A PL Model of Traps & Cycles

A Piece-wise Linear Model of Credit Traps and Credit Cycles: A Complete Characterization

By

Kiminori Matsuyama, Northwestern University, USA Iryna Sushko, Institute of Mathematics, National Academy of Science, Ukraine Laura Gardini, University of Urbino, Italy

Prepared for MDEF2012 7th Workshop “Modelli Dinamici in Economia e Finanza” September 20-22, 2012 Urbino, Italy 1

A PL Model of Traps & Cycles

1. Introduction  Matsuyama’s (AER 2007) regime-switching model of credit frictions, where  Entrepreneurs have access to heterogeneous investment projects  Due to credit frictions, entrepreneurs’ net worth affects their ability to finance different projects  A change in the current level of net worth causes credit flows to switch across projects with different productivity  This in turn affects the future level of net worth.  It was shown that this model generates a rich array of dynamic behavior.  Credit Traps  Credit Cycles  Leapfrogging  Reversal of Fortune  Growth Miracle  But, a complete characterization of the dynamic behavior was lacking.  Here, we offer a complete characterization for Cobb-Douglas production function, which makes the dynamical system piece-wise linear. 2

A PL Model of Traps & Cycles

2.A regime switching model of credit frictions: A quick review of Matsuyama (2007) A Variation of the Diamond OG model Final Good: Yt = F(Kt,Lt), with physical capital, Kt and labor, Lt. yt  Yt/Lt = F(Kt/Lt,1)  f(kt), where kt  Kt/Lt; f(k) > 0 > f(k).

Competitive Factor Markets: t = f(kt);

wt = f(kt)  ktf(kt)  W(kt) > 0.

Agents: A unit measure of homogeneous agents. In the 1st period, they supply one unit of labor, earn and save W(kt). In the 2nd period, they consume. Their objective is to maximize the 2nd period consumption.

3

A PL Model of Traps & Cycles

Investment Technologies: Agents can choose one (and only one) from J indivisible projects (j = 1,2, …J). Period t Project-j:

mj units of final good

mj: the (fixed) set-up cost,

Period t+1 

mjRj units of capital

Rj: the project productivity

To Invest or Not to Invest? By starting a project-j,

Ct = mjRjt+1  rt+1(mjwt),

By lending,

Ct = rt+1wt

Profitability Constraint:

Rjf(kt+1)  rt+1,

(j = 1,2,…,J)

(PC-j)

Credit Frictions (introduced by the pledgeability constraint a la Holmstrom-Tirole): Borrowing Constraint: λj:

λjmjRjf(kt+1)  rt+1(mj W(kt)),

(BC-j)

the pledgeable fraction of the project revenue 4

A PL Model of Traps & Cycles

Both (PC-j) and (BC-j) must be satisfied for the credit to flow into type-j projects. What is the maximal rate of return the borrowers can pledge to the lenders from running a type–j project? From (PC-j) and (BC-j),

rt+1/f'(kt+1)

(BC-j)

(PC-j) Rj rt 1  f ' (k t 1 ) max1, 1  W (k t ) / m j /  j 

Rj





λjRj

O

W(kt) (1–λj)mj

5

A PL Model of Traps & Cycles

Equilibrium Conditions

(1)

W(kt) = j(mjXjt).

(2)

kt+1 = j(mjRjXjt).

(3)

Rj rt 1  f ' (kt 1 ) max 1, 1  W (k t ) / m j /  j 





(j = 1, 2,…J)

where Xjt is the measure of type-j projects initiated in period t, and Xjt > 0 (j = 1, 2,…J) implies the equality in (3).

For each kt, we can rank the projects by the RHS of (3). Thus, generically, only one type of project, J(kt), gets all the credit; Xjt = 1 if j = J(kt), and Xjt = 0, otherwise. Hence, we call this “a regime-switching” model.

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A PL Model of Traps & Cycles

This means that eqs. (1)-(3) are simplified to:

(4)

k t 1  RJ ( k )W (k t ) . t

For k0 > 0, (4) determines the equilibrium trajectory. (W (k t ) is assumed to be concave, so that the dynamics would be simple without regime-switching.) 

With the Cobb-Douglas production, yt  Ak t  with 0    1, eq. (4) can be rewritten as a PWL system with a regime-dependent constant term: (5)

xt 1 =  Jˆ ( x )  xt t

by defining,

xt  log b k t ;

 Jˆ ( x )  log b (1   ) ARJ ( k ) t

t

where Jˆ ( xt )  J (k t ) and b is the base of the logarithm. 7

A PL Model of Traps & Cycles

Below, let us focus on the following case considered in Matsuyama (2007, Sec.4). R2 > R1 > λ2R2 > λ1R1, and m2/m1 > (1λ1)/(1λ2R2/R1).  Project 1 is less productive and less pledgeable than Project 2.  Project 1 requires the smaller set-up cost than Project 2.

R2 R1 λ2R2 λ1R1 O

W(kc)

W(kt) W(kcc)

8

A PL Model of Traps & Cycles

(8)

R2W(kt) if

kt < kc or kt > kcc

R1W(kt) if

kc < kt < kcc.

kt+1 =

kt+1

O

kt+1

kt kc k* kcc k**

O

kt+1

kt k* kc k** kcc

kt O

k* kc kcc k**

Figure 5a

Figure 5b

Figure 5c

Credit Trap or Leapfrogging or Reversal of Fortune

Credit Cycles

Cycles as a Trap or Growth Miracle 9

A PL Model of Traps & Cycles 

For the Cobb-Douglas Production; yt  Ak t  with 0    1:

 2  xt

if

xt  d1 and xt  d 2

1  xt

if

d 1  xt  d 2

xt 1 =

by defining, xt  log b k t ; 1  log b R1 (1   ) A   2  log b R2 (1   ) A ; d1  log b k c  < d 2  log b k cc , Three goods (final, capital, labor)  Two degrees of freedom in choosing units of measurement. We set 1 /(1 )  R2  A  1 / R1 (1   ) and b    , R  1 so that 1  0 and  2  1   . Then,

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A PL Model of Traps & Cycles

f L  xt   (1   )  xt

if

xt  d1 and xt  d 2

f R  xt   xt

if

d 1  xt  d 2

xt 1  f ( xt ) 

with the two discontinuity (or switching) points, d1  d 2 , given by:

  (2 / 1 )( R2 / R1 )  1  R1m2 ; d1  log b  (  /  )( R / R )( m / m )  1  2 1 2 1  2 1 d 2  log b 1  2 ( R2 / R1 )  R1m2  Note:  If the credit always flowed to the less productive type-1 projects, xt 1  f R  xt   xt , converging monotonically to x *R  0 .  If the credit always flowed to the more productive type-2 projects, xt 1  f L  xt   (1   )  xt , converging monotonically to x L*  1 .  Credit friction parameters, (1 , 2 , m1 , m2 ) , affect dynamics through ( d1 , d 2 ). Let us see how this PWL system changes with ( d1 , d 2 ). 11

A PL Model of Traps & Cycles

4.

Analysis

A Preview of the Results in the parameter space, ( d1  d 2 ), for   0.3 and   0.7

12

A PL Model of Traps & Cycles

Three Relatively Simple Cases: In all these cases, the dynamics globally converges to its unique steady state and the equilibrium trajectory changes the direction at most once. Case S-I: ( d1  0 & d 2  1 ), Orange; convergent to its unique steady state, x *R  0 .  If d 11    0 , monotone increasing for x0  (,0) and monotone decreasing for x0  (0, ) , as shown in the left Figure.

 If d 11    0 , monotone increasing for x0   n0 f Ln (d 1,0) & monotone decreasing for x0  (0, ) , as shown in Red in the right Figure. However, for x0  f Ln (0, d 11   ) , as shown in Green, it is monotone increasing for the first n periods and monotone decreasing afterwards.

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A PL Model of Traps & Cycles

Case S-II ( d1  d 2  0 ),Yellow; convergent to its unique steady state, x *L  1.

 monotone decreasing for x0  (1, ) .  monotone increasing for x0  (,1) . However, it is possible to have leapfrogging, i.e., x0  y0 and then xt  yt after

some periods, as shown in this Figure.

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A PL Model of Traps & Cycles

Case S-III (1  d1  d 2 ), Yellow; convergent to its unique steady state, x *L  1.  monotone increasing for x0  (,1) .

 monotone decreasing for the first (n+1) periods and then monotone increasing afterwards for x0  f n  f R1 ((0,1)  (d 1, d 2 )) .  monotone decreasing x0  (1, ) /  n 0 f n  f R1 ((0,1)  (d 1,d 2 )) .

(d 1 d 2  1)

(d 1 1  d 2 ) 15

A PL Model of Traps & Cycles

Cases A: Orange and Yellow Stripes ( d1  0  d 2  1): Both x *R  0 and x *L  1 are steady states Case A-I; (1   )  d1  d 2 (Above the line r). “Lower Steady State as a Trap” Two basins of attractions are simply connected and separated by d 2 ; B(0)  (, d 2 ) and B(1)  [d 2 , ) .  For x0  B (1) , monotone decreasing for x0  1 and monotone increasing for d 2  x0  1. Blue  For x0  B (0) ,  If d 11    0 , monotone increasing for x0  0 & monotone decreasing for 0  x0  d 2 .  If d 11    0 , monotone increasing for

x0   n0 f L n (d 1,0) & monotone decreasing for 0  x0  d 2 . Red . For x0  f Ln (0,d 11   ) , monotone increasing for the first n periods and monotone decreasing afterwards. Green 16

A PL Model of Traps & Cycles

Case A-II: (1   )  d1  d 2 , (Below the line r). “Reversal of Fortune” If x0  (d1 , d 2 ) , xt  x *R  0 ; If x0  d 2 , xt  x *L  1. If x0  d1 , x t converges to either x *R  0 or x *L  1. Blue indicates B(1) , the basin of attraction for x *L  1. Red indicates B(0) , the basin of attraction for x *R  0 . Two basins of attraction, B(1) & B(0) , are disconnected; they alternate and each accumulates to the origin in the space of k  b x .

17

A PL Model of Traps & Cycles

Case B: ( 0  d1  1  d 2 ); “Cycles for all initial conditions.” Neither x *R  0 or x *L  1 are steady states. For all x 0 , the path enters I  (d1 ,1    d1 ] in a finite time and continues fluctuating inside I. xt+1

Figure shows the 2-cycle

x0  f R  f L  x0  ; x1  f L  f R  x1  In symbolic dynamics (SD), LR =RL,

45 1 1‒α+αd1 1‒α

fL(xt)

It exists and is globally stable for: 1    d1   x0 , x1    ,    LR , 1   1     shown in Pink.

α fR(xt) αd1

 LR is symmetric around d1  0.5 . O

x0

d1

x1

xt 1 18

A PL Model of Traps & Cycles

Figure shows the 3-cycle of the form, x0  d1  x2  x1 x0  ( f R ) 2  f L  x0  , x1  f L  ( f R ) 2  x1  & x2  f R  f L  f R  x2 

xt+1 In symbolic dynamics (SD),

45 1

LR2 = R2L=RLR. It exists and is globally stable for:

fL(xt)

d 1   x0 , x 2 

 (1   ) 2 (1   )  =  ,    LR 3 3  1    1

1‒α 2

α fR(xt)

shown in Red on the left side of  LR .

O

x0 d1x2

x1

xt 1

19

A PL Model of Traps & Cycles

Figure shows the 3-cycle of the form, x1  x2  d1  x0 x0  ( f L ) 2  f R  x0  , x1  f R  ( f L ) 2  x1  & x2  f L  f R  f L  x2 

xt+1 In symbolic dynamics,

45 1

RL2 = L2R=LRL. It exists and is globally stable for:

fL(xt)

d1   x2 , x0   (1   ) 2 (1   )   1  , 1     RL , 3 3  1 1  

1‒α

2

α shown in Red on the right side of  LR .

fR(xt)

 LR &  RL are symmetric around d1  0.5 . 2

2

xt O

x1

x2 d1 x0

1

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A PL Model of Traps & Cycles

More generally,  The (n+1)-cycle of the form, x0  d1  xn  ...  x1 ; LRn =…=RLRn‒1 in SD x0  ( f R ) n  f L  x0  , ..., xn  ( f R ) n 1  f L  f R  x n  ;

exists and is globally stable if d1   x0 , xn 

 (1   ) n (1   ) n1  =  ,    LR . n 1 n 1 1  1  n

 The (n+1)-cycle of the form, x1  ...  x n  d 1  x 0 ; RLn =…=LRLn‒1 in SD x 0  ( f L ) n  f R  x 0  , …., x n  ( f L ) n 1  f R  f L  x n  ;

 (1   ) n (1   ) n 1  exists and is globally stable if d 1   x n , x 0 = 1  ,1     RL . n 1 n 1 1 1   n

The periodicity regions,  LR accumulates to d 1  0 The periodicity regions,  RL accumulates to d 1  1 . They are symmetric around d1  0.5 . n

n

21

A PL Model of Traps & Cycles

Cycles of the Higher Levels of Complexity Cycles of the form, RLn and LRn for n ≥ 1 are called the First Level of Complexity.

 (1   ) n (1   ) n  In the gap between  LR and  LR , i.e, d1   ,  for any integer n ≥ 1, n 2 n 1  1  1  there exist two infinite sequences of periodicity regions of cycles of the Second Level of Complexity, n 1

 LR n ( LR n1 ) m accumulating to  LR

n 1

n

and  LR n 1 ( LR n ) m

for each integer m ≥ 1.

and  LR , respectively. n

22

A PL Model of Traps & Cycles

To see this, define a new map on the interval (the gap between  LR and  LR ), as follows: (1   ) n n TL  xt   f R  f L  xt  if  xt  d1 1   n2 xt 1 = (1   ) n n TR  xt   f R  f L  f R  xt  if d1  xt < , n 1 1  xt+1 which can be rewritten as: B TL  xt   AL xt  B if  xt  d1 1  AR xt 1  B TR  xt   AR xt  B if d1  xt < , TL(xt) 1  AL where AL   n1 > AR   n 2 and B  (1   ) n . n 1

n

45

TR(xt) Therefore, following the same procedure, we can find: B/(1‒AR)

d1

B/(1‒AL) 23

xt

A PL Model of Traps & Cycles

 The (m+1)-cycle of the symbolic sequence, TL(TR)m  The [n+1+m(n+2)]-cycle of f with the symbolic dynamics LRn(RLR n)m,  Its periodicity region:  B(1  ARm 1 ) B[(1  ARm )  AL (1  AR ) ARm 1 ]   d1   , m m  (1  AR )(1  AL AR )  (1  AR )(1  AL AR )  accumulates to the right edge of  LR , as m →∞. n 1

 The (m+1)-cycle of the symbolic sequence, TR(TL)m  The [n+2+m(n+1)]-cycle of f with the symbolic dynamics, RLRn(LR n)m,  Its periodicity region:  B[(1  ALm )  AR (1  AL ) ALm 1 ]  B(1  ALm 1 )  d1   , m m  (1  AL )(1  AR AL ) (1  AL )(1  AR AL )   accumulates to the left edge of  LR , as m →∞. n

24

A PL Model of Traps & Cycles

 This procedure can be repeated infinitely many times. Thus, between the periodicity regions of the cycles of the kth-level of complexity, there are two infinite sequences of the periodicity regions of the cycles of the (k+1)th-level of complexity.  The union of all the periodicity regions thus constructed does not cover the entire interval of d1  (0,1) .  The set of d1 left is a set of measure zero. On this set, the trajectory is quasi-periodic, dense in the invariant set, which is a Cantor set.

25

A PL Model of Traps & Cycles

This Figure shows the periodicity regions for Case B (α = 0.7).

26

A PL Model of Traps & Cycles

This Figure shows how the periodicity regions for Case B change with α.

27

A PL Model of Traps & Cycles

Bifurcation Diagram, tracing the orbit of stable cycles as a function of d1  (0,1)

28

A PL Model of Traps & Cycles

The Rotation (Winding) Number: We can calculate, along the stable cycles, what fraction of the periods the economy is in an expansionary stage (that is, on the left side of d1). For the k-period cycles, along which the periodic orbit visits p times on the L side and k‒ p times, we can associate its rotation number, p/k. For example, On cycles of first level of complexity:



1 for LRn 1 n

&



n for RLn . 1 n

On cycles of second level of complexity between LRn+1 and LRn :



1 m for LRn(RLR n)m (1  n)  m( 2  n)

& 

1 m with RLRn(LR n)m, (2  n)  m(1  n)

and so on. More generally, 29

A PL Model of Traps & Cycles

Between the two periodicity regions of cycles whose rotation numbers, p1/k1 < p2/k2, are Farey neighhors, (i.e., they satisfy p1k 2  p 2 k1  1), we can find the periodicity regions of cycles with the rotation number: p1 p 2 p1  p 2 Farey composition rule: .   k1 k 2 k1  k 2 p p p p p p p p Since 1  1  2  2 and 1  2 is a Farey neighbor of both 1 and 2 , k1 k1 k2 k2 k1 k 2 k1 k2 this can repeat itself ad infinitum. Thus, the periodicity region for the rotation number equal to any rational number between 0 and 1 can be found, as shown by Farey tree.

30

A PL Model of Traps & Cycles

Furthermore, The rotation number can be expressed as a function of d1, ,  ( d1 ) . It is  continuous;  non-decreasing;  goes up from zero to one. Yet,  have zero derivate almost everywhere. That is, it is not absolutely continuous. A singular (Cantor) function, often referred to as the Devil’s staircase.

31

A PL Model of Traps & Cycles

Cases C: ( 0  d1  d 2  1); x *L  1 is the unique steady state. Furthermore,  All the stable cycles discussed in Case B survive as long as d2 is greater than the rightmost location along its orbit.  As soon as d2 collides with the orbit, the stable cycles are destroyed.  This explains the lower boundary of the periodicity regions, as shown in the Figure.

32

A PL Model of Traps & Cycles

In this Figure, Between the periodicity region of LR and LR2, A few regions of the cycles of the second level of complexity, can be seen.

33

A PL Model of Traps & Cycles

Case C-I: ( 0  d1  d 2  1 & 1    d1  d 2 ; Above the line, “r”). “Cycles as a Poverty Trap.” All the stable cycles survive and coexist with the steady state, x *L  1. Furthermore, two basins of attraction are simply connected, separated at d2. Blue: The basin of attraction for x *L  1. B(1)  [d 2 ,  )

Red: The basin of attraction for the cycles: B(C )  (, d 2 )

34

A PL Model of Traps & Cycles

Case C-II: ( 0  d1  d 2  1 & 1    d1  d 2  d1 ). “Growth miracle” If x0  d 2 , xt  x *L  1. If x0  d 2 , Case C-IIa: For some initial conditions, x0  d 2 , the path eventually crosses over d 2 and converges to x *L  1. For other initial conditions, the path fluctuate forever inside I  (d1 , d 2 ] . The periodicity regions are shown in the figure, the area below the line, r. B 0 (1)  (d 2 , ) ; The immediate basin of attraction for x *L  1.

B n (1)  f  n ((d 2 , d1  (1   ))) , The set of initial conditions from which, after n iterations, the path escape to the immediate basin of attraction for x *L  1. B(1)   n 0 B n (1) : The basin of attraction for x *L  1.

B(C )  R \ Cl ( B(1)) ; The basin of attraction for the cycles. Again, two basins of attraction are disconnected. 35

A PL Model of Traps & Cycles

Case C-IIb: (Yellow) For all x0  d 2 , the path eventually crosses over d 2 so that x *L  1 is globally attracting. That is, B(1)   n0 B n (1)  R

However,  The equilibrium trajectory changes its direction many times (unlike the other area of yellow, where it changes its direction at most once). Furthermore,  The structure of B n (1)  f  n ((d 2 , d1  (1   ))) can be quite complicated.

36

A PL Model of Traps & Cycles

This figure shows the co-existence of period-7 cycles and the intervals from which the orbit will eventually escape.

37

A PL Model of Traps & Cycles

In this figure,   0.7 and d 1  0.3252 and d 1  (1   )  0.52764 > d 2  0.5276 .

The 11-cycle exists. Furthermore, some paths escape above d 2 , as shown. The numbers of iteration required before the escape are indicated.

38

A PL Model of Traps & Cycles

Then, at d 2  0.525 , The 11-cycle no longer exist. All converges to x *L  1. Yet, some paths fluctuate for long time before the escape.

39

A PL Model of Traps & Cycles

This figure illustrates how the periodicity regions change with  for Cases C. ( d 2  0.8 ).

40

A PL Model of Traps & Cycles

Here’s the rotation number for Case C (α = 0.3; d 2  0.8 ).

Here’s its blow-up.

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A PL Model of Traps & Cycles

Here’s the rotation number for Case C (α = 0.7; d 2  0.8 ).

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A PL Model of Traps & Cycles

5. Some Concluding Remarks  A regime-switching model of credit frictions, by Matsuyama (2007), can display a wide array of dynamical behavior.  This paper showed that a complete characterization of the dynamic behavior on the parameter space is feasible for a PWL case. Among others, it showed: How stable cycles of any integer period can emerge. Along each stable cycle, how the economy alternates between the expansionary and contractionary phases. How asymmetry of cycles (the fraction of time the economy is in the expansionary phase) varies with the credit frictions parameters. How the economy may fluctuate for a long time at a lower level before successfully escaping from the poverty, etc.  The analysis was done for a restrictive set of assumptions (2 projects with 2 switching points), because it creates a rich array of dynamics with a relatively few parameters. With more projects, more switching points, generating even richer behaviors.  The discontinuity and piecewise linearity simplify the analysis. Similar results can be numerically obtained when the discontinuous, piecewise map is approximated by a continuous map with very steep slopes.  More generally, the analytical tool used in this paper should be useful for many other dynamic economic models. 43