The "Multiplier Effect" story goes like this: The state increases spending by £100. The person they pay this to saves 10% and spends the remaining £90. The next person saves 10% of that and spends the rest, and so on. An initial increase in state spending of £100 leads to an increase in GDP of £1000.
This is nonsense.
If it were true, all money would end up "saved" in vaults, and there would be no money circulating. In reality, money continues circulating indefinitely. Money that is saved bids up the price of money, causing other savers to withdraw their savings and spend. For every young person saving, there is an old person spending. There will be an equilibrium where a certain proportion of money is saved at any given time.
An alternative to saving is investment. Money that is spent on investment continues circulating.
The multiplier effect argument applies to all spending, not just government spending. Increases in state spending have to come from somewhere, whether it is taxation, borrowing or money creation. All three decrease purchasing power (i.e. spending) elsewhere.
A clearer way to look at things is to ignore money. An increase in government spending reallocates resources to the government. It does not increase the amount of resources available. It does not actually increase GDP.
So the above "multiplier effect" story is nonsense.
The real multiplier effect concerns productive, profitable investment. If an investment is profitable, it takes resources as input, and outputs a larger amount of resources. The multiplier factor is a measure of how profitable it is. Government spending, like private spending, only has a multiplier effect on GDP if goes into a successful, productive, profitable investment. If it goes into a foolish, loss-making investment project, or into consumption and spending, it has a demultiplier effect.