The multiplier effect – definition
The multiplier effect indicates that an injection of new spending (exports, government spending or investment) can lead to a larger increase in final national income (GDP).
This is because a proportion of the injection of new spending will itself be spent, creating income for other firms and individuals. These firms and individuals will also spend a proportion of their income, which itself creates income for other. This process continues until all no more extra income is left to be spent.
Example
If we assume that all recipients of new income save 25% (and have a margial propensity to save – mps – of 0.25) and 75% (and have a marginal propensity to consume – mpc – 0.75) noting that the sum of mps and mpc must be 0.1, then a new injection of 1000 would be allocated as follows:
New income [1]  Saving  Spending 
1000  250  750 
New income [2]  
750  187.50  562.50 
New income [3]  
562.50  140.63  421.87 
New income [4]  
421.87  105.47  316.40 
New income [5]  
316.41  79.10  237.30 
New income [6]  
237.30  59.33  177.98 
Running total  
3288.09 
Each ‘new income’ generates another level of new income, which is allocated toward savings, at 25%, and spending at 75%.
If we add up the running total so far, the initial injection of new income of 1000, and led to 3288.09 total new income [1000 + 750 + 562.50 + 421.87 + 316.41 + 237.30]. In fact, after 27 rounds of spending the cumulative total is 3996.99 – i.e. approaching 4000.
Of course, we do not need to go through this tortuous process as a simple formula will give us the final total, which is:
1 
1 mpc 
In this case, the mpc is 0.75, hence we have:
1 
1 – 0.75 
Which gives a multiplier of 4. Hence, the initial injection of 1000 will create 4000 of new income once all the rounds of spending are taken into account.
Hence, the size of the multiplier depends upon both the mpc and the mps. Of course, in a real economy there are more withdrawals, hence the mpc will be much lower, and hence the multiplier much smaller than in this simple example.

Read more on the fiscal multiplier