BOOMS, RECESSIONS AND FINANCIAL TURMOIL A FRESH LOOK AT by edk10782

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									BOOMS, RECESSIONS AND FINANCIAL TURMOIL:
  A FRESH LOOK AT INVESTMENT DECISIONS
       UNDER CYCLICAL UNCERTAINTY


                                     Yu-Fu Chen
                                    Economic Studies
                        School of Social & Environmental Sciences
                                   University of Dundee


                                   Michael Funke
                                Department of Economics
                                  Hamburg University



                                       June 2009

  Hamburg University                                                1
  Department of Economics
  Michael Funke
1. Introduction

Referring to the current financial crisis, Blanchard (2009) offers the
following observation.

“Crisis feeds uncertainty. And uncertainty affects behaviour, which feeds
the crisis. Were a magic wand to remove uncertainty, the next few quarters
would still be tough (…), but the crisis would largely go away”.




      Hamburg University                                                 2
      Department of Economics
      Michael Funke
2. A Model of (Partially) Irreversible Investment

The Cobb-Douglas production function is given by



(1)      Yt  K t N 1,       0    1,


where K is the capital stock, N is the constant employment level, and   is
                                                           
a parameter determining the shares between capital and labour in
production.




      Hamburg University                                                3
      Department of Economics
      Michael Funke
It is assumed that the firm faces a stochastic isoelastic demand function



(2)          p  Yt 1   Z t ,     1,



where p denotes the price, Z denotes the random demand shock, and  is
an elasticity parameter that takes its minimum value of 1 in a perfectly
competitive environment.




      Hamburg University                                                    4
      Department of Economics
      Michael Funke
Therefore, current profits, measured in units of output, are defined as


(3)            Z t K t1 N  2  Cost I t   xK t  wN ,

where  1    and  2  1     , I t is gross investment, x denotes
constant service expenses for capital, w is the constant real wage, and
Cost() are the total investment expenditures denoted by the following
functions:
                                      1 2
                             p K I t  2 I t   for I t  0
                            
(4)           Cost I t   0                   for I t  0
                                       1
                                
                             p K I t  I t2    for I t  0.
                                       2
      Hamburg University                                                  5
      Department of Economics
      Michael Funke
Purchase (resale) costs are the costs of buying (selling) capital. We
                               
assume that p K  p K  0 . Adjustment costs, I t2 2 , are continuous and
strictly convex in I, and  is a positive parameter. The adjustment of capital
over time is denoted by



             dKt
(5)               I t  K t ,
              dt

where  represents the constant depreciation rate.




      Hamburg University                                                    6
      Department of Economics
      Michael Funke
To capture probabilistic state transitions over time, Markov-switching
models popularized by Hamilton (1989, 1990) provide an attractive
analytical framework.

A key step in the modelling stage is the specification of the number of
regimes. Before embarking on the modelling exercise, we first try to detect
different regimes. In order to determine the number of regimes, we use the
Chicago Board Options Exchange (CBOE) volatility indices.




     Hamburg University                                                   7
     Department of Economics
     Michael Funke
                  Figure 1: CBOE Volatility Indices VIX and VBO



              100.00
               90.00
               80.00
               70.00
 volatility




               60.00
               50.00
               40.00
               30.00
               20.00
               10.00
                0.00
                       17/12/1996

                                    17/12/1997

                                                 17/12/1998

                                                              17/12/1999

                                                                           17/12/2000

                                                                                        17/12/2001

                                                                                                     17/12/2002

                                                                                                                  17/12/2003

                                                                                                                               17/12/2004

                                                                                                                                            17/12/2005

                                                                                                                                                         17/12/2006

                                                                                                                                                                      17/12/2007

                                                                                                                                                                                   17/12/2008
Hamburg University                                                                                                                                                                              8
Department of Economics
Michael Funke
(1) Although there is no perfect correspondence between cyclical phases
and regimes, apparently two regimes (booms vs. recessions) have existed
between 1996 and 2007.

(2) The depth of the current recession suggests the existence of a third
regime indicating episodes of financial turmoil and sharp contractions.

Given the evidence in Figure 1, we propose a nonlinear analytical
framework with three regimes (two recessionary states and one
expansionary state).




      Hamburg University                                                  9
      Department of Economics
      Michael Funke
We assume that the demand process follows the continuous-time
stochastic (geometrical Brownian motion) Markov switching processes



(6)          dZ t   i Z t dt   i Z t dWt , for i = 0, 1, 2


Where dWt         t dt denotes the increments of a standard Wiener process,
t is an i.i.d. sequence with mean zero and a standard deviation of unity,

. i is the drift parameter, and  i2 the variance parameter. It is assumed that
if the boom (state 2) occurs, the drift and the variance parameters  2 are
and  2 respectively;
      2




      Hamburg University                                                    10
      Department of Economics
      Michael Funke
if the recession state (state 1) emerges, they are  1 and  1 , respectively;
                                                             2


and if the financial turmoil state (state 0) occurs, the parameters are given
by  0 and  0 , respectively. It is expected that the value of the drift (growth
             2


of demand) of the state 2 is higher than the one of the states 1 and 0, i.e.
 2  1   0 holds. The corresponding volatility parameters are in the
opposite order       0  1   2 .
The next aim is to describe the connections between the phases of cycles.
We propose the following 3×3 transition matrix providing information on how
business cycle phases are related.

               1  t    0        t        
(7)             0      1  t      t        ,
                                              
                0 t
                        1t 1  0 t  1t 
                                               
      Hamburg University                                                      11
      Department of Economics
      Michael Funke
where        (  ) denotes the probability of changing from boom state to
          0       1
financial turmoil state (recession state). Correspondingly,  ( ) represents
the transition probability from recession state (financial turmoil state) to
boom state. It is assumed that there are no transitions between recession
state and the financial turmoil state. This constraint based on simple
economic considerations simplifies the analysis significantly.

Thus, the firm’s profit-maximisation problem is denoted by:



                                                    
                                                                         
(8) V  max E  Z t K t N  xK t  wN  Cost I t  e dt Z  Z 0 , K  K 0 
                      1  2                         rt
          It                                                              
              0                                                           

where r is the discount rate.



      Hamburg University                                                  12
      Department of Economics
      Michael Funke
Applying Ito’s Lemma, the stochastic nature of this optimization problem
requires the solution to the following Bellman equations for the states 0, 1
and 2:


                                 
             rV0  max ZK 1 N  2  xK  wN  Cost I   V0 K I  K 
                            I

                                                                                
(9)
                                       0 ZV0 Z   0 Z 2V0 ZZ 2   V2  V0  ,
                                                     2




                                 
              rV1  max ZK 1 N  2  xK  wN  Cost I   V1K I  K 
                            I

                                                                                     
(10)
                                       1 ZV1Z   12 Z 2V1ZZ 2   V2  V1  ,



       Hamburg University                                                       13
       Department of Economics
       Michael Funke
(11)
                      
       rV2  max ZK 1 N  2  xK  wN  Cost I   V2 K I  K 
                  I

                                                                                            
                                   2 ZV2 Z   2 Z 2V2 ZZ  0 V0  V2   1 V1  V2 
                                              1 2
                                              2                                             


where V0, V1, and V2 represent the value of the firm in the states 0, 1, and 2
respectively. The nature of the solution of this problem is now intuitive. The
investment policy that maximizes profits has a simple and intuitive form: the
q-type investment function for I for the states 0, 1 and 2 is denoted by


(12)             
               p K   I  qi ,

where for i = 0, 1, 2.


       Hamburg University                                                                 14
       Department of Economics
       Michael Funke
The solutions for q 0, q1 and,             q 2 all consist of particular solutions and gene-
ral solutions so that            q 0  q 0  q 0 , q1  q1P  q1 , q 2  q 2  q 2 .
                                         P      G                   G           P     G




                                        1 1       2       x
(13)                 P
                    q0      a0 ZK              N                ,
                                                           r 
                                                            x
(14)                 q1P  a1 ZK 1 1 N  2                     ,
                                                          r 

                                                            x
(15)                 q 2  a3 ZK 1 1 N  2 
                       P                                          ,
                                                          r 




       Hamburg University                                                                15
       Department of Economics
       Michael Funke
The general solutions for q 0,           q1, and q 2 represent the net value of options
and are

(16)            G
               q0        Ai ZK
                            i 1
                                 3
                                             
                                         1 1 i
                                                      6
                                                      Ai ZK
                                                     i 4
                                                                    
                                                                1 1 i



(17)            G
               q1        Bi ZK
                            i 1
                                 3
                                             
                                         1 1 i
                                                      6
                                                      Bi ZK
                                                     i 4
                                                                    
                                                                1 1 i




(18)            G
               q2
                             3
                         Ci ZK
                            i 1
                                             
                                         1 1 i
                                                      6
                                                      Ci ZK
                                                     i 4
                                                                    
                                                                1 1 i




where 1   2   3  0   4   5   6 and they are the six charac-
teristic roots of the following equation for ß.


       Hamburg University                                                           16
       Department of Economics
       Michael Funke
It can be noted that the positive ß terms of (16)-(18) are generally related to

investment options and the negative ß terms are linked to disinvestment
terms.


                                                                                                                     
      r      1  1   2    2    1  0  1  r      1  1   0    0    1   
                                      1 2                                                           1 2
                                     2                                                           2                   
(19)   r      1  1  1   1  12    1      0  r      1  1  1   1  12    1   
                                                                                                                       
                                       2                                                          2                    
                                                               
        1  r      1  1   0    0    1   .
                                             1 2
                                            2                  




        Hamburg University                                                                                    17
        Department of Economics
        Michael Funke
Note that the relationships between Ai, Bi, and Ci are according to the
following equations:

                                                          
          r      1  1   0    0    1   
                                        1 2
        
(20) Ai                                                     C , i  1,..., 6,
                                        2
                                  
                                                                i




                                                         
          r     1  1  1    12    1   
                                      1
        
(21) B                               2                     C , i  1,...,6.
                                  
      i                                                        i




      Hamburg University                                                      18
      Department of Economics
      Michael Funke
The set of boundary conditions that applies to this optimal stopping problem
is composed by the value matching conditions




       
(22) q Z  , A , A , A , A , A , A  q P Z   q G Z  , A , A , A , A , A , A  p  ,
      0 0     1   2   3   4   5   6   0    
                                          0     0   0     1   2   3   4   5   6   K

(23) q Z
      0
            
            0   , A , A , A , A , A , A   q Z   q Z , A , A , A , A , A , A   p ,
                  1   2    3    4   5   6
                                            P
                                            0
                                                
                                                0
                                                     G
                                                     0
                                                         
                                                         0   1   2       3       4       5       6
                                                                                                         
                                                                                                         K

(24) q Z
      1
          
          1     , B , B , B , B , B , B   q Z   q Z , B , B , B , B , B , B   p ,
                  1   2   3     4   5   6
                                            P
                                            1
                                                
                                                1
                                                     G
                                                     1
                                                         
                                                         1   1   2       3       4       5       6
                                                                                                         
                                                                                                         K

(25) q Z
      1
          
          1     , B , B , B , B , B , B   q Z   q Z , B , B , B , B , B , B   p ,
                  1   2   3     4   5   6
                                            P
                                            1
                                                
                                                1
                                                     G
                                                     1
                                                         
                                                         1   1   2       3       4       5       6
                                                                                                         
                                                                                                         K


(26) q Z   
                , C , C , C , C , C , C   q Z   q Z , C , C , C , C , C , C   p ,
                                            P       G                                                  
      2     2     1   2    3    4   5   6   2   21   2   2   1       2       3       4       5       6   K

(27) q Z
      2
            
            2   , C , C , C , C , C , C   q Z   q Z , C , C , C , C , C , C   p ,
                  1   2    3    4   5   6
                                            P
                                            1
                                                
                                                2
                                                     G
                                                     1
                                                         
                                                         2   1   2           3       4       5       6
                                                                                                         
                                                                                                         K




      Hamburg University                                                                                     19
      Department of Economics
      Michael Funke
and the corresponding smooth-pasting conditions

(28)
                    
                    
             q 0 Z 0 , A1 , A2 , A3 , A4 , A5 , A6        0,
                                    
                                 Z 0
                     
               q 0 Z 0 , A1 , A2 , A3 , A4 , A5 , A6    0
(29)                                                          ,
                                 Z 0

(30)                 
                     
               q1 Z 1 , B1 , B 2 , B3 , B 4 , B5 , B 6      0
                                                               ,
                                     
                                  Z 1

(31)                 
                     
               q1 Z 1 , B1 , B 2 , B3 , B 4 , B5 , B6       0
                                                               ,
                                     
                                  Z 1

(32)                 
               q 2 Z 2 , C1 , C 2 , C 3 , C 4 , C 5 , C 6   0
                                                               ,
                                     
                                  Z 2

(33)
                      
                        
               q 2 Z 2 , C1 , C 2 , C 3 , C 4 , C 5 , C 6   0
                                                               .
                                     
                                  Z 2

Making use of the value-matching and smooth-pasting conditions, we get
the boundary values that separate the space into two regions: one where it
is optimal to exercise the investment option and another where it is not.

       Hamburg University                                                   20
       Department of Economics
       Michael Funke
3. Model Simulations

The unit time length corresponds to one year. Our base parameters which
were chosen for realism are  0 = 0.35,  1 = 0.25,  2 = 0.15,  0 = -0.04,
         
. 1 = 0.01, 2         = 0.025,  = 0.07,  = 1.5,  = 0.3,  = 0.33,  = 0.15,
  
p K  1.0, p K  0.4 , r = 0.05, x = 0.1, K0 = N0 = 1.0, and  = 1.50. We set
             

the baseline standard switching probabilities  = 0.25,  = 0.4, 0  0.02
and 0  0.1, respectively. Hence, the expected duration for booms is
(1-  0 - 1 )/(  0+ 1 ) = 0.88/0.12 = 7.3 years. The expected duration of a
recession is (1- )/  = 0.6/0.4 = 1.5 years, and the expected duration of a
period of financial turmoil is (1- )/  = 0.75/0.25 = 3.0 years.




      Hamburg University                                                    21
      Department of Economics
      Michael Funke
                                              Figure 2: The Impact of  Upon the Z Thresholds



                         2                                                                                 0.8
                                                 Financial turmoil




                                                                               Dis-investment thresholds
                        1.9                                                                                0.7
Invesment tthresholds




                        1.8                                                                                0.6

                        1.7                                                                                0.5
                                                                                                                        Boom
                        1.6                                                                                0.4                                        Recession

                                                                Recession                                  0.3                    Financial turmoil
                        1.5
                        1.4                                                                                0.2
                                                                     Boom                                  0.1
                        1.3
                        1.2                                                                                 0
                              0.15       0.175      0.2       0.225     0.25                                     0.15     0.175    0.2       0.225     0.25
                                                                                                                                        




                                     Hamburg University                                                                                                       22
                                     Department of Economics
                                     Michael Funke
                                        Figure 3: The Impact of  Upon the Z Thresholds



                        1.9                                                                            0.8
                                             Financial turmoil




                                                                           Dis-investment thresholds
                        1.8                                                                            0.7
Invesment tthresholds




                        1.7                                                                            0.6

                        1.6                                                                            0.5          Boom

                        1.5                                                                            0.4                                          Recession
                                                                                                               Financial turmoil
                        1.4                                 Recession                                  0.3

                        1.3                                                                            0.2

                        1.2                                                                            0.1
                                                             Boom
                        1.1                                                                             0
                              0.25     0.3      0.35       0.4      0.45                                     0.25      0.3         0.35       0.4    0.45
                                                                                                                                         




                               Hamburg University                                                                                                               23
                               Department of Economics
                               Michael Funke
                                          Figure 4: The Impact of 0 Upon the Z Thresholds



                        1.9                                                                                0.8




                                                                               Dis-investment thresholds
                        1.8                   Financial turmoil                                            0.7
Invesment tthresholds




                        1.7                                                                                0.6

                        1.6                                                                                0.5
                                                             Recession                                               Boom
                                                                                                                                                 Recession
                        1.5                                                                                0.4
                                                                                                                             Financial turmoil
                        1.4                                                                                0.3

                        1.3                                       Boom                                     0.2

                        1.2                                                                                0.1
                              0        0.05      0.1        0.15         0.2                                     0    0.05   0.1        0.15      0.2
                                                       0                                                                          0




                                  Hamburg University                                                                                                    24
                                  Department of Economics
                                  Michael Funke
                                          Figure 5: The Impact of 1 Upon the Z Thresholds



                        1.9                                                                                  0.8
                                                Financial turmoil




                                                                                 Dis-investment thresholds
                        1.8                                                                                  0.7
Invesment tthresholds




                        1.7                                                                                  0.6

                        1.6                                                                                  0.5
                                                                                                                       Boom
                                                                                                                                                   Recession
                        1.5                                                                                  0.4
                                                                Recession
                                                                                                                               Financial turmoil
                        1.4                                                                                  0.3

                        1.3                                         Boom                                     0.2

                        1.2                                                                                  0.1
                              0         0.05       0.1        0.15         0.2                                     0    0.05   0.1        0.15      0.2
                                                         1                                                                          1




                                  Hamburg University                                                                                                      25
                                  Department of Economics
                                  Michael Funke
                                        Figure 6: The Impact of 0 Upon the Z Thresholds



                        2.2                                                                              1
                                                                                                        0.9




                                                                            Dis-investment thresholds
                         2                       Financial turmoil                                      0.8
Invesment tthresholds




                                                                                                        0.7
                        1.8                                                                             0.6
                                                                                                        0.5          Boom
                        1.6                                                                             0.4                                          Recession
                                                                                                                             Financial turmoil
                                                              Recession                                 0.3

                        1.4                                                                             0.2
                                                              Boom
                                                                                                        0.1

                        1.2                                                                              0

                              0.28   0.305     0.33        0.355     0.38                                     0.28   0.305       0.33        0.355    0.38
                                                      0                                                                                0




                                Hamburg University                                                                                                           26
                                Department of Economics
                                Michael Funke
                                        Figure 7: The Impact of 1 Upon the Z Thresholds



                         2                                                                               0.8
                        1.9




                                                                             Dis-investment thresholds
                                                                                                         0.7
Invesment tthresholds




                        1.8             Financial turmoil                                                0.6
                        1.7                                                                              0.5
                                                                                                                                    Boom
                        1.6                                                                              0.4
                                                                                                                      Financial turmoil                   Recession
                        1.5                                                                              0.3
                                                               Recession
                        1.4                                                                              0.2
                                                               Boom
                        1.3                                                                              0.1

                        1.2                                                                               0
                              0.18   0.205      0.23        0.255     0.28                                     0.18       0.205       0.23        0.255    0.28
                                                       1                                                                                    1




                                Hamburg University                                                                                                                    27
                                Department of Economics
                                Michael Funke
                                      Figure 8: The Impact of 2 Upon the Z Thresholds



                         2                                                                           0.8




                                                                         Dis-investment thresholds
                        1.9                                                                          0.7
                                                    Financial turmoil
Invesment tthresholds




                        1.8                                                                          0.6

                        1.7                                                                          0.5
                                                                                                                           Boom
                        1.6                                                                          0.4       Recession
                                                  Recession                                          0.3                           Financial Turmoil
                        1.5
                        1.4                                                                          0.2
                                                          Boom                                       0.1
                        1.3
                        1.2                                                                           0
                           0.125   0.15     0.175       0.2      0.225                                 0.125      0.15     0.175       0.2     0.225
                                                  2                                                                              2




                              Hamburg University                                                                                                       28
                              Department of Economics
                              Michael Funke
                                         Figure 9: The Impact of 2 Upon the Z Thresholds


                         2                                                                              0.8
                                             Financial turmoil




                                                                            Dis-investment thresholds
                        1.9                                                                             0.7
Invesment tthresholds




                        1.8                                                                             0.6

                        1.7                                                                             0.5
                                                                                                                                     Boom
                        1.6                                                                             0.4    Recession

                        1.5                                 Recession                                   0.3          Financial turmoil

                        1.4                                                                             0.2

                        1.3                                  Boom                                       0.1

                        1.2                                                                              0
                              0.02   0.026     0.032      0.038     0.044                                     0.02        0.026     0.032     0.038   0.044
                                                     2                                                                                  2




                                Hamburg University                                                                                                            29
                                Department of Economics
                                Michael Funke
Since the focus of the paper is investment, we next present a translation
from thresholds to investment and the capital stock and assess the impact
of the three regimes upon investment. In other words, we “reverse
engineer” time series for investment and the capital stock from our setup. In
this validation stage, we also test the ability of our model to replicate some
business cycle characteristics by using numerical simulations of the
dynamic system.

We specify a sequential iterations method that allows us to generate
discrete realizations of the nonlinear dynamical system and investigate the
oscillations, given the chosen levels of parameters.




      Hamburg University                                                   30
      Department of Economics
      Michael Funke
Equation (6) is proxied by the following discrete stochastic differential
equation – the Euler scheme,



(34)       Z t  t  Z t   i Z t t   i t Z t  t ,    t ~ N 0,1   for i = 0, 1, 2



where the normal random variables,              t , are generated via the central limit
theorem. As the time passes, the term Z t fluctuates according to the corres-
ponding stochastic processes and K will depreciate as long as Z t is staying
within the no-action area.




       Hamburg University                                                               31
       Department of Economics
       Michael Funke
If Z t hits the threshold            Z 1 in state 1 or the threshold Z 0 in state 0, the firm
                                                                        

will invest according to


(35)          It 
                       q i Z t   p K
                                      
                                            
                                                               
                                                qi Z t   q1 Z 1
                                                                           for i = 0, 1, 2
                                                        

After the level of investment is determined, the corresponding capital stock
is computed using the capital accumulation constraint


(36)            K t 1  K t  I t  K t           ,


which become the initial value of K for the time t+1, by which the new
thresholds are recomputed accordingly for the time t+1.

       Hamburg University                                                                    32
       Department of Economics
       Michael Funke
Up to now, we have interpreted the model as applying to a single firm.
Suppose that we re-interpret the model at the macroeconomic level, i.e. K
and I now represent economy-wide gross investment and the capital stock,
respectively, and the interpretation of q is likewise altered. Unlike micro-
economic data, aggregate investment series look smoother since micro-
economic adjustments are far from being perfectly synchronized. The
question arises as to whether aggregation eliminates all traces of infrequent
lumpy microeconomic adjustment. We again focus on investment (I), and
we model aggregate investment in terms of average investment of a large
number of individual firms indexed by i  [1,2000].




      Hamburg University                                                  33
      Department of Economics
      Michael Funke
Finally, we employ a “hybrid” model that endogenises the business cycle
turning points. Suppose that at each turning point, the proportion of firms
experiencing a peak (trough) at time t is assumed to be drawn at random
from a standard normal distribution around the predefined turning points in
Figure 10.




     Hamburg University                                                  34
     Department of Economics
     Michael Funke
Figure 10: A Sample Path of the Demand Shock (Z), the Z-Thresholds,
             Installed Capital (K), and Optimal Investment

                                                             2




                                       Z and Z thresholds
                                                            1.5
                                                                                                       Z0+
                                                                           Z1+

                                                             1
                                                                                                 Z2+
                                                                                          Zt
                                                            0.5
                                                                                                                                            Z2

                                                                                               Z0 Z1
                                                             0
                                                                  0              6        12           18                              24         30
                                                                                               time

    2.5                                                                                                                      2.4
                                                                                                                             2.2
                                                                                                                               2
     2
                                                                                                                             1.8




                                                                                                             Investment, I
                                                                                                                             1.6
    1.5                                                                                                                      1.4
                                                                                                                             1.2
K




     1
                                                                                                                               1
                                                                                                                             0.8
                                                                                                                             0.6
    0.5                                                                                                                      0.4
                                                                                                                             0.2
                                                                                                                               0
     0
          0        6       12          18                             24             30                                            0              6    12          18   24    30

                                time                                                                                                                        time


              Hamburg University                                                                                                                                             35
              Department of Economics
              Michael Funke
                 Figure 11: Aggregate Dynamics for Perfectly Synchronised
                               Business Cycle Turning Points

                                                                  (a) Investment Dynamics for  = 1
                                                   0.5


                                                   0.4


                                                   0.3




                                               I
                                                   0.2


                                                   0.1


                                                    0
                                                         0            6      12          18   24            30

                                                                                  Time


              (b) Investment Dynamics for  = 2                                                             (c) Investment Dynamics for  = 4
    0.5                                                                                           0.5


    0.4                                                                                           0.4


    0.3                                                                                           0.3




                                                                                              I
I




    0.2                                                                                           0.2


    0.1                                                                                           0.1


     0                                                                                             0
          0       6      12          18   24                 30                                         0        6     12          18   24      30

                              Time                                                                                          Time


                Hamburg University                                                                                                           36
                Department of Economics
                Michael Funke
                     Figure 12: Aggregate Dynamics for Heterogeneous
                              Business Cycles Turning Points


                        (a) Investment Dynamics for  = 1 and N(0, 0.00) around state-switching points
                                              0.5


                                              0.4


                                              0.3




                                         I
                                              0.2


                                              0.1


                                               0
                                                    0        6   12          18   24              30

                                                                      Time
(b) Investment Dynamics for  = 1 and N(0, 0.05)                                       (c) Investment Dynamics for  = 1 and N(0, 0.1)
          around state-switching points                                                          around state-switching points
    0.5                                                                                 0.5


    0.4                                                                                 0.4


    0.3                                                                                 0.3
I




                                                                                   I
    0.2                                                                                 0.2


    0.1                                                                                 0.1


     0                                                                                   0
          0     6       12          18   24             30                                    0        6   12          18   24    30

                             Time                                                                               Time

              Hamburg University                                                                                                 37
              Department of Economics
              Michael Funke
4. Summary and Conclusions

The focus of this article has been the incorporation of jump dynamics into
real options models in order to improve understanding of cyclical
investment behaviour, especially in the most volatile era.

The Markov-switching modelling approach allows the derivation of
analytical and numerical results on option pricing, taking into account that
firms not only either observe or infer the current state of the system but also
make predictions about future regime switches. The chief implication of the
model is that recessions and financial turmoil periods are important
catalysts for waiting.




      Hamburg University                                                    38
      Department of Economics
      Michael Funke

								
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