In the acceleration principle, the accelerator model states that if technology (and thus, the *capital-output ratio)* is held constant, an increase in output can be achieved only through an increase in capital stock.

(delta) *∆I = accelerator **´** ∆C*

Thus, *Accelerator = * ∆I / ∆C

## Capital – Output Ratio

The capital-output ratio shows that the demand for capital goods is not only derived from consumer goods but also from any direct demand for national output. The production of a given amount of output requires a certain amount of capital.

The required amount of capital can be calculated using the equation: **K = wY**, where **K** is the Capital Stock, **Y** is Level of Output, and **w** is the Capital Output Ratio or the Accelerator. Therefore, w equals K/Y and it is assumed constant in the accelerator theory.

For Example: If the capital-output ratio (w) is 3, then a capital stock (K) of Rs. 1,500 is required to produce output (Y) worth Rs. 500; Similarly, the capital stock of Rs. 1,800 is required to produce output worth Rs. 600.

## Change In Formula according to Time Period

The capital-output ratio formula changes with a change in the time period. For instance, Let us denote a particular time period by **t**, and preceding time periods by** t-1 & t-2.** Subsequent time periods become **t+1 & t+2.**

Assume that in the time period **‘t-1’,** the desired capital stock was sufficient to produce the level of output for the period. Thus, If the output rises from Yt-1 to Yt in period t, the desired capital stock will also rise to Kt, that is, **K****t-1**** = wY****t-1**

## The Acceleration Principle Formula or Working

- Capital-Output Ratio is 2 and is constant.
- The average life of the capital goods is assumed to be 10years and in each period, replacement investment is assumed to be 10% of the capital stock in period 1.
- There is the absence of excess capacity, and therefore, the investment will increase in response to an increase in demand for output.

As discussed in the assumptions, The capital-output ratio is 2 and the Replacement Investment is 10% of the capital in period 1, that is, 10% of 200 which equals **20**.

**In ****P1 & P2:** The output is 100 and the capital is 2 times the output which is 200. The net investment is 0 as it equals **w **(which is 2)** * difference in the capital of current and preceding year **(which is 200-00 = 0).

Gross investment is replacement investment + net investment, that is, **20 + 0 = 20**.

**In ****P3: **The demand for output rises by 10; therefore, gross investment rises from 20 to 40.

**In ****P4: **The demand for output rises by 10; however, gross investment remains at 40, as in P3. This is because, to maintain the gross Investment at a higher level after it has been increased, the output should continue to rise at the same absolute rate.

**In ****P5: **The absolute increase in output is higher than that from P3–P4; therefore, gross investment increases to 80.

**In ****P6:** Gross Investment falls to 60 despite the increase in output because the absolute increase in output from P5–P6 is lower than that from P4–P5. This decrease shows that a slight decline in the extent of absolute increase in the output leads to an absolute decrease in the gross investment.

**In ****P7: **When the output remains constant, the net investment becomes zero. Gross Investment does not fall to zero but reduces to replacement investment.

**In ****P8: **When the output falls by 10, the net investment falls by 20 and gross investment becomes zero. When the economy is moving downwards, the gross investment can fall only up to zero. The value of the accelerator during the downturn is limited by the inability of the demand for investment goods to fall below the value of the replacement demand.

Thus, **in P9:** Gross Investment falls to zero only. The excess of negative investment gives rise to excess capacity.

## Limitations of the Accelerator

- The capital-Output ratio is not constant.
- Excess Capacity
- The temporary rise in demand
- Non-availability of resources
- Time lag
- Non-availability of finance

## Super Multiplier Theory

The theory of super multiplier shows the interaction between the multiplier and accelerator. When the accelerator effect is combined with the multiplier effect, that is, the effects of changes in income (or consumption) and investment on each other are combined, we get a super multiplier. The super multiplier shows that an autonomous increase in the level of investment raises income by a multiple whose value is based on the multiplier.

An increase in income will lead to an increase in investment through the acceleration effect. The effect of the super multiplier is greater than that of a simple multiplier. The super multiplier indicates that fluctuations in employment, output and income are brought about by changes in investment or consumption.

## Super Multiplier Formulae

An increase in Autonomous Investment **∆****I****a** leads to an increase in Income through the multiplier, that is, ∆Y = k*∆Ia, which further leads to an increase in induced investment ∆Id through an accelerator (a): ∆Id = w*∆Y. This finally results in an increase in income & aggregate demand by a larger amount.

The above interaction between the multiplier and accelerator has been mathematically explained as **Y = C + I**, where “Y” is income, “C” is consumption and “I” is an investment.

Because investment can be divided into autonomous investment (Ia) and induced investment (Id), hence **Y = C + ****I****a**** + ****I****d **** **or **∆****Y = ****∆****C + ****∆****I****a**** + ****∆I****d **

Because a change in consumption (∆C) is equal to the marginal propensity to consume (or c) times a change in income, hence **∆C = C ∆ Y.**

Similarly, a change in induced investment is equal to accelerator times a change in income, that is, **∆I****d**** = w *** **∆Y.**

Substituting for **∆C **and **∆I****d** from the above equations, we get the following equation: **∆Y = C*∆Y + ∆****I****a**** + w*∆Y.**

The above equation for income shows that changes in income depend upon the values of MPC and the capital-output ratio (or accelerator) and changes in the amount of autonomous investment.