other prices decrease one at a time, GS implies that at each step the excess demand for the jth good decreases.
Another way to define GS, for differentiable excess demands, is that it must be $\partial \mathrm{z}{\mathrm{i}}(\mathbf{p}) / \partial \mathrm{p}{\mathrm{j}}>0$ if $\mathrm{i} \neq \mathrm{j}$. This again implies that the demand for a good decreases when the price of that good increases, because it implies $\partial \mathrm{z} / \partial \mathrm{p}{\mathrm{j}}<0$ : the proof is that the homogeneity of degree zero of excess demand functions and Euler’s theorem on homogeneous functions imply $\sum{i=1}^n\left(\mathrm{p}i \cdot \partial \mathrm{z}{\mathrm{j}} / \partial \mathrm{p}_{\mathrm{i}}\right)=0$, and all terms where $\mathrm{i} \neq \mathrm{j}$ are positive by assumption.

## 经济代写|微观经济学代写微观经济学代考|通过过度需求条件的唯一性

Sonnenschein-Mantel-Debreu结果在供求方法的追随者中引起了极大的不适，因为它显示了平衡过程结果中可能存在的不确定性。但是，如果能够找到能够避免多重均衡的假设，并且人们可以辩称，这些假设在实际经济中几乎总是能得到满足，那么损失就会大大减少。我继续列出目前发现的确保(即足以)唯一性的主要假设，然后评估它们的合理性

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