## 经济代写|微观经济学代写Microeconomics代考|ECON1010

2022年9月23日

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## 经济代写|微观经济学代写Microeconomics代考|Non-strictly Convex Preferences

In Chap. 4 it was shown that the certainty that a general equilibrium exists requires the continuity of $\mathbf{z}(\mathbf{p})$ for positive prices ( Sect. 4.7), and the absence of sudden ‘jumps’ of $\mathbf{z}(\mathbf{p})$ as some price becomes zero ( Sect. 4.15). Now let us go deeper into the problems raised by discontinuities of excess demand. It will take some time, as there are many issues to consider.
A first cause of discontinuities is non-convexity of indifference curves.
In the Edgeworth box in $\mathbf{- 1 g}$. $6.2$ consumer A has strictly concave indifference curves and as a result her offer curve or, as I prefer to call it, her choice curve consists of two separate segments on the axes, and then it can happen, as shown in the figure, that the two choice curves have no point in common: there is no equilibrium. There would then be a never-ending oscillation of the relative price: even if it were exactly at the discontinuity level where $\mathrm{A}$ is indifferent between two baskets, there would be either excess supply or excess demand for good 1 depending on A’s choice.

Exercise Draw an Edgeworth box complete with the indifference curves of the two consumers, in which both consumers have discontinuous choice curves owing to strictly concave indifference curves, and determine whether an equilibrium exists.

This difficulty is generally considered negligible for economies with many different consumers, because, it is argued, the oscillation of relative price will be around the discontinuity price, so the latter will still function as an indicator of the average of market price, which is the traditional role of equilibrium prices; furthermore the price oscillation will be very small, hence nearly unnoticeable, if the discontinuity concerns only one consumer out of many.

But there might be many consumers with similar endowments and tastes and therefore a discontinuity at the same relative price. If there are $\mathrm{k}$ identical consumers who at the discontinuity price vector are indifferent between two baskets, then the usual argument to minimize the relevance of this case is the following (cf., e.g. Hildenbrand and Kirman 1988, pp. 40-41): consider the segment joining the two vectors of excess demands corresponding to all consumers demanding the same basket, e.g. in the two-goods case, segment MK in $\boldsymbol{-}$ Fig. 6.3.

## 经济代写|微观经济学代写Microeconomics代考|Non-convex Consumption Sets

Even with strictly convex indifference curves the continuity of the excess demand of a consumer is not guaranteed unless the budget set is compact (i.e. closed and bounded) and convex. If the budget set is unbounded (because of some zero price), then there may be no definite choice that maximizes the consumer’s utility. What about convexity of the budget set? The budget set of the consumer is the intersection of the consumption set with the set of consumption vectors $\mathbf{x}$ that satisfy the budget constraint. If the budget constraint is not linear because the price of a good decreases with the quantity demanded, then the budget set need not be convex and discontinuities can arise (prove it graphically!) but nonlinear budget constraints are rare. But problems can be due to a non-convex consumption set. A consumer may find it possible to consume either good 1, or good 2, but no convex combination of them (e.g. hours spent in Paris or in Sydney in the same day). Then for given prices and given income to be spent on these two goods, the budget set restricted to these two goods consists of a single point in abscissa and a single point in ordinate, and a change in the prices of the two only the other good. 3

To overcome this problem it has been argued that the consumption set can include baskets including incompatible consumptions, because the consumption basket need not include the acts of consumption but only the availability for consumption, so one can purchase both a vacation in Australia and in Greece for the same day and then do only one of the two activities: the distinction between availability and actual consumption, plus a free disposal assumption,${ }^4$ makes this treatment possible.

Indivisible (or discrete) goods too prevent convexity of the consumption set. But the problems raised by indivisible consumption goods can reasonably be considered of secondary importance. For an indivisible good, what one can determine is the reservation price of $\mathrm{n}$ units of it, i.e. the maximum price that the consumer is ready to pay in order to purchase 1 , or 2 , or $3, \ldots$ units of the good, given all other prices (and income, if it is income that is given and not endowments). ${ }^5$ In between two reservation prices, the number of units of the good demanded by the consumer does not change. The problem is that at the reservation price for $\mathrm{n}$ units of the good, the consumer is indifferent between demanding n or $\mathrm{n}-1$ units of the good (for a slightly higher price she demands $\mathrm{n}-1$ units, for a slightly lower price she demands $\mathrm{n}$ units), and this affects her demands for the other goods which therefore have a discontinuity (two values) at each reservation price. Let us see this more formally.

# 微观经济学代考

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## MATLAB代写

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