The multiplicator accelerator model is an economic situation model. It wants to explain, why economic growth constantly does not run, but typically follows an economic cycle. The model can be developed from the development model of Harrod and Domar out, a special variant comes from Paul A. Samuelson and John Richard Hicks.

If one formulates a Harrod Domar model as temporally discrete model and introduces delays to the behavior equations, one arrives at the Samuelson Hicks model, which permits the illustration of business fluctuations.

Description

This concerns a keynesianisches model, i.e., the bottleneck is the demand. It assumed that each demand can be served, it gives no lack of offer. The demand sits down together from the demand of the workers for consumer goods and the demand of the enterprises for capital goods. The more largely the demand, the larger production, the stopped more workers, the more consumer goods will become inquired. The consumer demand is thus a certain part of C of production the preliminary period. The consumption reacts thus with delay (time-lay) to production. 1 (1-c) is "the multiplicator "("multipliers "), indicates around a how muchfold certain increase of a demand size, for example the investments, which total demand increases.

The demand for capital goods is somewhat more difficult to represent. First the enterprises try to get a capital stick, which itself proportionally to the demand changed (similarly as the number of the persons employed proportionally to production varies). This is "the desired capital stick ". It stands to the demand (reacts to the preliminary period, it thus with temporal delay) in a certain constant relationship v (v is "the accelerator "). In order to come from the actual capital stick to the desired capital stick, the difference must be invested. The enterprises inquire thus so many capital goods that the old capital stick, after he increased by the investments, equal to which desired capital stick is. However these investments increased the total demand as part of the demand, so that the desired capital stick became already again larger. The enterprises try to meet thus with the "desired Kapitalststock" a moving target. By the retarded reaction to the constantly changing demand the business fluctuations come off.

v is called therefore accelerator ("accelerators ") and not multiplicator, because the investments are not simply proportional to the total demand, but proportionally to the change of the demand. This accelerator is mathematically responsible for the cycles in connection with the temporal delay.

A definition by cases can be made depending upon size of the parameters v and C. the following of cases is conceivable:

• exponential growth
• exponential shrinking
• exploding cycles
• absorbed cycles
• as mathematical border line: constant cycles.

Criticism:

• one receives constant oscillations only for very special parameter values (v=1)
• emprische estimations suggest > 1, this to v lead however to explosive behavior
• at too largely selected periods (v>4) one does not receive yet times more oscillations
• Definition of one period is problematic (time for consumer adjustments needed reaction of investments to changes of demand to the income <)

Economic cycles

- Beginning

From these acceptance economic cycles result. If the demand rises for any reason, thereby also the goal value for the desired capital stick rises. It must be invested more. Thus a up-swinging upswing comes itself on, since these investments are also again demand, which increase overall economic demand thus, thus more investments make necessary, etc.

- upper turning point

Investments are however not only demand, it have also a capacity effect, it increase the capital stick. Finally the growth of the capital stick catches up the growth of the demand. The enterprises do not have to expand their investments further then any longer. If the all enterprise does, the demand for capital goods does not continue to rise. The capital stick grows however further, evenly at height of the still transacted investments. Shears now between far increasing capital stick opens and leaving demand growth. It lies a over investment and a over accumulation crisis forwards with production capacities under-working at full capacity. This crisis leads up to the wing over. The investments is thus inherent in a contradiction. A certain annual investment volume represents year by year a certain constant demand. It changed however year by year the offer capacities, it increases it annually by evenly its amount. One "offer in excess crisis "is thus programmed.

- Wing over

The capital stick is now too large, it is less invested. An intensifying wing over begins itself. , The smaller the overall economic demand, the lower the desired capital stick, is the less invested is the less invested. The investments are not sufficient finally, in order to adjust the wear of the production capacities. Production capacities are shut down. The capital stick shrinks. In addition a failure wave develops. The enterprises separating from the market contribute also to shrinking the capital stick.

- lower turning point

Finally shrinking the capital stick with the falling demand step holds. The failure wave continues, does not continue to expand however not. Thus the demand on low level is stabilized, while in consequence of the still continuous failure wave the production capacities shrink further. Shears now between far shrinking capital stick opens and itself stabilizing demand. Thus also a contradiction is inherent in to missing investments. They leave the demand unchanged, cause however that the offer capacities shrink annually around evenly the missing investments. One "insufficient supply crisis "is programmed, the investments releases, which lead up to the upswing.

- Upswing

The production capacities are now too highly working at full capacity. It must be invested again more. This leads to again rising demand, the investments rises therefore still more, it develops an intensifying upswing, sees itself further above.

Mathematically each mark the entire capital stick, which was developed in the upswing, in the wing over in a cleaning crisis is again destroyed.

Computational example of constant oscillations

First the equations generally:

Consumer function:

C_t = C_0 + C \ cdot Y_ {T-1}

Y_t = C_t + I_t

Investment function (adjustment at desired capital stick):

I_t = \ alpha \ cdot (K^*_ {T-1} - K_ {T-1})

Desired capital stick:

K^*_t = v \ cdot Y_t

K_t = K_ {T-1} + I_t

Now filled with numbers:

C_t = 700 + 0 {,} 3 \ cdot Y_ {T-1}

Y_t = C_t + I_t

I_t = 0 {,} 25 \ cdot (K^*_ {T-1} - K_ {T-1})

K^*_t = 3 {,} 1 \ cdot Y_t

Initial conditions:

I_0 = 500

C_0 = 700

Literature

• R.G.D. Everything: Macro Economic Theory: A Mathematical Treatment. - London, Melbourne, Toronto: Macmillan, 1968.

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