The Efficacy of the Invisible Hand in Simple Models

Transcription

1 Contemporary Issues and Ideas in Social Sciences December 2008 The Efficacy of the Invisible Hand in Simple Models Anjan Mukherji 1 Centre for Economic Studies and Planning Jawaharlal Nehru University, New Delhi Abstract The paper shows how specifying economic models enables us to obtain simpler forms of stability conditions. A general form of stability result obtained in Mukherji (2007) is applied to simple three good models with a single produced 1 I am indebted to Professor K. L. Krishna, who, in his capacity as the Editor of the Journal of Quantitative Economics, had provided the original impetus to prepare a survey of Non-Linear Dynamics in Economics for his journal and which appeared as Mukherji (2000). Professor Krishna was also kind enough to chair the Inaugural lecture delivered on February 17, 2006 where I had discussed a preliminary version of results reported here. A sketch of the main argument presented in this paper was prepared in response to Patnaik (2008) and presented at the Conference organized by IDEAs in August A more complete version was presented at the Conference organized by JNU-CIGI-NIPFP in Delhi in November The current version owes a lot to helpful comments from Subrata Guha, and a referee, which uncovered an omission; these are gratefully acknowledged. anjan@mail.jnu.ac.in

2 2 CIISS December 2008 good, labour and a numeraire commodity; such models are routinely used in many contexts. In particular it is shown that restrictive assumptions need not be employed to ensure stability. Further the meaning of stability discussions and their significance and relevance are discussed. 1 Introduction From the time of Adam Smith, there was a belief that the so called Invisible Hand, the market force, should be successful in attaining equilibrium or a matching of demand and supply. That this force was believed to be as mysterious and powerful as the Force which Luke Skywalker is supposed to have battled the odds with, is perhaps not very surprising. What is surprising however, is the almost complete neglect of this area by economic theorists. To briefly recapitulate, the problem was discussed by Walras in the 19th century (1874) but then he assumed the problem away; it was redefined by Hicks in Value and Capital in 1940 and later by Samuelson in the Foundations in 1950 s and subjected to searching and probing analyses by Arrow and Hurwicz and McKenzie in the early 1960 s 2 ; a masterly survey was produced by Takashi Negishi in 1962; but around the same time two papers, one by Scarf (1960) and the other by Gale (1963), showed that things were not as satisfactory as they might have been taken to be. Briefly the situation was as follows: if demand and supply do not match, it was entirely expected that prices would adjust in the direction of excess demand (which is demand - supply) and would thus remove any vestige of mismatch. Even to this day, a sudden spurt of demand is associated with a 2 The paper by McKenzie (1960), contained the most general treatment of the subject.

3 Mukherji: Invisible Hand 3 rise in prices. The intuitive appeal of this story is however far more strong than its actual effectiveness. In fact, a commentator no less eminent than Edmond Malinvaud in 1993, while celebrating 40 years of the proof of existence of competitive equilibrium would indicate that this was one of failures of economic theory. This tendency of considering economic policies based on the workings of the Invisible Hand had been questioned by theorists of the stature of Frank Hahn (1984), (p. 308): Now one of the mysteries which future historians of thought will surely wish to unravel is how it came about that the Arrow- Debreu Model came to be taken descriptively; that is as sufficient in itself for the study and control of actual economies. But now with the clear winner amongst economic doctrines being the competitive markets and their working, it was even more inexplicable that very few undertook a close scrutiny of the conditions under which competitive markets work; in fact economists such as Sen (1999) (p. 112) advocated, The need for critical scrutiny of the standard preconceptions and political-economic attitudes have never been stronger. And the strange and unresolved question of the stability of competitive equilibrium, the effectiveness of the Invisible Hand in Motion was never re-examined! There were several reasons why the study of stability problems was in disrepute. A strong contender for the primary position amongst them was the question of the manner in which the equation of motion of the invisible hand was written. The equation of motion was represented as a system of differential equations of the type: ṗ = Φ(Z(p)) where p, Z(p) represent prices and excess demands respectively and Φ(.) is

4 4 CIISS December 2008 some sign-preserving function. It was difficult to see whose behavior was being described by these equations and why that person or decision maker was behaving in such a fashion. The answer that this was how the Invisible Hand worked, although was found acceptable during the time of Smith did not find too many takers in recent times. Fortunately, the recent experiments of Charles Plott of Caltech and his associates tackled this question directly 3 : Experimentalists have discovered that classical models, which are based on an assumption of tatonnement, have remarkable explanatory power even when applied to the non-tatonnement, continuous, double auction markets. The classical tatonnement is precisely what we have defined just now. Consequently, given the results of the experiments conducted by Plott and his associates, we may address ourselves to the analysis of the classical tatonnement, even though we are not sure whose behavior is being analyzed; thanks to the experimental results, we know that this analysis will be a good predictor of what to expect. As we have mentioned above, the primary point of concern about the classical tatonnement lay in the paucity of results: these were difficult to come by and involved restrictions which appeared to be unwarranted. Technically speaking, what was at stake was the convergence of the solution to the system defined earlier 4. Before we start looking at the convergence, it is best to mention that the 3 See Anderson et. al.(2004) and Hirota et. al. (2005), Divergence, Closed Cycles and Convergence in Scarf Environments: Experiments in the Dynamics of General Equilibrium Systems, Caltech Working Paper. 4 I should say that the motion could be thought of as being in discrete time as well, but we shall confine ourselves to the continuous time version of the Invisible Hand: it makes good sense to keep the steps of the Invisible Hand invisible as well.

5 Mukherji: Invisible Hand 5 classical tatonnement was introduced by Walras and essentially involved two things: first that price movement was in the direction of excess demand and second that transactions occur only at equilibrium so that along the path of price adjustment, no transactions took place. We have shown elsewhere 5 that it is possible to relax both of these conditions and still consider the convergence to equilibrium. I shall confine attention to the classical tatonnment, below. It may be important to recall that during the mid-seventies, a series of papers by Sonnenschein and Debreu established that almost any set of functions which satisfied the restrictions of Walras Law and Homogeneity of degree zero could be obtained as excess demand functions for an appropriate economy. These two properties of excess demand functions were not restrictive enough. This was the first type of result which has been termed to belong to the class of Anything Goes results in the literature. But examples of instability were there from the very beginning. Thus it was not possible to infer anything meaningful from the assumptions of rational behavior by decision makers: the source of meaningful theorems that Samuelson had advocated in the Foundations, proved to be woefully inadequate in this respect. In fact, this lack of generality of results was not peculiar to the stability problem; many developments in different branches of economic theory were dependent on special forms and restrictions as well. The only difference that I can find is that in these other theoretical developments, no one ever owned up to the fact that their conclusions depended on there being a straight line demand curve or on a special form of the utility function; on the contrary, in the area of general equilibrium, which was primarily responsible for analysing 5 Mukherji (1995), (1990).

6 6 CIISS December 2008 the workings of the Invisible Hand, researchers were trying to show how special the theory was and how careful one must be to draw the correct inferences. This spirit led me to investigate whether any set of meaningful economic conditions could be identified as being necessary for the stability of equilibrium; such conditions were found and the fact that these are substantial requirements were noted 6 ; it is only relatively recently 7 that the result was rescued from its mimeographed and privately circulated status. There is another reason for general results being scarce; this has to do with the state of mathematics for the system of differential equations that we wrote down; a problem has been that economists generally conceive of their domain of discussion as one involving n variables; and to be respectable, one must not specify what n should be except to say that n 2. However, one must be careful with higher dimensions: this is because of the results of Stephen Smale, recipient of the Fields Medal 8 who showed that if the system of differential equations involved three equations, or more, then rather arbitrary behavior is possible in the sense that the attractors, the sets to which the solutions converge, could be arbitrary. This fact is often neglected by economists who treat dynamic results in lower dimensions with disdain. There is yet another problem that needs to be overcome viz. that of nonlinearity of the functions concerned; excess demand functions must necessarily be non-linear: this follows from the Walras Law. Even the straightforward forms of non-linearity in a differential equation system may lead to very complicated dynamics Consider, for example the equations defining the so 6 Mukherji (1989). 7 See, McKenzie (2002), p Smale was awarded a Fields Medal at the International Congress at Moscow in 1966.

7 Mukherji: Invisible Hand 7 Figure 1: The Rössler Attractor called Rössler Attractor: ẋ ẏ z = where σ, r, b are positive constants. y z x + σy rx bz xz The solution to the above system has a very complicated long-term behavior 9. See for example the Figure 1 where the values σ = 0.36, r = 0.4, b = 4.5 have been used. Notice that the extent of non-linearity is quite mild: the 9 Notice that there is no convergence to an equilibrium and limiting configurations are complicated in nature: such limiting configurations fall in the category of what are called strange attractors.

8 8 CIISS December 2008 non-linear term appearing only in one equation. This is a well known system and the attractor is called the Rössler Attractor. Thus on the basis of dynamical systems alone, we have another source of Anything Goes. On several counts, therefore, general results are not possible. It appears that these aspects have not been as well appreciated as they ought to have been. 2 Preliminary Considerations: The Single Market To motivate our analysis, we consider first of all, the case of two goods so that there is a single market. In a single market, the notion of a locally stable equilibrium is identified with the slope of the excess demand curve being negative at the equilibrium. But there does not seem to be any easy identification of global stability with any property even in such a simple setup. Consider the following example due to Gale (1963). There are two persons A,B with utility functions defined over commodities (x, y) as follows: U A (x, y) = min(x, 2y) and U B (x, y) = min(2x, y); their endowments are specified by w A = (1, 0), w B = (0, 1); routine computations lead to the excess demand function of the first good (x), Z(p), where p is the relative price of good x: Z(p) = p 1 (p + 2)(2p + 1) Thus the unique interior equilibrium is given by p = 1 10 ; now notice that if 10 There is an equilibrium at infinity; see for example, Figure 2.

9 Mukherji: Invisible Hand 9 Figure 2: Excess Demand - The Gale Example the adjustment on prices is given by ṗ = h(p) (1) where h(p) has the same sign as Z(p) and is continuously differentiable, then the solution to (1) say p t (p o ) is well defined for any initial point p o > 0. As Gale (1963) says, Arrow and Hurwicz have shown that for the case of two goods, one always has global stability... Nevertheless, some queer things can happen even in this case. To see the queer things referred to, consider the function 11 V (p t ) = (p t 1) 2 and notice that along the solution to the equation (1), we have V (t) > 0 for all t, if p o 1: so that the price moves further away from equilibrium and there is no tendency to approach the unique equilibrium. Consequently, even for two goods or the single market case, we need to identify conditions which will help us rule out such cases. Alternatively, a look at Figure 2 should convince us that the unique in- 11 Notice that this is just the square of the distance from equilibrium.

10 10 CIISS December 2008 Figure 3: Excess Demand under α and β terior equilibrium is unstable under the usual adjustment of prices. How do we rule out such situations? As an illustration of our analysis below, consider the following assumptions for the single market (two-good case) 12 : α: Z x (p), Z y (p) are continuous functions of p for all p > 0 and satisfies Walras Law viz., pz x (p) + Z y (p) = 0, p > 0; further lim p 0 Z x (p) > 0; similarly, lim p Z y (p) > 0. It is easy to check that condition α implies that the set of equilibria E = {p > 0 : Z x (p) = 0} is non-empty 13. Moreover, if Z x (p) is differentiable, then there must exist at least one p E such that Z x(p ) < 0: that is there must be at least one locally stable equilibrium. For global stability considerations however, we must proceed further. β: p E Z x(p) We shall write Z x (p), Z y (p) as the excess demand functions for the two goods x, y respectively with p being the relative price of good x. 13 See, for instance Mukherji (2002), p. 16.

11 Mukherji: Invisible Hand 11 Figure 4: Excess Demand for the Gale Example with a switch in endowments With conditions α, β however, we can show the following: first of all, there are only a finite number of points in E and secondly, for any p o > 0, p t (p o ) p E for some p E: this may be considered to be the counterpart of global stability when there are multiple equilibria. Thus the situation will be as in Figure 3, then. Thus one may say that overall the excess demand curve is downward sloping: since it begins from somewhere near the vertical axis and ends up below the horizontal axis. It should be noted that the boundary condition in α enforces the fact that excess demand functions have a negative slope most of the time; this is enough for global stability. We also know that the negative slope of excess demand function may be related to a Law of Demand; thus this condition, that excess demand functions be mainly downward sloping, may seem to

12 12 CIISS December 2008 be a generalization. We shall investigate below whether it will be possible to extend these conditions to cover the case when there are two or more markets. There is one other aspect of the Gale example we should note; suppose for example, we interchange the endowments i.e., A has (0, 1) while B has (1, 0); recomputing excess demand functions, we note that the situation is as in Figure 4: the unique interior equilibrium is now globally stable. One may therefore say that we had instability of the interior equilibrium because the distribution of purchasing power had not been right previously. Now, both α, β are met and excess demand curve becomes downward sloping. 3 General Global Stability Conditions In our search for natural or meaningful stability conditions, our discussion of the single market case leads us to the following conditions: If the excess demand curve is overall downward sloping, say as dictated by conditions α and β, in disequilibrium the prices will always move towards some equilibrium. Alternatively there may be some redistribution of purchasing power which might lead to a stable equilibrium. We shall see below how these can be extended to cover the two market case; recall that for larger dimensions, that is for more than two markets, stringent conditions need to be employed as indicated by Smale (1976). So to start afresh, in two dimensons, or on the plane, the basic adjustment process is given by, writing Z i (p) denotes the excess demand function for the

13 Mukherji: Invisible Hand 13 i-th good: ṗ i = φ i (Z i (p)), i = 1, 2 with p 3 1 (2) where φ i (Z i (p)).z i (p) > 0 whenever Z i (p) 0 and is 0 otherwise; thus a natural candidate for the function φ i (Z i ) = γ i Z i where γ i > 0 is a constant and is interpreted as the speed of adjustment in the i-th market. Also the choice of p 3 being fixed as unity reflects that the prices of goods in the first two markets are measured relative to the third good, the numeraire good. While considering the motion on the plane, a very strong result in differential equations theory becomes available: the Poincaré-Bendixson Theorem 14 which states that if the solution to (2) is bounded, i.e., remains within a bounded region, then it either approaches an equilibrium or cycles around an equilibrium. Basically then, if cycles are eliminated by some argument and if solutions are bounded, convergence to equilibrium follows. It is a pity that this result is valid only for motion on the plane or for our case, the situation with two markets. We mention here the results obtained in Mukherji (2007), which enable one to carry out the task described in the last paragraph. The condition 15 envisaged is the following: There is a differentiable function θ(p) : R 2 ++ R and such that div(θ(p)φ 1 (p), θ(p)φ 2 (p)) is not identically zero on R 2 nor does it change sign on R 2 where div(θ(p)φ 1 (p), θ(p)φ 2 (p)) = θ(p)φ 1(p) p 1 14 See, for example Hirsch and Smale (1974). 15 This condition is known as Dulac s Condition. + θ(p)φ 2(p) p 2.

14 14 CIISS December 2008 If one can find such a function θ, there can be no cycles. At an equilibrium therefore: θ(p)φ 1 (p) p 1 + θ(p)φ 2(p) p 2 = θ(p)[ φ 1(p) p 1 + φ 2(p) p 2 ] which has the sign of θ(p)(z 11 (p) + Z 22 (p)) this follows because of the sign restriction on the function φ i, the partial derivatives of the functions φ i and Z i should be identical at equilibrium. Recall too at this juncture that Z ii (p) is the same thing as the slope of the excess demand curve in the case of the single market and under conditions such as α and β there would be some equilibrium; moreover, there would be an equilibrium where the slope is negative. Further if in fact there is a function θ which satisfies the above, then so does θ and we may hence take θ to be positive, if indeed there is such a function. Consequently, the above condition would reduce to requiring that div(θ(p)φ 1 (p), θ(p)φ 2 (p)) < 0, everywhere: which is the generalization of the requirement that excess demand curve be downward sloping. Remember that we still need to keep the solution in some bounded region: this too be can be ensured by requiring that φ i (Z i (p)) Z i (p) for example and all requirements are met. The interested reader may refer to Mukherji (2007) for proofs. We note that this same paper contains an example of such a θ function which is used to prove stability in the case of Scarf s example, when endowments are perturbed. However notice that we have not made any specific assumptions about the nature of the economy which gives rise to such excess demand functions. What we shall do next is to make the model specific and as we hope to show, the nature of requirements become much simpler.

15 Mukherji: Invisible Hand 15 4 A Three Good Model with Production Let there be three goods: a consumption good denoted by x, labour l used to make the consumption good x via a production function x = f(l) where, to begin with, we shall assume that f (l) > 0, f (l) < 0 and a numeraire good y which is available in fixed quantity with household sector. There are two types of agents: households and firms; households as we have said own the numeraire and labor and they use these to buy the consumption good and numeraire; the firms produce the consumption good and spend their entire earnings on the numeraire. Thus firms maximize profits p.x wl subject to x = f(l) so that demand for labour is given by l d = f 1 (w/p), supply by x s = f(l d ) and profits by π = px s wl d. We may assume that the above is aggregate firm behavior with there being many firms, each having their own technology and similar looking labour demand and supply and profit functions. The households maximize: U h (x, y), an increasing strictly quasi-concave utility subject to the budget constraint p.x + y w.l h + y h o ; this leads to demand for the consumption good x h d(p, w) and hence the aggregate demand x h d (p, w) = x d (p, w) so that the excess demand in the consumption good market is given by Z x = x d x s ; in the labour market, the total supply of labour is l h = l is fixed so that excess demand in the labour market is given by Z l = l d l. Notice that the demand for the numeraire comes from households y h (p, w) and from firms π so that excess demand for the numeraire is given by Z y = h y h (p, w) + π h y h o. As a check observe that p.x h d(p, w)+y h (p, w) = w.l h +y h o h and that π = p.x s w.l d so that summing over all households and firms we have p. h x h d(p, w) + h y h (p, w) + π = w.( h l h l d ) + p.x s + h y h o p.z x + Z y + w.z l = 0 (p, w) > (0, 0). which

16 16 CIISS December 2008 of course is Walras Law. It should also be noted that the excess demand functions are homogenous of degree zero in all prices. We shall also require some restriction on the nature of these excess demand functions when (p, w) becomes very large and when they become very small; notice that this may be considered to be the counterpart of the condition (α) in the present context. If one may recall that apart from Walras Law and continuity of excess demand functions, and homogeneity of degree zero in all prices, their properties when relative prices approach zero, had to be specified. To do so now, it would be more convenient to use the notation (p 1, p 2, p 3 ) for the price system, where p 1 stands for p, p 2 for the wage rate w and p 3 for the price of the numeraire y; we shall choose the last to be unity in later considerations. (α ) Let P s = (p s 1, p s 2, p s 3) be some sequence such that for sp s i = 1 for some i and 16 P s + ; then Z i (P s ) +. Remark 1 Among other things, the above ensures that when some price (say p 1 ) goes to zero (and at least one does not), once can construct such a system P by dividing through out by p 1 ; the resultant P will become larger and larger, and excess demand for good 1 should then go to infinity. Reverting to our notation (p, w), (p 3 = 1) we may also conclude by virtue of the above that there is K 1 such that p > K 1 implies Z y (p, w) > 0; if no such K 1 exists, then we have a sequence (p s, w s ) such that p s + and Z y (p s, w s ) 0; consider the system P s = (p s, w s, 1) and this will directly contradict the assumption. We use the same argument to establish that there is some K 2 such that w > K 2 implies that Z y (p, w) > 0 and let K = max K i. We shall return to these considerations at a later point of time. 16 P = (p p p 2 3), if P = (p 1, p 2, p 3 )

17 Mukherji: Invisible Hand 17 Notice that if there is an equilibrium then Z l = 0 w/p = δ (3) since a downward sloping demand curve of labour reaches the level of full employment at a unique real wage. Consequently, the levels of w, p are then determined by the equilibrium in the goods market Z x = 0; further, in disequilibrium, we have: ṗ = γ 1 Z x (w, p) and ẇ = γ 2 Z l (w, p) (4) where γ i > 0, i = 1, 2. Notice that: Z x p Z xp = x dp x s p < 0; Z l w Z lw < 0; and Z lp > 0 assuming that consumption good is normal. Notice too that Z xw = x dw x s w where the second term is positive (including the - sign) while the first is positive. 4.1 Phase Diagrams We return to the general form of price adjustment equations contained in (2). We consider then, in the w p plane, the various combinations of w, p which would equilibrate the labour market i.e., Z l = 0 or w/p = δ. Notice too the regions where there is excess demand for labour (marked by +) and the region where there is excess supply of labour (marked with a -). A similar diagram for the goods market may also be constructed.

18 18 CIISS December 2008 w Figure 5: Excess Demand for Labour ø d + p Figure 6: Excess Demand for Goods + w Z x (w,p) = 0 ø p

19 Mukherji: Invisible Hand 19 When we put the two diagrams together, that is combine the Z l = 0 with Z x = 0 there are two possibilities, assuming that an equilibrium exits. Since the Z x = 0 is upward rising there may be two cases: one where at the intersection, the line Z x = 0 is steeper and one where the Z l = 0 line is steeper. We shall consider each of these cases below. However, first we need to establish that the equilibrium, if it exists will be unique. In the circumstances that equilibrium is non-unique there would be at least two points of intersection between the curves Z x = 0 and Z l = 0; or there would be two distinct configurations w 1, p 1 and w 2, p 2 such that w i /p i = δ and x d (w i, p i, 1) = f( l), the full employment output. Suppose then that p 1 > p 2 and consider the budget constraint: x + y p i = w i p i l h + yho p i = δl h + yho p i Thus we would expect that x h d(w 1, p 1, 1) < x h d(w 2, p 2, 1) for each h and hence it is not possible to have x d (w i, p i, 1) = f( l) and hence multiple equilibrium is not possible. Consider then putting Figure 5 together with Figure 6; recall that we have two possibilities. Consider first Figure 7, where the Z x = 0 curve is steeper than the Z l = 0 line at intersection. The arrows indicate the direction of price, wage movements in each sector of the diagram. Price movements are to read by the arrow parallel to the p axis and the wage movement is to be read by arrows parallel to the w axis. Notice that the region in between the two curves are such that price movement from these regions can never lead to prices outside the regions: consequently wage price revisions are such that equilibrium is attained. These regions are called absorbing states. From the remaining two regions, notice that wage price revisions lead to the absorbing states. Thus Equilibrium I depicts a globally stable equilibrium. It may

20 20 CIISS December 2008 Figure 7: Equilibrium I w Z x (w,p) =0 d p however be the case that we have the following Figure 8: Equilibrium II. Here the Z l = 0 line is steeper than the Z x = 0 line at equilibrium; the absorbing states disappear; in the regions trapped between the two curves, wage price movements are away from the equilibrium; however the wage price movements in the other two regions remain the same and wage price movement initiated from these regions take the economy to the region in between the two curves and hence away from equilibrium. However there is just one solution, from some initial points which lead to equilibrium. Equilibrium II depicts thus a saddle-point equilibrium; if initial conditions are right then equilibrium will be reached other wise solutions become unbounded. As we shall see below (α ) rules out such cases. To sum up, thus, without imposing additional restrictions of any kind, we have a globally stable equilibrium (Equilibrium I). In the case of

21 Mukherji: Invisible Hand 21 Figure 8: Equilibrium II w d Z x (w,p) =0 p Equilibrium II, the area above the δ ray and below the Z x = 0 curve contains the possibility of Z x < 0 when p 0 with w positive: but this is ruled out by α Firms are Privately owned Recall that in the above, the households did not receive income from profits and profit income was exclusively spent on the numeraire; while this may be of some interest, we need to consider the case when firms are privately owned so that each household h receives a share λ h π of profit income with λ h 0 h λ h = 1. With this modification, notice first of all that the budget constraint of each household changes to p.x + y w.l h + yo h + λ h π(p, w). Notice too this modification does not affect the claim that equilibrium is unique. This follows since along the δ-ray π(p, w) = pf( l) w l so that

22 22 CIISS December 2008 Figure 9: Excess Demands in the Goods Market w δ E p along this ray π/p is a constant. So the earlier argument for uniqueness may be repeated. We have then the following situation: Recall that along the δ-ray, π/p is constant; at E (p.w ), h x h (p.w ) = f( l); above E, p > p, h x h (δp, p) < f( l) so that Z x < 0 and similarly, below E, Z x > 0. This indicates the region where Z x = 0 must lie. But as should be clear this does not clinch matters; since we do not know whether the curve slopes up or down. It is possible to have Z xp > 0 or < 0; similarly for Z xw two possibilities arise; and thus there are four logically possible situations. This ambiguity stems from the fact that variation in p and w causes profit income to change and hence the phenomenon like the net sellers income effect comes into play. Clearly then the assumption α is not adequate. We need to impose some restriction in addition to deduce global stability. As we shall see, a version of overall downward sloping nature of excess demands will be helpful.

23 Mukherji: Invisible Hand 23 Remark 2 First we ensure that α is sufficient to ensure that the solution is bounded. As we see from Figure 9, irrespective of the position of the Z x = 0 line, along any solution to (4), neither p nor w can become zero. We proceed to demonstrate, following Mukherji (2007) that any solution to (4) must lie in a bounded region. Consider the arc drawn in Figure 9 to represent the equation p 2 /γ 1 + w 2 /γ 2 = M where M = 1 + K 2 /γ where K is as defined in Remark 1 and γ = min γ i ; then it must be the case that either p > K or w > K along the arc for otherwise p 2 /γ 1 + w 2 /γ 2 < M. Consider the function V (p, w) = p 2 /γ 1 + w 2 /γ 2. Thus along the arc, we must have V = 2[pZ x (p, w) + w.z l (p, w)] = 2Z y (p, w) < 0 the last inequality follows since along the arc, we have just shown that either p > K or w > K and then the conclusions noted in Remark 1 apply. Thus the solution must lie below the arc and hence in a bounded region. Consider next the following version of overall downward sloping excess demand: β : γ 1 Z xp (p, w) + γ 2 Z lw (p, w) is of the same sign on R 2 ++ and is not identically zero on R Remark 3 We do not restrict the sign of the expression in any manner; it could be non-positive on the entire region or non-negative over the entire region, and the only restrictions are that it is not identically zero nor does it change sign; as we shall soon see, this will imply that the sign must be negative at equilibrium. Which is why we refer to this as being a requirement that generally, excess demand curves are downward sloping. Now notice that we can immediately apply Dulac s criterion to conclude that there can be no cyclical orbit: choose θ = 1. The Poincaré-Bendixson Theorem now ensures that the only possibility is convergence to an equilibrium. This basically ensures that term in β must be negative at equilibrium:

24 24 CIISS December 2008 since otherwise we shall not be able to converge to it. This basically means that Z xp cannot be too positive, since Z lw < 0. Thus, as in the single market case, we have obtained a stability condition which may be interpreted in terms of slopes of excess demand functions. Specific examples may be constructed to show that even a weaker condition is enough by allowing choice of θ which is more complicated. If for example, Z xp < 0, as in the previous section notice that β is automatically satisfied. 4.2 The Case of Constant Returns to Scale In the last section, we assumed as is standard in such situations that the production function f(l) is strictly concave. What happens if the production occurs under constant returns to scale? To examine this variation, which is what Walras had originally assumed, we proceed below. In fact, because of constant returns to scale, profits are zero so that the scale of production is determined by the level of demand; so the notion of excess demand (demand - supply) is not really defined. Consequently, the earlier set-up describing the working of the Invisible Hand becomes inadmissible. Walras was naturally aware of this aspect and hence assumed that adjustment of prices took place in the factor markets. We shall show how this happens. Let f(l) = τl where τ > 0 is a constant; consequently profits are given by the expression p.τl wl = (p.τ w)l; thus w/p = τ to allow perfect competition to prevail with positive level of outputs; p = w/τ. There is no profit maximizing response and firms try to meet whatever is demanded. To determine demand, consider the budget constraint of each household, recalling that profits are zero: x + y/p = τl h ; or w.x + τy = w.τl h

25 Mukherji: Invisible Hand 25 specifying any w determines x h d(w) and hence the market demand x d (w) = h x h d(w) and hence the demand for labour is given by l d = x d (w)/τ; and excess demand in the labour market is now well defined by Z l = l d l; from the budget constraint, we can see, that assuming that the consumption good is normal, we have x h d (w) < 0 provided the income effect does not dominate; it follows that then Z l(w) < 0 and since we are back in a single market world, the equilibrium is globally stable. More importantly the condition that x h d (w) < 0 is the same as the condition for a single market equilibrium i.e., the net sellers income effect does not dominate. Thus consideration of constant returns to scale, makes matters simpler and we can conclude that the equilibrium is globally stable merely on the basis of say, the downward sloping nature of the excess demand curve for labour. Even the weaker α will also yield global stability. 5 Concluding Comments We have shown how the requirement of excess demand curve being on the whole downward sloping, in a single market (conditions α and β ) generalize to the case of two markets. We also show how imposing some structure like production eases conditions considerably 17 and we do not need to look around for some complicated function while searching for θ. At least in simple models, when profit income is spent entirely on the numeraire, a condition like α is sufficient to ensure that the Invisible Hand does work. But notice some such condition is always necessary. We have not covered the case of multiple (more than two) markets; this 17 This conclusion is in line with a study undertaken many years ago, Mukherji (1974).

26 26 CIISS December 2008 Figure 10: No Equilibrium I w d Z x (w,p) =0 p is because we know that this would require stringent restrictions due to the general conclusions by Smale (1976). However, we may report that in even in such cases, some progress has been achieved in arriving at restrictions on the distribution of purchasing power to ensure stability in Mukherji (2008). Two points need to be emphasized: one, a specific point relating to the production model and the other, a general point about any exercise of the type attempted here. First the specific item. It was taken for granted that an equilibrium exists in the production model of the last section. What happens if there is no equilibrium? Consider the Figure 10. Here the Z x = 0 lies below the Z l = 0; the arrows would indicate that wage price movements would lead to the middle region and which incidentally cannot be left. When the Z x = 0 lies above, (Figure 11) similarly one may show that that wage price movements go to zero or that the relative price of the numeraire becomes infinite.

27 Mukherji: Invisible Hand 27 Figure 11: No Equilibrium II w Z x (w,p) =0 d p These are the two situations when equilibrium does not exist. Basically in the first case, there is no reason for any one to hold the numeraire whereas in the second case there is no reason to hold any of the other goods and the model breaks down. In fact what we have is essentially a failure of condition α ; in the first case, as we have remarked above, dynamics is confined to the middle portion with both w, p increasing without bounds; thus even though the price of the numeraire (relative to any of the other goods) goes to zero, it has an excess supply. In the second case too, the middle region is attracting and here though p is approaching zero, there is excess supply. Notice then by insisting on a single condition α, we may rule out these cases. What happens if we try to identify the numeraire commodity with money? It is a fault of the model itself, if for example, one tries to identify the numeraire with money unless adequate safeguards (viz., condition α ) have

28 28 CIISS December 2008 been put in place to ensure that people wish to hold money as well as the other goods 18. Identifying the numeraire with money may require situating the model in a many period framework or alternatively ensuring that condition α is met. Secondly the general point about what does the process itself mean? Is it historical time that the Invisible Hand was supposed to function in? That is the time-derivatives that we see in say in equations (2), for example are derivatives with respect to historical time? This cannot be naturally; what then is t? It would be more sensible to consider this variable to be the number of bids by the unsuccessful agents when they realize that markets do not clear. In fact while solving such differential equations on the computer, convergence takes place quickly that is within a small number of bids, the difference between prices and equilibrium are negligible in convergent situations. Finally the experiments of Plott and associates should remove any doubt that one may have in analyzing such processes since the experiments have demonstrated beyond reasonable doubt, that the tatonnment is a very good predictor of how prices may move. Why this is so is not clear as yet. We hope an explanation will eventually emerge. 18 See, for example, the criticisms of Patnaik (2008) of mainstream economics. I would consider it more appropriate to direct the criticism towards inappropriate modeling.

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