Macroeconomics Extra Credit Algebra
Due Thursday, May 9
This is just for extra credit if you are into algebra and would like to see what is done in
intermediate macroeconomics. You will receive up to 5 points of extra credit based on the
accuracy of your answers. You should not copy someone’s work. If you turn in a correct
problem, you should be able to work a similar problem if I ask you to. If you turn in a copied
answer and are not able to work a similar problem in class, bad things will happen. This
assignment is not required and you will not see problems like it on any exam. The 5 points would
count as 1 of 2 extra credit articles. (To remind you of the extra credit articles—write up two
paragraphs of your opinion of a news article, economic speech, etc. Two paragraphs of your
opinion is probably an easier way to get extra credit than working through the algebra.)
Below is the algebra for our model of the economy. We can solve for the equilibrium level of
GDP just like we did graphically.
(1) AE = C + I + G + (X – IM) Definition of AE
C + MPC Y
(3) I = I
(4) G = G
(5) X = X
(6) IM = IM
(2) C =
Consumption Function
Investment
Government Purchases
Exports
Imports
Equilibrium Condition
(7) Y = AE
(a) Solve for the equilibrium level of GDP.
As an example, I have solved the model below for a simplified version, where
government and international trade have been left out.
AE = C + I
C + MPC Y
I= I
C=
Y = AE
Combining the first and last equations, since AE = C + I and since Y =
AE in equilibrium, then in equilibrium:
Y=C+I
Next, substitute in all of the middle equations:
Y=
C + MPC Y + I
Bringing Y terms to the left, factoring, and solving for Y:
Y – MPC Y =
C+ I
Y (1 – MPC) =
C + I
Y = ( C + I )/(1 – MPC)
Usually written as
Y = 1/(1 – MPC) x ( C + I )
with the multiplier written first being multiplied by all the bar terms.
To solve for the equilibrium level of GDP with the 7 original equations, I start out by
equating the first and last equation: If AE = C + I + G + (X – IM) and Y = AE in
equilibrium, then in equilibrium Y = C + I + G + (X – IM). Next, substitute in the
middle equations: Y = C + MPC Y + I + G + ( X – IM ). Next, bring the Y term
over to the left, factor out the Y, and divide so that you end up with just Y on the left
hand side. The equation is usually written as Y = the multiplier times the bar terms
( C + I + G + ( X – IM )). This is where our formula for the multiplier came from.
After solving the equations, plug in an MPC of .75 to calculate the multiplier.
(b) One key factor that has been left out of this algebraic model is the tax rate.
Consumption is really a function of your after-tax income, not your total income.
Thus, a more accurate model of the economy takes into account taxes. We would
write consumption as a function of income minus taxes:
(2a) C =
C + MPC(Y – Taxes)
We would also need an equation to determine the level of taxes. The main federal tax is
the income tax. The equation below says that taxes are a function of income: the tax rate
(t) times the level of income (or Y).
(2b) Taxes = t Y
Combining these two equations gives us a new equation 2:
(2) C =
C + MPC(Y – t Y)
Use this new equation 2 to re-solve the model for the equilibrium level of GDP. Note that
you now have three Y terms, so be careful in factoring and combining the Y terms. You
will also need to be careful with minus signs and parentheses. You need to end up with
Y by itself on the left. When you are done, again separate out the bar terms from the
new multiplier. What happens to the size of the multiplier? Plug in an MPC of .75 and a
tax rate of .2.
(c) Return to the original set of equations and now substitute in the following equation
for imports:
(6) IM = IM + d Y
This equation says that imports are positively related to the level of income (Y). As U.S.
income increases we buy more stuff, including stuff from other countries—imports
increase. Solve the system for a third time for the equilibrium level of output. Be careful
with the sign of dY as you bring this term to the left. What happens to the multiplier
(relative to part (a))? Use .1 to stand for “d”, the variable that measures the relationship
between changes in income and changes in imports. Again use an MPC of .75. What
happens to the size of the multipler?