Chaotic Price Dynamics, Increasing Returns & the Phillips Curve. by Graciela Chichilnisky, Columbia University Geoffrey Heal, Columbia Business School

Transcription

1 Chaotic Price Dynamics, Increasing Returns & the Phillips Curve by Graciela Chichilnisky, Columbia University Geoffrey Heal, Columbia Business School Yun Lin, Columbia University March 1992, Revised February 1993 Discussion Paper Series No. 646

2 Chaotic Price Dynamics, Increasing Returns & the Phillips Curve Graciela Chichilnisky Department of Economics Columbia University Geoffrey Heal Columbia Business School Yun Lin Department of Economics Columbia University March 1992 Revised February 1993 We are grateful to Jess Banhabib and Giulio Pianigiani for helpful comments. An earlier version of this paper was circulated as "Production Technologies and the Phillips Curve", Collaborative Paper CP-83-42, IIASA, Chichilnisky and Heal acknowledge support from the NSF.

3 I. Introduction We study the dynamics of price adjustment in economies with increasing returns to scale. The dynamics is given by a discrete Walrasian adjustment mechanism, so that prices adjust proportionally to excess demand in small discrete steps. It is shown in Heal (1982, 1991) and Chichilnisky and Heal (1987) that with non-convex production sets such economies may have a "stable disequilibrium price", i.e., a price vector that is locally stable under a continuous Walrasian adjustment process even though it is not a market clearing price. In this paper we extend the analysis to the global dynamics of a discrete Walrasian system. We explore the implications for wage and employment dynamics in economies with increasing returns. We establish the existence of a globally attracting set of prices within which the motion of the system is chaotic (Theorem 1). Long-run statistical properties of the system's behavior in this set are described by an ergodic measure. The attracting set contains a "stable disequilibrium price". Walrasian price dynamics drive the system into this region, and then chaotic motion takes over. This result is true for any specification of the economy where the discrete adjustment has step size greater than a specified minimum. In addition, for an open class of preferences and technologies showing complementarity between consumption and leisure, this is also true for any step size in the adjustment process. Our second result (Theorem 2) shows that when substitution in consumption is extensive, then the price dynamics converge to a period-two cycle, with the stable disequilibrium price located between the two limiting points. The "stable disequilibrium price", is a price vector at which the excess demand function of the economy is discontinuous. It is in fact the price vector that would clear markets and give a competitive equilibrium in the convex economy defined by replacing our nonconvex production sets by their convex hulls. It is one at which a firm's optimal choice is in the non-convex region of its production set in the original economy and responds in a "bang-bang" way to a small change in the prices (Heal, 1991). This is illustrated in figure 1. A distinctive technical feature of our analysis, is that the state transition function that determines the values of variables in period t as a function of their values in period (t-1) is a discontinuous map. The technical 1

4 argument builds in part on recent results due to Keener (1980) on chaotic behavior in piecewise continuous difference equations. The economic implications of this result are as follows. Within the attracting set of prices, there is always either excess supply or excess demand, which is accompanied by price changes. If we take the input to be labor and the price to be the real wage, then the price dynamics generate a time series of real wage changes and levels of unemployment: we show that this time series will have the statistical properties of a Phillips curve (Phillips( 1958), Sargan (1980)). Theorem 1 implies that wage changes and unemployment always have the opposite sign, so that there is a negative association between wage changes (inflation) and unemployment. Furthermore, this relationship is persistent in the sense that wage changes do not go to zero over time. In this framework, it is clear that a persistent negative relationship between wage changes and unemployment does not represent a locus of alternative equilibrium configurations. These are not alternative configurations between which a policy-maker can choose. They represent rather a stable limiting distribution of excess demand-price change pairs. The "Phillips curve" relationship therefore has no policy implication about a tradeoff between inflation and unemployment in this context: it is a byproduct of price dynamics in a non-convex economy. This derivation of the statistical relationship portrayed by the Phillips curve sheds new light on the possible implications of this relationship for the tradeoff between unemployment and inflation. We are able to predict from the parameters of the model whether there will be on average excess demand or excess supply in the very long run, as the system evolves within the attracting set and displays "Phillips curve-like" behavior. In a statistical sense, the economy will display chronic excess demand for or supply of labor, depending on the nature of technologies and preferences. The methodology of "chaotic systems" in economics is clearly reviewed in Day amd Pianigiani (1991). For an analysis of the methodological and conceptual issues associated with this type of system, the reader is refered to Baumol and Benhabib (1989).

5 n. The Economy We consider a simple general equilibrium model having a single input, labor, and a single output, a consumption good. There is one firm and one consumer. The firm's technology is given by the following production function, which defines a non-convex production possibility set: y = 0 if A(L-F) a otherwise where 0 < a < 1. There is a fixed cost introduced by a minimum input requirement of F (see figure 1): once this is met, output shows diminishing returns. F is assumed to be less than 1, which is the total labor supply. This production function will show the conventional U-shaped average cost curve. The price of the output is normalized to be one, and w is the wage rate. Hence profits TT are given by K = A(L-F)*-wL, LzF. ( 2 ) The first order conditions for profit maximization define the demand for labor, equation (3), which is discontinuous, as shown in figure 1. = { aa ' ^ (3) 0 otherwise The wage rate at which the demand for labor is discontinuous is found by substituting the labor demand function for w<w* into (2) and equating profit x to zero, which gives pi Labor demand at w* equals ^, which is independent of the scale parameter A. l-<x

6 Preferences are given by the CES utility function: where maximum labor supply is scaled to be one and Y is the worker's consumption level. The consumer maximizes utility, which gives the labor supply function If j8 is a large negative number, so that consumption and leisure are consumed in approximately fixed proportions, the labor supply curve is backward-bending: for # near unity, giving a high level of substitutability between consumption and labor, the supply curve has a positive slope. Hence the excess demand function for labor is 1 ( 2) L ) «- i i + j r i aa _ _ l + w H if w>w* (5) III. Dynamic Behavior The Walrasian price dynamics in discrete time is governed by the following equation: w, +l = w t +kz(w), (6) where X>0 is given. Price adjusts proportional to the excess demand of the current period. For convenience, define the map:

7 w, +1 = 8(w f ; A,a,$) (7) Proposition 1. There exists F^ = F'(A,ci >^)<:l-a > such that if F>F*, then no stable Walrasian equilibrium exists. Proof. From equation (3) we know that labor demand curve has two segments. The segment for w<w* is bounded below by L*(H>*)= p 1-a and shifts to the right along the L-axis as F increases. Choose F* such that the labor demand curve for w < w* and the labor supply curve have only one contact point and the former lies on the right hand side of the latter. Since labor supply is bounded F* above by 1, we have < 1 or F* < 1-a. It is therefore clear that labor demand curve for w < w* 1-a will not cross the supply curve if F>F\ For w> w*, labor demand is constant and equals to zero. Labor supply converges to zero only if w goes to infinity and /3<0. The assumption F* < F < 1-a will be maintained throughout our discussion. For F < F*, there always exists a stable Walrasian equilibrium. Dynamic adjustment process for that case will not be discussed here. If F>l-a, then the labor demand is either equal to zero or greater than the maximum labor supply. We split our discussion into two parts. In the first part (Theorem 1), we look at the case where preferences display complementarity between leisure and consumption (/3<0) and find that price dynamics demonstrate chaotic behavior which persists as X decreases. In the second part (Theorem 2), we look at the case where consumption and leisure are substitutes (0 < 0 < 1). Chaotic behavior may also be found: however, it disappears as X becomes smaller than some critical value X*, and is replaced by periodic behavior. Since chaotic behavior will not happen in a convex economy (i.e., F=0) for X small (this is shown by the arguments in Day and Pianigiani(1991), section 2), we may conclude that the fixed cost is responsible for the chaotic behavior that we found in our first case

8 We now establish the main result of the paper, which shows that for preferences displaying complementarity between leisure and consumption, the discrete Walrasian adjustment process (7) leads to chaotic behavior with an associated ergodic measure for any value of the adjustment parameter X. Theorem 1. Assume #<#, F>F*, any X, and a (0,1). Then for a sufficiently close to one: (i) There exist w, w and T>0, such that for any initial value w 0 and all t>t, w t Efw, w], (ii) Within the interval fw, w] the behavior of (7) is chaotic in the sense that there exists a unique measure \i. on fw, w] that is absolutely continuous with respect to the Lebesgue measure with the following property: for almost any initial conditions and any measurable subset S of fw, wj, n(s) is the average fraction of the total number of periods that a trajectory spends in S. Proof. A crucial step is to establish that w l+l =0(Wt) or w t+1 =^(Wt), for some integer i, is an expansive map (Day and Pianigiani (1991) page 45, Theorem 3). A map is expansive if the absolute value of its derivative is bounded above unity Lebesgue almost everywhere. Since the proof for /3 =0 is slightly different from that of j3e(-oo,0), we look at the two cases separately. case 1. 8 (-oo. 0). (see figure 2a) We show that (7) is an expansive map. For w > w\ ^ 1 dw t w P" 1 (8) dw x dw t is clearly greater than 1. For w < w,

9 1_ 2^ _J_ rfw, a-1 aa ' p-1 _1_ With some manipulation we get -a We need to show that the right-hand side is less than -1 or to show [ >>) [LM] 2 (Uw)-F)] < -2 w p-1 p-1 1- A sufficient condition would be We used several inequalities -fl_fl-i-l. > 2d (10) 1-a 1-a 4 p-1 X w < A, LJLw)-(L s (wyf < j, L^w)-F > F, 1 -a which can be derived easily. Inequality (10) will always hold if a is sufficiently close to unity. An application of Day and Pianigiani's theorem 3 now proves the theorem for /3E(-oo,o). Case 2. 3=0. (seefigure2b) In this case, labor supply equals 0.5 for all w and dw. dw t =1 for w>w. Theorem 3 of Day and Pianigiani does not apply. Instead, Corollaries 2 and 3 to their theorem 4 (Day and Pianigiani (1991), page 47) will be used. Basically, we need to find an integer i, such that the map w f+,=6'(w,; 4,a,p) is expansive.

10 Define w=w*-0.5x, and w=w+\z(w). Notice for any w t located in the range [w*,iv], the price adjustment will follow w t+1 =w t -0.5X, and after some finite number of periods, it will drop to the range [w,w*]. It is easy to see that for all initial prices located in [w*,iv], vv is the initial value from which the system will take the longest number of periods, say k, to reach a point lower than w*. Consider the map Differentiate with respect to w t and by the chain rule, we have t t+k-l t+k-2 dw t For any initial w t 6[w,vv], and its generated sequence {w t, w t+1,, w t+k }, there exists at least one w t+i which belongs to [w, w*], where the derivative is less than -1 under our assumption. So at least one of the terms on the right hand side of equation (11) is less than -1, all the other terms are either one, if w [w\w], or less than -1, if we[w,w*]. Their product in absolute value must be greater than 1. So the map w f+ *=8*(>v t ) is expansive. Theorem 1 now follows from corollaries 2 and 3 of Day and Pianigiani, page 47. This completes the proof of Theorem 1. This result has an important implication for the relationship between wage changes and unemployment. Consider a scatter diagram of wage changes against unemployment (the negative of excess demand). Theorem 1 implies that wage changes and unemployment always have the opposite sign, so that there is a negative association between wage changes (inflation) and unemployment. Furthermore, this relationship is persistent in the sense that wage changes do not go to zero over time. Formally, Corollary 1. Let {A w t, Z(wJ}, /=/,..,» be a sequence of wage change and excess demand pairs on a trajectory of (6). Then sign(awj=signz(wjfor all t. Furthermore, the sequence {\w t \} does not converge. 8

11 Proof. The proof of this result is immediate from Theorem 1. Corollary 2. The long run average wage is always greater than the stable disequilibrium price, w\ under the conditions of theorem 1. Proof. Denote [w,w*) region L and [w*,vv] region H (see figure 3). Consider first region L. Z'(w) is less than -1 for a close to unity. So for any w s EL, w s+1 > w*+z(w*), which implies that w would not spend more than one period in region L. Now suppose w t EH in period t. Since excess demand is negative, w t+1 will be lower than w t, after some finite periods, say k, it must drop down to region L. Price adjustment thus displays a cyclical pattern. During each cycle, the price will stay in region L only once and in region H at least once. If we can show the average price for each cycle is greater than w*, then it must also be true in the very long run. Consider one typical cycle, we see from figure (2a) and (2b) that in any period when the price drops to the L region, then in the next period it gains even more (this can be shown mathematically). Therefore the long run average price will be higher than w*. Under our parameter specifications, we would observe periodical fluctuations of the real wage (and therefore of unemployment). In the very long run, the average value of w along any time series of values of w is above w\ Hence there will on average be an excess supply of labor, i.e., unemployment. This completes the proof of Corollary 2. Finally we characterize the behavior of equation (7) describing the dynamics of wages for the case when 0<j3< 1, i.e., consumption and leisure are substitutes. Recall that Theorem 1 addressed the case of j8<0, and established that for any adjustment parameter X chaotic behavior is possible. With 0<#< 1 chaotic behavior is still possible, but only for large X. In this case, for small enough values of the adjustment parameters X, the system has a two period orbit which is both structurally and dynamically stable. Formally,

12 Theorem 2. For (3E(0,l) and F>F*(A,a,$), there exists a maximum adjustment parameter \ m = \*(A,a,f3), such that for adjustment parameters less than this maximum, i.e., 0<\<\*, there is a unique, globally attracting period two solution {w lt wj to the price adjustment process w r+/ =S(w p ' A, a, (3) such that w } < w* < w 2, where w* is the stable disequilibrium price. Furthermore, the solution is structurally stable. Proof. This theorem follows from discussions in section 3 of Keener (1980). We need, however, to reformulate our problem so that his results can be used. As in the case we can define a trapping region [w, w]. w will depend on X. Let \ x be the biggest value, such that w>0 for all X<X,. Now look at equation (8) and (9). For c*e(0,l) and 06(0,1), and for both equations, the derivative dw. dw t is uniformly bounded above by one. Since the derivative decreases monotonically with X for each case, there exists a maximum X* which is less or equal to X b such that for X<X*, the right-hand sides of equation (8) and (9) are both uniformly bounded below by zero for their corresponding domains of w. The following lemma follows from the definitions of F* and X*. Lemma 1. For any structural parameters A, a and 0, there exists F* and X*, such that iff> F* and 0<\<\\ then (i) 6(w; A,a,&) maps from [w, wj to itself, dw (ii) 0< ~-< i for w [w, w*) and wg (w\ w], dw t (Hi) d(w) > w* and 6(w) < vv\ Here (iii) is a direct result of (ii). The following two lemmas are taken from Keener's paper. Lemma 2. Suppose (iii) of lemma 1 is true, then 6(w) has a period two solution. (Keener, Lemma 3.2) 10

13 Lemma 3. With the assumptions of lemma 1, the period two solution in the above lemma is unique, globally attracting and structurally stable, (combination of Keener's lemma 3.1, corollaries 3.16 and 3.17) The above discussion together with lemmas 1-3 proves Theorem 2. It is of course still possible to prove Corollary 1 Theorem 1, which states that the relationship between wage changes and unemployment displays the characteristics of a Phillips curve. Corollary 2, stating that on average there will be unemployment, can also be derived with some specific assumptions on the structural parameters. IV. Concluding Remarks It is shown in Heal (1982) and Chichilnisky and Heal (1987) that an non-convex economy may have a price which is locally stable under a continuous Walrasian adjustment process but is not a marketclearing price. We have shown here that with a discrete Walrasian adjustment process the global dynamics are such that the price will be attracted to an attracting set which contains the stable disequilibrium price. Within the attracting set, the price dynamics may have chaotic or periodic behavior. 11

14 Output Demand Price Supply Demand Input Quantity Figure 1: a non-convex production set giving rise to supply and demand curves that do not cross but give a stable disequilibrium price p*. At prices above this level, supply exceeds demand, forcing prices down, and vice versa. There is a region around p* within which movement may be chaotic.

15 Figure 2 (a) (b)

16 H Figure 3

17 References Baumol, W. J. and J. Benhabib (1989). "Chaos: Significance, Mechanism And Economic Applications", Journal of Economic Perspectives, 3, Chichilnisky, G and G. M. Heal (1987). The Evolving International Economy. Cambridge University Press. Day, R. H. and G. Pianigiani (1991). "Statistical Dynamics and Economics". Journal of Economic Behavior and Organization, 16, Heal, G. M. (1982, 1991). "Stable Disequilibrium Prices". Economic Letters, Working Paper, Columbia Business School, Initially circulated 1982, revised Keener, P. James (1980). "Chaotic behavior in piecewise continuous difference equations". Transactions of the American Mathematical Society, Vol. 261, Number 2, October, Phillips, A. W. (1958). "The relationship between Unemployment and the Rate of Change of Money Wages in the United Kingdom, " Economica, XXV, November, Sargan, J. D. (1980). "A Model of Wage Price Inflation". Review of Economic Studies, XLVII,