This note develops a simple macro model where the pattern of wealth accumulation is determined by a credit multiplier and by the way households react to short run fluctuations. In this setup, long term wealth dynamics are eventually characterized by the presence of endogenous cycles.Credit constraints; Financial development; Consumer confidence; Endogenous business cycles; Nonlinear dynamics

Gomes, Orlando

English

Research Papers in Economics

 
 
Constraints on Credit, Consumer Behaviour and the 
Dynamics of Wealth 
 
Orlando Gomes
* 
 
 
Escola Superior de Comunicação Social [Instituto Politécnico de Lisboa] and  
Unidade de Investigação em Desenvolvimento Empresarial [UNIDE/ISCTE]. 
 
 
 
- January, 2007 - 
 
 
 
 
Abstract:  This  note  develops  a  simple  macro  model  where  the  pattern  of  wealth 
accumulation is determined by a credit multiplier and by the way households react to short 
run fluctuations. In this setup, long term wealth dynamics are eventually characterized by 
the presence of endogenous cycles.     
 
Keywords: Credit constraints, Financial development, Consumer confidence, Endogenous 
business cycles, Nonlinear dynamics. 
 
JEL classification: O41, E32, C61. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                 
* Orlando Gomes; address: Escola Superior de Comunicação Social, Campus de Benfica do IPL, 1549-
014  Lisbon,  Portugal.  Phone  number:  +  351  93  342  09  15;  fax:  +  351  217  162  540.  E-mail: 
ogomes@escs.ipl.pt. 
Acknowledgements:  Financial  support  from  the  Fundação  Ciência  e  Tecnologia,  Lisbon,  is  grateful 
acknowledged,  under  the  contract  No  POCTI/ECO/48628/2002,  partially  funded  by  the  European 
Regional Development Fund (ERDF). Constraints on credit, consumer behaviour and the dynamics of wealth  2 
 
1. Introduction 
 
The relation between collateral requirements or credit constraints and business 
cycles  is  a  relevant  theme  of  discussion  in  today’s  macroeconomic  analysis.  This 
subject has gained visibility with the contributions of Bernanke and Gertler (1989) and 
Kyotaki  and  Moore  (1997),  and  was  further  developed  with  the  work  of  Aghion, 
Bacchetta  and  Banerjee  (2001,  2004),  Demirguç-Kunt  and  Levine  (2001),  Amable, 
Chatelain  and  Ralf  (2004),  Aghion,  Angeletos,  Banerjee  and  Manova  (2005)  and 
Caballé, Jarque and Michetti (2006), among others. The basic intuition underlying the 
previous references is that markets where firms face some degree of credit constraints 
are  markets  where  investment  is  strongly  pro-cyclical  and,  thus,  main  economic 
aggregates  will  be  subject  to  amplified  volatility  that  tends  to  persist  over  time. 
Basically, the lesson one withdraws from this literature is that cycles are more likely to 
be observed for specific levels of financial development than for others.
1 For instance, 
Caballé, Jarque and Michetti (2006), hereafter CJM, conclude that stability is found for 
low and high levels of financial development, while for intermediate levels endogenous 
business cycles dominate. 
In this note, we recover the CJM model to present an alternative approach to the 
formation  of  endogenous  cycles  for  intermediate  levels  of  financial  development. 
Basically, we consider a same scenario as the previous authors, but with two important 
changes: first, we consider physical capital as the unique production input (we ignore 
the  country  specific  input  with  a  constant  supply  assumed  in  the  referred  model); 
second,  we  introduce  a  mechanism  through  which  households  respond  to  short  run 
wealth  deviations  from  a  potential  wealth  level.  Furthermore,  the  analysis  will  be 
undertaken under an endogenous growth framework, in the sense that it assumes an AK 
production function.  
This note is organized as follows. Section 2 presents the model’s structure, section 
3  discusses  local  dynamics;  section  4  characterizes  global  dynamics  and  section  5 
concludes. 
 
2. The Structure of the Model 
 
Consider a competitive economy populated by a large number of households and 
firms. Firms produce a tradable good under an AK production function, yt=Akt, with 
                                                 
1 The level of financial development is translated on the degree of constraints to credit. Constraints on credit, consumer behaviour and the dynamics of wealth  3 
 
A>0 a technology index and kt, yt the per capita levels of physical capital and output in 
moment t.
2 We assume that capital fully depreciates after one period and, hence, it=kt, 
with it per capita investment. Households have the possibility to lend their financial 
resources  directly  to  firms  if  the  marginal  productivity  of  capital  (A)  is  above  the 
economy’s nominal interest rate (r); hereafter, we impose this constraint on parameters: 
A>r. 
If  the  credit  market  is  subject  to  some  kind  of  imperfection,  firms’  financial 
resources (that we designate by wealth) will serve as collateral for the loans, and thus 
firms cannot borrow an amount over mwt, with wt the level of per capita wealth and m a 
credit  multiplier  that  reflects  the  degree  of  financial  development  of  the  economy. 
Households and firms agree on applying to the productive projects the largest amount of 
credit that can be subject to transaction, and thus investment in moment t corresponds to 
it=(1+m)×wt. Finally, the structure of the model is complete with a difference equation 
reflecting wealth dynamics, 
 
t t t t c w r y w - - = + m 1 ,  w0 given.  (1) 
    
Equation (1) states that wealth in moment t+1 corresponds to income in t, less the 
cost  of  debt  and  less  the  resources  allocated  to  consumption  (ct  is  per  capita 
consumption). In the CJM model, ct corresponds to a constant fraction of income less 
debt  payment.  We  generalize  this  assumption  by  considering  that  the  marginal 
propensity to consume depends on the observable difference between effective levels of 
wealth and expected or potential wealth. In practice, agents react to business cycles by 
adopting the following rule: the higher the level of last period’s wealth relatively to the 
benchmark level of wealth, the more optimistic households will be and, accordingly, the 
higher will be the share of consumption out of income. In other words, low (high) levels 
of  observed  accumulated  wealth  relatively  to  a  benchmark  level  will  imply  a 
precautionary (confident) behaviour that is translated on a higher (lower) savings rate. 
 Formally, we consider  ) ( ) ( 1 - × - × = t t t t w g w r y c c m , cÎ(0,1) and  ) ( t w g  a positive, 
continuous and differentiable function, with g’>0. Consider 
*
t w  as the potential level of 
wealth; this is supposed to represent a wealth trend that grows at a same rate, g, for all t. 
                                                 
2 To simplify, assume that population does not grow. Constraints on credit, consumer behaviour and the dynamics of wealth  4 
 
Thus, function g will be such that  1 | ) ( * =
= t t w w t w g ,  1 | ) ( * >
> t t w w t w g  and  1 | ) ( *<
< t t w w t w g . 
The following functional form fulfils the required properties: 
a
t
t
t w
w
w g  


 


= * ) ( , a>0. 
The reduced form of the dynamic system is straightforward to obtain given the 
previous information, 
 
[ ] t
a
t
t
t w
w
w
c r A A w ×








 


 


× - × × - + =
-
-
+ *
1
1
1 1 ) ( m   (2) 
 
Because production is subject to constant marginal returns, all relevant variables 
(kt, yt, it, ct and wt) grow at a constant positive rate in the steady state. Let this rate be g, 
and thus we define variable  t
t
t
w
w
) 1 (
ˆ
g +
º  and constant  t
t w
w
) 1 (
ˆ
*
*
g +
º . Effective wealth 
grows at a same rate as potential wealth in the steady state, but before this long run 
result is eventually accomplished, the growth rates might differ. Rewriting (2),  
 
t
a
t
t w
w
w
c
r A A
w ˆ
ˆ
ˆ
1
1
) ( ˆ
*
1
1 ×
















× - ×
+
× - +
=
-
+ g
m
  (3) 
 
Equation (3) has a unique equilibrium point: 
*
/ 1
ˆ
) (
1
1
1
w
r A A c
w
a
× 





 


 


× - +
+
- × =
m
g
. 
This result allows us to state the following proposition, 
 
Proposition  1:  The  wealth  model,  with  credit  constraints  and  consumption 
reaction to deviations from last period’s potential wealth, reveals that the higher is the 
level of financial development of the economy, the larger will also be the amount of 
accumulated wealth, in the steady state. 
 
Proof:  Take  the  steady  state  expression  for  the  wealth  variable  and  compute 
derivative 
m
m ¶
¶
=
w
w . The computation gives, Constraints on credit, consumer behaviour and the dynamics of wealth  5 
 
 
[ ]






× - +
- × +
× × 





 


 


× - +
+
- × × =
-
2
1
*
) (
) ( ) 1 ( 1
) (
1
1
1 ˆ
m
g
m
g
m
r A A
r A
c r A A c a
w
w
a
a
.  Because  this  is  a 
positive value, one infers that the accumulated level of wealth is positively correlated 
with financial development (measured by the credit multiplier parameter, m)￿ 
 
Note, relatively to the steady state value, that to guarantee  0 > w , the following 
inequality must hold:  g m + > × - + 1 ) ( r A A . This condition imposes a floor to the value 
of the credit parameter: 
r A
A
-
- +
>
g
m
1
 is the minimal requirement for the economy to be 
able to accumulate wealth. 
 
3. Local Dynamics 
 
In this section, we address the dynamics of equation (3) in the neighbourhood of 
point  w . This requires defining variables  w w w t t - º ˆ ~  and  w w z t t - º -1 ˆ ~ . With these 
variables,  we  turn  equation  (3)  into  a  two  equation  system  with  two  endogenous 
variables and just one time lag, 
 



 


=
- + ×















 +
× - ×
+
× - +
=
+
+
t t
t
a
t
t
w z
w w w
w
w z
c
r A A
w
~ ~
) ~ (
ˆ
~
1
1
) ( ~
1
* 1 g
m
  (4) 
 
Around the balanced growth path, system (4) takes the linearized form  
 






×








+
+ - × - +
× -
= 





+
+
t
t
t
t
z
w r A A
a
z
w
~
~
0 1
1
) 1 ( ) (
1
~
~
1
1
g
g m
  (5) 
 
Note that the steady state values of variables  t w ~  and  t z ~  are, in both cases, 0. Proposition 
2 synthesizes the local dynamics result. 
 Constraints on credit, consumer behaviour and the dynamics of wealth  6 
 
Proposition  2:  The  wealth  model  under  analysis  is  locally  stable  for 












-
- + × 




 +
-
- +
Î
r A
A
a
a
r A
A
) 1 (
1
;
1
g
g
m ;  when 
r A
A
a
a
-
- + × 




 +
=
) 1 (
1
g
m ,  the  system 
undergoes a Neimark-Sacker bifurcation. 
 
Proof:  The  Jacobian  matrix  in  (5)  has  a  positive  determinant, 
g
g m
+
+ - × - +
× =
1
) 1 ( ) (
) (
r A A
a J Det , and its trace is Tr(J)=1. Thus, stability conditions 
1-Tr(J)+Det(J)>0  and  1+Tr(J)+Det(J)>0  are  always  satisfied.  The  only  possible 
bifurcation occurs when the eigenvalues of the matrix are a pair of complex conjugate 
values  with  modulus  equal  to  one,  which  is  equivalent  to  say  that  Det(J)=1.  The 
equality expression in the proposition is determined by solving this last condition in 
order to m. Stability requires 1-Det(J)>0￿ 
 
From proposition 1, we have  concluded that the less constrained credit is, the 
larger is the amount of wealth the economy accumulates in the long run, while from 
proposition 2 one observes that there is a stability ceiling: if freedom to offer credit is 
too  high,  the  guarantee  that  the  steady  state  level  of  wealth  is  achieved  vanishes. 
Therefore, one can interpret this theoretical structure as indicating both the advantages 
of  financial  development  and  of  financial  responsibility,  in  the  sense  that  excessive 
credit may disrupt the financial system as agents fail to pay back the large amount of 
resources they have borrowed. 
Local dynamics conceal meaningful features of the model. First, cycles appear to 
be absent. The theory on nonlinear dynamics points to the eventual presence of cycles 
after a bifurcation. In our concrete system, we should expect to effectively encounter a 
fixed point in the stability area identified in proposition 2, and a-periodic motion after 
the bifurcation and before instability truly sets in. This becomes evident with the global 
analysis of the following section. Second, the global analysis of this specific model will 
show that some points of stability are present in the locally unstable area, a result that 
can be used to justify a same kind of conclusion as the one in the CJM model: financial 
instability  (cycles)  occurs  for  intermediate  levels  of  development,  while  stability  is 
found for low and high values of the credit market development.  
 Constraints on credit, consumer behaviour and the dynamics of wealth  7 
 
4. Global Dynamics 
 
To address global dynamics consider an array of reasonable parameter values:
3 [A; 
c;  g;  r;  w
*;  a]=[1;  0.75;  0.04;  0.03;  1;  0.7]  and  let  us  elect  m  as  the  bifurcation 
parameter. In this particular case, the system is stable for 0.041<m<1.573.  
Figure 1 shows the bifurcation diagram; a bifurcation, that occurs for m=1.573, 
separates an area of stability from an area where invariant cycles can be observed. After 
the region where endogenous cycles are evidenced, it follows a state where stability and 
instability alternate. Recall that  t w ~  is a variable that was modified twice: first, it was 
detrended and, then, adjusted to obtain a balanced growth path where the variable takes 
the value zero. In the long run, the original variable wt grows exponentially, with a 
detrended value equal to w . 
 
*** Figure 1 here *** 
 
To  understand  that  this  framework  produces  everlasting  endogenous  business 
cycles for specific values of the credit parameter, we present a time series of  t w ~  in 
figure 2. The Neimark-Sacker bifurcation, or Hopf bifurcation in discrete time, is able to 
generate  a  kind  of  dynamics  that  reproduces  considerably  well  real  world  business 
cycles, in the sense that several consecutive periods of increasing wealth are followed 
by some periods where there is a slowdown on the growth of wealth, and so on.  
 
*** Figure 2 here *** 
 
To close the graphical presentation, and taking the same value of m as in figure 2, 
we draw an attractor that reveals the long run relation between  1
~
- t w  and  t w ~  (figure 3) 
and  the  basin  of  attraction  that  furnishes  the  set  of  initial  points  that  allow  for  a 
convergence towards the long run attractor (figure 4). 
 
*** Figures 3 and 4 here *** 
 
5. Conclusions 
                                                 
3 In particular, observe that the marginal propensity to consume corresponds to 75% of income, that the 
growth rate is 4% and that the interest rate is 3%. Constraints on credit, consumer behaviour and the dynamics of wealth  8 
 
 
Following  the  literature  on  credit  constraints  and  business  cycles,  we  have 
considered  a  basic  AK  endogenous  growth  model,  where  financial  development  is 
addressed  through  a  credit  multiplier  and  where  economic  agents  take  consumption 
decisions by weighting last period’s difference between observed and potential wealth 
(if this difference is positive, consumers optimism rises and they will increase their 
consumption  share  of  income;  with  a  negative  difference,  households  become  less 
enthusiastic about consumption and they will prefer to increase the marginal propensity 
to save, as a precautionary measure). 
The proposed setup is able to elucidate about two important points: 
i) As the financial development level rises, per capita wealth, in the steady state, 
also increases; 
ii) Fixed point stability is found for low levels of financial development (and, thus, 
low levels of accumulated wealth); an intermediate level of the credit multiplier allows 
to identify a-periodic cycles generated through a Neimark-Sacker bifurcation; and high 
levels  of  financial  development  are  characterized  by  a  scenario  where  stability  and 
instability alternate. 
The instability result for high levels of the credit multiplier adds a new feature 
relatively to the CJM model: too high credit multipliers, associated to loans with no 
collateral, imply a high risk in the credit market, in the sense that borrowers may not 
carry out their debt payment obligations. In this case, instability may be interpreted as a 
scenario of financial crisis that imposes the need to restore a certain ceiling on the level 
of available credit. 
 
References  
 
Aghion,  P.;  G.  M.  Angeletos;  A.  Banerjee  and  K.  Manova  (2005).  “Volatility  and 
Growth:  Credit  Constraints  and  Productivity-Enhancing  Investment.”  NBER 
working paper nº 11349. 
Aghion,  P.;  P.  Bacchetta  and  A.  Banerjee  (2001).  “Currency  Crises  and  Monetary 
Policy in an Economy with Credit Constraints.” European Economic Review, vol. 
45, pp. 1121-1150.  
Aghion,  P.;  P.  Bacchetta  and  A.  Banerjee  (2004).  “Financial  Development  and  the 
Instability  of  Open  Economies.”  Journal  of  Monetary  Economics,  vol.  51,  pp. 
1077-1106.   Constraints on credit, consumer behaviour and the dynamics of wealth  9 
 
Amable, B.; J. B. Chatelain and K. Ralf (2004). “Credit Rationing, Profit Accumulation 
and Economic Growth.” Economics Letters, vol. 85, pp. 301-307. 
Bernanke,  B.  and  M.  Gertler  (1989).  “Agency  Costs,  Net  Worth,  and  Business 
Fluctuations.” American Economic Review, vol. 79, pp. 14-31. 
Caballé, J.; X. Jarque and E. Michetti (2006). “Chaotic Dynamics in Credit Constrained 
Emerging Economies.” Journal of Economic Dynamics and Control, vol. 30, pp. 
1261-1275. 
Demirguç-Kunt, A. and R. Levine (2001). Financial Structure and Economic Growth. 
Cambridge, MA: MIT Press. 
Kyotaki, N. and J. Moore (1997). “Credit Cycles.” Journal of Political Economy, vol. 
105, pp. 211-248. Figures 
 
Figure 1 – Bifurcation diagram (1.5<m m m m<2). 
 
 
Figure 2 – Time series (m m m m=1.891). 
 
 
Figure 3 – Attracting set (m m m m=1.891). Constraints on credit, consumer behaviour and the dynamics of wealth  2 
 
 
 
Figure 4 – Basin of attraction (m m m m=1.891). 
 
 

Constraints on credit, consumer behaviour and the dynamics of wealth

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