K_t = qY_t

where 'q' is known as the accelerator and represents a constant of proportionality between the two variables. If we restrict the definition of capital to only include business inventory stocks then we might note that:

0 < q < 1

Investment represents changes in these stocks such that:

I_t = K_t - K_t-1 + dK_t-1
  = qY_t - qY_t-1 + dK_t-1
  = qDY_t + depreciation.

In order to understand investment behavior in this model, we need to look at the determinants of income or equilibrium level of expenditure:

Y_t = C_t + I_t + G_t + NX_t

Defining consumption expenditure as being proportional to disposable income:

C_t = bY_d = b(1-t)Y_t -- where 'b' represents the marginal propensity to consume.

We can then substitute:

Y_t = b(1-t)Y_t + qY_t - qY_t-1 + dK_t-1 + G_t + NX_t

For simplicity, we will eliminate the depreciation term and solve for Y_t :

Y_t [1-b(1-t) - q] = A_t - qY_t-1 -- where A_t = G_t + NX_t

Thus: Y_t * = [1-b(1-t)-q]^-1(A_t - qY_t-1 )

The term in the brackets, [1-b(1-t)-q]^-1, is known as the multiplier which represents how a change in autonomous expenditure 'A_t ' affects the equilibrium level of income. We will replace this term with a simple variable 'a' such that:

Y_t * = a(A_t - qY_t-1 )

Using the following values for the parameters,

• the marginal propensity to consume -- b = 0.75
• income tax rates -- t = 0.20
• the accelerator -- q = 0.10

we can compute a value for the multiplier to be equal to 3.333 thus
Y_t = 3.333[A_t - 0.10Y_t-1]:

We find, in this example, in time periods 0 and 1 the economy is in equilibrium with an income level of $2500. In time period 2, there is an autonomous shock (an increase in government spending or perhaps an increase in net-export spending) of $100. Working through the multiplier, the $100 change in autonomous spending leads to a $300 increase in income (example of the multiplier process). Changes in income mandate increases in inventory to support this additional economic activity. However, after this initial shock, the economy oscillates towards a new equilibrium of $2750 by time period 6. Through time, with this cyclical behavior in income, there is also cyclical behavior in investment with additions taking being offset by reductions every other year. Eventually as the economy approaches the new level, the need for new investment approaches zero.

The nature of this cyclical behavior is sensitive to the values of the marginal propensity to consume, tax rates (which impacts the multiplier process), and the value of the accelerator (which affects both the multiplier and the rate of acceleration). For example suppose we increase the value of the accelerator to 20%. The new value of the multiplier will be equal to 5.0:

In this case the autonomous shock leads to an economy that oscillates forever between $2500 and $3000 with investment being +$100 every other year. It is the larger value of the multiplier relative to the size of the MPC that leads to this result.

Oscillations	Large q	Moderate q	Small q
High spending propensity (b>0.60)	Explosive	Cyclical	Damped
Moderate spending propensity (b<0.60)	Explosive	Explosive	Cyclical/Damped

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