OK, I'm stuck on this question if anyone can help me work through it.

Suppose the government decreases spending by $20 billion. How much
will this change equilibrium GDP if the MPC is 80 percent and the tax
rate is 1/6 (16.6 percent)?

Hi!!


First remember that change in equilibrium GDP equals the change in
total government spending times the multiplier:
change in GDP = (change in spending) * (multiplier) 

How the multiplier works?
Suppose the MPC for everybody is 0.8, this means that for every dollar
someone gets he will spend 80 cents and save 20 cents.
If you buy something and pay $100, the seller gets $100 and then he
will spend $80=(0.8*$100) in another thing; the second seller will
spend 0.8*(0.8*$100); the third seller will spend
0.8*(0.8*(0.8*$100)); etc.

The increase in total spending in this case will be:
  $100 + (0.8*$100) + (0.8^2*$100) + (0.8^3*$100) + ... =
= $100 * (1 + 0.8 + 0.8^2 + 0.8^3 + ...) =
= $100 * (1/(1-0.8)) =  this is a geometric series that converges
according to a known formula, then you will get the following result:
change in GDP = change in spending * (1/(1-MPC)) =

Then the multiplier is equal to (1/(1-MPC)) 

We found the formula without considering the taxes, if the taxes are
related to the income then if you get $100 you must save for taxes
T*$100, and from the resulting amount, (1-T)*$100, you will spend
MPC*(1-T)*$100, and the chain continues in the same mode than the
above example and the new factor in the formula is MPC*(1-T), so the
new multiplier is: (1/(1-MPC*(1-T))

Using the above result on your problem we have:
change in spending = -$20 billion
MPC = 0.8
T = 1/6

Multiplier = 1 / (1 - 0.8*(1-1/6) =
           = 1 / (1 - 0.8*5/6) =
           = 1 / (1 - 2/3) =
           = 1 / (1/3) =
           = 3

change in GDP = (change in spending) * (multiplier) =
              = -$20 billion * 3 = 
              = -$60 billion

The GDP will decrease by $60 billion.


For further references see the following pages:

"PRINCIPLES OF MACROECONOMICS - Lectures, Week 11":
Read the paragraph II. THE MULTIPLIER MODEL.
http://www.oswego.edu/~dighe/lstums11.htm

"Multipliers":
See at the bottom of page 2 "The Multiplier with taxes".
http://www.econ.umn.edu/~aevk/pdfs/teaching/2005spring/1102notes1corr.pdf

 
Search strategy:
gdp mpc tax formula
Government Spending mpc tax gdp


I hope this helps you. Feel free to use the clarification feature if
you find something unclear or incomplete. I will gladly respond your
requests for further assistance on this question.

Regards,
livioflores-ga