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fabio petri VDE6 existence 12/02/2016 p. 1 CHAPTER 6. EXISTENCE, UNIQUENESS AND STABILITY OF THE GENERAL EQUILIBRIUM OF PURE EXCHANGE, AND OF ACAPITALISTIC PRODUCTION AND EXCHANGE. The exchange economy: existence of equilibrium 6.1. The supply-and-demand approach to value and distribution argues that the competitive interaction of buyers and sellers pushes a market economy toward a situation of equality, or equilibrium, between supply and demand simultaneously on all markets. The theory must then show (i) that such a situation of general equilibrium exists, (ii) that the economy tends to it. This chapter presents and assesses the main results reached on these issues by the theory of general equilibrium for the pure exchange economy and for the ‘atemporal’ production economy without capital. In the next three chapters we will discuss the complications due to the presence of capital goods and of interest rates. In so doing we follow the usual sequence in the presentations of the neoclassical approach, the one adopted for example by Walras and by Wicksell. The traditional motivation of such a sequence is that the basic insights reached by considering the pure-exchange economy are not falsified by the successively added complications. We will find reasons to doubt such a thesis. The first part of the present chapter discusses the existence of solutions to the equations defining the general competitive equilibrium of an exchange economy(1). We have met the exchange economy at the end of Ch. 4. We remember its constituents. The economy consists of a finite number of consumers, indexed 1,..,h,..,H, each one with given preferences and a given n-vector of endowments ωh=(ω1h,...,ωnh). Consumers are price-takers and, given a vector of prices p, for each consumer utility maximization determines a set d h(p) of consumption bundles that maximize her utility under the budget constraint, a set that may contain 1 The problem of existence of solutions to the equations of general equilibrium is usually referred to as the problem of existence of a general equilibrium; but this way of naming the problem is misleading if interpreted to mean that the forces which according to this theory tend to determine value and distribution do exist and that the sole problem is to ascertain whether they can logically come to a state of rest. The reader should keep in mind that since in actual economies income distribution and quantities are determined by some force or mechanism, if the general equilibrium approach were to conclude to the non-existence of equilibrium the implication would be that the forces determining outputs and income distribution are other ones. But for brevity we will use the more concise terminology. fabio petri VDE6 existence 12/02/2016 p. 2 one element, or a finite or infinite number of elements. Any one of the elements in dh(p) can be(2) a vector of demands xh(p), to which there corresponds a vector of excess demands xh(p)–ωh. The function from p to the set of demands dh(p) is the demand correspondence of consumer h, or demand function if single-valued (in this case it coincides with xh(p)). The sum of all endowment vectors ωh is the aggregate endowment vector ω. A distribution of this aggregate endowment among the consumers is called an allocation. Thus an allocation {xh} is a vector of H vectors xh, h=1,...,H, such that Σhxh=ω. We can now define: Given for each consumer h an endowment ωh and a demand correspondence dh(p), an equilibrium of the exchange economy is a price vector p and an allocation {xh} such that xhdh(p), ∀h. Excess demand 6.2. The basic element in the search for equilibrium is the market excess demand function or correspondence, z(p)=(z1(p),...,zn(p)) where zi(p) is the vector (or set of vectors in the case of correspondence) of (market) excess demand for good i, obtained by summing the excess demands for that good of the H consumers. For brevity I will drop the ‘market’ specification in what follows when superfluous. For simplicity, perfect divisibility of goods is assumed. If individual excess demands are single-valued for all consumers, then z(p) is a function, z(p)=∑h(xh(p)–ωh); initially I assume that this is the case. The demand correspondence is guaranteed to be a function (i.e. uniquely determined for each p) only if one assumes that, for each price vector, the choice of each consumer is uniquely determined. As elementary microeconomics shows, this is only guaranteed if the indifference curves of consumers are strictly convex: if they are strictly concave or have strictly concave segments then there may be price vectors at which some consumer is indifferent between two disjoint consumption bundles (cf. Fig. 4.?? in Chapter 4); if indifference curves have flat segments, there may be price vectors at which for some consumer the budget constraint partly coincides with this flat section and then the consumer will be indifferent between a continuum of consumption bundles (I am assuming perfect divisibility of goods). In both cases z(p) is then a correspondence i.e. an application associating to each element of X a set of excess demand vectors, a set which may contain one element, a finite number of elements, or an infinite number. A zi(p)<0 means an excess supply of good i. If z(p)=0, there is equilibrium (excess demand 2. If dh(p) contains a unique element, the consumer chooses that unique element; if the consumer is indifferent among several possible consumption bundles which all equally maximize her utility, then how she will choose among these consumption bundles is a question not discussed in the literature, nor easily answerable. An assumption that would seem as good as any other is that the choice will be random. fabio petri VDE6 existence 12/02/2016 p. 3 zero means demand=supply) on all markets. But I assume that prices cannot become negative, hence a market where the price is zero and there is excess supply must also be considered in equilibrium, the pressure of excess supply is unable further to reduce the price so there is no tendency for the price to change. So equilibrium requires z(p)0 and a zero price on markets where excess demand is negative. Several properties of the market excess demand function z(p), derivable from the analysis of Ch. 4, are listed below. 1) It is homogeneous of degree zero in p, because for each consumer, since income is the value of the given endowments, the budget hyperplane passes through the endowment point and therefore the budget constraint does not move if all prices change in the same proportion. This makes it legitimate to restrict attention to relative prices, usually by setting their sum equal to 1 (this avoids the risk of choosing as numéraire a good whose price is zero). 2) It satisfies Walras’ law i.e. pz(p)=0 (as long as local non-satiation is assumed) because consumers’ budgets are balanced, hence the sum of the values of individual excess demands over all consumers is zero. This implies the important corollary that if all markets where price is positive are in equilibrium except one, then this last market too is in equilibrium. The qualification ‘where price is positive’ is important, there may be equilibrium on all markets where price is positive and excess demand on a market where the price is zero (cf. §4.2.2). 3) It is possibly not defined if some price is zero: if there is even only one consumer who is never satiated of a good with a zero price, then the demand for that good is infinite i.e. not defined. 4) It is bounded below, because consumers cannot supply goods in excess of their endowments and therefore the excess supply of any good cannot exceed the aggregate endowment of that good. 5) It is continuous on the interior of the price simplex (i.e. as long as prices are all positive) if preferences are continuous, strictly convex and locally nonsatiated over a choice set (the budget set) which is compact and convex. (dh(p) is an upper-hemicontinuous correspondence over the interior of the price simplex if the strict convexity of preferences is relaxed to simple convexity.) This was proven in Chapter 4, Section 4.??, "Demand and continuity". 6) assuming in addition that preferences are strongly monotone, if a sequence pn of price vectors, with pn>>0, pnSn-1, converges to p on the boundary of Sn-1 (that is, if the sequence converges to a price vector where some but not all prices are zero), then ║z(pn)║, the length of the excess demand vector, tends to +∞ (i.e. at least one excess demand tends to +∞). This last property needs proof. Before, I remember the meaning of a number of terms which have appeared in these sentences. Preferences are continuous if for all x in the consumption set the fabio petri VDE6 existence 12/02/2016 p. 4 set of consumption bundles weakly preferred to x and the set of consumption bundles to which x is weakly preferred are both closed sets. Preferences are weakly monotone if y>>x implies y≻x; this admits satiation for some goods but not for all goods simultaneously; preferences are strongly monotone if y≥x and y≠x imply yx. Preferences are locally non-satiated if, assuming all goods are perfectly divisible, for all x in the consumption set and for any >0 there is in the consumption set, within a distance from x not greater than , a consumption bundle y strictly preferred to x. This admits that some goods may be ‘bads’ i.e. have negative marginal utility. (Exercise 1: Suppose there are only two goods; if one assumes local non-satiation, can both goods be simultaneously 'bads' everywhere in the consumption set, that is, such that a decrease of the amount of either of them raises utility? Prove your answer.) From the continuity (or, for correspondences, upper hemicontinuity) of the excess demand of each consumer the continuity (respectively upper hemicontinuity) of the market excess demand follows trivially by summing over all consumers. Proof of property 6. We prove it for the excess demand dh(p) of a single consumer under the assumption that (i) her endowment has positive value at the prices p to which pn converges, and that (ii) there is a positive aggregate endowment of each good; note that because of (ii) there will always be at least one consumer in the economy for whom (i) holds, since in vector p the price of at least one good is positive, and at least one consumer has an endowment of that good; and if ║dh(pn)║ tends to +∞ for even only one consumer, then also ║z(pn)║ tends to +∞ because endowments are finite. (Is assumption (ii) defensible? To assume that there are prices for goods with zero endowment would cause problems to the existence of equilibrium: assume strictly monotone preferences including preference for a good with zero endowment, and assume that demand for this good is positive however high its price as long as some other good has positive price (this will be the case, for example, with a Cobb-Douglas utility function); then there is no equilibrium because excess demand for the good is always positive for any price of the good as long as some other good has positive price, while when the prices of all other goods are zero demand for them is infinite. However, it seems reasonable to assume that when the endowment of a good is zero people will discover it and will give up attempting to buy it; this justifies the assumption that the only goods demanded and with quoted prices are goods with nonzero endowments.) For simplicity we drop the superscript h. The proof of property 6 is by contradiction. We need first some intermediate results. Assume that ║d(pn)║ does not tend to +∞. Then there is a subsequence pn’ such that d(pn’) is bounded above, and hence bounded, for all n’ (remember that d(p) is bounded below: excess supplies cannot be greater than endowments). It is a well-known mathematical result that every bounded sequence in Rn contains a convergent subsequence. Therefore there is a subsequence pm of pn’ such that d(pm) converges to a finite limit x. fabio petri VDE6 existence 12/02/2016 p. 5 Now we use another result of consumer theory: Let the preference relation be continuous, convex and (weakly) monotone, let sequence pm converge to p, and let pω>0 where ω is the endowment of the consumer; if a sequence xmd(pm) converges to x as pm converges to p, then xd(p) i.e. no other element of the budget set B(p) is strictly preferred to it. The proof is as follows. Since pmxm=pmω and pm→p, xm→x, it follows that px=pω, i.e. that x is in the budget set B(p). Now consider any other x'B(p). From pω>0 it follows that, for any scalar in the open interval (0,1) it is px'<pω. Since pm→p, there is an m' sufficiently great such that for all m>m' it is pmx'<pmω; by the weak axiom of revealed preferences, since x' might have been chosen at the prices pm and on the contrary the consumer has chosen xm, this implies that for m>m', xm is weakly preferred to x'. The continuity of preferences then implies that x is weakly preferred to x'. Again because of the continuity of preferences, by making tend to 1 we obtain that x is weakly preferred to x'. Since this holds for any x'B(p), we obtain xd(p). On the basis of this result, the proof of property 6 continues as follows. If ║d(pn)║ does not tend to +∞, we obtain that the finite limit x is utility-maximizing at p (no other x'B(p) is strictly preferred to it). However this is absurd since at least one price in p is zero, and by increasing the amount of the corresponding good in some consumer's demand, her excess demand is still in the budget set but her utility is greater owing to the strong monotonicity assumption. This contradiction shows that it is not possible that ║d(pn)║ does not tend to +∞. █ Note that without the assumption of strongly monotone preferences for all consumers, it cannot be excluded that when prices tend to a price vector p on the boundary of the price symplex all consumers whose endowments have positive value at p are satiated with the goods whose prices tend to zero, and then market excess demand for these goods might tend to a finite limit as their price tends to zero, and jump discontinuously to + when a price becomes zero, and as a consequence it is possible that no equilibrium exists: we will see examples of this later. k O y Fig. 6.1 Note that if more than one price tends to zero at the same time, not all the excess demands for the goods whose prices are tending to zero need tend to +∞. Suppose there are three goods x, y, k and a consumer has utility function u(x,y,k) = x1/2 + (y+v(k))1/2 with v(k) such that the indifference curves between y and k are strictly convex but touch the y axis and that MUy/MUk>1 when k=0 fabio petri VDE6 existence 12/02/2016 p. 6 (see Fig. 8.1); this happens e.g. if v(k) = log(k + 1/2). Then if p>>0 and py/pk=1 the consumer maximizes her utility by demanding a zero quantity of good k. Then a price sequence {pn} where pn = (1, 1/n, 1/n) does not cause the demand for good k to grow although its price tends to zero. CONTINUITY 6.3. In Chapter 4 it was shown that important properties for the proof of the existence of at least one equilibrium are the continuity of z(p) for positive prices, and the absence of sudden 'jumps' of z(p) as some price becomes zero; and it was shown that, without these properties, an equilibrium might not exist. Now we go deeper into the problems raised by discontinuities. B’s origin Consumer A’s choice curve Consumer B’s choice curve endowment A’s origin Fig. 6.2. Edgeworth box with no equilibrium owing to concave preferences of consumer A. Non-strictly-convex preferences A first cause of discontinuities is non-convexity of indifference curves. Non-convexity is a problem, because the 'offer curve' or, as I prefer to call it, the choice curve of a consumer (the locus of points of tangency between budget constraint and indifference hypersurface as relative prices vary) may be discontinuous in this case. For example with two goods fabio petri VDE6 existence 12/02/2016 p. 7 and strictly concave indifference curves the consumer with a given endowment demands only one of the two goods, and jumps from demanding one to demanding the other as relative prices vary; also, there is one slope of the budget constraint which makes the consumer indifferent between consuming only good 1 or only good 2, hence at that relative price the demand of the consumer has two possible values: it is not a function but a correspondence. The discontinuity reappears in z(p), which is then a correspondence too; if this happens for several consumers at different price vectors, z(p) will have several discontinuities. But continuity is necessary for the existence of equilibrium to be guaranteed, as made clear by the Edgeworth box in Fig. 6.2 where consumer A has strictly concave indifference curves so her choice curve consists of the two disjoint red segments. Exercise 2: 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 because, it is argued, a sufficiently approximate quasi-equilibrium can all the same be plausibly assumed to exist. Here approximate quasi-equilibrium means a situation where disequilibrium is so small as to be negligible. If only one consumer has a discontinuity, the argument is that in all likelihood the discontinuity will be very small relative to the total quantities exchanged, and therefore negligible even if it causes excess demand never to be zero(3). If there are k identical consumers who at the discontinuity price vector are indifferent between two bundles, then the argument 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 bundle, e.g., in the two-goods case, segment MK in Fig. 6.3. Start from one extreme of the segment, e.g. in Fig. 6.3 the situation where all consumers are choosing the bundle where the amount of good 1 is smaller: market excess demand for good 1 is M. Now suppose that just one of the k consumers chooses the other bundle; then suppose that two consumers do so, then three, and so on; in this way we can make market excess demand correspond to either extreme of the segment, or to any of the k–1 intermediate equidistant points on the segment MK. Thus, even when the discontinuity is relevant for equilibrium (it will not be, if equilibrium is at a price vector where there is no discontinuity), there will exist a distribution of the k consumers between the two 3 No market is ever perfectly in equilibrium, so even if in a market the price is unable perfectly to stabilize because at the price at which the discontinuity arises excess demand is either positive or negative, the price oscillations due to a small discontinuity of excess demand will be unnoticeable, because very weak (probably the price will not even move, if it presents even a very limited viscosity) and swamped by the bigger oscillations due to the accidental and transitory changes that the market is anyway always undergoing. fabio petri VDE6 existence 12/02/2016 p. 8 bundles that makes market excess demand differ from zero by not more than if only one consumer presented the discontinuity – and therefore by a negligible amount. p1/p2 ● M ● ▼ 0 ● ● ● ● ◘ ● ● ● ● ● excess demand for good 1 ● ● K Fig. 6.3. Discontinuous excess demand for good 1 due to 14 identical consumers with concave indifference curves. At the price ratio where a consumer is indifferent between two choices, she can either have a small excess supply of, or a greater excess demand for, good 1. Point ◘ indicates the most probable excess demand at the critical price ratio if consumers choose randomly; point ▼ indicates the disequilibrium-minimizing distribution of choices (twelve consumers choose to be net suppliers, two choose to be net demanders of good 1). 6.4. The argument just presented is not convincing. The problem remains, that it is unclear what might ensure that the discrepancy-minimizing distribution of choices is indeed achieved: if each price-taking consumer chooses randomly which of the two bundles to demand – and it is unclear how else they can choose – then, with many consumers, the most probable outcome is that approximately one half of them will demand each bundle, in which case market excess demand will be approximately at the middle point of the segment, thus possibly non-negligibly different from zero (cf. Fig. 6.3); the pressure on price to change can then be non-negligible, but as soon as the price changes however slightly, all consumers jump to the same choice, and a significant price oscillation cannot be avoided. So the real way out of the problem is to postulate that consumers are not identical (a plausible assumption in many cases, but much less so in labour markets), so the discontinuities happen at different prices for different consumers and are then unnoticeable because very small relative to total demand, and therefore swamped by the irregularity and disequilibrium that anyway characterize real economies. The theory does not argue that market economies succeed in completely reaching an equilibrium, only that they tend to it in a world undergoing continuous shocks and attritions and accidental temporary variations, so that the equilibrium is a good fabio petri VDE6 existence 12/02/2016 p. 9 approximation to the average of market prices and quantities over some time span, and then if the tendency in a market is toward a small oscillation rather than toward a precise price and quantity this makes no difference to the explanatory and predictive power of the theory. A problem similar to the one just discussed arises if indifference curves are convex but not strictly convex, having flat segments: when the budget line coincides with a flat segment, the consumer is indifferent among all points of the segment; the difficulty then is not that an equilibrium will not exist(4); the difficulty is that, if at the equilibrium prices a consumer is indifferent between all points of a segment and only one of these points corresponds to the equilibrium, there is no guarantee that she will choose the equilibrium point, so there is no guarantee that equilibrium prices will ensure that equilibrium is actually attained; equilibrium prices are only compatible with equilibrium choices, but with probability 1 these choices will not be realized. But here too this indeterminacy of choice can be presumed to happen at different prices for different consumers, so at the equilibrium prices there will be at most one consumer in this situation and then the indeterminacy of excess demand will be small relative to total demand and an approximate quasi-equilibrium is ensured. On this basis, the discontinuities or indeterminacies associated with non-convex or nonstrictly-convex preferences do not appear to be a reason seriously to question the validity of the supply-and-demand approach. Non-convex consumption sets, and the survival assumption 6.5. As pointed out in Chapter 4, 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, or bounded but open, then there may be no definite choice that maximizes the consumer’s utility; then excess demand is not even defined. If the budget set is not convex, then the continuity of the excess demand function is not guaranteed (see below). Now, the budget set is the intersection of the consumption set with the set of consumption vectors xh that satisfy pxh≤pωh; the latter set is closed, convex(5) and bounded (we are assuming p>>0, ω≥0, x≥0), so the budget set is closed, bounded and convex if the consumption set Exercise 3: draw the choice curves of two consumers with such preferences in an Edgeworth box – these choice curves will be correspondences – and show that an equilibrium exists. 5 Exercise 5. Suppose that in a two-goods economy the price a consumer must pay for good 1 is not constant, but decreases (owing to discounts) if the demand for good 1 increases beyond a certain amount. Show graphically that the budget set can be non-convex and that in this case the demand function can be discontinuous. 4 fabio petri VDE6 existence 12/02/2016 p. 10 X is closed and convex. (The reason is clear: the intersection of convex sets is convex; the intersection of closed sets is closed; the intersection of a bounded set with any set is bounded.) No serious objection appears to exist against assuming that the consumption set is closed. Convexity is a different matter. Consider for example, in the two-good case, a consumer who for some reason (e.g. physical impossibility) can consume either good 1, or good 2, but no convex combination of them (e.g. hours spent on the Tour Eiffel in Paris or in the Metropolitan Museum in New York in the same day). Then for given prices and endowments (or income), the budget set consists of only two segments, one on the horizontal axis and one on the vertical axis, and the same kind of discontinuity of demand is possible as in the case of strictly concave indifference curves. (This difficulty is specific to intertemporal equilibria, which need to date every commodity, and does not arise if the equilibrium one is trying to determine is a long-period equilibrium. The latter type of equilibrium determines the average demands per period, not the demands on specific days, so one can always assume that the hours spent on the Eiffel Tower and in the Metropolitan Museum are not in the same day.) A way to overcome this problem has been proposed: it has been argued that the consumption set may well include bundles where incompatible consumptions appear, because the consumption bundle need not include the acts of consumption but only the availability for consumption, so one may well purchase both a vacation in Australia and in Greece for the same day, and then do only one of the two activities and waste the possibility to do the other: the distinction between availability and consumption, plus a free-disposal assumption(6), make this treatment possible. The same free-disposal assumption is used to justify the frequent assumption that all bundles greater than any bundle in the consumption set are also in the consumption set. Why might such an assumption be questionable? It is impossible to consume more than 365 days of vacation per year, or to sleep in more than one hotel in the same night; but with free disposal one can pay for too many vacations, and only enjoy some of them, or one can pay for seven hotel rooms for the same night, and leave six of them empty. Anyway the constraint that a good in the consumption set cannot be greater than a certain given quantity would not disturb the convexity of the budget set. To see why, draw the standard budget line of a consumer for a two-goods world and then assume that consumption of good 1 cannot exceed a certain amount: this may mean excluding part of the 6 . I.e. that one can get rid of goods at no cost. This assumption is formalized for firms as meaning that for each good there is available to all firms a process with that good as input and no output. To assume free disposal for consumers means to assume that consumers too have available for each good a ‘production process’ with that good as input and no output. We leave aside a discussion of the legitimacy of this assumption, and of the complications that arise if one does not make it. fabio petri VDE6 existence 12/02/2016 p. 11 previous budget set by drawing a vertical line that cuts it, but the budget set remains convex. Indivisible (or discrete) goods also 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 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,... units of the good, given all other prices (and income, if it is income that is given and not endowments)[7]. In between two reservation prices, the number of units of the good demanded by the consumer does not change. A problem arises because at the reservation price for n units of the good, the consumer is indifferent between demanding n and demanding n–1 units of the good (for a slightly higher price she demands n–1 units, for a slightly lower price she demands n units), and this affects her demands for the other goods which therefore have a discontinuity at each reservation price. Let us see this more formally. Assume that utility is u(x1,x2), good 1 comes in discrete units while good 2 is divisible, the consumer’s income is m, and the given price of good 2 is p2. The reservation price for 1 unit of good 1, let us indicate it as R1, is that price such that the utility from not consuming good 1 at all (and consuming therefore m/p2 units of good 2) is the same as the utility from consuming 1 unit of good 1 and dedicating to the purchase of good 2 the residual income m−R1: u(0, m/p2)=u(1, (m−R1)/p2) . Analogously, the reservation price R2 for the purchase of 2 units of good 1 is determined by u(1,(m−R2)/p2) = u(2, (m−2R2)/p2) and the reservation price for n units of good 1 is determined by u(n−1,[m−(n−1)Rn]/p2) = u(n,(m−nRn)/p2). Assuming good 1 is not a Giffen good, it is Rn>Rn+1 and for Rn>p1>Rn+1 the consumer demands n units of good 1 and (m−np1)/p2 units of good 2. This means that e.g. at prices (R2, p2) the consumer is indifferent between demanding (m−R2)/p2 and demanding (m−2R2)/p2 units of good 2; demand for good 2 has two values here, with a discontinuity, cf. Fig. 6.4. p1 R1 R2 7 The general definition of reservation price of a quantity x of a good (possibly perfectly divisible) is: the maximum price the consumer would be ready to pay for that quantity, given her income and the other prices. fabio petri VDE6 existence 12/02/2016 p. 12 x2 Fig. 6.4. Possible shape of the demand for good 2 (measured on the horizontal axis) as a function of the price of discrete good 1. But, like for the discontinuities caused by strictly concave indifference curves, these discontinuities, since in all likelihood they occur at different prices for different consumers, can be considered sufficiently small relative to total demand for good 2 as to be negligible. 2 2’ ωi ▪ xi2 xi O xi1 O’ 1 Fig. 6.5 Finally, the consumption of certain goods may be necessary to make it conceivable that certain other goods be consumed; or it may have the opposite effect. For example, in intertemporal choices, a choice of consumption at date t can have a stream of consequences which make it impossible to consume certain things at later dates, which on the contrary can be consumed with another choice at t. E.g. it is impossible to consume a swim in deep water if one has not learnt to swim; it is impossible to climb mountains without sufficient physical fitness, and the physical fitness depends on whether one chooses or not to do the needed previous physical exercise. Let us consider the shape of the consumption set restricted to the sole two goods "physical exercise at time t", good 1, and "mountain climbing at time t+1", good 2; and let us suppose that as long as physical exercise is less than the amount OO' in Fig. 6.5, no mountain climbing is possible, but physical exercise as great as OO' or greater allows any amount of mountain climbing the following period. If one excludes impossible consumption bundles from the consumption set, then the consumption set in Fig. 6.5 is not convex because it consists of the segment OO' plus the closed quadrant 1,O',2' (shown in gray). fabio petri VDE6 existence 12/02/2016 p. 13 Again, the need to exclude from the consumption set the bundles that include mountain climbing but not enough physical exercise is circumvented by arguing that, not mountain climbing as such should enter the consumption set but rather the clothes, shoes, socks, travel tickets, hotel nights etc. which make mountain climbing possible, and that these things can be available without being consumed (free disposal again!). This solution avoids the non-convexity of the consumption set, but raises another problem. In all likelihood the marginal utility of the goods useful for mountain climbing will not vary continuously, it will jump to zero when previous exercise for physical fitness falls below OO'; thus the consumer's demand for the goods only useful for mountain climbing may jump discontinuously from a positive amount to zero when the demand for physical exercise goes below OO'. We conclude that when a certain consumption level of a good is a pre-requisite for another good to have any utility, the assumption of continuity of preferences is unacceptable(8). A similar, albeit more tragic case is that of suicide. A consumer may consciously decide not to survive beyond a certain date depending on prices. Suppose that, if the utility of a consumer goes below a certain level, the consumer decides not to survive beyond the end of period 1: her supply of labour and her demands for goods in subsequent periods jump discontinuously to zero. 6.6. With this we arrive at the thorny issue of survival. Even excluding the possibility of suicide, there is the problem that a minimum level of consumption is necessary in order to survive: if one is considering intertemporal consumption sets, it seems illegitimate to admit consumption bundles including positive amounts of future goods and of labour supply together with amounts of current goods insufficient for survival. We can quote here a passage from Debreu (1959), which accompanies the drawing from which our Fig. 6.5 was taken: ...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 [O,O’] be the minimum quantity of the first commodity which that consumer must have available in order to survive until the end of 8 For most goods the resulting discontinuity in excess demand can be considered of secondary importance because, if consumers are numerous, the discontinuities are in all likelihood at different prices for different consumers, so they occur to one consumer at a time and hence are negligible vis-à-vis total demand, like the discontinuities due to strictly concave indifference curves. But the problem is nonetheless important because it shows that the universal assumption of continuous preferences (which is probably the reason why this possibility of discontinuity of demand, to the best of my knowledge, has never been discussed) is a much stronger assumption than is generally realized. fabio petri VDE6 existence 12/02/2016 p. 14 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 Xi [i.e. the consumption set of consumer i, F.P.] would therefore consist of the closed segment [O,O’] and a subset of the closed quadrant 1, O’, 2’. Such a set has the disadvantage of not being convex in general. However, if both commodities are freely disposable, the set Xi is the closed quadrant 1, O, 2, which is convex: if the consumer chooses (perhaps because he is forced to) a consumption xi in the closed strip 2, O, O’, 2’, it means that xi1 of the first commodity is available to him and he will actually consume at most that much of it, and that xi2 of the second commodity is available to him and he will actually consume none of it. ......The choice by the ith consumer of xi in Xi 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 is assuming no uncertainty, so consumers know the date when they will die, which may depend on one's choices and hence on prices; we discuss uncertainty in Chapter 9.) Once again, it is the distinction between availability and actual consumption (plus the free disposal assumption(9)) that avoids the non-convexity of the consumption set. But it does not avoid a 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 subsist beyond those periods, and their supply of labour and demand for goods for the subsequent periods drops discontinuously to zero(10). This problem motivates an assumption which is universally made in general equilibrium theory in order to prove the existence of equilibrium: the survival assumption, i.e. the assumption that each consumer has an endowment which is sufficient to survive without any trade. We comment on this assumption later in the chapter. Nowadays the standard practice in general equilibrium theory concerning the definition of consumption sets is different from Debreu's; the consumption set is defined as excluding the bundles which do not allow survival (cf. e.g. Mas-Colell et al., p. 19 Fig. 2.C.4, and p. 634 fn. 82). 9 But the legitimacy of the free disposal assumption is highly dubious in this case; the things, that people have not consumed by the time they die, are not disposed of, they pass as inheritance to other people; thus if survival depends on prices, one should admit that ownership of some goods in future periods may depend on prices, and since different owners will have different tastes, this will imply a discontinuity of demands at the prices at which the owner changes. (An analogous problem arises when one admits the possibility of bankruptcies in temporary equilibria, cf. Chapter ??) 10 This is a more serious problem than with mountain climbing (cf. fn. 6?? above), because as far as survival needs are concerned, many people are in a very similar situation. fabio petri VDE6 existence 12/02/2016 p. 15 For example in the case of Fig. 9.3 (referred now to Debreu's discussion and not to mountain climbing), the modern practice would be to define the consumption set as (possibly a subset of) the closed quadrant 1,O',2'. Then the survival assumption, which is also made, becomes simply the assumption that the endowment is in the consumption set. This way of defining the consumption set cannot be defended as aiming at excluding impossible choices. The consumer might find herself obliged to choose a bundle outside the consumption set so defined; she might not have enough income to survive. For example, in the case of Fig. 9.3 the consumer's endowment might be ωi (not present in Debreu’s original Figure); a sufficiently high relative price of good 1 will then make it impossible for the consumer to obtain at least OO' of good 1; but the consumer will still choose some consumption bundle, probably dedicating her entire income to buying as much as she can of good 1( 11); so it is not true that it is impossible for the consumer to choose a point outside the quadrant 1,O',2'. Note also that, since by definition the consumer can only choose bundles in the consumption set, if the latter excludes bundles incompatible with survival then one is excluding by assumption the possibility that a consumer may choose not to survive beyond a certain period by choosing bundles insufficient for survival (suicide by not eating). It has been concluded(12) that for intertemporal equilibria this definition of the consumption set coupled with the survival assumption implies "that every consumer survives in every competitive equilibrium, not merely for one period but over the whole (finite) Arrow-Debreu span. This is a breathtaking assertion". The minimum-income problem and the zero-income problem. 6.7. The discontinuities of excess demands caused by income falling below the subsistence minimum for some consumer are one kind of minimum income problem or minimum wealth problem. But another kind of discontinuity due to income falling too much is also possible. We call this other kind of minimum income problem the zero-income problem. To see the problem in isolation, let us assume that even a zero income ensures survival (the consumption set coincides with Rn ; the survival assumption is automatically satisfied). The zeroincome problem can arise when the endowment is on the frontier of Rn . We have already encountered this problem in Chapter 4, §4.11, cf. Fig. 4.14??, so we need not repeat the full 11 . She may hope that something will happen that will change things (e.g. a donation), so she does her best to survive in the meanwhile. But if really the consumer will die of hunger and knows it, then an outcome at least as plausible is that she drops out of the market and becomes a beggar or a criminal or perhaps a rebel joining a guerrilla army, so the continuity of her demand for good 1 is not ensured – see below in the chapter. 12 Peter Newman, entry "consumption sets", New Palgrave Dictionary of Political Economy I ed. fabio petri VDE6 existence 12/02/2016 p. 16 description here. It was shown there that it can happen that the excess demand for a good is negative as long as the price of the good is positive, jumping to +∞ if the price becomes zero. The reason is that if the endowment of a consumer with strongly monotone preferences consists solely of one good, then the consumer's demand for that good can never exceed the endowment as long as the price of the good is positive, but it can jump discontinuously to +∞ the moment the price of that good becomes zero(13). (This discontinuity may arise even without strongly monotone preferences, if the consumer becomes satiated with that good at a quantity greater than the endowment.) Exercise 4: Find out why the numerical example used in Chapter 4, §4.11 to illustrate Fig. 4.14?? would not have worked if we had assumed Cobb-Douglas preferences. Exercise 5: Draw an Edgeworth box where consumer A has the endowments and preferences of Fig. 4.14?? and consumer B has such endowments and preferences as to cause an equilibrium not to exist. (Hint: re-read the reasoning accompanying Fig. 4.14??. If you are unable to find a solution, consult Mas-Colell et al., 1995, p. 522.) Are you able to produce a similar case of non-existence of equilibrium if consumer A has such preferences that goods 1 and 2 are perfect complements (Lshaped indifference curves) in the proportion 1:1? Three possible alternative assumptions have been found that surmount the zero-income problem: A) monotonicity: preferences are strongly monotone for all consumers; B) interiority(14): all consumers have a strictly positive endowment of all goods; C) irreducibility: for any given allocation, whichever the way one divides the consumers into two non-empty subsets, the utility of at least one consumer in the first subset is increased if all the endowments of the consumers in the second subset are added to her/his allocation. Assumption A, monotone preferences, guarantees that if an equilibrium exists then the equilibrium price vector is strictly positive: the reason is simple, if a good had zero price the demand for it would be infinite and its excess demand too (because the endowment is finite). Therefore in the search for an equilibrium price vector one can restrict the search to the interior of the price simplex, i.e. to the strictly positive price vectors, where excess demand is continuous by assumption. Thus Assumption A does not prevent zero-income discontinuities but renders their possibility irrelevant; if an equilibrium at strictly positive prices does not exist, then there is no equilibrium independently of whether the zero-income problem causes discontinuities on the 13 It is not necessary that the endowment consists of only one good; but it is the more plausible case, because if the endowment consists of several goods, then income goes to zero only if all the prices of these goods go to zero. 14 . This name is due to A. T. Rizvi ("Specialisation and the existence problem in general equilibrium theory", Contributions to Political Economy, 1991, pp. 1-20, a highly recommended reading) and it refers to the fact that the endowment point is assumed to be in the interior of the consumption set. fabio petri VDE6 existence 12/02/2016 p. 17 frontier of the price simplex. Assumption B, strictly positive endowments, guarantees that the 'income' (the value of the endowment) of each consumer is always positive (because at least one price is positive). Assumption C, irreducibility(15), is considered less restrictive than the first two. The formal definition is as follows. Let {1,..,h,..,H} be the indices which distinguish the H consumers of the economy. Let us partition consumers into two mutually exclusive and exhaustive sets whose indices form two non-empty sets S1, S2 with S1∩S2= and S1∪S2={1,..,h,..,H}. The economy is irreducible if for any allocation {xh} and for any partition (S1, S2) there exists an h'S1 such that (ΣhS2ωh)+xh' is strongly preferred by consumer h' to xh'. Since the second group may consist of a single consumer, the irreducibility assumption implies that as the value of the endowment of any consumer approaches a zero value, the excess demand for at least one of the goods in this endowment becomes positive because there is always some other consumer ready to pay for it a positive price however small; so no consumer can have a zero income in equilibrium. Exercise 6. In your solution to Exercise 5, was irreducibility satisfied? Exercise 7. Produce in an Edgeworth box a graphical example of a two-goods two-consumers economy which is not irreducible and where nonetheless an equilibrium exists and with positive value of both endowments. (Hint: remember that irreducibility requires that for any initial allocation....) The generally accepted assessment of assumptions A (strict monotonicity) and B (interiority) is that "These are extremely restrictive hypotheses, as even a superficial observation of reality demonstrates" (Reichlin and Ventura, ??date, p. 75, our translation). For B this is obvious. For A, think of whether you would like to consume (i.e. not to re-sell, but instead to keep in your house) any amount of furniture: beyond a certain amount, too much furniture becomes a nuisance. Also, when we will get to production economies, we will have to include, in the goods among which a consumer chooses, the self-consumption of the services of the factors the consumer owns; now, there may well be factors of production (types of land, capital goods) whose services yield no direct utility to consumers. Irreducibility may look more acceptable, but a moment's reflection will show its limitations. This condition amounts to assuming that each consumer has in her endowment at least one good or 15 . It is also called resource relatedness (Arrow and Hahn) and indecomposability (Mas-Colell et al.). The term irreducibility (McKenzie, Rizvi) appears to be the more common one. The assumption is formalized slightly differently in different authors, we here follow Reichlin and Ventura. fabio petri VDE6 existence 12/02/2016 p. 18 service for which the remainder of the agents in the economy have some desire and are therefore ready to pay a positive price. But some consumers might have an endowment consisting only of the ability to perform types of labour no one cares for. Or, once one admits production and capital goods, then old people, too old to work, may have endowments consisting of ownership of capital goods specific to a certain production method, and technical progress may render that production method obsolete and those capital goods totally unwanted. However, in order to get a discontinuity at zero income the price, which by tending to zero causes the income of a consumer to tend to zero, must be the price of a good not desired by other consumers but desired by the consumer herself (otherwise her demand for the good would not jump up when its price becomes zero); so the discontinuity is avoided if one assumes that, when the prices of all the goods in the endowment of a consumer tend to zero because no one demands them, these goods are not desired by the consumer either: and this might well be a more plausible assumption than irreducibility. Thus, in the example of the old people whose assets earn nothing, the assets are factors of production without intrinsic desirability, so when their price becomes zero it is unlikely that the owners’ demand for them jumps up. As to the labour example, the supply of no type of labour remains positive even at a wage extremely close to zero; people prefer to ask friends for help, to scavenge in rubbish dumps, or to turn into beggars or criminals: these possible choices of consumers can cause discontinuities, but not at a zero price, as will be pointed out later. So it seems possible to conclude that the zero-income problem is not a serious problem for the theory. But the same cannot be argued for the survival problem, to which now we return. 6.8. Suppose now that survival requires a positive consumption of some goods. We have already seen that, if the endowment bundle does not allow survival, then if the value of the endowment bundle decreases sufficiently, survival may become impossible and discontinuities will arise in the demands for goods of subsequent periods. But even if one makes the survival assumption, and defines the consumption set as only including bundles allowing survival, still, if the endowment vector is on the frontier of the consumption set, discontinuities can arise as the income from the excess of the endowment over subsistence goes to zero, for reasons strictly analogous to those causing the zero-income problem. For example, consider a consumer with strongly monotone preferences, who has an endowment of 2 units of good 1 and 1 unit of good 2, and needs 1 unit of both goods for survival. Fig. 9.4 shows the consumption set (defined as excluding non-survival bundles) and the endowment of this consumer. As long as the price of good 1 is positive the consumer will always choose points on the budget line to the left of the endowment point; when p1=0 she will jump to demanding an infinite amount of good 1. fabio petri VDE6 existence 12/02/2016 p. 19 x2 consumption set ● 1 O 1 2 x1 Fig. 6.6 The three assumptions which surmount the zero-income problem must be reformulated as follows in order to surmount this more general minimum-income problem. A') monotonicity plus survival: all consumers have strongly monotone preferences, the survival assumption is satisfied, and the aggregate endowment of each good exceeds the aggregate subsistence need for that good; B') interiority plus survival: each consumer has an endowment where all goods are in excess of subsistence needs(16); C') irreducibility plus survival: the survival assumption is satisfied, and a modified irreducibility assumption holds, where in the definition under C the words "all the endowments of the consumers in the second subset" are replaced by "the excess of all the endowments of the consumers in the second subset over their aggregate subsistence needs". How these assumptions avoid the minimum-income problem should be clear on the basis of what we have seen à propos the zero-income problem. Exercise 8. Confirm that Assumptions A' or C' cannot dispense with the assumption that all goods must be available in excess of subsistence, by showing that no equilibrium exists in the following examples. Case I) There are two consumers; has an endowment of 2 units of good 2 and 1 unit of good 1, and her survival requires 1 unit of both goods; has the same survival requirements, and has endowments just sufficient for survival; both consumers have strongly monotone preferences, but has indifference curves like those of Consumer 1 in Mas-Colell et al., 16 The survival assumption on the contrary only requires that the endowment be sufficient for subsistence needs. fabio petri VDE6 existence 12/02/2016 p. 20 p. 522, Figure 15.B.10(a). Case II) Consumer is like in Case I, but now is identical to . Check whether in these examples the irreducibility assumption is satisfied. Again on the survival assumption. 6.9. It should be clear that all three assumptions A', B', C' are very restrictive. The most restrictive element in them appears to be the survival assumption itself(17). This assumption amounts to assuming the absence of the division of labour, or, to use the term proposed by Rizvi (1991), the absence of specialisation, a fundamental characteristic of modern economies. In a specialised economy, individuals need to exchange in order to survive; consumers capable of independent subsistence are a rare exception. Then, as already noted, discontinuities due to nonsurvival can arise at positive prices, and they are liable to arise for many individuals simultaneously (in particular for individuals whose endowment consists only of capacity to work(18)), and therefore cannot be considered negligible. The escape routes, in the general equilibrium literature, appear to be three: 1) the problem is not mentioned at all, evidently hoping that the reader will not notice it; 2) one makes the survival assumption; 3) one assumes that the endowments of consumers will never have such a low price as to endanger subsistence. Escape route 1 is of course scientifically unacceptable(19). Escape route 2 means that the theory can only pretend to apply to economies of farmers or hunters-gathererers capable of subsisting without exchanges. Escape route 3 means that one is assuming that certain prices will not take certain values, against the spirit of a supply-and-demand theory of prices which should be able to determine endogenously whether a price will or will not take certain values. We must conclude, it would seem, that general equilibrium theory is incapable of dealing with 17 However, while for the zero-income problem it can be argued that the discontinuity will seldom arise because if no one else desires the good in the consumer’s endowment then the consumer herself will in all likelihood not desire it, for the minimum-income problem in the presence of positive subsistence needs and of the subsistence assumption the same argument is less defensible, because subsistence goods are generally intrinsically desirable and therefore the demand for them will significantly increase when their price becomes zero. 18 If the worker is skilled but her skill is in excess supply, then she must try to supply unskilled labour, and then if the unskilled labour wage goes below subsistence one has the survival problem for skilled workers too. Thus the problem may arise simultaneously for workers who potentially are not homogeneous. 19 . Yet it is fundamentally the way out chosen by Mas-Colell et al. (1995) who assume for most of the analysis that the consumption set is the entire non-negative orthant (in spite of their Fig. 2.C.4, p. 19); only in their Appendix 17.BB.2 they assume (but the thing is made clear only in a footnote) that the consumption set embodies a survival assumption. fabio petri VDE6 existence 12/02/2016 p. 21 the generality of the situations that can arise in market economies[20]. This raises doubts as to whether the theory has correctly located the forces determining prices and distribution in the real world. Even if the assumptions necessary for the existence of general equilibrium were satisfied in 97% of the observed historical situations, one would have to explain what determined prices and quantities in the other 3% of situations; so one would have to suspect the existence of forces capable of determining prices and quantities, different from the ones postulated by general equilibrium theory. But then the question arises: why should these non-neoclassical forces be operative only when general equilibrium fails to exist? could not these forces be the true ones determining prices and quantities in general? Before leaving this issue we must ask how the problem presents itself in long-period equilibria. The discontinuities due to non-survival that we have discussed so far are those that arise in the usual framework of modern general equilibrium theory: the intertemporal equilibrium framework. We tried to follow the modern literature on general equilibrium as closely as possible, so the discontinuities we considered arise in the excess demands for dated goods pertaining to dates after the last period of survival(21). One may ask whether the cause of the problem is the intertemporal framework: perhaps discontinuities would be avoided in the framework of longperiod equilibria, where one tries to determine a normal, lasting situation of tranquillity? The answer is no; and the reason is that in a long-period equilibrium the data must have persistence, the supply of labour must be the normal one corresponding to a given population that is constant (or only slowly changing) and capable of supplying labour; and one needs to eat in order to be able to 20 In fact marginalist analysis, in order to explain wage formation in underdeveloped economies with a structural excess supply of labour, has had to assume that there are social mechanisms which insulate to an extent the modern sector, where supply and demand are assumed to operate, from the pressure of the unemployed so that the wage in the modern sector does not fall below subsistence. The usual ('dual economy') solution suggested by Lewis is to assume the existence, besides the modern sector, of a precapitalist sector where incomes are determined by forces other than the market interaction of supply and demand, and to assume that if a decent living standard cannot be achieved in the modern sector then workers return to the pre-capitalist sector: labour supply to the modern sector is assumed horizontal (infinitely elastic) at a real wage taken as basically given, equal to the wage level above which workers are attracted to the modern sector, and below which workers prefer to return to the pre-capitalist sector; the influence of the marginalist perspective is only in determining endogenously the demand for labour in the modern sector, owing to the assumed existence of a downward-sloping demand curve for labour. But in most countries there isn't a pre-capitalist sector to which one may go back, unemployment produces poverty and despair, and yet wages do not fall indefinitely. Evidently Adam Smith was right in assuming social mechanisms that prevent wages from falling below subsistence even in the presence of considerable unemployment. 21 If one divides time into sufficiently short 'days', then a consumer will survive at least for the first 'day' after her income has become zero. fabio petri VDE6 existence 12/02/2016 p. 22 work day after day. As already said, in modern economies, where many people only have their labour to offer, survival needs are liable to be very similar for large sections of the population; a real wage that goes below the survival needs of these people would entail a jump of the supply of labour from these people to zero, with a significant discontinuity in the labour excess demand function. An objection to the relevance of the survival problem points to an even graver problem of the approach - a problem that classical economics did not have. 6.10. All this may appear very artificial: the discontinuities due to non-survival – it may be observed – derive from an assumption of complete submission to the market: trade is implicitly assumed to be the sole way to get goods; no other alternatives are considered, such as becoming a beggar or scavenging in rubbish dumps; and this – it may be objected – is not how the world works. This objection is correct and important, but it does not eliminate the risk of non-negligible discontinuities, on the contrary it increases it. The discontinuity in the supply of labour may now arise, not because of starvation, but because of a refusal to continue to supply labour and a preference for non-market choices such as stealing or begging or – as shown by several historical examples – protests, riots, mobs attacking the property of the rich, armed rebellions, guerrilla activities. This possibility is not contemplated in the usual formalization of consumer choices, but it would be absurd to deny it – it has massive historical support. Feelings of violation of justice and of fairness will enter decisions of this type, and this will make it more likely that the refusal to continue to supply labour and the turn to other, perhaps violent activities may happen at the same level of the real wage for large numbers of people: social interaction tends to cause some agreement as to moral values and as to decisions among people in a similar situation. It seems clear that the neoclassical picture of a wage changing smoothly in response to excess demand would be inapplicable unless the danger of such events were remote: market societies develop institutions to avoid social problems for events of much smaller import, e.g. overproduction of agricultural products, so it is obvious that institutions, conventions and social mechanisms must be expected to exist to regulate labour markets in a way different from the one imagined by neoclassical theory, if it were felt that disasters such as wage reductions bringing half the population below minimally decent living standards could be easily produced by the neoclassical mechanism. It seems clear, then, that a necessary condition for general equilibrium theory, in whichever formulation, to be a plausible description of real economies is that the equilibrium real wage must be significantly above the minimum level felt by general consent to be necessary in order not to cause generalized disruptive behaviour. This indicates a lack of generality of the theory even greater fabio petri VDE6 existence 12/02/2016 p. 23 than the one highlighted by the survival problem. It is interesting then to note that the institutions, conventions and social mechanisms, that obviously would exist if the above danger were not remote, not only (i) can be suspected actually to exist, but also (ii) were taken for granted by the classical economists. On point (i): generally in market economies even when unemployment is considerable the real wage does not decrease to such low levels as to cause disruptive behaviour. For the marginalist economist, this is a problem in need of explanation(22); on the contrary the first observers of the capitalist system, i.e. the classical economists, considered such a situation as perfectly normal. They admitted persistent unemployment; a modern commentator, Mark Blaug, has written that "Ricardo assumed the existence of Marxian unemployment" and that "in an era when the number of individuals on public relief hovered steadily around [...] ten per cent of population [...] the existence of a hard core of surplus labour must have been taken for granted" (Ricardian Economics, 1958, p. 179, p. 58). What kept the real wage from falling indefinitely? According to the classical authors (cf. Chapter 1) the answer was social custom, social conventions, group solidarity, the respect of a minimum income “compatible with common humanity” (Adam Smith): they were thus implicitly admitting a risk that if “common humanity” were not respected, an orderly working of social relationships would not be possible (thus according to Smith wage labourers may have recourse "to the loudest clamour, and sometimes to the most shocking violence and outrage") and this would go against the interests of the capitalists themselves. This gives us a first reason to consider with attention the classical approach, whose different explanation of the real wage appears not to suffer from the lack of generality that seems inherent in a supply-and-demand explanation of wages. After this discussion of aspects too often neglected in the usual introductions to the theory of general equilibrium, let us return to presenting this theory in standard form. DEFINITION OF GENERAL EQUILIBRIUM OF PURE EXCHANGE 6.11. The definition of a ‘Walrasian’ price-taking general equilibrium, for a pure exchange economy with a given number of locally non-satiated consumers each one with a given endowment s of goods, is: Definition of Walrasian general equilibrium. An equilibrium is a vector of prices p* and a 22 Some proposed explanations are discussed in Chapter 12. fabio petri VDE6 existence 12/02/2016 p. 24 vector of demand vectors xs*, one for each consumer s, such that (i) xs* maximizes the utility of consumer s under her budget constraint, at the given prices p; (ii) z(p*)0. Note that if in equilibrium in some market there is excess supply the price in that market must be zero (we have assumed that prices cannot become negative), but it is unnecessary to specify this fact as part of the definition of equilibrium, because it derives from Walras’ law (which holds as long as local non-satiation is assumed, a universal assumption). Proof: from the definition of equilibrium zi(p*)0, i, and by Walras’ law p*z(p*)=0, a sum of non-positive terms because prices cannot be negative and equilibrium excess demands cannot be positive; and a sum of non-positive terms equals zero only if all terms are zero, so if zi(p*)<0 it must be pi=0. █ If preferences are strongly monotone then the equilibrium prices can only be all positive; there cannot be equilibrium with a zero price because the good with zero price would be demanded in infinite quantity and therefore the excess demand for that good would be positive. (If the demand for a good is infinite, the excess demand is also infinite because the total endowment of each good is finite.) EXISTENCE OF A PURE EXCHANGE GENERAL EQUILIBRIUM 6.12. We have already seen (Chapter 4, Section 4.2) that in the case of only two commodities at least one equilibrium certainly exists if either the economy-wide excess demand vector function z(p) is a continuous function on the entire price simplex Sn–1=(pRn+: Σipi=1), or if z(p) is a continuous function on the interior of the price simplex and zi(p) + when pi 0. The proof that an equilibrium exists for a higher (but finite) number of commodities can be obtained, but the less restrictive the assumptions, the longer the proof and the more complex the mathematical tools. To go over the several possible proofs at different degrees of generality and with different methods would take an entire book. We present two simple proofs under restrictive assumptions, and a less simple one to get a feeling of the kind of proof methods used in this area. The first proof is, to the best of my knowledge, new; it is very elementary mathematically, it extends to many commodities the proof method used for the two-commodities exchange economy with strongly monotone preferences (§4.2.2), but it needs an additional assumption, Assumption 2 in the following theorem: fabio petri VDE6 existence 12/02/2016 p. 25 Theorem 6.1: Consider an exchange economy with n goods where the consumption sets coincide with Rn (there is no minimum subsistence bundle), and market excess demand is a continuous function on the interior of the price simplex; assume: - Assumption 1: property 6 of z(p) is satisfied, i.e. if some price tends to zero, some excess demand tends continuously to +∞; - Assumption 2: having put the sum of prices equal to +1, for any good if its price tends to +1 then its excess demand sooner or later becomes and stays negative (before becoming zero owing to Walras' Law when the price becomes exactly +1); then this economy has an equilibrium. Proof. We can restrict the analysis to p>>0 owing to Assumption 1. Initially let us arbitrarily fix the ratios between the first n–1 prices: p2=a2p1, p3=a3p1, ..., pn-1=an-1p1, with a2,...,an-1 arbitrary positive scalars and p1>0; and let us vary pn/p1, or equivalently (since prices move in the unit simplex) pn; z(p) becomes a function of pn alone. Owing to the budget constraints, if a convergent sequence {pt} tends to the frontier of the unit simplex, the goods whose excess demands (by Assumption 1) tend to +∞ can only be goods whose price is tending to zero; this is because, if without loss of generality we let j=1,...,k be the indices of the goods whose price pj tends to a limit pj* different from zero, then the excess demand for any one of these goods is eventually(23) bounded above, for example, by the value n i i 1 2 p*j . Therefore, if we let pn tend to zero, since pn is the sole price tending to zero, for pn sufficiently close to zero it is zn(p)>0. Now let pn tend to 1: by Assumption 2, there is a pn<1 such that zn(p)<0. Therefore the sign of zn(p) changes in the open interval pn=(0,1) and, since z(p) is continuous, there is a pn* such that zn(p)=0. This is true for any ratios between the first n–1 prices. Now, pn* may not be unique; in this case, choose as pn* the smallest pn>0 for which zn is zero. Clearly, pn* is a function of a2,...,an-1, implicitly defined by the system of two equations Σipi=1, zn(p1,a2p1,a3p1,...,an–1p1, pn)=0; since both equations are continuous, by the implicit function theorem pn* is a continuous function of a2,...,an-1. Now let us maintain fixed only the ratios a2,...,an–2 and let pn-1/p1 vary, but obliging pn/p1 to vary at the same time in such a way as to maintain z n(p)=0. We have shown that this is possible, with pn/p1 so determined being a continuous function of pn–1/p1. As pn-1/p1 tends to zero, pn-1 tends to zero too; by Assumption 1 some excess demand tends to +∞ and it must be zn-1, because either pn- 23 We translate as 'eventually' the useful Italian mathematical term 'definitivamente' which means that there exists an n* such that a property is valid for all elements of a sequence {xn} for n>n*. fabio petri 1 VDE6 existence 12/02/2016 p. 26 is the sole price tending to zero, or pn tends to zero too but zn does not tend to + because it is zero. By Assumption 2, for values of pn-1 sufficiently close to 1 it is zn-1(p)<0. So zn-1(p) changes sign in the open interval pn-1=(0,1) and therefore there exist positive values of pn-1/p1 and pn* such that simultaneously zn–1=0 and zn=0, for any given ratios between the positive prices of the first n–2 goods. If the value pn–1/p1 so determined is not unique, choose its smallest value. For the same reason as before, these values are continuous functions of the ratios between the prices of the first n–2 goods. We can apply the same reasoning to pn–2, assuming fixed the ratios between the first n–3 prices, and varying pn–2/p1 imposing that pn–1/p1 and pn/p1 vary at the same time so as to maintain zn(p)=0 and zn-1(p)=0; again we can show that for any given ratios between the positive prices of the first n–3 goods there exist positive prices pn/p1, pn-1/p1, and pn-2/p1, such that zn–2=zn–1=zn=0. Repeating the reasoning, one concludes that there exist relative prices pn/p1, pn-1/p1,...,p2/p1, all positive, that render the last n–1 excess demands simultaneously equal to zero, and then by Walras' Law also z1(p)=0, so there exists an equilibrium. █ As shown above (property 6), Assumption 1 is verified if, for all consumers, preferences are complete, transitive, continuous, strictly convex over a choice set (the budget set) which is compact and convex, and strongly monotone (this of course implies local nonsatiation). Assumption 2 appears generally plausible, but I have only some partial answer to the question, which assumptions on preferences and endowments imply it[24]. 24 One way to justify Assumption 2 is by adding an interior endowment assumption (positive endowment of all goods for all consumers), together with strongly monotone and strictly convex preferences. Indeed, let the price of good n tend to 1 while the other prices remain positive. Consider a single consumer with a positive endowment of good n and (only for simplicity) a differentiable utility function u(x). Along the indifference hypersurface through the consumer’s endowment point ω the consumer’s demand xn for good n is a function of x1, x2, ... , xn-1 implicitly defined by u(x)=u(ω), with, at ω, negative partial derivatives ∂xn/∂xi for i=1,...,n-1. As pn tends to +1, all ratios pi/pn tend to zero, thus all possible budget hyperplanes necessarily end up having slopes -pi/pn greater (i.e. smaller in absolute value) than those partial derivatives; therefore sooner or later the marginal utility (at the endowment point) of the last unit of income spent on good n, (∂u(ω)/∂xn)/pn, becomes smaller than for all other goods and it becomes convenient for the consumer to exchange some part of the endowment of good n against each other good; thus for p n sufficiently close to 1 a consumer with a positive endowment of good n has a net supply of good n. If all consumers have a positive endowment of good n, Assumption 2 is valid for that good at the endowment point; but the same reasoning applies to any positive x that the consumer might reach by exchange, it suffices to consider it as an endowment point; thus an interior endowment assumption validates Assumption 2 for all goods. If not all consumers have a positive endowment of all goods, Assumption 2 can be derived from an assumption that for each good there is a level of its price sufficiently high (i.e. sufficiently close to 1) such that at any higher price those consumers who do not have a positive endowment of the good do not demand it, a rather fabio petri VDE6 existence 12/02/2016 p. 27 The next two proofs use a fixed-point theorem, the common procedure in the proofs of existence of general equilibrium. The method is to build an opportune continuous application g(p), from the price simplex S n–1 to the same simplex, which is connected to excess demands in such a way that if a p* exists such that g(p*)=p*, then the excess demands are all non-positive (and all zero, if p*>>0) and therefore p* is an equilibrium price vector; and then to apply a fixed-point theorem to prove that at least one p* exists such that g(p*)=p*. FIXED-POINT THEOREMS 6.13. I give the main theorems without proof. Brouwer’s fixed-point theorem n Let S n 1 x in R n : x j 1 be the unit simplex in R n , and let f : S n 1 Sn 1 be a j1 continuous function from the unit simplex onto itself; then f has a fixed point i.e. x S n 1 such that f x x . Generalized Brouwer’s theorem Let ARn be a non-empty, compact and convex set and let f:AA be a continuous function. Then f has a fixed point. f(x) 0 x 1 Fig. 6.7 The figure shows a graphical illustration of Brouwer’s theorem for the one-dimensional case f: [0,1][0,1]: any continuous function must touch the diagonal at least in one point. The generalized Brouwer theorem follows from the fact that the unit simplex is topologically equivalent plausible assumption. fabio petri VDE6 existence 12/02/2016 p. 28 to a non-empty, compact and convex set. It implies that any continuous deformation of a disk into the same disk (e.g. a disk of pizza dough), or any continuous remixing of a liquid in a bowl, must leave at least one point in the initial position (in real examples this is not going to happen because the particles composing the dough or the liquid are not infinitely small). Kakutani’s fixed-point theorem Let ARn be non-empty, compact and convex and let f:AA be an upper-hemicontinuous correspondence such that f(x) is a non-empty and convex set for all xA (f is then said convexvalued). Then f has a fixed point, in the sense that x such that xf(x). Exercise 9: show with drawings why a fixed point may not exist if the assumptions on f(x) in this theorem are violated. CASE OF EXCESS DEMAND EVERYWHERE CONTINUOUS 6.14. I give now a simple proof of existence of a general equilibrium of exchange under the assumption that market excess demand is continuous and bounded on the entire price simplex (i.e. the excess demand for a good tends continuously to a finite value when its price goes to zero), as a way to enter into the functioning of proofs of existence. Exercise 10. By modifying the proof of existence given above for the two-goods economy with strongly monotone preferences, prove (without fixed-point theorems) the existence of equilibrium in a two-goods economy with z(p) continuous and bounded on the entire price simplex. (What requires special attention is the possibility of equilibria with one price equal to zero.) In all existence proofs it is assumed that of all goods of which a price is quoted there is a positive aggregate endowment(25). Assume z(p) everywhere defined (i.e. finite) and continuous on the entire price simplex. Assume there are k goods. Of course zp : S k 1 R k satisfies Walras’law, pz(p)=0. Let the vector function g(p), g : S k 1 S k 1 be defined by: 25 . This assumption appears acceptable because it would become soon clear that there is no endowment of the good and then people would give up trying to obtain it. As long as they keep demanding the good there cannot be equilibrium because there will be excess supply of at least one of the other goods. fabio petri g j p VDE6 existence p j max 0 , z j p p. 29 12/02/2016 for j 1, ... , k k 1 max 0 , z h p h 1 Notice: g is continuous because every max 0,zj(p) is continuous since by assumption the functions zj(p) are continuous p max 0 , z p k g S k 1 k because g j1 j k j1 j j j1 k 1 max 0 , z h p 1 since at the numerator pj=1 and the h 1 other two sums coincide. (The function g raises the prices of the goods in excess demand, thus it is not economically counterintuitive; the denominator has the sole function to re-normalize the “new prices” g(p) so that their sum is unity.) Then by Brouwer’s theorem, since g(p) is a continuous function from the price simplex onto itself, there exists a price p* such that g(p*) = p*. It remains to show that if g(p*)=p* then z(p*)0 so that p* is an equilibrium price vector. From the definition of g, it follows that p * j 1 max 0 , z p p *j max 0 , z j p * k and passing the denominator * h 1 h p j p j max 0 , z h p p j max 0 , z j p k of the fraction to the other side: i.e. h 1 p j max 0 , z h p max 0 , z j p , j . Multiply both sides of this equality by zj(p*), and sum k h 1 k k up over all goods; on the left-hand side one obtains p j z j p max 0 , z h p which is j1 h 1 equal to zero because by Walras’ law the first sum equals zero. Hence we obtain 0 z j max 0 , z j p k which implies that all terms of the sum on the right-hand side must be h 1 zero since none of them can be negative; and this requires zj(p*)0, j. █ STRONGLY MONOTONE PREFERENCES 6.15.A less simple proof of existence is the following, for an exchange economy with strongly monotone preferences: this implies that z(p) is not defined on the boundary of the price simplex, so fabio petri VDE6 existence 12/02/2016 p. 30 the above proof is not applicable, and a more complex function of p must be used, in fact a correspondence, which requires Kakutani’s theorem for the proof that it has a fixed point. The proof assumes that there is no survival problem; the way to avoid the zero income discontinuity problem is solution A. The theorem to be proved is: Theorem ??: Let the consumption set for each consumer be Rn . A general equilibrium exists, and with all prices positive, if the aggregate endowment of each good is positive and if z(p) is a function, homogeneous of degree zero in p and: 1. Defined for all p>>0 belonging to the unit simplex S n 1 n n : p R : p j 1 j 1 2. Continuous for p>>0 3. Obeying Walras’ law, pz(p)=0 for all p 4. Bounded below i.e. s 0 : z j p s , j , p 5. Such that if a sequence pn tends to a p 0 with some pj=0, i.e. tends to the boundary of the price simplex, then Max (z1(pn),...,zn(pn))→ +∞. These are the relevant assumptions; the hypotheses on preferences are only relevant in so far as they justify these assumptions, e.g. strict convexity of preferences is required for z(p) to be a function and continuous; strong monotonicity of preferences has the role of ensuring assumption 5 (26). Assumption 4 simply means that endowments are finite, hence the maximum excess supplies of goods are finite (corresponding to the endowments), and it is therefore fully acceptable; s is any positive number greater than the greatest of the aggregate endowments of each good j. Proof. (This is a more user-friendly presentation of the proof in Mas-Colell et al. 1995 pp. 585-587.) The trick is again to find a continuous function, here a correspondence, of prices from Sn1 onto it, such that its fixed points are equilibria, i.e. a function which brings from p to the same p only if no excess demand is positive, and then to use a fixed-point theorem to prove that this function does have a fixed point. We distinguish three steps: 1. Choice of the correspondence f(p), in this case the union of two different correspondences depending on whether p is strictly positive or not. 2. Proof that the f(p) resulting from the union of these two correspondences is an upper hemicontinuous correspondence on a non-empty, compact, 26 . This was shown above as property 6: it was shown that strong monotonicity of preferences implies that, if some price tends to zero, ║z(p)║→ +∞, which is equivalent to Max (z1(pn),...,zn(pn))→ +∞. fabio petri VDE6 existence 12/02/2016 p. 31 convex set and that the image f(p) of p is a non-empty convex set for all pSn-1. Then by Kakutani’s theorem there exists at least one fixed point p*=f(p*). 3. Proof that a fixed point of f(p) is an equilibrium. Step 1. When p>>0, we use: fA(p) = qSn-1: qz(p) q’z(p), q’Sn-1 i.e. the correspondence with takes one from p to the price vectors q which maximize the value of z(p) (i.e. which assign price 1 to the good, or assign prices summing to 1 to the goods, whose excess demand is numerically greatest among those determined by p), and assign price 0 to all other goods. Notice that at least one such q exists. When pj=0 for some j, we use fB(p) = qSn-1: pq=0 = qSn-1: qi=0 if pi>0, i.e. if pj>0, then qj=0; if pj=0, qj can take any value as long as qSn–1; so here again at least one such q exists. The image of p according to fB(p) is therefore the subsimplex of the price simplex where all components corresponding to positive prices are zeros, and the sum of the components corresponding to zero prices equals 1. f(p) is the union of fA(p) and fB(p). Step 2. f(p) is defined on the price simplex which is non-empty, compact and convex. Also, each image f(p) is non-empty by construction, and it is convex because: (i) if p>>0, fA(p) is non-empty and convex because it is an h-dimensional unit simplex where h is the number of goods with the same greatest excess demand; (ii) if it is not p>>0, fB(p) is an r-dimensional unit simplex, where r is the number of goods with zero price in p. Now we prove that f(p) is upper-hemicontinuous, i.e. with closed graph and with bounded images of compact sets. This second thing is true by construction because qSn-1 is bounded. It remains to prove that, for every sequence qm converging to q with qmf(pm) where pm is a sequence converging to a price vector p, it is qf(p). This is the longest part of the proof. Let us distinguish the cases 2A: p strictly positive, and 2B: p not strictly positive. The case 2B will have to be subdivided in turn into two subcases, and one of these again into two subcases. 2A) If p>>0, then pm>>0 eventually, i.e. m*: pm>>0 for m>m*; then from qmz(pm)q’z(pm), q’Sn-1, and from the continuity of z(p), it follows that qz(p)q’z(p), q’Sn-1, i.e. qf(p) from the definition of fA(p). (Indeed if, however close qm gets to q, it is always qmz(pm)q’z(pm), q’Sn-1, then it cannot be that qz(p)<q’z(p) for some q’, because then it would also have to be, for pm sufficiently close to p, qmz(pm)<q’z(pm) for some q’Sn-1.) 2B) If p is not strictly positive, let us prove that the condition of closed graph is satisfied for each component of the vector q. Here we consider separately the components qi of qf(p) corresponding to positive prices in p, and the components qj corresponding to zero prices in p. fabio petri VDE6 existence 12/02/2016 p. 32 2B.1) Let us start with the components corresponding to positive prices in p. Take one of these positive prices, say pi. Then pim>0 eventually, hence pim is eventually bounded below by a positive lower bound, i.e. m*, >0 such that pim> for m>m*. Distinguish now two mutually exclusive subcases: either (2B.1.a) a subsequence pm’ of pm exists which is eventually on the boundary of the price simplex i.e. with some pjm=0 for m greater than some m*; or (2B.1.b) eventually pm>>0. 2B.1.a) In the first subcase, since eventually some pjm’=0, ji, eventually f(pm’)=fB(pm’) and qim’=0 eventually, hence qi=0 by convergence(27) and qifiB(p). 2B.1.b) In the second subcase it is pm>>0 eventually: then assumptions 4 and 5 in the theorem imply that it must be eventually zi(pm) < Max [z1(pm),...,zn(pm)], because, by assumption 5, Max[.] + while zi(pm) is bounded above because the demand for good i cannot be greater than when all income is employed in demanding good i, so cannot be greater than the maximum possible value of endowments divided by the minimum possible value of pim, where the maximum possible value of endowments is bounded above because no price can exceed 1, and pim is bounded below by >0; more formally: n z i (p ) m p mk k k 1 pi n i p mk k k 1 n i k 1 k ns , where is the lower bound to pim used above, and s is the upper bound to the excess supply of each good in assumption 4. This means that zi(pm) is eventually not the greatest excess demand; therefore fA(p) assigns eventually to pim the price qim=0, hence here too by convergence, qi=0 and therefore qifiB(p). 2B.2) There remain the zero prices in p, say, without loss of generality (it suffices to renumber the goods), that they are the first r ones in p. The above results imply that the sum of the corresponding (q1m,...,qjm,...,qrm) is eventually equal to 1, because qmSn-1 and we have seen that f(p) assigns eventually a price qim=0 to all goods whose price tends to a positive pi. By continuity therefore the sum of (q1,...,qr) is also 1 and therefore (q1,...,qr) are in fB(p); this completes the proof that qfB(p). The proof of step 2 is achieved. Therefore by Kakutani’s theorem f(p) has a fixed point. Step 3. Let p*=f(p*) be a fixed point of f(p). Then it is p*>>0: it is impossible that some pj*=0 because then f(p*)=fB(p*) and by construction it is always qfB(p)p because fB(p) assigns a zero qi to the positive prices in p and a positive qj to at least one of the zero prices in p, so we would 27 . A sequence is convergent only if all its possible subsequences converge to the same limit; hence p p and qm’q. m’ fabio petri VDE6 existence p. 33 12/02/2016 obtain f(p*)p*. It follows that it cannot be z(p*)0 because since p*>>0 and since by Walras’ law p*z(p*)=0, if one zj(p*) were different from zero there would have to be another zi(p*) different from zero and of opposite sign, i.e. at least one negative excess demand, say the one for good h, but then fA(p*) would assign to good h a qh*=0<ph* and again it would be f(p*)p*. So it must be z(p*)=0, i.e. (p*, z(p*)) is an equilibrium. █ Exercise 12. Produce examples of non-existence of equilibrium when only condition 3 or only condition 4 of the theorem are not satisfied. FURTHER READING 6.16. For those who are interested in further study of the issue of existence of general equilibrium, I recommend the whole of Hildenbrand and Kirman, Equilibrium Analysis, NorthHolland, 1988, and, if they can read Italian, the neat proofs of existence for the pure exchange economy without strongly monotone preferences, based on assumptions B (strictly positive endowments) or C (irreducibility), in Reichlin and Ventura, Equilibri competitivi ed economie dinamiche (Carocci 1998); this is a well organized and mathematically rigorous but not overly advanced book, useful also for a first introduction to equilibrium over infinite horizons and overlapping-generations economies, but it is disconcerting for its total neglect of issues of uniqueness and stability, as if existence were all that matters. (There is no discussion of uniqueness and stability also in G. A. Jehle and P. J. Reny, Advanced Microeconomic Theory, 2nd ed., Addison-Wesley, 2001.) In English an existence proof based on B can be found in M. C. Blad and H. Keiding, Microeconomics: Institutions, equilibrium and optimality, North-Holland 1990, pp. 157-161, and in Hildenbrand and Kirman, cit., pp. 108-111; as far as I am aware proofs based on irreducibility are not available in English-language textbooks, only in specialist articles and in Arrow and Hahn’s General Competitive Analysis, 1971, which cannot be considered a textbook. Mas-Colell et al., Appendix 17.BB.2, contains a proof of existence admitting correspondences but based on a "cheaper consumption condition" which requires directly that for each consumer there exists a consumption bundle in the consumption set (the latter is defined so as to guarantee survival) which costs less than her income, i.e. income must always be in excess of what is needed for survival; what might guarantee such a condition (our assumptions B' or C') is only briefly hinted at. A different proof technique is in Debreu, “Existence of competitive equilibrium” in Arrow and Intriligator, eds., Handbook of Mathematical Economics, vol. II, 1982; also important is L. W. McKenzie, "The classical theorem on existence of competitive equilibrium", Econometrica 1981. A fabio petri VDE6 existence 12/02/2016 p. 34 useful history of the evolution of general equilibrium theory including a discussion of the evolution of existence proofs is B. Ingrao and G. Israel, The invisible hand; unfortunately it neglects the existence of the long-period versions that we will discuss in Ch. 7. These references show that original research in this area requires a substantial mathematical background: most of the contributors in this area have a first degree in mathematics. For those who want to strengthen their maths without going all the way to becoming full-fledged mathematicians, I suggest B. Ellickson, Competitive Equilibrium, Cambridge UP, 1993. In order to approach the modern frontier, which is concerned with equilibrium over infinite horizons, C. D. Aliprantis, D. J. Brown, O. Burkinshaw, Existence and optimality of competitive equilibria, Springer-Verlag, 1990, is mathematically more complete than Reichlin and Ventura. fabio petri VDE6 existence 12/02/2016 p. 35 UNIQUENESS AND STABILITY UNIQUENESS OF THE GENERAL EQUILIBRIUM OF PURE EXCHANGE The non-uniqueness of equilibrium in general. Possibility of several locally stable equilibria 6.17. As a premise, uniqueness of equilibrium means a unique equilibrium allocation and a unique equilibrium relative price vector. Below we take it for granted that a numéraire has been chosen, so 'different price vectors' means non-proportional price vectors. We have seen in Chapter 4, Fig. 4.5??, an Edgeworth box with three equilibria. We reproduce that Figure here for easier reference. intl. 13 B 2A A 1A Fig. 6.8. Three equilibria. Consumer A's choice curve is the thick continuous line, consumer B's choice curve is the thick broken line. The three slopes of the budget line which cause it to cross one of the three equilibria indicate the three equilibrium relative prices. intl. 13 B 2A fabio petri A VDE6 existence 12/02/2016 p. 36 1A Fig. 6.9. A case of tangent choice curves causing an even number of equilibria. Consumer A's choice curve is the thick continuous line, consumer B's choice curve is the thick broken line. It is also possible that the two choice curves be tangent to each other, in the endowment point (then the equilibrium is necessarily unique if demands are single-valued), or elsewhere, cf. Fig. 6.9; or that the two choice curves overlap over an entire stretch (cf. Fig. 6.14?? below), in which case there is a continuum of equilibrium price and quantity vectors. The Figures show that in a two-goods economy, if the choice curves cross each other, the equilibria alternate in character: in the Edgeworth box of Fig. 6.8, the central equilibrium is such that for relative prices only a little different from the equilibrium relative prices, the excess demands are such as to push the economy away from it, while the opposite is true for the other two equilibria. We see therefore that there may be many locally stable equilibria, i.e. equilibria such that the forces of demand and supply push the economy toward one or another one of them depending on where prices start from(28). Each locally stable equilibrium has an 'area of attraction' i.e. an open set of relative prices such that if initially prices are in that set, they tend to that equilibrium, but if they are outside that set, they do not. Exercise. Draw a 6×6 Edgeworth box. Both consumers have perfect-complement preferences; consumer A demands 2 units of good 1 for each unit of good 2; consumer B demands 2 units of good 2 for each unit of good 1. A’s endowment is (5,1). Show that there are three equilibria, of which the interior one is unstable. The consequences for the theory are potentially very serious, even assuming that the interplay of supply and demand causes the economy to converge to one of the locally stable equilibria( 29). Suppose that in an economy initially in equilibrium there is a change in the data, which changes the set of equilibria: the economy will find itself in disequilibrium and it is not generally possible to know in advance in which ‘area of attraction’ it will be, hence a risk of indeterminateness of predictions, and a great difficulty with comparative statics. For example if a new tax or labour immigration alter the equilibria, we may be unable to predict the effect of the tax or of labour immigration because we may not know which of the new equilibria will be reached after the 28 This definition of local stability is for the moment only intuitive, based on the idea that when excess demand is positive (negative) for a good, the good's price tends to rise (to decrease). Later in the chapter we will be more precise on the adjustment mechanism. 29 We will see later that there may be cases in which the price adjustment process is unable to reach an equilibrium even when equilibria do exist. fabio petri VDE6 existence 12/02/2016 p. 37 introduction of the tax. If the equilibria are very numerous and close to one another, it is also likely that the economy may move from one equilibrium to another not because of a change in the data but simply because of transitory accidents which cause it to be pushed temporarily into disequilibrium. This indeterminateness, if it cannot convincingly be excluded or limited to very unlikely occurrences, may raise the suspicion that the theory has not correctly grasped the forces determining average prices, average quantities and in particular income distribution in actual market economies, where these magnitudes, and income distribution in particular, appear to be well determined, and to change slowly. Regular economies and finiteness of the number of equilibria: a sceptical assessment 6.18. A particularly bad case for the theory would be if there were a continuum of equilibrium allocations and of associated equilibrium prices, because then the indeterminateness of the equilibrium would be total within that continuum. Theorists have explored the likelihood of a continuum of exchange equilibria, and the following result has been proved. In the Proposition below, ∂z(p) is the Jacobian matrix of z(p), with elements ∂zij ≡ ∂zi(p)/∂pj; and an equilibrium price vector p (normalized in the unit simplex) is locally isolated (or locally unique) if there is an ε>0 such that for all p'≠p such that ║p'–p║<ε it is not z(p')≤0, i.e. there is no other equilibrium price vector in a sufficiently small neighbourhood of an equilibrium price vector. Proposition 6.1. Regular economies imply a finite number of equilibria. Assume that the economy is one where each consumer's preferences are continuous, strictly convex and strongly monotone, and where the market excess demand function z(p) is continuously differentiable on Rn . Define regular this economy if its Jacobian matrix z(p) has rank n–1 at all equilibrium price vectors. For every regular economy the equilibrium price vectors (normalized in the unit simplex) are finite in number and hence locally isolated. Proof. We only give an intuitive proof. Note that n–1 is the maximal rank of the Jacobian matrix of z(p), because from the homogeneity of z(p), differentiating both sides of the equality z(tp)=z(p) with respect to t we obtain z(tp)p=0 and if we perform this calculation in t=1 we obtain z(p)p=0; so z(p) is always singular. The relevant property is that if the aggregate demand vector is considered a function of relative prices, then its (n–1)×(n–1) Jacobian matrix must be of full rank at all equilibrium relative price vectors. The interpretation of this condition is as follows. Put pn=1 (we have the right to choose a numéraire because we are only interested in strictly positive price fabio petri VDE6 existence p. 38 12/02/2016 vectors(30)), consider the aggregate demand vector a function of the first n–1 prices, z = z^(p^) where p^=(p1,...,pn–1), and let ∂z^(p^) be its Jacobian matrix; if p^* is an equilibrium relative price vector, and if ∂z^(p^*) is of full rank, then there is no nonzero vector of variations of relative prices dp^ such that ∂z^(p^*)∙dp^=0. Thus any infinitesimal change of p^ causes the economy no longer to be in equilibrium; each equilibrium relative price vector is locally isolated. Hence the set of equilibrium relative price vectors is countable, and so is also the set of equilibrium price vectors normalized in the unit simplex. Because of property 6 of z(p) (chapter 4, Section ??), equilibrium price vectors are bounded away from zero, so, going back to prices normalized in the unit simplex, there is a scalar s such that if z(p)=0, then for all pi it is pi>s; therefore there is a closed (and obviously bounded) subset of the unit simplex {p Rn : Σipi=1 and pi≥s for all i=1,...,n}, that contains all equilibrium price vectors. A countable set of points of a closed and bounded (i.e. compact) subset of Rn is finite unless it has some accumulation point; but an accumulation point of equilibrium prices would have to be itself an equilibrium price by the continuity of z(p), and this would contradict the local uniqueness of each equilibrium. ■ In the case of two goods for example, the rank of the Jacobian matrix is non-maximal i.e. zero if at a point where z(p)=0 it is z1(p)/p1=0, cf. points A and B in Fig. 9.3. z1 A 0 B p1 at both equilibria A and B 1 it is z1(p)/p1=0 Fig. 6.10.. Excess demand in a non-regular 2-goods economy with p1+p2=1. Nothing would change with a different normalization; in particular, if ∂z(p) is of rank n–1, then whichever the chosen numéraire that renders aggregate demand a function of n–1 relative prices the corresponding Jacobian will also be of rank n–1, and the sign of its determinant is independent of the numéraire chosen, cf. MathApp?? 30 fabio petri VDE6 existence 12/02/2016 p. 39 The above result has been greeted with great interest because of the following Fundamental Property of Regular Economies (which is intuitive enough to allow us to dispense with proving it): if under the assumptions of Proposition 6.1 an economy is not regular, then by an arbitrarily small change in initial endowments, keeping preferences fixed, we obtain a regular economy with probability 1 (Hildenbrand and Kirman, p. 223); therefore regular economies are the generic case among economies with continously differentiable z(p) (where a property is generic on a set if the property holds for 'almost all' the elements of the set, i.e. for all elements of the set except at most a subset of Lebesgue measure zero, so that, if an element in the set is selected at random with a nonatomic probability distribution, the probability of selecting an element for which the property does not hold is zero). Hence, it is argued, the probability of encountering a non-regular economy among the economies with continuously differentiable z(p) is zero. 6.19. Proposition 6.1, in view of the genericity of regular economies (among economies with continuously differentiable excess demands), has been interpreted to mean that there is no need to worry about actual occurrences of the strong indeterminacy associated with a continuum of equilibria. However, as noted in chapter 3, if excess demand is very close to zero in a rather ample neighbourhood of a locally isolated equilibrium price vector (the case of practically indeterminate equilibria), for the purposes of the theory the situation in that neighbourhood is akin to a continuum of equilibria – indeterminacy – because the forces tending to change prices are extremely weak. Furthermore, the assumptions of Proposition 6.1 essentially imply that all consumers always demand all goods. This is because as we now show differentiability of z(p) requires that for each consumer the goods in positive demand are the same at all prices, that is, it must never happen that a change in prices causes the demand for a good to go from positive to zero, or vice-versa. This striking and seldom stressed implication derives from the fact that, except for flukes, differentiability of the excess demand function will not obtain if a consumer's demand for a good is positive at certain prices and zero at other prices (an extremely common occurrence). The reason is that at the price, at which the consumer’s demand for the good passes from positive to zero, there will generally[31] be a kink in her demand function; then the market excess demand too will not be differentiable at that price (unless one assumes a continuum of infinitesimal and not identical ‘Generally’, because it is possible that when indifference curves touch the axes they are tangent to the axes, so the demand function does not have a kink when the demand for a good becomes zero (for example, assume that indifference curves are the south-west quarters of circles inscribed in the axes). But clearly the general case is kinks. 31 fabio petri VDE6 existence p. 40 12/02/2016 consumers). A simple example confirms this. Assume that in a two-goods economy a consumer has quasilinear utility u= ( x1 1 4 ) 1 2 x2 , and an endowment consisting of 1 unit of good 1. The strictly convex indifference curves are all parallel vertical displacements of any one of them, and they touch the vertical axis with slope –1; so the consumer has positive demand for good 1 only if p1/p2<1 (the reader is invited to draw a picture of the indifference curves and budget lines of this consumer at different relative prices). Let us normalize prices by setting p2=1; then for p1<1 the demand for 1 1 good 1 is given by the solution of ( x1 1 4 ) 2 (the reader should make sure she understands 2 p1 why it is so). At p1=1 it is x1=0, the right-hand derivative of x1(p1) in p1=1 is zero, the left-hand derivative is 1/2, so x1(p1) is not differentiable in p1=1. Now, Proposition 6.1 also assumes strongly monotone preferences; this prevents a good from having zero marginal utility and therefore ensures that, if the consumer's income is positive, each good will be demanded if its relative price becomes low enough. Therefore if the goods in positive demand must be the same at all prices, then, since each good will be demanded for sufficiently low price, the assumption of z(p) differentiable essentially implies that one is assuming that all consumers always demand all goods. B K H Y A X Ω Fig. 6.11. The indifference curves of consumer A are drawn as continuous thin curves, those of consumer B as dotted thin curves. Ω, the endowment point, is the lower right-hand corner. All budget lines between the thick broken straight lines ΩH and ΩK represent equilibrium relative prices. fabio petri VDE6 existence 12/02/2016 p. 41 In an Edgeworth box (pure exchange), one implication of this assumption is that the equilibria are not on the edges of the box. Then, since all equilibria are interior, a continuum of equilibria requires, either that the initial allocation is itself an equilibrium allocation and the consumers’ indifference curves have kinks at that point so that the slope of the budget line ensuring equilibrium is not uniquely determined, or that the choice curves overlap for a stretch in the interior of the box (in this latter case not only equilibrium prices but also equilibrium allocations form a continuum); it is intuitive that both cases can only be flukes, generically destroyed by any small change in endowments or preferences. Suppose instead that indifference curves touch the axes. Then the case of Fig. 6.11 can happen, where there is a continuum of equilibrium relative prices: all slopes of the budget line included in between segments ΩH and ΩK represent equilibrium relative prices. In this case the existence of a continuum of equilibrium relative prices is robust with respect to a small modification of preferences, and also with respect to a small change in endowments as long as the latter consists, for each consumer, of a small change in the sole endowment of the good in positive endowment. It might be objected that the latter is a change of probability zero among all possible endowment changes; but this is true only if all directions of change of the endowment vectors are possible, which may well not be the case: a consumer may be restricted (e.g. because of geographical constraints) to have endowments from which certain goods are absent. Exercise: Suppose that in an Edgeworth box the consumers’ indifference curves are strictly convex but all have a kink along the set of Pareto-efficient allocations or Pareto set, which is an interior curve (except at the two origins). Show graphically that if the endowment point is in the Pareto set then it is the unique equilibrium allocation but it may be associated with a continuum of equilibrium relative prices. Also show graphically that if the endowment point is not Paretoefficient the equilibrium allocation need not be unique, in which case each equilibrium allocation is associated with different relative prices. Explain why these cases can be considered flukes. The Sonnenschein-Mantel-Debreu result 6.20. A further objection to the relevance of regular economies is that, even granting the continuous differentiability of z(p), Proposition 6.1 only proves that the number of equilibria is finite. There might still be hundreds of equilibria very close to one another – which would mean nearly the same degree of indeterminateness as with a continuum of equilibria. Can we exclude this possibility? Not without special assumptions, as shown by the following result: there is no other property of the market excess demand function derivable from wellbehaved(32) strictly convex preferences, except homogeneity of degree zero, continuity, Walras' 32 I.e. complete, reflexive, transitive and continuous. fabio petri VDE6 existence 12/02/2016 p. 42 law, and (if strong monotonicity is assumed) the boundary behaviour described in §6.2 as property 6 of z(p). More precisely the result is the following: Proposition 6.2 ( Sonnenschein – Mantel – Debreu theorem): any function z(p)Rn, defined everywhere on a compact subset of the interior of the unit price simplex Sn-1 (i.e. everywhere on Sn–1 except for a lower positive bound on each price, which can be made as small as one likes), and with the properties of continuity on S n1 Rn Walras’s law: p z p 0 , p S n1 Rn can be obtained as the market excess demand function of a pure exchange economy with n goods and no less than n consumers with continuous, strictly convex and strongly monotone preferences. We omit the proof (cf. Hildenbrand-Kirman 1988, or Shafer-Sonnenschein 1982). Thus if we neglect prices very close to zero, as long as consumers are at least as numerous as the number of goods there will always exist a set of preferences and of endowments generating any continuous market excess demand function we may choose. So, however many and however close to each other we stipulate the positive-price equilibria to be, there will be sets of well-behaved preferences and of endowments generating them: practically anything is possible. The risk of indeterminacy is therefore almost as high as with a continuum of equilibria (Fig. 6.12). z1 the Sonnenschein-Mantel-Debreu theorem implies that it is for example possible that the excess demand 0 p1 1 function in a two-goods economy has this shape Fig. 6.12 One might hope that a very high number of equilibria is possible only if preferences and endowments are very peculiar, e.g. differing among consumers in special ways. But recent research has shown that even with consumers having 1) the same preferences, 2) co-linear endowments, and 3) endowments differing in pre-assigned amounts, still the Sonnenschein-Mantel-Debreu result holds (Hildenbrand and Kirman, p. 208). Thus for positive prices the standard assumptions on preferences (continuity, convexity, fabio petri VDE6 existence 12/02/2016 p. 43 monotonicity) impose no restriction at all on the possible form of the market excess demand function (apart from homogeneity, continuity and Walras’ law), and hence on the number and proximity to each other of the equilibria. Uniqueness through conditions on excess demand. The index theorem. 6.21. 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 perhaps 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. We proceed to list the main assumptions so far discovered that ensure (i.e. are sufficient for) uniqueness, and then to assess the plausibility of their validity in actual economies. Two types of assumptions have been explored in this respect: assumptions on z(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 6.3. Gross substitutes. If the market excess demand function of a pure exchange economy satisfies the gross substitutes (GS) condition, then if there is an equilibrium it is unique. The Gross Substitutes condition, for brevity GS, states: if the price of only one good varies, the excess demands of all other goods vary in the same direction, and this is true whichever the price that varies. Formally: the goods are all gross substitutes if, for any two price vectors p and q (not normalized to be in the unit simplex) such that pi = qi except for one good, j, for which qj>pj, then zi(q)>zi(p), ij. This condition can be specified for individual excess demand functions, or – as here – for the market excess demand function. (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[33] GS applied to the case of z(p) homogeneous of degree zero (because income derives from endowments) implies 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 33 Or in the case of a consumer whose income derives from given endowments. fabio petri VDE6 existence 12/02/2016 p. 44 excess demands, the increase of the sole j-th price has the same effect on excess demands as a decrease in the same proportion of all other prices; and if we keep the j-th price constant and let the other prices decrease one at a time, GS implies that at each step the excess demand for the j-th good must decrease. The converse is not true: that the own-price effects be all negative does not imply that if a price rises, all excess demands for the other goods increase. Another way to define GS for differentiable excess demands is that it must be zi(p)/pj>0 if ij. This again implies that the demand for a good decreases when the price of that good increases, because it implies ∂zj/∂pj<0: the proof is that the homogeneity of degree zero of excess demand functions and Euler’s theorem on homogeneous functions imply n (pi∙∂zj/∂pi) = 0, and all terms i 1 where i≠j are positive by assumption. The proof that GS implies uniqueness in pure-exchange economies is not difficult if one assumes strongly monotone preferences[34]. Let p*Sn-1 be an equilibrium price vector with {xs*} the associated equilibrium allocation with xs* the vector of equilibrium demands of consumer s, s=1,...,S where S is the number of consumers. Let p>>0, pSn-1 be another price vector; we want to show that it cannot be an equilibrium price vector. Since p*>>0 we can define t = max j pj/pj* > 0; suppose t=pi/pi*. Consider the vector tp*; it is tpj*pj, with tpi*=pi. Because of the homogeneity of z(p) it is z(tp*) = z(p*) = 0. Now in successive steps let us lower each price tp j*tpi* until it becomes equal to pj. There is at least one such step, otherwise it would be p=p*. At each step only one price decreases and GS implies that the excess demands for all other goods decrease, among them the excess demand for good i which therefore becomes negative. When the equality to p has been achieved for all prices, it is zi(p)<0 which is incompatible with equilibrium since p>>0; hence p cannot be an equilibrium price vector. █ GS is additive across excess demand functions; so if each consumer’s excess demand function satisfies it, then the market excess demand function does as well; the proof is trivial, if as a price 34 . The general proof is based on the fact that GS implies that in equilibrium p*>>0, whether preferences are strongly monotone or not; the proof of the latter result is rather complex and we omit it, and for simplicity we have preferred to obtain p*>>0 from an assumption of strongly monotone preferences. We know from Chapter 4 that strongly monotone preferences imply p*>>0, but we give here a simple direct proof. Suppose the contrary i.e. that p* contains a zero price, say pk; let ek be the k-th unit vector with as many elements as there are goods, ek =(0,…,1,…0) with 1 in the k-th place; let xs* and ωs represent the equilibrium demand vector and the endowment vector of consumer s; then p*(xs*+ek) = p*xs* = p*ωs ; thus consumer s might afford demanding a vector xs*+ek, obtaining a higher utility, against the hypothesis that at p* the vector xs* is an optimal choice. fabio petri VDE6 existence p. 45 12/02/2016 changes the sign of the change in individual excess demands for a good is the same for all individuals then the sign of the change of market excess demand for that good is the same. The universal assessment of the GS assumption is that it is "very restrictive" (Mas-Colell et al., 1995, p. 611), and rightly so. Let us first consider a consumer with given income, and let us consider the import of GS on how her choice between goods 1 and 2 is affected by an increase in the price of good 1: Fig. 6.13b shows that GS is satisfied only if the new optimal choice is on the segment BC, a result which is only guaranteed if good 2 is inferior or neutral. Now, more in the spirit of general equilibrium, let us assume that the consumer’s income is the value of a given endowment, and let us consider Fig. 6.13a, that represents possible choices of a consumer with endowment in the two-goods case. If the consumer’s choice changes from A to B as the budget line becomes less steep, this is a violation of GS because the rotation of the budget line can be seen as caused by an increase of the sole price of good 2, in which case GS requires that the demand for good 1 increases. Analogously if the consumer’s choices were C at the old prices and D at the new prices, this is a violation of GS because the rotation of the budget line can be seen as due to a decrease of the sole price of good 1 in which case GS requires that the demand for good 2 decreases. The same reasoning applies with more than two goods. good 2 good 2 ● A C •B Ω● B A • D C● good 1 Fig. 6.13a good 1 Fig. 6.13b Thus GS at the single consumer level excludes a perfectly possible (indeed plausible) occurrence that requires nothing particular about preferences once income comes from given endowments: choice curves with upward-sloping sections[35]. And if GS does not hold at the 35 Thus the absence of Hicksian complementarity is not sufficient for GS to hold: for example when there fabio petri VDE6 existence 12/02/2016 p. 46 individual consumer level there is no guarantee that it will hold at the aggregate level: there is no guarantee that if some consumers increase their demand for a good when the price of another good decreases, other consumers will more than compensate for it by sufficiently decreasing their demand for the first good. 6.2.2. A second result is relevant, as we shall see, for production economies too. It holds if one assumes that the Weak Axiom of Revealed Preferences holds for the economy-wide consumers’ excess demand vector function z(p). The WARP was defined in ch. 4 with reference to the value of demand vectors, but it has an immediate translation into an axiom on excess demands if one assumes that the income utiized to demand goods derives from given endowments. The WARP (as usually formulated, i.e. assuming single-valued consumer choices) says that if a demand vector is revealed preferred to a different demand vector, then when the second demand vector is chosen the first one must not be affordable; formally, if at prices p the demand vector is x(p) and px(p)≥px(p') with x(p)≠x(p'), then p'x(p)>p'x(p'). Under a balanced budget and a given endwment vector ω it is px(p)=pω, p'x(p')=p'ω, hence px(p)≥px(p') can be written p[x(p')–ω]≤0, and p'x(p)>p'x(p') can be written p'[x(p)–ω]>0; therefore the WARP can be formulated in terms of the consumer's excess demand function z(p): WARP when income derives from given endowments, : if pz(p')≤0 and z(p')≠z(p), then p'z(p)>0. If one assumes that the economy-wide (or market) excess demand function z(p) satisfies this condition one is making the assumption called WA in ch. 4, only reformulated now in terms of excess demand. [36] An implication of WA, to be used when we come to the study of stability, is the following: are only two goods these are necessarily substitutes in the Hicksian sense, and yet GS need not hold, as Fig. 6.6a shows. For completeness I mention that GS implies two mathematical properties of the Jacobian of z(p) each one of which suffices to ensure uniqueness of equilibrium for an exchange economy, Diagonal Dominance and Negative Definiteness; these do not imply GS so they are slightly less restrictive than GS, but still highly restrictive because, like GS, both imply that the own-price effect of a change in the price of a good upon its market excess demand is always negative, a property difficult to justify unless the choice curves of all consumers never have upward-sloping portions, a very restrictive condition; the aggregate demand for leisure, for example, must never be an increasing function of the real wage (i.e. of the price of leisure). Also, no economically meaningful condition implying these properties has been found apart from GS. Therefore it does not seem worthwhile to discuss these properties. 36 Note that for WA to hold the two price vectors p and p' must be non-proportional (i.e. relative prices must be different) because if p’=αp with α a positive scalar then z(p)=z(p’) because of the homogeneity and single-valuedness of z(p). fabio petri VDE6 existence 12/02/2016 p. 47 WA Lemma: under WA, if p' is an equilibrium price vector and p is such that z(p)≠z(p'), then p'z(p)>0. Proof: by the definition of equilibrium it is z(p')≤0 and therefore pz(p')≤0 for any p≥0, so the premise of WA is satisfied. █ This implies that under WA if p' is an equilibrium price vector, then any non-equilibrium z(p) has positive value at prices p'. It further implies that if p and p' are both equilibrium price vectors but z(p)≠z(p') (e.g. because in one equilibrium a good is in excess supply with price zero, while in the other equilibrium it has positive price), then WA does not hold. Thus we have: Corollary of the WA Lemma: non-uniqueness of (relative) equilibrium prices is compatible with WA only if z(p) is the same at all equilibrium prices. (However, if one assumes free disposal then equilibrium excess demands can always be assumed to be identically zero.) On this basis one can prove: Proposition 6.4. Quasi-uniqueness under WA. Under WA the set of equilibrium price vectors is convex. Proof. Suppose p' and p" are two (non-proportional) equilibrium price vectors i.e. z(p')≤0, z(p")≤0. Let p=αp'+(1-α)p" for 0<α<1. We must prove that p is an equilibrium price vector, that is, z(p)≤0. By Walras' Law 0=pz(p)=αp'z(p)+(1-α)p"z(p) so it is not possible that both terms on the right-hand side are positive, hence either αp'z(p)≤0 or (1-α)p"z(p)≤0 or both. If αp'z(p)≤0 then also p'z(p)≤0; then the only way for the WA Lemma not to be contradicted is that z(p)=z(p')≤0. If (1α)p"z(p)≤0 then for the same reason z(p)=z(p")≤0. So certainly z(p)≤0 and p is an equilibrium price vector. In fact because of the Corollary of the WA Lemma it is z(p)=z(p')=z(p"). █ Convexity of the set of equilibrium price vectors implies that either there is a continuum of relative equilibrium price vectors, or there is at most one. Thus if the economy is regular, WA implies that if there is an equilibrium then it is unique. The case with three equilibria of §6.17, Fig. 6.8, is incompatible with WA. On the contrary the case of the Edgeworth box of Fig. 6.14?? is compatible with WA. intl. 13 B β α 2A fabio petri VDE6 existence A 12/02/2016 p. 48 1A Fig. 6.14??. Continuum of equilibria: all price lines in between the α and the β lines are equilibrium price vectors. Consumer A's choice curve is the continuous line, consumer B's choice curve is the broken line. Exercise: Show what precisely is incompatible with WA in the case with three equilibria of Fig. 6.8 (in §6.17). How plausible is WA? In Ch. 4 it was seen that the fact that the WARP holds at the level of individual consumers is no guarantee that it will hold for the sum of their demands. So WA will easily not hold for the market excess demand, unless there is a representative consumer with wellbehaved (i.e. complete, reflexive, transitive, continuous and non-satiated) and strictly convex preferences. In which case, since we are excluding production, for this representative (aggregate) consumer the endowment point is the sole possible equilibrium allocation[37], and the equilibrium price vector can be non-unique only if the representative consumer's indifference hypersurface has a kink at the endowment point. Which brings us to our third result on uniqueness: Proposition 6.5. Single consumer. The equilibrium is unique if there is a representative aggregate consumer with a strictly quasi-concave and differentiable utility function. Of course the existence of a representative consumer for the purposes of general equilibrium theory requires, as shown in Chapter 4, extremely restrictive conditions: if endowments are arbitrary, preferences must be the same for all consumers and furthermore homothetic; if endowments are co-linear, preferences must be homothetic; if endowments are identical, preferences must be either the same, or homothetic. A fourth result is the following: Proposition 6.6: If preferences are strongly monotone and strictly convex and if the initial allocation of endowments is itself an equilibrium, then that is the sole exchange equilibrium allocation (the equilibrium price vector need not be unique). The proof is postponed to ch. 14 because it uses the First Fundamental Theorem of Welfare Economics which we will discuss there. An intuition can be obtained from the two-consumers two- 37 Exercise: Assume an economy with a single consumer with locally nonsatiated but not monotone preferences. Show that even in this case her endowment point is the only possible equilibrium allocation, but possibly with some price equal to zero. fabio petri VDE6 existence 12/02/2016 p. 49 goods Edgeworth-box case: if the endowment point is an equilibrium and if indifference curves are strictly convex, the choice curves of the two consumers are tangent at the endowment point and separated by the equilibrium budget line, so they never cross (draw the diagram and check it!). Note that only the equilibrium allocation is guaranteed to be unique, if the indifference curves of both consumers have kinks at the endowment point the equilibrium relative price is not unique. The index theorem 6.22. For regular exchange economies the results based on GS and WA[38] can be all derived (cf. Hildenbrand and Kirman, pp. 222-228) from the index theorem, a theorem based on differential topology, which we state without proof: For a regular exchange economy, let p* be an equilibrium price vector and assign to p* an index as follows: consider the determinant of –∂z(–h)(p), the Jacobian matrix of z(p) with inverted signs and deprived of any one row and of the corresponding column, and assign to p* index +1 if this determinant is positive, –1 if it is negative; assume also that zi(p)>0 if pi=0 (this implies p*>>0); then the sum of the indices of the finite number of equilibria is +1. The implication of this theorem for uniqueness analysis is that if it can be shown that all possible equilibria have index +1, the equilibrium is unique. (For a proof and considerations on the usefulness of the index theorem cf. the chapter by T. Kehoe in Kirman ed., Elements of General Equilibrium Analysis.) The index theorem does not greatly add to our knowledge of the conditions guaranteeing uniqueness, for two reasons. First, it is only applicable if the economy is regular, and we have argued that this is a very restrictive assumption. Second, it states a mathematical condition which remains economically obscure until conditions implying it are found: GS and WA are the known conditions that do, and they had all been found before the formulation of the index theorem. However, the index theorem can be useful in order to produce examples of non-uniqueness, because if one proves that under the assumptions of the theorem there is an equilibrium with index –1, then there are at least two more equilibria. 6.23. How restrictive are these conditions guaranteeing uniqueness? The conditions of Propositions 5 and 6 are obviously extremely restrictive. As to GS, it excludes the possibility that the demand for a good increases when its price increases: this is much more restrictive than the usual, and probably acceptable, assumption that goods are not Giffen goods, because in the 38 And Diagonal Dominance and Negative Definiteness too. fabio petri VDE6 existence 12/02/2016 p. 50 exchange economy the consumer’s income derives from her endowments and then, when the consumer is a net supplier of a good, the likelihood that her demand for the good increases with its price is quite high. WA is the sole one, of the conditions ensuring uniqueness, that allows the excess demand for a good to increase when its price increases[39]; but we have seen that WA too can easily be violated, owing to different preferences even when consumers have identical endowments, or owing to different endowments even when consumers have identical (but not homothetic) preferences. We must conclude that non-uniqueness of equilibrium does not require implausible conditions. Conditions on the distribution of characteristics. 6.24. In more recent times some work has gone into trying to find assumptions, on the distribution of characteristics among agents, which might ensure the uniqueness of equilibrium. Some cases have been found in which a particular distribution of endowments and preferences ensures uniqueness. The literature is mathematically very advanced so I only summarize some results. Hildenbrand (1983) opened the way, by showing that one can obtain WA for an exchange economy with a continuum of infinitesimal consumers by assuming that 1) consumers have colinear endowments (then the relative value of their endowments is independent of prices, and choosing the endowment basket as numéraire one can consider demands to depend on prices p and income m rather than on prices and endowments), 2) consumers have the same demand function x(p,m), and 3) having ordered consumers according to their income, if we divide them in income classes then equal increases in income embrace decreasing numbers of consumers, i.e. the consumers with income from, say, 0 to 10 are more numerous than the consumers with income from 10 to 20, who are more numerous than those with income from 20 to 30 and so on: Hildenbrand assumes a continuum of infinitesimal consumers, so this is an assumption of decreasing density of the distribution of individual total expenditure, i.e. income. All three assumptions are very restrictive: 1) that endowments should have price-independent relative values is a very unrealistic assumption, think e.g. of the effect of changes of wages; 2) consumers will have the same demand function f(p,m) only if they have the same preferences; 3) the third assumption is generally empirically false for low income classes. After Hildenbrand there have been other papers attempting to find other, possibly less 39 Thus WA does not imply GS. The converse is also true, GS may hold when WA does not, cf. MasColell et al. (1995, Example 17.F.1, p. 611, and pp. 613-14). fabio petri VDE6 existence 12/02/2016 p. 51 restrictive, assumptions ensuring uniqueness of equilibrium (e.g. Grandmont, 1992; Quah, 1997, 2000; Jerison, 1999); but no significant relaxation of the assumptions has been achieved; no reason exists to expect the assumptions to hold in the majority, let alone the quasi-totality, of cases[40]. 6.25. The situation appears therefore to be, that not only the theory is unable to yield a high likelihood of uniqueness, but it is also unable to yield a low likelihood that equilibria may be close to each other and therefore that the economy may easily jump from one equilibrium to another for accidental and transitory reasons, and that a small change in one of the equilibrium's data may result in considerable indeterminateness of which new equilibrium the economy will settle in. It is universally admitted that this situation creates problems to the method of comparative statics. But the problem is deeper: these results if confirmed for more realistic models (i.e. including production and capital goods) raise doubts about the entire supply-and-demand approach. The real question is indeed not that we can be certain that the economy goes to some equilibrium(41) and that the multiplicity of equilibria only renders somewhat indeterminate where the economy ends up; the question is, whether the theory has correctly grasped the forces determining prices and quantities, since reality does not seem to exhibit the indeterminacy, or sudden jump to new persistent positions owing to accidental transitory disturbances, that multiplicity of equilibria should cause us to observe[42]. STABILITY 6.26. A caveat: stability is difficult to study because it is impossible to reach a universally valid model of detailed adjustments, given the wide differences in arrangements in different markets, and the possible different behaviours of agents, who may decide to wait, or bargain, or 40 Cf. de Villemeur (1998, 1999) for such an assessment. For example, Jerison (1999, p. 16) can only claim that his result holds “in a broad class of economic models” (‘a broad class’ is far from meaning ‘a majority’). Another example is Quah (1997) who assumes, among other things, a) that consumers can be divided into classes, each class with such a dispersion of preferences that “each class will behave approximately like an agent with a homothetic preference” (p. 1423), b) that endowments are so randomly distributed that it is as if each one of these representative consumers had the economy’s average endowment. In other words the economy must behave very much as if there were a finite number of consumers each one with homothetic preferences and all with the same endowment. In this case a representative consumer exists by Eisenberg’s theorem. But there is no reason to consider these assumptions as even only vaguely plausible. 41 . In fact as we will see as we come to stability, even this is not guaranteed in general. 42 . “... our experience with real economies gives little support to the notion that there are multiple equilibria” (Gintis, 2007, p. 1299). fabio petri VDE6 existence p. 52 12/02/2016 give in immediately, etcetera. Also, disequilibria in one market can have repercussions in other markets; e.g. suppose that in order to buy, one must have money and that an agent arrives at the market with corn to sell and the intention to buy cloth, but with no money, intending first to sell the corn and thus to get the money to buy the cloth; suppose the price of corn is too high, not all the supply of corn is sold, and our agent's supply of corn remains unsold; then she cannot buy the cloth, and the excess supply on the corn market may cause excess supply on the cloth market too. Further complications are due to the difference that the precise specification of adjustments can make, e.g. the importance of lags. A standard example is the following, based on partialequilibrium analysis of a single market. First-year textbooks indicate that if the demand curve in a market is downward-sloping and the supply curve is upward-sloping, then the tendency of price to increase if demand exceeds supply and to decrease if supply exceeds demand will cause a tendency toward the equilibrium price: the equilibrium is stable. But suppose there is some lag in the adjustment. For example suppose that supply adjusts with a lag, because we are speaking of an agricultural product, whose production requires a year or more to be altered. We can then have the so-called hog cycle or cobweb. Suppose a periodic market, say once a year, with D(pt)=S(pt-1); each time the market opens up supply is given, determined by production decisions taken the year before, based on expectations: price is expected to be at t the same as at t–1; and suppose a price adjustment sufficiently rapid in each market as to reach the equilibrium price on the basis of the given demand curve and of the given supply, with negligible ‘false price’ transactions. p p S(pt-1) C pt D B A D(pt) St+1 St q (a) q (b) Fig. 6.14. A stable cobweb, and a cobweb tending to a limit cycle. Then Fig. 6.14(a) tells the story: Suppose at t supply is St= S(pt-1). Then pt, determined by D(pt)=S(pt-1), determines St+1=S(pt), so pt+1 is determined by D(pt+1)=St+1=S(pt) and so on; joining fabio petri VDE6 existence 12/02/2016 p. 53 the corresponding points on the supply and demand curves, one obtains a cobweb-like spiral which is convergent if the demand curve is less steep (in absolute terms) than the supply curve, and divergent in the opposite case: the reader is invited to draw the latter case and to verify the instability. In the latter case, the equilibrium is unstable in spite of the ‘well-behaved’ slopes of the demand curve and of the supply curve. If the curves are non-linear one may even obtain a limit cycle, as shown by Fig. 6.14(b). (The name ‘hog cycle’ derives from the observation of cycles in the price of hogs, that were attributed to peasants slaying young hogs if the price of hogs was too low, and letting them reach maturity in the opposite case, thus generating a cobweb cycle.) However, this type of instability does not appear to be a reason to expect more than some temporary instability which will then disappear, because in all likelhood learning processes will modify expectations in a direction favouring stability: suppliers, having observed that the price oscillates, will no longer expect pt+1=pt but rather pt+1>pt if they observe that pt<pt-1, and will expect pt+1<pt if they observe pt>pt-1, and will become more cautious in altering their supply, which will greatly decrease the likelihood of instability; for example, suppose that in the situation of Fig. 6.14.(a) producers from time t onwards decide no longer to plan St+1=S(pt) but instead St+1=St+½[S(pt)-St]; in words, they only change their supply by one half of the change suggested by static price expectations; then in all likelihood an unstable cobweb becomes stable (the reader is invited to check this graphically). This shows the possible diversity of disequilibrium behaviours depending on how people react. 6.27. The more modern developments have started with, and have been dominated by, Paul Samuelson’s proposal to formalize the Walrasian tâtonnement as a dynamical process where dp i/dt = aizi(p(t)), with ai a positive scalar and zi(p) the ‘Walrasian’ excess demand for good i. This may be a plausible adjustment rule for the elementary textbook analysis of partial equilibrium of a single market where the demand function and the supply function are given. But for the analysis of general disequilibrium it is highly criticisable because it assumes that demands derive from incomes equal to the value of endowments, while when there is disequilibrium generally a consumer’s purchasing power will not equal the value of her endowments unless the consumer finds purchasers for all her net supplies, which can be true for all consumers only in equilibrium. Very often one must first sell in order to be able to buy, so inability to sell as much as one hoped on a market will imply inability to buy as much as one wanted on other markets. So the excess demands actually manifesting itself on the several markets when prices are p cannot be generally assumed to be indicated by z(p). For example, suppose that one of the goods exchanged in an exchange economy is labour services and that many consumers count on selling them in order to fabio petri VDE6 existence 12/02/2016 p. 54 be able to buy food: if there is excess supply of labour services it would be absurd to assume that the demand for food will be the one corresponding to the ex-ante intentions of consumers based on finding purchasers for all the net supply of labour services; and yet this is the assumption implicitly made by Samuelson’s formalization. This difficulty has been dealt with, not by changing the formalization, but by imagining that adjustments proceed in a very unrealistic institutional structure (first proposed in the later editions of Walras's Eléments d'Economie Politique Pure) capable of justifying the formalization – a rather unconvincing procedure, one must say. It is imagined that economic activity is suspended, all agents meet as if in a big stadium (or connect through Internet), and there is a central institution, called the auctioneer, which proposes prices for all goods; at these prices the agents write down promises of excess demands which become binding only if they generate an equilibrium, and pass these bons (translatable as ‘pledges’[43]) to the auctioneer which sums up the excess demands for each good and then, if a general equilibrium has not been achieved, declares the promises not valid and proceeds to announce new prices according to Samuelson’s rule. (The continuous adjustment implied by Samuelson's formalization entails a further idealization, prices cannot be changed by infinitesimal amounts in the real world, so difference rather than differential equations would appear more correct, but this can be considered a secondary issue.) This process is repeated until an equilibrium is reached. Only then do promises become binding and exchanges are allowed and, in an economy with production, production processes are started[44]. This adjustment process is nowadays referred to as a Walrasian tâtonnement(45). Since in such a scenario the promises to sell or to purchase only become binding in equilibrium, agents have the right to count on an income equal to the value of their endowments. There remains a problem. If there is stability but the adjustment process only asymptotically converges to an equilibrium (the normal case with a Samuelsonian tâtonnement) then the equilibrium is never actually reached. Taken literally, the story implies that exchanges (and production) never start! The only plausible way to interpret the tâtonnement then seems to be that it tries to mimick a process where economic activity – exchanges and production – goes on during the adjustments, and simply (if there is stability) becomes closer and closer to equilibrium activity as This translation, proposed by Walker, appears superior to Jaffe’s ‘tickets’. so some assumption is indispensable that the tâtonnement stops and economic activity is authorized when equilibrium is sufficiently approached; but when is disequilibrium sufficiently small? and what are the effects of the residual disequilibrium? These problems strongly suggest the different interpretation of the tâtonnement as trying to mimick a repetition of actual markets, to be shortly suggested in the text. 45 . French word for ‘search by groping as if blindfolded’. Walras’s description of the tâtonnement process is different from Samuelson’s, but we cannot discuss this issue here. 43 44, fabio petri VDE6 existence 12/02/2016 p. 55 time goes on. For an exchange economy the picture might be something like the following. Imagine an exchange economy, an isolated valley where families have small farms producing different goods and once a month meet in a one-day-long market fair in the central village to exchange their perishable net supplies of consumption goods. Suppose a unique equilibrium and that the tâtonnement would converge to it if it were possible to carry it out according to the auctioneer fairy tale. Unless the economy is stationary and had already reached the equilibrium in past fairs so that the prices that will rule at the fair can be confidently and correctly predicted, it is utterly unrealistic to assume that in a market fair the equilibrium prices corresponding to the endowments of the several families will be immediately hit upon: families will easily mispredict the average prices that will rule at the fair, and regret afterwards their decisions as to how much of their endowments to take to the fair as net supplies to be offered for sale. At the fair itself there will be tentative settings of prices, exchanges soon afterwards regretted, people unable to buy because they were unable to sell, prices changing during the market day, and so on. The transactions and the price changes will alter the value of the changing basket of goods at the disposal of each participant in the fair, so that even assuming that each participant had come to the fair with its entire endowment, after some exchanges its endowment would be different and of different value, so that convergence to the equilibrium corresponding to the initial endowments becomes impossible. But then the determination, via the tâtonnement, of that equilibrium is of little use: it does not yield a correct description of the outcome of that fair[46]. But suppose that the data (agents, endowments and tastes) remain unaltered for a number of successive fairs, and concentrate on what will plausibly happen over a repetition of market fairs to the opening prices and to the flows of sales for the first hours of each fair; to each one of these fairs the same participants arrive starting with the same endowments and tastes so their excess demand functions are the same, and their experience of previous fairs suggests them what opening prices to expect and, accordingly, what amounts of goods to take to the fair and what amounts of goods they can plan to buy: if for example in one fair some sellers of a certain good are unable to sell and by the end of the day have made losses (by accepting a very low price toward the end of the day to get rid of their perishable supplies), in the next fair many sellers of that good can be expected to start with a lower price than in the previous fair. What happens during each fair cannot be detailed, it depends on too many accidents; but one can presume that if the initial average price of a good is too 46 The problem can only be made worse by the existence of production and of savings and investment, with a consequent possibility of significant underutilization of resources due to aggregate demand problems. fabio petri VDE6 existence 12/02/2016 p. 56 high then, during the fair, sellers realize that they are selling less than they hoped, and in the next fair they start with a lower price on average. The adjustment of opening prices and of net supplies and of intended net demands can then be considered to depend on the 'excess demands' (concretely, on whether sales were more or less than expected in the first hours of the fair) experienced at the previous fairs; since at each fair the participants' endowments are the same by assumption, and presumably temporary credits based on the value of past realized sales will allow surmounting part of the problem that many participants may need to sell first their wares in order to become in turn demanders, one can perhaps consider the process of adjustment of prices and quantities over the repetition of fairs to be reasonably approximated by the tâtonnement formalization. The fact, that during each fair disequilibrium transactions and price changes alter the value of the endowments of participants does not make convergence to the equilibrium based on the initial data impossible, because these data remain the same over the repetition of fairs, so that by learning and trial and error prices can tend to those that ensure that equilibrium. The tâtonnement appears therefore perhaps not so absurd an idealization of the adjustment processes contemplated in the marginal approach, if it is taken to describe the tendency of prices over a repetition of market interactions during which the equilibrium's data do not change[47] – and then the equilibrium aims at describing a persistent position and is therefore the exchange-economy equivalent of a long-period equilibrium. The question then arises, why isn't the tâtonnement justified in these terms, rather than in terms of the obviously absurd fairy tale of an imaginary 'auctioneer' stopping everything and collecting bons in a congealed-economy situation? The very important answer is that, once production is introduced, repetition of market interactions must admit not only disequilibrium exchanges but also disequilibrium production decisions, but these will not change the equilibrium's data only if these data do not include given endowments of the several capital goods. Most capital goods are rapidly consumed by use – circulating capital goods, for example, disappear in a single utilization – and are rapidly produced, so their endowments can be very quickly changed by production. As will be explained in chs. 7 to 9, modern general equilibrium theory follows Walras precisely in including among the data of equilibrium given endowments of the several capital goods. Then admitting repetition of market interactions and disequilibrium productions becomes incompatible with studying the stability of the equilibrium; the question of convergence to the equilibrium corresponding to given data becomes nonsensical if the 47 Or change slowly enough relative to the presumable speed of convergence toward equilibrium, as to justify treating them as unchanging. For example, in a general equilibrium with production, and labour as one of the factors of production, generally labour supply will not be strictly constant because population will be growing, but its rate of change is so slow that one can assume labour supply to be given. fabio petri VDE6 existence 12/02/2016 p. 57 adjustment processes themselves alter the equilibrium by altering those data. The only way to study the stability of this type of equilibrium is by assuming adjustment processes that do not involve actual productions. The reason, why Walras in the 4th edition of his Eléments introduced this very unrealistic assumption of a tâtonnement proceeding on the basis of provisional pledges while exchanges and production are suspended, was precisely that he finally realized that he could not allow disequilibrium productions because these would alter the economy’s endowments of capital goods, endowments which he was including among the data determining the equilibrium. However, the usefulness may be doubted of a notion of equilibrium whose stability can only be studied on the basis of adjustment processes having no correspondence with reality. But these issues are complex and we cannot anticipate. For the moment, we must be content with understanding the reason why at least for the general equilibrium of exchange, and for the one of production and exchange but without given endowments of the several capital goods among its data, the tâtonnement has appeared to most neoclassical economists a defensible way to study the stability of equilibrium and accordingly has been extensively investigated, and we proceed to summarize the most important results, for the moment for the case of exchange. We need a few mathematical notions. 6.28. Consider a subset S of Rn. A dynamical system F, x(t) on S is a way of associating a vector xS to a scalar t, usually interpreted as time, through a rule F: S×R→S which starting from x(t) and hR yields x(t+h). If x changes continuously as t increases, given an initial point x(0) the dynamical system can be viewed as describing a trajectory of the point x=x(t,x(0)) in S as time passes. I shall only discuss dynamical systems consisting of systems of differential equations dx 1 t dt d x f xt , t where x x t . . The function f is known, but the functions xi(t) are . dt dx t n dt not; so what must be found are functional forms of x1(t), ... , xn(t) that satisfy the system of equations. Such functional forms are called a solution of the dynamical system. Given an initial position x(0), the solution(48) determines all subsequent values of the state variables x1,..., xn as t grows. I shall only use qualitative considerations on the character of the solutions, so no familiarity 48 Under certain conditions generally satisfied in economic applications, the solution is unique apart from a dependence on initial conditions. fabio petri VDE6 existence 12/02/2016 p. 58 will be needed with the techniques to find solutions of differential equations. What do we mean by stability of a dynamical system? Many meanings have been distinguished by mathematicians. Here we are concerned with the asymptotic stability of equilibria. An equilibrium of a dynamical system of the type we are concerned with is a point x* such that d x t 0 for all t, and therefore if the system is in x* it remains there. f(x*,t) = 0, i.e. such that dt It is also called a fixed point or a zero of the dynamical system. An equilibrium, or fixed point, x* is Liapunov stable if, intuitively speaking, all solutions with initial conditions sufficiently close to x* (i.e. in a neighbourhood of x*) remain forever close to x*. Formally: An equilibrium x* of a dynamical system is said to be Liapunov stable if for any scalar ε>0 and any t° there exists a scalar δ>0 such that any solution x: R→Rn for which ||x(t°)–x*||≤δ, t°≥0, has the property that for all t≥t° it is ||x(t; x(t°), t°))–x*||≤ε, where δ depends on ε and on t°. If δ can be chosen independently of t° then x* is said to be uniformly Liapunov stable. An equilibrium x is locally asymptotically stable if (i) it is Liapunov stable and (ii) lim xt x for all x(0) in a neighborhood of x* (49); it is uniformly asymptotically stable if in (i) t Liapunov stability is replaced with (i') uniform Liapunov stability; it is globally asymptotically stable if (i) holds and (ii) is replaced by lim x(t ) x*, x(0) S , where S is the set of possible t states x of the dynamical system, called the state space; it is uniformly globally asymptotically stable if (i) is replaced with (i'). In the remainder of this chapter by stability I shall mean asymptotic uniform stability unless otherwise expressly indicated. With this meaning, a globally stable equilibrium x* is a point to which all trajectories converge, independently of where they start from. An equilibrium x* that is globally stable is unique: any other equilibrium would be a point from which the dynamical system would not tend to x*, contradicting the global stability of x*. Below I use the so-called Liapunov's second method or direct method for the study of the stability of an equilibrium x* of a dynamical system. This method consists of looking for a scalar function V(x(t)) of the state vector x(t), which a) is indirectly a continuously differentiable function of t, for any x; b) always decreases for x≠x* as t increases, i.e. its total derivative with respect to t is always negative for x≠x*; c) reaches a minimum in x*. 49 It is possible to produce examples in which (ii) is satisfied but not (i); see Takayama (1974, p. 349). fabio petri VDE6 existence p. 59 12/02/2016 If such a function can be found, then the dynamical system is globally stable because any trajectory reaches lower and lower values of V as t increases, so if V has a lowest value the dynamical system tends to it, and the lowest value of V is reached at point x*. More formally, suppose we are given a dynamical system(50) F,x(t), xS with S compact (so the values that x can take are bounded), and suppose we know that x* is an equilibrium; suppose we can find a scalar function of x, V: SR , i.e. a function v(t)=V(x1(t),...,xn(t)), with continuous partial derivatives ∂V/∂xi, and with the following properties: a) it is positive except at most in x* and it reaches a minimum in x* and only there (without loss of generality this minimum can be assumed to be zero); V dxi 0, x x * , while at x* it is =0 : i.e. as t increases, whichever the b) V ( x(t )) xi dt x(0)x* in S from which the system starts V(x(t)) always continuously decreases, except at x*. Such a function V(x(t)) is called a Liapunov function[51] and it can be proved that, if such a function exists, then x* is globally stable.(52) An intuitive sketch of the proof is as follows(53). Given that V has a unique minimum in x*, the positiveness and continuous partial differentiability of V imply the existence of a continuum of closed level curves (or more generally hypersurfaces) of V(x) in S around the equilibrium point, corresponding to lower and lower values of V as one approaches x*. For any x(0) the function V(x(t,x(0)) viewed as a function of t is decreasing and bounded below by zero so it must tend to a limit as t→∞. It follows that lim V ( x(t , x(0)) 0 but t since V is negative for x≠x*, this can only occur if x(t,x(0))→x* as t→∞. 6.29. With these minimal mathematics we can prove: Under WA, if the equilibrium of the exchange economy is unique, then it is globally stable with respect to a tâtonnement adjustment process. Proof. Let p° and p* be any two different and non-proportional(54) price vectors (not necessarily normalized in the price simplex). Assume p* to be an equilibrium price vector; then by the assumption of equilibrium uniqueness p° is not an equilibrium price vector; then as proved earlier, WA implies p*z(p°)>0: the value at equilibrium prices of any non-equilibrium excess 50 The dynamical system of differential equations given above in the text is one such system. Liapunov was a Russian mathematician. His name is also written Lyapounov. 52 If in condition b the inequality <0 is replaced with the weak inequality ≤0, then x* is Liapunov stable but not necessarily asymptotically stable. 53 Cf. D. K. Arrowsmith and C. M. Place, Ordinary Differential Equations, London: Chapman and Hall, p. 200. Also cf. Gandolfo 1971, pp. 369-75. 54 That is, it is not the case that p°=cp* for c a positive real scalar. 51 fabio petri VDE6 existence p. 60 12/02/2016 demand vector is positive. Assume now that the price adjustment rule is the following (as usual, a dot over the symbol of a variable means the derivative of that variable with respect to time): (**) p i ( t ) ai zi p ( t ) (with ai>0 scalar); it can be shown that it makes no difference to the qualitative behaviour of the dynamical system if we assume simply all coefficients ai to be equal to 1: (***) p( t ) z p( t ) . These price adjustment rules imply the following. If an initial price p(0) has been chosen and d n 1 p(t) is given by (**), then from ( (p i ( t )) 2 ) = ∑i[(2/ai)pi(t)·dpi(t)/dt] = 2∑i[pi(t)zi(p(t))] = 0 dt i1 a i , where the last equality follows from Walras’ law, we derive that the quantity n 1 a i 1 (p i (t )) 2 i remains constant as t increases(55). Therefore if we want to enquire whether an equilibrium is stable, we must choose a normalization for the equilibrium price vector p* once p(0) is chosen, or n 1 for p(0) once p* is chosen, which respects this constancy, i.e. such that (p i (0)) 2 = i 1 a i n 1 (p i *) 2 . Since all that matters for excess demands is relative prices, this normalization i 1 a i condition is no restriction from the economic point of view. If with this normalization we can prove that p* is globally stable, the implication will be that a non-normalized p(t) converges on the equilibrium ray {p=αp*}, with α any positive scalar. The adoption of this normalization has on the other hand an advantage: for t≥0, it cannot be simultaneously p(t)≠p* and p(t) equi-proportional to p*, so we are certain that for p(t)≠p* it is p*z(p(t))>0. Let us then adopt the price adjustment rule (**) with the above normalization of p(0). I show that the function Vp p i p i / a i n i 1 2 is a Liapunov function for our dynamical system. a) V(p) > 0 for p p*, and V(p) = 0 only if p = p*, hence V(p) reaches its minimum in p* and only there. n n dV n 2 p i p i p i 2 p i p i z i p 2 p i z i p p i z i p dt i1 a i i 1 i 1 b) 2 ( p zp p zp ) 2p zp 0 0 byWalras'law where the last inequality follows from p*z(p)>0. So the system has a Liapunov function and therefore p* is globally stable.(56) █ 55 If all coefficients ai are chosen equal to 1, then p(t) moves on the positive portion of a spherical hypersurface. 56 . It is immediate that the proof still holds if ai=1, i. So assuming (***) in place of (**) makes no difference. If the assumption, additional to WA, that equilibrium is unique (an assumption often omitted in other treatments, cf. e.g. Varian, 1992, p. 399) is dropped, and the possibility of a continuum of equilibrium prices is admitted, then V can be still defined in the same way and it remains true that it reaches a minimum fabio petri VDE6 existence 12/02/2016 p. 61 As we know, the only relevant case where we can be assured that WA holds is when there exists a representative consumer with strictly convex preferences. Another case of global tâtonnement stability for exchange equilibria is GS. One can adopt for the proof in this case exactly the same Liapunov function, because it can be proved that GS too implies that if p* is an equilibrium price vector then p*z(p)>0 for all p≠p* and non-proportional to p*; therefore the proof that Vp p i p i / a i is a Liapunov function is the same. The proof n i 1 2 that GS implies p*z(p)>0 is omitted (cf. e.g. B. Beavis and I. G. Dobbs, Optimization and Stability Theory for Economic Analysis, Cambridge U. P., 1990, pp. 195-6).[57] Unstable equilibrium and limit cycle in the 3-goods price simplex (the triangle with vertexes (1,0,0), (0,1,0), (0,0,1) in R3). ▪p * Fig. 6.15 6.30.1. Of course an exchange economy can have a unique equilibrium without satisfying any of the conditions sufficient for uniqueness that also imply tâtonnement stability. In this case tâtonnement stability is not guaranteed: numerical examples have been produced since 1960 by Herbert Scarf proving that, as long as there are at least 3 goods (hence a state space of relative prices of dimension 2) and three consumers, there are cases (not of measure zero, i.e. robust to small perturbations of the equilibrium’s data) where the equilibrium is unique but unstable, and the at p* and only there, but in step b) it is no longer guaranteed that p*z(p)>0 because p might be another equilibrium price vector, in which case p*z(p)=0 and dV/dt=0; however, it remains true that V(p(t)) will go on decreasing as long as no equilibrium price vector is hit; therefore as t increases it remains true that p(t) reaches lower and lower level curves of V(p), so if it does not hit another equilibrium price it tends to p*. Since any other equilibrium price vector might take the place of p*, what is proved is that p(t) reaches lower and lower level curves of V(p) whichever the equilibrium price vector used to define V(p), that is, it asymptotically approaches the set of equilibrium price vectors. 57 I mention without proof that the other two conditions known to ensure uniqueness also ensure global tâtonnement stability: for Diagonal Dominance the interested reader may consult e.g. Arrow and Hahn (1971, p. 295), for Negative Definiteness cf. Hildenbrand and Kirman (1988, p. 237 and p. 220) or Gandolfo (1971, pp. 377-80). For the possibility that stability may depend on the choice of numéraire, and how to surmount this economically unacceptable result, cf. Parrinello ??. fabio petri VDE6 existence 12/02/2016 p. 62 tâtonnement tends to a limit cycle: the time path of prices asymptotically approaches a closed orbit (cf. Fig. 6.15), the auctioneer would never be able to stop! With four or more goods, and hence a state space of dimension 3 or greater, a dynamical system in continuous time can produce a path that wanders irregularly in the state space without converging to any point or limit cycle: the dynamical system is then called chaotic; examples have been produced of chaotic tâtonnements ??REFS. 6.30.2. These possibilities do not seem to be strictly due to the specific formalization of the 'Walrasian' (actually Samuelsonian) tâtonnement. Some recent studies report experiments with real participants mimicking a multi-good exchange economy that tries to reach an equilibrium without the auctioneer, and convergence is not always obtained: sometimes there is instability, sometimes cycling without apparent equilibration (Anderson et al. 2004; Noussair et al. 2004, p. 50 fn. 1). In these experiments there is a repetition of trading periods, in each one of which the participants propose exchanges and prices to other participants via a ‘double auction’ mechanism[58], with exchanges taking place continuously, hence generally at non-equilibrium prices. Because of this, the endowments of consumers change during trading in ways incompatible with the reaching of the equilibrium based on the original endowments; whether there is convergence toward the general exchange equilibrium[59] associated with the original endowments is studied by looking at what happens over a repetition of trading periods with a re-setting of endowments at the initial values before each trading period, exactly the picture suggested above in §6.27 of what the tâtonnement should be seen as trying to formalize for the exchange economy[60]. This experimental evidence 58 “Trades in a double auction, in contrast, take place at discrete instances, and often at disequilibrium prices. ‘Double auction’ refers to the fact that price changes can come from both buyers and sellers of each commodity. At any time, an agent may tender a bid or an ask, accept another agent’s bid or ask, or have a bid or ask accepted. Newly tendered bids and asks must improve (be higher or lower than, respectively) upon those in the market and cancel any previous bids or asks. The price discovery process can be observed as the prices at which trades are executed fluctuate through time. Allocations change with each transaction and can change no more only when the market closes due to expiration of trading time. The markets studied here have a number of such trading periods: after time expires and payoffs are calculated based on final allocations, endowments are reset to their initial values and another trading period begins. Within a period, trades depend on the patterns of bids, asks and acceptances. Therefore, there are within-period shifts in holdings that could cause deviations from the price adjustment paths predicted by tâtonnement models.” Anderson et al. 2004 p. 213 59 In order to examine one problem at the time, the economy is constructed so that equilibrium is unique. 60 An older literature on non-tâtonnement models (the main names are Hahn and Negishi) studied what can be expected to happen in a single trading period if transactions go on as long as there is room for profitable exchanges; not surprisingly, the result is that – if no time constraint exists – some situation will fabio petri VDE6 existence 12/02/2016 p. 63 confirms that the stability of exchange equilibria is far from being guaranteed. A recent study by Gintis (2007), which also considers production, claims that the introduction of (i) 'private prices'[61], (ii) a need to choose at each instant whether to exchange, or to wait so as to collect more information about prices proposed elsewhere, and (iii) imitation of the more successful agents, powerfully makes for stability, for example produces stability in the Scarf exchange example where the tâtonnement converges to a limit cycle. Gintis's results are derived from simulations of agent-based models, i.e. of complex dynamical models with very many equations (the behaviour of each agent must be formalized); with these studies one has verification problems worse than those arising with econometric studies, because the problem with the latter is the truthfulness of the calculations reported, but econometric papers indicate the equations and the methods adopted, so in theory it would be enough to have the data set and any other scholar should be able to replicate the calculations; the problem is fundamentally one of discovering cheating. With agent-based models the problem arises with the model itself, only available as a computer program normally not made available, which means that the reader of the paper reporting the agentbased study does not know how precisely the several assumptions were formalized, and even if she were given the program she would need a great expertise at programming (and a long time) to understand the formalization choices, and therefore cannot generally evaluate the significance of the simulations; the problem is not one of a danger of dishonesty but instead that it is very difficult to have an in-depth understanding of all the choices made in the study, or to discover mistakes made in the translation of the assumptions into a program – and even the best economists can make debatable choices, or mistakes. Therefore in spite of having the greatest respect for professor Gintis's correctness and competence I think that his results are in need of further verification; anyway I will argue later in this book that, contrary to what Gintis appears to believe, his claims, even if accepted for the cases (the only ones he examines) of pure exchange and of production without capital, would not apply to the really relevant case, production with capital goods. 6.30.3. When equilibrium is not unique, if the several equilibria are locally isolated there may be more than one locally stable equilibrium[62]. In this case, the most one can hope for is what is sooner or later be reached where no room for profitable exchanges remains, at least as long as agents are sufficiently informed on exchange possibilities. The analysis is anyway strictly limited to exchange economies. 61 That is, prices proposed by the agents in each transaction, with only imperfect information of prices proposed or accepted in other transactions. 62 . I skip the discussion of conditions guaranteeing the local stability of equilibria because they seem to have little interest. Gintis (2007) avoids the problem by making assumptions that ensure uniqueness of fabio petri VDE6 existence 12/02/2016 p. 64 called global quasi-stability or (Mas-Colell et al., p. 623) system stability. An economy is globally quasi-stable or system-stable if for any starting point the dynamical adjustment takes the economy to some equilibrium (generally a different one depending on initial conditions). If this condition holds and if furthermore the equilibria are few and far apart, then one can hope that transitory shocks and disequilibrium accidents do not push the economy into the ‘basin of attraction’ of another equilibrium, and also that small changes in the data (e.g. due to taxation, immigration, innovations) cause only a small shift of the equilibrium the economy was at, so the economy remains in its ‘basin of attraction’, and comparative statics remains possible. Unfortunately when the conditions guaranteeing uniqueness are not satisfied there seem to be no plausible general conditions ensuring that the equilibria be only few and far apart. Also, tâtonnement global quasistability is not guaranteed: Scarf’s example suffices to show it[63]. 6.30.4. A further problem mentioned earlier, and definitely possible although generally neglected in the literature, is that of practically indeterminate equilibria: the ones where in an ample neighbourhood of the equilibrium price vector the excess demands remain very close to zero and therefore even if the equilibrium is locally stable the forces tending to push toward it are very weak: then the equilibrium cannot be considered a good indication of the average of market realizations, because the tendency toward equilibrium cannot be presumed sufficiently to correct the deviations due to temporary accidents, nor to operate faster than the slow changes of the equilibrium itself over time due e.g. to population growth or taste changes. One has here another cause of indeterminateness of the theory’s predictions. 6.31. In conclusion, the existence of general equilibria of pure exchange does not appear to encounter serious difficulties, except for the survival problem: the latter is a serious difficulty, that points to an inability of the theory to deal with the generality of possible situations, and prompts equilibrium. 63 A superpowerful central computer with complete information about the partial and cross derivatives of z(p) would be able to find a path of prices converging to equilibrium even when other adjustments were unstable, through the use of global Newton methods: these, by taking into account the effect of the change in each price on all excess demands, are able to calculate the direction of change of relative prices required to bring all excess demands closer to zero simultaneously (cf. Mas-Colell et al. 1995 p. 624, Smale 1976, Saari and Simon 1978); but the amount of information needed by these methods is enormous, all quantitative direct and cross price effects must be known; even a central planner would be unable to collect this information; in a market economy there is no incentive for prices to behave in the way required for convergence. fabio petri VDE6 existence 12/02/2016 p. 65 one to consider with interest the very different approach to wage determination of the classical economists. The uniqueness of equilibrium, which traditionally was considered essential to the solidity of the approach, is guaranteed only under very restrictive hypotheses. As to stability, the results on tâtonnement stability are just as negative (tâtonnement stability is not guaranteed even when equilibrium is unique; tâtonnement quasi-stability too may not obtain); the problem of practically indeterminate equilibria is neglected but should not be; the attempts to study nontâtonnement adjustments that explicitly allow for time-consuming adjustments involving disequilibrium exchanges (and hence – in order to make it possible to discuss the stability of the equilibrium corresponding to the initial endowments – must consider a repetition of markets with unchanged data) do not always obtain convergence even when equilibrium is unique (and would become incompatible with repetition of markets with unchanged data if applied to economies with production and capital goods). THE LIKELIHOOD OF UNIQUENESS AND STABILITY 6.32. The difficult and delicate question becomes then, whether the lack of uniqueness and of stability undermines the plausibility of the entire supply-and-demand approach. Although the discussion can only refer to the exchange economy at this stage, it is not premature because very similar problems arise for the general equilibria of the production economy to be studied in the second part of the present chapter, and of the intertemporal economy with capital goods to be studied in ch. 8. Indeed one view – which I do not share, for reasons that will become clear in subsequent chapters – is that production and capital add very little to what can be understood on the basis of the pure exchange model. The practice, of all those economists in applied fields (e.g. macroeconomics, growth theory, international trade theory, applied welfare economics) who in spite of these results continue to assume a unique and stable neoclassical equilibrium, appears to imply a judgment that although a guarantee of uniqueness and stability is impossible to achieve, nonetheless there is reason to presume that multiple equilibria close to one another, or absence of global quasi-stability, are very unlikely events; and that this is all that the applied theorist needs, in order to have the right to go on assuming uniqueness and stability and hence the legitimacy of comparative statics. However, an explicit statement and defence of this position in recent decades does not seem to exist. The tendency in textbooks and treatises is either (i) not to mention the problem at all(64), or (ii) to 64 . The textbook by Reichlin and Ventura, the one by the one on infinite-horizon economies by Aliprantis, Brown and Burkinshaw, do not discuss uniqueness nor stability. The books which do discuss fabio petri VDE6 existence 12/02/2016 p. 66 express some discomfort with the results which we have summarized, then briefly to hint at the need for some theory of equilibrium selection or for better theories of disequilibrium, and to leave it at that. If one turns to older authors, one does find some discussion of the problem, that concludes that one need not worry; but we are still waiting for good studies critically re-examining these arguments. What seems clear is that until recently non-uniqueness and instability were not a cause of widespread worries. Certainly an important role in this regard must be attributed to Hicks’s position, presented in the very influential Value and Capital (1939).(65) Hicks starts by discussing the exchange of two goods; in that case it suffices to look at the excess supply for only one of the two goods; Hicks distinguishes the net suppliers or sellers from the net demanders or buyers of the chosen good, and then writes: “A fall in price [of the chosen good, starting from the equilibrium price] sets up a substitution effect which increases demand and diminishes supply; this therefore must increase excess demand. It sets up an income effect through the buyers being made better off and the sellers worse off. So long as the commodity is not an inferior good for either side, this means that the income effect will tend to increase demand and increase supply. Thus the direction of the income effect on excess demand depends on which of these two tendencies is the stronger. If the income effect on the demand side is just as strong as the income effect on the supply side, then the income effect on excess demand will cancel out, leaving nothing but the substitution effect. In this case.... equilibrium must be stable.” (V&C 64) Hicks admits that this is not necessarily the case and that “The sort of difficulty which does arise in such cases is that there may be more than one position of stable equilibrium” (p. 65; note the admission that multiple equilibria are a “difficulty”, although it is not explained why). But he has set the premises for an argument like: for normal goods, income effects tend always at least partially to efface one another, because they tend to increase both the net supply and the net demand for the commodity; hence, the residual net income effect can be presumed generally to be fairly weak, and therefore, even when stronger for the net suppliers than for the net demanders, to be generally overpowered by the substitution effect. (As for inferior goods, they never reappear in the these issues usually only contain an exposition of some of the results also reported here, but with little attempt to discuss the implications. A partial exception is Hildenbrand and Kirman, cf. e.g. p. 239. 65 . On this issue I have found it very useful to read S. M. Fratini, Il problema della molteplicità degli equilibri da Walras a Debreu, Ph.D. thesis, Università di Roma La Sapienza, 2001. fabio petri VDE6 existence 12/02/2016 p. 67 remainder of the argument, perhaps because Hicks thought that they were of secondary importance.) In fact when Hicks goes on to discuss the exchange with more than two goods, he argues that “There are just the same reasons ...[as in the two-goods case] for supposing that the income effect on excess demand will often be very small (since it consists of two parts which probably work in opposite directions)” (p. 68), and proceeds with “Thus, if (as an approximation) we neglect the income effect...” (ibid.). On this basis, he concludes that, if the market for a commodity X is stable “taken by itself (that is to say, a fall in the price of X will raise the excess demand for X, all other prices being given)”, then it is very unlikely that it be rendered unstable by reactions through the markets for other commodities. He then adds that if the market is unstable when “taken by itself”, then it is very unlikely that it be rendered stable by indirect reactions; but this case is again considered unlikely on the basis of the weakness of income effects. So he concludes: “To sum up the negative but reassuring conclusions which we have derived from our discussion of stability. There is no doubt that the existence of stable systems of multiple exchange is entirely consistent with the laws of demand. It cannot, indeed, be proved a priori that a system of multiple exchange is necessarily stable[66]. But the conditions of stability are quite easy conditions, so that it is quite reasonable to assume that they will be satisfied in almost any system with which we are likely to be concerned. The only possible ultimate source of instability is strong asymmetry in the income effects. A moderate degree of substitutability among the bulk of commodities will be sufficient to prevent this cause being effective.” (pp. 72-73) Hicks also argues that production systems can be treated along much the same principles, and indeed concludes that “the general equilibrium of production will be stable in most ordinary circumstances” (p. 104); but I postpone a discussion of his arguments on production to after the discussion of the general equilibria of production economies. No doubt the little importance given for decades by the profession to the possibility of multiple and unstable equilibria must have been based on this type of argument; but how solid is it? 6.33. Hicks reaches his “reassuring conclusion” on the basis of a number of half-implicit assumptions which deserve more careful examination. Are inferior goods really so negligible? Can . [Notice how Hicks considers ‘stable’ only a system with a single stable equilibrium. It is very clear that he considered the proof that equilibrium can be reasonably assumed to be unique as essential to the plausibility of the theory. – F. P.] 66 fabio petri VDE6 existence 12/02/2016 p. 68 we really assume that the income effects of suppliers and demanders largely efface each other? Is the substitution effect generally considerable? And finally, even if it could be plausibly argued that uniqueness and stability obtain in “almost all systems with which we are likely to be concerned” or “in most ordinary circumstances”, would that be sufficient for the plausibility of the theory? I attempt now some consideration on the second of these questions: whether income effects can be considered generally incapable of causing multiple equilibria. In §6.17 we saw an Edgeworth box with three equilibria. Another drawing of this kind is e.g. in Mas-Colell et al., p. 521. What is it in these drawings that makes it possible to obtain the “strong asymmetry in the income effects” which results in multiple equilibria? Do these drawings inadvertently exclude even that “moderate degree of substitutability” which according to Hicks would prevent multiple equilibria? The answer to the last question is no; Hicks underestimated the extent to which differences in endowments can result in “strong asymmetry in income effects” in spite of considerable substitutability. This can be shown as follows. Assume a two-good world, take a consumer with her map of indifference curves, and draw her choice curves (offer curves), assuming first that her endowment consists only of one good, then only of the other good. The resulting choice curves can be seen as the choice curves of two consumers with the same preferences but with different endowments. In Fig. 6.15 the reader can check, by drawing a map of indifference curves, that indifference curves admitting considerable substitutability may well be such that curve A is the choice curve when the endowment consists of the quantity ω1 of good 1, and curve B is the choice curve when the endowment consists of quantity ω2 of good 2. If now we form an Edgeworth box with consumer A having endowment (ω1, 0) and consumer B having endowment (0, ω2) and both having the preferences of Fig. 6.15, we obtain Fig. 6.16 where the endowment point is the lower right-hand corner, and the choice curve of consumer B is choice curve B of Fig. 6.15, rotated 180°. The two choice curves cross three times. Thus Hicks was wrong to think that income effects would largely cancel out and would be prevented from causing multiple equilibria by even only “a moderate degree of substitutability”. Identical preferences admitting considerable substitutability can cause “strong asymmetry in income effects” if endowments are different. And in real economies endowments are considerably different. fabio petri VDE6 existence p. 69 12/02/2016 ω2 A B ω1 Fig. 6.15 fabio petri VDE6 existence p. 70 12/02/2016 Ω Fig. 6.16 What is the implication of this refutation of Hicks? Hicks appears to admit that the theory would be in trouble if an argument could not be advanced that, apart from very rare cases, uniqueness of equilibrium obtains. Now, we have seen that his argument to that effect is unconvincing. Then we cannot find it surprising that in recent times, owing to an increasing realization of the difficulties of any such argument, a number of general equilibrium theorists have produced very negative statements on the state of health of general equilibrim theory. For example, Alan Kirman, after writing "The intrinsic limits of modern economic theory. The emperor has no clothes", Economic Journal, 1989, where he stressed the lack of uniqueness of equilibrium as a grave problem, in a subsequent paper (“The future of economic theory”, in A. Kirman and L-A. Gérard-Varet, Economics beyond the Millennium, Oxford U. P. 1999, p. 8) has written: fabio petri VDE6 existence 12/02/2016 p. 71 ...we are at a turning point in economic theory. Much of the elegant theoretical structure that has been constructed over the last one hundred years in economics will be seen over the next decade to have provided a wrong focus and a misleading and ephemeral idea of what constitutes an equilibrium. But not even Kirman has gone into great details on why non-uniqueness is a grave problem for the theory. Let us try to make the issue clearer with an example; in particular, let us try to point out why non-uniqueness should be highly exceptional in order not to question the entire theory. Microeconomic theory shows that a ‘backward-bending’ supply of labour is a possibility, and we know that it can cause multiple and unstable equilibria. Then arguments to the effect that empirically a backward-bending supply of labour is not a very frequent occurrence do not save the theory, because an infrequent occurrence still occurs, albeit infrequently; and since there have been enough different economic situations in two centuries of capitalism in dozens of different economies, even an infrequent occurrence must have occurred several times; so it is practically certain that at least sometimes it should have caused multiple or practically indeterminate equilibria; and therefore the indeterminacy associated with “practically indeterminate” equilibria, or the phenomena of sudden jumps in competitively determined wages because of multiple and unstable equilibria, should have been observed at least a few times historically – but the jumps do not seem to have been observed, and as to the practically indeterminate equilibria, something has anyway determined wages and maintained them considerably steady; the inference is, that we need to admit forces different from the ones postulated by the supply-and-demand approach in order to understand what determines wages at least in some situations: but then is it plausible to restrict the determining role of these forces only to the situations where the supply-and-demand theory clearly fails? isn't it more plausible to look at these forces as the basis for a theory capable of that generality that the supply-and-demand approach is unable to achieve? The possibility of absence even of quasi-stability has added to the unease; on these issues the last chapter of B. Ingrao and G. Israel, The Invisible Hand, MIT Press, 1990, especially pp. 358-62, definitely deserves a reading. So nowadays some GE specialists concur with opponents of the neoclassical approach to value and distribution on the need for a fundamental re-orientation of the analysis(67); unfortunately 67 Already in 1981 Frank Hahn had written (reviewing a book by M. Beenstock, in the Economic Journal, 1981, p. 1036): "I have always regarded Competitive General Equilibrium analysis as akin to the mock-up an aircraft engineer might build....theorists all over the world have become aware that anything based on this mock-up is unlikely to fly, since it neglects some crucial aspects of the world, the recognition of which will force some drastic re-designing. Moreover, at no stage was the mock-up complete; in fabio petri VDE6 existence 12/02/2016 p. 72 not all GE specialists are aware of the already existing alternative research programmes. For example, Ingrao and Israel have written: “There is another alternative: to formulate a completely new research program and conceptual approach. As we have seen, this is often spoken of, but there is still no indication of what it might mean.” (p. 350). Perhaps this is true of the writings of general equilibrium specialists, but otherwise it is quite false, alternatives are not lacking, as chs. 1 and 2 have started to show and the chapters from 10 onwards will confirm. particular, it provided no account of the actual working of the invisible hand". Hahn’s last sentence will become clearer when we come to capital and the neo-Walrasian models. fabio petri VDE6 existence 12/02/2016 p. 73 PART II GENERAL EQUILIBRIUM OF PRODUCTION AND EXCHANGE EXISTENCE ??joint production big problem 6.34. The results on existence valid for the pure exchange economy are also valid for the atemporal general equilibrium of production and exchange, if the derived excess demand function for factors ζ(v) (cf. chapter 5, §5.27) has the same properties as the excess demand function z(p) of the pure exchange economy[68]. One needs that ζ(v) be continuous on the interior of the factor rental simplex, plus some condition avoiding discontinuities on the boundary of that simplex or making them irrelevant by guaranteeing that, in a neighbourhood of the boundary, excess demands are certainly positive so that the equilibrium, if it exists, has positive factor rentals. Since ζ(v)=x(p(v),v)–X(p(v),v), the two terms on the right-hand side must be continuous on the interior of the factor rental simplex; we show now that this is the case with assumptions no more restrictive than for the pure exchange economy, and therefore existence poses no additional problems. X(p(v),v) is a continuous function of p and v under the same hypotheses as for the pure exchange economy; p is a continuous function of v if minimum average costs are continuous functions of v, which is guaranteed by the properties of the average cost function(69). As to x(p(v),v), it is a continuous function of p and v if the quantities produced and hence the demands for 68. Like in chapter 5, in order to make an intuitive grasp of the arguments easier I neglect joint production, except in §6.35 whose theorems are stated and proved so that they are also valid with joint production. With joint production the derived excess demand for factors ζ(v) cannot be obtained as simply as we obtained it in chapter 5, because generally the price of jointly produced commodities cannot be determined independently of demand. More on this will be said in chapter 10. 69. The fully general proof is mathematically complex, because some borderline cases (e.g. Cobb-Douglas production functions when one factor rental is zero) create complications, and we omit it. (Mas-Colell et al. omit it as well: on p. 140 they send the reader back to the study of the consumer’s expenditure function, but on p. 59, Proposition 3.E.2, they state but do not prove that the expenditure function is continuous). However, in order to make the complications disappear it suffices realistically to assume that the subset of Rn+ in which the isoquant hypersurface has negative slopes is bounded, i.e. that the marginal product of any factor becomes zero if the employment of the factor is sufficiently increased; then the economically relevant part of the isoquant is bounded, and one derives the continuity of minimum average cost as a straightforward application of the Theorem of the maximum, which states: Let f(x,), with x a vector and a vector of parameters, be a continuous function with a compact range and suppose that the constraint set G() is a nonempty, compact-valued, continuous correspondence of ; then M(), i.e. max f(x,) such that xG(), is a continuous function and x() is an upper-hemicontinuous correspondence. This theorem yields the result we need with –f(x,) = x∙α, with xG() the inputs and the rental vector, which if constrained to be in the unit simplex renders the range of f(x,) compact because (see below) G(α) is compact; f(x,) is obviously continuous in x and ; G() is the set of input vectors satisfying the constraint that production must equal 1, non-empty by the assumptions of continuity and of CRS of the production function, compact-valued by our assumption on marginal products because then only a compact region of the isoquant is admissible, and a continuous correspondence of because independent of α. fabio petri VDE6 existence 12/02/2016 p. 74 consumption goods are continuous functions of p, which will be the case under the same hypotheses as for the pure exchange economy, and if technical coefficients are continuous functions of v, which is guaranteed by the assumption of strictly convex isoquants. The last condition shows that in order to avoid ζ(v) being a correspondence we must assume strict convexity of isoquants; this hypothesis appears more restrictive than for consumers’ indifference curves, in that it is often considered realistic to assume that the production possibilities of firms are of the activity analysis type; but the result, that without strict convexity of isoquants factor demands are upper hemicontinuous correspondences, only obliges one to have recourse to the kind of more sophisticated proofs of existence, already necessary for the pure exchange case if strict convexity of preferences is not assumed. On the contrary the discontinuities caused by totally or partially strictly concave indifference curves have no parallel with firms as long as one assumes CRS and perfectly divisible factors. Concave technological isoquants do not cause discontinuities in factor demands, because here too we must distinguish technological from economic isoquants, and in the economic isoquants the strict concavities disappear, for the reason illustrated in §5.8.2 which we remember here for the reader’s convenience. Suppose then that in between two input vectors x and y on the same isoquant the technological isoquant is strictly concave: but because of CRS and divisibility the firm can produce the same quantity as with x or with y by using any convex combination ax+(1–a)y, 0a1, of the two production processes x and y, so the economically relevant isoquant is the convexification of the technological one, with the strictly concave portions replaced by straight segments, as in the Figure below. Factor demands are then correspondences, not functions, but are not discontinuous. factor 2 X y factor 1 If CRS do not hold at the plant or firm level, in competitive industries they still hold fabio petri VDE6 existence 12/02/2016 p. 75 (approximately) at the industry level owing to firm or plant replication, and then the discontinuities are as negligible as the non strict validity of CRS. Therefore factor demands never present relevant discontinuities due to technical choices, the discontinuities can only derive from discontinuities in the demand for the products for whose production factors are demanded. However, the problem with demand correspondences noticed for the exchange economy, that nothing guarantees that at the equilibrium prices the equilibrium demand will be chosen if it is a point in a correspondence, arises for factor demands too, and it is more serious than for consumer demand because, owing to the tendency of firms in an industry to adopt the same technical choices, it concerns an entire industry, not a single consumer. Thus if a large industry (e.g. the automobile industry) has an isoquant with a flat segment and the isocost is just tangent to that flat segment, this can mean a significant indeterminacy in the demand for certain kinds of labour. Since realism requires us to assume only approximate optimization by agents, the fact that a perfect tangency would be a fluke of zero probability does not eliminate the problem, because approximate tangency would still create a practical indeterminacy. I am not aware of discussions of this problem in the literature; one finds here, it would seem, another reason to nurture some doubts on the adequacy of the supply-and-demand approach for the determination of income distribution. As to the danger of discontinuities on the boundary of the factor rental simplex, these cannot be due to firm choices: the demand for an input never jumps up discontinuously when the input rental becomes zero; if a certain employment of the input is convenient when the input rental is zero, then optimal employment will be already at that level, or going continuously to that level, for an input rental very close to zero. By the way, the possibility that the demand for a factor goes to +∞ as the factor rental goes to zero because the marginal product of the factor never goes to zero (as in Cobb-Douglas production functions) is of course totally unrealistic and therefore can be disregarded; but anyway even then the demand for the factor would increase gradually, without discontinuities. Discontinuous jumps in the excess demand for a factor when its price becomes zero can only be due to consumer choices: its supply may jump to zero, or the demand for a good produced by that factor alone may jump discontinuously to +∞, owing to minimum income problems. Therefore the assumption of (i) strongly monotone preferences, or (ii) strictly positive endowment vectors, or (iii) irreducibility, will guarantee the existence of an equilibrium (if one neglects the survival problem). Therefore we need not spend time giving a formal proof of the existence of the general equilibrium of production and exchange under one of the three assumptions (i), (ii), (iii) just listed and assuming survival does not cause problems. The reason why in most textbooks the need is felt for an explicit proof is the different formalization explained in §5.25, based on the assumption of a fabio petri VDE6 existence 12/02/2016 p. 76 given number of (potential) firms with decreasing returns to scale, that determines well-defined firms’ excess supply functions and non-zero profits which enter the consumers’ budgets. We have argued in that chapter (cf. especially §5.21.3) that this different formalization is indefensible. A number of general equilibrium theorists, for example McKenzie and Morishima, fully agree with us on this issue as shown by their assuming without hesitation constant-returns-to-scale technologies. We complete this Section by giving an example that confirms that in the absence of one of the three assumptions that avoid the zero-income problem an equilibrium need not exist in the production economy[70]. The example will be presented in an informal way, the reader may try making it mathematically fully rigorous as an Exercise. Assume a production economy where two consumption goods, food and diamonds, are produced by labour and either one of two specialized lands, food-land and diamonds-land, with fixed coefficients: the production of one unit of food requires lf units of labour and mf units of foodland; the production of one unit of diamonds requires ld units of labour and md units of diamondland. Food is the numéraire. Because of competition, firms producing food or diamonds set prices equal to average cost. Population is divided in two groups, labourers who own no land, and landowners who supply no labour. Utility only depends on consumption of food and of diamonds; labour supply and supply of either land is rigid at all prices and factor rentals (including at zero rentals). The labourers demand no diamonds when their income is low. Assume the full utilization of both lands allows employing less labour than the given labour supply; then a necessary condition for equilibrium is a zero wage. Assume that consumers have a satiation level for food consumption, and that the satiation level is such that total demand for food from landowners is always less than the food production (call it F) obtainable with the full utilization of food-land, while total satiation demand for food (i.e. the maximum one coming from both landowners and labourers), call it S, is greater than F. Then when the real wage (measured in terms of food) is zero, if the price of food is positive demand for food is less than F because it comes only from landowners, while if the price of food is zero (because the rental of food-land is zero too) demand for food is S>F: there is no equilibrium. The reason is the discontinuity in food demand at a zero wage, demand for food jumps discontinuously up from the situation with positive food price (where there is excess food supply) to the situation of zero food price where there is excess food demand. This discontinuity arises because none of the three conditions that would prevent it is satisfied. Monotonicity of utility functions and interiority of endowments are clearly not satisfied; irreducibility too is not satisfied, because when land is fully utilized the excess labour supply yields no additional utility to anyone else (clearly, in a production economy where endowments consist of potential supply of factor 70 The example is taken from the Ph.D. thesis of Burak Unveren (Siena, 2011). fabio petri VDE6 existence 12/02/2016 p. 77 services, irreducibility must mean that the utilization, rather than the transfer, of the factor endowment of any agent increases the utility of at least one other agent). UNIQUENESS 6.35. We prove now that if WA holds for the market excess demand function derived from the sole choices of consumers, then for a production economy the set of equilibrium price vectors is convex; furthermore, the equilibrium aggregate consumption vector is unique. Note that the market excess demand function z(∙) of consumers is the sum of the individual vector excess demands of consumers for consumption goods and for factor services, it is not the sum of direct and indirect excess demands for factors ζ(v) defined in chapter 5. The result does not need the simplifying assumptions made in chapter 5, of no joint production and no overlapping between factors and consumption goods; thus it applies to the intertemporal reinterpretation of the model too, where some produced goods (e.g. sugar) can be demanded both by consumers for consumption purposes, and by firms for use as inputs; therefore we premise the proof with the indication of how the equilibrium equations of the production economy can be formulated in order to allow for this interpretation. But we still write the equations without dating goods. ‘Goods’ must now be interpreted as referring both to commodities, and to services of factors. There are n goods; prices of goods are indicated by vector π. For the production decisions we use the netput notation, so for example in the no-joint-production case the production of a quantity qn of good n through the employment of quantities a1nqn,...,aknqn of inputs (with k<n) is represented by the vector y=(y1,...,yk,0,...,0,yn,) where y1= –a1nqn, ... , yk= –aknqn, yn=qn. Consumers’ individual excess demands for each good j – their demand for the good minus their endowment of the good – are summed up to yield the consumers’ market excess demand scalar function zj(π); we call the vector function z(π)=(z1(π),...,zn(π))T the consumers’ market excess demand. The firms’ decisions can be summed up too to yield a market netput vector y; since in netput notation inputs are negative and outputs are positive, a negative yj indicates that the aggregate of firms demands good j (firms use more of it as an input than the amount they produce of it), and a positive yk indicates that the aggregate of firms supplies good k. Therefore for given z(π) and y the market excess demand is z(π)–y. Note that z(π)–y≤0 means that with the available production methods it is possible to satisfy the desired consumption demands with the desired factor supplies; since we assume free disposal, in this case there will also be some y in the production possibility set Y such that z(π)=y i.e. z(π)Y. With this notation, equilibria can be characterized in an extremely simple way as long as there are constant returns to scale for industries: fabio petri VDE6 existence 12/02/2016 p. 78 A price vector * is an equilibrium price vector for an economy with CRS production possibility set Y (that satisfies free disposal) if and only if (i) *y0 for all yY (no possibility of positive profits), and (ii) z(*)Y . Proof. Only if part: If * is an equilibrium price vector, then (i) is necessary for well-defined profit maximization when there are constant returns to scale, and (ii) is implied by market clearing. If part: we must prove that if (i) and (ii) hold then *, x(*) and y*=z(*) constitute an equilibrium, i.e. satisfy market clearing and profit maximization: indeed y*=z(*) implies market clearing, and y* is profit maximizing because from Walras' Law *y*=*z(*)=0 and so from (i) we obtain *y*≥*y for all y in Y. The reason why one writes y*=z(*) even when – as might be the case with joint production – some goods are produced or some factors are supplied in excess of demand is the assumption of free disposal. ■ As a further premise, we note that because of the consumers’ balanced budgets we obtain what can be called a Walras’ Law restricted to consumers’ excess demands, π’∙z(π’)=π”∙z(π”)=0, so the assumption we call WA (weak axiom holding in the aggregate), that states that for any two different non-proportional price vectors π’ and π” such that z(π')z(π"), if π’∙z(π’)≥π’∙z(π”) then π”∙z(π”)<π”z(π’), can be reformulated for the production economy too as: WA: if π’∙z(π”)≤0 and z(π')z(π"), then π”z(π’)>0. This can also be stated as follows: if π’z(π”) 0 and π”z(π’) 0, then z(π’)=z(π”). The result we intend to prove is: Theorem of equilibrium quasi-uniqueness under WA. Assume preferences are continuous, strictly convex and strongly monotone. Let z(π) stand for the (continuous) excess demand function of consumers. Let Y stand for the set of production possibilities of the several industries, that satisfies free disposal. I) Suppose that z(π) is such that, for any constant returns technology Y, the economy formed by z(π) and Y has a unique (normalized) equilibrium price vector π. Then z(π) satisfies WA. II) Conversely, if z(π) satisfies WA then, for any constant returns convex technology Y, the set of equilibrium price vectors is convex (and so unique if finite) and the equilibrium aggregate consumption vector and aggregate production vector are unique. Proof of I). We prove the converse i.e. that if z() does not satisfy WA then there is a fabio petri VDE6 existence p. 79 12/02/2016 constant-returns-to-scale Y for which equilibrium is not unique. Suppose z() violates WA i.e. there exist two normalized price vectors ’, ” such that z(’)z(”), ’z(”)0 and ”z(’)0. Then there exists a CRS convex production possibility set such that both ’ and ” are equilibrium prices: this is Y*=yRn: ’y0 and ”y0. Y is not empty since it contains the two vectors z(’) and z(π") now interpreted as netput vectors, because ’z(’)=0 and thus the condition ’z(’)0 is satisfied, and the same holds for z(”); but this means that both conditions (i) and (ii) for ’ and ” to be equilibrium price vectors in the (z(),Y*) economy are satisfied. Proof of II). Suppose that ’ and ” are two different equilibrium price vectors, i.e. z(’)Y, z(”)Y and for any yY it is ’y0 and ”y0. Let =’+(1–)” for 0<<1. We show that is an equilibrium price vector. By construction y=’y+(1–)”y0 for all yY; it remains to show that z()Y. By Walras’ law restricted to consumers it is 0=z()=’z()+(1–)”z(); not both terms on the right-hand side can be positive so it is either ’z()0 or (1–)”z()0. Suppose ’z()0, then also ’z()0; but since z(’)Y and we have shown that πy0 for all yY, it is also z(’)0, and WA is violated unless z()=z(’). If it is not ’z()0, then it is (1– )”z()0 and the same reasoning concludes to z()=z(”). Since either z()=z(’) or z()=z(”) (or both), it is z()Y and we have shown that the set of equilibrium prices is convex. In addition, since the proof that either z()=z(’) or z()=z(”) holds for any (0,1), and since z() is continuous, it must be z(’)=z(”), otherwise there would be a discontinuity of z(∙) somewhere as α varied over the interval [0,1]; so all equilibrium price vectors are associated with the same z() and hence also with the same aggregate production vector y[71]. ■ In the exchange economy, the situations in which a unique equilibrium market excess demand does not uniquely determine the equilibrium relative prices (if one leaves aside the extreme flukes of zero probability in which the changes in the choices of one consumer are exactly compensated by the changes in the choices of other consumers) are due to 'kinks' in indifference curves, or to corner solutions, at equilibrium. For example, if the equilibrium is a no-exchange equilibrium (everybody satisfied with her initial endowment) and there are kinks in the indifference curves of all consumers at their endowment point, the equilibrium prices may be indeterminate within an interval (draw the picture in an Edgeworth box!); the same may happen if the equilibrium entails for a certain good a corner solution for all consumers. But since either of these cases must concern all consumers, it is an extremely unlikely occurrence. With production, the situation is even less likely: leaving again 71 . Note that this does not mean that the equilibrium allocation is unique: different equilibria might be associated with different excess demands of the several consumers, which happen to sum up to the same consumer market excess demand vector. fabio petri VDE6 existence 12/02/2016 p. 80 aside the zero-probability case of perfect compensation of the changes in factor supplies and consumption demands by one consumer with the changes by the other consumers, the equilibrium must be either a no-production equilibrium (again with kinks in the indifference curves at the endowment points) or an equilibrium where factor supplies are rigid within a certain range of relative factor rentals (owing to kinks or corner solutions), and where furthermore consumption demands too are rigid for the corresponding range of relative product prices. When factor supplies are given but relative factor prices (and hence, in general, relative consumption goods prices) vary, a constant demand for consumption goods obtains in general only if for all consumers all consumption goods are perfect complements (or almost perfect complements, i.e. with kinks of the indifference curves) in the same proportions, so that the composition of the demand of all consumers is the same and remains unaltered as relative consumption goods prices vary. Since these cases are extremely unlikely, it seems possible to conclude that, in the pure exchange economy as well as in the production economy, under WA the equilibrium is almost certainly unique. 6.36. We have argued that it is unclear why WA should hold apart from the case when there exists a representative consumer. Interestingly, outside this case the WA assumption is even more restrictive for a production economy than for the exchange economy, for a reason highlighted by Hildenbrand and Grodal(72): they have shown that, in production economies with at least two factors and where factors of production do not yield direct utility to consumers (they are then called pure factors of production and their net supply coincides with their endowment – note that this excludes monotone preferences), if one takes the utility functions of consumers and the aggregate factor endowments as given, then the set of factor endowment allocations among the consumers which cause the market excess demand function z() to satisfy WA is a negligible subset (i.e. a subset of Lebesgue measure zero) of the set of possible factor endowment allocations; in other words, if a certain distribution of factor endowments determines a z() which satisfies WA, then any very small redistribution of factor endowments among the consumers will cause the new z() to violate WA with probability 1; therefore if the factor endowment distribution is randomly chosen among all the possible ones, the probability that z() obeys WA is zero. Without going into the topological arguments necessary rigorously to prove this result, we can grasp the basic reason by noticing the following Fact. . W. Hildenbrand, “The Weak Axiom of Revealed Preference for Market Demand is Strong”, Econometrica, 1989, pp. 979-985; B. Grodal and W. Hildenbrand, “The Weak Axiom of Revealed Preference in a Productive Economy”, Review of Economic Studies, 1989, pp. 635-639. 72 fabio petri VDE6 existence 12/02/2016 p. 81 Let the first m goods be the consumption goods, with prices p, while v stands for factor rentals. Call zC(p,v) the m-vector of the first m components of z(p,v), the ones which refer to consumption goods; and let stand for the n-vector of aggregate endowments of factors; then it is pzC(p,v)=v because of the balanced budget assumption. In an economy where factor supplies coincide with factor endowments whatever the prices, if two different factor rental vectors v and v’ are such that v=v’, then the validity of the WARP for the consumers’ market excess demand requires that for any given (non-normalized) p it is zC(p,v)=zC(p,v’). Proof. This is because v=v’ implies pzC(p,v)=pzC(p,v’); since z(p,v) is the vector with zC(p,v) as its first m components and – as its last n components, it is (p,v)z(p,v') = pzC(p,v')–vω = pzC(p,v)–vω = 0 = (p,v')z(p,v), hence it is both (p,v)z(p,v’)0 and (p,v’)z(p,v)0, and the only way for WA to hold is that z(p,v)=z(p,v’). ■ So for WA not to be violated in this economy with rigid factor supplies, all changes in factor prices that leave the aggregate value of endowments unaltered must leave all market demands for consumption goods unaltered. But an unchanging aggregate value of endowments can always be obtained, simply by choosing the aggregate endowment vector as numéraire[73], so what is needed is that changes in relative factor prices leave all market demands for consumption goods unaltered. This obtains, apart from the case when there exists a representative consumer, only in two special cases: 1) there is only one good demanded by consumers; 2) changes in relative factor prices cause no redistribution of wealth, that is, the endowment vectors of consumers are co-linear, a case of measure zero among all allocations of factor endowments among consumers. A point apparently not noticed by Hildenbrand and Grodal, who assume that all factors are pure factors, is that the above Fact – which is the key to their result – only needs that at least two of the factors be pure factors; the Fact survives unaltered, if the excess demands for the other factors, the ones for which there can be demand, are included in the vector zC(p,v), their rentals are included in the vector p, and and v are understood to refer only to the pure factors. Thus the existence of a variable reservation demand for labour (i.e. for leisure) does not render Hildenbrand and Grodal’s result irrelevant, because the result only needs that there be at least two other factors which yield no direct utility to their owners, e.g. inhospitable lands of different types useful for production but unattractive for direct enjoyment, or – in capitalistic economies – pure capital goods; and this is 73 . As noticed in the Ph.D. thesis of S. M. Fratini, Il problema della molteplicità degli equilibri da Walras a Debreu (Università di Roma 3, 2001), Ch. 7. fabio petri VDE6 existence 12/02/2016 p. 82 very likely. Thus recourse to the WA assumption in order to ensure uniqueness of equilibrium is even less justifiable in production economies than in pure exchange economies. 6.37. The other main assumption guaranteeing uniqueness of equilibrium in the pure exchange case is gross substitutability. But assuming GS for consumer choices does not guarantee the uniqueness of equilibrium in the production economy. Here is an example. Consider the twogoods, two-factors economy with fixed coefficients (no technical choice) discussed in Chapter 3 to clarify the indirect factor substitution mechanism. Assume that workers (all identical) are the sole suppliers of labour, landowners (all identical) are the sole suppliers of land, and for all relative factor rentals workers demand the labour-intensive good in a greater proportion to the landintensive good than landowners (cf. ch. 3, footnote 39). If factor supplies are rigid and the composition of demand of the two classes is also rigid, a decrease of the wage rate relative to the rent rate causes a decrease of the labour-land ratio in factor demand, hence an upward-sloping demand for labour if land is assumed fully employed (cf. ch. 3, Fig. 3.6), with the possibility of three equilibria of which the full-employment one is unstable, while in the other two locally stable equilibria one factor is not fully employed and gets a zero rental. So far this example does not satisfy GS for consumer choices; but we can modify the assumption about preferences by assuming that all consumers have a very small direct demand for both factors and that there is a very small variability of consumer choices, such that if the sole price of one consumption good increases then all consumers increase by a tiny amount their demands for the other consumption good and their direct demands for the two factors, and if the sole rental of one factor increases then the owners of that factor decrease their direct demand for that factor (i.e. increase its supply) by a tiny amount, increase their demand for both consumption goods (by a common percentage), and increase by a tiny amount their direct demand for the other factor, while the owners of the other factor decrease by a tiny amount their direct demand for the first factor and increase by a tiny amount their demands for the two consumption goods and their direct demand for their own factor. Now GS holds for z(p,v), but the variations in factor supply and in the composition of consumer demand relative to the rigid case can be made as small as one likes, hence the sign of the dependence of the composition of factor demand on relative factor prices can be made not to be affected, thus if before this modification of preferences there were three equilibria there will still be three equilibria after the modification. The point is that in order to guarantee uniqueness the GS property should hold for the excess demands of the reduced pure-exchange equivalent economy i.e. for ζ(v), and its holding for consumer choices does not suffice for that. The excess demand for a factor includes the firms' fabio petri VDE6 existence 12/02/2016 p. 83 demand for it, which does not obey GS; in our example when the wage goes up firms, in order to adapt production to consumption demand at zero-profit prices, need more instead of less leisure/labour. With the exception of WA, the assumptions known to ensure uniqueness (and stability) for exchange economies (GS, Diagonal Dominance, Negative Definiteness) do not do the same for the production economy unless the number of firms is given and firms have sufficiently decreasing returns to scale so that the aggregate netput vector is a single-valued function of (p,v) – no doubt an important reason for the continuing acceptance of this approach by general equilibrium theorists. But as argued in ch. 5 this approach is untenable, so we omit the proof of these results[74]. If free entry and hence CRS industries are admitted, then only very special sufficiency results are known ensuring uniqueness, e.g. that ζ(v) satisfies GS if the economy is regular and both all utility functions and all production functions are Cobb-Douglas or super-Cobb-Douglas (75). THE TÂTONNEMENT IN THE PRODUCTION ECONOMY 6.38. For the production economy as formalized here, i.e. with free entry and hence CRS at least for the industries, in the tâtonnement the supplies of price-taking firms (and the associated demands for factors) are infinite or zero or indeterminate depending on whether profits are positive, negative or zero. The infinite or zero production decisions indicate to the auctioneer the presence of positive or negative profits and oblige it to find and announce, together with the vector v of factor prices, zero-profit product prices (i.e. prices obeying equations (A) of ch. 5, if the simplifying assumptions there made are accepted); then in order to derive factor demands one must assume that production intentions adjust to demands for products (as in equations (B) of ch. 5). In other words, the tâtonnement must be assumed to operate as if the economy were the reduced pure exchange economy of factors illustrated in ch. 5, with excess demand function ζ(v). (This is e.g. admitted by Arrow and Hahn, 1971, p. 317). The resulting tâtonnement operates only on factor rentals and accordingly has been called a factor tâtonnement (Mandler 2005). Connecting the tâtonnement to real adjustment processes becomes then even more difficult than for the exchange economy. I point out just two of the many problems additional to the problem already pointed out that consumer incomes cannot equal the value of their endowments in 74 Cf. Hahn 1982. Cf. Mas-Colell, “On the uniqueness of equilibrium once again” in W. Barnett et al., Equilibrium Theory and Applications, Cambridge UP, 1991, pp. 275-296. A function h: Rs+R is super-Cobb-Douglas if at every x0 there is a Cobb-Douglas function hx: Rs+R such that hx(x)=h(x) and hx(x’)h(x’) for all x’ in a neighbourhood of x. Essentially this means at least as much factor substitutability as with a Cobb-Douglas. 75 fabio petri VDE6 existence 12/02/2016 p. 84 disequilibrium (because not all their factor supplies find purchasers). Even if one assumes realworld adjustments of produced quantities and product prices to be faster than adjustments of factor rentals, the difficulty remains that productions cannot be equal to consumer demands in disequilibrium because some of the factors will not be available, but then product prices will not be zero-profit prices and therefore consumer demands cannot be the corresponding ones; and a firm, when constrained in the amount it can obtain of one factor, will determine its demand for other factors taking that constraint into account. So there is no reason why firms' demands for factors should be the ones, or close to the ones, appearing in ζ(v). Indeed nowadays there is very little attempt to argue that the tâtonnement mimics, however remotely, the way adjustments operate in real-world markets. The tâtonnement is admitted to be an ideal construction, that must be conceived to operate with the auctioneer and provisional 'pledges'. However, then a problem arises which seems to have escaped attention so far. In the traditional analyses, that were aimed at determining equilibria endowed with persistence (longperiod equilibria), it was assumed that adjustments took time, that disequilibrium productions were actually produced, that therefore the amount an industry could produce was limited by the amounts of factors it could actually get hold of; thus the supply of an industry could not be infinite, and realistically it was further limited, during the adjustments toward the long-period equilibrium, by the short-period availability of the specialized factors needed by that industry: specific lands, specialized labour, fixed plants and other durable capital goods requiring time for their construction. The marginal products of the unspecialized factors could therefore be assumed to be decreasing, and this would impose a finite optimal employment of them by the industry once their rentals and the product price were given; this would generate an upward-sloping supply curve as the product price increased relative to the rentals of variable factors[76]. If the specialized factors included capital goods, the short-period supply curve would shift in time as the number and quality of these capital goods was gradually altered by their utilization and their production. The long-period adjustment of production to the level of demand forthcoming at a price equal to minimum average cost MAC was therefore a legitimate assumption. But in the auctioneer-guided tâtonnement, there is no actual disequilibrium production; there are only announcements of intentions to produce under a hypothesis that one can get all the factors one is demanding (the plans transmitted to the auctioneer become binding only if they come out to be equilibrium plans); this makes it possible for desired 76 The rentals of the unspecialized factors could be assumed to change at a much slower pace that the product price, as long as only a small portion of the total demand for these factors came from the industry under examination. This was the situation authorizing the use of Marshallian partial equilibrium analysis, where the rentals of variable factors are treated as given. fabio petri VDE6 existence 12/02/2016 p. 85 product supplies (and for the associated factor demands) to shoot up to infinity the moment profits are not zero in an industry[77]; because of this the auctioneer is obliged to restrict the product prices he calls to zero-profit prices. But then at these prices firms are indifferent as to whether to produce or not, so the number of firms in each industry, and even the size of firms if firms have CRS production functions, are indeterminate. Then how can equilibrium be reached? The announcement of the equilibrium prices leaves product supplies and factor demands indeterminate. The possible and very minor indeterminacies we noticed in the demand for consumption goods owing to nonstrictly-convex indifference curves, and the analogous (although less minor) indeterminacies in factor demands (when the quantities to be produced are given) due to non-strictly-convex isoquants, could be set aside with some justification(78); here we have a certainty of very grave indeterminacy, product supplies (and the associated factor demands) can take any value between zero and +∞. Thus the auctioneer story in the production economy obliges one to assume that the auctioneer acts like a true central planner, imposing how many and possibly also how big the firms must be in each industry so as to render production equal to equilibrium demands; if firms chose randomly whether to enter or not, and chose randomly their production if endowed with CRS production functions, equilibrium would never be reached. The market as depicted in the auctioneer fairy tale needs a central planner! The way out is to admit that disequilibrium productions must actually take place, but this means admitting that disequilibrium adjustments take considerable time and, when we come to production with capital goods, this entails that during the disequilibrium adjustments the quantities in existence of the several capital goods are affected by the disequilibrium actions of agents and cannot therefore be unchanging during the process which should bring the equilibrium about – with grave consequences, to be discussed in Ch. 8, for the meaningfulness of neo-Walrasian equilibria. 6.39. If one still wants to study the tâtonnement in the (acapitalistic) production economy, there seems to be little alternative to accepting that the auctioneer acts like a true planner, by announcing, together with factor prices, product prices equal to minimum average costs and by somehow ensuring the adjustment of the quantities that firms promise to produce to the quantities demanded at those zero-profit prices, so that the tâtonnement is actually a tâtonnement on factor markets only, it acts on the rental vector v and alters vj depending on the sign of ζj(v). 77 Cf. §5.25.3 for a refutation of the argument that the short-period nature of the situation in which the tâtonnement operates justifies a given number of firms with given fixed factors. 78 . Although less justification for factors than for consumption goods, as noticed earlier. fabio petri VDE6 existence 12/02/2016 p. 86 Only one sufficient condition of clear economic interpretation is known that ensures the global stability of this tâtonnement: WA holding for the consumers excess demand z(∙). A complication is that ζ(v) is not necessarily single-valued, i.e. it can be a correspondence: if firms can choose among a finite number of fixed-coefficients methods in order to produce a good, then isoquants are piecewise linear (‘kinked’), and it can happen that at the given v two methods are both cost-minimizing for the production of a certain good; in this case a given output of that good can be produced with either method or with a linear combination of the two methods, hence factor demands are not functions, but correspondences of factor prices and consumer demands. It is possible to prove that under WA the tâtonnement is convergent even in this case; the proof, due to Mandler (2005), uses results (from the theory of Liapunov stability for dynamical systems with correspondences) that we cannot explain here; a simplified version of the proof, that assumes strictly convex isoquants, is postponed to ch. 8 because Mandler makes assumptions about technology that make the proof more intuitively interpretable as applying to an intertemporal equilibrium. (Other sufficient conditions for tâtonnement stability are known only for the unacceptable production economy with a given number of decreasing-returns firms.) It may be opportune then to remember that WA is a highly restrictive assumption, practically certainly violated if at least two factors are in rigid supply: now, when one comes to the intertemporal re-interpretation of the production economy necessary to admit the presence of capital goods in the same formal structure of equations (cf. ch. 8), among the initial endowments there will be stocks of many capital goods, for most of which the assumption that they yield no direct utility to consumers and hence that their supply is rigid (coinciding with the endowment) is the natural one; hence WA cannot plausibly be assumed to hold. AGAIN ON THE LIKELIHOOD OF UNIQUENESS AND STABILITY 6.40. We can conclude that the conditions that guarantee the uniqueness or stability of equilibrium in the production economy are even fewer and more restrictive than for the pure exchange economy. One can be certain that the equilibrium is unique and tâtonnement stable only if the consumer side behaves as if there were a representative consumer; lacking that, uniqueness and stability are the more likely, the more elastically factor proportions react to changes in relative factor rentals, thus rendering more probable that technical substitution overpowers any plausible ‘perverse’ income effects. However, it was pointed out that some economists argue that insistence on a certainty that equilibrium be unique and stable is excessive. We must therefore ask, whether uniqueness and stability are at least likely, and how likely, for the production economy. fabio petri VDE6 existence 12/02/2016 p. 87 It is not easy to find explicit discussions of this issue. Hicks in Value and Capital (1946 p. 104) suggests that there is less reason to worry about multiplicity and hence instability of equilibria in the production economy than in the exchange economy, because in production choices there is nothing analogous to the income effects which are the cause of instability in the pure-exchange economy: the technical choices of industries know only a ‘substitution’ effect (a tendency to decrease the proportion in which the factor, whose relative rental has increased, is combined with other factors), which always works in the ‘right’ direction and hence strengthens the likelihood of uniqueness and stability. Here too, Hicks's argument is not convincing. First, even if it were true that there is less reason to worry, would that mean that one need not worry? There is no guarantee that technical substitution will always succeed in counterbalancing the tendencies to instability deriving from consumer choice: the elasticity of substitution in production might be limited. Hicks's optimism largely derives from his persuasion that already at the level of consumer choice one need not greatly worry about income effects, but we have seen that his argument on this issue is not persuasive. Second: although income effects are not present at the firm level, production introduces a new route through which income effects can work against uniqueness and stability at the aggregate level. The indirect factor substitution mechanism illustrated in chapter 3 is not guaranteed to work in the ‘right’ direction, as noticed in ch. 3 fn. 39, and above in §6.37. Inferior goods may also cause problems, but the more likely new 'perverse' case is when the demand for a good comes from the owners of the factor more intensively used in the production of that good. There is a productionmediated ‘income effect’ in the demand for labour due to the change in income of the two social groups[79], that affects the firms' demand for labour owing to the change in the demand for corn and for iron. If the indirect substitution mechanism works in the ‘wrong’ direction, the demand curve for labour, derived under an assumption of full employment of the other factors, may therefore be at least partly upward-sloping rather than downward-sloping, with a high likelihood of multiple equilibria some of which would be unstable. This new possibility of non-uniqueness and instability, which apparently escaped Hicks, goes counter the claim that the introduction of production reduces the likelihood of non-uniqueness and instability. 79 . Exercise: Try to define absence of income effects. (The usual answer is, income effects are absent when one considers compensated price variations, i.e. price variations accompanied by lump-sum taxes or subsidies which keep each consumer on the same indifference curve she was on at the start. But this only applies to changes from an initial situation. When would you say that an economy is free of income effects?) fabio petri VDE6 existence 12/02/2016 p. 88 However, even this problem has been considered of limited relevance. In one of the very rare discussions of the likelihood of multiple equilibria in the production economy, Harry Johnson(80), after presenting the neoclassical two-goods two-factors production economy[81] with rigid factor supplies, admits that there may be multiple equilibria and comments: “It can be shown, however, that the conditions required for multiple equilibrium are extremely restrictive. For multiple equilibrium there must be an unstable equilibrium. This requires that an increase in the price of a commodity produces an excess demand for it[82]. The transformation effect in production and the normal compensated substitution effects in consumption must tend to produce an excess supply [of the commodity]. Hence instability must depend on a redistribution effect coming through the change in factor prices associated with the rise in the price of the product; and for that redistribution effect to tend to create excess demand, the factor used relatively intensively in producing the good whose relative price has risen must have a relatively stronger preference at the margin for that good than does the other factor, i.e. must have a higher marginal propensity to consume it out of income. Further, since these marginal propensities to consume are both fractional (assuming both goods normal, neither inferior in consumption) the redistribution of income must be greater than the increase in the value of output of the good whose price has risen. Now, if the elasticities of substitution in production for the two goods are both unity, the amount of income redistributed will be only (approximately) the difference between the shares of the factors in the total income generated by the two industries multiplied by the change in output, and since these shares are fractional so will be the income redistributed as a fraction of the output increase in the industry where the price has risen. For that fraction to be greater than unity, the elasticities of substitution must in some average sense be less than unity, so that the factor whose price increases obtains an increased share of the income generated in each industry.” (p. 61). Formalizing these lines can be a useful Exercise. I limit myself to highlighting the main point raised by Johnson. He too discusses stability by concentrating on a single market while assuming equilibrium on all other markets, but he chooses the market of a consumption good; however, the connection between change in the product price and change in income distribution is the one we 80 Harry G. Johnson, The Theory of Income Distribution, London: Gray-Mills, 1973. . In his model one of the factors is capital, but treated as perfectly analogous to labour or land, so, since capital presents special problems to be examined in another chapter, it is better now to interpret his analysis as referring to labour and land producing corn and iron. 82 . [This is only true if the economy is regular, i.e. all equilibria are locally either stable or unstable, no equilibrium is one-sided unstable and disappears owing to an infinitesimal redistribution of endowments – F. P.] 81 fabio petri VDE6 existence 12/02/2016 p. 89 already know from the corn-iron example: the commodity rises in (relative) price because income distribution has changed in favour of the factor used relatively more intensively in its production; to help intuition, let us refer to our example and suppose the commodity whose price rises is corn and its production is labour intensive. The key element of Johnson's argument is the ‘transformation effect in production’, that is the tendency of the production vector to change along the production possibility frontier of the economy (under an assumption of factor substitutability and of given fully employed factor supplies) in favour of the product which has become relatively more expensive: the rise in wages induces a decrease of the labour-land ratio in both industries, which makes it possible and indeed necessary, in order for the full employment of both factors to be maintained, to expand the corn industry at the expense of the iron industry. The point made by Johnson is that this expansion of the corn industry when the wage rises may well suffice to make the increase in labour income too low for an increase in the demand for corn greater than the increase in its supply to be plausible or even possible. Thus assume the wage rises 10%; the price of corn rises too, say by 4%; hence the labourers' demand for corn increases by 6% at most, and since they are not the sole demanders and the demand for corn from landowners decreases, if full-employment corn production increases by 6% or more then an excess demand for corn is impossible; this, Johnson argues, is what will happen except for very low elasticities of substitution. A numerical example may help to illustrate his point. Suppose the initial labour-land ratio is 4 in the corn industry and 2 in the iron industry, and a 10% wage rise induces a decrease of both ratios by 5% (an elasticity of substitution equal to 1/2) i.e. respectively to 3.8 and to 1.9. Assume initially 10 units of land were employed to produce corn and 20 units to produce iron, hence total labour employment is 80. The full employment of 30 units of land and 80 units of labour with the new labour-land ratios requires the employment of land in corn production to rise to 12.1 units, and the employment of labour to rise to 46 units, with a percentage increase in corn production of approximately 10%. Thus, Johnson is right that in order for upward-sloping factor demand curves to arise owing to this cause, not only there must be a considerable difference in the composition of demand for consumption goods out of incomes from different factor rentals and this composition must be 'selfintensive', but also factor proportions must be rather insensitive to changes of relative factor rentals. But is this combination of characteristics “extremely restrictive”? There are many economists who consider technical choices to be little affected by realistic changes in income distribution. And the composition of demand of different social groups with different factor ownership and income levels is different. Also, as argued in ch. 3 the plausibility of the neoclassical approach needs factor demand curves to be not only downward-sloping but also considerably elastic, in order to minimize the risk fabio petri VDE6 existence 12/02/2016 p. 90 of multiple equilibria or of practically indeterminate equilibria due to backward-bending factor supply curves, as well as the risk of implausible changes in income distribution due to small changes in factor endowments. Now, even if 'self-intensive' demand is unable to overpower the consumption and technical substitution effects, it nonetheless has the effect of decreasing the elasticity of the factor demand curves, thereby increasing the risks just remembered. There appears therefore to be insufficient reason to think that the presence of (acapitalistic) production renders the supply-and-demand approach more plausible than in the pure exchange case. The pessimistic conclusion reached in §6.11?? is not weakened by the consideration of production. fabio petri VDE6 existence 12/02/2016 p. 91 Appendix 1 - Alternative proof of stability under GS A sketch of an alternative proof of global stability under GS for the exchange economy is supplied here to show that a result can often be reached in many different ways. Assume for simplicity (***) of §6.29 as the price adjustment rule and consider the function h(p) = Max (z1(p)/p1,…,zn(p)/pn), defined for all positive p. The ratios zi(p)/pi are the time rates of percent variation (i.e. the growth rates) of prices. Therefore if zi(p)/pi=h(p), and if equilibrium has not been achieved, then, first, pi is growing because zi(p) must be positive; second, no other price grows faster than pi and at least one grows slower (in disequilibrium at least one excess demand is positive and at least one is negative or zero). Let us choose pi as numéraire, then at least another price pj is decreasing; by GS, zi(p)/pj>0, therefore zi(p) is decreasing through time, and therefore zi(p)/pi is decreasing too, since pi is increasing. If in the course of time one zj(p)/pj becomes greater than zi(p)/pi, let us adopt good j as numéraire and repeat the reasoning. Therefore h(p) decreases through time as long as one of the excess demands is positive, and since the greatest of the excess demands always decreases, all positive excess demands must sooner or later decrease, and therefore they all tend to zero i.e. to the equilibrium; and in equilibrium h(p) reaches a minimum because only there no excess demand is positive. Finally, for the assumed continuity of z(p), h(p(t)) is a continuous and differentiable function of t, and therefore it qualifies as a Liapunov function. Appendix 2 – WA holding for z(p,v) does not imply WA holding for ζ(v). We have seen that if the weak axiom is satisfied by aggregate demand, it is also satisfied by market excess demand, so I can limit myself to discussing aggregate demand. Consider an economy with only one produced consumption good C with price p equal to unit cost, produced by two factors X and Y (whose rentals are w and v) according to a differentiable constant-returns-to-scale production function common to all firms. The variable technical coefficients (inputs per unit of output) are ax(w/v), ay(w/v). The two factors yield utility and therefore there is an aggregate direct demand for them (x,y) by consumers. We can fix p=1. When the consumers’ direct aggregate demand vector is (c(w,v), x(w,v), y(w,v)) and the technical coefficients chosen by firms are ax, ay, then the aggregate reduced demand vector for factors is D(w,v)=(Dx,Dy)=(x+axc, y+ayc). In this economy WA means: For two different price vectors (p=1,w,v), (p=1, w’,v’), if (c,x,y) is demanded when prices are (1,w,v) and the different vector (c’,x’,y’) is demanded when prices are (1,w’,v’): fabio petri VDE6 existence 12/02/2016 p. 92 (†) c+w’x+v’y ≤ c’+w’x’+vy’ implies c’+wx’+vy’>c+wx+vy Let ax=ax(w/v), ax’=ax(w’/v’) and similarly for ay; when prices are (1,w’,v’) the demand for a quantity c of the consumption good implies a derived demand (ax’c, ay’c) for the factors, and the value of c equals the value of the derived factor demand, c=w’ax’c+v’ay’c, hence WA translates to reduced (direct plus derived) factor demands as follows: (††) w’(x+ax’c)+v’(y+ay’c) ≤ w’(x’+ax’c’)+v’(y’+ay’c’) implies w(x’+axc’)+v(y’+ayc’) > w(x+axc)+v(y+ayc). This is not the same as the assumption that the weak axiom holds directly for the reduced exchange economy. In such an economy the weak axiom would state that, if at prices (w’,v’) a certain reduced demand for factors (Dx,Dy) different from the chosen one (Dx’,Dy’) is affordable, then when (Dx,Dy) is chosen at prices (w,v), (Dx’,Dy’) must not be affordable: (†††) w’Dx+v’Dy ≤ w’Dx’+v’Dy’ implies wDx’+vDy’>wDx+vDy. In order for (Dx,Dy) to be chosen at prices (w,v) it must be Dx=x+axc, Dy=y+ayc, and in order for (Dx’,Dy’) to be chosen at prices (w’,v’) it must be D x’=x’+ax’c’, Dy’=y’+ay’c’, which means that (†††) can be rewritten as: (††††) w’(x+axc)+v’(y+ayc) ≤ w’(x’+ax’c’)+v’(y’+ay’c’) implies w(x’+ax’c’)+v(y’+ay’c’) > w(x+axc)+v(y+ayc). The technical coefficients on the left-hand sides of the two inequalities in (††††) are different from the ones in (††). This means that it is possible that WA holds for z(1,w,v) but does not hold for ζ(w,v). fabio petri VDE6 existence 12/02/2016 p. 93 EXERCISES AND CONTROL QUESTIONS Explain in what cases GS implies that the demand for a good is decreasing in the good’s own price. FURTHER READING 6.??. For those who are interested in further study of the issue of existence of general equilibrium, I recommend the whole of Hildenbrand and Kirman, Equilibrium Analysis, NorthHolland, 1988, and, if they can read Italian, the neat proofs of existence for the pure exchange economy without strongly monotone preferences, based on assumptions B (strictly positive endowments) or C (irreducibility), in Reichlin and Ventura, Equilibri competitivi ed economie dinamiche (Carocci 1998); this is a well organized and mathematically rigorous but not overly advanced book, useful also for a first introduction to equilibrium over infinite horizons and overlapping-generations economies, but disconcerting for its total neglect of issues of uniqueness and stability, as if existence were all that matters. (There is no discussion of uniqueness and stability also in G. A. Jehle and P. J. Reny, Advanced Microeconomic Theory, 2nd ed., Addison-Wesley, 2001.) In English an existence proof based on B can be found in M. C. Blad and H. Keiding, Microeconomics: Institutions, equilibrium and optimality, North-Holland 1990, pp. 157-161, and in Hildenbrand and Kirman, cit., pp. 108-111; as far as I am aware proofs based on irreducibility are not available in English-language textbooks, only in specialist articles and in Arrow and Hahn’s General Competitive Analysis, 1971, which cannot be considered a textbook. Mas-Colell et al., Appendix 17.BB.2, contains a proof of existence admitting correspondences but based on a "cheaper consumption condition" which requires directly that for each consumer there exists a consumption bundle in the consumption set (the latter is defined so as to guarantee survival) which costs less than her income, i.e. income must always be in excess of what is needed for survival; what might guarantee such a condition (our assumptions B' or C') is only briefly hinted at. A different proof technique is in Debreu, “Existence of competitive equilibrium” in Arrow and Intriligator, eds., Handbook of Mathematical Economics, vol. II, 1982; also important is L. W. McKenzie, "The classical theorem on existence of competitive equilibrium", Econometrica 1981. A useful history of the evolution of general equilibrium theory including a discussion of the evolution of existence proofs is B. Ingrao and G. Israel, The invisible hand; unfortunately it neglects the existence of the long-period versions that we will discuss in Ch. 7. fabio petri VDE6 existence 12/02/2016 p. 94 These references show that original research in this area requires a substantial mathematical background: most of the contributors in this area have a first degree in mathematics. For those who want to strengthen their maths without going all the way to becoming full-fledged mathematicians, I suggest B. Ellickson, Competitive Equilibrium, Cambridge UP, 1993. In order to approach the modern frontier, which is concerned with equilibrium over infinite horizons, C. D. Aliprantis, D. J. Brown, O. Burkinshaw, Existence and optimality of competitive equilibria, Springer-Verlag, 1990, is mathematically more complete than Reichlin and Ventura. On uniqueness and stability a good critical discussion is in B. Ingrao and G. Israel, The Invisible Hand; the Handbook of Mathematical Economics edited by Arrow and Intriligator (1982) contains in Vol. II a chapter on the existence of general equilibrium (by G. Debreu), one on stability (by F. Hahn), and one on regular economies (by E. Dierker) but no chapter specifically on uniqueness. Rizvi Kirman in EJ and in Medio Saari and Simon