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Y=C(y,r)+I(r)+G

Consumption is a function of disposable income (income minus tax. national income is also the same as GDP) y which is y=Y-T and the interest rate r. Investment is also a function of the interest rate r.

Now let's assume that the economy is in a recession and the government wants to boost GDP by either a) tax cuts or b) more government spending. Which one will be more effective?

Totally differentiating the IS curve we have dY=CydY-CydT+Crdr+Irdr+dG.

Cy=the derivative of consumption wrt to y, Cr is the derivative of consumption wrt to r, etc.

Now how does GDP respond to more government spending i.e. what is dY/dG?

Assuming taxes stay the same and interest rates don't move

We have dY=CydY+dG or dY/dG=1/(1-Cy) Cy is also known as the marginal propensity to consume which is how much people spend out of their entire incomes. The marginal propensity to consume of course 0<Cy<1.

Let's assume that the marginal propensity to consume for Americans is .9. Then if the government spends 1 dollar GDP will increase by 10. 1/1-Cy is also known as the multiplier for government spending.

Now let's look at what happens when the government cuts taxes. Assuming interest rates don't change and government spending don't change the totally differentiated IS equation becomes

dY=CydY-CydT

So how does GDP respond to tax cuts, i.e. what is dY/dT? Using simple algebra we find dY/dT=-Cy/(1-Cy). Assuming marginal propensity to consume is still .9 then for every dollar that the government cuts taxes dY/dT=9!!!!

(the negative sign goes away since we are talking about cutting taxes). -Cy/(1-Cy) is also known as the tax multiplier.

Thus dollar for dollar increasing government spending/bigger government is better at stimulating the economy in a recession than tax cuts. But of course politicians won't tell you this since Americans think tax cuts are always better.