Competitive business cycles in an overlapping generations economy with productive investment

JOURNAL

OF ECONOMIC

THEORY

46, 45-65 (1988)

Competitive Business Cycles in an Overlapping Generations Economy with Productive Investment BRUNO JULLIEN * GSAS,

Department of Economies. Harvard University, Cambridge, Massachusetts O-7138

Received June 23, 1986; revised June 25. 1987

This paper examines the emergence of long-run competitive periodic equilibria in a two-period overlapping generations model with capital accumulation. The existence of periodic equilibria is a consequence of the existence of an aggregate bubble (money) and of a “non-linear” behaviour of aggregate savings. The technique used is to show that the problem can be reduced to the study of a dynamical map from an interval into itself, although it is originally two dimensional. This will allow us to study the occurrence of cycles from a global point of view. Journal of Economic Literature Classification Numbers: 021. 023. cr 1988 Academic Press, Inc.

INTRODUCTION

Since D. Gale [ 131 pointed out that equilibrium cycles may arise in overlapping generations (OLG) models, it is known that a perfectly competitive economy can exhibit persistent fluctuations under “laissez-faire.” The emergence of cycles in an OLG model with no capital accumulation was studied recently by Grandmont [ 151. The object of the present paper is to study the dynamics of a two-period OLG model when the technology requires both capital and labor. (See woodford [ 191 for a good survey on OLG models.) The existing literature on the subject studies the appearance of cycles near the steady state through bifurcation theory. Both Farmer [IZ] and Benhabib and Laroque [4] show that fluctuations can occur if savings decrease with the interest rate or outside money is negative. Reichhn [17] shows how there can be cycles when there is no outside money but enough complementarities in the technology. Due to the nature of the bifurcation * I am very indebted to Jean-Michel Grandmont for introducing me to the subject and for the constant interest he has shown in my work. I am grateful to Philippe Aghion, Patrick Bolton, Bernard Caillaud, Andreu Mas-Colell, Eric Maskin, and Jean Tirole for useful discussions and comments.

45 0022-0531/88 $3.00 CopyrIght !I? 1988 by Academtc Press, Inc All rights of reproducfmn in any form reserved

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JULLIEN

theory, all the results are local: fluctuations are nearby the steady state and occur when the parameters of the model are close to some “critical value.” In this paper an alternative approach is proposed. We will provide a global analysis of the dynamics of the model. Under an assumption of substitutability of the production function, we will show that the long-term behaviour of the economy is captured by a one-dimensional dynamical system, although the original problem is two dimensional (prices and capital). The advantage is that global results are available on the cyclical behaviour of one-dimensional maps. The theory developed is based on the saving-investment relation in the context of a monetary economy. It will be shown that if the response of savings to the interest rate is nonmonotonic, the dynamics of the economy may induce excessive movements of investment and periodic equilibria may exist. The phenomenon described is monetary: the existence of some nominal asset is necessary to relax the link between investment and aggregate saving, and generate cycles through self-fulfilling expectations on returns. In the present paper this will be done through inflation with a fixed quantity of money, but the approach is valid for any kind of agregate bubble. The paper is organised as follows. Sections 1 and 2 expose the model and define the equilibria. The model used is very close to Diamonds original OLG model [ 111, and to its version developed by Tirole [18] for the study of asset bubbles. Section 3 examines the dynamics and shows how to reduce the dimension of the problem. In particular it exhibits an invariant curve to which all periodic equilibria must belong and studies its properties. Section 4 studies the emergence and the nature of cycles.

1. THE

MODEL

The model is an extension of Diamond’s version [ 1 l] of the OLG model. At each date t (t goes from 0 to infinity) a single-good is produced. This good can be consumed during the period or stored as an input (capital good) for future production. Each generation lives two periods and reproduces identically. The young generation sells one unit of labor inelastically at a real wage u’,, consumes the quantity Ci, of the good in the first period, and saves the real quantity S, for next period consumption by holding money and capital. The old generation spends all its savings from the previous period. A typical consumer is characterized by his utility function U(C,,, C,,) and faces the following intertemporal maximization problem:

ENDOGENOUS BUSINESS CYCLES

47

s.t.c,, + s, < w, C,, G R, +1. S, c;, z 0, i= 1, 2. R I+, is referred to as the real rate of interest between t and t + 1. Notice that we have assumed away labor substitution. (For an analysis of intertemporal labor substitution, see Barro and King [ 1 I.) Under standard assumptions, the consumer’s decision problem has a unique solution characterized by the savings function S( w,, R, + 1). We assume that the savings function S is derived from a nicely behaved utility function and verifies: Assumption (1.a). O
S(W, R)=

differentiable,

increasing with W, and R . S(w, R) is

+a,

lim RS(w, R)= + co. R- +m Production is made through a neoclassical constant return to scale technology. Output per capita is a function of capital intensity, ~,=f(k,), where f is a gross production function including depreciated capital. Assumption (Lb). f is increasing, strictly concave on R + , and C2 on R*, , lim S’(k) E [0, l[, k+ +cc lim kd

kf’(k)

f(k) - kf’(k) = +co,

+m

lim f’(k) = + co, k-0

lim f(k) - kf’(k) = 0, k-0

is non-decreasing.

The first conditions are standard.’ The last one will be used to insure some regularity of the dynamical system (namely it will be characterized by a monotonic map).’ In our model it means that profits do not decrease ’ The assumptions on the limit properties of the wage and the rate of interest are mainly technical. We could drop them and the main results of the paper would remain the same, but the mathematics would become much more complicated. * This assumption could be relaxed but would become less intuitive. We want the revenue from monetary savings to increase with the rate of interest (.f’(k)(S(~t./‘(k))-k) decrease with k). 642/4611-4

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when the amount of investment increases. This seems reasonably to assume at an aggregate level, and it requires that there are not too many complementarities in production.3 For example, it is verified by the CobbDouglas production functions and by the CES production functions with an elasticity of substitution larger than 1. The competitive behaviour of firms leads to the equalization of the marginal productivity of each factor to its cost:

iv, = f(k,)

- k,f’(k,)

= W(k,).

We will refer to W(k) as the wage function. It is well known that under Assumption ( 1.b) this function is C’ on RT , increasing, and maps R + into itself. There is a fixed nominal amount of money M available in the economy. The capital assets are assumed to be sold at their market fundamental value.4

2. PERFECT FORESIGHT (PF) EQUILIBRIA At date 0, the economy is endowed with a fixed quantity of capital k, inherited from the past and a quantity of money M. A perfect foresight equilibrium is a sequence of prices (p,),,,, capital stocks (k,),zO, interest which achieves a competitive temand real wages (u’,),>~ rates (RILdO, porary equilibrium with perfect foresight at each date. We characterize such an equilibrium by using the capital stock and the real quantity of money M/p, denoted m,. A PF equilibrium is then a sequence (k,),,,, verifying, at each date t 3 0, (m,),,. m,+k,+,

(2.1 f

=S(Wk,),f’(k,+,))

m,,., =f’(k,+l)m,,

k, > 0, m, 2 0, k, given.

(2.2)

Equation (2.2) expresses that the interest rate on money p,/p,+ 1 is equal to the interest rate on capital f”(k,+ ,). Equation (2.1) then equalizes the demand and supply of assets, given that the labor market is in equilibrium. We call an equilibrium non-monetary or monetary according to m, = 0 or m,>O. 3 The case of complementarity has been studied by Reichlin [ 171. 4 This is mainly for convenience. As it will appear from system (2.1), (2.2) of Section 2, the model just expresses that there is an aggregate bubble in the economy. For a discussion of bubbles, see Tirole [IS].

ENDOGENOUS

BUSINESS

49

CYCLES

It appears from the system above that, when the savings function is not monotonic, the equilibria cannot be characterized by using a forward dynamic map (i.e., the equilibrium values at date t + 1 are not functions of the equilibrium values at date t or less). On the contrary, (2.1), (2.2) induce a well-defined backward dynamics. Combining the two equations, we can replace (2.1) by

Under Assumptions (1 .a) and (l.b), the right-hand side of (2.3) increases from 0 to infinity with k,. So we can invert the relation and express the current capital stock as a function of the future capital stock and the future real quantity of money. We denote this relation

The function g is defined for k, + , > 0 and m, + ,3 0. LEMMA (2.1). g is C’, increasing in each of its arguments, and g(k, m) tends to 0 (resp. infinity) when k goes to 0 (resp. infinity) with m fixed.

Proof S(Wk,h f’(k,+,))~f’(k,+l)-m,+,-k,+lf’(k,+l) is C’, increasing with k,, and decreasing with k, + , , m, + , , so the first part follows from the implicit function theorem. When k,, 1 goes to 0, S( W(k,), f’(k,+ , )) .f’(k,+ 1) stays bounded, which is only possible if k, tends to 0. W(k,)>S>k,+, implies that k, tends to infinity when k,, I goes to infinity. Q.E.D. Now define the C’, increasing map F as F(k, m) = (g(k, m), m/f’(k)) Then a PF equilibrium t>O

for

k > 0 and

is a sequence (k,, m,),,,

m > 0.

such that

(k,,m,)=F(k,+,,m,+,) k, > 0, k, given m, > 0 if monetary equilibrium m, = 0 if non-monetary

equilibrium.

(2.4)

Steady States We make no assumption about the non-monetary economy contrary to Diamond [ 111 and Tirole [ 181. We assume that a monetary steady state exists (the Samuelson case in the terminology of Gale [ 131). We refer to

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this steady state by using the notation by

X* = (k*, m*). It is uniquely defined

f’(k*)

m*=S(W(k*),

= 1

(2.5)

1)-k*.

(2.6)

From Assumption (l.b), k* is well defined. In order for a monetary steady state to exist we need to assume: Assumption

(2.a).

S( W(k*),

1) > k*.

LEMMA 2.2. Under Assumptions (La), (l.b), and (2.a) there exist a unique monetary steady state X* and at least one inefficient non-monetary steady state (k > k*).

Proof: The first point has already been shown. A non-monetary steady state is a level k of capital stock which verifies S( W(k), f'(k)) = k. It is inefficient if k > k*. It is straightforward to show that W(k) < k for k large enough. So S( W(k), f’(k)) < W(k) < k, for k large enough. The result follows by continuity of S( W(k), f’(k)). Q.E.D. Among all the inefficient non-monetary steady states, the less capital intensive one will be of special interest. We will refer to it as k, = inf{k > k*/g(k, 0) = k}. 3. BACKWARD

DYNAMICS

AND CYCLES

A periodic PF equilibrium (cycle) in a periodic sequence (k,, m,),30 which verifies (2.4). The function g(k, 0) being increasing, the only periodic non-monetary PF equilibria are the steady states. The set of cycles is in bijection with the set of periodic orbits of the map F: if (k,, mt),bO is a cycle with period p, {(k, +p _ , , m f +p _ ,), .... (k,, m,)} is a periodic orbit of F with period p. The problem of the existence of periodic orbits of twodimensional map is in general studied by using the bifurcation theory.5 The drawback is that the theory only provides a local analysis. The strategy adopted in the present paper is to exploit the monotonicity of F to reduce the dimension to one. This is the purpose of the present section. We will s A bifurcation occurs when the stability of a family of maps changes for some value of the parameters. It is associated with the emergence of an invariant set nearby the steady state. A Hopf bifurcation is characterized by the emergence of an invariant circle and happens when the eigenvalues at the steady state are complex and cross the unit circle. A Flip bifurcation is characterized by the emergence of a period 2 cycle and happens when one eigenvalue crosses -1. (See Guckenheimer and Holmes [16] for an exposition of the theory, and Benhabib and Nishimura [S, 61 and Dana and Malgrange [g] for earlier applications to economics.)

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show that all the periodic orbits must belong to a C’ invariant curve. As a consequence, we will be able in the subsequent section to focus on the dynamics reduced to the curve and to derive a global analysis of the cycles. Define F”(X) = (k,(X), m,(X)), where X= (k, m). Notice that, from the budget constraint, the image of F is bounded by the relation W( g(k, m)) > m/f’(k) + k, which expresses that the young generation saves less than its wage income. It follows that

Wk,(W) >m,(X).

(3.1)

This property, along with the monotonicity of F, allows us to derive the results summarized in the following diagram:

k*

k,.

k

The real quantity of money increases to infinity along the orbit of a point greater than (k*, m*), and decreases to 0 along the orbit of a smaller point (this is because the interest rate remains smaller than 1 in the former case and larger than 1 in the latter case). Between these two behaviours it is possible to exhibit a set of points with an orbit bounded away from 0 and infinity. This set is a C’ curve. THEOREM 3.1. There exist a compact set Kc Rr2 and a function h, decreasing and C’, from R*, to R*, , such that if we define the sets

f = {(k, m)ER:‘/k=h(m)) I-+ = {(k,m)~R:*/k>h(m)) I-

= {(k,m)ERT2/k
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then {r, r+ , C } ts . an F-invariant partition of R*,‘, and XEr,o

lim n-+x

m,(X) = +cc

XEro

lim ,z++r

m,(X) = 0

XE~QF”(X)EK ProojI

,for n large enough.

See Appendix.

Modified versions of the theorem can be found for different sets of assumptions. We give here two of them, skipping the proof for simplicity. It can be obtained by slight modifications of the original proof. If f’(k) remains bounded when k goes to 0, the function h is defined for rn in a bounded interval. If W(k) remains bounded when k goes to infinity, F”(X) will eventually not exist for XE r+ . The useful and robust result is that all cycles belong to the invariant curve. In fact, for cyclical points, we can find some additional information on the derivative of h. PROPOSITION 3.1. Let X= (k, m) be a cyclical point of order p. Then X belongs to r and ( 1, h’(m)) is the eigenvector of DFP(X) associated to its smallest eigenvalue (it has two distinct real eigenvalues).

Proof: Let X= (k, m) be a cyclical point of order p (p is the smallest integer such that FP(X) = X). Then X belongs to r because its orbit is bounded. Define V= (1, h’(m)), V is tangent to r at X. Since r is C’ and F-invariant, DFr(X) . V is also tangent to r at X. Therefore there exists c( that DFP(X) . V= CIV. V is an eigenvector of DFP( X). But DFP(X) has all its elements positive. It has two distinct real eigenvalues. The coordinates of the eigenvector associated to the largest eigenvalue have the same sign. The coordinates of the eigenvector associated to the smallest eigenvalue are of opposite signs. h’(m) is negative, so a must be the smallest eigenvalue. Q.E.D.

The next step is the characterize the limit properties of h( .). Since the equilibria are bounded by the budget constraint relation (3.1), we are only interested in the behaviour of h(m) when m becomes small. PROPOSITION

3.2. h(m) tends to k, when m goes to 0.

Proof. Let k0 be the limit of h(m) as m goes to 0 (+cc allowed). Then k,> k* = h(m*). Suppose first that k,< k,. As k, is the smallest k> k* such that g(k, 0) = k and, by Assumption (2.a), g(k*, 0)
ENDOGENOUS

BUSINESS

53

CYCLES

therefore F”(li,, 0) <(k*, m*). By continuity of F, for m small enough, F”(h(m), m) < (k*, m*), which is impossible. Suppose now that k, > k,, then for m small enough (h(m), m) > (k,, 0). This is true for all the iterates of (h(m), m), which implies Vn>O

m,(h(m),

m) > ml(f’(k,))“.

As f’(k,)
when n goes to Q.E.D.

Not surprisingly we find that when m goes to 0, h(m) tends to a nonmonetary steady-state stock of capital. Notice that in the case of a nonmonetary economy a forward dynamics is well defined by k, = g(k,+ , , 0) since g is increasing. In this dynamics k, is a locally stable stationary equilibrium. The global stability of the model is a delicate question because of the inexistence of a forward map which characterizes the equilibria. The role of the curve I- is illustrated in the last result. THEOREM

(3.2).

one of the following

is a competitive PF equilibrium, two situations must prevail:

If (A’,),,,

then at least

* X, converges to r as t goes to infinity, * X, converges to an inefficient non-monetary steady state as t goes to irzfinity. ProqJ:

See Appendix. 4.

PERIODIC

EQUILIBRIA

We have seen in the previous section that all cycles must belong to the curve I-. This allows us to restrict the dimension of the map to one by focusing on the dynamics on IY So we can define the new function6

m + m/f ‘(h(m)). It has the following properties: 4 is C’, d(m) - m is positive for m < m*, negative for m > m*. There is a &invariant compact set K4 of R: such that for all m, d”(m) belongs to K4 for n large enough (K, is the projection of K on the second axis). h In order to show the similitude of the results with an economy without investment, one could express the dynamics as follows. Define m(R) = h- ‘[(/‘)-‘(R)]/R, m(R) is the monetary savings when the equilibrium belongs to f, and the rate of interest is R. An equilibrium on f is characterized by m(R ,+ ,) = R,m(R,). m( .) has all the usual properties of a savings function.

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The cycles of 4 are equivalent Proposition 3.1) m#O,

JULLIEN

to the cycles of F in the sense (from

F”(k,m)=(k,m)ok=h(m)

and

f(m)

= m.

This result shows that there is a systematic relation between the real quantity of money and the stock of capital along any periodic equilibrium. If we use output y, rather than the stock of capital, we find the relation yr =f(hh)). So on any cycle, the real quantity of money is a decreasing function of output. In order to clarify what happens on a cycle, let us examine the case of a period 2 cycle. A boom is characterized by a high level of output and wage due to previous period large accumulation. The consumption of the old generation is low because this generation has saved during the previous recession when the wage and the interest rate were low. The equilibrium is reached by stimulating aggregate consumption. If the wage is not very sensitive to output, its effect on consumption will be insufficient. If in addition the wealth effect is predominant in the savings decision, the adjustment is made by an increase of the interest rate (or expected deflation). The result is a low level of investment. There is a small excess saving which is compensated by a contraction of the real quantity of money. Therefore the boom is characterized by a high inflation as anticipated by the generation born during the recession. It is clear that this requires a strong wealth effect in aggregate savings. Notice that the cycle can be broken if the new equilibrium is reached by an increase of investment rather than private consumption, and this is the reason of the multiplicity of temporary equilibria at each date. 4.1. The Emergence of Cycles One of the main results on the dynamics of maps on the interval is that cycles can only appear in a specific order. This is known as Sarkovskii’s theorem (see Collet and Eckmann [7] or Grandmont [ 143). We can apply the theorem first to the restriction of 4 to K,, and next to F. Consider the following ordering of the positive integers: 3>5>7> .‘. >2,3>2.5>2.7>

...

> 2” .3 > 2” .5 > 2” . I > . . ..

.

>2”>.

. >4>2>1.

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55

THEOREM 4.1. If a cycle of order p exists, then there exists a cycle of order n for any positive integer n such that p > n in the above order.

This theorem has important implications. First it shows that a necessary condition for the existence of some cycle is the existence of a period 2 cycle. Thus period 2 cycles are of particular interest and will be analysed in greater detail. Second, except when there is no cycle, multiple periodic equilibria coexist. In particular a sufficient condition for the existence of cycles of all orders is the existence of a period 3 cycle. Between these two limiting cases, the model is able to present any configuration of periodic allowed by Sarkovskii’s order. Let us mention that the existence of a period 3 cycle is also associated with chaotic behavior of some trajectories. This will not be analysed in the paper. (For studies of erratic behaviors in different economic set-up, see Benhabib and Day [2, 31, Day [9, IO].) 4.2. Cycles of Order 2

We know that for m sufficiently small 4’(m) is greater than m. Hence if the derivative of 4’ evaluated at m* is greater than one there will exist a period 2 cycle. LEMMA

(4.1).

d’(m*) is the smallest eigenvalue of DF(k*, m*).

Proof. &, verifies Vm > 0, (h(#m)), 4(m)) = F(h(m), m). By differen tiating with respect to m, we obtain @(m*) h’(m*) d’(m*)

The result comes from Proposition

(4.1

= DF(k*, m*) .

(3.2).

Q.E.D.

Looking directly at DF(k*, m*) we see easily that b’(m*) is less than 1 (it is a positive matrix with 1 as a diagonal term). Going back to the savings and production functions, the following condition of existence is obtained: THEOREM (4.2). order 2 is that

A sufficient condition for the existence of a cycle of

S(W(k*),l)-kk*+2S’,(W(k*),1)+2k*S’,(W(k*),

I)-2/f”(k*)
(4.2)

Proof: A direct calculus shows that Eq. (4.2) is equivalent to d’(m*) < -1. For m sufficiently small, 4’(m) is equivalent to m/f’(k3) and is greater than m. qb’(m*)2> 1 so that for m smaller than but close to m*, b2(m) cm. By continuity there exists a point m between 0 and m* such that d’(m) = m. Q.E.D.

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The condition is rather strong but not implausible. It is satisfied when savings decrease enough with the real rate of interest at the stationary equilibrium. More precisely the elasticity of the savings function with respect to the interest rate evaluated at the monetary steady state must be at least strictly less than -4. If this condition is fulfilled, we will be able to find a production function such that a period 2 cycle exists (see Example 1). An elasticity less than - f is the sufficient condition of existence of a period 2 cycle found by Grandmont [ 151 in the case of a one-factor technology.’ It extends to the present model because we assume enough substitutability. Allowing more complementarity would relax this requirement. The restrictions on the production function imposed by (4.2) can be expressed as restrictions on its concavity. As a general consequence of the existence of a monetary steady state, we know that W(k*) > k*. This is equivalent to the fact that the elasticity of production with respect to the capital stock is less than t : k*f’(k*)/f(k*)

< $.

(4.3)

Condition (4.2) implies that in addition this elasticity be less than half the elasticity of the interest rate with respect to the capital stock: k*j-‘(k*)
.f(k* 1

-k*f”(k*) 2 f’(k*)

(4.4)

’

Condition (4.4) characterizes the set of technologies for which we can find a utility representation such that (4.2) is verified (see Example 2). Since, from Assumption (l.b), the right-hand side is less than t, condition (4.4) is stronger than condition (4.3). For example, if the production function is Cobb-Douglas, f(k) = Ak”, it requires CI< f. Examples of the Existence of a Cycle of Order 2

1: FIXED UTILITY FUNCTION. such that the savings function verifies EXAMPLE

Let f, be the production f,(k)=k”/(l

Let the utility

function

be

function -,)(‘P%r,

O
‘In fact the condition is on monetary savings, If m(u: R) is the non-productive savings function (m= S-k, where k is seen as a function of R), (4.2) can be rewritten m* + 2(&1/6R)* - ~(&I/cSW)*(&V/~R)* < 0, where the asterisk means evaluated at the steady state. Under this form the condition is the same for a one-factor or a two-factor technology.

ENDOGENOUSBUSINESSCYCLES

57

Then the steady-state capital stock is k, = a/( 1 - CI) with a wage of 1. Assumptions (1.b) and (2.a) are verified for CI small enough. For these production and savings functions, the condition (4.2) is S(1, 1)+2&(1,

l)+s;,.(l,

l)a/(l

-cr)+2cr/(l

-U)‘
When CI is close to 0, the third and fourth terms vanish and there exists a period 2 cycle. What happens is that when o! goes to 0 we find the one-factor linear technology as a limit case. To see that, the best is to consider the interest rate R =f’(k) rather than k, because k, tends to 0 when c( goes to 0. The dynamics is given by S(R;“l

l’,R,+,)R,+,=m,+,+R;$‘,-“‘a/(l-a) m,=m,+llR,+l.

For a fixed (R,, , , m,+,),ifS(l,R,+I)R,+I#m,+,,thenR,willtendto 0 or + cc when c( goes to 0. This means that the curve I- rewritten in terms of (R, m) will tend to S( 1, R)R = m. On I’, for a finite number of iterations and o! small, the dynamics will be close to the dynamics of the one-factor technology (CX= 0) S(l, R,+,)=S(l,

414

m,+, =mA+,. So the analysis made by Grandmont [ 151 can be extended to the case where c1is small. In particular if the one-factor technology has a cycle of order n, (R,, .... R,), there will exist a period n cycle (R,,, .... R,,) close to (R,, .... R,) for CI small, with a cyclical fluctuation of output of the first order in a given by (1 - a Lor RLzr .... 1 - ~1Log R,,,). EXAMPLE

2:

FIXED

verifying Assumption

TECHNOLOGY. Let f be a production function (1.b) and (4.4). Let b be a positive constant such that

W(k*) > b > -k* -2/f

“(k*) 2 k*.

We choose the separable utility function ~(C,,C,)=C;-6’/(1-q)+b”2.C~-“2/(1-a,) cJi>O,

i= 1,2.

Now, fix all parameters except (T* and consider what happens when (T* goes to infinity. The savings function is given by [IV - S]“’ = l/R . [ RS/b]“*.

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S( W(k*), 1) is a continuous function of cz and tends to b when (TVgoes to + co. Since b is greater than k*, a monetary steady state exists for ez large enough. The derivatives of S at this steady state are Sl,( W(k*), 1) =

GI CT,+ az( W(k*) - S)/S

Si( W(k*), 1) =

(1 -az)(Wk*)-S) cr, + a*( W(k*) - S)/S’

S’, tends to 0, while SR tends to -b. At the limit condition b > -k*

(4.2) becomes

- 2/‘f”(k*).

Thus a period 2 cycle exists for e2 large enough. This example confirms that cycles will emerge when the so-called “Arrow-Pratt measure of relative risk aversion” on second period consumption is large compared with first period consumption (see Grandmont [ 15 ] ). 4.3. Cycles of Order 3 In order to prove the existence of a period 3 cycle, we have to find a point between 0 and m* such that #3(m) k*, m* > m > m,(X).

Proof: Let X be such a point. Suppose that k> h(m), then X> (h(m), m). This implies that m3(X) > m,(h(m), m). By definition of 4, m* > m > d3(m). Suppose now that k -Ch(m), then h-‘(k) exists and is less than m*. It verifies (k, h-‘(k))>X, so k< k3(X)< k,(k, h-‘(k)). By combining with h-l, we obtain m* > h-‘(k) > qS3(h-l(k)). The existence of a period 3 cycle follows by continuity of q5since for m small, d3(m) > m. It is immediate that the condition is necessary. If (ki, m,),= ,. 3 is a period 3 cycle, then f ‘(k,) f’(k,) f’(k,) = 1 so that for some i, ki > k* and mi = h-‘(k,) cm*. The point (ki, m,) verifies the conditions of the theorem. Q.E.D. This result can be used in two ways. First one can find such points through computer simulations. Second it allows us to exhibit formal examples of existence of a period 3 cycle. The second approach is used here.

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Example of the Existence of a Cycle of Order 3 The example is derived from the second example period 2 cycle. It shows that if in addition to a large production function has a low elasticity with respect then a period 3 cycle will exist. The result presented Cobb-Douglas production function with an elasticity

of the existence of a concavity of U,, the to the capital stock, remains valid for a small enough.

COROLLARY. Let the production function be f(k) = 1 + x + Log(k), for k > exp( -x), and the utility be U(C,, C,) = Log C, + 2”C4p0/( 1 - a). When x>x*, where x* is the largest root of (x - 1) exp(3 - x) = 1, a cycle qf order 3 exists for o large enough.

,Prooj

As long as k. > exp( -x), k, =exp(k,(l

the backward dynamics is given by

+m,)-x+k,[(l

+m0)/2]“)

m, =m,k,. The steady-state capital stock is k* = 1 and 1 + m* + [( 1 + m*)/2]” = x. When x > 2, m* tends to 1 when c goes to infinity (notice that x* > 2). We choose the initial point such that m,k,= 1. The successive iterates are then k, =exp(k,+

1 -x+k,[(l

+m,)/2]“),

m, = 1,

k,=exp(3.k,-x), k,=exp(k,(l

m2=ki, +k,)-x+k,[(l

+k,)/2]“),

m,=k,

exp(3.k,-x).

When x > .Y*, by choosing k, close enough to x - 1 but larger, the iterates will verify l
lim k,= 0’ + I,

+oo,

1 >m,=

l/k,>btrflX

m3.

Q.E.D.

CONCLUSION

We have shown how endogenous fluctuations may occur in an OLG economy with investment and money. By choosing a one-sector model with an inelastic supply of labor and a well-behaved production function we emphasized the role of the financial sector. It is clear from the results that what is really determinant is the demand for unproductive assets (money). This demand results jointly from savings and investment decisions. This suggests that it may be fruitful to relax the assumptions made on the productive sector. In particular what happens in a multisector model is an open question.

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The main technical tool was the possibility to reduce the problem from two dimensions to one. For that purpose, we used a result close to the stable manifold theorem. The main difference is that it is global and applies even when both eigenvalues are unstable. This result relies mainly on the monotonicity of the map and its range of applicability should go much beyond the framework of the model studied.

APPENDIX

Proof of Theorem (3.1) IS,

When there will be no ambiguity we will note F”(X) = (k,, m,). Define = {c-x*), IS- = {x
A.l.

XEZS,

(resp. IS-)*m,(X)+,,, IS,

and IS-

+m +oo (re.sp.0).

are F-invariant.

ProoJ: Let X E IS,, then for all n > 0, F’“(X) % X*, so belongs to IS,. m n+1 =m,/f’(k,) > m,,, by k,, > k+. So the sequence m, is increasing. Suppose it has a finite limit nz > m *, then f’(k,) must tend to 1 and k, to k*. This implies that g(k*, m) = k* and so m = m*, which contradicts m > m*. By the same argument, if X < X*, m, is decreasing and tends to 0. Definition Let

and Properties of the Compact Set K

Kl={W~‘(m*)
*, W(k*)), m*/f ‘( W- ‘(m*)) d m d m* )

K=KluZQ. LEMMA

A.2.

(Vn > 0, F”(X) E CC) * (3N/Vn 2 N f “(X) E K),

if

XE K,

N=O. Proof We first show that there exists n such that F”(X) E K. Let us investigate two exhaustive cases: (a) Ina l/m, am*, k,m, so W(k*)>m, and W-‘(m*) c k,. F”(X)E Kl. (b) Vn 2 1 m, d m*, k, 2 k*. As for all n, f ‘(k,) d 1, the sequence m, is non-decreasing and bounded above. So it converges, which is only possible if F”(x) tends to X* (see proof Lemma A.l). As the intersection of

ENDOGENOUS

a small neighborhood enough.

BUSINESS

of X* and CC is in K, F”(X)

We now show that F”(X)E K=>F”+‘(X)e (a) (b)

61

CYCLES

is in K for n large

K.

If m,,+, >m* and k,+l k*.

If F”(X) E Kl then m n+, >m*/f’(W‘(m*)) and k,,+] m,~m*/f’(W~‘(m*)) and k*
g(k*,

W(k*)).

LEMMA

F”(X’)

F”+ ‘(X) E K2.

A.3. If X> x’, belong to CC.

it is impossible

that for

all n 3 0, F”(X)

and

ProojI From Lemma A.2 and the fact that F is increasing we can restrict to (X, xl) E K2. Let X> x’ and F”(X) E CC and F”(Y) E CC, for all n. When n > 0, F”(X) 9 F”(X’) and both belong to K. Then

m,,, I lm L+l =(m,lm:,)(f’(k~)lf’(kn))>mnlml,~m,lm;=17>

1.

Let 6 be the minimum on {(k, m) E K, (k’, m’)E K, m/m’> IT) of g(k’, PI)- g(k’, m’): k,+l -kL+, ,> g (kk, m,) - g(kL, ml) 3 6 > 0. Let 6, be the minimum on {(k, m) E K, (k’, m’) E K, k -k’ 2 6) off’(k’)/“(k): 6, > 1 This ratio must tend to infinity, which is and m,,+ Ill+ I >8,(m,/mL). impossible since K is bounded. Q.E.D. Existence

of h

We construct h(m) as a limit of a functional Let the sequence (p,), be given by

dp,(m),

sequence.

ml = pn- 1Cm/!‘(pn(m))13

p,(m) = k*.

This defines a sequence of C’, decreasing functions (for n > 0) which has the property k = p,,(m) o k,(k, m) = k*. Let the sequence (l,), be given by

L(kYf’(k)=L,Cg(k

L(k))l,

I,(k) = m*.

As before we have defined a sequence of C’, decreasing functions which this

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BRUNO JULLIEN

time has the property m = f,(k) o m,(k, m) = m*. It is straightforward derive that lim Z,(k)=

to

tco,

k-0

either lim, _ + ~ l,(k) = 0, or lim,, +m l,(k) = (lim,, +a3 f,-,(k)) . (lim k- + a f’(k)). e minimum of 1, tends to 0 when n goes to infinity. For -T-h m positive, p,,(m) and (Z,)-‘(m) exist for n large enough. LEMMA

A.4.

For n large enough and s non-negative,

infCp,(mh C’(m)1 Q Pn+s(m)yL+!.(m) ~wCp,(m),

C’(m)l.

Proof If k k* and m,(k, m) 0, F”(k, m) E CC. Lemma A.3 shows that for a fixed m there can be at most one such k. Therefore the two sequences converge to the same limit. We define h(m)= The CC, with and

lim p,(m)= n-t +a,

lim l;‘(m). n+ +m

proof of Lemma A.4 shows also that the orbit of (k, m) will remain in end up in IS, or IS, according to k=, >, of
Continuity

of h

LEMMA AS. Let [a, b] be a closed interval of R*, , then p,, is uniformly convergent on [a, b].

Proof: Let E>O, Then Vn 2 N,

and N be such that s~pr~,~, Il;l(m)-p,(m)\

GE.

su~Ih(m)-p,(m)l~~~~,l1"~(m)-p.(m)l Cash1

4

< sup II,‘(m)

- p,(m)1 GE.

CabI

As a uniform limit

of continuous

functions, h is continuous.

Q.E.D.

ENDOGENOUS

Differentiahilit~~

BUSINESS

63

CYCLES

of’ h

We first show the differentiability on the intersection of r (graph of h) and K. Lemma A.2 shows that I-n K is F-invariant. Let XE f n K, and (X,),, be a sequence in I-, converging to X, such that (X,Y- X)//l X,Y- XII has a limit V when s goes to infinity. We show that there is only one possible limit V, up to the sign. For all n >, 0, F”(X,y) - F”(X)

IIx., - XII

z

DF”(X). V.

Since h(m) is decreasing, V must satisfy Vn 3 0 DF”(X) the cone { .YY< 0). Define DF(F”-

‘(X)) = M,, = a” I c,

. VE Co, where C, is

” 4, I

For all n, F”(X) E K so that a,,, b,, c,, d,, are bounded above and below by positive numbers. P, = M, M, , . . . M, and all its components are positive. The reciprocal of Co by P,, is the closed cone C,: (X/JJ + B,/A,I) (-K/Y + DJC,) < 0. We write this cone as -oln 6 x/y < --(T~,~, where g,,, - oZn = I B,,/A,, - D,/C,,l. As M,, is positive, C, + , c C,,. So 0, C, is reduced to a line if G,,~- c2,, tends to 0 when n goes to + co. B n+ I ---

A l7+1

Dnt, C n+l

I I

B = ‘-2 A, B,, < --I An

lan+,4+,-bn+~c,+,I A,C,, C,, (a,+,A.+b,+,C,,)(c,+,A.+d,+lC,) D, sup(c,+,An>dn+,Cn) CnI c,+,An+dn+,C,,

D

Notice that since A,,/C,, = (a,, A,, ~ , + b,, C,, ~ 1)/( c, A, ~ I + d,, C, ~ I ) is bounded, cn+,AnIdn+, C, is bounded above and below. Therefore there exists some fl< 1 such that 0 < 0, n+ 1- (T~,~+, < /?(a,, - (T~,~).crln - oZ,, tends to 0 and 0, C, is reduced to a hne. As VE n, C,, V is unique up to the sign. Continuity

of the Derivative

Let o,,,(X) X= continuity of and a&X), DE’“(X),

and a,,,(X) be the boundaries of the cone C, associated to (k, m). Since F is C’ these two functions are continuous. The h’(m) is an immediate consequence of the continuity of a,,(~) the inequality a,,(X) < -h’(m)< a,,(X), and the fact that

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BRUNO JULLIEN

a,,(X) - az,(X) tends to 0 when n goes to infinity (the limit of two adjacent sequences of continuous functions is continuous). So Tn K is C’. The orbit of any point (h(m), m) of r ends up in the interior of f n K. In a neighbourhood of m, h verifies k,(h(m’), From the implicit function decreasing everywhere.

m’) = h(m,(h(m’),

m’)).

theorem,

this insures that h(m) is C’ and

(X,),=

(k,, m,),, where t goes from 0 to

Proof of Theorem (3.2)

We denote an equilibrium infinity. LEMMA

A.6.

All equilibria are bounded away from infinity.

All equilibria verify W(k,) > k,, , +m,. This implies that + m,. Since for k large enough W(k) < k, W’(k,) is bounded. > k, + , Q.E.D.

Proof:

W’(kJ

LEMMA

A.7.

If, for all t, X, 2 X* (resp. < ), then (X,), converges to X*.

Proof Suppose that for all t, X, > X*. Then f ‘(k,) < 1 and m, is nonincreasing. As it is bounded below, m, has a limit which is only possible if X, converge to X* (see Lemma A.1). The proof is symmetric for X, < X*. Q.E.D. LEMMA

A.8.

If X = (k, m) is an adherencevalue of (X,),, either XE r or

m = 0. Proof: From Lemma A7, we can assume with no loss of generality that X0 E CC. As X0 = F’(X,), this implies that

inf(p,(m,),

1, ‘(m,)) d k, 6 sw(p,(m,)~

As pt and 1; l converge uniformly,

t’(ml)).

if m # 0, X must belong to r.

LEMMA A.9. If X= (k, 0) is an adherence value of (X,),, monetary steady state, k > k,, and (X,), converges to X.

Q.E.D. X is a non-

Proof Suppose that for some T, k,< k,. As (k,, 0) is a fixed point, for all t 3 T, k, d k,. The invariance of IS,, 1%) and Lemma A.7 implies that after some time X, E CC. We can assume X,E CC, t 20. But then inf( p,(m,), 1, ‘(m,)) d k, G k,, which implies that k = k,. Assume now X,, close to (k,, 0) so that X,, 4 ZSUK. A corollary of the proof of Lemmas A.1 and A.2 is that ISUK is F-invariant. So, for t 3 0, X,$ ISUK. But K has

ENDOGENOUS

BUSINESS

CYCLES

65

been built such that if k, k* for t large and k 2 k,. It follows that m, is decreasing and converges to 0. If k,, , = g(k,+2, m,,,) > k, = then k,+?>k,+, (since m,+,