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Ergodic fluctuations in deterministic economic models
Journal
of Economic
ERGODIC
Behavior
and Organization
FLUCTUATIONS
Richard University
8 (1987) 339-361.
North-Holland
IN DETERMINISTIC MODELS*
H. DAY
and
of SouthernCaliJornia,
Wayne
ECONOMIC
SHAFER
Los Angeles, CA 90089, USA
Received July 1986, final version
received
February
1987
Nonperiodic business fluctuations are shown to exist with positive measure generically and to behave like stationary stochastic processes in the standard dynamic macro model. The implication is that so called ‘chaotic’ fluctuations are probable for random initial conditions in a large class of recursive economies.
1. Introduction Nonperiodic trajectories have been investigated for dynamic models in a variety of scientific fields. But the mere existence of ‘mathematical chaos’ does not tell how important irregular fluctuations are likely to be - even if the equations in question were accurate analogs of some real world processes. This is because in nonlinear dynamics long run behavior depends on initial conditions: although chaotic trajectories may exist for an unaccountable number of initial conditions this ‘scrambled’ set may have zero measure. If this were true then ‘most’ randomly chosen initial conditions would lead to periodic cycles.’ But suppose nonperiodic behavior did result for initial conditions from a set with positive measure. When trajectories have no finite representation in terms of limit cycles, but wander indefinitely among never repeated values, can they be characterized in some finite manner, and is that characterization structurally stable? An example is provided by the standard macro-model with Robertsonian lag developed by Mezler (1940) Modigliani (1944) and Samuelson (1948). Elsewhere we showed that this model exhibited chaotic fluctuations when *Work by the first author on the present study was performed at the Netherlands Institute for Advanced Study while that of the second was carried out at the University of Southern California. The computer graphics were prepared by T.Y. Lin. ‘This issue has been called the ‘observability’ or the ‘measurability’ question by Melese and Transue (1985). Benhabib and Day (1982) suggested looking at the distribution of a trajectory in their initial studies but did not present an analysis of the problem. In the meantime, however, Day and Shafer (1985) and Grandmont (1985) have presented studies in which conditions for measurable nonperiodic behavior are given. See also Day and Kim (1987) 0167-2681/87/$3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
340
R.H. Day and W Shafer, Ergodicjluctuations
in economic models
induced investment was strong enough [Day and Shafer (1985)]. Here we provide conditions under which irregular fluctuations have positive measure and behave like stationary stochastic processes.’ The general model is described in section 2. It can be reduced to a piecewise monotonic map on the unit interval which has one or two turning points with two or three monotonic pieces respectively. When the pieces are linear and only one turning point occurs especially sharp results can be obtained. Indeed, in this special case it is possible to give a complete answer to the questions of interest in this paper. Although this is not a very complicated business cycle model and is perhaps the simplest conceivable nonlinear map the possible patterns of dynamic behavior are rich: stable cycles of all orders, chaos and ergodic, stationary stochastic fluctuations with a unique probability distribution. These results are illustrated with numerical examples in section 3. In sections 4 and 5 we review the theory of dynamical systems which can be applied to general recursive economies with a single state variable. In section 5 that theory is used to show that the simulations of section 3 are not misleading, but reflect exact theoretical results. We then briefly explore the more general cases that arise in the business cycle model under consideration in section 6. In order to see what we are headed for the reader may be reminded of a few basic concepts. Suppose B is a continuous map on the unit interval. A point X= e(X) is a stationary state. Periodic or cyclic points of period k are fixed points of the kth iterated map generated by 0; i.e., if x=Bk(x) and x # V(x) except for t mod k =0 then x is a k period cycle. Suppose 8 is differentiable at a stationary state X. If (e’(x)1 < 1 then X is locally asymptotically stable. If 0’(x) < - 1 then X is locally cyclically unstable and fluctuations may persist. In this event trajectories may converge to limit cycles of some period k or they may wander within the interval in a nonperiodic, more-or-less erratic fashion. One way to characterize this wandering is to construct the density or cumulative distribution of values of the sequence converges as n-tco. If X, e(x), 02(x), . . . , P(x) and determine if this distribution this happens and that distribution function has certain regularity properties then the ‘long run’ character of fluctuations can be characterized by probabilities that future values of GNP will fall in given intervals. We will show that the normalized sample means of model generated values obey the central limit theorem. We also show that the support of the distribution is the union of a finite set of intervals. These intervals are periodic sets. Thus if there are k intervals IO,. . . ) Ikm ’ with finite support, then ek(l’) = I’. This means that the long run %J a preliminary version of this work (1983) we presented a few results about the existence positive measures for nonperiodic behavior. In this paper we are able to go much further providing rather complete answers to the above questions.
of in
R.H. Day and W Shafer, Ergodicfluctuations
behavior of fluctuations shocks from a stationary
in economic models
is like a stable k-cycle superimposed distribution of finite range.
341
with random
2. A nonlinear business cycle model The business cycle model we use consists of monetary and real sectors. The monetary sector is represented by the demand for money, Dm(r, Y), where r is the interest rate and Y is real national income; the supply of money, Sm(r, Y; M), where M is a money supply parameter; and a market clearing equation, Dm(r, Y)=S”(r, Y; M). The latter implicitly defines the LM curve, Y= L( Y; M), which gives the market-clearing (temporary equilibrium) interest rate.3 The real sector is represented by an induced consumption function C= C(r, Y), and an induced investment function, I=l(r, Y). Autonomous investment, government and consumption expenditure will be denoted by the parameter ‘A’. Substituting the LM function for interest in the consumption and investment functions we obtain respectively the consumption-income (CY) function, G( Y; M): = C(L( Y; M), Y), and the investment-income (IY) function, H( Y; M): = Z(L( Y; M), Y). Assuming that current consumption and investment demand depend on lagged income we get the difference equation ~+,=8(~;11,M,A):=G(~;M)+~LH(~;M)+A,
(1)
where ~20 is a parameter measuring the ‘strength’ or ‘intensity’ of induced investment. The model is relevant in the ‘Keynesian Regime’ i.e., for YE [0, YF] where YF is the highest level of income compatible with available capacity, the supply of labor and the supply of money. Under standard assumptions the CY curve is a continuous, monotonically increasing function with G(O)=0 and the IY curve is a more-or-less bell shaped curve which eventually falls as increasing transactions crowd the money market and interest rates rise which in turn reduces investment demand. Consequently, aggregate demand 0(Y) has the cocked-hat or titled-z profiles shown in fig. 1. Note that the nonlinearity becomes more pronounced when induced investment is important, i.e., when 1-1is ‘large’.4 Let Y be the largest stationary state. If 0’(Y) < - 1 it is unstable and fluctuations are perpetuated. Notice that there is a local minimum Ymi” and a local maximum Ymax such that Ymi”< Y< Ymax. Let I/: = [ Ymi”, Ym”“]. Evidently fluctuations ‘A4 is usually the money supply itself so that ,Sm(r, Y;m)=M. In all of the examples discussed below we make this assumption, but it is by no means necessary. 41f G( .) has the linear form usually assumed and p =0 then (1) is the standard Kahn-Keynes multiplier equation. When p is slightly positive this picture is not changed much but as n becomes large the strong nonlinearity emerges. This aspect of aggregate demand often comes as a surprise to those who think about it for the first time. For an elaborate discussion see Day and Lin (1986).
342
R.H. Day and W Shafer, Ergodicjlluctuations in economic models
Y’
7
(a)
I,” 7
(cl
7
(b)
:a Y'
Y2
7
(d)
Fig. I. Profiles for aggregate demand. [The diagrams show the possible qualitative shapes of aggregate demand and the number and location of stationary states on the interval (0, Y ‘).I
occur only in this trapping set. Within it increases in the aggregate demand for goods, accompanied by rising labor and money demand is followed by a decline in aggregate demand for goods, labor and money and so on.’ We confine attention to those cases where bounded oscillations are perpetuated. Three distinct cases can be identified which depend on the relation of the maximum ‘overshoots’ O(Ymi”) and t3(Ymax) to the boundary values Ymi” and Ymax of 1/: Let Y* and Y ** be the minimizer and maximizer respectively of 0 on 1/: i.e., Ymin=O( Y*), Ymax=O( Y**). Case I: If 51n fig. la there is a single, stable stationary state. In figs. lb and Id there are two and three stationary states respectively, but fluctuations are possible only around the largest of these in the interval K In fig. lb the smaller stationary state, say Y’ is stable from the left and unstable from the right. If Y”‘“> Y’ then V will trap all trajectories with initial condition Yn> Y’. Otherwise, some or almost all trajectories would converge to Y’. In fig. Id the smallest stationary state Y’ is stable while the middle one, say Y *, is unstable. If Y”‘“z Y2 the interval V traps trajectories for all initial conditions YO> Y’.
R.H. Day and W Shqfer, Ergodicjluctuations
in economic models
343
Ymin< Y** < Y* < Ymax then V traps all the fluctuations. Case II: If Y* > Ymax then the maximum overshoot is tl( Ymax) and I/‘: = [Ymax, O(Ym”“)] traps all fluctuations. Case III: If Y** < Ymi” then O(Ymi”) is a maximum overshoot and V2: = [ Ymi”, O(Ymi”)] traps all fluctuations. But Cases II and III are equivalent. Thus, if 0 is a Case III map the map h- .z. h will be Type II where h(Y) = 1 - Y Therefore, Cases I and II can serve as the two canonical maps for further investigation. See fig. 2. They exhibit a property crucial to the subsequent analysis, namely that they are piecewise smooth and piecewise monotonic with kinks or turning points which give the points of maximum overshoot or maximum undershoot of the maximum stationary state. max Y
mox Y
1’
’
J’ I’, I\
min Y
Y
min
, 1 1 ‘1 \ \ \ \ \ \
\
L!‘z! Y
(b)
(a)
Y (cl
Case
**
II
max.
Y*
Case
min
I
O(y
Y (d)
Case
min
1
III
Fig. 2. Constructing the trapping set. [The minimal trapping set is determined by the local maximum Ymax and minimum Ymi” and the maximum ‘overshoot’ of the stationary state e i.e., O(Y “I”) and O(Y”““). The smooth lines show the general piece-wise smooth maps, and the dotted lines give the piece-wise linear example.]
344
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
3. Model 1: Piecewise linear aggregate demand Linear demand functions for goods and money provide an important special case of the model under consideration, important because complete results can be obtained and because it is so frequently used in policy discussions [cf. Hall (1975)] and for pedagogical purposes [for example the texts of Gordon (1978) or Morley (1983)]. We depart from these standard works only by adding nonnegativity restrictions that are usually ignored or left implicit and by expressing the model in explicit dynamic form. First is the demand for money, P(r, Y) = Lo - h + kY, where Lo is a constant, and 2 and k parameters. Given M, the supply of money, the LM curve can be written
0,
osYzY**
(k/i-)( Y - Y**),
Y**sYsM/k,
r=L”(Y;M):=
where Y** =(M - L’)/k. Assume induced consumption is ctY where cx is the marginal propensity to consume and let investment demand be I(r, Y): = max {O,p( Y - Y’)- yr), where Y’ is a threshold above which the direct effect of income on investment is positive. With these assumptions the adjustment equation for GNP is
B+bY,,
Y’S I;< y**
C-cx:,
Y**IYIY* _ t-
A+a&
r*sk;gyF,
E;+,=8(Y):=
(2)
where A is autonomous consumption and B= A-p/W’, c=po--cr, a=yk/,l-/3, [(yk/ll) Y** - pY’]/o, Y** = (M - L’)/k. If GNP and Y* locally minimizes GNP. We aggregate demand that is of interest, and occur.6
investment expenditure, b = c(+ p/3, y*= C=A+puoY*, and c > 0 then Y** locally maximizes therefore get the tilted-z profile for all of the cases shown in fig. 1 can
‘In the intervals [0, Y’] and [Y*, YF] the ordinary Kahn-Keynes multiplier describes the process because the interaction of monetary and real sectors is inoperative. Between Y’ and Y** increasing income stimulates induced investment but interest rates are unaffected. Between Y** and Y interest rates rise as the transaction demand for money begins to ‘crowd’ the money markets. Above Y* induced investment is eliminated by high interest rates and only autonomous investment is left, which is included in the autonomous expenditure parameter A. These discrete regimes in aggregate demand can be thought of as approximations of the smooth, nonlinear transitions to be analyzed below.
R.H. Day and K Shafer, Ergodicfluctuations
in economic models
345
If g > 0 then for all p > (1 + c()/u the parameter c = .DCJ - a > 1. Hence, if there is a stationary state in the third, interest sensitive regime it is unstable and bounded oscillations are perpetuated. In this locally unstable situation the trapping sets are non-degenerate so any given map will be equivalent to one of the following maps on the unit interval: Case I.
ymin< y** < y* < ymax
r
T(y):= i
I
( y _ ymin)/( y-
l--by**+bY,
YE cO?Y**l
1 +cy**-cy,
YE L-y**,Y*l
--ay**+ccy,
where we note that Y** -y* Case II.
y=
_ ymin),
(3)
YE cy*> 11,
= l/(c).
ymin < y** < ymax < y*
>
y = [Y - e( Ym=)]/[ ymax- 8( Ymax)],
1 +by**+by,
YE[o,y**=l-l/c]
1+ cy** - cy,
YE cy**,
T(y): =
Case III.
Y** <
T(y): =
Ymin <
!
cY*
Y* < Ymax,
-
CY,
- my* + ccy,
YE
yz
[
co, Y* =
YE cy*,
y
(4)
11. -
Y”‘“]/[&
pi”)
- Ym'"],
l/cl
11.
(5)
turning points Y* and Y** Note that y* and y** are the transformed respectively. Recall that by setting 1 -y = x and substituting in (5) we get (4) except that b is replaced by c1 and y by x. We can therefore confine attention to Cases I and II. Case II is the simplest case, a map with two piecewise linear segments and a single turning point. The complete characterization of trajectories for this case, which we shall derive in section 5, can be expressed graphically as shown in fig. 3. The shaded area gives the combination of parameter values that yield stable, periodic cycles. The unshaded area shows where erogodicity prevails, that is, where empirical frequency distributions of iterated values converge to a limit distribution function (or probability measure) whose support is the union of a finite set of nondegenerate intervals. This measure can be represented by a unique, differentiable cumulative distribution function or equivalently by a unique continuous density function. Fig. 3 reflects the qualitative behavior of the model for all parameter combinations leading to the Case II map. It can be interpreted in terms of
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
the economic parameters underlying the demand for money and goods. Thus b =a+@ where CI is the marginal propensity to consume and p/3 is the marginal propensity to invest. Also, c =pyk/;lb, where pry is the marginal effect of the interest rate on investment demand, where k is the marginal effect of income on the demand for money and ;1 is the marginal effect of interest on the demand for money.’ In fig. 4 numerical comparative dynamic simulation has been performed for a slice through the (b, y**) space of fig. 3, for b=+. The horizontal axis in ‘Note that b +c=pyk/l. Suppose we fix b. Then by varying c we are in effect varying yk/l. Because p appears in b the three ‘independent’ economic parameters c are determining y, k and 1. Hence, for any given value of b the variation in y ** implies a corresponding variation in yk/l.
(a) ‘Short’ run behavior
(b) ‘Long’ run behavior Fig. 4. Bifurcation
diagrams
for the Case II.
348
R.H. Day and WI Shafer, Ergodicfluctuations
in economic models
fig. 4 is y** = 1 - l/c. The parameter c is varied in increments of 0.1 from one to 11, giving 1,000 values of y ** between zero and 1- l/l 1 on the horizontal axis. For each value of c the Case II map was iterated 250 times and the values plotted on the verticle axis. In panel (a) the first 125 iterates are shown giving a picture of how the ‘short run dynamics’ evolve and how the pattern of behavior shifts with y** (and hence with $/A). In panel (b) the next 125 iterates are plotted giving an approximate picture of the ‘long run’ dynamics. In fig. 5a a histogram and in fig. 5b the cumulative distribution have been constructed for 10,000 iterates of (4) for given parameter value corresponding to point A in fig. 3 using 150 subintervals of the unit interval. The sequence of model generated values can be regarded as the realization of a stochastic process which obeys the central limit theorem. Thus, successive runs of say n values can be thought of as samples from this process, the means of which are distributed normally. Fig. 5c shows the computed histogram of the sample means. The smooth curve is the normal distribution whose moments are equated with those of the numerical distribution.
(a) Histogram
of GNP
(b) Cumulative
values
(c) Histogram Fig. 5. Ergodic
of sample means and normal behavior
curve
for the piecewise linear map.
distribution
R.H. Day and W Shafer, Ergodicfluctuations
4. Dynamical
in economic models
349
systems theory
We shall now give a formal summary of the concepts needed to establish the results that we have just illustrated. Consider the class of all recursive economies whose dynamics on a domain D c R can be represented by a continuous nonlinear map&D-D. The state of the economy in a given period t is given by a value X, ED. The succeeding state is generated by the difference equation xL+
1 =
KG 4 = W,),
(6)
where IZ is a vector of parameters for the function 8. We take D to be a closed bounded interval which means that attention is confined to globally stable models. (This is usually accomplished in economic models by ‘detrending’ the variables.) Define the iterated map 0”: D-+D by 0”(x) =x and 0”(x) = Then the sequence ~(x)--(@‘(x)),~~ is called the 000’-‘(x), n-1,2,3 ,.... trajectory from the initial condition x. The orbit from x, is the set y(x)-{@‘(x)In~O}. Th e asymptotic behavior of a trajectory is described by the limit set w(x) of the trajectory z(x); o(x) is defined to be the set of all limit points of z(x). Note that w(x) is closed and @o(x)) =0(x). An attractor for 8 is a closed set F c D such that w(x) = F for x in a set of positive Lebesgue measure. Attractors represent the asymptotic behavior of solutions for a nontrivial set of initial conditions. The two types of attractors which were illustrated in the numerical examples and which are typical for the more general economies we consider in this paper are those corresponding to stable periodic points and to invariant, ergodic measures absolutely continuous with respect to Lebesgue measure. We now describe these. A point XE D is a periodic point of at least period n if Q”(X)=X and @(X) #Y for j< n. Note in this case that w(@(X)) = y(X) for any j. A periodic point X and the corresponding orbit y(X) are called locally stable if there is a nondegenerate interval V containing X such that w(x)=y(X) for all XE I/: In this case y(X) ix an attractor. It is not unusual for a map to have many periodic points. In fact 0 will have infinitely many periodic points of different periods whenever some power 6”’ of 0 has a periodic point of odd least period n2 3. In this case however, it is possible that none of the periodic points is locally stable. It is in this situation that ergodic theory can be used to describe the asymptotic behavior of trajectories. Let p denote a probability measure on the Bore1 subsets of D with p(D)= 1. The support of p, denoted suppp, is the smallest closed subset D’ of D for which p(D’) = 1. The measure p is said to be invariant for 8 if p(Q- ‘(B)) = p(B) for all Bore1 subsets B of D. It is called ergo&c for 8 if F’(B)=& implies p(B) is 1 or 0 where B is measurable. The simplest example of an invariant ergodic measure for 0 is a measure p which assigns probability l/n to each point in the orbit y(X) of a periodic point of least period n. However,
350
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
this is not a very interesting case. More interesting is when there is an invariant ergodic measure for 8 which is absolutely continuous (with respect to Lebesgue measure). Let A denote Lebesgue measure. An example is the map %(x)=min{2x,l-2x)
for
D=[O,l].
(7)
In this case it is easy to show that for any interval J c [0,l],n(%%‘(J)) = n(J) so that 3, is an invariant measure for 8. It can also be shown to be ergodic. The language of stochastic processes can be used to interpret the significance of the existence of an absolutely continuous invariant ergodic measure for %. If ,U is a Bore1 probability measure on D, (D,p) can be viewed as a probability space, and each H”:D+D as a random variable. We write {%“,p} as the resulting stochastic process, with the joint distribution P, %%‘(B,)). of {%“,%I,. . . ) %“- ‘} given by P,(x:%‘(x) E&, i=O,. . , n- 1)~p(n;Z,’ Each realization (%‘(x)),,, of {H”,p} is, of course, just the trajectory r(x) of 8. If p is invariant for 8, then {%“,n} is a stationary stochastic process, and if in addition p is ergodic, then this process is a stationary ergodic stochastic process. To describe an implication of this, define the indicator function A by A(x, B) = 1 if x E B and 0 if x # B. The function s, is defined by s,(x, B)
=t”$;A(%‘(.+B),
(8)
I
which gives the fraction of points in {x, e(x),. . ., d’-‘(x)} that lie in B. The function s,(x, .) is the probability measure on D which describes the empirical frequency distribution of the first IZ points of r(x). Now suppose p is an absolutely continuous invariant ergodic measure for 8. Then i(supp 11)> 0, and by the Birkhoff ergodic theorem and the continuity of 8 [see Parthsarathy (1967)], the following holds: s,(x;)%p
for
1
a.e.
x,fi
%-‘(suppp). i=O
Here 5 denotes convergence in distribution (or weak convergence of probabability measures). One implication of this is that w(x) = supp p for a nontrivial set of initial conditions x, i.e., suppp is an attractor. But (9) says more: it implies a certain statistical regularity to the trajectories. Indeed ergodicity is a weak form of asymptotic independence for {F,pu). In the example of section 3 we saw that this asymptotic independence appeared to be sufftciently strong to imply Central Limit Theorem results. Define S,:D+R to be the centered sample sum, i.e., n-1
S,(x) =
izoe’(x)-n JWW).
(10)
R.N. Day and W Shafer, Ergodicfluctuations
in economic models
351
If p is an absolutely continuous invariant ergodic measure for 0, with m=Jx~(dx)=lim,,, l/nx;-’ Q’(x), we say that the stochastic process (@‘,p) satisfies the Central Limit Theorem Hypothesis if there is a number a* such that
5 %N(0, 02). $
(11)
It is well known that (7) provides an example. Situations often arise in which we have an invariant measure p for 6’ which is not ergodic. In this case it is often possible to find a finite ergodic decomposition of (0,~) i.e., a finite number of invariant ergodic measures pi,. . , pm for 0 and positive numbers a,, . . . , LX, such that xjaj= 1 and p= cjcxjpj. .One case when this can arise follows. Suppose ,u is an invariant ergodic measure for t9 and the support of p is the union of a finite number of intervals IO,. . , I” 1 which have at most end points in common. Suppose e I modm(li) = I(i+r) modm, Then p is invariant for P, but not ergodic, because empi) = Ii, i = 1,. . . , m. Define ,uj by p,(B)=p(B A Zj)/p(rj). Then each pj is invariant for em, and if each pj is ergodic, then the pj forms an ergodic decomposition of (P’, p). If in addition each {emk,~jjk20 satisfies (1 l), then the trajectories of 0 will appear as random perturbations about a cycle of length m. The computer simulation shown in fig. 5a appeared to have this type of behavior, and our theoretical results will give justification to this interpretation.
5. The basic theorems We have shown that for the business cycle model under consideration, there exists a range of parameters on which fluctuations exist and are perpetuated on a trapping set I/: For this range the map 8, restricted to the trapping set, i.e., 8,: I/+ I/ can be transformed to a map on the unit interval T: Z-+1: = [0,11. We also showed that these maps were piecewise monotonic. In what follows we will restrict the analysis to the class of such maps whose segments are sufficiently smooth. Formally we adopt the following standing assumptions (although they are sometimes stronger than necessary). Basic assumptions (BA) (a) T is continuous. (b) there is a finite number of points (yi}~~~, with y’< yif ‘, i=O,. . . , n, and with y’=O, y”+‘= 1, such that (i) T is C3 on each [y’, y’+‘], and T’(x) #O on (y’, y’+‘),i=O,. . . ,n. (ii) sgn T’(y’- E) + sgn T’(y’+ E)=0 for sufficiently small E>O, for each i=l , . . . , n, that is, y’, i= 1, . . . , n is a turning point of i7
352
R.H. Day and W. Shafer, Ergodicfluctuations
in economic models
As suggested in the previous section, attractors for T often correspond to stable orbits or absolutely continuous invariant ergodic measures. In deriving conditions for these two types of attractors the following definition plays an important role. Let (a, b) be an interval on which T’(x) #O. Then the Schwartzian derivative of T on (a,b) is the map ST:(a,b)-+R given by ST(x)= T”‘(x)/T’(x) - 3/2(( T”(x)/T’(x))‘. Remark 1. In some of the results below the condition ST(x) 5 0 is imposed: this is equivalent to the requirement that 1/1TI”* is a convex function on (a, b). [Misiurwicz (1980, p. 18).] The first theorems of interest will enable attractors are stable, periodic cycles.
us to characterize
models
whose
Theorem I. If ST(x) 50 on each (y’, y'+ ‘), then every stable orbit of T attracts a point in {y’}lf,‘, i.e., if y(X) is a stable orbit then o(y’) =y(X) for some i. ProoK Singer (1978) proved this theorem for the case where T is everywhere C3 and ST ~0. However, his method of proof actually establishes the more general result stated. Q.E.D. Theorem 2. Suppose on each (y’,y’+‘), either ST(x)i=o n+l is attracted to a stable orbit, then almost every XEI is attracted to a stable orbit. Pro05 The hypotheses imply T has no sinks outside a neighborhood of the stable orbits and the set {y’}lf,‘. Given this fact the result follows from Lemmas 1.3 and Proposition 2.1 in Misciurwicz (1981). Q.E.D. The next three theorems concern the second kind of attractors, those that are uncountable and have associated with them absolutely continuous invariant measures. The first theorem due in part to Lasota and Yorke (1973) and Kowalski (1975) [see also Pianigiani (1978)] and in part to Hofbauer and Keller (1982a, b) is concerned with maps that are not differentiable at their turning points and which are expansive, that is, whose derivatives, whereever they are defined, are greater than one in absolute value. The piece-wise linear example we looked at in section 4 falls in this class. Theorem 3. Suppose there exists an integer rnz 1 such that infxplITm’(x)l> 1. (By T’(y’) we always mean both one sided derivatives.) Then there exists a
R.H. Day and WI Shafer, Ergodicfluctuations
finite that (i) (ii) (iii)
number of absolutely
continuous
invariant
in economic models
measures
(ii)
(iii)
pI,. . . , pk for 7; such
The support of each ,ui is a finite union of disjoint closed intervals, and k does not exceed the number of turning points of T For i almost every x E I, w(x) = supp pi for some i. There exists an integer 1 such that for each i= 1,. . , k the dynamical system {T’, pi> possesses an ergodic decomposition { T1,,ttij}g~ 1 such that each {T’“, pi,} satisfies the central limit theorem hypothesis.
ProoJ: Lasota and Yorke and Kowalski demonstrate m= 1. The above generalization follows by applying T” and then observing (i)
353
(i) and (ii) for the case their result to the map
if 1; is an invariant ergodic measure for T”, then p= l/m~~z~o’ fi. T-’ is an invariant ergodic measure for p. since each turning point y’ is a local max or min of 7; two distinct invariant ergodic measures for T whose supports are a finite union of intervals cannot have the property that their supports have a turning point in common. Q.E.D. result (iii) has been established by Hofbauer and Keller.
Remark 2. Note that the hypotheses existence of stable orbits.
of Theorem
3 explicitly
rule out the
The hypotheses of this theorem also rule out the possibility that the one sided derivatives of T at a turning point y’ are the same, i.e., that T’(y’) =O. The following result allows for this possibility. In particular they enable us to establish comparable result for nonexpansive maps. Theorem
4.
Suppose
(a) ST(x)50 on each (y’,y’+‘). (b) If Tm(x) -2, then IT”‘(x)1 > 1. (c) There is a neighborhood U of {y’}lf,’ such that T”(x)~{y~}~~d for every x E {y’)r=+,’and m 2 0. (d) For each y’ either T’(y’) # 0, or T”(y’) # 0 or T”‘(y’) #O. Then there exists a finite pi,. . , pn for 7; such that (i) (ii) (iii)
number of absolutely
continuous
invariant
u[I\U]
measures
The support of each pi is a finite union of disjoint closed intervals, and k does not exceed the number of turning points of T. For almost every x E I, o(x) = supp ,U~for some i. There exists an integer 1 such that for each i= 1,. . . , k the dynamical system { T’,pi} possesses an ergodic decomposition { T’,uij}$L 1 such that each {T’“, pi,} satisfies the central limit theorem hypothesis.
R.H. Day and W Shafer, Ergodicfluctuations
354
Proof (1985).
Misierewicz Q.E.D.
(1981) established
6. Ergodic fluctuations
in economic models
(i) and (ii) while (iii) is due to Ziemian
for the piecewise linear business cycle model
Reconsider the Type II canonical cycle model of section 3.
map
for the piecewise
linear
business
Theorem 5. Let T be a Type II canonical map and let kz 1 be the minimum integer such that Tk-‘(0) < y** and Tk(0) 2 y**. If bkc < 1 there exists a unique stable orbit of least period k+ 1 which attracts almost every XE I. If bkc> 1 there exists a unique absolutely continuous invariant ergodic measure p for T which attracts almost every x E I i.e., supp p is an attractor). Furthermore, there exists an integer p such that (TP, p) has an ergodic decomposition (ul,. . , , p,,,) such that each (TP,pi) satisfies the Central Limit Theorem Hypothesis. Proof: Choose k as in the theorem. By continuity, there is a z~(O,min(T(0),~~**)] such that Tkm ‘(z)=y**. Note that Tk+l(z)=O. Define F(x)= Tk+l(x)-X. Then F(0)=Tk+l(O)>O and F(z)=-zy**. Thus ITk+“(X)I=bkc. By definition of k, for any x E I, T’(x) E (y**, 1] for at least one je (0,1,.. . , k}. Thus if b < 1, then infx,,lTk+’ ‘(x)1= (Tk+ “(X)l= bk c. If b> 1, IT”‘(x)~> 1 for all x and k. Thus if bkc> 1, we obtain the second part of the Theorem from Theorems 3 and 5 of section 5. Suppose bkc < 1. Then X is stable and must attract a turning point or an endpoint for T by Theorem 1. But for this T, we have T(y**) = 1, T( l)=O, so if X attracts one of (0, y**,‘l}, it must attract all three. Thus X is unique and by Theorem 2 must attract almost every XEI. Q.E.D. Fig. 3 in section 3 is obtained from this proof as follows. Consider the case O
f(b)-
k
Furthermore,
l-bk l-bk-’ l_bk
y** < 1 - bk -gk(b).
(14 bkc < 1 is equivalent
to: (13)
R.H. Day and W Shafer, Ergodicfluctuations
355
in economic models
In fig. 3, the region S, + , is the set of parameter values (b,y**) satisfying (12) and (13), and Ek+i is the set of parameter values satisfying (12) and satisfying (13) with the inequality reversed. It is not possible to show on the figure but nevertheless easy to check that for every k 2 1, S, + i # $4and Ek+ 1 # $3. A similar complete characterization for the two (or more) turning point case involves an argument of much greater complexity. So far we have established the following partial results for the Case I canonical map. Let y’ and ys be the unique preimages in [y**,y*] of y** and y* respectively, i.e., T(y’) = y** and T(ys) = y*. In table 1 the inequality expressions are given in terms of these points, the turning points, the stationary state j, the end points and their images. Recall that given T(0) (or T(1)) these can be expressed in terms of the slopes of the linear segments (b,c,cc) which in turn derive from the underlying economic parameters of demand B, y, k, 1, p. In the table S, indicates the existence of stable two period cycles while E, indicates the existence of an absolutely continuous invariant ergodic measure. The results are obtained by constructing the map TZ( .) and exploiting the arguments developed in section 5 and Theorem 5. In cells 21 and 12, the two cases enclosed in heavy square lines, 7”( .) restricted to the intervals [0, yJ] and [y’, l] are Case II and Case III canonical maps respectively so that Table Characterization
of dynamics
1
for the Case I canonical
(1) .I
y"
(2) ;
map.
(3)
y"
(4)
T(O)
I
UOPye
T( l)
T2[ye,ll+[ye,11 yields Case III so Th. 5 applies
TWyf T2:IO,Y~I+IO,Y~I yields Case II canonical map and Th. 5
y*'T(l)
?
?
356
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
Theorem 5 applies directly. A complete characterization as shown in fig. 3 is therefore available in these special situations. For the other cells in the table sufficient conditions for ergodicity can be established for special cases, but the number of situations that has to be examined is large and the calculations tedious. What we have done so far is sufficient for the reader to see what is involved.
7. The piecewise nonlinear business cycle model So far our analysis has exploited piecewise linearity, a property that has enabled us to obtain precise and rather complete answers to our basic questions. But the business cycle model under consideration encompasses more general functional forms as indicated in figs. 1 and 2. Suppose, for example, that the demand for money D”‘(r, Y):=%/(r-r’) + kY do that the LM curve is r = r’ + I,/( M - kY). Suppose also that investment demand is OZYSY'
0, I(r,Y): =
f
bC(Y - W(SYSPWrY
3
YZ Y',
where b, t, p, fl and y are parameters and where Y’ is full capacity and Y’ a threshold above which the Kaldorian multiplier effect of increasing income on investment is positive. Let induced consumption demand be cry and let A be the level of autonomous consumption investment and government demand combined. Then the Mezler-Modigliani-Samuelson model (1) becomes OSYSY
A+a& r,+,=O(Y):= i A+crY,+pLB(Y,-
Y’)f[r’+2/(M-kx)leY,
Y’s Y sp/k,
(14) where B= bpY(4Yf)mP is a constant. If /I and y are both larger than unity then this function has a smooth, ‘cocked-hat’ shape as shown by the solid lines in the diagram of fig. 1. In particular the slope of aggregate demand @( .) is zero at the local maximum and minimum Y** and Y* which exist when p is large enough. Hence, Theorem 3 which requires expansivity throughout cannot be exploited. Instead Theorem 4 must be invoked. Unfortunately, establishing condition a-c of the theorem has so far not been accomplished. In the absence of such analytical conditions we have resorted to computations. In fig. 6a profiles of the map Q(.) restricted to the trapping set and transformed to the unit interval are shown for several values of p. We have computed the criterion l@(y)\* which has the required convex segments on the intervals corresponding to the monotonic segments of the map as shown in fig. 6b. See Remark 1
R.H. Day and W Shafer, Ergodicfluctuations
357
in economic models
(a) The map for several values of p.
(b) The criteria Fig. 6. Qualitative
for the Schwartzian
Derivative
profile of the Type I canonical
Condition
piece-wise
(see Remark
nonlinear,
1)
smooth
map.
above. A bifurcation diagram for p is shown in fig. 7. These results are similar to the ones presented earlier. The regions where stable cycles appear are interspersed with regions where the density of values is apparently dispersed in finite intervals very much like that shown in the piecewise linear case. This suggests that Condition 4b is satisfied. Moreover, the initial conditions for all of the trajectories was y, =O. As T’(y**) = T(y*) = 0
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
359
hypothesis 4c appears to be satisfied in these ‘chaotic’ regions. As 4d is readily verified it seems reasonable to conjecture that Theorem 4 is applicable and that (14) possesses an absolutely continuous invariant measure and behaves like a stationary stochastic process for a large number of parameter values. These conjectures are illustrated in fig. 8 which gives the computed density function for the trajectory and the sample sums.
I _______‘/Y LL
(a) Histogram
of GNP
values
(c)Histogram
(b) Cumulative
of sample means
Fig. 8. Ergodic
behavior
and the normal
for the smooth
distribution
curve
map.
7. Conclusion Our primary purpose has been to provide a complete comparative dynamic analysis in a familiar economic setting, one in which the major questions of interest could be posed precisely and answered definitively. This we have been able to do. Although similar results may be difficult to derive for less manageable models the analysis and numerical illustration presented here suggest, nonetheless, how carefully designed computational studies can
360
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
be interpreted in less tractable cases. We therefore think our study has implications beyond the mere exegesis of ergodic dynamics in general or of the Keynesian business cycle model in particular. So far as the latter is concerned Day and Lin (1986) have shown that plausible parameter values for the piecewise linear model of sections 3 and 6 lead to results that further illustrate the ergodic theory applied here. In particular they present bifurcation results for various policy parameters (supply of money, tax rate, government expenditure) which show that changes in such controls can trigger qualitative changes in the behavior of aggregate fluctuations, thus introducing a new source of uncertainty to the effects of macroeconomic instruments.
References Benhabib, J. and R.H. Day, 1982, A characterization of erratic dynamics in the overlapping generations models, Journal of Economic Dynamics and Control 4, 37-55. Branson, W.H., 1979, Macroeconomic theory and policy, 2nd ed. (Harper and Row, New York). Collett, P. and J.P. Eckmann, 1980, Iterated maps on the interval as dynamical systems (Birkhauser, Basel). Day, R.H. and K.H. Kim, 1987, A note on nonperiodic demoeconomic fluctuations with positive measure, Economics Letters 23, no. 3. Day, R.H. and T.-Y. Lin, 1986, Irregular fluctuations and comparative policy dynamics in the Keynesian regime, MRG working paper no. M8604 (University of Southern California, Los Angeles, CA). Day, R.H. and W. Shafer, 1983, Keynesian chaos, MRG working paper no. 8306 (University of Southern California, Los Angeles, CA). Day, R.H. and W. Shafer, 1985, Keynesian chaos, Journal of Macroeconomics 7, 277-295. Dunford, N. and J. Schwartz, 1958, Linear operators. Part I: General theory (Interscience, New York). Gordon, R.J., 1978, Macroeconomics (Brown, Boston, MA). Grandmont, J., 1985, On endogenous competitive business cycles, Econometrica 53, 99551045. Hall, R.E., 1977, Investment, interest rates and the effects of stabilization policies, Brookings Papers on Economic Acitivity 1, 61-121. Halmos, P., 1949, Measurable transformations, Memoirs of the American Mathematical Society 55, 1015-1034. Hofbauer, F. and G. Keller, 1982a, Equilibrium states for piecewise monotonic transformation, Ergodic Theory and Dynamical System 2, 2343. Hofbauer, F. and G. Keller, 1982b, Ergodic properties of invariant measures for piecewise monotonic transformation, Mathematische Zeitschrift XX, 119-140. Kan, I., 1984, A chaotic function possessing a scrambled set of positive Lebesgues measure, Proceedings of the American Mathematical Society 92, 4549. Kowalski, Z.,-1975, Invariant measures for piecewise monotonic transformation, in: Z. Ciescelski, U. Urbanik and W.A. Wovczinski. eds., Probabilitv-winter school (Springer-Verlag, Berlin). Lasota, A. and J. Yorke, 1973, On the existence of invariant measurks’for -piecewise monotonic transformations, Transaction of the American Mathematical Society 186, 481488. Lasota, A. and J. Yorke, 1977, On the existence of invariant measures for transformation with strictly turbulent trajectories, Bulletin de I’Academie Polonaise des Sciences 25, 233-238. Li, T. and J. Yorke, 1978, Ergodic transformations from an interval into itself, Transactions of the American Mathematical Society 235, 183-192. Mezler, L., 1941, The nature and stability of inventory cycles, The Review of Economic Studies. Misiurewicz, M., 1980, Absolutely continuous measures for certain maps of an interval, Publications Mathematiques 53, 17-51.
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
361
Modigliani, F., 1944, Liquidity preference and the theory of interest and money, Econometrica 12, 45-88. Morley, S.A., 1983, Macroeconomics (The Dryden Press, Chicago, IL). Parasarathy, K., 1967, Probability measures on metric spaces (Academic Press, New York). Pianigiani, G., 1978, On the existence of invariant measures, Applied Nonlinear Analysis, 299307. Pianigiani, G. and J. Yorke, 1979, Expanding maps on sets which are almost invariant: Decay and chaos, Transactions of the American Mathematical Society 252, 351-366. Pohjola, M., 1981, Built-in flexibility of progressive taxation and the dynamics of income: Stability, cycles or chaos? (Department of Economics, University of Tampere, Tampere). Samuelson, P.A., 1948, Foundations of Economic Analysis (Harvard University Press, Cambridge, CA). Singer, D., 1978, Stable orbits and bifurcation of maps of the interval, SIAM Journal of Applied Mathematics 35, 260-267. Smital, J., 1984, A chaotic function with a scrambled set of positive Lebesgues measure, Proceedings of the American Mathematical Society 92, 5&54. Ziemian, K., 1985, Almost sure invariance principle for some maps of an interval, Ergodic Theory and Dynamical Systems 5, 6255640.
of Economic
ERGODIC
Behavior
and Organization
FLUCTUATIONS
Richard University
8 (1987) 339-361.
North-Holland
IN DETERMINISTIC MODELS*
H. DAY
and
of SouthernCaliJornia,
Wayne
ECONOMIC
SHAFER
Los Angeles, CA 90089, USA
Received July 1986, final version
received
February
1987
Nonperiodic business fluctuations are shown to exist with positive measure generically and to behave like stationary stochastic processes in the standard dynamic macro model. The implication is that so called ‘chaotic’ fluctuations are probable for random initial conditions in a large class of recursive economies.
1. Introduction Nonperiodic trajectories have been investigated for dynamic models in a variety of scientific fields. But the mere existence of ‘mathematical chaos’ does not tell how important irregular fluctuations are likely to be - even if the equations in question were accurate analogs of some real world processes. This is because in nonlinear dynamics long run behavior depends on initial conditions: although chaotic trajectories may exist for an unaccountable number of initial conditions this ‘scrambled’ set may have zero measure. If this were true then ‘most’ randomly chosen initial conditions would lead to periodic cycles.’ But suppose nonperiodic behavior did result for initial conditions from a set with positive measure. When trajectories have no finite representation in terms of limit cycles, but wander indefinitely among never repeated values, can they be characterized in some finite manner, and is that characterization structurally stable? An example is provided by the standard macro-model with Robertsonian lag developed by Mezler (1940) Modigliani (1944) and Samuelson (1948). Elsewhere we showed that this model exhibited chaotic fluctuations when *Work by the first author on the present study was performed at the Netherlands Institute for Advanced Study while that of the second was carried out at the University of Southern California. The computer graphics were prepared by T.Y. Lin. ‘This issue has been called the ‘observability’ or the ‘measurability’ question by Melese and Transue (1985). Benhabib and Day (1982) suggested looking at the distribution of a trajectory in their initial studies but did not present an analysis of the problem. In the meantime, however, Day and Shafer (1985) and Grandmont (1985) have presented studies in which conditions for measurable nonperiodic behavior are given. See also Day and Kim (1987) 0167-2681/87/$3.50
0
1987, Elsevier Science Publishers
B.V. (North-Holland)
340
R.H. Day and W Shafer, Ergodicjluctuations
in economic models
induced investment was strong enough [Day and Shafer (1985)]. Here we provide conditions under which irregular fluctuations have positive measure and behave like stationary stochastic processes.’ The general model is described in section 2. It can be reduced to a piecewise monotonic map on the unit interval which has one or two turning points with two or three monotonic pieces respectively. When the pieces are linear and only one turning point occurs especially sharp results can be obtained. Indeed, in this special case it is possible to give a complete answer to the questions of interest in this paper. Although this is not a very complicated business cycle model and is perhaps the simplest conceivable nonlinear map the possible patterns of dynamic behavior are rich: stable cycles of all orders, chaos and ergodic, stationary stochastic fluctuations with a unique probability distribution. These results are illustrated with numerical examples in section 3. In sections 4 and 5 we review the theory of dynamical systems which can be applied to general recursive economies with a single state variable. In section 5 that theory is used to show that the simulations of section 3 are not misleading, but reflect exact theoretical results. We then briefly explore the more general cases that arise in the business cycle model under consideration in section 6. In order to see what we are headed for the reader may be reminded of a few basic concepts. Suppose B is a continuous map on the unit interval. A point X= e(X) is a stationary state. Periodic or cyclic points of period k are fixed points of the kth iterated map generated by 0; i.e., if x=Bk(x) and x # V(x) except for t mod k =0 then x is a k period cycle. Suppose 8 is differentiable at a stationary state X. If (e’(x)1 < 1 then X is locally asymptotically stable. If 0’(x) < - 1 then X is locally cyclically unstable and fluctuations may persist. In this event trajectories may converge to limit cycles of some period k or they may wander within the interval in a nonperiodic, more-or-less erratic fashion. One way to characterize this wandering is to construct the density or cumulative distribution of values of the sequence converges as n-tco. If X, e(x), 02(x), . . . , P(x) and determine if this distribution this happens and that distribution function has certain regularity properties then the ‘long run’ character of fluctuations can be characterized by probabilities that future values of GNP will fall in given intervals. We will show that the normalized sample means of model generated values obey the central limit theorem. We also show that the support of the distribution is the union of a finite set of intervals. These intervals are periodic sets. Thus if there are k intervals IO,. . . ) Ikm ’ with finite support, then ek(l’) = I’. This means that the long run %J a preliminary version of this work (1983) we presented a few results about the existence positive measures for nonperiodic behavior. In this paper we are able to go much further providing rather complete answers to the above questions.
of in
R.H. Day and W Shafer, Ergodicfluctuations
behavior of fluctuations shocks from a stationary
in economic models
is like a stable k-cycle superimposed distribution of finite range.
341
with random
2. A nonlinear business cycle model The business cycle model we use consists of monetary and real sectors. The monetary sector is represented by the demand for money, Dm(r, Y), where r is the interest rate and Y is real national income; the supply of money, Sm(r, Y; M), where M is a money supply parameter; and a market clearing equation, Dm(r, Y)=S”(r, Y; M). The latter implicitly defines the LM curve, Y= L( Y; M), which gives the market-clearing (temporary equilibrium) interest rate.3 The real sector is represented by an induced consumption function C= C(r, Y), and an induced investment function, I=l(r, Y). Autonomous investment, government and consumption expenditure will be denoted by the parameter ‘A’. Substituting the LM function for interest in the consumption and investment functions we obtain respectively the consumption-income (CY) function, G( Y; M): = C(L( Y; M), Y), and the investment-income (IY) function, H( Y; M): = Z(L( Y; M), Y). Assuming that current consumption and investment demand depend on lagged income we get the difference equation ~+,=8(~;11,M,A):=G(~;M)+~LH(~;M)+A,
(1)
where ~20 is a parameter measuring the ‘strength’ or ‘intensity’ of induced investment. The model is relevant in the ‘Keynesian Regime’ i.e., for YE [0, YF] where YF is the highest level of income compatible with available capacity, the supply of labor and the supply of money. Under standard assumptions the CY curve is a continuous, monotonically increasing function with G(O)=0 and the IY curve is a more-or-less bell shaped curve which eventually falls as increasing transactions crowd the money market and interest rates rise which in turn reduces investment demand. Consequently, aggregate demand 0(Y) has the cocked-hat or titled-z profiles shown in fig. 1. Note that the nonlinearity becomes more pronounced when induced investment is important, i.e., when 1-1is ‘large’.4 Let Y be the largest stationary state. If 0’(Y) < - 1 it is unstable and fluctuations are perpetuated. Notice that there is a local minimum Ymi” and a local maximum Ymax such that Ymi”< Y< Ymax. Let I/: = [ Ymi”, Ym”“]. Evidently fluctuations ‘A4 is usually the money supply itself so that ,Sm(r, Y;m)=M. In all of the examples discussed below we make this assumption, but it is by no means necessary. 41f G( .) has the linear form usually assumed and p =0 then (1) is the standard Kahn-Keynes multiplier equation. When p is slightly positive this picture is not changed much but as n becomes large the strong nonlinearity emerges. This aspect of aggregate demand often comes as a surprise to those who think about it for the first time. For an elaborate discussion see Day and Lin (1986).
342
R.H. Day and W Shafer, Ergodicjlluctuations in economic models
Y’
7
(a)
I,” 7
(cl
7
(b)
:a Y'
Y2
7
(d)
Fig. I. Profiles for aggregate demand. [The diagrams show the possible qualitative shapes of aggregate demand and the number and location of stationary states on the interval (0, Y ‘).I
occur only in this trapping set. Within it increases in the aggregate demand for goods, accompanied by rising labor and money demand is followed by a decline in aggregate demand for goods, labor and money and so on.’ We confine attention to those cases where bounded oscillations are perpetuated. Three distinct cases can be identified which depend on the relation of the maximum ‘overshoots’ O(Ymi”) and t3(Ymax) to the boundary values Ymi” and Ymax of 1/: Let Y* and Y ** be the minimizer and maximizer respectively of 0 on 1/: i.e., Ymin=O( Y*), Ymax=O( Y**). Case I: If 51n fig. la there is a single, stable stationary state. In figs. lb and Id there are two and three stationary states respectively, but fluctuations are possible only around the largest of these in the interval K In fig. lb the smaller stationary state, say Y’ is stable from the left and unstable from the right. If Y”‘“> Y’ then V will trap all trajectories with initial condition Yn> Y’. Otherwise, some or almost all trajectories would converge to Y’. In fig. Id the smallest stationary state Y’ is stable while the middle one, say Y *, is unstable. If Y”‘“z Y2 the interval V traps trajectories for all initial conditions YO> Y’.
R.H. Day and W Shqfer, Ergodicjluctuations
in economic models
343
Ymin< Y** < Y* < Ymax then V traps all the fluctuations. Case II: If Y* > Ymax then the maximum overshoot is tl( Ymax) and I/‘: = [Ymax, O(Ym”“)] traps all fluctuations. Case III: If Y** < Ymi” then O(Ymi”) is a maximum overshoot and V2: = [ Ymi”, O(Ymi”)] traps all fluctuations. But Cases II and III are equivalent. Thus, if 0 is a Case III map the map h- .z. h will be Type II where h(Y) = 1 - Y Therefore, Cases I and II can serve as the two canonical maps for further investigation. See fig. 2. They exhibit a property crucial to the subsequent analysis, namely that they are piecewise smooth and piecewise monotonic with kinks or turning points which give the points of maximum overshoot or maximum undershoot of the maximum stationary state. max Y
mox Y
1’
’
J’ I’, I\
min Y
Y
min
, 1 1 ‘1 \ \ \ \ \ \
\
L!‘z! Y
(b)
(a)
Y (cl
Case
**
II
max.
Y*
Case
min
I
O(y
Y (d)
Case
min
1
III
Fig. 2. Constructing the trapping set. [The minimal trapping set is determined by the local maximum Ymax and minimum Ymi” and the maximum ‘overshoot’ of the stationary state e i.e., O(Y “I”) and O(Y”““). The smooth lines show the general piece-wise smooth maps, and the dotted lines give the piece-wise linear example.]
344
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
3. Model 1: Piecewise linear aggregate demand Linear demand functions for goods and money provide an important special case of the model under consideration, important because complete results can be obtained and because it is so frequently used in policy discussions [cf. Hall (1975)] and for pedagogical purposes [for example the texts of Gordon (1978) or Morley (1983)]. We depart from these standard works only by adding nonnegativity restrictions that are usually ignored or left implicit and by expressing the model in explicit dynamic form. First is the demand for money, P(r, Y) = Lo - h + kY, where Lo is a constant, and 2 and k parameters. Given M, the supply of money, the LM curve can be written
0,
osYzY**
(k/i-)( Y - Y**),
Y**sYsM/k,
r=L”(Y;M):=
where Y** =(M - L’)/k. Assume induced consumption is ctY where cx is the marginal propensity to consume and let investment demand be I(r, Y): = max {O,p( Y - Y’)- yr), where Y’ is a threshold above which the direct effect of income on investment is positive. With these assumptions the adjustment equation for GNP is
B+bY,,
Y’S I;< y**
C-cx:,
Y**IYIY* _ t-
A+a&
r*sk;gyF,
E;+,=8(Y):=
(2)
where A is autonomous consumption and B= A-p/W’, c=po--cr, a=yk/,l-/3, [(yk/ll) Y** - pY’]/o, Y** = (M - L’)/k. If GNP and Y* locally minimizes GNP. We aggregate demand that is of interest, and occur.6
investment expenditure, b = c(+ p/3, y*= C=A+puoY*, and c > 0 then Y** locally maximizes therefore get the tilted-z profile for all of the cases shown in fig. 1 can
‘In the intervals [0, Y’] and [Y*, YF] the ordinary Kahn-Keynes multiplier describes the process because the interaction of monetary and real sectors is inoperative. Between Y’ and Y** increasing income stimulates induced investment but interest rates are unaffected. Between Y** and Y interest rates rise as the transaction demand for money begins to ‘crowd’ the money markets. Above Y* induced investment is eliminated by high interest rates and only autonomous investment is left, which is included in the autonomous expenditure parameter A. These discrete regimes in aggregate demand can be thought of as approximations of the smooth, nonlinear transitions to be analyzed below.
R.H. Day and K Shafer, Ergodicfluctuations
in economic models
345
If g > 0 then for all p > (1 + c()/u the parameter c = .DCJ - a > 1. Hence, if there is a stationary state in the third, interest sensitive regime it is unstable and bounded oscillations are perpetuated. In this locally unstable situation the trapping sets are non-degenerate so any given map will be equivalent to one of the following maps on the unit interval: Case I.
ymin< y** < y* < ymax
r
T(y):= i
I
( y _ ymin)/( y-
l--by**+bY,
YE cO?Y**l
1 +cy**-cy,
YE L-y**,Y*l
--ay**+ccy,
where we note that Y** -y* Case II.
y=
_ ymin),
(3)
YE cy*> 11,
= l/(c).
ymin < y** < ymax < y*
>
y = [Y - e( Ym=)]/[ ymax- 8( Ymax)],
1 +by**+by,
YE[o,y**=l-l/c]
1+ cy** - cy,
YE cy**,
T(y): =
Case III.
Y** <
T(y): =
Ymin <
!
cY*
Y* < Ymax,
-
CY,
- my* + ccy,
YE
yz
[
co, Y* =
YE cy*,
y
(4)
11. -
Y”‘“]/[&
pi”)
- Ym'"],
l/cl
11.
(5)
turning points Y* and Y** Note that y* and y** are the transformed respectively. Recall that by setting 1 -y = x and substituting in (5) we get (4) except that b is replaced by c1 and y by x. We can therefore confine attention to Cases I and II. Case II is the simplest case, a map with two piecewise linear segments and a single turning point. The complete characterization of trajectories for this case, which we shall derive in section 5, can be expressed graphically as shown in fig. 3. The shaded area gives the combination of parameter values that yield stable, periodic cycles. The unshaded area shows where erogodicity prevails, that is, where empirical frequency distributions of iterated values converge to a limit distribution function (or probability measure) whose support is the union of a finite set of nondegenerate intervals. This measure can be represented by a unique, differentiable cumulative distribution function or equivalently by a unique continuous density function. Fig. 3 reflects the qualitative behavior of the model for all parameter combinations leading to the Case II map. It can be interpreted in terms of
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
the economic parameters underlying the demand for money and goods. Thus b =a+@ where CI is the marginal propensity to consume and p/3 is the marginal propensity to invest. Also, c =pyk/;lb, where pry is the marginal effect of the interest rate on investment demand, where k is the marginal effect of income on the demand for money and ;1 is the marginal effect of interest on the demand for money.’ In fig. 4 numerical comparative dynamic simulation has been performed for a slice through the (b, y**) space of fig. 3, for b=+. The horizontal axis in ‘Note that b +c=pyk/l. Suppose we fix b. Then by varying c we are in effect varying yk/l. Because p appears in b the three ‘independent’ economic parameters c are determining y, k and 1. Hence, for any given value of b the variation in y ** implies a corresponding variation in yk/l.
(a) ‘Short’ run behavior
(b) ‘Long’ run behavior Fig. 4. Bifurcation
diagrams
for the Case II.
348
R.H. Day and WI Shafer, Ergodicfluctuations
in economic models
fig. 4 is y** = 1 - l/c. The parameter c is varied in increments of 0.1 from one to 11, giving 1,000 values of y ** between zero and 1- l/l 1 on the horizontal axis. For each value of c the Case II map was iterated 250 times and the values plotted on the verticle axis. In panel (a) the first 125 iterates are shown giving a picture of how the ‘short run dynamics’ evolve and how the pattern of behavior shifts with y** (and hence with $/A). In panel (b) the next 125 iterates are plotted giving an approximate picture of the ‘long run’ dynamics. In fig. 5a a histogram and in fig. 5b the cumulative distribution have been constructed for 10,000 iterates of (4) for given parameter value corresponding to point A in fig. 3 using 150 subintervals of the unit interval. The sequence of model generated values can be regarded as the realization of a stochastic process which obeys the central limit theorem. Thus, successive runs of say n values can be thought of as samples from this process, the means of which are distributed normally. Fig. 5c shows the computed histogram of the sample means. The smooth curve is the normal distribution whose moments are equated with those of the numerical distribution.
(a) Histogram
of GNP
(b) Cumulative
values
(c) Histogram Fig. 5. Ergodic
of sample means and normal behavior
curve
for the piecewise linear map.
distribution
R.H. Day and W Shafer, Ergodicfluctuations
4. Dynamical
in economic models
349
systems theory
We shall now give a formal summary of the concepts needed to establish the results that we have just illustrated. Consider the class of all recursive economies whose dynamics on a domain D c R can be represented by a continuous nonlinear map&D-D. The state of the economy in a given period t is given by a value X, ED. The succeeding state is generated by the difference equation xL+
1 =
KG 4 = W,),
(6)
where IZ is a vector of parameters for the function 8. We take D to be a closed bounded interval which means that attention is confined to globally stable models. (This is usually accomplished in economic models by ‘detrending’ the variables.) Define the iterated map 0”: D-+D by 0”(x) =x and 0”(x) = Then the sequence ~(x)--(@‘(x)),~~ is called the 000’-‘(x), n-1,2,3 ,.... trajectory from the initial condition x. The orbit from x, is the set y(x)-{@‘(x)In~O}. Th e asymptotic behavior of a trajectory is described by the limit set w(x) of the trajectory z(x); o(x) is defined to be the set of all limit points of z(x). Note that w(x) is closed and @o(x)) =0(x). An attractor for 8 is a closed set F c D such that w(x) = F for x in a set of positive Lebesgue measure. Attractors represent the asymptotic behavior of solutions for a nontrivial set of initial conditions. The two types of attractors which were illustrated in the numerical examples and which are typical for the more general economies we consider in this paper are those corresponding to stable periodic points and to invariant, ergodic measures absolutely continuous with respect to Lebesgue measure. We now describe these. A point XE D is a periodic point of at least period n if Q”(X)=X and @(X) #Y for j< n. Note in this case that w(@(X)) = y(X) for any j. A periodic point X and the corresponding orbit y(X) are called locally stable if there is a nondegenerate interval V containing X such that w(x)=y(X) for all XE I/: In this case y(X) ix an attractor. It is not unusual for a map to have many periodic points. In fact 0 will have infinitely many periodic points of different periods whenever some power 6”’ of 0 has a periodic point of odd least period n2 3. In this case however, it is possible that none of the periodic points is locally stable. It is in this situation that ergodic theory can be used to describe the asymptotic behavior of trajectories. Let p denote a probability measure on the Bore1 subsets of D with p(D)= 1. The support of p, denoted suppp, is the smallest closed subset D’ of D for which p(D’) = 1. The measure p is said to be invariant for 8 if p(Q- ‘(B)) = p(B) for all Bore1 subsets B of D. It is called ergo&c for 8 if F’(B)=& implies p(B) is 1 or 0 where B is measurable. The simplest example of an invariant ergodic measure for 0 is a measure p which assigns probability l/n to each point in the orbit y(X) of a periodic point of least period n. However,
350
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
this is not a very interesting case. More interesting is when there is an invariant ergodic measure for 8 which is absolutely continuous (with respect to Lebesgue measure). Let A denote Lebesgue measure. An example is the map %(x)=min{2x,l-2x)
for
D=[O,l].
(7)
In this case it is easy to show that for any interval J c [0,l],n(%%‘(J)) = n(J) so that 3, is an invariant measure for 8. It can also be shown to be ergodic. The language of stochastic processes can be used to interpret the significance of the existence of an absolutely continuous invariant ergodic measure for %. If ,U is a Bore1 probability measure on D, (D,p) can be viewed as a probability space, and each H”:D+D as a random variable. We write {%“,p} as the resulting stochastic process, with the joint distribution P, %%‘(B,)). of {%“,%I,. . . ) %“- ‘} given by P,(x:%‘(x) E&, i=O,. . , n- 1)~p(n;Z,’ Each realization (%‘(x)),,, of {H”,p} is, of course, just the trajectory r(x) of 8. If p is invariant for 8, then {%“,n} is a stationary stochastic process, and if in addition p is ergodic, then this process is a stationary ergodic stochastic process. To describe an implication of this, define the indicator function A by A(x, B) = 1 if x E B and 0 if x # B. The function s, is defined by s,(x, B)
=t”$;A(%‘(.+B),
(8)
I
which gives the fraction of points in {x, e(x),. . ., d’-‘(x)} that lie in B. The function s,(x, .) is the probability measure on D which describes the empirical frequency distribution of the first IZ points of r(x). Now suppose p is an absolutely continuous invariant ergodic measure for 8. Then i(supp 11)> 0, and by the Birkhoff ergodic theorem and the continuity of 8 [see Parthsarathy (1967)], the following holds: s,(x;)%p
for
1
a.e.
x,fi
%-‘(suppp). i=O
Here 5 denotes convergence in distribution (or weak convergence of probabability measures). One implication of this is that w(x) = supp p for a nontrivial set of initial conditions x, i.e., suppp is an attractor. But (9) says more: it implies a certain statistical regularity to the trajectories. Indeed ergodicity is a weak form of asymptotic independence for {F,pu). In the example of section 3 we saw that this asymptotic independence appeared to be sufftciently strong to imply Central Limit Theorem results. Define S,:D+R to be the centered sample sum, i.e., n-1
S,(x) =
izoe’(x)-n JWW).
(10)
R.N. Day and W Shafer, Ergodicfluctuations
in economic models
351
If p is an absolutely continuous invariant ergodic measure for 0, with m=Jx~(dx)=lim,,, l/nx;-’ Q’(x), we say that the stochastic process (@‘,p) satisfies the Central Limit Theorem Hypothesis if there is a number a* such that
5 %N(0, 02). $
(11)
It is well known that (7) provides an example. Situations often arise in which we have an invariant measure p for 6’ which is not ergodic. In this case it is often possible to find a finite ergodic decomposition of (0,~) i.e., a finite number of invariant ergodic measures pi,. . , pm for 0 and positive numbers a,, . . . , LX, such that xjaj= 1 and p= cjcxjpj. .One case when this can arise follows. Suppose ,u is an invariant ergodic measure for t9 and the support of p is the union of a finite number of intervals IO,. . , I” 1 which have at most end points in common. Suppose e I modm(li) = I(i+r) modm, Then p is invariant for P, but not ergodic, because empi) = Ii, i = 1,. . . , m. Define ,uj by p,(B)=p(B A Zj)/p(rj). Then each pj is invariant for em, and if each pj is ergodic, then the pj forms an ergodic decomposition of (P’, p). If in addition each {emk,~jjk20 satisfies (1 l), then the trajectories of 0 will appear as random perturbations about a cycle of length m. The computer simulation shown in fig. 5a appeared to have this type of behavior, and our theoretical results will give justification to this interpretation.
5. The basic theorems We have shown that for the business cycle model under consideration, there exists a range of parameters on which fluctuations exist and are perpetuated on a trapping set I/: For this range the map 8, restricted to the trapping set, i.e., 8,: I/+ I/ can be transformed to a map on the unit interval T: Z-+1: = [0,11. We also showed that these maps were piecewise monotonic. In what follows we will restrict the analysis to the class of such maps whose segments are sufficiently smooth. Formally we adopt the following standing assumptions (although they are sometimes stronger than necessary). Basic assumptions (BA) (a) T is continuous. (b) there is a finite number of points (yi}~~~, with y’< yif ‘, i=O,. . . , n, and with y’=O, y”+‘= 1, such that (i) T is C3 on each [y’, y’+‘], and T’(x) #O on (y’, y’+‘),i=O,. . . ,n. (ii) sgn T’(y’- E) + sgn T’(y’+ E)=0 for sufficiently small E>O, for each i=l , . . . , n, that is, y’, i= 1, . . . , n is a turning point of i7
352
R.H. Day and W. Shafer, Ergodicfluctuations
in economic models
As suggested in the previous section, attractors for T often correspond to stable orbits or absolutely continuous invariant ergodic measures. In deriving conditions for these two types of attractors the following definition plays an important role. Let (a, b) be an interval on which T’(x) #O. Then the Schwartzian derivative of T on (a,b) is the map ST:(a,b)-+R given by ST(x)= T”‘(x)/T’(x) - 3/2(( T”(x)/T’(x))‘. Remark 1. In some of the results below the condition ST(x) 5 0 is imposed: this is equivalent to the requirement that 1/1TI”* is a convex function on (a, b). [Misiurwicz (1980, p. 18).] The first theorems of interest will enable attractors are stable, periodic cycles.
us to characterize
models
whose
Theorem I. If ST(x) 50 on each (y’, y'+ ‘), then every stable orbit of T attracts a point in {y’}lf,‘, i.e., if y(X) is a stable orbit then o(y’) =y(X) for some i. ProoK Singer (1978) proved this theorem for the case where T is everywhere C3 and ST ~0. However, his method of proof actually establishes the more general result stated. Q.E.D. Theorem 2. Suppose on each (y’,y’+‘), either ST(x)
R.H. Day and WI Shafer, Ergodicfluctuations
finite that (i) (ii) (iii)
number of absolutely
continuous
invariant
in economic models
measures
(ii)
(iii)
pI,. . . , pk for 7; such
The support of each ,ui is a finite union of disjoint closed intervals, and k does not exceed the number of turning points of T For i almost every x E I, w(x) = supp pi for some i. There exists an integer 1 such that for each i= 1,. . , k the dynamical system {T’, pi> possesses an ergodic decomposition { T1,,ttij}g~ 1 such that each {T’“, pi,} satisfies the central limit theorem hypothesis.
ProoJ: Lasota and Yorke and Kowalski demonstrate m= 1. The above generalization follows by applying T” and then observing (i)
353
(i) and (ii) for the case their result to the map
if 1; is an invariant ergodic measure for T”, then p= l/m~~z~o’ fi. T-’ is an invariant ergodic measure for p. since each turning point y’ is a local max or min of 7; two distinct invariant ergodic measures for T whose supports are a finite union of intervals cannot have the property that their supports have a turning point in common. Q.E.D. result (iii) has been established by Hofbauer and Keller.
Remark 2. Note that the hypotheses existence of stable orbits.
of Theorem
3 explicitly
rule out the
The hypotheses of this theorem also rule out the possibility that the one sided derivatives of T at a turning point y’ are the same, i.e., that T’(y’) =O. The following result allows for this possibility. In particular they enable us to establish comparable result for nonexpansive maps. Theorem
4.
Suppose
(a) ST(x)50 on each (y’,y’+‘). (b) If Tm(x) -2, then IT”‘(x)1 > 1. (c) There is a neighborhood U of {y’}lf,’ such that T”(x)~{y~}~~d for every x E {y’)r=+,’and m 2 0. (d) For each y’ either T’(y’) # 0, or T”(y’) # 0 or T”‘(y’) #O. Then there exists a finite pi,. . , pn for 7; such that (i) (ii) (iii)
number of absolutely
continuous
invariant
u[I\U]
measures
The support of each pi is a finite union of disjoint closed intervals, and k does not exceed the number of turning points of T. For almost every x E I, o(x) = supp ,U~for some i. There exists an integer 1 such that for each i= 1,. . . , k the dynamical system { T’,pi} possesses an ergodic decomposition { T’,uij}$L 1 such that each {T’“, pi,} satisfies the central limit theorem hypothesis.
R.H. Day and W Shafer, Ergodicfluctuations
354
Proof (1985).
Misierewicz Q.E.D.
(1981) established
6. Ergodic fluctuations
in economic models
(i) and (ii) while (iii) is due to Ziemian
for the piecewise linear business cycle model
Reconsider the Type II canonical cycle model of section 3.
map
for the piecewise
linear
business
Theorem 5. Let T be a Type II canonical map and let kz 1 be the minimum integer such that Tk-‘(0) < y** and Tk(0) 2 y**. If bkc < 1 there exists a unique stable orbit of least period k+ 1 which attracts almost every XE I. If bkc> 1 there exists a unique absolutely continuous invariant ergodic measure p for T which attracts almost every x E I i.e., supp p is an attractor). Furthermore, there exists an integer p such that (TP, p) has an ergodic decomposition (ul,. . , , p,,,) such that each (TP,pi) satisfies the Central Limit Theorem Hypothesis. Proof: Choose k as in the theorem. By continuity, there is a z~(O,min(T(0),~~**)] such that Tkm ‘(z)=y**. Note that Tk+l(z)=O. Define F(x)= Tk+l(x)-X. Then F(0)=Tk+l(O)>O and F(z)=-z
f(b)-
k
Furthermore,
l-bk l-bk-’ l_bk
y** < 1 - bk -gk(b).
(14 bkc < 1 is equivalent
to: (13)
R.H. Day and W Shafer, Ergodicfluctuations
355
in economic models
In fig. 3, the region S, + , is the set of parameter values (b,y**) satisfying (12) and (13), and Ek+i is the set of parameter values satisfying (12) and satisfying (13) with the inequality reversed. It is not possible to show on the figure but nevertheless easy to check that for every k 2 1, S, + i # $4and Ek+ 1 # $3. A similar complete characterization for the two (or more) turning point case involves an argument of much greater complexity. So far we have established the following partial results for the Case I canonical map. Let y’ and ys be the unique preimages in [y**,y*] of y** and y* respectively, i.e., T(y’) = y** and T(ys) = y*. In table 1 the inequality expressions are given in terms of these points, the turning points, the stationary state j, the end points and their images. Recall that given T(0) (or T(1)) these can be expressed in terms of the slopes of the linear segments (b,c,cc) which in turn derive from the underlying economic parameters of demand B, y, k, 1, p. In the table S, indicates the existence of stable two period cycles while E, indicates the existence of an absolutely continuous invariant ergodic measure. The results are obtained by constructing the map TZ( .) and exploiting the arguments developed in section 5 and Theorem 5. In cells 21 and 12, the two cases enclosed in heavy square lines, 7”( .) restricted to the intervals [0, yJ] and [y’, l] are Case II and Case III canonical maps respectively so that Table Characterization
of dynamics
1
for the Case I canonical
(1) .I
y"
(2) ;
map.
(3)
y"
(4)
T(O)
I
UOPye
T( l)
T2[ye,ll+[ye,11 yields Case III so Th. 5 applies
TWyf T2:IO,Y~I+IO,Y~I yields Case II canonical map and Th. 5
y*'T(l)
?
?
356
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
Theorem 5 applies directly. A complete characterization as shown in fig. 3 is therefore available in these special situations. For the other cells in the table sufficient conditions for ergodicity can be established for special cases, but the number of situations that has to be examined is large and the calculations tedious. What we have done so far is sufficient for the reader to see what is involved.
7. The piecewise nonlinear business cycle model So far our analysis has exploited piecewise linearity, a property that has enabled us to obtain precise and rather complete answers to our basic questions. But the business cycle model under consideration encompasses more general functional forms as indicated in figs. 1 and 2. Suppose, for example, that the demand for money D”‘(r, Y):=%/(r-r’) + kY do that the LM curve is r = r’ + I,/( M - kY). Suppose also that investment demand is OZYSY'
0, I(r,Y): =
f
bC(Y - W(SYSPWrY
3
YZ Y',
where b, t, p, fl and y are parameters and where Y’ is full capacity and Y’ a threshold above which the Kaldorian multiplier effect of increasing income on investment is positive. Let induced consumption demand be cry and let A be the level of autonomous consumption investment and government demand combined. Then the Mezler-Modigliani-Samuelson model (1) becomes OSYSY
A+a& r,+,=O(Y):= i A+crY,+pLB(Y,-
Y’)f[r’+2/(M-kx)leY,
Y’s Y sp/k,
(14) where B= bpY(4Yf)mP is a constant. If /I and y are both larger than unity then this function has a smooth, ‘cocked-hat’ shape as shown by the solid lines in the diagram of fig. 1. In particular the slope of aggregate demand @( .) is zero at the local maximum and minimum Y** and Y* which exist when p is large enough. Hence, Theorem 3 which requires expansivity throughout cannot be exploited. Instead Theorem 4 must be invoked. Unfortunately, establishing condition a-c of the theorem has so far not been accomplished. In the absence of such analytical conditions we have resorted to computations. In fig. 6a profiles of the map Q(.) restricted to the trapping set and transformed to the unit interval are shown for several values of p. We have computed the criterion l@(y)\* which has the required convex segments on the intervals corresponding to the monotonic segments of the map as shown in fig. 6b. See Remark 1
R.H. Day and W Shafer, Ergodicfluctuations
357
in economic models
(a) The map for several values of p.
(b) The criteria Fig. 6. Qualitative
for the Schwartzian
Derivative
profile of the Type I canonical
Condition
piece-wise
(see Remark
nonlinear,
1)
smooth
map.
above. A bifurcation diagram for p is shown in fig. 7. These results are similar to the ones presented earlier. The regions where stable cycles appear are interspersed with regions where the density of values is apparently dispersed in finite intervals very much like that shown in the piecewise linear case. This suggests that Condition 4b is satisfied. Moreover, the initial conditions for all of the trajectories was y, =O. As T’(y**) = T(y*) = 0
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
359
hypothesis 4c appears to be satisfied in these ‘chaotic’ regions. As 4d is readily verified it seems reasonable to conjecture that Theorem 4 is applicable and that (14) possesses an absolutely continuous invariant measure and behaves like a stationary stochastic process for a large number of parameter values. These conjectures are illustrated in fig. 8 which gives the computed density function for the trajectory and the sample sums.
I _______‘/Y LL
(a) Histogram
of GNP
values
(c)Histogram
(b) Cumulative
of sample means
Fig. 8. Ergodic
behavior
and the normal
for the smooth
distribution
curve
map.
7. Conclusion Our primary purpose has been to provide a complete comparative dynamic analysis in a familiar economic setting, one in which the major questions of interest could be posed precisely and answered definitively. This we have been able to do. Although similar results may be difficult to derive for less manageable models the analysis and numerical illustration presented here suggest, nonetheless, how carefully designed computational studies can
360
R.H. Day and W Shafer, Ergodicfluctuations
in economic models
be interpreted in less tractable cases. We therefore think our study has implications beyond the mere exegesis of ergodic dynamics in general or of the Keynesian business cycle model in particular. So far as the latter is concerned Day and Lin (1986) have shown that plausible parameter values for the piecewise linear model of sections 3 and 6 lead to results that further illustrate the ergodic theory applied here. In particular they present bifurcation results for various policy parameters (supply of money, tax rate, government expenditure) which show that changes in such controls can trigger qualitative changes in the behavior of aggregate fluctuations, thus introducing a new source of uncertainty to the effects of macroeconomic instruments.
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in economic models
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Modigliani, F., 1944, Liquidity preference and the theory of interest and money, Econometrica 12, 45-88. Morley, S.A., 1983, Macroeconomics (The Dryden Press, Chicago, IL). Parasarathy, K., 1967, Probability measures on metric spaces (Academic Press, New York). Pianigiani, G., 1978, On the existence of invariant measures, Applied Nonlinear Analysis, 299307. Pianigiani, G. and J. Yorke, 1979, Expanding maps on sets which are almost invariant: Decay and chaos, Transactions of the American Mathematical Society 252, 351-366. Pohjola, M., 1981, Built-in flexibility of progressive taxation and the dynamics of income: Stability, cycles or chaos? (Department of Economics, University of Tampere, Tampere). Samuelson, P.A., 1948, Foundations of Economic Analysis (Harvard University Press, Cambridge, CA). Singer, D., 1978, Stable orbits and bifurcation of maps of the interval, SIAM Journal of Applied Mathematics 35, 260-267. Smital, J., 1984, A chaotic function with a scrambled set of positive Lebesgues measure, Proceedings of the American Mathematical Society 92, 5&54. Ziemian, K., 1985, Almost sure invariance principle for some maps of an interval, Ergodic Theory and Dynamical Systems 5, 6255640.