© 1999-2020, Douglas A.Ruby (06-02-2020)

A different approach to investment compared to the profit-maximizing model is that of the accelerator model. This model begins with the notion that a certain amount of capital is necessary to support a given level of economic activity. We can define this relationship as being proportional to GDP:

K_{t}= Θ Y_{t}

where ' Θ' 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 < Θ < 1

Investment represents changes in these stocks such that:

I_{t}= K_{t}- K_{t-1}+ Δ K_{t-1}

= Θ Y_{t}- Θ Y_{t-1}+ Δ K_{t-1}

= Θ Δ Y_{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}+ Θ Y_{t}- Θ Y_{t-1}+ Δ K_{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) - Θ = A_{t}- Θ Y_{t-1}

where A_{t} = G_{t} + NX_{t}. Thus:

Y_{t}* = [1 - b(1 - t) - Θ ]^{-1}(A_{t}- Θ Y_{t-1})

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

Y_{t}* = α (A_{t}- Θ Y_{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 -- Θ = 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}]:

time |
0 |
1 |
2 |
3 |
4 |
5 |
6 |

Y_{t} |
2,500 | 2,500 | 2,833 | 2,720 | 2,760 | 2,744 | 2,750 |

A_{t} |
1,000 | 1,000 | 1,100 | 1,100 | 1,100 | 1,100 | 1,100 |

I_{t} = Θ Δ Y_{t} |
0 | 0 | 33.33 | -11.30 | 4.00 | -1.60 | 0.60 |

We find, in this example, in time periods 0 and 1 the economy is in equilibrium with an income level of $2,500. 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 $2,750 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:

time |
0 |
1 |
2 |
3 |
4 |
5 |

Y_{t} |
2,500 | 2,500 | 3,000 | 2,500 | 3,000 | 2,500 |

A_{t} |
1,000 | 1,000 | 1,100 | 1,100 | 1,100 | 1,100 |

I_{t} = Θ Δ Y_{t} |
0 | 0 | +100 | -100 | +100 | -100 |

In this case the autonomous shock leads to an economy that oscillates forever between $2,500 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.

This relationship between the accelerator and the MPC (or marginal propensity to spend when taxes are considered) helps explain the cyclical behavior of economic activity. We can summarize this relationship below:

Oscillations |
Large Θ |
Moderate Θ |
Small Θ |

High spending propensity (b > 0.60) |
Explosive |
Cyclical |
Damped |

Moderate spending propensity (b < 0.60) |
Explosive |
Explosive |
Cyclical/ Damped |

You can practice with the Accelerator model below:

Another approach to understanding investment decisions is in using an expression known as Tobin's-q. This q-value represents a ratio between the market value of existing capital and its replacement cost:

q = Market Value / Replacement Cost

= {(Revenue)[1 - (1+r)^{-n}] / r} / [P_{K}K]

0 < q < 1

If this value is greater than one, then the value of capital is greater than replacement costs and it would make sense to add or invest in more capital. It the q-value is less than one then the market value is less than replacement costs and business firms will allow existing capital to depreciate resulting in negative levels of investment. Given the inverse relationship between asset prices and market interest rates we can note that when these interest rates rise the market value of installed capital falls relative to replacement costs -- disinvestment will occur.

- Accelerator
- Autonomous Expenditure
- Autonomous Shock
- Marginal Propensity to Consume
- Multiplier
- Tobin's Q