Multiplier accelerator model

The multiplier-accelerator model (also sometimes called the Samuelson model or Samuelson-Hicks model ) is a business cycle model . It aims to explain why economic growth is not monotonous, but typically follows an economic cycle . The model can be made of the growth model of Harrod and Domar develop out a special variant comes from Paul A. Samuelson and John Richard Hicks .

If a Harrod-Domar model is formulated as a discrete time model and delays are introduced into the behavioral equation, one arrives at the Samuelson-Hicks model, which allows the mapping of economic fluctuations and connects national income with consumption and investment.

A simple model of this kind could look like this:

• ${\displaystyle Y_{t}=C_{t}+I_{t}}$
• Consumer function : with ${\displaystyle C_{t}=C_{a}+c\cdot Y_{t}}$ ${\displaystyle c\in [0,1]}$
• Investment function : with v greater than 0 ${\displaystyle I_{t}=v(Y_{t-1}-Y_{t-2})}$

where the production or the total demand, the consumption at time t, the investment and stands for the "autonomous", ie constant consumption. ${\displaystyle Y_{t}}$ ${\displaystyle C_{t}}$ ${\displaystyle I_{t}}$ ${\displaystyle C_{a}}$

In principle, the time lags in the investment function can also be made more complicated, and consumption can also react to income with a delay.

description

It is a Keynesian model, so the bottleneck is demand. It is assumed that any demand can be met, there are no supply bottlenecks. The demand is made up of the demand of the workers for consumer goods and the demand of the companies for capital goods. The greater the demand, the greater the production, the more workers are hired, the more consumer goods are in demand. The consumer demand is thus a certain part c of the production and that of the previous period. Consumption therefore reacts to production with a delay (time lag). 1 / (1-c) is the " multiplier " ("multiplier"), which indicates how many times a certain increase in a demand variable, for example investment, increases total demand.

Similar to the Harrod-Domar model , the demand for capital goods can either be interpreted as an adjustment to a desired capital stock or as an investment function in which the investments are determined by the change in demand.

In the case of the “desired capital stock”, this is related to the demand (the previous period, so it reacts with a time lag) in a certain constant ratio v (v is the “ accelerator ”). In order to get from the actual capital stock to the desired capital stock, the difference must be invested. So the companies ask for so many capital goods that the old capital stock, after having increased by the investments, is equal to the desired capital stock. However, as part of the demand, these investments have increased the overall demand, so that the desired capital stock has already increased again. So companies are trying to hit a moving target with the "desired capital stock". By a lay Time delayed reaction to the ever-changing demand, cyclical fluctuations take place.

Investments smaller than zero are to be interpreted as a reduction in the capital stock. In the simple model, symmetry is assumed, investments can be arbitrarily larger or smaller than zero. The latter assumption would have to be corrected if the reduction in the capital stock is limited by depreciation .

Therefore, v is called an accelerator and not a multiplier, because the investments are not simply proportional to the total demand, but proportional to the change in demand. This accelerator is mathematically responsible for the cycles in connection with time lags , i.e. with time delays.

Increasing, dampened and stable vibrations

A case distinction can be made depending on the size of the parameters v and c. The following cases are possible:

• exponential growth
• exponential shrinkage
• exploding cycles
• dampened cycles
• as a mathematical borderline case: constant cycles.

Business cycles

Beginning

Business cycles result from these assumptions. If, for whatever reason, demand increases, the target value for the desired capital stock also increases. More needs to be invested. This sets in motion an upswing, since these investments are also demand again, i.e. increase overall economic demand, i.e. make more investments necessary, etc.

Upper turning point

Investments are not just demand, they also have a capacity effect, they increase the capital stock. Ultimately, the growth of the capital stock catches up with the growth of demand. The companies then no longer have to expand their investments. If all companies do this, the demand for capital goods will not increase any further. However, the capital stock continues to grow, in line with the investments that are still being made. There is now a gap between the growing capital stock and the slowing growth in demand. There is an overinvestment and overaccumulation crisis with underutilized production capacities. This crisis leads to the downturn. There is therefore a contradiction inherent in investments. A certain annual investment volume represents a certain constant demand year after year. However, it changes the supply capacities year after year, it increases them annually by the same amount. An "oversupply crisis" is thus programmed.

Downturn

The capital stock is now too big and less is invested. A self-reinforcing downturn begins. The less is invested, the lower the overall economic demand, the lower the desired capital stock, the less is invested. After all, the investments are no longer sufficient to compensate for the deterioration in production capacities. Production capacities are shut down. The capital stock is shrinking. There is also a wave of bankruptcies. The companies leaving the market are also contributing to the shrinking capital stock.

Lower turning point

After all, the shrinking capital stock keeps pace with falling demand. The bankruptcy wave continues, but does not expand any further. This stabilizes demand at a low level, while production capacities continue to shrink as a result of the ongoing bankruptcy wave. There is now a gap between the further shrinking capital stock and stabilizing demand. There is also a contradiction in the lack of investment. They leave the demand unchanged, but have the effect that the supply capacities shrink every year due to the lack of investment. An “undersupply crisis” is programmed, which triggers investments that lead into the upswing.

Boom

The production capacities are now too busy. More must be invested again. This leads to increased demand again, investments therefore increase even more, a self-reinforcing upswing develops, see above.

Mathematically, every time the entire capital stock that was built up in the upswing is destroyed again in a downswing in a cleaning crisis.

Mathematical example

System of equations

Consumer function :

${\displaystyle C_{t}=C_{a}+c\cdot Y_{t}}$

${\displaystyle Y_{t}=C_{t}+I_{t}}$

Investment function :

${\displaystyle I_{t}=v\cdot (Y_{t-1}-Y_{t-2})}$

Case distinction

The following distinction can be made for this system of equations:

• v <s: damped vibrations
• s = v: constant vibrations
• s <v <4s: exploding vibrations
• v> or = 4s: steady growth

Other case distinctions would result for other equations.

• 

  Damped oscillation: s = 0.6, v = 0.5

• 

  Exploding vibrations: s = 0.45, v = 0.5

• 

  Steady growth: s = 0.125, v = 0.5

Numerical example (case of constant oscillation s = v)

Figure for numerical example

For this system of equations, constant oscillations result if the following values ​​are used for c and v:

${\displaystyle C_{t}=50+0{,}5\cdot Y_{t}}$

${\displaystyle Y_{t}=C_{t}+I_{t}}$

${\displaystyle I_{t}=0{,}5\cdot (Y_{t-1}-Y_{t-2})}$

Initial condition:

${\displaystyle I_{0}=29{,}5}$

For is 129.5 and for 159.0. ${\displaystyle C_{0}}$ ${\displaystyle Y_{0}}$

The capital stock K is obtained by adding up the investments I:

${\displaystyle K_{t}=K_{t-1}+I_{t}}$

Version adaptation to desired capital stock

Instead of using the investment function, the model can also be interpreted as an adjustment to a desired capital stock.

The desired capital stock is:

${\displaystyle K_{t}^{*}=v\cdot Y_{t}}$

${\displaystyle K_{t}^{*}-K_{t}}$is the deviation of the desired capital stock from the actual capital stock. If this value is invested, the gap between wish and reality can be closed, so to speak. The multiplier-accelerator model assumes a reaction of the investments with a time lag :

${\displaystyle I_{t}=K_{t-1}^{*}-K_{t-1}}$

If you set the initial value , you get exactly the same investments mathematically as would result in the above model, where an investment function that directly depends on the change in production Y was applied. Otherwise, the result is different by a constant amount. So there is no mathematical difference between the two models, apart from one constant. ${\displaystyle K_{0}=20{,}5}$

If the system of equations is specified differently, other conditions arise for constant oscillations.

Other uses

The multiplier-accelerator model can also be used to depict similar processes, for example a macroeconomic inventory cycle, if it is assumed that a certain level of inventories compared to the gross domestic product, or if a certain level of national debt is aimed at compared to the gross domestic product . In the latter case, the state debt would correspond to the capital stock, the financial balance of the state would correspond to the investments, whereby, similar to the investments are part of the gross domestic product, the state financial balance changes the gross domestic product.

literature

• Roy GD Allen: Macro-Economic Theory. A Mathematical Treatment. Macmillan, London et al. 1968.

Individual proof

1. ^ Allen: Macro-Economic Theory. 1968, p. 335.