ECON 100 Tutorial: Week 15
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Keynesian Cross Diagram
Last week we calculated GDP using the expenditure method where
GDP = Aggregate Expenditure (AE) = C + I + G + NX.
By definition, GDP is total output, i.e. GDP = Y.
So we call “Y = C + I + G” the product market equilibrium because we are
setting total production equal to aggregate expenditure.
The Keynesian cross diagram graphs total output and aggregate expenditure
as two separate lines.
The 45° line is where GDP = Y
Or GDP = total Output,
Some texts call this the Aggregate Supply curve.
AE
GDP = AE = C+I+G = f(Y)
GDP = Aggregate Expenditure
(note: sometimes we say Aggregate Demand, AD)
Where the two lines cross, Y*, is the
equilibrium level of income in the
economy.
Y*
The first half of Chapter 33 in Mankiw & Taylor corresponds with this tutorial material.
Why is the Keynesian Cross Diagram relevant?
Keynes published work in 1936, that tried to explain short-run economic
fluctuations in general and the Great Depression in particular. His primary
message is that recessions and depressions can occur because of inadequate
aggregate demand for goods and services. So, he advocated policies that
increased aggregate demand, in particular government spending on public
works.
The Keynesian Cross Diagram and the Keynesian Multiplier build on classical
economics theory to illustrate his theory.
Read through Chapter 33 of the Mankiw and Taylor textbook for more info.
Question 1 (a)
The economy is described as follows:
Y=C+I+G
G = 150
C = 60 + 0.6(Y – T)
T = 100
I = 250
Derive the equilibrium level of income for the economy.
For problems like this, the first step is usually to plug values into the
equilibrium equation:
Y=C+I+G
Y = 60 + 0.6(Y-T) + 250 + 150
Y = 60 + 0.6(Y – 100) + 250 + 150
Y = 460 + 0.6Y – 60
Y = 400 + 0.6Y
This is the equation for the AD line that we graph in (b).
(note: the coefficient on Y in the AD equation is the MPC)
0.4Y = 400
Y = 1000
This is the equilibrium value of Y (or Y*)
where the two lines will cross.
Question 1(b)
Illustrate the equilibrium on an appropriate diagram.
This graph is called the Keynesian Cross (Fig. 33.1 Mankiw)
The 45⁰ line
connects all points
where consumption
spending is equal to
national income.
Aggregate
Expenditure
Y = 0.56Y + 450
The economy is in
equilibrium where
the C+I+G (GDP=AE)
line cuts the 45⁰
(GDP=Y) line.
1(c) Find the equilibrium value of Y if G = 200
Y=C+I+G
G = 150
Y=C+I+G
C = 60 + 0.6(Y – T)
G = 200
Y = 60 + 0.6(Y – T) + 250 + 200 I = 250
T = 100
Y = 60 + 0.6(Y – 100) + 250 + 200
Y = 510 + 0.6Y – 60
Y = 0.6Y + 450
This is the AD line that we will graph.
Y – 0.60Y = 450
0.40Y = 450
Y = 1,125
This is Y*
Question 1(d)
What is the value of the multiplier in the economy?
We increased G from 150 to 200 and as a result, Y went
from 1,000 to 1,125.
First, how do we find the multiplier?
Multiplier = ∆Y / ∆G
=(1125 – 1000)/(200 – 150)
= 125/50 = 2.5.
Also, as a check,
Multiplier = 1/(1 – MPC)
= 1 /(1 – 0.6)
= 1/0.4
= 2.5.
A side note:
MPC: marginal propensity to consume.
The fraction of extra income that a
household consumes rather than saves.
i.e. if MPC = .6,
then for every £1 earned, £0.60 is spent.
How do we find MPC?
It is the coefficient on Y in the equation
of the Aggregate Demand line.
Question 1(e)
What would be the effect on equilibrium income if the government
reversed policy and introduced an “austerity” budget in which it
reduced G to equal T?
Y=C+I+G
G=T
C = 60 + 0.6(Y – T)
I = 250
T = 100
Again, the first step is to plug values into the equilibrium equation:
Y=C+I+G
Y = 60 + 0.6(Y-T) + 250 + 100
Y = 60 + 0.6(Y – 100) + 250 + 100
Y = 410 + 0.6Y – 60
Y = 0.6Y + 350
0.4Y = 350
Y = 875
The new equilibrium income would be 875.
Question 1(e)
What would be the effect on equilibrium income if the
government reversed policy and introduced an “austerity”
budget in which it reduced G to equal T?
A second, alternate, way to find Y is to use the
multiplier.
Income falls from original income by the drop in G
times the multiplier.
Initial G was 150 and Y was 1000
Final G was 100, a fall of 50 from the initial G.
So income falls by 50*2.5 = 125
So final Y = 1000 – 125 = 875
Question 1(f)
What are the limitations of the analysis using this model?
The model ignores a number of features of the economy. These
include prices, which are implicitly assumed fixed in the model. This
is an obvious weakness, despite Friedman’s view that the realism of
assumptions should not matter. Friedman himself, of course, thought
this assumption was too much.
Interest rates are not included in the model, an assumption that will
be addressed in later lectures. Also, government spending has to be
financed. This is not allowed for in either a) or c) where the
government is running a deficit.
The long-term implications of such a (deficit) policy are not
considered. Nor does the model include a supply-side. Everything
goes through demand and capacity constraints are ignored.
Some of these limitations are addressed in developing the model
beyond this simple representation.
Question 2: Deflationary Gap
If there is a deflationary gap, then that means that there is spare capacity in the
economy that unemployment will rise. (here: Y1 < Yf )
The deflationary gap is the difference between full employment output and the
expenditure required to raise the equilibrium employment output to the point that it
equals full employment output. To remove or get rid of the deflationary gap, we then
have to raise government spending in order to raise the equilibrium employment
output.
So what we want to do here is find the multiplier. To remove the deflationary gap, the
government should increase government spending by the deflationary gap amount
divided by the multiplier.
For more on deflationary gap and inflationary gap:
Figure 33.1 and pgs. 707 and 708 in Mankiw & Taylor 2nd Ed.
Question 2 ctd.
The deflationary gap in an economy is calculated to be $700 billion.
MPS = 0.1
The marginal propensity to save is 0.1
MPM = 0.15
The marginal propensity to import is 0.15
MPT = 0.1
The marginal rate of taxation is 0.1.
By how much would the government need change its spending on goods and services to
eliminate the deflationary gap?
First, let’s find MPC:
MPC = 1-(marginal propensity to save +
marginal propensity to import + marginal rate of taxation)
MPC = 1-(MPS + MPM + MPT)
MPC = 1 – (0.1 + 0.15 + 0.1)
MPC = 1 - 0.35
We can plug MPC into our equation for the value of the multiplier:
multiplier = 1/(1-MPC)
1/(1- (1- 0.35) = 1/0.35 = 2.857
So, the rise in G required is
700/Multiplier =
700/2.857 = $245 billion.
Question 2 ctd.
The deflationary gap in an economy is calculated to be $700 billion.
The marginal propensity to save is 0.1
MPS = 0.1
The marginal propensity to import is 0.15
MPM = 0.15
The marginal rate of taxation is 0.1.
MPT = 0.1
By how much would the government need change its spending on goods and services to eliminate the
deflationary gap?
Alternatively, pgs. 711 – 713 of Mankiw talks about the Marginal Propensity to Withdraw, MPW.
MPW = MPS + MPT + MPM
The multiplier is = 1/MPW
So, we can ignore all of the 1- ‘s in the previous slide and just use MPW:
MPW = MPS + MPT + MPM
MPW = 0.1 + 0.1 + 0.15
MPW = 0.35
Multiplier = 1/MPW
Multiplier = 1/0.35
Multiplier = 2.857
So, to remove the deflationary gap, the government should increase government spending by the
deflationary gap divided by the multiplier:
Spending increase = 700/ 2.857
Spending increase = $245 billion.
Question 3
The marginal propensity to consume in an economy is equal to 0.8. It
also equals the average propensity to consume. In addition, I = 300; X =
100; M = 150 and G = 100. What is the equilibrium level of national
income in the economy?
Let’s start with Y = C + I + G + X - M
We have I, G, X and M, so we can plug those in.
To find C, the hint here is that we are told about the Average propensity
to consume. It equals MPC. So, C = average propensity to consume * Y,
so C = MPC *Y = 0.8Y.
So we can plug in to Y = C + I + G + X - M
Y = 0.8Y + 300 + 100 + 100 – 150
Y = 0.8Y + 350
0.2Y = 350
Y = 1,750
Question 3
If the full-employment level of national income is 2,000, what
would the government need to do to its spending to achieve this
level of national income?
We found Y* = 1,750, and want to increase it to 2,000.
To do so, we need Government spending to increase by the
deflationary gap ÷ the multiplier.
The deflationary gap is the difference between equilibrium income
and the full-employment level of income, 250.
The multiplier is 1 /(1 – MPC). MPC = 0.8 (from prev. slide)
So, Multiplier = 1/0.2 = 5
So to raise income to 2,000 requires an increase of 250. Thus G
needs to increase by:
deflationary gap ÷ the multiplier
= 250 ÷ 5 = 50.
Question 4
Consider the following model
Y=C+I+G
C = 50 + 0.8(Y-T)
T = 0.2Y
I = 100
G = 120
where Y, C, I, G and T respectively denote income,
consumption, planned investment, government spending,
and tax revenues.
The equilibrium level of income is:
a)
b)
c)
d)
700
750
800
850
Question 4
Consider the following model
Y=C+I+G
C = 50 + 0.8(Y-T)
T = 0.2Y
I = 100
G = 120
where Y, C, I, G and T respectively denote income, consumption,
planned investment, government spending, and tax revenues.
Y=C+I+G
Y = 50 + 0.8(Y-0.2Y) + 100 + 120
Y = 270 + 0.8Y-0.16Y
This is the equation for the AD line, and from here,
Y = 270 + 0.64Y
the coefficient on Y gives us our MPC = 0.64
0.36Y = 270
Y = 270/0.36
Y = 750
Question 5
Consider the model in question (4)
above, but suppose that G rises to 125.
The new equilibrium level of income is:
a)
b)
c)
d)
714
728
764
880
Y=C+I+G
C = 50 + 0.8(Y-T)
T = 0.2Y
I = 100
G = 120
G = 125
Question 5
Y = 50 + 0.8(Y-0.2Y) + 100 + 125
Y = 275 + 0.8Y-0.16Y
Y = 275 + 0.64Y
0.36Y = 275
Y = 275/0.36
Y = 764
Y=C+I+G
C = 50 + 0.8(Y-T)
T = 0.2Y
I = 100
G = 120
G = 125
Question 6
Using the results obtained in questions (4) and (5)
above, the multiplier is: Q(4)
Q(5)
Y=C+I+G
G = 125
a) 1.5
C = 50 + 0.8(Y-T)
Y* = 764
T = 0.2Y
b) 2.3
I = 100
c) 2.6
G = 120
d) 2.8
Y* = 750
Question 6
Y=C+I+G
C = 50 + 0.8(Y-T)
T = 0.2Y
I = 100
Q(4)
G = 120
Y* = 750
Q(5)
G = 125
Y* = 764
Multiplier = Change in Y/Change in G
= (764 – 750)/(125 – 120)
= 14/5
= 2.8.
Alternatively:
Multiplier = 1/(1-MPC) = 1/(1 – 0.64) = 1/0.36 = 2.78
So an increase in G of 5 increases Y by 5*2.78 = 14
Question 7
A change in an injection causes a change in
income over time in an economy as follows:
+100 + 70 + 49 + 34.3 + …
What is the value of the multiplier?
a)
b)
c)
d)
0.3
1.43
3
3.33
Question 7
A change in an injection causes a change in income over time in an
economy as follows: +100 + 70 + 49 + 34.3 + …
What is the value of the multiplier?
Hmm, So the pattern is:
100 + 0.7*100 + 0.7*0.7*100 + 0.7*0.7*0.7*100 + … =
100 + 0.7*100 + 0.72 *100 + 0.73 *100 + … =
(An example of what is happening here is: I get £100 buy something
from you for £70, You take the £70 I paid you, spend 0.7 of it on
something your neighbor is selling, they take the £49 you paid them,
and spend 0.7 of that on something, etc. )
So, 0.7 is the MPC.
So our equation says the multiplier is:
1/1-MPC = 1/(1-0.7) = 1/.3 = 10/3 = 3.33
Next Week
Check Moodle for a worksheet and work through it
before coming to tutorial.
Read through Chapter 33 in Mankiw & Taylor.