Let me be careful about multipliers, because they are a controversial topic. Barro has an extremely hostile view of multipliers from spending by the Federal government, which is on display in
I believe that his view is widely shared; economists want to get paid, and sucking up to people who rabidly want to cut back government is a successful strategy these days.
For the Keynesian mulitipliers (a la (G+I + k)/(1-c)), let /\ denote change.
So if we have /\G =40, then, under the Keynesian assumptions, and we let M denote our multiplier, we have /\Y = M*/\G. If the numerical value of M were 2, as used to be assumed, then /\Y = 80.
My own feeling is that the harsh view of multipliers is misleading. Suppose the government builds a road where there was none before. My impression, which could be verified with an intensive search that I can't promise to do now, is that in Barro's view the spending increases from the road in the 1st year ONLY are counted as a response to the government spending. But of course the road promotes spending for a number of years, and all the spending should be discounted and put in to get an honest answer. You keep getting things like this - when West Germany absorbed East Germany to reunify the German state, my recollection is that someone calculated some multiplier involved as 0.2, conveniently overlooking that the West German government paid off the (worthless) debts of the East in legitimate marks.
On the occasions when I pretend to be a "respectable" economist, I do regional analysis. The work usually involves making some use of IMPLAN, one of the few input-output programs of the necessary size around these days. The key assumption behind IMPLAN is that you have constant returns to scale and no bottlenecks, and multipliers wind up from 1.7 to a little over 2 for that sort of work.
To return to Barro, his assumptions fall very easily from his assumptions about aggregate supply. If you assume aggregate supply is nearly perfectly inelastic with regard to any government spending, then you get all his results about crowding out and inflation and the rest. This aggregate supply assumption is of course not Keynesian; it is the opposite, that there really is no slack. It is difficult to develop an unambiguous test for whether there is slack or not.
A good example of standard thought about multipliers now, not that I necessarily agree with it, is in Price Fishback, "How Successful Was the New Deal? The Microeconomic Impact of New Deal Spending and Lending Policies in the 1930s", Journal of Economic Literature, December 2017, Vol. 55, No. 4, 1435 - 1485.