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Random shocks in a simple growth model
Random shocks in a simple growth model
Richard Stone
The variables in my earlier paper [4] oscillated around smooth paths and did not exhibit the irregular movements characteristic of economic timeseries. As illustrated in this note, irregularities make their appearance if a random term is introduced into the model, and the results provide another confirmation of Wicksell’s conjecture that random shocks serve to maintain cycles which would otherwise die away. Keywords: Simulation;
Economic models; Random shocks
Professor Sir Richard Stone is Emeritus Professor of Finance and Accounting in the University of Cambridge. Final manuscript received 1 February 1984.
The introduction of random shocks To the pure economist who is willing to go along with the assumptions in Stone [4] the diagrams in the appendix of that paper will look all right; but to the applied economist who has observed the irregularities of the world we live in, there will seem to be something missing. Economic time-series are not composed simply of trend and cyclical components, but show irregular variations around these basic components of movement. The purpose of this note is to illustrate the consequences of introducing a random term into the model given in Stone [4]. This can be done in a very simple way by extending the system as follows. Let us assume that each year the economy receives from the rest of the world a transfer, E say, which, when positive, enables it to augment its own production by imports and, when negative, requires it to export correspondingly. Thus output, q say, must now be distinguished from income, y. In fact
y=q+&
(1)
which implies that
q=c+v-&
(2)
Apart from this the equations are the same as before. In successive time intervals the transfer, E, is a random drawing from a normal distribution with mean zero and variance equal to 10% of q.
The consequences The consequences of introducing this modification into standard runs with l3 = 50 and fi = 0.5 are shown in Figures 1 and 2, which can be compared with the corresponding diagrams in the earlier paper. It should be noted that, in Figure 1, t runs from -50 to 27.5 and not, as in Figure 2 and the main diagrams of the earlier paper, from -50 to 100. As before, the simulations were carried out by Frederick van der Ploeg on a ‘quarterly’ basis, that is to say there are four to each ‘year’. As can be seen the regular time paths of the earlier diagrams now give way to irregular ones. These results confirm the conjecture of Wicksell [5] and the demonstrations of Slutsky [3], Frisch [2] and Adelman [l] that random shocks can serve to maintain cycles which would otherwise
0264-9993/B4/030277-04
$03.00 @ 1994 Butterworth
& Co (Publishers) Ltd
277
Random shocks in a simple growth model: R. Stone panelI
0.f
0.0’
i
-60
I -40
1
-20
Time
0
20
I
I
0.7 40
-60
I
-40
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I
30
0
20
40
0
20
40
Flmel3 I
0.6 -
0.4 -
s
QOo
-60
-40
0
-20
20
Time
Figure 1.
40
-60
-20
Tii
A highly stable version of the standard run incorporating a random error term.
@= 50; other parameters = standard values.
278
ECONOMIC MODELUNG July 1984
Random shock-sin a simple growth model: R. Stone Fmel2 I
0.6
P
,v:
I
0
Time
50
loo
0.7 ’ -50
0
60
loo
50
0
Time
panel 4
Panel 3
0.4
0.1
-J
-50
0
50
Q(
DO
Time
Figure 2.
A stable version of the standard run incorporating
0 Time
a random error term.
fi = 0.5; other parameters = standard values.
ECONOMIC MODEUING
July 1984
279
Random shocks in a simple growth model: R. Stone
die away. In the present case, and this is naturally most noticeable in Figure 2, the cycles tend through time to diminish in amplitude but in the end the time paths will fluctuate at random round the equilibrium values of the stationary state. The exact time paths of the variables shown in the diagrams depend on the particular sequence of values chosen for E; had another sequence been used the time paths would have been different, but their general tendencies would have been similar and they would have ended by fluctuating around the same ‘equilibrium’ values.
A list of works cited 1 Irma Adelman and Frank L. Adelman, ‘The dynamic properties of the K.z;oldberger model’, Econometrica, Vol XXVII, No 4, 1959, pp 2 Ragnar Frisch, ‘Propagation problems and impxlse problems in dynamic economics’, in Economic Essays in Honour of Gustav Cassel, Allen and Unwin, London, 1933. 3 Eugen Slutsky, ‘The summation of random causes as the source of cyclic progress’, Problems of Economic Conditions (Moscow), Vol3, No 1, 1927. Revised English version, Econometrica, Vol 5, No 2, 1937, pp 105-46. 4 Richard Stone, ‘Model design and simulation’, Economic Modelling, Vol 1, No 1, January 1984, pp 3-23. 5 Knut Wicksell, ‘Krise-mas g&a’, Stats0konomisk Tidsskri,ft, Vol 21, No 4, 1907, pp 255-84. English translation (‘The enigma of business cycles’) in International Economic Papers, No 3, London, Macmillan, 1953.
ECONOMICMODELUNG July 1994
Richard Stone
The variables in my earlier paper [4] oscillated around smooth paths and did not exhibit the irregular movements characteristic of economic timeseries. As illustrated in this note, irregularities make their appearance if a random term is introduced into the model, and the results provide another confirmation of Wicksell’s conjecture that random shocks serve to maintain cycles which would otherwise die away. Keywords: Simulation;
Economic models; Random shocks
Professor Sir Richard Stone is Emeritus Professor of Finance and Accounting in the University of Cambridge. Final manuscript received 1 February 1984.
The introduction of random shocks To the pure economist who is willing to go along with the assumptions in Stone [4] the diagrams in the appendix of that paper will look all right; but to the applied economist who has observed the irregularities of the world we live in, there will seem to be something missing. Economic time-series are not composed simply of trend and cyclical components, but show irregular variations around these basic components of movement. The purpose of this note is to illustrate the consequences of introducing a random term into the model given in Stone [4]. This can be done in a very simple way by extending the system as follows. Let us assume that each year the economy receives from the rest of the world a transfer, E say, which, when positive, enables it to augment its own production by imports and, when negative, requires it to export correspondingly. Thus output, q say, must now be distinguished from income, y. In fact
y=q+&
(1)
which implies that
q=c+v-&
(2)
Apart from this the equations are the same as before. In successive time intervals the transfer, E, is a random drawing from a normal distribution with mean zero and variance equal to 10% of q.
The consequences The consequences of introducing this modification into standard runs with l3 = 50 and fi = 0.5 are shown in Figures 1 and 2, which can be compared with the corresponding diagrams in the earlier paper. It should be noted that, in Figure 1, t runs from -50 to 27.5 and not, as in Figure 2 and the main diagrams of the earlier paper, from -50 to 100. As before, the simulations were carried out by Frederick van der Ploeg on a ‘quarterly’ basis, that is to say there are four to each ‘year’. As can be seen the regular time paths of the earlier diagrams now give way to irregular ones. These results confirm the conjecture of Wicksell [5] and the demonstrations of Slutsky [3], Frisch [2] and Adelman [l] that random shocks can serve to maintain cycles which would otherwise
0264-9993/B4/030277-04
$03.00 @ 1994 Butterworth
& Co (Publishers) Ltd
277
Random shocks in a simple growth model: R. Stone panelI
0.f
0.0’
i
-60
I -40
1
-20
Time
0
20
I
I
0.7 40
-60
I
-40
I
I
30
0
20
40
0
20
40
Flmel3 I
0.6 -
0.4 -
s
QOo
-60
-40
0
-20
20
Time
Figure 1.
40
-60
-20
Tii
A highly stable version of the standard run incorporating a random error term.
@= 50; other parameters = standard values.
278
ECONOMIC MODELUNG July 1984
Random shock-sin a simple growth model: R. Stone Fmel2 I
0.6
P
,v:
I
0
Time
50
loo
0.7 ’ -50
0
60
loo
50
0
Time
panel 4
Panel 3
0.4
0.1
-J
-50
0
50
Q(
DO
Time
Figure 2.
A stable version of the standard run incorporating
0 Time
a random error term.
fi = 0.5; other parameters = standard values.
ECONOMIC MODEUING
July 1984
279
Random shocks in a simple growth model: R. Stone
die away. In the present case, and this is naturally most noticeable in Figure 2, the cycles tend through time to diminish in amplitude but in the end the time paths will fluctuate at random round the equilibrium values of the stationary state. The exact time paths of the variables shown in the diagrams depend on the particular sequence of values chosen for E; had another sequence been used the time paths would have been different, but their general tendencies would have been similar and they would have ended by fluctuating around the same ‘equilibrium’ values.
A list of works cited 1 Irma Adelman and Frank L. Adelman, ‘The dynamic properties of the K.z;oldberger model’, Econometrica, Vol XXVII, No 4, 1959, pp 2 Ragnar Frisch, ‘Propagation problems and impxlse problems in dynamic economics’, in Economic Essays in Honour of Gustav Cassel, Allen and Unwin, London, 1933. 3 Eugen Slutsky, ‘The summation of random causes as the source of cyclic progress’, Problems of Economic Conditions (Moscow), Vol3, No 1, 1927. Revised English version, Econometrica, Vol 5, No 2, 1937, pp 105-46. 4 Richard Stone, ‘Model design and simulation’, Economic Modelling, Vol 1, No 1, January 1984, pp 3-23. 5 Knut Wicksell, ‘Krise-mas g&a’, Stats0konomisk Tidsskri,ft, Vol 21, No 4, 1907, pp 255-84. English translation (‘The enigma of business cycles’) in International Economic Papers, No 3, London, Macmillan, 1953.
ECONOMICMODELUNG July 1994