This proof is no a rebuttal but a tautology.

Equation I = (1-b)(Y-T) is actually derived from the two identities:
(1) Y = C + I + G
(2) Y - T = C + S
and from the behavioral equation:
(3) C = b Y

Once you found the expression for I, you can of course plug it into (1)-(3) but this does not make any sense.

To understand what's going on, you have to look the other way.
(1) Start from your previous result (expression for Y): Y = 1/(1-b) + (G - bT)/(1-b)
(2) Set G = T (government spending is entirely financed by taxes)
You get:
Y = 1/(1-b) + G

The multiplier does not disappear but is no longer depending on the marginal propensity to consume: it is equal to 1.
Steve Kates rediscovered the Haavelmo theorem. Well done!

The equality T = G simply means that in a closed economy, if you exclude the possibility of financing the deficit by creating money (the LM part has been excluded from the outset) and issuing bonds (otherwise, equation (2) becomes Y - T + transfers = C + S), it must necessarily be financed by taxes. How surprising it is!

Best,
Aurélien Saïdi

> Le 31 août 2018 à 19:04, Fred Foldvary <[log in to unmask]> a écrit :
> 
> Steve Kates <[log in to unmask]> seeks a rebuttal to the Keynesian multiplier.
> 
> 
> 
> The Keynesian multiplier is 1/(1-b).
> 
> As I see it, the derivation is:
> 
> T is taxes that pay for G, government spending.
> b is the portion of income Y that is spent for consumption C.
> I is economic investment.
> 
> Y = C + I + G[
> C = b(Y-T)  
> Y = bY -bT + I + G
> Y-bY = I + G - bT
> Y(1-b) = I + G -bT
> Y=(I+G-bT)/(1-b)
> 
> That assumes constant I.
> If investment I is a function of savings, then
> I = (1-b)(Y-T)
> Substitute into the third equation:
> Y = bY -bT + (1-b)(Y-T) + G
> Which concludes with
> G=T
> 
> The multiplier disappears, and there is no determination of Y from 1/(1-b)
> 
> Fred Foldvary
> 



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