Why does a reduction in taxes have a smaller multiplier affect?

Question

Why does a reduction in taxes have a smaller multiplier affect than an increase in government spending of an equal amount?

Answer 1

Because G=1 and T=.1 usually.

If the government spends money, that money goes straight into the economy.

If someone is taxed that money comes out of the economy and then comes back to the economy later.

Answer 2

Mathematically (I think this is how you will be asked to show it)....
Let's assume a Keynesian consumption function:
C=c0+c1(Y-T) [autonomous consumption plus marginal propensity to consume times disposable income, income minus taxes)
Assume government spending is exogenous and given as G- reasonable as only the gvt decides on how much it spends.
Assume Investment is given by:
I=b0+b1Y-b2r [autonomous investment and income=positive effects, real interest rate=negative effects]
We know Y=C+I+G
Y=c0+c1(Y-T)+b0+b1Y-b2r+G
Rearranging gives:
Y-c1Y-b1Y=c0-c1T+b0-b2r+G
Y= [1/(1-c1-b1)]{c0-c1T+b0-b2r+G}
Differentiate wrt T gives:
dY/dT= -c1/(1-c1-b1)
Differentiate wrt G gives:
dY/dG= 1/(1-c1-b1)

An increase in G by 1 offsets the decrease in T by 1 by:
1-c1/(1-c1-b1). This is due to the above assumptions that government spending is decided exogenously by the government and not by anything else. However taxation only affects output through the consumption function (disposable income). Because the level of consumption is also dependent upon the marginal propensity to consume, the probability a consumer will spend more if disposable income increases, it means the effect of taxation upon output is less than that for government spending, which is just assumed to be a given value.