Economics 51D
Due 12 October 2007
Problem Set 5 Answers
1.
Time on the Keynesian Cross
A.
The shortrun equilibrium GDP is found where Y = Y
E
, or actual GDP = Planned
Expenditure.
Planned expenditure is given by Y
E
= C + I + G + X – M in the basic case, but C
depends on disposable income (YT) and T may depend on Y as well (if
it’s an income tax), and
M may depend on Y, too.
In the problem, we’ll only have C depend on YT and we’ll write C =
a
C
+ b
C
*(YT), so that the expenditure function is Y
E
=
a
C
+ b
C
*(YT) + I + G + X – M.
Then we
can set Y = Y
E
, and collect the terms with Y in them on the lefthand side, so we get
Y*(1 b
C
) = a
C
 b
C
*T + I + G + X – M.
Finally, dividing both sides by (1 b
C
) gives
C
C
C
b
M
X
G
I
T
b
a
Y


+
+
+

=
1
*
.
Putting the numbers in from the problem gives
92
.
1
400
400
600
460
550
*
92
.
100
*


+
+
+

=
Y
= 8175 billion or 8.175 trillion.
The calculations are also done on the accompanying spreadsheet.
Note that on the spreadsheet,
C = a
C
+ b
C
*(YT), T = a
T
+ b
T
*Y, and M = a
M
+ b
M
*Y.
Thus, a lumpsum tax of 550 has a
T
=
550 and b
T
= 0, and lumpsum imports of 400 have a
M
= 400 and b
M
= 0.
B.
The graph is presented on the accompanying spreadsheet.
C.
If taxes are raised by $100 billion, then we can simply replace the numbers in the answer to
Part A and recalculate:
92
.
1
400
400
600
460
650
*
92
.
100
*


+
+
+

=
Y
= 7025 billion or 7.025
trillion.
This is a fall of 1150 billion, or 1.15 trillion.
D.
Now let’s use the tax multiplier to calculate the change in GDP.
The tax multiplier is found
by solving for GDP in terms of changes.
Let the greek letter Δ in front of a letter represent the
change in that quantity.
Therefore, the equilibrium in terms of changes is given by ΔY =
Δa
C
+
b
C
*( ΔY ΔT) + Δb
C
*(Y – T) + ΔI + ΔG + ΔX – ΔM. Now, the only variables that are changing
are Y and T, so all the other “Δ” variables are equal to 0.
Setting these equal to 0 and collecting
terms gives ΔY*(1 b
C
) =  b
C
*ΔT or
C
C
b
T
b
Y

∆

=
∆
1
=  (92 / .08) =
1150, so equilibrium GDP
will fall by 1150 billion.
Note that the simple tax multiplier here is
C
C
b
b


1
= .92 / .08 = 11.5.
E.
The diagram graphs the mediumrun and longrun effects of this increase in lumpsum taxes.
In the short run, we know that equilibrium GDP falls by 1150 billion.
This means that the AD
curve shifts to the right by 1150 billion.
Since this is the only change to the graph, the MRAS
and LRAS curves are initially unchanged.
In the medium run, the fall in AD will lead to lower
GDP and lower prices, though the mediumrun GDP is higher than the shortrun GDP because of
1