Download Business Cycles Analysis: A Model to Study the Fluctuations of

Business Cycles Analysis: A Model to Study
the Fluctuations of Output Around the Steady
State
A simple multiplier-accelerator model
Ct  a  a Y
0 1 t 1
0  a 1
1
(1)
It  b  b Y Y 
0 1 t 1 t 2 
b 0
1
(2)
Yt  Ct  It  Gt  PCAt
Gt  PCAt  C
0
(3)
Ct is consumption,Yt is output,It is investmentGt is
government spending,PCAt is the current account balance.
Substituting (1) and (2) in (3) we have
Yt  a  a Y  b  b Y Y   C
0 1 t 1 0 1 t 1 t 2  0
Yt  a  b   a  b Y  b Y  C
0 0  1 1Macroeconomic
t 1 1 t 2Theme:
0 11
1
Steady State and Transitional Dynamics
In the steady state: Yt  Y
t 1
a b
Y   0 0
1 a 

1

Y Y
t 2
(4)
We need to solve for a complementary (homogenous) solution to find the
transitional dynamics
Yt  K t  K t
1
2
Yt   a  b Y  b Y
0
1 t 1 1 t 2
 1
K t   a  b  K t 1  b K t 2  0
1
1 1
1 1
 1
Iterating forward by two periods and dividing both sides by K1
t 2   a  b t 1  b t  0

1 1
1
 2   a  b   b  0

1
1
1
Macroeconomic Theme: 11
2
Solution for the Transition Path
Two roots of this equation are:
2
a  b  a  b  4b
1 1
1
 ,  1 1
1 2
2b
1
















Three possible solutions exist for this problem
2
a  b  4b
1 1
1
2


a b 
b. repeated root case if  1 1  4b1


2


a b 
c. complex root case  1 1  4b1
a. real and distinct root case if
Complementary solution








Yt  K t  K t still requires
11
2 2
K and K to make it definite. The initial values of
1
2
Y
output to find the definite solution 0 and Y1 .
values of
Macroeconomic Theme: 11
3
A Numerical Example for Complementary
Let a = 1.8 and b =1 Solution
1
1
2
a  b    a  b   4b
1
1
 1
1, 2  1 1
2b
1
1.8 1 
1.8 1  4








1, 2 
 2.37,0.42
2



Yt  K t  K t  Y  K  2.37 t  K  0.42t
t
1 1
2 2


1
2
Using two initial conditions
0
0
Y  K  2.37   K  0.42  K 2  Y0  K1
0
1
2



Y  K  2.37   K  0.42 Y1  K1 2.37   Y0  K1  0.42 


1
1
2
Y  0.42Y
1
0
K

 1
1.85
Y  0.42Y
1.43Y Y
K Y  K
1
0
0 1

2
0
1  K 2  Y0 
1.85
1.85
Macroeconomic Theme: 11
4
Complete Time Path for Income
Complete solution includes both particular and
complementary solutions
1.43Y Y 
Y  0.42Y 


t
t




1
0
0
1



Yt  Y 
2.37  
0.42


1.85 
1.85 










The first term shows the output in the steady state and the
second part shows the transitional dynamics around that
steady state. The time path of the variable also depends on
the initial values of output.
Macroeconomic Theme: 11
5
Real Business Cycle Model Impact of Productivity Shock in
Labour Demand, Price and Money Supply
Labour Supply
a
Real wage rate
B
Labour demand 1
Labour demand 2
Employment
A negative productivity shock lowers labour demand from a to b. wage rate as well as labour supply
declines. This causes a reduction in output. Prices rise, real money balance decrease.
LM 2
LM1
r
C
d
IS-curve
FY2
FY2Theme: 11Output
Macroeconomic
Prices are perfectly flexible and economy is always in full employment. Full employment level of
output changes as people keep changing their labour supply in response wage rates and productivity
6
Simple specification of RBC
Model
1a
A L 
a
Y

K
Output t
with 0  a  1
t  t t 
Capital accumulation Kt 1  Kt (1 d )  It
 Kt (1 d )  Yt  Ct  Gt
t 1
Wage rate is according to the marginal productivity
of labour
a
 K 

a


wt  ¶Y  1a  Kta  At Lt  At  1a  t  At
A L 
¶L
 t t
K
Interest rate is paid according to the marginal product
of capitals
1a
A L 
1

a


rt  ¶Y  aKta 1 At L 
a  t t 
 K 
¶K
t


Macroeconomic Theme: 11
5
7
Shocks to the technology and Government
Expenditure
ln At  A  gt   t
0
where the term  t represents shock to the technology.
How is  t determined ? It depends on its past values
 t  
t 1
 vt
 t is a white noise disturbance mean-zero shock and
0   1 is an autocorrelation term less than one.
All these things imply that the effect of technological
shock disappears over time.
Macroeconomic Theme: 11
8
Shocks to the Government Expenditure
Similar to technological shock some RBC models
includes shock due the government demand, which
depends on exogenous government spendingG0 ,
growth rate of population (n) and the economy (g).
Gt  G  n  g t  t
0
where  t is a shock to government spending. It is
determined by its past values:
 t  
t 1
 vt
Macroeconomic Theme: 11
9
Intertemporal Substitution and Labour-Leisure
Choice
Max U C1 , l1 , C 2 , l 2   ln C1   ln 1  LS 1   e   ln C 2   ln 1  LS 2 
C
w LS
C1  2  w1 LS1  2 2
1 r
1 r
C
w LS 

L  ln C1   ln 1  LS1   e   ln C 2   ln 1  LS 2    C1  2  w1 LS1  2 2 
1 r
1 r 

¶L


 w1 or
¶l1 1  LS1 

w
¶L
e 

  2 or
¶l 2 1  LS 2 
1 r
Eliminating


w1 1  LS1 
e    1  r 

w2 1  LS 2 
1  LS1 
w2
term in above equations we get 1  LS  
w1e   1  r 
2
Macroeconomic Theme: 11
10
Procyclical, Anticyclical and Acyclical
Variables: Leading and Lagging indicators
Procyclical variables move along with the GDP.
GDP components move together and Inflation
Anticyclical variables move in the opposite direction of the GDP
Unemployment rate
Acyclical variables does not have any relation with GDP and
amount of water under the Humber river.
In order to identify which variables are pro, anti or a-cyclical RBC
analysts use correlation coefficient between the GDP and
concerned variable, X
Macroeconomic Theme: 11
11
Use of Correlation Coefficient to determine
lagging and leading indicators
 Y  Y X  X 
;
 Y  Y   X  X 
T
 y,x 
covYt , X t 
varYt  varX t 

t
t 0
t
2
t
2
t
Variable is pro-cyclical if  y, x  0 ;
counter-cyclical if  y , x  0
and acyclical if  y , x  0 ;
lagging procyclical if  y , x
leading procyclical if  y
t
t 1
 0;
t 1 , xt
0
Macroeconomic Theme: 11
12
Reference
Bhattarai and Jones (2000) Macroeconomic Fluctuations in the UK Economy”, Hull
Economics Working Paper.
Burda and Wyplosz (2002) Chapter 14.
Barro, R. J. (1976), “Rational Expectations and the Role of Monetary Policy” Journal of Monetary
Economics, 2 (January): 1-32
Burns, A and W. Michell, (1946), “Measuring Business Cycles” NBER, New York.
Holly, Sean and Martin Weal (2000), (eds), “Econometric Modelling: Techniques and Application
Cambridge University Press.
Mankiw NG (1989) Real Business Cycles: A New Keynesian Perspective, Journal of Economic
Perspectives- 3:3: 19-90.
Prescott, E.C. (1986), “Theory Ahead of Business Cycle Measurement,” Federal Reserve Bank of
Minneapolis, Quarterly Review; Fall.
Quah, D.T., (1995), “Business Cycle Empirics: Calibration and Estimation,” The Economic Journa
105 (November) 1594-1596
Romer D. (2002) Advanced Macroeconomics McGraw Hill.
Macroeconomic Theme: 11
13